@prefix vivo: . @prefix edm: . @prefix ns0: . @prefix dcterms: . @prefix skos: . vivo:departmentOrSchool "Applied Science, Faculty of"@en, "Civil Engineering, Department of"@en ; edm:dataProvider "DSpace"@en ; ns0:degreeCampus "UBCV"@en ; dcterms:creator "Gohl, W. Blair"@en ; dcterms:issued "2011-01-27T00:44:11Z"@en, "1991"@en ; vivo:relatedDegree "Doctor of Philosophy - PhD"@en ; ns0:degreeGrantor "University of British Columbia"@en ; dcterms:description """The analysis of the dynamic response of pile foundations to earthquake shaking is a complex problem and has been treated using concepts developed from the theory of elasticity, applicable to low level shaking, and to models incorporating non-linear soil response appropriate for stronger shaking intensities. A review of available field reports indicates that due to the lack of complete instrumental recordings describing the response of full scale pile foundations to earthquake loading, the above analysis techniques are in large measure unchecked. To provide a reliable data base suitable for checking various models of dynamic pile foundation response, a series of small scale model tests on single piles and pile groups embedded in dry sand foundations were carried out on shaking tables at the University of British Columbia. A similar series of tests were carried out using a geotechnical centrifuge equipped with a base motion actuator located at the California Institute of Technology. Under the centrifugal forces acting on the model, full scale stress conditions are simulated in the sand foundation. Since soil behaviour is stress level dependent, the centrifuge tests are considered to provide a more realistic simulation of full scale pile foundation behaviour. Both the shake table and centrifuge single pile tests were carried out using both sinusoidal and random earthquake input motions over a range of shaking intensities. From the data, details of soil-pile interaction were elucidated. This provided a basis for improvement in methods of estimating required input parameters used in the dynamic analysis of pile foundations. Prior to each test, shear wave velocity measurements were made throughout the prepared sand foundations using piezoceramic bender elements. This technique has proved particularly useful in the centrifuge environment since the bender element source and receivers could be triggered remotely from off the centrifuge arm while the model was in flight. The shear wave velocity data were used to compute small strain, elastic shear moduli in the soil which have been found to be in close agreement with predictions made using an equation proposed by Hardin and Black (1968). Elastic compression wave velocities were also identified from the bender element responses recorded during the shake table tests. The single pile tests demonstrated that significant non-linearity and strain softening occurs in near field soil response, which is responsible for reductions in fundamental vibration frequency and pile head stiffness parameters with increasing amplitudes of lateral pile vibration. An analysis technique developed to estimate average effective strains around a single pile leads to predictions of large modulus reduction around the pile, depending on the amplitude of pile vibration. Soil reaction pressures (p) due to relative horizontal movement between the soil and the pile (y) were deduced from the test data for various cycles of shaking, or so-called p-y curves. The cyclic p-y curves developed show clearly the non-linear, hysteretic near field response near the pile head. Approximately linear elastic p-y response occurs at greater depth. Backbone p-y curves computed using procedures recommended by the American Petroleum Institute (API) are in poor agreement with the experimental shake table and centrifuge measurements. Material damping inferred from the area within the p-y hysteresis loops increases, in general, with increasing pile deflection level. The experimental p-y hysteresis loops were reliably simulated using a Ramberg-Osgood backbone curve and the Masing criterion to model unload-reload response. Comparing the flexural response observed on single piles during the shake table and centrifuge tests, the depth of maximum bending moment relative to the pile diameter has been observed to be greater in the shake table tests. This can be anticipated from the laws of model similitude. Cyclic p-y curves developed from the shake table and centrifuge tests also show substantial differences, with the shake table p-y curves being stiffer than predicted using the API procedures, while the opposite behaviour was found in the high stress, centrifuge environment. Damping in the low stress level environment of the shake table has been found to be greater than under full scale stress conditions in the centrifuge. Two-pile tests, where the piles have been oriented inline, offline or at 45 degrees to the direction of shaking, indicate that pile to pile interaction is very strong for inline and 45 degree shaking, and is relatively minor for offline shaking. Interaction effects observed under low and high intensities of shaking die off with increasing pile separation distance at a quicker rate than predicted using elastic interaction theory. Interaction effects for inline and offline cyclic loading may be neglected for centre to centre pile spacings of about six and three pile diameters, respectively. For close pile separations during inline shaking, elastic theory underpredicts the extent of interaction. Similar conclusions were reached from the shake table and centrifuge tests conducted. Based on the experimental data and data available from the literature, modifications to elastic pile interaction coefficients have been suggested. Predictions of single pile response to earthquake shaking have been made using an uncoupled, sub- structure approach incorporating non-linear pile head springs and equivalent viscous dashpots (foundation compliances) derived from the test data. The foundation compliances account for the deflection level dependent stiffness and damping characteristics of the below ground soil-pile system. The measured free field surface motions have been used as the input excitation. Agreement between computed and measured pile responses was found to be excellent. A fully coupled analysis using the commercially available program SPASM8, where the below ground portions of the pile are directly considered in the numerical discretization of the problem has also been used. Interaction between the soil and vibrating ground is accounted for using a Kelvin-Voight model which includes non-linear Winkler springs and equivalent viscous dashpots to simulate radiation damping. Free field ground motions deduced from an independent free field response analysis using the computer program SHAKE are applied to the free field end of the soil-pile interaction elements. Using this full coupled model, the possible effects of kinematic interaction are accounted for. Results from the analysis show that SPASM8 underpredicts pile flexural response. A key difficulty in using an analysis of this kind is the accurate determination of free field input motions to be used along the embedded length of the pile. A computer program, PGDYNA, has been developed to analyse the uncoupled response of a superstructure supported by a group of foundation piles, taking into account non-linearity of the pile head compliances and the effects of pile group interaction. Interaction factors developed from the experimental test program were used to calculate deflection level dependent pile head stiffnesses. Preliminary testing of the program indicates that use of the free field surface motions as input, neglecting the effects of kinematic interaction, leads to an overestimate of pile group response."""@en ; edm:aggregatedCHO "https://circle.library.ubc.ca/rest/handle/2429/30882?expand=metadata"@en ; skos:note "R E S P O N S E O F P I L E F O U N D A T I O N S T O S I M U L A T E D E A R T H Q U A K E L O A D I N G : E X P E R I M E N T A L A N D A N A L Y T I C A L R E S U L T S V O L U M E II B y W . B L A I R G O H L B . Eng. (Civi l ) M c G i l l University 1976 M . Eng. (Civi l ) M c G i l l University 1980 A THESIS S U B M I T T E D IN P A R T I A L F U L F I L L M E N T O F T H E R E Q U I R E M E N T S F O R T H E D E G R E E O F D O C T O R O F P H I L O S O P H Y in T H E F A C U L T Y O F G R A D U A T E STUDIES CIVIL E N G I N E E R I N G We accept this thesis as conforming to the required standard T H E U N I V E R S I T Y O F BRITISH C O L U M B I A July 1991 © W . B L A I R . G O H L , 1991 R E S P O N S E O F P I L E F O U N D A T I O N S T O S I M U L A T E D E A R T H Q U A K E L O A D I N G : E X P E R I M E N T A L A N D A N A L Y T I C A L R E S U L T S V O L U M E I B y W . B L A I R G O H L B . Eng. (Civil) M c G i l l University 1976 M . Eng. (Civi l) M c G i l l University 1980 A THESIS S U B M I T T E D IN P A R T I A L F U L F I L L M E N T O F T H E R E Q U I R E M E N T S F O R T H E D E G R E E O F D O C T O R O F P H I L O S O P H Y in T H E F A C U L T Y O F G R A D U A T E STUDIES CIVIL E N G I N E E R I N G We accept this thesis as conforming to the required standard T H E U N I V E R S I T Y O F BRITISH C O L U M B I A July 1991 © W . B L A I R G O H L , 1991 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of Civil Engineering The University of British Columbia Vancouver, Canada Date ,luly ??, 1991 DE-6 (2/88) Abstract The analysis of the dynamic response of pile foundations to earthquake shaking is a complex problem and has been treated using concepts developed from the theory of elasticity, applicable to low level shaking, and to models incorporating non-linear soil response appropriate for stronger shaking intensities. A review of available field reports indicates that due to the lack of complete instrumental recordings describing the response of full scale pile foundations to earthquake loading, the above analysis techniques are in large measure unchecked. To provide a reliable data base suitable for checking various models of dynamic pile foundation response, a series of small scale model tests on single piles and pile groups embedded in dry sand foundations were carried out on shaking tables at the University of British Columbia. A similar series of tests were carried out using a geotechnical centrifuge equipped with a base motion actuator located at the California Institute of Technology. Under the centrifugal forces acting on the model, full scale stress conditions are simulated in the sand foundation. Since soil behaviour is stress level dependent, the centrifuge tests are considered to provide a more realistic simulation of full scale pile foundation behaviour. Both the shake table and centrifuge single pile tests were carried out using both sinusoidal and random earthquake input motions over a range of shaking intensities. From the data, details of soil-pile interaction were elucidated. This provided a basis for improvement in methods of estimating required input parameters used in the dynamic analysis of pile foundations. Prior to each test, shear wave velocity measurements were made throughout the pre-pared sand foundations using piezoceramic bender elements. This technique has proved ii particularly useful in the centrifuge environment since the bender element source and receivers could be triggered remotely from off the centrifuge arm while the model was in flight. The shear wave velocity data were used to compute small strain, elastic shear moduli in the soil which have been found to be in close agreement with predictions made using an equation proposed by Hardin and Black (1968). Elastic compression wave ve-locities were also identified from the bender element responses recorded during the shake table tests. The single pile tests demonstrated that significant non-linearity and strain softening occurs in near field soil response, which is responsible for reductions in fundamental vibration frequency and pile head stiffness parameters with increasing amplitudes of lateral pile vibration. An analysis technique developed to estimate average effective strains around a single pile leads to predictions of large modulus reduction around the pile, depending on the amplitude of pile vibration. Soil reaction pressures (p) due to relative horizontal movement between the soil and the pile (y) were deduced from the test data for various cycles of shaking, or so-called p-y curves. The cyclic p-y curves developed show clearly the non-linear, hysteretic near field response near the pile head. Approximately linear elastic p-y response occurs at greater depth. Backbone p-y curves computed using procedures recommended by the American Petroleum Institute (API) are in poor agreement with the experimental shake table and centrifuge measurements. Material damping inferred from the area within the p-y hysteresis loops increases, in general, with increasing pile deflection level. The exper-imental p-y hysteresis loops were reliably simulated using a Ramberg-Osgood backbone curve and the Masing criterion to model unload-reload response. Comparing the flexural response observed on single piles during the shake table and centrifuge tests, the depth of maximum bending moment relative to the pile diameter has been observed to be greater in the shake table tests. This can be anticipated from iii the laws of model similitude. Cyclic p-y curves developed from the shake table and centrifuge tests also show substantial differences, with the shake table p-y curves being stiffer than predicted using the A P I procedures, while the opposite behaviour was found in the high stress, centrifuge environment. Damping in the low stress level environment of the shake table has been found to be greater than under full scale stress conditions in the centrifuge. Two-pile tests, where the piles have been oriented inline, offline or at 45 degrees to the direction of shaking, indicate that pile to pile interaction is very strong for inline and 45 degree shaking, and is relatively minor for offline shaking. Interaction effects observed under low and high intensities of shaking die off with increasing pile separation distance at a quicker rate than predicted using elastic interaction theory. Interaction effects for inline and offline cyclic loading may be neglected for centre to centre pile spacings of about six and three pile diameters, respectively. For close pile separations during inline shaking, elastic theory underpredicts the extent of interaction. Similar conclusions were reached from the shake table and centrifuge tests conducted. Based on the experimental data and data available from the literature, modifications to elastic pile interaction coefficients have been suggested. Predictions of single pile response to earthquake shaking have been made using an uncoupled, sub- structure approach incorporating non-linear pile head springs and equiv-alent viscous dashpots (foundation compliances) derived from the test data. The founda-tion compliances account for the deflection level dependent stiffness and damping char-acteristics of the below ground soil-pile system. The measured free field surface motions have been used as the input excitation. Agreement between computed and measured pile responses was found to be excellent. A fully coupled analysis using the commercially available program SPASM8, where the below ground portions of the pile are directly iv considered in the numerical discretization of the problem has also been used. Interac-tion between the soil and vibrating ground is accounted for using a Kelvin-Voight model which includes non-linear Winkler springs and equivalent viscous dashpots to simulate radiation damping. Free field ground motions deduced from an independent free field response analysis using the computer program S H A K E are applied to the free field end of the soil-pile interaction elements. Using this full coupled model, the possible effects of kinematic interaction are accounted for. Results from the analysis show that SPASM8 underpredicts pile flexural response. A key difficulty in using an analysis of this k ind is the accurate determination of free field input motions to be used along the embedded length of the pile. A computer program, P G D Y N A , has been developed to analyse the uncoupled re-sponse of a superstructure supported by a group of foundation piles, taking into account non-linearity of the pile head compliances and the effects of pile group interaction. In-teraction factors developed from the experimental test program were used to calculate deflection level dependent pile head stiffnesses. Preliminary testing of the program indi-cates that use of the free field surface motions as input, neglecting the effects of kinematic interaction, leads to an overestimate of pile group response. v Table of Contents Abstract i i List of Tables x i i List of Figures x v Acknowledgement x x x v i 1 Statement of Research 1 1.1 Behavioural Aspects of Single Pile Response to Cycl ic Lateral Loading . 1 ] .2 Observations of Full Scale Pile Response Dur ing Earthquake Loading . . 7 1.3 Experimental Observations of Pile Group Interaction 9 1.4 Numerical Modell ing of Single Pi le Response to Earthquake Loading . . . 15 1.5 Numerical Modell ing of Pile Group Behaviour to Static and Dynamic Loading 25 1.5.1 Low Frequency, Quasi-Static Loading 25 1.5.2 Higher Frequency Dynamic Loading 28 1.6 Scope of Study 30 2 Shake Table Test Procedures 35 2.1 U B C Shaking Table Characteristics 35 2.2 Foundation Sand Characteristics 40 2.3 Sand Foundation Preparation 46 2.4 Single Pile Characteristics and Model Layout 49 vi 2.5 Pile Group Characteristics and Model Layout 54 2.6 Instrumentation and Measurement Resolution 58 2.7 Elastic Wave Velocity Measurements on the Shake Table 60 2.8 Accuracy of Elastic Wave Velocity Measurements 68 3 Centrifuge Test. Procedures 75 3.1 The Principles of Centrifuge Modelling 75 3.2 Description of Caltech Centrifuge and Base Motion Actuator 80 3.3 Pile Characteristics and Model Layout 83 3.4 Pile Group Characteristics and Model Layout 86 3.5 Instrumentation and Measurement Resolution 89 3.6 Foundation Sand Characteristics 91 3.7 Sand Foundation Preparation and Test Procedures 93 3.8 Elastic Wave Velocity Measurements on the Centrifuge 95 3.9 Accuracy of Shear Wave Velocity Measurements 97 4 Centrifuge Test Results 103 4.1 Introduction 103 4.2 Shear Wave Velocities 104 4.2.1 Measured Wave Arrivals 104 4.2.2 Theoretical Bender Response 106 4.2.3 Wave Velocity Distributions 108 4.3 Base Motion Excitation of Single Piles 110 4.3.1 Low-level Sinusoidal Shaking 113 4.3.2 Random Earthquake Excitation 115 4.4 Soil-Pile Interaction 129 4.4.1 Introduction 129 vi i 4.4.2 Earthquake Excitation 144 4.4.3 Low Level Sinusoidal Shaking 157 4.4.4 Near Field Hysteretic Damping 169 4.5 Equivalent Visco-Elastic Soil Resistance 172 4.5.1 Computed Lateral Winkler Stiffness and Material Damping . . . . 176 4.6 Non-Linear Modelling of P -Y Hysteresis Loops 190 4.7 Base Motion Excitation of Pile Groups 201 4.7.1 Introduction 201 4.7.2 Low Level Shaking - Two Pile Groups 203 4.7.3 Pile Group Interaction Analysis 220 4.7.4 Base Motion Excitation of a 2 x 2 Pile Group 230 4.7.5 Summary 240 5 Shake Table Test Results 244 5.1 Introduction 244 5.2 Elastic Wave Velocities 245 5.3 Natural Frequency Tests 255 5.3.1 Introduction 255 5.3.2 Test Procedures 257 5.3.3 Single Pile Tests in Loose Sand 260 5.3.4 Single Pile Tests in Dense Sand 286 5.3.5 Natural Frequency Tests - 2 Pile Groups in Dense Sand 306 5.4 Base Motion Excitation of Single Piles 323 5.4.1 Free Field Response 323 5.4.2 Single Pile Flexural Response - Shake Table vs. Centrifuge Results 324 5.4.3 Shake Table Test Results 333 v i i i 5.5 Soil-Pile Interaction 348 5.5.1 Hysteretic Damping 356 5.5.2 Non-Linear Modelling of P -Y Hysteresis Loops 358 5.6 Base Motion Excitation of Pile Groups 366 5.6.1 Introduction 366 5.6.2 High Level Shaking - Two Pile Groups 368 5.6.3 Pile Group Interaction Analysis 380 5.6.4 Base Motion Excitation of 2 x 2 Pile Group 384 5.6.5 Summary 390 5.7 Cyclic Axial Load Behaviour of Model Piles 393 6 Single Pile Response to Earthquake Excitation 396 6.1 Introduction 396 6.2 Uncoupled Non-Linear Analysis 398 6.2.1 Pile Head Stiffnesses 398 6.2.2 Pile Head Damping 400 6.2.3 Coupled Versus Uncoupled Analytical Solution 404 6.2.4 Uncoupled Equations of Motion - Single Pile 410 6.2.5 Prediction of Ringdown Test Results 412 6.2.6 Prediction of Pile Response to Free Field Ground Motions . . . . 414 6.3 Coupled Dynamic Pile Analysis 440 6.3.1 Introduction 440 6.3.2 Methodology 441 6.3.3 Results of Analysis 446 6.3.4 Summary 457 ix 7 Uncoupled Dynamic Solution for a Pile Group 470 7.1 Introduction 470 7.2 Preliminary Testing of P G D Y N A 472 7.3 Dynamic Analysis Results 476 7.3.1 Shake Table Test - Four Pile Group Subjected to Strong Sinusoidal Shaking 476 7.3.2 Centrifuge Test - Four Pile Group Subjected to Earthquake Exci-tation 483 7.4 Summary 486 8 Summary and Suggestions for Future Work 490 8.1 Introduction 490 8.2 Single Pile Test Results 492 8.3 Pile Group Test Results 499 8.4 Suggestions for Future Work 502 Bibliography 503 A Shake Table Tests - Instrumentation and Data Acquisition 533 A . l Strain Gauges 533 A.2 Displacement. Transducers (LVDT's) 536 A.3 Accelerometers 538 A.4 Data Acquisition 540 A.5 Spectral Analysis and Waveform Aliassing 541 A. 6 Digital Filtering 543 B Centrifuge Tests - Instrumentation and Data Acquisition 547 B. l Strain Gauges 547 X B.2 Displacement Transducers 549 B.3 Accelerometers 550 B.4 Data Acquisition 551 B. 5 Data Processing 552 C Strain Fields Around Laterally Loaded Piles 554 C. l Introduction 554 C.2 Navier's Equations of Motion in Three Dimensions 555 C.3 Simplifications to the Three Dimensional Equations of Motion 555 C.4 Solution to Navier's Equations of Motion - Plane Displacement Case . . . 557 C.5 Solution to Navier's Equations of Motion - Plane Strain Case 560 C.6 Comparison of Plane Strain Analytic Solution With Non- Linear Finite Element Solution 563 C. 7 Comparison of Strain Fields - Plane Displacement versus Plane Strain Solutions 567 D Static Laterally Loaded Pile Solutions 570 D. l Introduction 570 D.2 Winkler Modulus Proportional to the Square Root of Depth 573 D. 3 Winkler Modulus Linearly Proportional to Depth 579 E Calculation of Soil Resistance - Lateral Pile Displacement Curves 580 E. l Methodology 580 E. 2 Comparison of Method With Cubic Spline Differentiation 583 F Single Pile Response in a Winkler Medium to Base Motion Excitation588 F. l Equations of Motion 588 F.2 Free Vibration Response 590 X i F. 3 Forced Vibration Response 592 G Uncoupled Solution for a Pile - Structural Mass System 596 G. l Equations of Motion 596 G.2 Free Vibration Response 596 G.3 Forced Vibration Response 599 G. 4 Pile Head Stiffnesses 601 H Finite Element. Solution for a Pile - Structural Mass System 604 H. l Equations of Motion 604 H. 2 Solution of Equations of Motion 609 I Shake Table Tests - Low Level Shaking of 2-Pile Groups 612 I. 1 Test Data 612 1.2 Effects of Group Interaction on Lateral Soil Stiffness 621 J Uncoupled Dynamic Analysis of a Pile Group 627 J . l Finite Element Discretization 628 J.2 Dynamic Solution Methodology 633 x i i List of Tables 2.1 Summary of Pile Cap Structural Properties - Shake Table Tests for Two and Four Pile Groups 57 2.2 Instrument Noise Levels After Digital Filtering (Shake Table Tests) . . . 59 3.1 Centrifuge Scaling Relations 79 3.2 Summary of Model Pile and Pile Cap Structural Properties Used in Cen-trifuge Tests) 85 3.3 Summary of Pile and Pile Cap Structural Properties - Centrifuge Tests on Two Pile Group 87 3.4 Summary of Pile and Pile Cap Structural Properties - Four Pile Group (Centrifuge Tests) 89 3.5 Centrifuge Instrument Noise Levels 91 4.1 Parameters used in dynamic analysis of bender response to a travelling shear wave 108 4.2 Single Pile Test Characteristics - Centrifuge 112 4.3 Fundamental Frequencies of the Pile and Free Field 115 4.4 Winkler Model Predictions of Single Pile Deflections - Centrifuge Tests . 150 4.5 Computed Relative Soil-Pile Stiffnesses, KT, for Low Level Shaking . . . 179 4.6 Hyperbolic Stress-Strain Parameters Used in Finite Element Analysis . . 182 4.7 Computed vs. Measured Deflections and Bending Moments Using Equiv-alent Elastic Winkler Moduli - L A T P I L E Analysis 194 4.8 Cyclic p-y Curves - Masing Loop Parameters 201 xi i i 4.9 Pile Group Fundamental Natural Frequencies 207 4.10 Pile Group Test Data 210 4.11 Average Forces and Deflections of Centrifuged Pile Groups - Low Level Shaking .217 4.12 Pile Group Interaction Analysis - Centrifuge Tests -Low Level Shaking . 225 4.13 Measured and Computed Deflections in a 2 x2 Pile Group 236 5.1 Frequency Sweep Test Data - Free Field Response in Loose Sand 266 5.2 Frequency Sweep Test Data - Pile Response in Loose Sand 281 5.3 Frequency Sweep Test Data - Free Field Response in Dense Sand . . . . 290 5.4 Frequency Sweep Test Series III - Pile Response in Dense Sand 302 5.5 Frequency Sweep Test Series IV - Pile Response in Dense Sand 302 5.6 Ringdown Test Measurements - 2 Pile Groups in Dense Sand 315 5.7 Single Pile Test. Characteristics for Moderate Shaking - Shake Table vs. Centrifuge 327 5.8 Single Pile Test Characteristics'for Strong Shaking on the Shake Table . 334 5.9 Winkler Model Predictions of Single Pile Deflections - Shake Table Tests 351 5.10 Backbone P-y Curve Parameters for Shake Table Test 23 363 5.11 Pile Group Test Data - High Level Shaking 374 5.12 Average Forces and Deflections - High Level Shaking 378 5.13 Pile Group Interaction Analysis - Strong Shaking 383 6.1 Pile and Soil Properties Used in Test. Case 407 6.2 Superstructure Response to Harmonic Base Motion - Coupled Analysis (Test Case) 408 6.3 Superstructure Response to Harmonic Base Motion - Uncoupled Analysis (Test Case) 409 xiv 6.4 Ringdown Analysis Parameters - Tests R-L5 and R-D2 414 6.5 Uncoupled Analysis Parameters - Centrifuge Tests 431 6.6 Uncoupled Analysis Parameters - Shake Table Tests 432 6.7 Computed Versus Measured Pile Response 458 7.1 Computed Natural Frequencies - P G D Y N A vs. M A C E Solution 473 7.2 Computed Natural Frequencies With and Without Group Interaction Ef-fects - P G D Y N A 475 7.3 Computed Vs. Measured Pile Group Response for Strong Sinusoidal Shak-ing - Four Pile Group on the Shake Table 483 7.4 Computed Vs. Measured Peak Pile Group Response for Moderate Level Earthquake Shaking - Four Pile Group on the Centrifuge 486 C . l Elastic Soil Properties Used in Plane Strain Analytic Solution 564 C.2 Parameters Used in Plane Strain/Plane Displacement Analyses 568 1.1 Pile Group Test Data - Low Level Shaking 618 1.2 Average Forces and Deflections - Low Level Shaking 623 XV List of Figures 1.1 Cycl ic lateral load - displacement characteristics of soil in soft clay (a) zone of unconfined response (b) confined response (after Bea et al , 1980) 3 2.1 Shake table pump vibration recorded using a sampling rate of 1 k H z per channel (a) Shake table A - measured table accelerations, (b) Shake ta-ble A - com puted Fourier spectra, (c) Shake table B - measured table accelerations, (d) Shake table B - computed Fourier spectra 37 2.2 Typica l sinusoidal input base motions recorded using a sampling rate of 303 H z per channel - shake table A : (a) table accelerations - moderate in-tensity shaking (b) table accelerations - high int. ensity shaking (c) Fourier spectrum - moderate intensity shaking (d) Fourier spectrum - high inten-sity shaking 39 2.3 Typica l earthquake input base motions recorded using a sampling rate of 303 Hz per channel - shake table A (a) measured table accelerations (b) Fourier spectrum 40 2.4 Gradation curve for C-109 Ottawa sand used in shake table tests and comparison wi th Toyoura sand (after Tatsuoka and Fukushima, 1984) . . 41 2.5 Peak friction angles versus void ratio for C-109 Ottawa sand 42 2.6 Normalized secant, shear modulus versus cyclic shear strain for C-109 Ot-tawa sand (a) medium dense sand (Dr = 50 percent) (b) loose sand (DT = 30 percent) (c) dense sand (DT = 90 p ercent) 45 2.7 Single pile used in shake table tests showing instrumentation layout . . . 53 xvi 2.8 Two pile group used in shake table tests showing instrumentation layout 55 2.9 Four pile group used in shake table tests showing instrumentation layout 56 2.10 Body wave propagation from a bender source, showing the dependance of receiver location on measured body waves 64 2.11 Piezoceramic bender elements (a) single element (b) general layout of source and receivers 66 2.12 Electrical layout of bender elements (a) source (b) receiver 67 3.1 Side view of Caltech centrifuge (after Allard, 1983) 76 3.2 Vertical effective stress distribution in the sand taking into account g-gradients in the centrifuged soil model 77 3.3 Schematic drawing of centrifuge arm (after Scott, 1979) 81 3.4 Single pile used in centrifuge tests showing instrumentation layout . . . . 84 3.5 Two pile group showing instrumentation layout 86 3.6 Four pile group showing instrumentation layout 88 3.7 Gradation curve for Nevada 120 sand used in centrifuge tests 92 3.8 Peak friction angles versus void ratio after consolidation from drained tri-axial tests on Nevada 120 sand 93 4.1 Source and receiver bender element voltage outputs 105 4.2 Lumped mass mechanical model used to describe bender element response to a travelling wave pulse 107 4.3 Theoretical bender element response to a travelling shear wave 109 4.4 Shear wave velocities during centrifuge flight at 60 g (a) loose sand (b) dense sand I l l 4.5 Measured accelerations - centrifuge test 17 (a) input base accelerations (b) free field surface accelerations (c) pile cap accelerations 116 xvii 4.6 Measured accelerations - centrifuge test 41 (a) input base accelerations (b) free field surface accelerations (c) pile cap accelerations 117 4.7 Measured displacements parallel to shaking direction - centrifuge test 17 118 4.8 Measured displacements parallel to shaking direction - centrifuge test 41 119 4.9 Computed Fourier amplitude ratios ( A P H / A F F ) - test 17 120 4.10 Computed Fourier amplitude ratios ( A P H / A F F ) - test 41 121 4.11 Measured bending moment time histories - centrifuge test 17 (a) strain gauge 1 (b) strain gauge 3 (c) strain gauge 6 122 4.12 Measured bending moment distribution during steady state excitation (t = 16.5 sec.) - centrifuge test 17 123 4.13 Measured bending moment distribution during steady state excitation (t. = 17.0 sec) - centrifuge test 41 124 4.14 Measured accelerations - centrifuge test 12 (a) input base accelerations (b) free field surface accelerations (c) pile cap accelerations 127 4.15 Computed Fourier amplitude ratios - test 12 (a) pile amplitude ratio ( A P H / A F F ) (b) free field amplitude ratio ( A F F / A B ) 128 4.16 Measured displacements parallel to shaking direction - centrifuge test 12 129 4.17 Measured bending moment time histories - centrifuge test 12 (a) strain gauge 1 (b) strain gauge 4 (c) strain gauge 7 130 4.18 Measured bending moment distribution during peak pile displacement (t = 12.0 sec) - centrifuge test 12 131 4.19 Measured accelerations - centrifuge test 14 (a) input, base accelerations (b) free field surface accelerations (c) pile cap accelerations 132 4.20 Computed Fourier amplitude ratios - test 14 (a) pile amplitude ratio ( A P H / A F F ) (b) free field amplitude ratio ( A F F / A B ) 133 4.21 Measured displacements parallel to shaking direction - centrifuge test 14 134 xvi i i 4.22 Measured bending moment time histories - centrifuge test 14 (a) strain gauge 1 (b) strain gauge 3 (c) strain gauge 6 135 4.23 Measured bending moment distribution during peak pile displacement, (t = 11.0 sec) - centrifuge test 14 136 4.24 Measured accelerations - centrifuge test 15 (a) input base accelerations (b) free field surface accelerations (c) pile cap accelerations 137 4.25 Computed Fourier spectra - test 15 (a) pile amplitude ratio ( A P H / A F F ) (b) free field amplitude ratio ( A F F / A B ) 138 4.26 Measured displacements parallel to shaking direction - centrifuge test 15 139 4.27 Measured bending moment time histories - centrifuge test 15 (a) strain gauge 1 (b) strain gauge 3 (c) strain gauge 6 140 4.28 Measured bending moment distribution during peak pile displacement, (t = 10.9 sec) - centrifuge test 15 141 4.29 Lateral soil reaction distribution at peak pile deflection (t = 12.0 sec) -centrifuge test 12 145 4.30 Cyclic p-y curves at three different times during shaking at the 3 pile diameter depth - centrifuge test 12 146 4.31 Cyclic p-y curves at various depths during the shaking cycle when peak pile deflection occurred (t = 11.72 - 12.78 sec) and comparison with API curves - centrifuge test. 12 148 4.32 Equivalent lateral stiffnesses versus depth derived from experimental and A P I p-y curves - centrifuge test 12 149 4.33 Computed bending moment distribution during peak pile deflection (t = 12.0 sec) using lateral stiffnesses from experimental and A P I p-y curves -centri fuge test 12 151 xix 4.34 Cyclic p-y curves at various depths and times, and comparison with A P I p-y curves - centrifuge test 14 153 4.35 Equivalent lateral stiffnesses versus depth derived from experimental and A P I p-y curves - centrifuge test 14 154 4.36 Computed bending moment distribution during peak pile deflection (t = 11.0 sec) using secant lateral stiffnesses from experimental and API p-y curves - centrifuge test 14 155 4.37 Cyclic p-y curves at different times during shaking at the 3 pile diameter depth - centrifuge test 15 156 4.38 Cyclic p-y curves at various depths during shaking cycle when peak pile deflection occurred (t = 10.82 - 11.58 sec) and comparison with API p-y curves - centrifuge test 15 158 4.39 Equivalent lateral stiffnesses versus depth derived from experimental and A P I p-y curves - centrifuge test 15 159 4.40 Computed bending moment distribution during peak pile deflection (t = 10.9 sec) using secant lateral stiffnesses from experimental and A P I p-y curves - centrifuge test 15 160 4.41 Cyclic p-y curves in loose sand during sinusoidal shaking at various depths and comparison with A P I p-y curves - centrifuge test 17 162 4.42 Cyclic, p-y curves in very dense sand during sinusoidal excitation at various depths and comparison with A P I p-y curves - centrifuge test 41 163 4.43 Secant lateral stiffnesses versus depth derived from experimental and A P I p-y curves - centrifuge test 17 165 4.44 Computed bending moment distribution during steady state shaking using secant lateral stiffnesses derived from experimental and A P I p-y curves -centri fuge test 17 166 XX 4.45 Equivalent lateral stiffnesses versus depth derived from cyclic p-y curves -centrifuge test 41 167 4.46 Computed bending moment distribution during steady state shaking using secant lateral stiffnesses derived from experimental and A P I p-y curves -centri fuge test 41 168 4.47 Frictional damping ratios, D, versus dimensionless pile deflection y /d (a) test no. 17 (b) test no. 12 (c) test no. 15 173 4.48 Proposed relationship between 8 and Kr for various values of H/2r0 at zero frequency and comparison with other researcher 's relationships, (after Kagawa and Kraft, 1980a) 178 4.49 Plane strain finite element model of a rigid translating disc 181 4.50 Computed lateral load - deflection relationships for plane strain translation of a rigid disc in a no-tension soil (a) loose sand (b) dense sand 183 4.51 Computed lateral Winkler stiffnesses, k^, for plane strain translation of a rigid disc in a no-tension soil (a) loose sand (b) dense sand 185 4.52 Computed variation of proportionality constant 8 versus dimensionless pile deflection y/d in a no-tension soil 186 4.53 Computed relationship between zone of influence factor Ie and lateral pile deflection 188 4.54 Results of elastic analysis - test 17 (a) effective shear strains (b) effective shear moduli (c) Winkler moduli . 1 9 1 4.55 Results of elastic analysis - test 14 (a) effective shear strains (b) effective shear moduli (c) Winkler moduli 192 4.56 Results of elastic analysis - test 41 (a) effective shear strains (b) effective shear moduli (c) Winkler moduli 193 4.57 Construction of unloading and reloading curves based on the Masing rule 195 xxi 4.58 Computed Masing loops versus measured p-y hysteresis loops - test 12 (a) z/d = 1 (b) z/d = 2 (c) z/d = 3 (d) z/d = 5 (e) z/d = 7 199 4.59 Computed Masing loops versus measured p-y hysteresis loops - test 15 (a) z/d = 1 (b) z/d = 2 (c) z/d = 3 (d) z/d = 5 (e) z/d = 7 200 4.60 Typical input base and free field surface motions - centrifuge test 39 (a) base accelerations (b) free field accelerations 203 4.61 Pile cap response - inline shaking test 39 (s/d = 4) (a) pile cap accelerations (b) pile cap displacements in direction of shaking 205 4.62 Comparison of Fourier spectra - inline test 39 (s/d = 4) (a) pile cap ac-celerations (b) free field accelerations 206 4.63 Bending moment, vs. depth in a two pile group for s/d = 2 (a) offline shaking (b) inline shaking 208 4.64 Bending moment vs. depth in a two pile group - inline shaking - s/d = 6 209 4.65 Pile cap displacements (a) parallel and (b) perpendicular to the direction of shaking - s/d = 2 (8 = 45°) 212 4.66 Steady state pile cap displacements versus pile spacing ratio 213 4.67 Comparison of predicted and measured bending moments using a Winkler model for offline shaking (a) s/d = 2 (b) s/d = 4 (c) s/d = 6 216 4.68 Comparison of predicted and measured bending moments using a Winkler model for inline shaking (a) s/d = 2 (b) s/d = 4 (c) s/d = 6 218 4.69 Comparison of predicted and measured bending moments using a Winkler model - 8 ~ 45° (a) s/d = 2 (b) s/d = 4 (c) s/d = 6 219 4.70 Experimental versus Randolph and Poulos' interaction factor -quv (a) 3 = 0 degrees (b) 3 = 45 degrees (c) 3 = 90 degrees 22 9 xx i i 4.71 2 x 2 pile group response to low level sinusoidal shaking - test 43 (a) input base accelerations (b) free field accelerations (c) pile cap acceleration s (d) displacements at top of mass 233 4.72 2 x 2 group response for low level sinusoidal shaking - test 43 (a) strain gauge 1 (b) strain gauge 3 (c) strain gauge 7 (d) dynamic axial load . . . 234 4.73 Bending moment vs. depth for both directions of shaking in a four pile group - low level sinusoidal shaking 235 4.74 2 x 2 group response during earthquake shaking - test 46 (a) base acceler-ations (b) free field accelerations (c) pile cap accelerations (d) top of mass displacements 238 4.75 Computed Fourier spectra during earthquake shaking of a four pile group (a) base accelerations (b) free field accelerations (c.) pile cap accelerations 239 4.76 2 x 2 group response to earthquake base motion - test 46 (a) strain gauge 1 (b) strain gauge 3 (c) strain gauge 7 (d) dynamic axial load 241 4.77 Bending moment vs. depth at peak pile cap deflection during earthquake shaking of a four pile group 242 5.1 Bender element voltage outputs recorded on the shake table (a) receiver R - l (b) receiver R-2 (c) receiver R-3 247 5.2 Bender element voltage outputs - confirmatory test in dense vibrated sand (a) source (b) receiver 248 5.3 Receiver response during P-wave velocity measurements in dense sand using a hammer plate source and an accelerometer receiver 249 5.4 Shear wave velocities in loose sand foundations with no cyclic pre-strain on the shake table 251 5.5 Shear wave velocities in vibrated dense sand foundations on the shake table.252 xxi i i 5.6 Compression wave velocities in loose sand foundations with no cyclic pre-strain on the shake table 253 5.7 Compression wave velocities in vibrated dense sand foundations on the shake table 254 5.8 Typical hammer impact test (a) input, base accelerations, (b) Fourier spec-trum 258 5.9 Typical frequency sweep test at a predominant input frequency of 12 Hz (a) input base accelerations, (b) Fourier spectrum 261 5.10 Measured foundation surface settlements during natural frequency tests in loose sand 263 5.11 Hammer impact test in loose sand (a) Foundation surface accelerations, (b) the Fourier spectrum 265 5.12 Frequency sweep test in loose sand: normalized Fourier spectra of free field surface accelerations and comparison with S H A K E output 267 5.13 Ringdown test R - L l in loose sand (a) pile head displacements ( L V D T 2), (b) pile head displacements (LVDT 1), (c) bending moments (strain gauge 3) 27 1 5.14 Ringdown test R-L3 in loose sand (a) pile head acceleration, (b) pile head displacement (LVDT 1), (c) pile head displacement (LVDT 2), (d) bending mome nt (strain gauge 3) 274 5.15 Ringdown test R-L4 in loose sand (a) pile head acceleration, (b) pile head displacement (LVDT 1), (c) pile head displacement (LVDT 2), (d) bending mome nt (strain gauge 3) 276 5.16 Ringdown test R-L5 in loose sand (a) pile head acceleration (b) peak bend-ing moment distribution in the first cycle of ringdown (c) pile deflections in the first cycle of ringdown 278 xxiv 5.17 Hammer impact test H - L l in loose sand (a) pile head acceleration, (b) foundation surface acceleration, (c) pile head displacement (LVDT 2), (d) bending moment (strain gauge 3) 280 5.18 Frequency sweep test in loose sand: (a) normalized Fourier spectra of pile head accelerations; (b) normalized peak bending moment at strain gauge 3 2 83 5.19 Hammer impact test. HI-Dl in dense sand (a) Foundation surface acceler-ations (b) the Fourier spectrum 288 5.20 Hammer impact test HI-D2 in dense sand (a) Foundation surface acceler-ations (b) the Fourier spectrum 289 5.21 Frequency sweep test in dense sand: normalized Fourier spectra of free field accelerations and comparison with S H A K E output 291 5.22 Effect of location in sand container on measured free field surface response 294 5.23 Hammer impact test HI-Dl (a) Pile head accelerations (b) Fourier spec-trum of pile head accelerations (c) pile head displacements - L V D T 1 . . 296 5.24 Hammer impact test HI-D2 (a) Pile head accelerations (b) pile head dis-placements - L V D T 1 (c) Bending moment distribution along the pile at the start o f free vibration 298 5.25 Ringdown test R - D l (a) pile head accelerations (b) Fourier spectrum of pile head accelerations (c) pile head displacement - L V D T 1 299 5.26 Ringdown test R-D2 (a) pile head accelerations (b) pile head displace-ments - L V D T 1 (c) peak bending moment distribution during first, cycle of ringdown 301 5.27 Fourier spectra from frequency sweep test in dense sand (a) pile accelera-tions normalized with respect to base accelerations - test series III and IV ( b) pile accelerations normalized with resp ect to free field accelerations - test series III 304 x x v 5.28 Frequency sweep test in dense sand - normalized peak bending moments 305 5.29 Ringdown test P G R l - inline shaking at s/d = 2 (a) accelerations at the centre of gravity of mass (b) bending moments at the peak of the first cycle of ringdown 309 5.30 Ringdown test PGR6 - offline shaking at s/d = 2 (a) accelerations at the centre of gravity of mass (b) bending moments at the peak of the first cycle o f ringdown 310 5.31 Free body diagram of forces acting on a two pile group for (a) inline shaking (b) offline shaking 312 5.32 Pile head accelerations measured during a ringdown test on a single pile for comparison with offline pile group tests 314 5.33 Typical input base motions at 10 Hz used in pile group frequency sweep tests (a) low level shaking (b) strong shaking 316 5.34 Frequency sweep tests for a two pile group subjected to offline shaking (a) low level shaking (b) strong shaking 318 5.35 Frequency sweep tests for a two pile group subjected to inline shaking (a) low level shaking (b) strong shaking 321 5.36 Fourier amplitudes of free field accelerations - loose sand tests (a) base (b) mid-height (c) soil surface 325 5.37 Fourier amplitudes of free field accelerations - dense sand tests (a) base (b) mid-height (c) soil surface 326 5.38 Comparison of pile flexural response observed on the centrifuge and the shake table - moderate level shaking in loose sand 329 5.39 Comparison of pile flexural response observed on the centrifuge and the shake table - moderate level shaking in dense sand 330 xxvi 5.40 Theoretical versus measured bending moment distribution - shake table test 15 332 5.41 Input base accelerations (a) test 14 (b) test 25 335 5.42 Computed Fourier spectra of input base accelerations (a) test 14 (b) test 25336 5.43 Free field surface accelerations (a) test 14 (b) test 25 (c) test 23 338 5.44 Computed Fourier spectra of free field surface accelerations (a) test 14 (b) test. 25 (c) test 23 339 5.45 Pile head mass accelerations (a) test 14 (b) test 25 (c.) test 23 340 5.46 Computed Fourier spectra of pile head mass accelerations (a) test 14 (b) test 25 (c) test 23 341 5.47 Bending moment time histories - shake table test 14 (a) strain gauge 1 (b) strain gauge 4 (c) strain gauge 6 343 5.48 Peak bending moment distribution - shake table test 14 344 5.49 Peak bending moment distribution - shake table test 23 345 5.50 Peak bending moment, distribution - shake table test 25 346 5.51 Pile head mass displacements (a) test 23 (b) test 25 (c) test 14 347 5.52 Cyclic p-y curves - shake table test 14 (a) z/d = 1 (b) z /d = 3 (c.) z /d = 5 (d) z/d = 10 (e) z /d = 15 349 5.53 Computed secant lateral stiffnesses from p-y curves during steady state shaking - shake table test 14 350 5.54 Cyclic p-y curves - shake table test 25 (a) z/d = 1 (b) z /d = 3 (c) z /d = 5 (d) z/d = 10 (e) z /d = 15 353 5.55 Computed secant lateral stiffnesses from p-y curves during low and peak amplitude shaking - shake table test 25 354 5.56 Cyclic p-y curves - shake table test 23 (a) z/d = 1 (b) z /d = 5 (c) z /d = 10 (d) z /d = 15 (e) z /d = 25 357 xxv i i 5.57 Cyclic p-y curves at the 1 pile diameter depth for constant amplitude shaking - shake table test 23 358 5.58 Computed secant lateral stiffnesses from p-y curves during low and peak amplitude shaking - shake table test 23 359 5.59 Cyclic, p-y curves - shake table test 30 (low level shaking) (a) z/d = 1 (b) z/d = 3 (c) z/d = 5 (d) z/d = 10 (e) z/d = 15 360 5.60 Frictional damping ratios, D, versus dimensionless pile deflection y /d in loose sand (shake table test 23) 361 5.61 Frictional damping ratios, D, versus dimensionless pile deflection y / d in dense sand (shake table tests 14, 25 and 30) 362 5.62 Computed versus measured p-y hysteresis loops at various dimensionless depths - shake table test 23 (a) z/d = 1 (b) z/d = 5 (c) z /d = 10 (d) z /d = 15 365 5.63 Pile group response - high level, inline shaking (s/d = 2) (a) base acceler-ation (b) free field surface acceleration (c) acceleration at the e.g. of ma ss (d) pile cap displacement (LVDT 1) 36 9 5.64 Computed Fourier spectra - high level, inline shaking (s/d=2) (a) base acceleration (b) free field acceleration (c) acceleration at the e.g. of mass 370 5.65 Bending moment, vs. depth in a two pile group - high level shaking (s/d = 2) (a) offline shaking (b) inline shaking 372 5.66 Bending moment vs. depth in a two pile group - high level, inline shaking (s/d = 8) 373 5.67 Comparison of Fourier spectra - inline shaking (a) input base accelerations (b) pile cap accelerations (s/d = 4) (c) pile cap accelerations (s/d = 6) . 375 5.68 Influence of pile spacing on pile cap displacement for high level shaking. . 376 xxviii 5.69 Single pile test 34 - bending moments vs. depth for strong shaking and comparison to Winkler model 378 5.70 Comparison of predicted and measured bending moments using a Winkler model for high level, offline shaking (a) s/d = 2 (b) s/d = 4 (c) s/d = 6 . 379 5.71 Comparison of predicted and measured bending moments using a Winkler model for inline, high level shaking (a) s/d = 2 (b) s/d = 4 (c) s/d = 6 (d) s/d 8 381 5.72 Shake table interaction factor, 77\"\", and comparison with other experimen-tal and theoretical (Randolph- Poulos) results (a) inline shaking (b) offline shaking 385 5.73 2 x 2 pile group response - high level shaking (s/d = 3) (a) base acceleration (b) free field surface acceleration (c) acceleration at the e.g. of mas s (d) pile cap displacement (LVDT 1) (e) pi le mass displacement (LVDT 2) . 387 5.74 Computed Fourier spectra (a) base acceleration (b) free field acceleration (c) acceleration at the e.g. of mass 388 5.75 2 x 2 pile group response - bending moment and axial load time histories (a) strain gauge 1 (b) strain gauge 3 (c) strain gauge 5 (d) dynamic axial lo ad -. 389 5.76 Average bending moment vs. depth in the four pile group and comparison with Winkler model 391 5.77 Cyclic axial loading test on a single model pile in dense sand 395 6.1 Computed bending moment distribution - coupled versus uncoupled ana-lytic solution 405 6.2 Computed pile and free field displacements relative to base motion - cou-pled analytic solution 406 x x i x 6.3 Computed versus measured ringdown time histories - test R- L5 (a) dis-placements at L V D T 2 (b) bending moments at the soil surface 415 6.4 Computed versus measured ringdown time histories - test R- D2 (a) dis-placements at L V D T 2 (b) bending moments at the soil surface 416 6.5 Experimental variation of the logarithm of pile head stiffness Kld ver-sus pile deflection y0 and comparison with A.P.I, recommendations - cent rifuge test 12 418 6.6 Normalized shear modulus-shear strain attenuation relationships used in S H A K E analysis of centrifuge tests 419 6.7 S H A K E analysis of free field test 10 (a,b) input base accelerations and computed Fourier spectra (c,d) computed versus measured Fourier spectra at found ation mid- depth and sand surface, respect ively 421 6.8 Fourier spectra of free field surface accelerations computed using S H A K E versus measured spectra - shake table test 14 423 6.9 Fourier spectra of free field surface accelerations computed using S H A K E versus measured spectra - shake table test 23 (a) measured spectra (b) computed spectra 424 6.10 Fourier spectra of free field surface accelerations computed using S H A K E versus measured spectra - centrifuge test 41 425 6.11 Fourier spectra of free field surface accelerations computed using S H A K E versus measured spectra - centrifuge test 12 (a) measured spectra (b) com-puted spectra 426 6.12 Free field surface accelerations computed using S H A K E versus measured accelerations - centrifuge test 12 (a) measured accelerations (b) computed accele rations 427 XXX 6.13 Computed pile response using exp't. pile head stiffnesses versus measured pile response - C.T. 41 (a) pile head accelerations (b) pile head displacemen ts (c) shear force at soil surface (d) ben ding moment at soil surface . . . 435 6.14 Computed pile response using exp't. pile head stiffnesses versus measured pile response - C.T. 12 (a) pile head accelerations (b) pile head displacemen ts (c) shear force at soil surface (d) ben ding moment at soil surface . . . 436 6.15 Computed pile response using A.P.I, pile head stiffnesses versus measured pile response - C.T. 12 (a) pile head accelerations (b) pile head displacemen ts (c) shear force at soil surface (d) ben ding moment at soil surface . . . 437 6.16 Computed pile response using exp't. pile head stiffnesses versus measured pile response - S.T. 14 (a) pile head accelerations (b) pile head displacemen ts (c) shear force at soil surface (d) ben ding moment at soil surface . . . 438 6.17 Computed pile response using exp't. pile head stiffnesses versus measured pile response - S.T. 23 (a) pile head accelerations (b) pile, head displacemen ts (c) shear force at soil surface (d) ben ding moment at soil surface . . . 439 6.18 Structural model used in SPASM8 analysis (a) centrifuge tests (b) shake table tests 442 6.19 Free field surface accelerations computed using S H A K E versus measured accelerations - centrifuge test 41 (a) computed (b) measured 447 6.20 Free field surface accelerations computed using S H A K E versus measured accelerations - centrifuge test 15 (a) computed (b) measured 448 6.21 Free field surface accelerations computed using S H A K E versus measured accelerations - shake table test 23 (a) computed (b) measured 449 6.22 Relative free field displacements computed using S H A K E - centrifuge test 41 (a) mid-depth of sand layer (b) top of sand layer 450 xxxi 6.23 Relative free field displacements computed using S H A K E - centrifuge test 15 (a) mid-depth of sand layer (b) top of sand layer 451 6.24 Relative free held displacements computed using S H A K E - centrifuge test 12 (a) mid-depth of sand layer (b) top of sand layer 452 6.25 Relative free field displacements computed using S H A K E - shake table test 23 (a) mid-depth of sand layer (b) top of sand layer 453 6.26 Computed shear force at the soil surface (cent, test 12) using radiation damping coefficients computed using the equations of Gazetas and Dobry (1984) and Lysmer and Richart (1966) 455 6.27 Computed shear force at soil surface using A.P.I, p-y curves and compar-ison with measured response - centrifuge test 12 459 6.28 Computed bending moment at soil surface using A.P.I, p-y curves and comparison with measured response - centrifuge test 12 459 6.29 Computed shear force at soil surface using experimentally derived p-y curves and comparison with measured response - centrifuge test 12 . . . . 460 6.30 Computed bending moment at soil surface using experimentally derived p-y curves and comparison with measured response - centrifuge test 12 . 460 6.31 Computed displacement at the top of the structural mass using A.P.I, p-y curves and comparison with measured response - centrifuge test 12 . . . . 461 6.32 Computed displacement at the top of the structural mass using exper-imentally derived p-y curves and comparison with measured response -centrifuge test 12 462 6.33 Computed shear force at soil surface using A.P .L p-y curves and compar-ison with measured response - centrifuge test 41 463 6.34 Computed bending moment at soil surface using A.P.I , p-y curves and comparison with measured response - centrifuge test 41 463 xxxi i 6.35 Computed shear force at soil surface using experimentally derived p-y curves and comparison with measured response - centrifuge test 41 . . . . 464 6.36 Computed bending moment, at soil surface using experimentally derived p-y curves and comparison with measured response - centrifuge test 41 . 464 6.37 Computed shear force at soil surface using A.P.I, p-y curves and compar-ison with measured response - centrifuge test 15 465 6.38 Computed bending moment at soil surface using A.P.I, p-y curves and comparison with measured response - centrifuge test 15 . . . . 466 6.39 Computed shear force at soil surface using experimentally derived p-y curves and comparison with measured response - centrifuge test 15 . . . . 466 6.40 Computed bending moment at soil surface using experimentally derived p-y curves and comparison with measured response - centrifuge test 15 . 467 6.41 Computed shear force at soil surface using A.P.I, p-y curves and compar-ison with measured response - shake table test 23 467 6.42 Computed bending moment at soil surface using A.P.I, p-y curves and comparison with measured response - shake table test 23 468 6.43 Computed shear force at soil surface using experimentally derived p-y curves and comparison with measured response - shake table\" test 23 . . . 468 6.44 Computed bending moment at soil surface using experimentally derived p-y curves and comparison with measured response - shake table test 23 . 469 7.1 Plane frame structural model for two pile group subjected to inline shaking on the shake table 474 7.2 Sample earthquake record used as input into P G D Y N A 476 7.3 Structural model for four pile group subjected to inline shaking on the shake table ' 481 xxx i i i 7.4 Pile group response computed using P G D Y N A versus measured response for a 4-pile group subjected to inline shaking on the shake table 482 7.5 Structural model for symmetrical four pile group tested on the Caltech centrifuge 487 7.6 Pile group response computed using P G D Y N A versus measured response for a symmetric 4-pile group subjected to moderate level, earthquake shak-ing on the c entrifuge 488 A . l Half bridge configuration to measure pile bending strains 534 A.2 Half bridge configuration to measure pile axial strains 535 A.3 Waveform aliassing: (a) sampling a lower frequency waveform at interval At (b) sampling a higher frequency waveform at interval At (afte r Sheriff and Geldart, 1983) 542 A. 4 Test case showing the effect of a low pass filter on a sine series (a) spectral input (b) comparison of unfiltered and filtered time series data compute d using a 50 Hz cutoff filter 546 B. l Chevron bridge circuit used in pile strain gauging 548 C. I Computed maximum shear strains around a vibrating disc - analytic versus finite element solution (a) 9 = 10 degrees (b) 6 = 40 degrees (c) 9 = 80 degrees 565 C.2 Pile displacement profiles yP(z) used in analysis of strain field around vi-brating pile 567 C.3 Computed maximum shear strains around a vibrating disc - plane dis-placement versus plane strain solution (a) 6 = 0 degrees (b) 9 = 40 degr ees (c) 9 = 90 degrees 569 xxxi v D. l Pile deflection, rotation, shear force and bending moment positive sign conventions 575 E . l Lateral loading of a cantilever beam - test case 584 E.2 Bending moment distribution along laterally loaded beam - test case . . . 586 E . 3 Distribution of lateral reaction force p(x) along a laterally loaded beam -test case 587 F. l Single pile model 589 G. l Single pile model - uncoupled analysis 597 H. l Structural layout assumed for finite element discretization of pile-structural mass system (uncoupled analysis) 605 I. 1 Pile group response - low level inline shaking (s/d = 2) (a) base accelera-tion (b) free field surface acceleration (c) acceleration at the e.g. of mas s (c) pile cap displacement (LVDT 1) 613 1.2 Computed Fourier spectra - low level inline shaking (s/d = 2) (a) base acceleration (b) free field acceleration (c) acceleration at the e.g. of mass 6 14 1.3 Bending moment vs. depth in a two pile group - low level shaking for s/d = 2 (a) offline shaking (b) inline shaking 616 1.4 Bending moment vs. depth in a two pile group - low level, inline shaking -s/d = 6 617 1.5 Influence of pile spacing on pile cap displacement for low level shaking. . 620 1.6 Pile group test PG4C - bending moments vs. depth for low level offline shaking (s/d = 6) and comparison to Winkler model 622 XXXV 1.7 Comparison of predicted and measured bending moments using a Winkler model for low level, offline shaking (a) s/d = 2 (b) s/d = 4 624 1.8 Comparison of predicted and measured bending moments using a Winkler model for inline, low level shaking (a) s/d = 2 (b) s/d — 4 (c) s/d = 6 . . 626 J . l Three dimensional beam element and local nodal numbering system . . . 631 J.2 Three dimensional lumped mass element and local nodal numbering system632 J.3 Three dimensional foundation spring element and local nodal numbering system 634 J.4 Global coordinate system 635 xxxvi Acknowledgement The author wishes to thank his research advisor, Professor Liam Finn, for having made arrangements permitting the author to carry out centrifuge testing at the California Institute of Technology and for many helpful discussions during preparation of this thesis. Special thanks are extended to Professor Ron Scott and Mr. John Lee at Caltech without whose assistance the successful completion of the centrifuge work would not have been possible. The author would also like to acknowledge the technical staff m the Department of Civil Engineering at U B C for their assistance with constructing the experimental equipment used in the shake table tests; in particular, Mr. Mark Abraham and Mr. Max Nazar. Numerous helpful discussions were held with fellow graduate students and professors in the Department, of Civil Engineering over the course of the present research. The author would like to acknowledge Professors Byrne, Campanella and Vaid for their helpful comments in the areas of soil dynamics, elastic wave velocity measurements and experimental soil mechanics. The structural dynamics courses taught by Professor S. Cherry proved particularly helpful in developing the analysis procedures used in the present thesis. Fellow graduate students A. Fileautrault and D. Dolan were particularly generous with their time and assistance with the shake table tests conducted. Ms. D. Cheveldeaw spent long hours assembling final figures into the thesis. Her patience and skill is gratefully acknowledged. Financial assistance for the present research was made possible through grants from the Canadian National Science and Engineering Research Council, Imperial Oil Limited and Franki Canada Limited. Hardy B B T Limited, the author's employer, provided com-puter facilities and time off from day to day consulting activities to permit the author to xxxvi i complete the present thesis. The above support is most appreciated. Finally, the author would like to acknowledge his wife Yvonne and his parents for their love and support over the course of preparation of this thesis. This thesis is dedicated to them and our new arrival, Christina. x x x v i i i Chapter 1 Statement of Research The response of piles to dynamic excitation, whether resulting from forces applied at their top due to wind, wave and machine loadings or from seismic waves propagating through the surrounding soil, has been the subject of considerable interest and research in recent years. Most of these studies have focused on the problem of pile response to cyclic pile head loading. However, many onshore and offshore pile supported structures are required to be designed for large earthquake loads for which there are few instrumental records detailing strong motion response of pile foundations. Because of the scarcity of instrumental data it has not been possible to validate certain available numerical models which are used to compute pile response to earthquake shaking. 1.1 Behavioural Aspects of Single Pile Response to Cycl ic Lateral Loading A number of researchers have carried out pile loading tests where the piles were subjected to cyclic lateral loading at the pile head. In these tests, the amplitudes of lateral pile motion were sufficient to induce significant non-linearity in the soil adjacent to the pile (the near field). Both full scale field tests (Matlock, 1970; Reese et al, 1974; Scott et al, 1982a; Blaney and O'Neill, 1986a; Han and Novak, 1988) and centrifugal modelling tests (Scott, 1979; Scott et al, 1982b, Barton, 1982; Prevost and Abdel-Ghaffar, 1982) have been carried out. These studies have shown that non-linear soil-pile interaction dramatically influences pile head response. In particular, the reduction in measured 1 Chapter 1. Statement of Research 2 natural frequency and increase in system damping with increasing dynamic load level has been observed, indicative of a strain softening system whose stiffness and hysteretic damping characteristics are load level dependent. It is worthwhile summarizing certain aspects of near field soil response observed during cyclic pile head loading to clarify the mechanisms responsible for strain softening. Soil response to cyclic pile head loading in saturated clay has been adequately summarized by Bea (1980) and Sangrey (1977). Figure 1.1a shows the net lateral resistance p of a soft clay to lateral pile motions y in the so-called unconfined zone where during a sufficient number of cycles of loading soil-pile separation (gapping) and soil remoulding occurs. The confining stresses in the soil near the pile head are not sufficient to close the gaps that may develop between the soil and the pile during the cyclic loading, resulting in the dog-boned hysteresis loops shown in the figure. The importance of this gap formation on dynamic pile response has been documented by Blaney and O'Neill. Figure 1.1b shows the hysteretic response of the soil in the zone of confined response. Here, soil confining pressures are sufficient to close any gaps. The cyclic response in this zone is similar to that derived from cyclic triaxial or simple shear tests. The softening of lateral resistance p with further load cycles having constant displacement amplitude y is indicative of pore pressure build-up during undrained loading. With sufficient number of cycles of loading, the clay may approach its critical state condition, at which point continued build-up in pore pressure and reduction in undrained strength and modulus ceases. Thus, there is a limit to the deterioration of lateral soil resistance. The concept of limited deterioration of soil resistance within the confined zone is in agreement with recommendations made by Matlock (1970) and Reese et al (1975) for construction of p-y curves in soft and stiff clays below a critical depth. The critical depth has been defined by Matlock with respect to the undrained strength of the clay and the Figure 1.1: Cyclic lateral load - displacement characteristics of soil in soft clay (a) zone of unconfined response (b) confined response (after Bea et al, 1980) Chapter 1. Statement of Research 4 width of the pile and corresponds to the starting point of confined lateral pile response. Above the critical depth, more severe soil degradation occurs due to the combined pro-cesses of pore pressure build-up, gapping, soil-water flow and soil erosion. According to Matlock's criteria for soft clay, the lateral resistance will reach a limiting value after a number of load cycles, although the lateral stiffness of the soil may be very low or non-existent until contact is regained between the pile and the soil during the cyclic loading. For brittle, overconsolidated or sensitive clays greater reductions are possible and allowed for in design using the recommendations of Reese et al. Near field soil response in saturated medium dense to dense sands has been described by Ting (1987) in an analysis of full scale shaker tests carried out by Scott et al (1982a), and by Barton (1982) in slow cyclic pile head loading tests carried out using the Cam-bridge geotechnical centrifuge. It is noted that pore pressure build-up around the pile was not significant during Barton's tests since they were carried out using a very slow rate of cycling. Ting's analysis indicates significant soil non- linearity occurs in the near field p — y response at shallow depths as a result of strain softening, soil-pile gapping and pore pressure build-up which caused transient liquefaction of the sand. The latter was substantiated by the presence of sand boils around the pile. At greater depths, near linear p — y response was observed. Barton's centrifuge tests showed that during the initial half cycle of load, sand near the head of the pile undergoes shear and rupture as the pile pushes into it. Behind the pile, the soil cannot sustain tensile stresses and so the pile breaks away leaving a cavity which can stand for short periods of time due to the transient suctions created in the sand. On reversal of the loading direction, with free water present, the previously ruptured material now becomes liquefied and flows into the cavity around the pile. The fluidized sand later becomes compacted as the pile moves across in the reversed direction. With subsequent Chapter 1. Statement of Research 5 load cycles, more material flows down around the pile and becomes densified at some depth below the surface. The result is that the lateral response becomes stiffer with number of cycles until a steady state is reached. This stabilization of pile head deflection after a number of cycles of loading has also been noted by Reese et al (1974) during full scale pile load tests. The combined processes of soil-water flow and densification result in the pile being pushed in the direction corresponding to the initial direction of loading. After the load is removed, the pile remains in a distinctly distorted shape and residual bending moments are induced. It is noted that Barton's tests were carried out in dense saturated sand so that the sand response would be expected to be dilative, allowing the sand response to stabilize after a number of loading cycles. In looser sands where liquefaction occurs due to the cyclic shearing and the processes of soil-water flow, such stabilization might not occur. The near field soil response to cyclic loading depends fundamentally on whether the soil is a contractive or dilative material. During cyclic shear loading of a saturated soil under undrained conditions, contractive soils such as normally consolidated clays and loose sands accumulate positive excess pore pressure. This leads to a reduction in stiffness of the material and strains progressively accumulate. Depending on the magnitude and number of cycles of loading and the initial state of effective stress in the soil, the cumulative pore pressure rise may cause the state of stress to approach a transient critical value K„ = q/p, where q represents the maximum shear stress in the soil and p is the mean effective confining stress in the soil. Based on cyclic undrained triaxial testing, Vaid and Chem (1982) have noted that for loose sands with relative densities less than about 45 percent, large increases in strain and pore water pressure occur when q/p reaches K„ which is less than the peak stress ratio at failure. Prior to this time, cumulative strain and residual pore pressure build-up is small. Depending Chapter 1. Statement of Research 6 on the initial static shear stress on the horizontal plane of loading, the residual pore pressures may increase to the point that effective confining stresses are zero, causing complete liquefaction. Continued cyclic shear loading to large enough strains results in dilation and a state of stress lieing along the static failure envelope. This leads to a reduction in pore pressure and a consequent build-up in shear strength. For overconsolidated clays and medium dense to dense sands which exhibit dilative behaviour during shear, cyclic undrained loading may cause negative excess pore pres-sures to accumulate, or, depending on the density of the sand (or pre-consolidation of the clay) and the magnitude and number of load cycles, the cyclic shear loading may induce residual positive pore pressure response (Sangrey, 1977). Vaid and Chern have shown for medium dense to dense sands (relative densities greater than approximately 50 percent) when q/p > K„ the rate of shear strain development increases and dilation occurs. The dilation counteracts any positive pore pressures which may have developed and rapidly stabilizes the deformation. This type of behaviour has been described as 'cyclic mobility' by Castro (1975). With shear stress reversal to levels below this critical threshold there is an increase in rate of pore pressure development. Pore pressure development is therefore facilitated by the absence of initial shear stress on the plane of loading such that rever-sal in sign of cyclic shear stress occurs. With continued cycling, pore pressure build-up and strain accumulation occurs. Ultimately, a transient state of liquefaction may result which promotes large strain development. After initial liquefaction, the sample regains its strength by dilating when it is further sheared, thus limiting additional deformation within the loading cycle. In summary, three distinct zones of soil behaviour can be identified for cyclic lateral loading of piles in saturated soils. At the pile head over approximately the 0 to 1 pile diameter depth (zone 1) there is an area of rupture and erosion, accompanied by the Chapter 1. Statement of Research 7 formation of a no tension cavity. Below this is a zone of plastic yielding, accompanied by gap development between the soil and the pile, pore pressure build-up and in the extreme liquefaction (zone 2). For clays, the undrained strength of the material may be sufficient to allow physical gaps between the soil and the pile to stay open below the zone of erosion. In the case of sands, it is necessary for transient suctions to be set up in the soil mass for gapping to occur. It is more probable that the creation of liquefied zones of sand around the pile having very low shear resistance would be interpreted from p-y curves developed from cyclic load tests as indicative of a 'gap'. At greater depths in excess of 10 pile diameters (zone 3) where lateral pile displacements are relatively small, the soil response is approximately linear elastic. 1.2 Observations of Ful l Scale Pile Response Dur ing Earthquake Loading Significant damage to pile supported onshore structures has occurred during major earth-quakes such as the Niigata and Alaska earthquakes of 1964 and the San Francisco earth-quake of 1906 (Kagawa, 1980). For example, a number of pile foundations for bridge structures suffered damage during these earthquakes. Liquefaction was the major cause of this damage, resulting in failure and slides of the foundation soils which caused un-acceptably large displacements of pile foundations. The buckling failure of steel pipe piles supporting the piers of the Showa River bridge during the Niigata earthquake was reportedly due to the resonance of the bridge as a result of soil-pile-structure interaction. During pile resonance, large lateral pile displacements occurred which allowed the uncon-nected simple spans to fall off the piers of the Showa bridge. An excellent summary of pile damage during earthquakes having occurred up to 1978 has been made by Kishida (1966), Ross et al (1969), Margason (1977) and Sugimura (1981). Unfortunately, no Chapter 1. Statement of Research 8 instrumental recordings detailing pile response to strong earthquake shaking exist. From the available pile damage reports, the following observations of pile performance during strong shaking have been made: 1. If the surrounding soils near the head of the pile fail by liquefaction or undergo excessive softening, the loss of lateral soil support combined with the presence of large structural inertia loads will cause large lateral pile displacements and racking of the superstructure. 2. If the surrounding soils along the pile shaft undergo excessive softening due to pore pressure build-up then damaging pile settlements could result if the remaining available shaft and tip resistance is insufficient to resist the imposed axial loads. For piles completely embedded in loose sands where the earthquake causes liquefaction below the pile tip, pile bearing failure and large building settlement results. 3. If the soil fails by shearing along an interface between soft and stiff layers, the pile will be subjected to potentially damaging bending. 4. If the soil does not fail, bending damage to the piles is governed by pile ductility rather than flexural rigidity. Maximum pile curvatures occur just below the pile cap for fixed head piles or at slightly deeper locations below the ground surface for pinned piles, and in locations where rapid changes in soil modulus occur such as near the boundaries between soft and stiff soil layers. There have been several case histories published in recent years documenting the field response of pile foundations to low level earthquake shaking, where free field surface accelerations have been less than about 10 percent of gravity (Hamada and Ishida, 1980; Esashi and Yoshida, 1980; Oda et al, 1980; Sugimura, 1977; Ohira et al, 1984; Abe et Chapter 1. Statement of Research 9 al, 1984). The available data cover a broad range of soil conditions where, in general, pile bending and axial strains, pile accelerations, free field accelerations and shear wave velocity distributions have been measured at various depths. The available instrumental data have confirmed observations based on pile damage reports; i.e. of large bending strains at the pile cap level for fixed head piles or at greater depths well removed from the influence of structural inertia forces due to the free field ground motions. Typically these larger bending moments have occurred at the transition between softer and stiffer layers. The following observations have also been made: 1. Pile response during earthquake shaking occurs over a broad range of frequen-cies with peak responses corresponding to the natural frequencies of the pile-superstructure system and of the ground. This reflects the combined influence of structural inertia loadings and free field ground motions on pile bending. 2. Dynamic axial strains are of significance in pile groups and occur in response to dynamic moments imposed by the superstructure on the pile group. The axial strain response occurs, generally, at predominant frequencies corresponding to the natural frequency of the superstructure. 3. At those depths along the pile well removed from the influence of pile head struc-tural loading and in absence of significant softening or shearing in the near field soils due to the earthquake motion, flexible piles generally follow the free field ground motion (Sugimura, 1977). Curvatures in the pile can therefore be estimated sat-isfactorily by performing a free field response analysis of the site. However, stiffer piles may not move entirely in phase with the soil. This has been documented by Tajimi (1979) who has noted that larger diameter piles have smaller accelerations than the free field soil at corresponding depths. Chapter 1. Statement of Research 10 1.3 Experimental Observations of Pile Group Interaction Since pile foundations generally consist of groups of closely spaced piles pile interaction effects must be considered. Interaction consists of two components: (a) alteration of soil stress states and physical characteristics when piles are installed in close proximity to each other (installation effects), and, (b) superposition of strains and alteration of failure zones in the soil mass due to the simultaneous loading of two or more neighbouring piles (stress overlap effects). These components are interdependent although most interaction analyses do not consider the influence of item (a) above. The effects of interaction for low frequency shaking would be expected to result in a reduction in the combined stiffness of the group relative to that computed from single pile stiffnesses, summed appropriately over all piles in the group. Unfortunately, the above instrumental data are insufficient to determine whether interaction effects between piles are significant during earthquake excitation since single pile data are not available for comparison with the measured pile group response. Available data documenting pile group interaction during cyclic pile head loading are described subsequently. Pile group interaction studies have been carried out during slow cyclic pile head loading using full scale testing (Matlock et al, 1980; Bogard and Matlock, 1983; Janes and Novak, 1985; Brown et al, 1987, 1988; Ochoa and O'Neill, 1989) and centrifuge modelling (Barton, 1982; Ting and Scott, 1984). Where structural inertia forces dominate pile group response, these tests provide insight into the significance of pile to pile interaction during low frequency, earthquake shaking. Higher frequency pile head shaking tests were also carried out by El Sharnouby and Novak (1984) using a group of 102 piles. The latter tests were carried out at model scale to provide a check of the ability of pile group interaction factors derived using the theory of elasticity to predict the combined stiffness of the group Chapter 1. Statement of Research 11 and its dynamic response. Unfortunately, a fundamental study of interaction between two piles for higher frequency shaking was not carried out, which would have facilitated analysis of the larger group. The extent of pile to pile interaction is strongly influenced by non-linear soil response and soil type. The cyclic loading tests conducted in soft clay by Matlock and his co-workers show that as cycling proceeds, vertical heave and soil-pile separation occurs near the tops of the piles. As a result, the lateral resistance is significantly reduced, reducing the stress transferred to the surrounding soil. For relatively large pile spacings where the gaps between the soil and the pile do not overlap, this effect minimizes group interaction. Where the piles are closely spaced and the gaps around each pile in the group overlap, the effects of group interaction will be very pronounced. At depths where the soil is sufficiently confined so that soil movement is essentially horizontal, group interaction occurs via a non-linear intact medium, similar to the interaction that occurs during static loading. Bogard and Matlock suggest that within the confined zone, the zones of plastic flow around each pile in a group of widely spaced piles limit interaction. For closely spaced piles, the plastic zones overlap which leads to more significant interaction. Cyclic pile head loading tests were carried out by Brown et al (1987) on a nine pile (3 x 3) group arranged at centre to centre spacings equal to three pile diameters. The piles were driven into stiff, saturated clay and subjected to two way, slow cyclic loading. The study showed that group deflections and maximum bending moments in any pile in the group were significantly greater than that of a single pile loaded to the same average load per pile in the group. The latter occurred in spite of the fact that gapping and scour occurred down to an unspecified depth around the head of each pile. The effects of group interaction were observed to increase with increasing magnitude of load. This implies that the zones of plastic flow around each pile in the group enlarged with increasing Chapter 1. Statement of Research 12 load level, leading to more pronounced interaction. Significant non-uniform load sharing was also observed among the piles in the group. The greatest portion of shear on the group was transferred to the leading row of piles in the group for a particular direction of loading, with successively less shear distributed to the middle and back rows. This effect is known as 'shadowing'. Brown et al (1988) also load tested the same group of piles described above after excavating the natural clay over the top 2.9 m and replacing it with sand fill. The sand was compacted around the piles to a medium dense state and subsequently flooded. As expected, the group as a whole was observed to deflect significantly more than an isolated single pile when loaded to a similar average load per pile for a given load cycle. Similar non- uniform load sharing was observed as described above, with leading piles for a particular direction of loading sustaining the greatest shear load. The shadowing effect was observed, however, to be more significant for the tests in sand than those in the stiff clay. It was also noted that the group as a whole was not softened by cyclic loading to the same degree as was the group in clay. This was attributed to densification of the sand around the piles. Ochoa and O'Neill (1989) in an analysis of the above test results have derived inter-action factors to express the increase in lateral pile deflection of one pile due to shear loading of an adjacent pile. Interaction factors were derived for the case of inline loading where the direction of lateral shear loading coincided with a line connecting the centres of the two piles. Ochoa and O'Neill note that the computed interaction coefficients were dependent on the load cycle considered, the lateral load level and whether inline interac-tion was computed for a pile located behind or in front of the loaded pile. An increase in interaction with increasing load level was observed which is consistent with the hypothe-sis of an expanding plastic zone around each pile as load level increases. Interaction was Chapter 1. Statement of Research 13 observed to decrease as number of cycles of loading increased. Interaction coefficients were also observed to be larger when computed for a pile located behind a loaded pile. The reason for this difference appears to be that the forward movement of the leading pile releases confinement from the soil providing passive resistance to the trailing pile, thereby increasing the flexibility of the trailing pile. For the case of offline loading, i.e. where a pile is located perpendicular to the direction of loading of a neighbouring pile, interaction was found to be substantially smaller relative to the case of inline loading. The pile group tests described by Janes and Novak (1985) are also particularly in-structive. The tests were carried out on small diameter (101 mm), closely spaced piles embedded in a multi- layered, natural soil profile. The lateral load behaviour of the piles would have been predominantly influenced by the presence of a surficial 0.4 m thick layer of loose topsoil overlying a 1.0 m thick layer of compact silty sand. The tests were carried out to derive factors expressing the effects of pile to pile interaction for various centre to centre pile spacings, loading directions and methods of loading. During initial monotonic loading, interaction between piles decreased with increas-ing pile separation, increasing pile deflection, and as the angular orientation (8) of an unloaded pile (pile 2) with respect to the direction of loading of an adjacent pile (pile 1) went from 0 through 180 degrees. For 8 = 0 degrees, pile 2 leads the loaded pile and therefore interaction effects are augmented by the fact that the loading of pile 1 is transmitted to pile 2 via compressional soil loading. For 8 = 180 degrees, pile 2 trails pile 1 and since cohesionless soil cannot sustain tension, the effects of loading pile 1 on pile 2 were found to be approximately half of those observed for 8 = 0 degrees. For offline loading (8 = 90 degrees), pile to pile interaction was found to be small and about the same as for 8 = 180 degrees. For very small deflections, less than about 0.5 percent of the pile diameter, pile to pile interaction was found to be about the same as predicted by Chapter 1. Statement of Research 14 elastic theory. With increasing pile deflection as plastic yielding of the soil around each pile in the group occurs, pile interaction interaction effects were found to be significantly less than predicted using elastic theory for centre to centre pile spacings in the range of 3 to 6 pile diameters. This suggests that the effect of plastic yield around piles in a group is to attenuate pile to pile interaction. Following initial monotonic loading, the piles were unloaded and then reloaded to provide an indication of the effects of cyclic loading on pile group interaction. Interaction effects were found to be significantly reduced from those observed during initial loading provided deflection levels during reloading did not exceed those produced during initial loading. Barton (1982) has also carried out slow cyclic loading of pile groups in dense sand using the Cambridge geotechnical centrifuge. The tests resulted in conclusions similar to those described above and showed that non-linear soil behaviour has a significant effect on pile interaction. The measured pile group response was also compared with predictions using pile interaction factors derived from elastic theory. The comparison showed that for piles at very close spacings (centre to centre pile spacings less than or equal to 3), elastic analysis underpredicted the extent of interaction. For larger spacings, elastic theory tended to overpredict interaction because soil yielding caused the influence of loading of one pile on another to be attenuated more quickly than predicted using elastic theory. Little data is available concerning the extent of group interaction during higher fre-quency pile head loading. Ting and Scott (1984) have subjected two and four pile groups embedded in saturated sand to dynamic pile head loading using the Caltech geotechnical centrifuge. Excitation frequencies in the range of 2 Hz were examined. They have noted little interaction for offline shaking over a range of pile spacings while interaction was more pronounced for inline shaking for centre to centre pile spacings of up to about 6 Chapter 1. Statement of Research 15 pile diameters. O'Neill (1983) has summarized available test data involving static and cyclic lateral loading of model pile groups under 1 g conditions. The data suggest little pile to pile interaction for offline loading where centre to centre pile spacings exceed 2.5 pile diame-ters. For inline loading interaction has been found to be unimportant for centre to centre pile spacings greater than 6 to 8 pile diameters. Blaney and O'Neill (1986a) suggest that limited interaction occurs for larger pile spacings due to inelastic soil deformation. At larger spacings, soil deformation during cyclic loading is caused primarily by propagation of elastic (low strain) body and surface waves. Their effect on an adjacent pile would be secondary for situations where relatively large amplitude pile vibration is of concern due to the small soil deformations that such waves produce. This is not necessarily the case for pile supported machine foundations where the response of the piles to low amplitude vibration is of greater importance. 1.4 Numerical Model l ing of Single Pile Response to Earthquake Loading The response of single piles to earthquake ground motion has been modelled theoretically using three main approaches: (a) elastic continuum models applicable to low level exci-tation, (b) finite element methods, and (c) lumped mass-spring models using empirically or analytically derived Winkler type springs and viscous dashpots. To obtain realistic estimates of pile response, the following aspects should be taken into account: 1. the change in soil properties with depth. 2. the non-linear hysteretic stress-strain behaviour of the near field soil which will be modified during shaking due to strain softening, pore pressure rise and soil-pile gapping. Chapter 1. Statement of Research 16 3. the variation of ground motion with depth in the soil profile. 4. the three dimensional displacement pattern of the soil as the pile pushes into the soil. 5. the influence of radiative energy losses. Elastic continuum models advanced to date cannot incorporate item 2 above with any degree of rigour. The work of Novak, Nogami and their co-workers (Novak, 1974; Nogami and Novak, 1977; Novak and Aboul-Ella, 1978; Novak et al, 1978; Novak and Sheta, 1980,1982) is particularly significant in advancing solutions to the problem of an elastic beam vibrating in a homogeneous or multi- layered elastic isotropic continuum subjected to dynamic pile head loadings. The principal problem in using these methods lies in selecting appropriate equivalent elastic moduli compatible with strains occurring around the pile during shaking. Such 'small strain' solutions have been used to provide pile head impedance parameters which can be defined as the transfer functions describing the ratios between the displacement response at the head of the pile, which is complex valued, and the surface harmonic exciting force (see chapter 6). These are often used as foundation spring and dashpot parameters in an uncoupled analysis of a superstructure subjected to the free field surface motions induced by the earthquake. There are two assumptions inherent in the above approach. The first is that the soil stiffness near the head of the pile and the pile head response is dominated by soil strains induced by structural inertia loads rather than from the influence of the free field ground motions. In absence of liquefaction or significant softening of the ground near the soil surface, the softening of the ground due to the earthquake is likely to be more important at greater depths. The second assumption used in an uncoupled analysis is that one may use the free field surface accelerations as input into the base of the superstructure, Chapter 1. Statement of Research 17 thereby neglecting the influence that the relative soil-pile stiffness may have on ground motions in the vicinity of the pile head at the ground surface. The latter is referred to as kinematic interaction. The influence of kinematic interaction on differences in pile acceleration at the ground surface relative to that in the free field has been studied by Gazetas (1984). He concludes based on dynamic elastic finite element analyses that the effects of kinematic interaction are relatively minor for excitation frequencies up to about 1.5 times the fundamental frequency of the free field. Similar conclusions have been made by Waas and Haartman (1984). Provided the free field surface motions are dominated by frequencies in this range, the neglect of kinematic interaction in an uncoupled superstructure analysis appears to be valid. The neglect of kinematic interaction generally results in an overestimate of dynamic pile cap motions transmitted to the superstructure. Pile head impedances derived using elastic theory are most appropriate for low level shaking where the dynamic pile head forces induce essentially elastic strains in the soil around the pile of the order of I O - 2 percent or less (Ishihara, 1982). The derivation of impedance functions using elastic theory also demonstrate their sensitivity to assump-tions used in the analysis. A number of issues have been addressed in detail by Novak, Nogami and their co-workers including the importance of out of plane soil response on lateral soil resistance at a particular depth z, and the significance of rotational soil resis-tances due to angular rotation of the pile and axial load on the dynamic response of the pile. Novak (1979) has noted that for slender piles the rotational soil resistance does not greatly influence the dynamic pile response and that incorporation of axial load effects is important only for slender piles in very soft soils. In the majority of cases, the latter effects may be neglected and prediction of soil reaction forces is made by considering a translational pile displacement mode. Chapter 1. Statement of Research 18 Theoretical visco-elastic soil resistances have been based on analytic solutions to Navier's equations of motion assuming either plane strain or plane displacement modes of vibration in an infinite visco-elastic medium. In the latter case, lateral soil response at a particular depth is affected by all six strain components although a simplification is made in the formulation that soil displacement is confined to the horizontal plane of loading. With the plane strain assumption only the three in plane strains are considered. This is analogous to the Winkler hypothesis which states that soil resistance at any point is due solely to soil deformation at that point. The mathematical formulations are expressed in terms of complex shear moduli, the real component of which represents the secant elastic stiffness of the soil and the imag-inary component accounts for frictional (hysteretic) damping. Consequently, the com-puted dynamic soil resistance has both a real and imaginary component. The real com-ponent represents the elastic stiffness of the soil while the imaginary part is indicative of energy losses due to wave propagation away from the pile (radiation damping) and hysteretic damping in the soil. For low strain excitation, hysteretic damping is small and system damping is dominated by radiative energy losses. The real component of the plane strain elastic stiffness (lateral force per unit length divided by unit lateral dis-placement) has been found to equal zero at zero frequency. This may be anticipated from the two dimensional case of a line load on the surface of a semi-infinite elastic half space where displacements at the point of loading are predicted to go to infinity. For higher frequency loading, the plane strain soil stiffnesses are very similar to analytical predictions using the plane displacement assumption (Novak, 1974; Nogami and Novak, 1977). The imaginary component of the plane strain stiffness, kl™, has been found to in-crease approximately linearly with increasing frequency, similar to predictions using the plane displacement solution. This is indicative of velocity proportional, radiative energy Chapter 1. Statement of Research 19 losses. Differences do exist, however, in the amount of radiation damping predicted by the two solutions especially for more flexible piles in the low frequency range. Equivalent viscous dashpot coefficients, cv , may be defined from the imaginary stiffness coefficients to represent radiation damping using cv = kx™/u>. Frequency dependent and frequency independent values of this dashpot coefficient have also been proposed by Gazetas and Dobry (1984) and Lysmer and Richart (1966), respectively. It has been found that the dashpot coefficients proposed by Lysmer and Richart, which absorb P and S waves and increase in direct proportion to the pile radius, provide a reasonable match of values computed using the plane displacement solution (Kagawa and Kraft, 1980b). Based on the above comparisons, Novak and Nogami have suggested that plane strain soil compliances can be used provided low frequency corrections are applied. In the low frequency range, the soil stiffnesses are assigned static values derived from three dimensional theory. Nogami and Novak use static stiffnesses when the dimensionless frequency a 0 = wr0/Vs is less than about 0.3, where to is the excitation frequency, r 0 is the pile radius and Va is the effective shear wave velocity in the depth range where maximum pile bending occurs. Either plane displacement (Nogami and Novak, 1977) or Mindlin's static solution for the displacement of a point resulting from a lateral force applied inside an elastic medium (Penzien, 1970) may be used to define the static stiffnesses. Radiation damping is theoretically predicted to be zero below the fundamental frequency of the soil layer using plane displacement theory since a very weak progressive wave is generated below resonance of the layer (Nogami and Novak, 1977). This characteristic is not predicted using a plane strain solution. Using plane strain soil reactions with the appropriate low frequency corrections, No-vak (1974) has derived pile head impedance functions for a homogeneous elastic medium. Chapter 1. Statement of Research 20 Nogami and Novak (1977) derived similar pile head impedances using the plane displace-ment soil reactions. The latter have also compared with three dimensional, elastic finite element solutions for the case of pile head loading (Kuhlemeyer, 1979; Blaney et al, 1976). The results were found to be similar, suggesting that plane strain models of soil response can provide reasonably accurate predictions of pile head impedance. It is apparent that the use of plane strain soil reactions allows specification of non-uniform moduli along the pile since soil response at one depth depends only on soil properties at that depth. This technique has been adopted by Novak and Aboul-Ella (1978) in their development of an efficient computer program (PILAY) to compute pile head impedance in a multi-layered viscoelastic medium. This program is extensively used in the elastic analysis of pile response to low amplitude vibration. Novak and Nogami's work has been extended by Kagawa and Kraft (1980a) to the case where pile excitation is due to seismic base motion. This work has examined whether neglect of the free field ground motions in solving the dynamic equations of motion of the pile has a significant effect on the computed dynamic soil reactions. Homogeneous, elastic soil response has been assumed and soil reactions around the perimeter of the pile determined by solving simultaneously Navier's equations of motion for a viscoelastic medium and the dynamic equations of beam flexure. The solution has included soil inertia terms involving the input base accelerations, which have been neglected in the work of Nogami and Novak. Kagawa and Kraft have simplified the problem by making an assumption that soil displacements due to the lateral pile motion occur in a horizontal plane. The plane displacement solutions have been found to be in good agreement with three dimensional finite element solutions where vertical displacements are considered (Kagawa and Kraft, 1980b). The plane displacement solution demonstrates that the elastic soil stiffness (i.e. Chapter 1. Statement of Research 21 the real component of soil reaction per unit lateral pile displacement) at a point has some dependence on pile curvature since gradients of lateral displacement in the vertical direction are considered. This has been expressed by Kagawa and Kraft in terms of a dependence on the relative soil-pile stiffness, Kr, defined as Kr = EI/EarQ where EI is the pile's flexural rigidity, Ea is the equivalent elastic Young's modulus of the soil and r 0 is the pile radius. A similar three dimensional effect has been examined for the case of static lateral pile head loading by Poulos (1980). The solution for lateral soil stiffness, while limited by the assumption of linear elastic response, enables comparisons to be made with solutions considering only the case of cyclic pile head loading. The lateral soil stiffness computed by Kagawa and Kraft has been found to vary over the length of the pile. Kagawa and Kraft (1980b) have averaged the real component of the lateral soil stiffness, k™, over the pile length using an averaging procedure based on considerations of elastic strain energy. This gave k™ = 6\\ Es where the proportionality factor Si was found to vary with Kr and pile length to diameter ratio. A similar averaging procedure was carried out for the imaginary component of soil stiffness, kx™, giving = 82Ea. Here S2 is an average proportionality constant which increases in an approximately linear manner with increasing frequency. The computed proportionality constant Si was found to be moderately higher for the case of earthquake excitation relative to the case where only cyclic pile head loading was considered. The extent of the difference was dependant on the value of KT and the frequency range of interest. For extremely flexible piles (Kr = 1 0 - 7 or less) in the frequency range up to about 5 times the fundamental frequency of the free field soil, Kagawa and Kraft found no effect of method of pile excitation on the computed proportionality constant. For rigid piles (Kr = 1), Si was up to 34 percent less than values computed where only pile head loading was considered, identical to the case examined by Chapter 1. Statement of Research 22 Nogami and Novak. Computed radiation damping was found to be fairly similar for the case of pile head and earthquake excitation for Kr in the range of 10 - 4 to 10 - 7 and pile length to diameter ratios of up to 70. Larger discrepancies were predicted to occur for higher frequencies and for short, stiff piles. The radiation damping for the case of seismic loading was found to be reasonably well simulated using the viscous dashpot coefficients proposed by Lysmer and Richart. The results of Kagawa and Kraft indicate that the method of pile loading can have an effect on the lateral soil response and provide a theoretical basis for estimating the importance of relative soil-pile stiffness, pile length to diameter ratio and excitation frequency on dynamic elastic compliance. It is noted that the lateral soil compliances can be assumed to remain constant during shaking using equivalent elastic soil parameters or could be varied for each time step during a step by step integration of the equations of motion of the pile. The results of Kagawa and Kraft also suggest that the elastic dynamic stiffness is reasonably independent of frequency in the low frequency range. They note that during an earthquake, the frequency of most interest for onshore and offshore pile supported structures is about 0.1 to 2 Hz, which corresponds to the likely range of structural fundamental frequencies, although typical earthquake records include frequency components up to about 25 Hz. Therefore, we are most concerned with soil-pile interaction during low frequency shaking. Elastic analyses can be applied provided one can estimate equivalent elastic soil prop-erties appropriate to strains occurring in the near field soil during pile vibration. As a pile pushes into the soil, shear strains in the near field soil are very high immediately adjacent to the pile and decay rapidly with distance from the pile. At a sufficient dis-tance from the pile, the shear response of the soil is unaffected by the pile vibration and is governed by the intensity of ground shaking in the free field. Given the large strain Chapter 1. Statement of Research 23 gradients around the pile, it is difficult to estimate an average effective modulus for use in analyses based on concepts of homogeneous elasticity. Attempts have been made by Kagawa and Kraft (1981a) to calculate an effective shear strain around the pile based on the computed lateral pile deflection and estimates of an appropriate zone of influence around the pile in which to carry out the strain averaging. Their methodology esssentially follows that proposed by Skempton (1951) to estimate the settlement of spread footings. Novak and Sheta (1980), recognizing the large strains existing in close proximity to the pile, proposed a 'two zone' visco-elastic plane strain model to represent the dynamic soil response. In this model, an inner annulus of elastic material was chosen to represent the softened near field soil and was surrounded by an outer elastic region representing the free field. The model was proposed for the case of cyclic pile head loading but could be modified for the effects of base motion excitation. The uncertainties in selecting the thickness and properties of the near field zone make this model difficult to apply and therefore dynamic pile head loading tests are used to calibrate parameters used in the analysis (Han and Novak, 1988). To bypass the difficulties associated with selecting equivalent elastic moduli, various researchers have suggested that the lateral soil stiffness be modelled along the pile using a series of non-linear Winkler springs derived from full scale test measurements or non-linear finite element solutions (Yegian and Wright, 1973; Arnold et al, 1977; Matlock et al, 1978a,b; Bea et al, 1984; Nogami and Chen, 1987). The stiffness of these springs represents the combined stiffness of the strain softened, near field soil and the exterior free field soil whose properties are governed by the intensity of the earthquake ground motions. The soil stiffness for a particular depth is defined using as input a non-linear soil reaction versus lateral pile deflection (p-y) curve where y represents the relative deflection between the pile and the moving ground during shaking and p is the net soil resistance Chapter 1. Statement of Research 24 to the pile motion. The type of p-y curve input is dependent on the method of pile analysis chosen. For example, using equivalent static methods of analysis where the effects of the earthquake are replaced by pile head loads to represent superstructure inertia loadings and a dis-tribution of lateral ground displacements to represent the free field motion (Byrne and Janzen, 1984), a set of degraded p-y curves are input. These represent an envelope of soil resistance following shaking and do not account for the step-by-step degradation of the soil as shaking proceeds. Based on the distribution of p-y curves input along the pile, a secant stiffness distribution ks = p/y is computed which is compatible with the pile deflection level computed. Thus, a number of iterations are required until convergence is achieved between the assumed lateral secant stiffness and the computed pile deflection. Using a step by step solution of the dynamic pile response, it is possible to simulate the load-unload response of the soil to the lateral pile motions as shaking proceeds. For example, the dynamic pile analysis program, SPASM8 (single pile analysis with support motion), developed by Matlock et al (1978a) to model single pile response to earthquake shaking simulates the change in soil stiffness during loading or unloading by computing a soil stiffness kt tangent to a point along the input p-y curve compatible with the computed lateral deflection. To carry out this analysis, one inputs a backbone p-y curve which is appropriate for monotonically increasing lateral load either in the positive (+p, +y) or negative (— p, — y) sense. Unload-reload behaviour is simulated by reversing the prescribed backbone curve. For each complete reversal of lateral deflection the user can prescribe a degradation factor which reduces the ultimate lateral resistance by a certain percentage from the previous load cycle. The SPASM8 algorithm also allows one to simulate the formation of gaps between the soil and the pile. Other schemes to simulate unload-reload behaviour incorporating Masing behaviour are also possible (Kagawa and Chapter 1. Statement of Research 25 Kraft, 1981a). It is noted that by using a tangent soil stiffness approach hysteretic damping in the near field soil is automatically accounted for. Radiation damping is simulated by using equivalent viscous dashpots placed along the length of the pile derived from the elastic analyses previously described. Specification of the mathematical form of the backbone p-y curves for both static and cyclic pile head loading of piles in sand and clay are available from several sources (American Petroleum Institute, 1979; Stevens and Audibert, 1979; Gazioglu and O'Neill, 1984; Murchison and O'Neill, 1984). These recommendations have come from the results of full scale pile head loading tests. The cyclic p-y curves derived from simulated wave loading on piles represent an envelope of p-y behaviour after shaking but are often used as backbone p-y curves in earthquake analysis of single piles. The assumption implicit in their use is that the pile response is dominated by the effects of loading at the pile head and that the lateral soil response is reasonably independent of frequency in the low frequency range. Nogami and Chen (1987) have suggested a procedure in which the input low frequency p-y curves, derived from full scale test measurements or non-linear finite element analysis, can be adjusted to account for higher frequency loading. The procedure involves decomposing the input p-y curves into their static near field and free field components, with the free field properties derived from an appropriate free field response analysis. For most earthquake response analyses, however, this frequency correction would not appear to be necessary. The empirically based p-y curves suggested above are based on a relatively limited set of test data and are specific to local soil conditions. The most commonly used set of spec-ifications for constructing p-y curves is based on the recommendations of the American Petroleum Institute. The validity of their use for earthquake analysis of single piles has not been verified. Kagawa and Kraft (1980a,1981a) have also proposed that backbone p-y Chapter 1. Statement of Research 26 curves be constructed using more fundamental soil parameters with a mathematical form based on the hyperbolic or Ramberg-Osgood equation. These equations are defined from (a) the initial slope of the p-y curve using the measured low strain stiffness properties of the soil and results from elastic analysis of dynamic pile response, and (b) the ultimate lateral resistance of the soil to a translating pile derived from the theory of plasticity and based on the shear strength properties of the soil. Unload-reload p-y response is generally modelled assuming Masing behaviour. Similar approaches could be applied based on in situ determination of soil properties from pressuremeter tests (Atukorala et al, 1986). 1.5 Numerical Model l ing of Pile Group Behaviour to Static and Dynamic Loading 1.5.1 Low Frequency, Quasi-Static Loading Numerical studies of pile group interaction under static axial and lateral load have used three dimensional boundary integral and finite element analysis methods. The boundary integral formulations are based on Mindlin's solution which relates soil reactions at one point on a pile to the displacement produced at another point along an adjacent pile assuming homogeneous elastic soil conditions (Poulos, 1971). Imposing displacement compatibility at the soil-pile interface and utilizing the governing static beam bending equation for a pile embedded in an elastic medium, Poulos and Davis have solved for the soil reactions and displacements along the piles, subject to the appropriate pile head and tip boundary conditions. Poulos has presented interaction factors between two piles for various pile spacings, angular orientations and relative soil-pile rigidities for end-bearing or floating piles. Anal-ogous interaction factors have been proposed by Randolph (1981) and El Sharnouby and Chapter 1. Statement of Research 27 Novak (1985). These interaction factors relate the increase in deflection (or rotation) of one pile at the soil surface caused by pile head loading of an adjacent pile. Under elastic conditions, these interaction effects may be superimposed to yield a system of pile head flexibility equations for either free head or fixed head pile groups. The pile group flexibility matrix may also be inverted to give the static stiffness of the group. The solu-tion to the flexibility equations yields the distribution of pile head forces and moments throughout the group and pile head deflections provided the unit 'elastic' flexibility of a single pile is known. The latter procedure forms the basis of a static pile group analy-sis program developed by Randolph (1980) known as P I G L E T . The program has been developed for use in the microcomputer environment and is suitable for routine design. It is emphasized that the use of elastic interaction factors imposes restrictions on the range of validity of the above analyses since soil non-linearity has been shown to have a significant effect on the extent of pile interaction and predictions of load sharing in a pile group. Approximations to the model have been proposed by Poulos that permit the variation of elastic moduli with depth, soil-pile yielding, and soil-pile separation. This generally results in a reduction in the extent of interaction predicted. Focht and Koch (1973) incorporated the Poulos interaction factors to predict the pile head response with a non-linear p-y approach for determining bending moments and deflections along the piles in the group. The method involves computing single pile flexibilities appropriate to average load levels in the group using a non-linear single pile analysis, writing the flexibility equations for the pile group using elastic interaction factors, and solving for loads and deflections of the group at the soil surface. The bending response of the most heavily loaded pile in the group is then solved for using the single pile analysis, incorporating p- y curves along the pile which are softened sufficiently using 'y- multipliers' until a match of the computed group deflection is achieved. Chapter 1. Statement of Research 28 An extension of the Focht and Koch procedure has been described by O'Neill et al (1977) who used a structural frame analysis of battered pile groups in three dimensions, incorporating non-linear lateral and axial soil reactions along each pile. The non-linear axial soil response was modelled using t-z curves which express the soil resistance t to axial pile deformation z. Lateral soil response was modelled using p-y curves recommended by the American Petroleum Institute. The pile group response, including the distribution of lateral and axial loads along each pile, was initially solved neglecting pile to pile interaction. Successive iterations using Mindlin's elastic solution were used to compute added deflections due to the influence of adjacent piles at various depths. The approach represents an improvement on the computer program developed by Randolph which neglects the influence of interaction when computing bending moment distributions along each pile in a group. However, the use of non-linear p-y curves to account for non- linearity in the near field combined with Mindlin's elastic solution to predict interaction between piles is clearly inconsistent. The use of static elastic 2-pile interaction factors developed for the case of pile head loading may be used to estimate pile group stiffness for low frequency, quasi-static earth-quake shaking (Novak and Aboul-Ella, 1980; E l Sharnouby and Novak, 1985). With this approach, it is assumed that pile group interaction is dominated by pile head inertia forces induced by accelerations of the superstructure. Novak and Aboul-ella suggest that this is a reasonable assumption. The question also arises as to whether interaction factors derived for the case of pile head loading can be applied to the case where pile groups are subjected to earthquake shaking. Kagawa (1983) has examined this question using a dynamic Winkler analysis of pile groups, accounting for pile to pile interaction using a plane strain model. Based on this study, Kagawa has concluded that the use of interaction factors derived for the Chapter 1. Statement of Research 29 case of pile head loading may be used in the analysis of pile groups subjected to seismic excitation. 1.5.2 Higher Frequency Dynamic Loading Pile group interaction factors have also been developed for the case of dynamic pile head loading using three dimensional finite element and boundary element procedures assuming homogeneous and inhomogenous elastic soil response (Wolf and Von Arx, 1978; Waas and Hartmann, 1984; Kaynia and Kausel, 1982; Banerjee and Sen, 1987). Simpli-fications to these methods have been proposed assuming that interaction between two piles is primarily a plane strain phenomenon (Kagawa, 1983; Nogami, 1985b). Compar-ison of computed pile group response with that computed assuming three dimensional interaction have shown that this is a reasonable assumption. Wolf and Von Arx, Waas and Hartmann, and Kaynia and Kausel have also determined the elastic response of pile groups to earthquake excitation. Considering the case of dynamic pile head loading, two pile interaction factors have been developed over a range of pile spacings, angular orientations and forcing frequencies by Kaynia and Kausel, and also by Nogami. Simplified analytic expressions have been proposed by Dobry and Gazetas (1988). These enable one to adjust the static interaction factors for the fact that for higher frequency loading the waves propagating from a loaded pile in the group may be moving out of phase at the location of an adjacent loaded pile. The occurrence of this phase shift depends on the wavelength of surface and body waves emitted and the distance between piles. A phase shift results in seemingly anomolous 'negative' interaction coefficients which suggests that the stiffness of a dynamically loaded pile group can in fact be higher than the combined stiffness of a single pile multiplied by the number of piles in the group. The dynamic interaction factors also have an Chapter 1. Statement of Research 30 imaginary component which can be used to adjust single pile damping coefficients for group interaction. Pile group stiffnesses and damping factors have been computed by Kaynia and Kausel for a 4 x 4 pile group using the superposition approach of Poulos and dynamic two pile interaction factors. The results were compared with a fully coupled analysis of the pile group and were found to be in reasonable agreement. From this Kaynia and Kausel have concluded that the superposition approach using 2-pile interaction factors can be used to predict pile group stiffness for both static and dynamic pile head loading. The above dynamic elastic interaction analyses predict that pile interaction is sig-nificant for large pile separation distances. This may be true for low amplitude body wave propagation and is considered important in the design of machine foundations and in determination of radiation damping coefficients for pile groups. However, for large amplitude shaking of pile groups, the experimental data previously cited suggests that incorporation of soil non-linearity will significantly attenuate the extent of interaction and hence predictions of pile group stiffness for largely spaced pile groups. Novak and Sheta (1982) have demonstrated using 'two zone' elastic approaches that incorporation of a weakened elastic zone around each pile in the group reduces the group effect on stiffness but may increase the group effect on material damping. Seismic response studies of 2 x 2 and 10 x 10 pile groups carried out by Wolf and Von Arx assuming elastic soil response show that differences between amplitudes of soil displacement in the free field do not differ significantly from motions at the heads of the piles for excitation frequencies less than the natural frequency of the soil profile. Similar conclusions have been made by Kaynia and Kausel, Waas and Hartmann, and Tazoh et al (1988). This implies that the dynamic analysis of a pile group can be made using the motion of the free field as input to the base of the superstructure at the pile cap level for Chapter 1. Statement of Research 31 low frequencies of excitation. For higher frequencies, the effects of kinematic interaction cause pile cap motions to be less than those in the free field. To analyse the response of the group, substructuring methods may be used in which the superstructure is uncoupled from the pile foundation using a series of pile head stiff-nesses modified to account for the effects of soil non-linearity and pile to pile interaction (Randolph and Poulos, 1982; Novak and El Sharnouby, 1985). This is the most popular method of evaluating group effects since it is easier to use than a fully coupled analy-sis and is readily applicable because elastic interaction factors are available in the form of charts and formulae. The theoretical elastic interaction factors can also be adjusted based on empirical measurements to account for the effects of soil non-linearity. 1.6 Scope of Study The preceding review has summarized a number of computational procedures used to analyse the response of pile foundations to earthquake loading. Behavioural aspects of soil-pile interaction during dynamic lateral loading were also discussed which have been based mainly on studies of cyclic pile head loading. There are also instrumental records available which illustrate pile foundation response to low level earthquake excitation. Detailed measurements of pile behaviour are not available for stronger shaking so that it has not been possible to validate those analytical procedures which propose to account for soil nonlinearity. For foundations consisting of closely spaced piles, the available data are generally insufficient to determine whether pile group interaction effects are significant during earthquake loading or whether currently used elastic models of interaction can account for observed behaviour during low level shaking. Since data from instrumented pile foundations are not available for larger earthquakes, the capability of these elastic Chapter 1. Statement of Research 32 models in describing pile group response during strong shaking is also unknown. To supplement the existing data base, centrifuge and shake table model tests were carried out using single piles and pile groups embedded in dry sand and subjected to simulated earthquake loading. Data were obtained for a range of shaking intensities which were then compared with the predictions of selected numerical models used to predict single pile and pile group response to earthquake loading. A series of centrifuge tests were carried out using the geotechnical centrifuge located at the California Institute of Technology (Caltech). Preliminary testing was done in December, 1985 to determine if earthquake simulation tests could be successfully carried out on centrifuged model piles using the base motion actuator developed at Caltech (Finn and Gohl, 1987). More extensive testing was performed during March and April, 1986. Test data were obtained for single piles and two and four pile groups embedded in dry sand and subjected to low to moderately strong shaking. The two pile tests were carried out using low level shaking to ensure approximately linearly elastic soil response, permitting the checking of pile group interaction factors derived using elastic soil models. A limited number of four pile tests were also carried out using low level and moderately strong shaking to determine whether interaction factors determined from the previous two pile tests could be superimposed to give a reasonable estimate of the pile group displacements. An extensive series of tests were also carried out using shaking tables at the University of British Columbia (UBC). Unlike centrifuge tests, shake table tests cannot simulate prototype pile foundation behaviour since full scale stress conditions in the foundation are not achieved. The experimental results on small scale model behaviour can be used in the validation of computational models provided soil and pile characteristics can be determined at the low stress levels operative on the shake table. Natural frequency and forced vibration tests were carried out on both single piles and pile groups embedded in Chapter 1. Statement of Research 33 dry sand, subjected to a range of shaking intensities. Data from the shake table tests have also been compared with centrifuge test results to determine stress-level dependant differences in pile foundation response and whether pile to pile interaction effects observed on the centrifuge were similar to those observed on the shake table. Following completion of the above testing, behavioural aspects of soil-pile interac-tion were examined by computing p-y curves from the centrifuge and shake table test measurements over several cycles of shaking. Secant stiffness and hysteretic damping distributions were determined based on the cyclic p-y curves computed. Computation of single pile response to base motion excitation was also determined using non-linear methods of analysis and compared with measured centrifuge and shake table response. The commercially available computer program SPASM8 was used to compute the dynamic response of the model piles using a fully coupled analysis where the pile was linked to the input free field ground motions along the length of the pile. Elastic moduli were selected compatible with estimates of average strains in the free field soil and used to derive radiation damping parameters along the pile. Non-linear cyclic p-y curves have been used to model the interaction between the soil and the pile during shaking. These have been derived from the test measurements and comparisons made with p-y curves recommended by the American Petroleum Institute. Single pile and pile group response to earthquake shaking was also determined by uncoupling the free standing portion of the pile(s) and the superstructure from the em-bedded pile foundation. The latter was replaced by a series of frequency independent, non-linear springs which were found to depend on force levels applied to the pile head and, in the case of pile groups, interaction between the piles. The superstructure (i.e. those portions of the pile(s) and structural mass above the soil surface) were discretized using finite elements. The free field surface motions were then applied to the free field end Chapter 1. Statement of Research 34 of the foundation compliances and the uncoupled response of the superstructure solved for. Single pile head stiffnesses were determined appropriate to the shaking intensities used based on Winkler moduli distributions derived from back analysis of bending moment distributions measured during the tests. The computed stiffnesses were also compared with those estimated based on p-y curve distributions recommended by the American Petroleum Institute. Equivalent viscous dashpot coefficients at the pile head were as-sumed to be constant during shaking and were computed by integrating radiation and hysteretic damping contributions along the embedded portion of the pile using the com-puter program PILAY. Radiation damping was estimated based on shear wave velocity distributions computed from the free field response analysis. Hysteretic damping ratios along the pile were derived from cyclic p-y curves determined from the test measurements. Pile group response was solved for by modifying the single pile compliances for group interaction using low frequency interaction factors derived from the pile group tests. These were appropriately modified at higher frequencies using various charts proposed by Kaynia and Kausel. From the pile group interaction factors and single pile head flexibilities, pile group stiffness and damping parameters were derived. Chapter 2 Shake Table Test Procedures 2.1 U B C Shaking Table Characteristics Model pile tests were carried out using two different shake tables available in the civil en-gineering laboratories at U B C . The first series of experiments used the U B C Earthquake Laboratory shaking table (shake table A) which measures 3.0 m by 3.0 m. Subsequent tests used a smaller shaking table (shake table B) measuring 1.2 m by 2.4 m, located in the Soil Dynamics Laboratory. The principles of shake table operation are described below. Table motions are input using a hydraulic shaker which consists of a piston-cylinder, a servo-valve, a fluid pump and a driving electric motor. Hydraulic fluid (oil) is pressurized and pumped into the cylinder through a servo-valve which regulates the flow of oil to the piston and thereby controls its motion. The servo-valve is moved by means of a linear torque motor, which is driven by an external electrical excitation. This excitation comes from an M T S controller which is driven by a sine wave function generator or other computer controlled input. An L V D T monitors the displacement of the piston and sends the signal back to the controller. The controller compares the input signal supplied by the external function generator with the feedback signal. If a difference occurs, the controller will send a correction signal to attempt to make the two signals identical as described in detail by Ramsay (1982). The table hydraulics, actuator and feedback control system generate a continuous uni-directional shaking of the table and are capable of inputting a 35 Chapter 2. Shake Table Test Procedures 36 wide variety of sinusoidal or random earthquake motions into a test model. Hydraulic actuators are most suitable for heavy load testing and can be operated over a range of frequencies. Large displacements (stroke) are possible only at low frequencies. Although any general excitation can be input into the controller, faithful reproduction of these signals is virtually impossible because of distortion and higher order harmonics introduced by the high noise levels that are common in hydraulic systems. Ambient vibrations recorded by the table accelerometer when the hydraulic pump pressure for table A was at its maximum are shown in Figure 2.1a. Analogous data are shown for shake table B in Figure 2.1c. The acceleration data was recorded using a high speed data acquisition system which interfaced with an IBM PC microcomputer, described in Appendix A. A sampling rate of 1 kHz per channel was used for this series of tests so that frequencies of up to 500 Hz could be reliably discerned. Fourier spectra computed from the recorded accelerations are shown in Figure 2.1b and d, respectively. The figure shows that the table hydraulics create low amplitude accelerations of up to 0.03 g and 0.005 g for Tables A and B, respectively, and contain a number of higher frequency components. As discussed in section 5.2, some of these frequencies overlap the natural frequencies of the sand foundation, whose fundamental frequency has been found to be in the 40 to 60 Hz range depending on foundation density and the intensity of shaking. Where the frequency content of the pump vibration matches the natural frequencies of the sand, amplified accelerations at the soil surface have been observed. The first mode frequency of the model pile, on the other hand, lies in the 10 to 25 Hz range (see section 5.2). It has been found that the high frequency accelerations contained in the pump vibration are not significantly amplified by the pile since they are well removed from its resonant frequency. The shake table input was calibrated using sinusoidal base motion over a range of Chapter 2. Shake Table Test Procedures 37 (c) 0.05 H MEASURED TABLE B PUMP VIBRATION INPUT BASE ACCELEROMETER 0.04 -| 0.03 A 0.02-j o.oi A 0.00 J | - 0 . 0 H -0.02 A -0.03-1 -0.04 A -o.os 4-400 600 Time (msec) (d) Fourier Amplitude Pump Vibration - Shake Table B Frequency (Hz) Figure 2.1: Shake table pump vibration recorded using a sampling rate of 1 kHz per channel (a) Shake table A - measured table accelerations, (b) Shake table A - computed Fourier spectra, (c) Shake table B - measured table accelerations, (d) Shake table B -computed Fourier spectra Chapter 2. Shake Table Test Procedures 38 input frequencies varying between 5 and 70 Hz. This was done by varying the amplitude of table displacement (span) for a given input frequency and measuring peak table accel-erations using a storage oscilloscope. The feedback control system was fine tuned using procedures described in the table operations manual so that the shape of the input sine wave was optimized. Typical sinusoidal base motions produced by shake table A at an input frequency of 10 Hz and recorded using a sampling rate of 303 Hz per channel are shown in Figure 2.2. Average peak accelerations of 0.2 g and 0.6 g are shown in Figures 2.2a and c, respectively. Fourier spectra computed from the recorded base accelerations are shown in Figures 2.2b and d, respectively. The figure shows that higher frequency overtones are present in the table input and that these are more pronounced for lower intensities of shaking. Similar performance data were obtained using shake table B. Shake table A was also used to simulate earthquake excitation of the model pile. An earthquake record with a suitable frequency content was chosen from the U B C tape library. The time interval between acceleration data points was then scaled to shift the frequency content of the record so that it coincided with the natural frequency of the pile, thereby achieving maximum pile response. The scaled acceleration time history was converted to a binary format using an appropriate computer program which was used as input by the M T S controller. The binary record was then converted by the controller to an analogue signal which drives the servo-valve and controls the flow of oil to the table actuator. Peak acceleration levels delivered by the actuator were controlled by adjusting the span control on the shake table. The feedback control system was also adjusted to give optimum base motion characteristics. Typical table accelerations resulting from the earthquake input and the Fourier spec-trum computed from the recorded accelerations are shown in Figures 2.3 a and b, Chapter 2. Shake Table Test Procedures 39 (a) 3000 1000 5000 TLne I mi. 11 lsec I Figure 2.2: Typical sinusoidal input base motions recorded using a sampling rate of 303 Hz per channel - shake table A: (a) table accelerations - moderate intensity shaking (b) Fourier spectrum - moderate intensity shaking (c) table accelerations - high intensity shaking (d) Fourier spectrum - high intensity shaking Chapter 2. Shake Table Test Procedures 40 respectively. The earthquake selected for use was the vertical acceleration component of surface motions recorded at the Taft School recording station during the 1954 Kern County earthquake. The time interval between acceleration data points was reduced by a factor of 1/4, resulting in a predominant frequency content between 0 and 20 Hz. A high frequency component of motion also occured in the 40 to 50 Hz range which was due to mechanical vibration generated by the table actuator and feedback control system. 2.2 Foundation Sand Characteristics The model piles were embedded in a dry sand foundation which was prepared using C-109 Ottawa sand. This sand was chosen for the model tests since an extensive amount of laboratory test data exists at U B C characterizing its mechanical properties. Ottawa sand is a naturally occurring, aeolian deposit from Ottawa, Illinois. It has rounded particles with a minimum and maximum void ratio of 0.50 and 0.82, respectively (Negussey, 1984). Its mineral composition is primarily quartz with a specific gravity of 2.67. Pile tests were carried out in both loose and dense sand. A gradation curve for the sand is shown in Figure 2.4 which indicates an average particle size D50 of 0.40 mm and a coefficient of uniformity (DQO/DW) of 1.5. To estimate the sand's ultimate resistance to lateral pile motion and its shear modulus-shear strain attenuation characteristics, the peak effective friction angle of the sand must be known (Barton, 1982; Reese et al, 1974; Seed and Idriss, 1970). This has been determined from drained (CID) triaxial tests over a range of sand densities. Drained triaxial tests have been carried out by Wijewickreme (1982) on saturated sand prepared to different void ratios and isotropically consolidated using a confining stress of 200 kPa. Two additional tests were carried out in the present study using sand specimens Chapter 2. Shake Table Test Procedures 41 Figure 2.3: Typical earthquake input base motions recorded using a sampling rate of 303 Hz per channel - shake table A (a) measured table accelerations (b) Fourier spectrum Chapter 2. Shake Table Test Procedures 42 0.01 0.1 1 Grain Size , (mm.) Figure 2.4: Gradation curve for C-109 Ottawa sand used in shake table tests and com-parison with Toyoura sand (after Tatsuoka and Fukushima, 1984) Chapter 2. Shake Table Test Procedures 43 prepared to higher densities than achieved in Wijewickreme's tests. Densification was carried out by strongly vibrating the sand while in a rigid mold used to prepare the triaxial specimen. Vertical load was also applied to facilitate densification. In the first test, a saturated sand specimen was prepared to a relative density of 81 percent. Higher densities were not easily achieved using saturated sand. Therefore, a dry sand specimen was prepared using the above densification technique in which a relative density of 100 percent was obtained. The sand specimens were then isotropically consolidated using a confining stress of 50 kPa and sheared while allowing pore water (or pore air) to escape. Data from the above tests have been used to derive the peak effective friction angles, fimaxi °f ^ n e s a n ( i - These have been plotted versus initial void ratio in Figure 2.5 and show an approximately linear relationship. The above triaxial tests were carried out using an effective confining stress of 50 kPa or greater, which is approximately ten times larger than exist at the mid-depth of the sand foundation placed on the shake table. Peak drained friction angles are expected to be higher at the low stress levels operative on the shake table due to dilation effects. Vaid and Chern (1982) have summarized the conditions required for dilative behaviour to occur in C-109 Ottawa sand over a range of relative densities and confining pressures. Their data suggest that for the low stress conditions operative on the shake table, dilative behaviour will occur in the sand at larger strains for relative densities of about 35 percent or greater. At the larger confining pressures used in the above triaxial tests, the dilative tendencies of the sand are suppressed and peak friction angles may be less than would be measured at lower stress levels. The importance of stress level dependent changes in friction angle may be surmised from tests carried out by Negussey (1984) on C-109 sand prepared to a relative density of 50 percent and using confining pressures in the range of 50 to 250 kPa. There was Chapter 2. Shake Table Test Procedures 44 42 CO 40 Q) _U 38 Q) C < c o D Q_ 36 H :E 34 32 H A Wijewickreme, 1985 X Gohl, 1990 Avg. Line 30 H 28 0.5 0.6 0.7 Void Ratio 0.8 Figure 2.5: Peak friction angles versus void ratio for C-109 Ottawa sand Chapter 2. Shake Table Test Procedures 45 little change in peak friction angle observed over the above stress range, suggesting that stress level dependent changes in friction angle may not be significant for this particular sand. Drained triaxial and plane strain compression tests at confining stress levels as low as 0.03 kgf/sq. cm (approximately 3 kPa) have been reported by Tatsuoka et al (1986) on saturated and air dried Toyoura sand. This is an angular to subangular quartz sand having the gradation shown in Figure 2.4. It is considered that an angular sand will exhibit greater changes in dilation rate and hence peak friction angle with change in confining stress level compared to those in a more rounded sand (Vaid, 1987). Particular care was taken by Tatsuoka and his co- workers to account for the effects of sample self-weight and membrane force on computed stresses within the sample. Negligible change (less than 1 degree) in peak friction angle was found in the consolidation stress range of 3 to 50 kPa over a range of sand densities. Based on the above data, it seems reasonable to neglect stress level dependent changes in friction angle for the C-109 Ottawa sand in the low stress range of interest. The analysis of shear wave propagation through sand during earthquake excitation requires a knowledge of the shear stiffness properties of the sand. Low strain stiffness properties at a relative density of 50 percent have been measured using resonant column tests by Negussey (1984). These were carried out at confining stress levels of 50 kPa or greater, and in the strain range of 10\"6 to 10~4. It is noted that resonant column tests subject the sand sample to thousands of cycles of low amplitude shaking, and result in moduli stiffer than would be measured for fewer loading cycles. Negussey's data show that the low strain shear modulus, Gmax, is proportional to mean effective confining stress raised to the power of 0.47 which is consistant with other researcher's results (Hardin and Drnevich, 1972; Seed and Idriss, 1970). The secant shear modulus, G , normalized with respect to Gmax has been plotted versus cyclic shear strain amplitude in Figure 2.6 Chapter 2. Shake Table Test Procedures 46 for an effective confining stress of 50 kPa. Negussey's original data were given in terms of Young's modulus and cyclic axial strain which were related to shear modulus and shear strain using elasticity relationships and a Poisson's ratio, v, of 0.20. The latter value of Poisson's ratio has been determined from a comparison of compression and shear wave velocities measured during shake table testing (see section 5.1). The above data are compared with normalized secant shear modulus (G/GMAX) versus shear strain relationships proposed by Hardin and Drnevich (1972) for a relative density of 50 percent (see Figure 2.6a). A peak friction angle of 33 degrees, a KQ value of 0.4, and a confining stress of 50 kPa was used in the calculations. The computed curves have been found to be reasonably insensitive to slight changes in friction angle and are in good agreement with Negussey's experimental data in the strain range examined. Since data are not available for low stress levels typical of those in the sand foundation on the shake table, the Hardin and Drnevich relationships have been used to estimate shear modulus-shear strain attenuation curves at the 10 and 100 mm depths (see Figure 2.6a). The curves show that shear moduli will soften with increasing shear strain at a faster rate than at higher confining stresses. Similar curves have been computed for loose (relative density = 30 percent) and dense sand (relative density = 90 percent) in Figures 2.6b and c, respectively, for the stress range of interest in the shake table tests. Peak friction angles used in the calculations are based on triaxial test data described previously. These curves will be referred to in later sections for purposes of estimating strain dependent modulus reduction in the sand. Chapter 2. Shake Table Test Procedures 47 (a) 0.0001 0.001 0.01 0.1 Shear Strain (%) (b) (0 0.0001 0.001 0.01 0.1 Shear Strain (%) o.oooi o.ooi o.oi o.i Shear Strain (%) Figure 2.6: Normalized secant shear modulus versus cyclic shear strain for C-109 Ottawa sand (a) medium dense sand (DR = 50 percent) (b) loose sand (DR = 30 percent) (c) dense sand (DR = 90 percent) Chapter 2. Shake Table Test Procedures 48 2.3 Sand Foundation Preparation A level dry sand foundation was prepared in a rigid container bolted to the shake table. Two 25 mm thick styrofoam pads were placed at each end of the container to minimize wave reflection from the sides of the box perpendicular to the direction of base motion. The plan area of the soil container after the above inserts were installed was 463 mm by 463 mm with a total depth of 635 mm. Placement of large volumes of sand to achieve a uniform foundation density is a difficult task. A number of methods were considered at the start of the laboratory investigation including air pluviation through a series of screens (Miura and Toki, 1982) or using a travelling sand hopper to deposit the sand from a given drop height (Dean and Lee, 1984). These studies have shown that placement densities are sensitive to drop height and rate of flow. Vaid and Negussey (1984) have suggested that for air pluviation, placement densities could be significantly affected by the height of pouring, especially in the low range of drop heights. The question arises as to how sensitive lateral pile behaviour is to slight variations in initial sand density? The literature suggests that lateral pile behaviour will not be greatly affected by small variations in average density, since existing procedures to esti-mate lateral soil resistance distinguish the influence of sand density in a rather approx-imate manner (American Petroleum Institute, 1979). This is borne out by the present experimental investigation, as discussed in chapter 5. The method of pile installation after foundation placement may be a more important consideration since this affects soil behaviour in close proximity to the pile. Full scale field tests and centrifuge model studies have shown that pile behaviour under axial loading is significantly affected by the method of pile installation (Meyerhof, 1976; Ko et al, Chapter 2. Shake Table Test Procedures 49 1984; Craig, 1985). The influence of this factor on lateral load response appears to be less important (Craig, 1985). This is because flexible pile response is dominated by soil reactions in the upper part of the sand where installation induced stress changes are lowest. The nature of lateral pile loading also modifies the stress regime around the pile to a greater extent than in axial loading. It is postulated that the method of installation affects lateral pile behaviour for small vibration amplitudes during the first few cycles of loading. For larger vibration amplitudes and increasing numbers of cycles of loading, the sand structure in the near field is altered by the cyclic lateral loading so that the influence of method of installation is gradually erased. Experimental data from the present study show that the method of foundation densification and pile installation can affect lateral pile response to a small degree but that the intensity of shaking after installation is a more significant factor. Foundation preparation methods were selected that gave reproducible average foun-dation densities with a minimum of variation. Loose sand with an initial average void ratio of 0.72, or an average relative density of 31 percent, was prepared by spreading constant weights of dry sand uniformly over a perforated screen placed on the base of the container or on top of the previous sand layer. The newly placed sand was levelled by hand and then bulked by pulling the screen up through the sand. The elevation of the surface of the sand was measured at nine points with respect to a reference cross bar using a tape measure with 0.79 mm (1/32 in) graduations. An average thickness was then computed for each layer placed, allowing the average dry density and void ratio of the layer to be calculated. Using this procedure, void ratio variations from the average of ± 0 . 0 3 were computed, corresponding to a relative density variation of ± 1 2 percent. During sand placement a vertical array of piezoceramic bender elements was installed to measure shear wave velocities, described later. Instrumented model piles were then Chapter 2. Shake Table Test Procedures 50 pushed into the loose sand foundation by hand. The distance between the centre of the pile (or pile group) and the boundary of the soil container was 36 pile diameters. The distance between the soil surface and the underside of the pile head mass was also measured using a scale attached to the pile. Four lightweight settlement plates were then placed a minimum of 20 pile diameters from the centre of the pile or pile group to measure surface settlement. The settlement is due to the cyclic shear strains generated by the base motion. Following a series of single pile tests in loose sand, a dense sand foundation was achieved by densifying the loose sand using high frequency vibration with the pile in place (foundation densification method A). The loose sand foundation was shaken for several minutes using sinusoidal input with frequencies of 25 to 30 Hz and peak acceleration amplitudes of 0.5 g or greater. The large amplitude shaking was necessary to achieve significant sand densification which is believed due to the small confining stresses acting in the sand mass. It is expected that the large number of cycles of shaking of the originally normally consolidated sand has increased its K0 value, based on experimental data provided by Youd and Craven (1975). The model pile was restrained from movement in the sand during the vibration process by fixing the head of the pile using piano wire. An average void ratio of 0.57 or less was achieved, corresponding to relative densities of 78 percent or greater. From the settlement plate measurements, average void ratio changes in the dense sand foundations were found to be negligible during base motion excitation. A series of natural frequency tests on single piles (see section 5.2) were also carried out where the sand was placed in layers having constant weight and each layer densified using high frequency vibration (foundation preparation method B). In this way, care was taken to ensure the sand in the free field contained no loose zones. A relative density Chapter 2. Shake Table Test Procedures 51 of approximately 100 percent was achieved. The pile was then pushed into the soil by hand, assisted by high frequency vibration. This method of pile installation may have caused some loosening of the sand in the near field since the soil dilated strongly when the pile was pushed into it. These tests, when compared with similar tests using vibration method A, show the effect of differences in method of pile installation on lateral pile response. In the case of testing of pile groups, centre to centre pile spacings and pile group orientations were varied between tests. The tests were carried out using dense sand foundations prepared using vibration method A. After each test, the piles were extracted from the sand, the pile group configuration altered, and the piles pushed back into the sand assisted by high frequency vibration. The vibration process was continued for several minutes to ensure that the sand was fully densified around the piles. Following foundation placement, an accelerometer was placed in the surface of the sand a distance of 180 mm (28 pile diameters) from the centre of the model pile (or pile group) with the sensitive axis of the accelerometer oriented in the direction of shaking. Another accelerometer was placed on the base of the soil container to measure input accelerations. 2.4 Single Pile Characteristics and Model Layout Model studies are often planned using principles of dimensional analysis (Sabnis et al, 1983). Provided complete model similitude is achieved then the model test results, ex-pressed in terms of suitable dimensionless variables, are expected to simulate prototype behaviour. Given the low stress levels acting on the shake table model, complete simili-tude is very difficult to achieve. This may be seen by carrying out a dimensional analysis Chapter 2. Shake Table Test Procedures 52 of the laterally loaded pile problem subject to ground motion excitation. The dynamic lateral deflection, y, of the pile is assumed to be a function of the following variables: where b is the pile diameter, 1 is the pile length, pa and pp are the soil and pile mass density, respectively, g is the gravitational constant, EI is the flexural rigidity of the pile, Ga is the depth and strain level dependant shear stiffness of the soil, u> and UQ are the frequency and amplitude, respectively, of the input base motion, and m 0 is the structural mass at the top of the pile. Using principles of dimensional analysis one may show that the following functional relationship, K, applies: According to the principles of dimensionless analysis if the above dimensionless groups agree between model and prototype then the model test results can be scaled to give an accurate representation of prototype behaviour. It can be seen that the second dimen-sionless group in the above equation is satisfied if the same mass densities for the soil and the pile are used for the model and prototype test. The scaling law represented by the fourth dimensionless group is also satisfied if input base accelerations agree between model and prototype. Defining the model scale factor A as A = lp/lm, where subscript p refers to the pro-totype and subscript m refers to the model, the first dimensionless group is satisfied if «o ,p /u 0 , m equals A. This can be easily achieved but implies that the input frequencies used in the model test must be higher than occur in the prototype if input base accel-erations are to agree between model and prototype. Considering the fifth dimensionless group shown above and assuming that the mass density of the pile head mass is identical y = F(b, I, pa, pp, g, EI, Ga,u, u0, m 0) (2.1) (2.2) Chapter 2. Shake Table Test Procedures 53 between model and prototype then it is seen that this scaling law is also satisfied since Vm u: * - (2.3) where V refers to the volume of the pile head mass. To satisfy the third scaling law it is necessary that Ga,m Elmu0itn Assuming that the soil's stress-strain properties are represented by an equivalent elastic shear modulus, Ga, and that this is proportional to stress level for a particular level of shearing strain then it can be assumed that GSiP/GStTn is approximately equal to A. Thus, The above relationship is very difficult to satisfy since if one uses the same structural material for the model and prototype pile so that Ep = Em then the moment of inertia of the model must be extremely small relative to that of the prototype. For example, if one uses a model scale factor of 100 and assuming that the area moment of inertia I is proportional to b4 then Ip/Im = 108. This violates equation 2.5. A possible solution would be to use a model pile with an extremely thin pile wall or alternatively to use a material of lower modulus than used for the prototype. In the latter case, the scaling law involving the ratio of soil and pile mass densities would probably be violated. The above analysis indicates the difficulty in achieving perfect model similitude on the shake table. Instead, the approach has been taken of viewing the shake table test has a prototype event in itself. The shake table test results are then used to provide a data base against which the validity of selected computational models can be checked. The test results may, however, be expressed in terms of dimensionless variables of the Chapter 2. Shake Table Test Procedures 54 form suggested by Matlock and Reese (1960) or Franklin and Scott (1979) to allow comparison with full scale or centrifuge test results. The use of these dimensionless variables essentially eliminates the dependency of the test results on shaking intensity and confining stress level. The structural characteristics of the model pile and pile head mass used in the shake table tests were chosen to maximize flexure of the pile during lateral vibration so that one could clearly define bending moment distributions along the pile. It was also desirable to use a pile of sufficient length to ensure that pile tip reactions did not significantly influence the pile head response. This is important if one wishes to compare model test with full scale test results using appropriate dimensionless variables since a \"long pile\" boundary condition is achieved in the majority of field tests (Reese, Cox and Koop, 1975; Scott et al, 1982). Provided a foundation of sufficient thickness is used, displacement gradients are more pronounced over the depth of the sand foundation. The influence (if any) of free field displacements on pile response can then be determined. The model pile was constructed of hollow aluminum tubing having an outside diameter of 6.35 mm (0.25 in) and a wall thickness of 0.89 mm (0.035 in). The flexural rigidity (EI) of the pile was measured using transverse dead loading of the pile while it was clamped at one end and free at the other. From measurements of pile deflection versus applied transverse load, the pile EI value was computed to be 4.65 x 106 N-mm 2 (1621 lb-in2) using static beam bending formulae. A rigid pile head mass with a weight of 7.78 N (1.75 lb) was clamped to the head of the pile to simulate the effect of a superstructure (Figure 2.7). From the geometry and weight of the mass, the centre of gravity of the mass has been computed to be 25.35 mm (0.998 in) above its base. The mass moment of inertia for rotation about the centre of gravity of the mass has been computed to be 0.307 N-sec2-mm (0.0027 lb-sec2-in). The Chapter 2. Shake Table Test Procedures 55 distance from the underside of the mass to the tip of the pile is 610 mm (24.0 in). A sensitive ± 5 g, lightweight accelerometer and two displacement transducers (LVDT's) were mounted on the pile head mass to measure accelerations and displacements of the mass. Displacements were measured with respect to the moving base of the soil container. Knowing the vertical separation distance between the LVDT's, the rotation of the mass was also computed. In earlier tests, the accelerometer was placed at the top of the mass and measured accelerations somewhat different from those at the centre of gravity due to the influence of rotational acceleration. In later tests, the accelerometer was placed at the centre of gravity which proved convenient for calculation of the shear force applied to the pile at the soil surface. Seven pairs of calibrated strain gauges were placed at various points along the outside of the pile to measure peak bending strains (see Figure 2.7). The electrical leads from the gauges were brought up the inside of the pile through holes drilled through the tube wall. The wires were then passed through a slot in the pile head mass and connected to bridge completion and amplifier circuitry. Including the contribution of the strain gauges and electrical lead wires, the pile cross-section has a weight per unit length of 0.0005 N/mm (0.0028 lb/in). 2.5 Pile Group Characteristics and Model Layout Pile group tests were carried out on the shake table using a range of shaking intensities to examine pile group interaction effects and to determine whether currently used elastic interaction theories can be applied to determine pile group response to strong shaking. The piles were placed at various centre to centre spacings and arranged in two pile and four pile (2x2) groups, as shown in Figures 2.8 and 2.9, respectively. Two instrumented piles were used in the tests to permit estimates of load distribution in the group to Chapter 2. Shake Table Test Procedures Pile Head Mass L C G . + Strain Gauge Locations V ^No.3 No.4 No.5 No.6 No.7 Pile Head Accelerometer — LVDT No.2 LVDT No.3 /Free Field Accelerometer 80mm \"\"---Sand Foundations Model Pile Pile 610mm Long -7ZWX7777777777777777777777777, 7 77777? ^-Rigid Base of /-Table { Accelerometer Shaking Table 20mm Figure 2.7: Single pile used in shake table tests showing instrumentation layout Chapter 2. Shake Table Test Procedures 57 be made. The first instrumented pile (pile A) was used for the single pile tests. The second instrumented pile (pile B) was constructed from the aluminum tubing described previously. Pile A was instrumented with an additional strain gauge located near the mid-point of the pile to measure average axial load caused by rocking of the pile foundation during shaking. Five strain gauges were placed at various depths along pile B to measure peak bending strains. The piles in the group were rigidly attached to a pile cap whose design allowed the pile spacing to be varied. An additional mass was then bolted to the pile cap to simulate the effects of a superstructure. The pile cap and mass assembly was instrumented with an accelerometer placed at the centre of gravity and two LVDT's as shown in Figures 2.8 and 2.9. Pile cap dimensions and weights are given in Table 2.1 for both the two pile and four pile groups. From these, the location of the centre of gravity of the pile cap/mass assembly and its mass moment of inertia about both axes of shaking have been calculated as given in the table. An examination of Table 2.1 shows that the weights of the pile cap and mass on a per pile basis for the two pile group are the same as for the single pile. The locations of the centre of gravity of mass are also within 5 percent of that calculated for the single pile. Therefore, the effects of pile group interaction may be seen by direct comparison of pile group with single pile test data, for similar intensities of shaking. The centre of gravity of mass for the four pile group is nearly identical to that of the two pile group. Data from these tests have been used to determine whether group interaction factors determined from the two pile tests can be superimposed to yield the combined stiffness of the four pile group. The group stiffness may then be used in an analysis of the response of the group to earthquake excitation. Chapter 2. Shake Table Test Procedures 58 50.8 LVDT No. I (VARIABLE LOCATION) PLEXIGLASS BALL oo O' PILE CLAMP 1 0 S.G. 2-1 -777 7T7 \" 77-7 S.G. 2-2-S.G.2-3-BENDING STRAIN GAUGES -\\ (PILE No.2) \\ S.G. 2-4-S.G. 2-5 PILE TIP 386mm BELOW LINE A-A. BASE OF SAND ~6IOmm BELOW SOIL SURFACE. S.G. I-1 PILE CAP MASS(83.8 mm LONG) CG. (ACCELEROMETER) CABLE WITH STRAIN GAUGE LEAD WIRES PASSED THROUGH SLOT IN PILE CAP MASS TO BRIDGE COMPLETION BOX LVDT No.2 (VARIABLE LOCATION) PILE CAP (76.2mm LONG) A SOIL SURFACE S.G. I- 2 BENDING STRAIN GAUGES (PILE No. I) S.G.I-3 •PILE No. I (PILE No.2 IN LINE WITH PILE No. I IN OUT OF PLANE DIRECTION WITH VARIABLE SEPARATION DISTANCE) S.G. 1-4 0 1— S.G.I-5 SCALE 20mm —1 AXIAL STRAIN GAUGE 165mm BELOW S.G.I-5 Figure 2.8: End view of two pile group used in shake table tests showing instrumentation layout Chapter 2. Shake Table Test Procedures 59 76 .2 LVDT No. I P L E X I G L A S S B A L L STRAIN GAUGE L E A D WIRES PILE C A P ( 7 6 . 2 m m LONG) A PILE C A P M A S S ( 96 .5mm LONG) C G . ( A C C E L E R O M E T E R ) SHAKING DIRECTION S.G. I-I BENDING STRAIN GAUGES S.G. 1-2 S.G. 1-3 P ILE No. I S.G.I-4 * N O T E : DUMMY P ILES LOCATED IN OUT OF P L A N E DIRECTION AT 3 P ILE D IAMETER SPAC ING. S.G.I-5 AXIAL S T R A I N G A U G E 165mm BELOW S.G. 1-5 LVDT No. 2 SOIL S U R F A C E S.G. 2-1 —7-77 S.G. 2-2 S.G. 2-3 \"777-- C E N T R E TO C E N T R E PILE SPACING = 3 P ILE D IAMETER S.G. 2-4 PILE No. 2 S.G. 2-5 0 2 0 m m S C A L E P I LE TIP 3 8 9 mm BELOW LINE A - A . BASE OF SAND ~ 6 I O m m BELOW SOIL S U R F A C E . Figure 2.9: End view of four pile group used in shake table tests showing instrumentation layout Chapter 2. Shake Table Test Procedures 60 Pile Group Item Dimensions (mm) Weight (N) 2x1 Pile cap mass Pile cap Pile head clamps (2) Piles (2) 83.6 x 50.8 x 50.8 (LxWxH) 76.2 x 38.1 x 12.7 (LxWxH) Area = 12.5 x 18.5 Height = 5.1 Length = 412 Diam. = 6.35 13.39 2.19 0.035 ea. 0.298 ea. 2x2 Pile cap mass Pile cap Pile head clamps (4) Piles (4) 74.2 x 97.5 x 55.0 (LxWxH) 75.0 x 75.0 x 12.7 (LxWxH) Area = 19.5 x 19.5 Height = 5.5 Length = 412 Diam. = 6.35 26.60 5.16 0.053 ea. 0.298 ea. 1. Centre of gravity ( C G . ) of mass for two pile group is 24.2 mm above line A - A ' (Figure 2.8). 2. Centre of gravity of mass for four pile group is 24.4 mm above line A - A ' (Figure 2.9). 3. Mass moment of inertia with respect to C G . for shaking about the long axis (of-fline), Ixx = 0.89 N-sec2-mm (two pile group). 4. Mass moment of inertia with respect to C G . for shaking about the short axis (inline), / w = 1.43 N-sec2-mm (two pile group). 5. Mass moment of inertia with respect to C G . for shaking about the longitudinal axis, Ixx = 2.69 N-sec2-mm (four pile group). 6. Mass moment of inertia with respect to C G . for shaking about the short axis, Iyy = 1.70 N-sec2-mm (four pile group). Table 2.1: Summary of Pile Cap Structural Properties - Shake Table Tests for Two and Four Pile Groups Chapter 2. Shake Table Test Procedures 61 2.6 Instrumentation and Measurement Resolution The pile foundation models were instrumented with strain gauges, accelerometers and displacement transducers (LVDT's), as shown in Figures 2.7, 2.8 and 2.9. A description of the instrumentation, calibration procedures, and the analogue to digital (A/D) data acquisition systems used during testing may be found in Appendix A . Basic principles of digital sampling and the digital filtering procedures used to increase the signal to noise ratio of the data are also described. Prior to testing, a series of checks were run to determine the amplitudes of instrument line noise which, in turn, affect the resolution with which the engineering quantities of interest ( e.g. bending moment, acceleration and displacement) can be measured. The line noise is due to a number of factors including inadequate grounding or shielding of lead wires and power supplies, and in the case of the LVDT's, internal signal conditioning circuitry designed to convert A C to DC voltage output. Analogue filters were not applied to the amplified instrument signals since it was felt that these might remove parts of the output that were due to mechanical vibration. Experimental observations combined with analysis of the lateral vibration response of the pile have shown that it vibrates primarily in its first mode at a frequency of about 25 Hz or less. Preliminary tests have shown that the majority of line noise from the strain gauges is at 60 Hz or greater, while electronic noise from the LVDT's occurs over a broad frequency range. Line noise from the accelerometers is of the order of 0.01 g or less and may be neglected. Therefore, a 50 Hz low pass filter (see Appendix A) was applied to the strain gauge and L V D T outputs to improve the signal to noise ratio. Filtering was found to be particularly necessary for low amplitude signals. The noise levels after digital filtering, expressed in terms of the engineering quantities of interest, are given in Chapter 2. Shake Table Test Procedures 62 Instrument Noise Level Bending strain gauges Axial strain gauge Accelerometers LVDT no. 1 LVDT no. 2 ± 2 to 8 N-mm ± 2 N ±0.01 g or less ±0.02 mm ±0.05 mm Table 2.2: Instrument Noise Levels After Digital Filtering (Shake Table Tests) Table 2.2. Lateral deflections, y(z), of the pile measured with respect to the base of the sand container have been computed from the measured bending moment distributions, M(z), using the following equation derived from moment-curvature relations, The maximum error in pile deflection at the top of the pile, £[j/o]5 has been estimated using the above equation taking into account potential errors in measured bending mo-ment. In these calculations it has been assumed that the error, £ m 0 x , is distributed uniformly over the distance, /, where significant bending moments were measured along the pile. Using the values £ m a x = 5 N-mm, EI = 4.65 x 106 N-mm2 and / = 279 mm, E[yQ] is computed to be 0.04 mm. The error in computed deflection may be less than the above value since the integrated effect of errors in measured bending moment are partially self cancelling. The above error estimate is seen to be within the range of accuracy of the LVDT's. 2.7 Elastic Wave Velocity Measurements on the Shake Table Following preparation of the sand foundation, installation of the model pile(s), and con-nection and checking of all instrumentation, shear and compression wave velocities were Chapter 2. Shake Table Test Procedures 63 measured in the sand prior to shake table testing. Computation of the free field motions, which are used as input in the analysis of pile response to earthquake loading (Matlock et al, 1978a), requires a knowledge of shear wave velocity distributions in the foundation soils. Since stress levels in model pile tests using a shake table are much lower than cor-responding stresses in full scale tests, it is necessary to measure soil properties at these low confining stress levels. During the wave velocity measurements, it was important to use a source that would generate waves of an amplitude that would not affect the basic soil fabric and would remain well within the elastic range of the material. It is considered that provided shear strain amplitudes generated during body wave propagation do not exceed 10~4 percent, then elastic behaviour of the soil may be presumed (Ishihara, 1982). For the larger strain excitation that occurs during earthquake shaking, the moduli are reduced below their low strain values (Seed and Idriss, 1970; Hardin and Drnevich, 1972). The low strain moduli then represent reference values which are the starting point for the analysis of larger strain dynamic excitation. Low strain, elastic moduli may be calculated from the shear (Vs) and compression (Vp) wave velocities using the relationships, G m„ = PV,2 (2.7) E _ = ^ ( 1 + ^(1-2.) ( 2 8 ) where p is the mass density of the soil, u is Poisson's ratio, Gmax is the low strain shear modulus and Emax is the low strain Young's modulus. From the above equations, Vs and Vp are related as, V? l - 2 i / Chapter 2. Shake Table Test Procedures 64 To insure low amplitude wave propagation, a shear wave source consisting of a piezo-ceramic bender element was used during testing. Shirley and Hampton (1978) have described the use of these elements for measuring shear wave velocities in soil sediments. Subsequently, Schultheiss (1981) has described their use when mounted in an odome-ter or triaxial apparatus. The Norwegian Geotechnical Institute (NGI, 1984) measured shear wave velocities in triaxial specimens using both bender elements and the more conventional resonant column technique. Within the limits of experimental error both methods gave similar results. From their results it can be concluded that shear strain amplitudes induced by the bender source are of about the same amplitude as occur dur-ing sample vibration using a resonant column device. The NGI study suggests that the bender element provides a clean source of shear (S) waves, and that the wave source is relatively uncontaminated by higher frequency compression (P) waves. This study shows that waves travelling at the compression and shear wave velocity are generated by the bender source. The bender element technique has been used to measure shear and compression wave velocities in the container of sand mounted on the shake table. This represents an im-provement in shake table testing as it allows an accurate measure of the moduli necessary to correlate model test results with analytical predictions of pile response to earthquake shaking. The bender element consists of a sandwich of two piezoceramic plates rigidly bonded together. The polarization of the ceramic material in each plate and the electri-cal connections are such that when a driving voltage is applied to the element, one plate elongates and the other shortens so that the element bends as in Figure 2.11a. One edge of the element is attached to a lightweight, electrically conductive bearing plate through which all electrical connections are made. The bearing plate also allows the element to be oriented as desired in the sand and provides fixity to one end of the element. The Chapter 2. Shake Table Test Procedures 65 bearing plate dimensions are approximately 50 mm by 50 mm, and each plate and bender element weighs about 0.10 N (10 gmf). During testing, the element was pushed tip first into the soil surface to a depth of 12.1 mm to achieve good coupling between the soil and bender. When driven by an A C voltage, the bender vibrates as a cantilever beam and acts as a source of body waves which propagate vertically down into the foundation soil. It is noted that the direction of shear wave propagation used in these tests is the same as occurs during shaking of the sand foundation. This is important if one considers that the soil could be cross-anisotropic with respect to its shear stiffness characteristics. The source bender element was connected to a function generator which supplied a ± 3 0 volt square wave to the bender, causing it to vibrate. The frequency of vibration was varied between 0.5 and 30 Hz to examine whether measured velocities were sensitive to input frequency. Subsequent testing showed that the computed wave velocity was insensitive to this factor. Alternatively, a wave pulse was generated by lightly tapping a shear plate placed on top of the soil in a horizontal direction using a hammer. When the hammer made contact with the shear plate an electrical circuit was completed, generating a voltage pulse to signal the start of wave propagation. The latter method was used at the beginning stages of testing and was discontinued when it was found that both methods of source excitation gave similar results. The source bender element (or shear plate) was connected to a high speed digital storage oscilloscope (Figure 2.12a) having a maximum sampling speed of 1 microsecond per data point. The driving signal recorded by the oscilloscope was used to determine the time of departure of the wave pulse. The rise time of the pulse was found to be essentially instantaneous. Cherry (1962) has carried out a theoretical study of three dimensional wave propa-gation resulting from a horizontal shear source applied dynamically to the surface of an Chapter 2. Shake Table Test Procedures 66 elastic half-space. This model simulates the action of the bender source and assumes that the system vibrates at a steady state frequency u, thereby neglecting its initial transient response. Despite this shortcoming, the analysis illustrates certain directional characteristics of wave propagation from a horizontal shear source. The analysis shows that compression (P), shear (S) and Rayleigh surface waves are generated by the source. It is known that shear and compression waves arrive at a receiver prior to the surface waves and that the latter attenuate rapidly with depth (Richart et al, 1970). Since the shear and compression wave velocities are used in the present study to determine the soil moduli, we shall not be concerned with the identification (if present) of surface waves. If receivers are located directly under the source and in the same vertical (x-z) plane, the analysis shows that displacements due to shear waves (SH and SV) dominate the receiver response. If the receiver is offset perpendicular to the x-z plane (> 0 degrees) and is located at an angle 0 > 15 degrees measured with respect to the vertical z axis (see Figure 2.10), P waves have an increased effect on receiver output. This shows that^the vertical alignment of the receiver relative to the source can affect whether a receiver will pick up a significant percentage of P-waves relative to S-waves. Sanchez-Salinero (1983) has carried out a transient analysis of wave propagation from a point source embedded in an elastic two and three dimensional space. The Green's func-tions used in their analysis do not satisfy the free surface boundary conditions necessary for a rigorous analysis of wave propagation from a shear source at the soil surface. Their analysis is useful in that it illustrates certain aspects of receiver response to transient excitation and the dependence of this response on distance from the source. Considering the case of wave propagation from a horizontal shear source, Sanchez-Salinero's analysis shows that for small source-receiver separation distances two wave arrivals can be distinguished from the receiver output. The first wave has a relatively Chapter 2. Shake Table Test Procedures 67 HARMONIC SHEAR SOURCE 0 r0' Jwt ELASTIC MEDIUM (G,v, V s , V p ) 7r27z2\" 0 ' \\ Ap= p- WAVE DISPLACEMENT VERTICAL VIEW PLAN VIEW Ap = tp r 0 2 x Sin 0 x F( Q, h, V p > V s) x e' G R h = w (wt-hR) RADIATION PATTERN SOBARS OF CONSTANT F Figure 2.10: P- wave propagation from a bender shear source, showing the dependance of receiver location on measured P- waves (after Cherry,1962) Chapter 2. Shake Table Test Procedures 68 low amplitude and travels at the compression wave velocity of the medium. The initial wave does not represent a compressional deformation of the soil, but is due to shear distortion. A second larger amplitude wave then arrives at the receiver, travelling at the shear wave velocity of the medium and also represents shear distortion of the soil. The initial low amplitude \"P-wave\" arrival is a near field effect and attenuates with distance. According to Sanchez-Salinero's analysis, at distances greater than about two shear wavelengths, A, where A = Vs/f and f is the dominant frequency of the shear wave, the amplitude of the initial P-wave will be negligible. From this analysis, one concludes that even if one has perfect vertical alignment of the source and receiver so that the receiver is not affected by true compressional wave disturbances, the receiver output may show a P-wave arrival at small source-receiver separation distances. A corollary of the above analysis is that if one reverses the direction of the shear impulse, the \"P-wave\" resulting from the near field effect and the shear wave both reverse their polarity. This effect is confirmed experimentally in section 5.1. The technique of reversing the direction of the shear impulse to distinguish P-waves from S-waves (Stokoe and Hoar, 1978; Robertson et al, 1986) may therefore be misleading if one is carrying out shear wave velocity measurements over small source-receiver separation distances. Given the variety of low amplitude waves emitted from the bender source it is nec-essary to use a sensitive receiver with wide frequency response. The bender element de-scribed previously satisfies these requirements and was used in the present study. When a body wave arrives at the tip of the receiver, the element undergoes a slight lateral movement which generates an electrical signal. The bender has its greatest output for lateral vibration about its weak axis of bending. A series of receiver elements were buried in a vertical line with their tips pointed up beneath the source, as shown in Figure 2.11b, but slight variations in vertical alignment of Chapter 2. Shake Table Test Procedures 69 the receivers did occur. The receiver bender elements were rigidly attached to electrically conductive bearing plates through which all electrical connections were made. The length of an element above its base plate varied between 8.3 and 12.3 mm. The receiver bender elements were buried in the sand during foundation preparation and their tip to tip separation distances measured with respect to a reference cross bar using a tape measure with 0.79 mm (1/32 in) divisions. Since separation distances between the source and receivers were as much as 530 mm and amplitudes of particle displacement in the far field attenuate approximately as 1/R with distance from the source (Cherry, 1962), the electrical outputs from the receivers were amplified. Signals were then transmitted to the storage oscilloscope (Figure 2.12b). Wave velocity measurements were made by applying a square wave voltage pulse to the source element and measuring the voltage response of the array of receivers. Voltage traces were stored in the permanent memories of the oscilloscope for future analysis and plotting. By using the cursor controls on the oscilloscope, the compression and shear wave arrival times were determined directly off the screen. Test data and interpretations of wave arrival times are described in section 5.1 for both the loose and dense sand cases. The distance between the tips of two adjacent bender elements divided by the interval transit time of the wave between them gives the average wave velocity over the depth interval in question. From these, low strain moduli have been computed using equations 2.8. 2.8 Accuracy of Elastic Wave Velocity Measurements The accuracy of the measured wave velocities is limited by the accuracy with which the distances between the tips of the bender elements can be measured. The initial positions Chapter 2. Shake Table Test Procedures 70 (a) Direction of element tip and soil particle movement ^ • Direction of shear wave propagation Soil specimen (b) Driving voltage from shear wave source Time Amplified voltage signal from receiver 4 - Time Lead wires ^—// {< III W III W I' IWII'W-IIIWI»W -tl Leads to high speed A/D converter Source Bender element Bearing plate 'III W> III W H I \\ S / / I \\ W I I ) W I I W-IIIV Receiver Bender elements Soil container • Energy absorbing Boundaries •Rigid base Figure 2.11: Piezoceramic bender elements (a) single element (b) general layout of source and receivers Chapter 2. Shake Table Test Procedures 71 FUNCTION GENERATOR PIEZOELECTRIC SOURCE (a) o-T 6 o SCOPE CH2 TRIGGER Figure 2.12: Electrical layout of bender elements (a) source (b) receiver Chapter 2. Shake Table Test Procedures 72 of the elements during placement in the loose sand can be measured to an accuracy of ± 0 . 8 mm but the separation changes during shaking of the sand foundation. Therefore, wave velocity measurements are presented prior to shaking of the loose sand. In the case of dense sand, the benders were buried in the originally loose sand prior to sand densification and their final location determined at the end of testing during excavation of the bender elements. Surface settlements were found to be negligible during shaking of the dense sand so that bender separation distances were assumed to be the same as those measured at the end of testing. The potential error in average interval wave velocity between any two receivers R l and R2 can be estimated by writing the vertical location of the tips of two adjacent receivers with respect to a fixed reference (ZRI and ZR2, respectively) as, ZRI = ZR1 ± t\\ ZR2 = Z*R2±e1 (2.10) where Z* refers to the measured location of a receiver and e\\ refers to the potential error in the measured location of the receiver. The travel time of the wave from the source, s, to receiver R l (or R2) is given as, I ? 1 = i J i ± e2 ± * (2-11) where T^\\ is the measured travel time from the source to the receiver, e2 is the resolution error of the oscilloscope and c 3 is the potential error in measurements of the wave arrival time. Therefore, the incremental travel time A T ^ f from receiver R l to R2 is, A T * 2 = TR2_Tm = TZ-IZ (2.12) Chapter 2. Shake Table Test Procedures 73 since errors e2 and € 3 tend to be self- cancelling using an incremental travel time procedure (Stokoe and Hoar, 1978). The incremental wave velocity V ^ 2 over a sub-layer from receivers R l and R2 is then given as, VR2 _ -Zfll ~ ZR2 m - A T * 2 = A T * 2 ±ATM ( 3 ) where E\\ = ic^ =F t\\. This error also tends to self-cancel. To estimate the maximum measurement error, it is conservatively assumed that error terms t\\ are additive. The maximum error, E m a x , in computed wave velocity may then be written as, E m a x = —-±- (2.14) min where A T m ; n is the minimum travel time between receivers R l and R2. From the above equation it can be seen that the shorter the separation distance between adjacent re-ceivers the greater the potential measurement error since wave arrival times are reduced. The minimum separation distance used during testing was between the bender source and the top receiver which had separation distances as small as 10 mm. P-wave travel times of about 100 /xsec were measured over this distance and assuming ei = 0.8 mm gives a maximum measurement error of 16 m/sec, or a possible 16 percent error in com-puted velocity. The magnitude of errors in computed shear wave velocity are less since incremental travel times are larger. Separation distances between adjacent receivers at greater depths varied between 40 and 100 mm. Potential errors in the computed velocity are therefore substantially less than occur near the surface. Scatter in measured wave velocities is not due solely to measurement error but can occur due to slight variations in sand density or confining stress level. The latter is Chapter 2. Shake Table Test Procedures 74 dependent on the K0 value in the sand which is affected by the method of sand placement and boundary constraints. In the case of dense vibrated sand, cyclic stressing of the deposit leads to increases in KQ from its original normally consolidated value (Youd and Craven, 1975; Bhatia, 1980). The increase in K0 depends on the previous intensity of shaking and could vary within the soil mass. Differences in sand fabric have also been shown to have an effect on measured elastic wave velocities (De Alba et al, 1984). Chapter 3 Centrifuge Test Procedures This section describes experimental procedures used to carry out model pile tests on the Caltech geotechnical centrifuge. The Caltech centrifuge has been in operation for approximately 25 years under the direction of Professor Ron Scott. During this time, Professor Scott and his co-workers have developed extensive expertise in centrifuge testing and have carried out a variety of static and dynamic pile tests involving both axial and lateral pile head loading. The Caltech research group has recently developed a base motion actuator capable of inputting a prescribed shaking motion to the base of the soil container while in flight at the end of the centrifuge arm (Roth et al, 1986). The actuator was used in the present series of tests to simulate earthquake loading of the centrifuged pile foundations. 3.1 The Principles of Centrifuge Modelling Many fields of engineering use scaled models of large objects to study physical phenom-ena. Scaled models of geotechnical structures under normal gravity, however, seriously lack similitude because the stress levels in the model do not match those in the full scale prototype. By placing the model in an increased gravitational field, thus making the model material heavier, one obtains prototype stress levels in the model. The centrifuge (see Figure 3.1) achieves this increased gravitational field from the centripetal acceler-ation acting on the centrifuge bucket, where the soil model is placed, and spun at the end of the centrifuge arm. The arm has a fixed radius, r, and rate of revolution, u>. The 75 .5 0 i 1.0 FEET Figure 3.1: Side view of Caltech centrifuge (after Allard, 1983) centripetal acceleration, a r , acting at distance r from the axis of revolution is given as: ar — to2r (3.1) The centripetal acceleration creates an artificial gravity field in the vertical (z) direc-tion of the model, N times that of normal gravity. If a model of the prototype structure is built whose dimensions are reduced by a factor 1/N, an acceleration field of N times gravity will lead to stresses in the model the same as those in the prototype structure, neglecting the normal gravitational force in the x-direction (Figure 3.1). Since each model is of finite size, different parts of the model are at different radii 76 from the rotational axis of the centrifuge. Therefore, at any speed different parts of the model will be subjected to different gravitational intensities. The greater the radial distance of the model compared with the dimension of the model in the direction of the centrifuge arm, the more uniform the acceleration field across the model. In the present series of tests, which were carried out using a rotational speed of 242.5 rpm, calculations indicate that the acceleration varies linearly from 55 g at the surface of the model to 68.5 g at the base. The variation in vertical effective stress over the height of the model taking into account this g-gradient is shown in Figure 3.2. This may be compared to the stress distribution computed using an average centrifuge scale factor, N, equal to 60. The differences in computed stress distribution are not large so that a constant scale factor of 60 has been used in converting model test quantities to prototype scale. The other scaling relationships used in centrifugal modelling studies are summarized in Table 3.1. Reduction of length by model factor 1/N means area and mass reductions as shown in the table, assuming identical mass densities between model and prototype. It is noted that parameters such as the cross sectional moment of inertia (I) of the model pile is scaled by a factor iV 4 to correspond to the prototype pile I value. Since the Young's modulus, E , of metal is not affected by stress level, the prototype E value of the pile is the same as that of the model. If the prototype soil is used for the model and if the model experiences the same stresses as the prototype, then it may be assumed that the nonlinear stress-strain be-haviour of the model material is the same as that of the prototype. Hence, dimensionless strains will be the same at corresponding points within the earth mass. While this state-ment deserves some qualification, as discussed later, it describes the basic advantage of testing on a centrifuge. The scaling laws operative on a centrifuge under dynamic loading have been described by Schofield (1981) and may be explained by considering a harmonic 77 160 -\\ 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 50 100 150 Vert. Effective Stress (kPa) Figure 3.2: Vertical effective stress distribution in the sand taking into account g-gradients in the centrifuged soil model 78 motion, which in the prototype (denoted by subscript 'p') is: xp = AP sin uitp dx „ -j^- = io AV C O S LOtp d?x = -u2Apsmcotp (3.2) where to refers to the prototype frequency of excitation, AP is the displacement magnitude, coAp is the velocity magnitude and — LO2AP is the acceleration magnitude. The model must experience the prototype strain (x/L)p and since the prototype length Lp is reduced in the model by a factor 1/N so also the displacement of the prototype xp must be reduced in the model by a factor 1/N. To experience correct stresses, the model accelerations due to the imposed dynamic loading must be increased by a factor N. Hence, the motion that the model must experience is: (Ap/N) sin Ncotm dx, dtn cPx„ = io Ap cos Nu>tm = -J2 APN sin Ncotm (3.3) where AP/N is the displacement magnitude, ooAp is the velocity magnitude and —NLO2AP is the acceleration magnitude. From the above, it is seen that model frequencies and accelerations of input motion must be increased by a factor of N times relative to the prototype. Model velocities agree with those of the prototype. The influence of average grain size {D^Q) relative to pile width (b) may be inferred from centrifuge tests carried out by Oveson (1975) and summarized by Cheney (1985). These tests examined the influence of the ratio of average particle grain size to footing width on the settlement of footings placed in dry dense sand. They noted that there was 79 Quantity Full Scale Centrifuge Model (Prototype) (at N g's) Linear Dimension 1 1/N Area 1 1 /N 2 Volume 1 1/N 3 Time (Dynamic Events) 1 1/N Velocity 1 1 Acceleration 1 N Mass 1 1 /N 3 Force 1 1/N 2 Energy 1 1 /N 3 Stress 1 1 Strain 1 1 Mass Density 1 1 Frequency 1 N Table 3.1: Centrifuge Scaling Relations no significant influence of grain size on the load-settlement behaviour for model b/D5o ratios of 30 to 180. In the present series of centrifuge tests, a fine sand with a D50 of 0.13 mm known as Nevada sand was used. The model pile diameter was 9.5 mm (0.375 in). This corresponds to a b/Dso ratio of 73, which is considered adequate to minimize the effects of particle size on pile behaviour. Centrifuge modelling provides a powerful means of satisfying model similitude. Ap-plying the centrifuge scaling laws to the dimensionless variables described in section 2.4, it can be shown that the model dimensionless variables are equal to those of the pro-totype. This demonotrates that a properly scaled centrifuge model can approximately simulate the behaviour of a full scale soil-structure model when subjected to a specific set of environmental loads. There are some qualifications to the above statement. Since the stress-strain be-haviour of soil is dependant on not only the state of stress but also on the previous stress 80 history of the material, it is impossible to simulate exactly the stress-strain behaviour of a natural soil deposit using a centrifuge (Scott,1979). The method of placement of the soil in the centrifuge container followed by consolidation of the soil under the increased gravity field of the centrifuge results in stress- strain behaviour unique to the conditions of the test. The boundaries of the centrifuge soil container may also influence test results. It is therefore better to refer to a centrifuge model test as a small-scale test with stress levels similar to those in full- scale situations, which provides data indicative of full-scale behaviour. The current trend in centrifuge modelling is to use this data to check the validity of computational models rather than to construe that the properly scaled data will give exact correspondance with full scale behaviour. 3.2 Description of Caltech Centrifuge and Base Mot ion Actuator The Caltech centrifuge is a Model A1030 Gensico \"G- accelerator\", which consists of a 2.03 m (80 in) diameter aluminum alloy arm rotating in the horizontal plane (Figure 3.3). The centrifuge is rated at a 10,000 g-lb (44.48 g-kN) payload capacity. At each end of the arm is located a 460 x 560 mm (18 x 22 in) magnesium swing basket in which a soil container is mounted. In the tests described, the machine was operated at 60 g, corresponding to a radius of 0.91 m (36 in) at the midheight of the soil container. The soil container in which the model pile(s) and sand was placed has a plan area of 0.56 x 0.18 m (21.9 x 7.0 in) and is approximately 0.25 m (10 in) deep. Two 12.5 mm (0.5 in) thick styrofoam pads were placed at each end of the soil container to act as energy absorbing boundaries along the sides of the box perpendicular to the direction of base motion. Two glass plates were fitted on the other two sides of the sand box to minimize side friction during the tests. The plan area of the sand box after the above inserts were installed was 0.530 x 0.176 m (20.88 x 6.94 in). 81 Figure 3.3: Schematic drawing of centrifuge arm (after Scott, 1979) Electrical and hydraulic power, air pressure and signals to and from the rotating arm or basket are conducted through electrical sliprings and rotating unions. For accurate determination of the rotational speed, there is located on the main drive shaft of the centrifuge a 600 tooth gear wheel, which, via a magnetic pickoff, produces 600 pulses per revolution. The pulses are read by an electronic counter which converts them to an L .E .D . display of R P M accurate to 0.1 rpm. The centrifuge arm is housed in an aluminum enclosure with all the controls and instrumentation located remotely. To input a continuous base motion to the soil container, a hydraulic actuator con-trolled by a servo-mechanical feedback system is used to provide the side to side motion of the soil container. The system is analogous to those used on the U B C shaking tables, with the exception that the shaker on the centrifuge must be capable of inputting very high frequency signals up to about 500 Hz. This requires a rapid delivery of hydraulic pressure to the system, which is complicated by the fact the system is subjected to high centrifugal forces. The shaker is described in detail by Allard (1983). A brief description 82 of the system follows. The soil container is attached to a special mounting frame. On each side of it, there is one row of four vertical cylindrical rods to support the test container, and at the centre of the rectangular base is located the hydraulic piston with a servo-valve underneath it. To suspend the soil container on its mounting frame, four horizontal bars on top of the container are screwed to the vertical rods of the mounting frame. This prevents all back and forth and up and down movements of the bucket relative to its support, and permits only side to side movement. A Moog control valve, subject to signals from the controller, regulates the flow of hydraulic fluid into the piston. In turn, the controller is driven by the signal generator which may deliver a sinusoidal or random earthquake signal. With an adequate supply of hydraulic fluid and rapid hydraulic response, a continuous shaking from side to side of the soil container is obtained. The above electro-hydraulic system is driven by an external pump and motor. The flow capacity of the pump and the seals on the rotating hydraulic unions are not adequate to drive the hydraulic actuator at the rates required for centrifuge modelling. Thus, two small accumulators are used which are pressurized reservoirs that can deliver a few gallons of hydraulic fluid very quickly. These are fixed underneath the centrifuge arm inside the enclosure and deliver the hydraulic fluid at a rapid rate directly to the servo valve and piston. In addition, the connecting lines from the accumulator to the valve are very short, so that line flexibility and dissipation are minimized. The controller is used to drive the servo-valve in the dynamic tests, with a feedback system supplied by an L V D T connected to the hydraulic actuator. The L V D T monitors the displacement of the actuator, and sends the signal back to the controller. The con-troller compares the input signal supplied by an external function generator or computer, and the feedback signal. If a difference occurs, the controller will send a correction signal 83 to attempt to make the two signals identical. An electrical analogue signal generator under the control of a micro-computer is used to generate the input signal. When a sine wave base motion input is desired, the sig-nal generator is used to send analogue signals directly to the controller, with a relay \"start-stop\" switch controlled by the computer. The frequency of the input sine wave is controlled by the signal generator while the amplitude of the signal is modulated using the span control on the controller. When an earthquake base motion is the desired input, a digital signal is given by an appropriate computer program. The digital signal is con-verted to an analog signal using a digital to analog convertor, which is then transmitted to the controller. Peak displacements (and hence accelerations) of the input motion are controlled by adjusting the span on the controller. 3.3 Pile Characteristics and Mode l Layout A scaled drawing of the pile used in the single pile tests is shown in Figure 3.4. The model pile was made of 9.52 mm (0.375 in) outside diameter stainless steel tubing having a 0.25 mm (0.010 in) wall thickness. The pile was strain gauged with 8 pairs of foil type strain gauges mounted on the outside of the pile to measure peak bending strains at the locations shown in Figure 3.4. The strain gauge signals were calibrated using transverse dead loading of the test pile when it was clamped at one end and free at the other (Appendix B). From the strain gauge output versus applied load, the peak bending strain and flexural rigidity of the pile at the location of the strain gauges were computed using static beam theory giving an average flexural rigidity of 13.26 N-m 2 . Lead wires from each strain gauge pair were threaded through holes drilled in the tube and brought up inside the tube to the pile head. The wires were then passed through a slot in the pile head mass and connected to 84 Pile Head Clamp and Mass | 6 5 Soil Surface C.G.- 22.2 —-Accelerometer No. 2 ™ + Location of' Strain Gauges ,No. 4 vNo. 6 No. 8 Base of Centrifuge Bucket No. I No. 3 No. 5 -Axial Strain Gauge No. 7 le Tip SCALE' 0 20mm Figure 3.4: Single pile used in centrifuge tests showing instrumentation layout bridge completion and amplifier circuitry mounted on the centrifuge arm. A mass was screwed to a clamp attached to the head of the pile to simulate the influence of a superstructure (Figure 3.4). The items that contributed to the pile and pile head mass are described in Table 3.2. From the dimensions and weights of the pile head mass, the location of the center of gravity of the mass and the mass moment of inertia with respect to the centre of gravity has been computed and are given in the table. The pile head mass was instrumented using a non-contact photovoltaic displacement transducer manufactured by United Detector Technology Inc. and an Entran minia-ture accelerometer (Appendix B). The locations of the accelerometer and light emitting diode (L.E.D.) used by the displacement sensor are shown in Figure 3.4. Pile head 85 Item Dimensions Weight (mm) (N) Pile Head Mass Diam. - 43.7 2.356 Height = 23.1 Pile Head Insert Height = 9.5 0.016 Diam. = 9.3 Pile Head Clamp Area = 19.0 x 19.0 0.044 ea. Height = 5.08 Conical Pile Tip Diam. — 9.6 (nominal) 0.114 Height = 10.9 Pile (includes Length = 209.5 0.114 weight of strain Diam. = 9.52 gauges and lead wires) 1. Centre of gravity of mass 16.5 mm above the base of the pile head clamp. 2. Mass moment of inertia about the centre of gravity, Ixx = 0.0683 N-sec2-mm. Table 3.2: Summary of Model Pile and Pile Cap Structural Properties Used in Centrifuge Tests) displacements were measured with respect to the moving base of the soil container. 3.4 P i l e G r o u p Character ist ics and M o d e l Layout Tests on pile groups, consisting of two piles located at various spacings, were conducted to evaluate pile interaction effects. The centrifuge tests were similar to those described using the shake table with the exception that shaking intensities, at prototype scale, were lower on the centrifuge than were used on the shake table. A typical pile group configuration is shown in Figure 3.5. Both piles were instrumented to measure bending strains as described earlier so that load distributions in the group could be determined. From the strain gauge calibrations, the flexural rigidity of the second instrumented pile was found to be essentially identical to the first pile. In addition, one pile was instrumented with 86 L.E.D. 7.9 ,Pile Clamp 24.6 'l» m /W-AccelerometerTw> No. 2 N No. 3 0.4 No.5 Axial Strain Gauge ( No.8)--^' No.6 No.7 Pile No.l\" Pile Cap Mass le Cap ^No. I HI » i »>y» it ni >» No.2 I No.3 Soil Surface No. 4 No.5 \"Variable Centre to Centre Pile Separation Distances No.6 Pile No.2 Pile Tip p — A 7 v ^ IIIWIIWIAWAWIAWWIAWIf.' Base of Centrifuge Box W W SCALE: O 20 mm Figure 3.5: Two pile group showing instrumentation layout a pair of strain gauges to record axial strains caused by rocking of the pile foundation during shaking. The piles in the group were rigidly attached to a pile cap whose design allowed the pile spacing to be varied. An additional mass was then bolted to the pile cap to simulate the effects of a superstructure as in the case of a single pile. Pile cap dimensions and weights, the location of the centre of gravity above the base of the pile cap, and the mass moments of inertia about both axes of shaking are given in Table 3.3. From the table, the total weight of the pile cap mass is seen to be 4.97 N (1.12 lb) which gives a weight per pile within 3 percent of that used in the single pile tests. The locations of the centre of gravity of mass are also within 3 percent of each other. This allows the effects of pile group interaction to be seen by direct comparison of the single pile and pile group tests 87 Item Dimensions Weight (mm) (N) Pile Cap Mass 108.0 x 47.8 x 14.9 (LxWxH) 4.123 Pile Cap 101.6 x 37.8 x 9.7 (LxWxH) 0.728 Pile Head Inserts (2) Height = 9.5 0.016 ea. Diam. = 9.3 Pile Head Clamps (2) Area = 19.0 x 19.0 0.044 ea. Height = 5.08 Piles (2) Length = 209.5 0.114 ea. Diam. = 9.52 1. Centre of gravity (CG. ) of mass 16.9 mm above the pile cap base. 2. Mass moment of inertia with respect to the C G . for shaking about the short axis (inline), Iyy = 0.510 N-sec2-mm. 3. Mass moment of inertia with respect to the C G . for shaking about the longitudinal axis (offline), Ixx = 0.141 N- sec2-mm. Table 3.3: Summary of Pile and Pile Cap Structural Properties - Centrifuge Tests on Two Pile Group for similar intensities of shaking. The pile cap mass assembly was instrumented with an accelerometer and displacement L .E .D . as shown in Figure 3.5. A series of tests were also run using a four pile (2 x 2) group with a centre to centre pile spacing equal to 2 pile diameters. These tests were carried out to determine whether group interaction factors derived from the previous tests could be superimposed to yield the combined stiffness of the four pile group. The group stiffness may then be used in an analysis of the response of the group to earthquake excitation. The pile group layout is shown in Figure 3.6. Two of the four piles in the group were instrumented as described earlier. The piles in the group were rigidly clamped to a pile cap and four cylindrical masses bolted to the cap at the locations shown in Figure 3.6. Pile cap dimensions and weights, 88 G_ PILE CAP MASS NO.2 (DISPLACEMENT LED LOCATED ON TOP OF MASS) CYLINDRICAL MASS BOLTED TO PILE CAP (14.3 mm 0, 38.1 mm HIGH) SHAKING DIRECTION PILE CAP r L.E.D. CG. •f S.G.I-I 777 S.G.I-2 S.G.I-3 PILE NO.I — 1 r C_P ILE CAP MASS NO. 3 PILI THI E CAP(9.5mm ~ I CK) -ACCELEROMETER SLOTS TO ALLOW VARIABLE PILE SPACINGS 0 20mm i 1 SCALE S.G. 2-1 —>/> S.G.2-2 S.G. 2-3 —PILE NO.2 PILE CAP MASSES -PILE HEAD CLAMP •ACCELEROMETER SOIL SURFACE SIDE VIEW Figure 3.6: Four pile group showing instrumentation layout REMAINING STRAIN GAUGES AS SHOWN IN FIG. 3.5 89 Item Dimensions Weight (mm) (N) Pile Cap Cylinders (4) Diam. = 28.6 1.98 ea. Height = 38.1 Pile Cap 98.8 x 98.8 x 9.6 (LxWxH) 1.86 Pile Head Inserts (4) Height = 9.5 0.016 ea. Diam. = 9.3 Pile Head Clamps (4) Area = 19.0 x 19.0 0.044 ea. Height = 5.08 Piles (4) Length = 209.5 0.114 ea. Diam. = 9.52 1. Centre of gravity (CG. ) of mass 25.6 mm above the pile cap base. 2. Mass moments of inertia with respect to the C G . about x- x axis of shaking, Ixx = 0.92 N-sec2-mm. 3. Mass moments of inertia with respect to the C G . about y- y axis of shaking, / w = 0.92 N-sec2-mm. Table 3.4: Summary of Pile and Pile Cap Structural Properties - Four Pile Group (Cen-trifuge Tests) the location of the centre of gravity above the bottom of the pile cap, and the mass moments of inertia are given in Table 3.4. A pile cap accelerometer and displacement L . E . D . were placed at the locations shown in the figure. 3.5 Instrumentation and Measurement Resolution The model pile foundations were instrumented with strain gauges, accelerometers and a displacement transducer, as shown in Figures 3.4, 3.5 and 3.6. A description of the instrumentation , calibration procedures, and the data acquisition systems used during testing may be found in Appendix B. Data acquisition is more difficult in the centrifuge environment compared to that of 90 the shake table since amplified signals must be transmitted from the model while in flight to an external data acquisition system. Data transmission was achieved using electrical slip rings located at the centre of rotation of the centrifuge. A problem with the use of slip rings is that static electricity on the rings may introduce electronic noise into the signal. Noise may also result from inadequate grounding or shielding of lead wires and power supplies, and from ambient vibration resulting from wind forces acting on the centrifuge model. The amplitudes of instrument line noise have been determined from signals recorded during the quiescent period prior to or after shaking of the model. The amplified strain gauge and accelerometer signals obtained during testing were filtered using a 10 kHz analogue filter, as described in Appendix B. The filtering reduces the high frequency noise associated with centrifuge testing. The frequency transfer function of the filter is such that frequency components below 3 kHz are not attenuated by the filter. Using a centrifuge scale factor of 60, this corresponds to a prototype frequency of 50 Hz. This is considerably higher than the first mode natural frequency of the pile which has been found to be in the range of 1 to 2 Hz (see chapter 4). Since the pile vibrates mainly in its first mode, removal of these high frequency components has no discernable effect on the principal components of pile response. Similarly, the measured free field response is not significantly affected by the analogue filtering since base motion inputs used in the testing contained prototype frequencies of at most 10 Hz. The noise levels recorded after analogue filtering, expressed in terms of the engineering quantities of interest at prototype scale, are given in Table 3.5. Since the frequency content of input base motions used in the centrifuge testing did not exceed 10 Hz, a 10 Hz low pass digital filter was also applied to the instrument signals to improve the signal to noise ratio. The digital filtering is described in Appendix A . The instrument noise levels after digital filtering are also given in the table. 91 Instrument Noise Level (no digital filter) Noise Level (with 10 Hz digital filter) Bending strain gauges Axial strain gauge Accelerometers Displacement transducer ± 5 - 10 kN-m ±11 kN ±0.01 g or less ±0.8 mm ± 3 - 6 kN-m ± 7 kN ±0.006 g or less ±0.5 mm Table 3.5: Centrifuge Instrument Noise Levels Using equation 2.6, the maximum error in pile deflection at the top of the pile, i?[y0]j has been estimated taking into account potential errors in measured bending moment. Following the procedures described in section 2.6, and using the prototype values £max — 5 kN -m, EI = 172614 k N - m 2 and / = 12.6 m, E[y0] at prototype scale is computed to be 1.1 m m . The above error estimate is conservative and is wi thin the range of accuracy of the photovoltaic displacement transducer. 3.6 Foundation Sand Characteristics The model piles were embedded in a dry sand foundation which was prepared using a fine grained sand known as Nevada 120 sand. Nevada sand is a processed material derived from a friable bedded sandstone located in Overton, Nevada and is composed almost entirely of silica quartz. The sand has sub-rounded to sub-angular particles with a minimum and maximum void ratio of 0.56 and 0.90, respectively, determined using the A S T M procedure D2049. The sand has a specific gravity of 2.67. Pile tests were carried out in both loose and dense sand to examine the influence of significant density differences on pile response. A gradation curve is shown in Figure 3.7 which indicates an average particle size D$o of 0.13 mm and a coefficient of uniformity (^6o/-^io) °f 1-6. 92 c D C C 0 is the angular frequency of the travelling wave and t is time. At t = t + At, where At = l/Vs, the shear wave arrives at mass point 2. The ground acceleration at mass point 2 is assumed to have the same harmonic form as above, except shifted in time by an amount At. Using the parameters listed in Table 4.1 the natural frequency of the receiver has been estimated to be 3.7 kHz. This is similar to the approximate value of 4 kHz quoted by N.G.I. (1984). The dynamic equations of motion given in equation 4.1 have been solved using a time step integration procedure, based on a constant average acceleration method (Clough and Penzien, 1975). The bender response to the travelling shear wave, expressed in terms of the relative displacement between mass point 1 and 2, is shown in Figure 4.3. When the shear wave arrives at the receiver the bender output increases quickly. As the Chapter 4. Centrifuge Test Results 108 Parameter Value m0 4.9 x IO - 7 N-sec2/mm EI 16000.0 N-mm2 1 12 mm Gmax 15.4 N/mm 2 vs 100 m/sec t 0.10 A 10.0 mm/sec2 Wo 628.0 rad/sec Table 4.1: Parameters used in dynamic analysis of bender response to a travelling shear wave wave travels over the length of the bender, the bender output reaches a maximum and then undergoes a gradual decay as the net loading due to the ground motion reaches an approximately constant value. The computed response of the bender is similar to that measured and shows that the bender output depends on the properties of the surrounding soil and the characteristics of the incoming ground wave. In particular, the measured decay of the bender voltage output with time after the initial wave arrival shows energy losses due to radiation and material damping should be considered in modelling the bender response. 4.2.3 Wave Velocity Distributions The distribution of shear wave velocities determined in the loose sand foundations both prior to and after shaking is shown in Figure 4.4a. It is evident from measurements made before and after each shaking test that the very small void ratio changes or changes in lateral stress which occur during shaking have had little effect on the measured shear wave velocity. Low strain shear moduli Gmax have been computed versus depth using an equation proposed by Hardin and Black (1968) and related to the shear wave velocity by Chapter 4. Centrifuge Test Results 109 i 1 1 1 i 1 1 1 i 1 1 1 i 0.4 0.5 0.6 0.7 Time (msec) Figure 4.3: Theoretical bender element response to a travelling shear wave. Gmax = pVs2- The Hardin and Black equation is given as: Gmax = 3230 ( 2 - 9 1 7 ^~ ^ Vm)1\" (4-2) where e0 is the in-situ void ratio of the sand and cr^ is the mean effective confining stress in kPa. The mean confining stress has been calculated from the vertical effective stress, l + 2K0 , (4.3) Using e0 = 0.78 which corresponds to the average void ratio in the centrifuged sand foundation and K0 = 0.4 in the preceding formula, shear wave velocities in the loose sand have been computed versus depth. Figure 4.4a shows that the measured shear wave velocities and those computed using the Hardin and Black equation are in very good agreement. The data show that the low strain shear moduli vary as the square root of the depth. Corresponding data for dense sand foundations are shown in Figure 4.4b. The data again confirm that the Hardin and Black equation using e0 = 0.57 and K0 = 0.6 provide Chapter 4. Centrifuge Test Results 110 a reasonable estimate of shear wave velocity distribution in the sand and that the shear stiffness varies as the square root of the depth. 4.3 Base Motion Excitation of Single Piles Data from five centrifuge tests carried out on single piles embedded in loose sand (tests 12 and 17) and dense sand (tests 14,15 and 41) will be presented to illustrate typical aspects of single pile response to base motion excitation. All data are presented at prototype scale, using a centrifuge scale factor of 60. Tests 17 and 41 were subjected to low level sinusoidal base motion with a peak acceleration of 0.04 g while low level, random earthquake excitation having a peak base acceleration of 0.05 g was used in test 14. In tests 12 and 15, the pile was subjected to moderately strong, random earthquake excitation with a peak base acceleration of 0.15 g. Loose and dense sand foundations were prepared as described in section 3.7. Following preparation of the sand foundations, the model pile was pushed into the sand under 1 g conditions. Pertinent single pile test characteristics are given in Table 4.2. Following installation of the model pile and checking of all instrumentation, the cen-trifuge was brought up to a 60 g centrifugal acceleration level. Shear wave velocity measurements were then made from which the low strain shear moduli of the founda-tion sand were determined. A series of input motions was then applied to the base of the model using the Caltech shaker. Input base, free field and pile head accelerations, displacements of the pile head mass and bending moments along the pile were measured during the tests. For a given test set-up, up to three tests were carried out sequentially using progres-sively stronger shaking. Tests 17, 14 and 41 followed a test where 10 cycles of sinusoidal base motion were applied to the pile model using peak accelerations of 0.02, 0.025 g and Chapter 4. Centrifuge Test Results 111 (a) LOOSE SAND 6-8 -10 \\ A X D BEFORE TEST 11 BEFORE TEST 12 AFTER TEST 12 BEFORE TEST 16 BEFORE TEST 17 BEFORE TEST 18 'A \\ Theory (Hordin 8 Block, 1968 I 1 ' 1 ' I 0 100 200 300 400 Shear Wave Velocity (m/sec) (b) DENSE S A N D o o 3 \"o CO _o o < 8 o c _o b 6-\\ ^ ^ Pile Top V -Soil Sur face - ^ \\ -; / / - ' A / -/ / > 1 1 1 1 1 1 1 1 1 I 1 1 1 1 I 1 1 1 1 75 -25 0 25 50 Bending Moment (kN-m) 100 Figure 4.12: Measured bending moment distribution during steady state excitation (t = 16.5 sec) - centrifuge test 17 Chapter 4. Centrifuge Test Results Q -i — _CD cL~ CD > o < CD O c _o CO Q 13 12 11 10 9 8 7 6 5 4 • 3-2 1 0 ^ Ground s urface T j • Experimental -100 -75 -50 -25 0 25 50 Bending Moment (kN—m) Figure 4.13: Measured bending moment distribution during steady state excitation (t 17.0 sec) - centrifuge test 41 Chapter 4. Centrifuge Test Results 125 the pile at a time when maximum pile head deflection occurs (t=12.0 sec) is shown in Figure 4.18. Similar data are shown for centrifuge tests 14 and 15 carried out in dense sand (see Figures 4.19 to 4.23 and 4.24 to 4.28, respectively). From the data, the following observations may be made: 1. The pile head and free field peak accelerations were magnified relative to the input base acceleration. 2. The predominant period of the pile head response is longer than that of the free field ground surface response due to the presence of the structural mass on the pile. This suggests that the pile will filter out the high frequency components of the ground motion. 3. Bending moments along the pile have the same general frequency content as the pile head accelerations, showing that pile head inertia forces dominate the pile response. 4. For test 12, bending moments increase to a maximum near strain gauge 4 at a depth of 4.4 diameters. For test 15 carried out using a similar shaking intensity, the point of maximum bending occurs at a slightly shallower depth equal to 3.5 pile diameters due to the greater lateral stiffness of the dense sand. The point of maximum bending during test 14 occurred at the 2.6 pile diameter depth which is due to the lateral soil stiffness being higher for the lower shaking intensities used in test 14. Bending moments decrease to zero at depth for all tests. 5. The spatial variation of bending moments along the pile shows that all points along the pile experience the same sign of bending moment at any instant in time, suggesting that the pile is vibrating in its first mode. 6. Small residual bending moments exist at depth along the pile. These are more pronounced for centrifuge test 12, carried out using a loose sand foundation, and Chapter 4. Centrifuge Test Results 126 indicates a slight distortion in the shape of the pile has occurred due to cyclic shakedown of the sand. This is confirmed by the small residual.displacements of the pile head mass (Figure 4.16). The estimated fundamental frequencies of the sand foundation (free field) and the pile are given in Table 4.3 based on an examination of Fourier amplitude ratios computed from the measured accelerations. The fundamental frequencies of the pile from centrifuge test 15 and 12 are estimated to be 1.7 and 1.0 Hz, respectively. Since the tests were carried out using identical base motion excitation, the fundamental frequency is seen to decrease with decreasing lateral soil stiffness, as would be expected. The fundamental frequency determined from centrifuge test 14 is approximately 2.0 Hz. In comparing this with the result from centrifuge test 15, it can be seen that as the excitation intensity decreases the fundamental frequency increases. The natural frequencies of the free field were estimated to be 2.75 Hz and were similar for tests 12, 14 and 15. This suggests that for similar intensities of shaking during tests 12 and 15 the rate of modulus reduction with shear strain level is more rapid in the dense sand. This counteracts the higher low strain stiffness of the dense sand relative to that of the loose sand. When comparing tests 14 and 15 the natural frequency results suggest that despite the differing intensities of excitation, and hence different effective shear strains in the sand, the effective shear stiffness of the sand is similar. This implies that the secant shear modulus-shear strain relation for the sand is relatively flat over the strain range induced by the shaking. Free field response analyses carried out using the computer program S H A K E (see chapter 6) suggest that maximum shear strains in the range of 0.01 to 0.04 percent occurred during test 14 and increased to values between 0.02 to 0.08 percent during test 15. Chapter 4. Centrifuge Test Results 127 (a) CD c o 0.4-1 0.2-O 0.0 o -0.4 A .^rtlll.AiiJIlll'lillllillllll J /I /^ ArflAjArliAA A A - - - -10 15 20 Time (seconds) 25 30 (b) CD C o a> o o < 0.4 -0.4 10 15 20 Time (seconds) (c) 0.4 CD C q \"o i_ _a> a) o o < -0.4 10 15 20 Time (seconds) Figure 4.14: Measured accelerations - centrifuge test 12 (a) input base accelerations (b) free field surface accelerations (c) pile cap accelerations Figure 4.15: Computed Fourier amplitude ratios - test 12 (a) pile amplitude ratio ( A P H / A F F ) (b) free field amplitude ratio ( A F F / A B ) Chapter 4. Centrifuge Test Results 129 100 Time (seconds) Figure 4.16: Measured displacements parallel to shaking direction - centrifuge test 12 £ i c £ o CD _C TJ C a> CD 4 0 0 2 0 0 - 2 0 0 -- 4 0 0 5 10 15 20 25 Time (seconds) 30 .(b) i z £ o cn c T> C Q> m 4 0 0 2 0 0 - 2 0 0 - 4 0 0 10 15 Time (seconds) (c) 4 0 0 - 1 - 4 0 0 Time (seconds) Figure 4.17: Measured bending moment time histories - centrifuge test 12 (a) strain gauge 1 (b) strain gauge 4 (c) strain gauge 7 Chapter 4. Centrifuge Test Results 130 . G r o u n d surface ^Interpolated • • Experimental A -400 -200 0 200 Bending Moment (kN-m) Figure 4.18: Measured bending moment distribution during peak pile displacement (t = 12.0 sec) - centrifuge test 12 Chapter 4. Centrifuge Test Results 131 (a) o.io cn 0.05-Tlme (seconds) Figure 4.19: Measured accelerations - centrifuge test 14 (a) input base accelerations (b) free field surface accelerations (c) pile cap accelerations Chapter 4. Centrifuge Test Results i (a) ! 8-132 a rr XI \"a. E < 2H f = 1.8 Hz A A 10 Frequency (Hz) (b) o TJ Frequency (Hz) Figure 4.20: Computed Fourier amplitude ratios - test 14 (a) pile amplitude ratio ( A P H / A F F ) (b) free field amplitude ratio ( A F F / A B ) Chapter 4. Centrifuge Test Results 133 - 50 E E Time (seconds) Figure 4.21: Measured displacements parallel to shaking direction - centrifuge test 14 Chapter 4. Centrifuge Test Results 134 !(c) 2 cn c . _) T ! C CO CO 200 • 150 -3 100 5CH 0 -50 -100 -150 -200 0 10 15 20 Time (seconds) 25 30 Figure 4.22: Measured bending moment time histories -gauge 1 (b) strain gauge 3 (c) strain gauge 6 centrifuge test 14 (a) strain Chapter 4. Centrifuge Test Results 135 13-12-r z 10-°- 9-> o JO 8-S o i l S j r f ace 0 ) u c b 7-6--50 A Exp't. T - ' 0 50 100 150 Bending Moment (kN-m) 200 Figure 4.23: Measured bending moment distribution during peak pile displacement (t = 11.0 sec) - centrifuge test 14 Chapter 4. Centrifuge Test Results 136 o (_ 0) _ ) CO O O cn Time (seconds) !(b) o (_ CO _ ) CD O O C C 0.5 0.4 0.3 0.2 0.1 0.0 - 0 . 1 - 0 . 2 - 0 . 3 -0.4 - 0 . 5 •\\irtJli^illjlil ilk 10 15 20 Time (seconds) 25 30 (c) o c CO _) CO o o cn Figure 4.24: Measured accelerations - centrifuge test 15 (a) input base accelerations (b) free field surface accelerations (c) pile cap accelerations Chapter 4. Centrifuge Test Results 137 • ( b ) o or CD x i 4 6 Frequency (Hz) 4 6 Frequency (Hz) 10 Figure 4.25: Computed Fourier spectra - test 15 (a) pile amplitude ratio ( A P H / A F F ) (b) free field amplitude ratio ( A F F / A B ) v Chapter 4. Centrifuge Test Results 138 100 Figure 4.26: Measured displacements parallel to shaking direction - centrifuge test 15 Chapter 4. Centrifuge Test Results (a) 139 (b) •(c) 6 I cn c . .J TJ C ca ea I cn c ._) T J C CO m £ I Z c CO e o sr cn c ._> TJ C CO CD 600 600 400 H 200 0 -200 -400' -600-10 15 20 TLme ( s e c o n d s ) 15 20 TLme ( s e c o n d s ) 10 IS TLme ( s e c o n d s ! 20 25 30 Figure 4.27: Measured bending moment time histories - centrifuge test 15 (a) strain gauge 1 (b) strain gauge 3 (c) strain gauge 6 Chapter 4. Centrifuge Test Results 13 140 i i -100 100 300 500 700 Bending Moment (kN-m) Figure 4.28: Measured bending moment distribution during peak pile displacement (t 10.9 sec) - centrifuge test 15 4.4 Soil-Pile Interaction 4.4.1 Introduction The interaction between soil and pile during earthquake loading is commonly described using linear or non-linear Winkler springs which, despite the lack of shear coupling be-tween adjacent springs, have been found to adequately model pile response. These Win-kler models fall into two main classes; one where the soil is idealized as an equivalent linear, viscoelastic medium (Novak and Aboul-Ella, 1978; Novak and Sheta, 1980; Nogami and Novak, 1980; Kagawa and Kraft, 1980a,b), and another where the soil is modelled as a non-linear, hysteretic material using as input soil resistance - lateral deflection ('p-y') curves (Matlock et al, 1978a,b; Kagawa and Kraft, 1981a; Nogami and Chen, 1987). Here, y represents the relative lateral deflection between the pile and the free field ground motions since interaction forces can only be set up when relative movements occur. The equivalent linear analyses require as input the secant shear modulus distribution in the soil, and may distinguish between the soil stiffness in the near and far field (Novak and Chapter 4. Centrifuge Test Results 141 Sheta, 1980). Since the shear moduli are strain level dependent, the secant moduli are selected to be compatible with average shear strains in the soil and are then used to compute the lateral Winkler stiffness, kh. Approximate three dimentional solutions show that kh depends on the relative soil-pile stiffness, the pile length to diameter ratio and the method of pile loading (Kagawa and Kraft, 1981). The Winkler stiffness may also depend on the frequency of pile excitation for higher frequencies since inertia forces set up in the soil are frequency dependent and cause a reduction in lateral stiffness as excitation frequencies increase (Novak et al, 1978; Nogami and Novak, 1977). Empirically based, non-linear p-y curves may also be used to derive secant lateral stiffnesses compatible with average pile deflections during shaking. Once the secant Winkler stiffnesses have been computed, they are used in an equivalent linear analysis of the pile's dynamic response using uncoupled or coupled analysis procedures. More rigorous, step by step solutions of a pile's response to earthquake loading are obtained using non-linear, tangent lateral soil stiffnesses (Matlock et al, 1978a,b). These are defined from the input 'backbone' p-y curves which represent the non-linear response of the soil to monotonic lateral loading of the pile. Unloading p-y response may be mod-elled assuming Masing or other mathematical descriptions of unload-reload behaviour, with or without the influence of gapping between the soil and the pile. A state of the art dynamic solution incorporating the non-linear and hysteretic behaviour of the lateral soil stiffness has been developed by Matlock and his co-workers and is embodied in the computer program SPASM8. SPASM8 is based on a coupled analysis wherein the pile is fully coupled to the free field ground motions along the pile length. Various methods based on full scale test measurements have been proposed to compute non-linear p-y curves for both static and slow cyclic lateral loading. The cyclic p-y curves have been derived for conditions simulating wave loading on piles. No accounting is generally made for the effect of the frequency of loading since this effect is minor for low frequencies of excitation. Where higher frequency loading is of interest, Nogami and Chen (1987) have proposed that the frequency effect be accounted for by computing Chapter 4. Centrifuge Test Results 1 4 2 the lateral soil stiffness resulting from the combined effects of a frequency independent, non-linear 'near field' spring with that of a linear, frequency dependent 'far field' spring. Procedures for computing frequency independent, back-bone p-y curves in sand have been summarized by Murchison and O'Neill (1984) and include the method recommended by the American Petroleum Institute (API, 1979) for static and cyclic loading . This is based on one full scale test in sand and is described by Reese et al, 1974. The recom-mended cyclic p-y curves represent an envelope of behaviour and do not consider the change in lateral soil stiffness as cycling proceeds. Murchison and O'Neill, in their evalu-ation of the accuracy of the API procedures using available full scale test data, concluded that predictions of lateral pile response were rather inaccurate using the API procedures. They recommended an alternate method for estimating static and cyclic p-y backbone curves and that additional research be carried out to evaluate the nature of soil-pile in-teraction during lateral pile loading. Ting (1987), in his evaluation of p-y curves during cyclic lateral loading of a full scale pile in saturated sand, concluded that the API rec-ommendations overestimated the available resistance of the soil system. Below about the 5 pile diameter depth, the dynamic p-y relation was found to be linear, but still considerably softer than the API recommendations. Ting's data also suggested that for a given lateral deflection level, the influence of rate of loading on secant lateral stiffness was relatively minor in the frequency range examined from 2 to 7 Hz. Care must be taken to distinguish between the effect of the rate of loading on lateral stiffness due to differences in pore pressure dissipation versus frequency dependent, inertia effects. For slow cyclic loading in sands where drainage has time to occur, the lateral stiffness is higher than if rapid cyclic shearing occurs and pore pressures do not have time to dissipate. Where significant pore pressures develop around the pile during shaking, the p-y curves should be corrected for the effects of pore pressure build-up (Finn and Martin, 1979b; Kagawa and Kraft, 1981a). In the present series of Chapter 4. Centrifuge Test Results 143 tests carried out in dry sand, pore pressure effects are of no concern. The rate of loading was concentrated in the 0 to 5 Hz range so that inertia effects are also considered to be unimportant, based on calculations using the viscoelastic soil model proposed by Novak (see section 6.2). To determine the nature of interaction between the soil and the pile during base mo-tion excitation, cyclic p-y curves were derived from the single pile data using procedures described by Ting (1987) and given in Appendix E . The method involves characterizing the unknown soil reaction distribution along the pile using a 7th order polynomial and the method of Lagrange multipliers. Discrete bending moments measured along the pile are used in the evaluation of the polynomial, subject to the constraint that the net soil reaction p is zero at the soil surface. Lateral pile deflections along the pile were evaluated by fitting a piecewise cubic spline through the measured bending moments and then double integrating the cubic spline to compute the pile deflection. The accuracy of the numerical integration was checked by computing deflections at the top of the pile head mass where displacements were measured. Discrepancies of 0.5-1.0 mm (at prototype scale) occurred owing to the limited number of bending moment data points employed in the cubic spline fit and inherent measurement error. Therefore, computed deflections were scaled a constant amount along the pile such that a match of the measured pile head deflection was achieved. The lateral pile deflections are computed with respect to the moving base. These are not necessarily the same as the relative deflection between the pile and the free field, which is the form the p-y curves should be expressed in for earthquake loading. The experimental data shows that bending moments greater than about the 10 pile diameter depth are small, suggesting that pile curvatures due to the free field ground motions are similarly small. Furthermore, it will be shown that measured bending moment distribu-tions along the pile are well predicted when one considers that only structural inertia Chapter 4. Centrifuge Test Results 144 forces contribute to pile bending. Computed pile head deflections due to the pile head inertia forces, which represent pile deflections relative to the free field displacements, also agree closely with pile head deflections measured relative to the moving rigid base. Thus, for all practical purposes, the measured bending moment data can be used to develop soil reaction pressures (p) due to relative movement between the pile and free field (y). 4.4.2 Earthquake Excitation P-y curves were computed for a number of different loading cycles covering a range of shaking intensities during random earthquake excitation. This was done to determine whether significant changes in lateral soil stiffness occurred during shaking. Soil reaction pressures computed during centrifuge test 12 at peak pile deflection (t=12.0 sec) are shown in Figure 4.29 to illustrate typical presssure distributions. Cyclic p-y curves computed at the 3 pile diameter depth for three different loading cycles are shown in Figure 4.30. The peak to peak loading (p) amplitude was reasonably constant for each of the three cycles but the secant lateral stiffness of the soil, defined as the slope of the line passing through the endpoints of the hysteresis loop, gradually increased with the number of shaking cycles. The secant stiffness increased by a factor of approximately two over the three cycles and indicates that progressive densification and hardening of the loose sand occurred. Similar behaviour was observed up to about the 5 pile diameter depth. Cyclic p-y curves have been computed for test 12 during the shaking cycle when maximum pile deflection occurred and are shown for a range of depths in Figure 4.31. The computed p-y curves up to about the 5 pile diameter depth are non-linear and exhibit hysteresis loops typical of an inelastic, highly damped system. The p-y response is linear past about the 7 pile diameter depth. Below this depth p-y curves were not computed since pile deflections could not be reliably discerned. No signs of gapping between the Chapter 4. Centrifuge Test Results 145 14-1 o I 1 1 1 1 i 1 1 1 1 I 1 1 1 1 i 1 1 1 1 ! -100 -50 0 50 100 Pressure (kN/m) Figure 4.29: Lateral soil reaction distribution at peak pile deflection (t centrifuge test 12 = 12.0 sec) -Chapter 4. Centrifuge Test Results 146 Figure 4.30: Cyclic p-y curves at three different times during shaking at the 3 pile diameter depth - centrifuge test 12 Chapter 4. Centrifuge Test Results 147 sand and the pile are evident from the curves. The computed p-y curves are also compared to the cyclic p-y curves recommended by the API. The API p-y curves were computed using a peak friction angle of 35 degrees, derived from triaxial test measurements, and an value of 6750 k N / m 3 . The latter defines the initial slope of the p-y curve and was based on values recommended for loose dry sand. The computed API curves bear little resemblance to the experimental p-y curves past the one pile diameter depth and are seen to be considerably stiffer. The secant lateral soil stiffnesses derived from the p-y curves are plotted versus di-mensionless depth in Figure 4.32. Their variation over the three loading cycles examined is also shown. The greatest changes in lateral soil stiffness occurred around the 5 pile diameter depth which is near the point of maximum pile bending. The lateral stiffnesses for a particular cycle of shaking are seen to increase with depth. The stiffness distribu-tion computed during peak pile deflection may be approximated by a linear with depth Winkler modulus distribution given as k^ = n^z, where = 2500 k N / m 3 . Peak bending moments measured along the pile during peak pile deflection are shown in Figure 4.33. To check the accuracy of the modulus distribution, bending moments have been com-puted using the static laterally loaded pile model described in Appendix D. The latter incorporates a linear versus depth Winkler modulus distribution and assumes the pile is subjected to moment and shear loading at the soil surface resulting from accelerations of the structural mass. The influence of the free field ground motions and accelerations of the pile cross-section are therefore neglected. Preliminary calculations using a simplified analytic solution for the case of a single pile supporting a structural mass and subjected to base motion excitation have been carried out to check the reasonableness of the above approximation (see section 6.2). The analytic solution suggests that the flexural response of the model pile is dominated by structural inertia forces rather than by the free field ground motions and that provided one has a reasonable estimate of pile shear forces and Chapter 4. Centrifuge Test Results 148 CO • IA 1/1 0) 100-25 0 -25 -50 -75 -100 Depth = 7 Pile Diam. K A.P.I. Curve / » Time = 11.72-12.78 sec -25 -20 -15 -10 -5 0 5 10 Pile Displacement Y (mm) 15 20 25 Figure 4.31: Cyclic p-y curves at various depths during the shaking cycle when peak pile deflection occurred (t = 11.72 - 12.78 sec) and comparison with API curves - centrifuge test 12 Chapter 4. Centrifuge Test Results 0-t 149 N CL 0) Q V) V) c o \"t/1 C 4 . E Q 5 -6-VAH HA-< Exp't. KH = 2500 z A.P.I. V~A< \\ 5000 10000 15000 KH (kPa) 20000 Figure 4.32: Equivalent lateral stiffnesses versus depth derived from experimental and API p-y curves - centrifuge test 12 bending moments at the soil surface, one can use a simplified static analysis to obtain a reasonable estimate of bending moments along the pile. The moment and shear force used in the above calculations were derived from bending moments and structural mass accelerations measured during centrifuge test 12. The computed bending moment distribution is seen to agree closely with measured bending moments. Pile deflections and rotations at the soil surface were also computed using the Winkler model. Knowing the pile deflections and rotations at the soil surface and the moment and shear force at the pile top, a static structural analysis of the free standing portion of the pile was carried out. This yielded the deflections, yt, and rotations, 0t, of the pile at the top of the pile. Assuming a rigid connection between the pile and the mass, deflections at the top of the structural mass were then computed as A* = yt + Ah sin 9t Chapter 4. Centrifuge Test Results 150 Test No. Winkler Model Vo Mo A x A 2 A m (kN) (kN-m) (mm) (mm) (mm) 12 kh = nhz (nh = 2500 kN/m 3 ) 93.0 215.0 62.9 43.6 68.0 14 h = nhz (nh = 4900 kN/m 3 ) 49.0 92.0 19.9 6.9 22.0 15 h = nhz (nh = 4800 k N / m 3 152.0 270.0 61.2 27.6 72.5 17 £/i = n^z (n f c = 5600 kN/m 3 ) 18.3 48.0 9.9 8.4 10.5 41 = a z 1 / 2 (a = 20,000 kN/m 5 / 2 ) 25.0 57.0 7.5 4.6 8.0 V0 = applied shear force at soil surface. MQ = applied bending moment at soil surface. A i = computed deflection at top of pile head mass using Winkler stiffness derived from experimental p-y curves. A 2 = computed deflection at top of pile head mass using Winkler stiffness derived from API p-y curves. A m = measured deflection at top of pile head mass. Table 4.4: Winkler Model Predictions of Single Pile Deflections - Centrifuge Tests where Aft is the vertical distance between the pile top and the location of the displacement L . E . D . on top of the structural mass. The computed and measured deflections are given in Table 4.4 and agree to within 7 percent, confirming the dominance of the structural inertia forces on the pile response and the reasonableness of the inferred modulus distribution. The API p-y curves were also used in the finite difference computer program L A T P I L E to predict the lateral pile response (Byrne and Jantzen, 1984). Using the same pile head moment and shear force as used in the above calculations, the static pile response was solved iteratively using finite differences until the lateral soil stiffness kh = p/y was compatible with the pile deflections. Bending moments computed using L A T P I L E are shown in Figure 4.33 and are less than those measured over the full length of the pile. Chapter 4. Centrifuge Test Results 13 151 -100 100 300 500 Bending Moment (kN—m) Figure 4.33: Computed bending moment distribution during peak pile deflection (t = 12.0 sec) using lateral stiffnesses from experimental and API p-y curves - centrifuge test 12 Computed Winkler moduli are shown in Figure 4.32 and are higher than the moduli deduced from the experimental p-y curves. This has resulted in computed deflections at the top of the pile head mass which are 36 percent less than those measured (Table 4.4). Similar p-y data for centrifuge tests 14 and 15 carried out in dense sand are shown in Figures 4.34 to 4.36 for test 14 and 4.37 to 4.40 for test 15, respectively. Centrifuge test 14 was carried out using low level earthquake excitation while test 15 was carried out using stronger shaking. Data from these tests were used to determine whether approximately linear p-y response would occur when a pile is shaken in a dense sand foundation sand and over what range in shaking intensities this would occur. Chapter 4. Centrifuge Test Results 152 Cyclic p-y curves computed for test 14 for two different load cycles including the cycle when maximum pile displacement occurred are shown for the 1 to 5 pile diameter depth-in Figure 4.34. The p-y data are seen to exhibit approximately linear elastic response over the range of depths and shaking intensities examined. The lateral Winkler stiffnesses derived from the p-y data are plotted versus depth in Figure 4.35. The data may be described by a linear versus depth stiffness distribution given as kh = rihZ where nh = 4900 k N / m 3 . Bending moments have been computed along the pile using the above linear with depth Winkler modulus distribution. Bending moments have again been computed along the pile using the static laterally loaded pile solution given in Appendix D. The computed bending moments are compared with those measured and are seen to agree closely (Figure 4.36). Deflections at the top of the pile head mass were also computed (Table 4.4) and are within 10 percent of measured deflections. Cyclic p-y curves computed using the API procedures are also shown in Figure 4.34. A peak friction angle of 45 degrees derived from triaxial test data and a nh value of 61,000 k N / m 3 was used in the calculations. The value of nh is based on recommendations made by the API for very dense dry sands. The API p-y curves are again seen to be substan-tially stiffer than those derived from the experimental measurements. Bending moments along the pile (Figure 4.36) and pile head mass deflections (Table 4.4) computed using the API curves and the program L A T P I L E are therefore less than those measured. Cyclic p-y curves computed for three different load cycles over a range of loading amplitudes have been developed from centrifuge test 15 and are shown in Figure 4.37. Non-linear hysteretic p-y response is seen for the stronger shaking used in the test. The secant slopes of each hysteresis loop are reasonably constant, suggesting that little change in global soil stiffness occurred during shaking. It would normally be expected that the secant stiffnesses would decrease for higher loading amplitudes, indicative of Chapter 4. Centrifuge Test Results 153 Depth = 0.57 m (1 pile diam.) Depth = 1.14 m (2 pile diam.) I 25 I • Time = 4.61-5.51 soc • Time = 10.81 - 11.31 sec A.P.I. -10 -5 0 5 10 15 Pile Displacement Y (mm) 1 25 Oepth = 2 Pile Diam. / . A\" * Time = 4.61-5.51 sec - Tim© = 10.81 - 11.31 sec A.P.I. -10 -5 0 5 10 Pile Displacement Y (mm) CYCLIC P-Y CURVES - CENTRIFUGE TEST 14 Depth = 1.71 m (3 pile diam.) Depth = 3 Pile Diam. ?f » Time = 4.61-5.51 sec • Time = 10.81 - 11.31 sec A.P.t. -10 -5 0 5 10 Pile Displacement Y (mm) 1 Depth = 2.28 m (4 pile diam.) Depth = 4 Pile Diom. * ** r • Time = 4.61-5.51 sec • Time = 10.81 - 11.31 sec 4.P.I. -10 -5 0 5 Pile Displacement Y (mm) Depth = 2.85 m (5 pile diam.) a. o < CP o c _o V) b 10 8 -7-6-5 -\\ Soil S jrface / Y i / / / / » i . ' A ExpM. / KH = 4900 z / 1 A.P.I. - 5 0 0 50 100 150 Bending Moment (kN-m) 200 Figure 4.36: Computed bending moment distribution during peak pile deflection (t = 11.0 sec) using secant lateral stiffnesses from experimental and API p-y curves - centrifuge test 14 Chapter 4. Centrifuge Test Results 1 5 6 Pile Displacement Y (mm) Figure 4.37: Cyclic p-y curves at different times during shaking at the 3 pile diameter depth - centrifuge test 15 strain softening response. The fact that this did not occur indicates that the soil in close proximity to the pile underwent large strain softening and subsequent load transfer to regions of soil farther away from the pile. Cyclic hardening of the sand may also have occurred. The load transfer and cyclic hardening has apparently counteracted strain softening of the sand adjacent to the pile, resulting in about the same effective stiffness for the range of shaking intensities examined. Cyclic p-y curves computed during the shaking cycle when maximum pile deflection occurred are shown for a range of depths in Figure 4.38. The computed curves are non-linear and hysteretic up to about the 5 pile diameter depth. The p-y response is linear beyond the 7 pile diameter depth. The secant lateral stiffnesses increase with depth (Figure 4.39) and may also be approximated by a linear with depth Winkler modulus Chapter 4. Centrifuge Test Results 157 distribution using = 4800 kN/m 3 . This effective stiffness distribution is not much different from that derived from test 14 due presumably to the effects of load transfer described above. Bending moments have again been computed along the pile using the linear with depth Winkler model. The computed bending moments are compared with those measured and are seen to agree closely (Figure 4.40). Deflections at the top of the pile head mass were also computed (Table 4.4) and are within 16 percent of measured deflections. Cyclic p-y curves computed using the API procedures are shown in Figure 4.38. The API p-y curves are again substantially stiffer than those derived from the experimental measurements. Bending moments along the pile (Figure 4.40) and pile head mass de-flections (Table 4.4) computed using the API curves and the program L A T P I L E are less than those measured. 4.4.3 Low Level Sinusoidal Shaking Cyclic p-y curves have also been computed for centrifuge tests 17 and 41 where the pile was subjected to relatively low level sinusoidal shaking. Test 17 was carried out using a loose sand foundation. It will be recalled that p-y curves developed from centrifuge test 12 showed significant non-linear response for the moderately strong shaking used in the test. It is therefore of interest to determine whether linear p-y response occurs for the lower shaking intensities used in test 17. Cyclic p-y data for test 41 were also developed and checked for linearity since test 41 was used in the analysis of pile group interaction where linear elastic soil response was assumed. During sinusoidal shaking, p-y curves were computed for the initial cycle of transient excitation when maximum pile response occurred and were then compared with curves evaluated during steady state excitation. Cyclic p-y curves computed during test 17 carried out in loose sand are shown in Figure 4.41. The p-y curves have been computed Chapter 4: Centrifuge Test Results 158 Depth = 0.57 m (1 piie diam.) Depth = 1.14 m (2 pile diam.) Pile Displacement Y (mm) Pile Displacement Y (mm) Depth = 1.71 m (3 pile diam.) Depth = Z85 m (5 pile diam.) Pile Displacement Y (mm) Pile Displacement Y (mm) 125 100 7 5 0 - 2 5 -- 5 0 Depth = 3.99 m (7 pile diam.) D e p t h = 7 P i l e D i a m . J A . P . I . C u r v e . * • 1 = 5 . 1 5 - 5 . 9 S s e c • t = 1 0 . 8 2 - 1 1 . 5 8 s e c - 3 0 - 2 0 - 1 0 0 10 2 0 Pile Displacement Y (mm) 3 0 Figure 4.38: Cyclic p-y curves at various depths during shaking cycle when peak pile deflection occurred (t = 10.82 - 11.58 sec) and comparison with API p-y curves - centrifuge test 15 Chapter 4. Centrifuge Test Results 159 KH (kPa) Figure 4.39: Equivalent lateral stiffnesses versus depth derived from experimental and API p-y curves - centrifuge test 15 Chapter 4. Centrifuge Test Results 160 13 4 I « 1 ' 1 • 1 ' r— -100 100 300 500 700 Bending Moment (kN-m) Figure 4.40: Computed bending moment distribution during peak pile deflection (t = 10.9 sec) using secant lateral stiffnesses from experimental and API p-y curves - centrifuge test 15 Chapter 4. Centrifuge Test Results 161 up to the 5 pile diameter depth since below this depth lateral pile deflections could not be reliably discerned. During the initial transient excitation, relatively large lateral pile deflections occurred. The computed p-y curves are non-linear and exhibit hysteretic behaviour even for the low shaking intensities used in the test. No signs of gapping between the sand and the pile are evident from the curves. Cyclic p-y curves computed during steady state excitation over one cycle of shaking (t = 16.0 - 18.0 sec) are also shown. The lateral pile deflections are less than those that occurred during the initial transient excitation, although similar non-linear p-y behaviour was observed. The p-y response is approximately linear beyond the 5 pile diameter depth. Similar data is also plotted in Figure 4.42 for centrifuge test 41. The higher density of the sand used in this test resulted in smaller lateral pile deflections. The p-y data indicate a near linear response during the initial transient and. steady state excitation. The use of elastic models to simulate soil-pile interaction would therefore appear to be justified. The exception to this occurred at the 1 pile diameter depth when a small amount of non-linearity occurred during the initial transient shaking. Examination of Figures 4.41 and 4.42 shows that the secant stiffness for a particular shaking cycle increases with depth, as was observed during tests 12, 14 and 15. For test 17, the secant stiffness during steady state shaking is clearly larger than during the stronger transient excitation, indicating that strain hardening of the loose sand has occurred under the lower amplitude repetitive shaking. For test 41, no significant change in secant soil stiffness was observed for the range of shaking intensities examined. Cyclic p-y curves computed using the API procedures are also shown on the figures. These again provide a poor match of the experimental p-y curves and significantly overestimate the lateral soil stiffness for large pile deflections. The secant stiffnesses computed during steady state shaking for test 17 are plotted Chapter 4. Centrifuge Test Results 162 Depth = 171 m (3 pile diam.) Depth - 2.85 m (5 pile diam.) Pile Displacement Y (mm) Pile Displacement Y (mm) Depth = Z28 m (4 pile diam.) -50 | i i i i | i i i i i i i i i i i i i i i i i i i | i ' ' ' i ' • ' ' i ' ' ' • i 1 1 1 1 I ' • 1 1 -25 -20 -15 -10 -5 0 5 10 15 20 25 Pile Displacement Y (mm) Figure 4.41: Cyclic p-y curves in loose sand during sinusoidal shaking at various depths and comparison with API p-y curves - centrifuge test 17 Chapter 4. Centrifuge Test Results 163 j : 25 Depth = 0.57 m (1 pile diam.) Depth = 1 Pile Diam. • 1 = 0.32-1.28 sec • t = 16.91-18.76 sec A.P.I. -2.5 0 2.5 Pile Displacement Y (mm) Depth = 1.14 m (2 pile diam.) 50 Q. Depth = 2 Pile Diom. /.* •i * • 1 = 0.32-1.28 sec - 1 = 16.91-18.76 sec A.P.I. - 2 - 1 0 1 2 3 Pile Displacement Y (mm) Depth = 1.71 m (3 pile diam.) Depth = 2.28 m (4 pile diam.) 50 j : 25 -50 Depth = 3 Pile Diam. / 1\"' • t = 0.32-1.28 sec - 1 = 16.91-18.76 sec A.P.I. - 2 - 1 0 1 2 Pile Displacement Y (mm) so 4 \" z / /•• • 1 = 0.32-1.28 soc - 1 = 16.91-18.76 sec A.P.I. - 2 - 1 0 I 2 3 Pile Displacement Y (mm) Figure 4.42: Cyclic p-y curves in very dense sand during sinusoidal excitation at various depths and comparison with API p-y curves - centrifuge test 41 Chapter 4. Centrifuge Test Results 164 versus dimensionless depth in Figure 4.43. The data points for test 17 may be ap-proximated by a linear versus depth Winkler modulus distribution kh(z) = n^z where n/j = 5600 kN/m 3 . Peak bending moments during steady state shaking are shown in Figure 4.44. To check the accuracy of the modulus distribution, bending moments and pile deflections at the top of the pile head mass have been computed using the linear with depth Winkler model. The computed bending moment distribution shown in Figure 4.44 and pile head mass deflection (Table 4.4) agree closely with measured values. Using the API p-y curves for loose sand and program L A T P I L E , the bending moment distribution along the pile is closely predicted. This implies that for relatively small pile deflections the discrepancies between the experimental and API p-y curves are not overly significant. This has resulted in lateral soil stiffnesses that are in reasonable accord with stiffnesses derived from the experimental p-y curves (Figure 4.43). Deflections computed at the top of the pile head mass are 20 percent less than those measured. The secant modulus distribution deduced from test 41 is shown in Figure 4.45. The modulus distribution may be approximated by a square root of depth variation given as kh = az1/2, where a = 20, 000 k N / m 5 / 2 . Using the static pile model given in Appendix D for the case of a Winkler stiffness increasing proportional to the square root of depth, bending moments along the pile have been computed during steady state shaking (Figure 4.46). Pile deflections at the top of the pile head mass have also been computed and are given in Table 4.4. Computed deflections and bending moments agree closely with measured quantities. Using program L A T P I L E and the API curves for dense sand, bending moments along the pile and pile head deflections are seen to be significantly underpredicted. Chapter 4. Centrifuge Test Results 165 Figure 4.43: Secant lateral stiffnesses versus depth derived from experimental and API p-y curves - centrifuge test 17 Chapter 4. Centrifuge Test Results 166 Bending Moment (kN-m) Figure 4.44: Computed bending moment distribution during steady state shaking using secant lateral stiffnesses derived from experimental and API p-y curves - centrifuge test 17 Chapter 4. Centrifuge Test Results 167 25000 50000 75000 100000 KH (kPa) Figure 4.45: Equivalent lateral stiffnesses versus depth derived from cyclic p-y curves -centrifuge test 41 Chapter 4. Centrifuge Test Results 168 13 12 11 E 9 4} > O 3 8 CD 2 7 b X Pile Top -Soil Surface - / s '? / - • / ' • ' / / 1 1 1 / / 1 / A Exp't. - / KH = 20000 Z**1/2 -, , , ,1 ' ' 1 1 i A.P.I. -25 0 25 50 75 100 Bending Moment (kN-m) Figure 4.46: Computed bending moment distribution during steady state shaking using secant lateral stiffnesses derived from experimental and API p-y curves - centrifuge test 41 Chapter 4. Centrifuge Test Results 169 4.4.4 Near Field Hysteretic Damping The equivalent visco-elastic models of Novak and other researchers approximate the pre-vious non-linear, hysteretic p-y curves using a Kelvin-Voight model. Here, a linear spring with stiffness kh. is placed in parallel with a viscous dashpot to represent frictional en-ergy losses during one cycle of shaking within the near field soil (Seed and Idriss, 1970; Roesset, 1980). The area enclosed within a p-y curve for one cycle of shaking is equal to the energy expended due to frictional damping. For convenience of analysis, it is common to equate the work done by a frictionally damped system over one load cycle (AWd) to the energy absorbed by a viscously damped system. One can then define a viscous dashpot coefficient (ceq) or, alternatively, the fraction of critical damping ( is the angular frequency of vibration and y is the peak amplitude of lateral pile vibration relative to the free field ground motions. Substituting c/m = 2£ujn gives where m is the mass and u)n is the natural frequency of the idealized single degree of freedom system. Defining a damping energy ratio, ER^, as AWd = iraoy2 (4.4) AWd = 2irLzuuny2 (4.5) ERd = AWd (4.6) Chapter 4. Centrifuge Test Results 170 where We\\ = 0.5/c/j?/2 is the elastic strain energy contained within the p-y curve over one load cycle, one can show that ERd = 4 < — (4.7) Since the dynamic response of a single degree of freedom system is most sensitive to the value of damping at resonance, we define a damping ratio D as the fraction of critical damping for LO = u>n. This gives D = ^-ERd (4.8) 47T As seen by equation 4.8, the damping ratio can be defined from the area within the hysteresis loop (AWd) and the elastic strain energy (Wei). The damping ratio of sands is normally assumed to be independent of the frequency of cyclic shearing. Damping increases with the level of cyclic shear strain and decreases with the number of cycles of shaking (Hardin and Drnevich, 1972). Damping ratios of 25 to 30 percent are commonly adopted during large strain excitation of dry sand when shear strains exceed approxi-mately 1 percent. The average strain in the near field soil may be related to lateral pile displacement, based on analyses described in Appendix C. Material damping ratios have therefore been determined from the experimental p-y curves using equation 4.8 and plotted versus dimensionless displacement y/b for the depth under consideration. Here, y is the peak to peak lateral displacement divided by two and b is the pile diameter. Damping ratios for the loose sand deduced from centrifuge test 17 during initial shak-ing (t = 0.78 - 1.95 sec) are in the range of 20 to 30 percent, approaching an asymptotic value of about 30 percent for large pile deflections. Damping is observed to increase with increasing pile deflection over the range examined (Figure 4.47a). Dimensionless dis-placements y /d exceed 0.01 for the data plotted, suggesting that relatively large strains Chapter 4. Centrifuge Test Results 171 existed in the soil adjacent to the pile. The computed damping ratios are therefore con-sistent with large strain sand behaviour. Damping ratios derived during steady state shaking after approximately 10 loading cycles are also shown in the figure and are seen to be in the range of 10 to 15 percent. The amplitudes of pile deflection are less than those that occurred during transient shaking. This plus cyclic densification of the loose sand is believed responsible for the reduced damping in the near field soil. Damping ratios have also been derived from centrifuge test 12 carried out in loose sand. Damping ratios have been derived from the p-y curves obtained during two load cycles at times t = 4.62 -5.60 sec and t = 11.72 - 12.78 sec (Figure 4.47b). The computed damping ratios are in the range of 20 to 27 percent and show no significant increase with increasing pile deflection over the range examined. Average shear strains around the pile resulting from the lateral pile movement have been computed to be in excess of 1.0 percent over the 0 to 5 pile diameter depth using the analytical models described in section 4.4. The above damping ratios are consistent with large strain values normally assumed for sands. Damping ratios have also been derived for a later cycle of shaking (t = 17.06 -17.91 sec) and are in the range of 11 to 15 percent above the 4 pile diameter depth. These are lower than for previous cycles due presumably to the effects of cyclic densification. Similar trends have been observed by Hardin and Drnevich. Data obtained at the 5 pile diameter depth, marked with an asterisk in the figure, shows large strain damping ratios to be substantially higher than for shallower depths. This may suggest that as pile vibration amplitudes decrease with depth, cyclic densification and the reduction in Dmax is not as pronounced. Damping ratios derived from test 15 carried out in dense vibrated sand where pile vibration levels were high enough to induce hysteretic p-y response are presented in Figure 4.47c. Computed damping ratios are in the range of 8 to 12 percent, and are similar to values deduced from test 12 after a large number of shaking cycles. The damping Chapter 4. Centrifuge Test Results 172 ratios are reasonably constant over the range of lateral pile deflections examined and demonstrate that as the number of shaking cycles increases and cyclic densification of the sand occurs, frictional damping decreases. 4.5 Equivalent Visco-Elastic Soil Resistance The lateral soil resistance to relative movement between the pile and the free field soil during low level shaking is often described using the theory of visco-elasticity using equivalent elastic soil properties compatible with average strain levels around the pile. The real component of stiffness is described in terms of a lateral Winkler stiffness, kh, which equals the secant slope of the p-y hysteresis loops described in section 4.3. The Winkler stiffness is theoretically determined from the equivalent elastic shear (or Young's) modulus of the near field soil. The p-y curves have been seen to exhibit significant non-linearity depending on the strength of shaking and the density of the foundation soil. It is therefore not clear whether equivalent elastic models can be used to predict the variation of lateral Winkler stiffness inferred from the computed p-y curves. The shear stiffness of the near field soil depends on confining stress and shear strain levels around the pile which vary with angular orientation, 0, and distance, r, from the pile. It is assumed that the shear modulus in the near field soil depends on the maximum shear strain in the horizontal r — 9 plane. Selection of an effective shear modulus, Gnf, therefore depends on estimating shear strains and a zone of influence around the pile over which to compute average strains. Procedures to compute shear strains around the pile and strain averaging procedures are described in Appendix C. These are based on analytic solutions to the problem of a pile vibrating in a homogeneous elastic medium. The elastic solutions are derived from Navier's equations of motion and simplified using plane displacement and plane strain assumptions of soil deformation. A comparison of Chapter 4. Centrifuge Test Results 173 ;(a) O a a. E A Initial Transient Shaking x Steady State Shaking 0 . 0 0 0.01 0 . 0 2 0 . 0 3 Dimensionless Defl. y/cl (b) .2 2 0 H o a: o> c E o Q A t = 4.62 - 5.60 sec x t = 11.72 - 12.78 sec • t = 17.06 - 17.91 sec 0.01 0 . 0 2 0 . 0 3 Dimensionless Defl. y/g^ 0 . 0 4 (C) 3 0 Q .2 2 0 o or O) c o Q A t = 5.15 - 5.95 sec X t = 10.82 - 11.58 sec 0 . 0 0 0.01 0 . 0 2 0 . 0 3 Dimensionless Defl. y/^ Figure 4.47: Frictional damping ratios, D, versus dimensionless pile deflection y /d (a) test no. 17 (b) test no. 12 (c) test no. 15 Chapter 4. Centrifuge Test Results 174 the solutions is given in the appendix. This shows that the plane strain assumption of soil behaviour predicts strains less than those predicted using the plane displacement assumption. It has also been shown that provided one can estimate a strain compatible modulus for use in the homogeneous elastic solution, computed strains are in reasonable agreement with those computed using non-linear finite element solutions. Provided one can estimate the effective zone of influence, average maximum shear strains,7maa., and moduli in the near field soil, Gnf, can be computed. The zone of influ-ence used in the calculations is based on finite element studies described subsequently. Relationships between imax and GnfJGmax have been based on the hyperbolic model proposed by Hardin and Drnevich (1972) where Gmax is the low strain shear modulus de-termined from the bender element measurements. The Hardin and Drnevich relationships are given for cohesionless soils as, Gnf 1 Gmax 1 + 7 max 17r (4.9) where lr = ^ (4.10) {^max f /1 + A-Q , . A 2 (1-K0 A 2 \\ 1 / 2 Tmax = U Y~ 0-vSmJ — <7VJ j (4.11) The imaginary component of the lateral subgrade stiffness represents energy losses due to radiation and hysteretic damping in the near and free field soil. The distinction between the near and free field is made to indicate that the near field response is dom-inated by the effects of localized pile vibration relative to the free field ground motions. The stiffness and damping properties of the free field soil are governed by the strength of shaking generated by the earthquake base motions and are considered to be unaffected by the presence of the pile. Chapter 4. Centrifuge Test Results 175 Current engineering practise evaluates radiation damping in the free field based on the results of one or two-dimensional wave propagation analyses. Radiation damping due to shear wave propagation away from the pile is proportional to the velocity of shear wave propagation (see section 6.2) and is estimated based on strain level dependent shear moduli from which the effective shear wave velocities are computed. Since shear strains are higher in the near field soil, the effective shear wave velocity in the near field is less than the corresponding velocity in the free field. It may therefore be concluded that radiation damping will be dominated by soil properties in the free field. Hysteretic damping, on the other hand, is mainly dependent on shear strains in the near field where shear strains and material damping ratios are highest. Radiation damping is only fully effective if a progressive wave is capable of being generated which occurs if the frequency of excitation lies above the fundamental frequency of the free field soil layer (Nogami and Novak, 1977). The presence of the walls of the soil container on the centrifuge may also reduce radiation damping if the styrofoam boundaries are not capable of preventing wave reflections. Hysteretic damping in the near field is described using a frictional damping ratio D (see section 4.3.4) which may be estimated using the following relationship modified slightly from the original equation proposed by Hardin and Drnevich to give a non-zero damping value for small (approximately zero) strains: to equation 4.9, Dmax is the large strain damping ratio and J9 m , n is the small strain damping ratio of dry sand. The experimental p-y data indicates Dmax has values of 25 to 30 percent for loose sand and decreases to values of 10 to 15 percent for vibrated dense sand. (4.12) where Gnf/Gmax depends on the average shear strain level in the near field according Chapter 4. Centrifuge Test Results 176 Having estimated frictional damping ratios, viscous damping coefficients due to hys-teretic damping, CH, have been derived from equations 4.4 to 4.8 and are given as, Ch = (4.13) where co is the frequncy of excitation. The above equation was derived assuming an equivalent viscously damped system and predicts that as the frequency of excitation increases the dashpot coefficient decreases. When carrying out dynamic pile analysis in the time domain, it is necessary to choose an average to for the frequency range of interest. Since pile response is most sensitive to damping around resonance of the system, to is generally set equal to the first mode resonant frequency of the soil-pile system. 4.5.1 Computed Lateral Winkler Stiffness and Material Damping Equivalent visco-elastic models of lateral soil response are most often applied to low level shaking of piles where average strains in the near field are sufficiently low that equivalent elastic secant moduli can be computed to reasonable accuracy. For larger strain excitation when lateral soil response is dominated by the ultimate lateral resistance of the soil, computations of equivalent elastic moduli using the Hardin and Drnevich relationships are inaccurate and predict near zero secant stiffnesses for average shear strains in excess of about 1 percent(Ishihara, 1982). For stronger shaking, the lateral soil response is, in general, characterized using non-linear p-y curves where secant lateral stiffnesses are selected to be compatible with pile deflections based on the input p-y curve. Equivalent visco-elastic models have therefore been used to compute lateral Winkler stiffnesses, kh(z), during low level shaking of the model piles. The lateral stiffness is dependent on the following factors based on analytical and finite element solutions to the problem of a pile in a homogeneous, visco-elastic medium subjected to base motion excitation (Kagawa and Kraft, 1981b): Chapter 4. Centrifuge Test Results 177 • the visco-elastic properties of the near field soil (Gnf, v, D) which depend on the pile vibration amplitudes. • the relative soil-pile flexibility KT = EI/En/r^ where Enf = 2G n / ( l + v) and ro is the pile radius. • the pile slenderness ratio H/2rQ where H is the pile length. • the frequency (u) of base motion excitation. • the depth z. Kagawa and Kraft's solution provides a three dimensional solution for the complex valued lateral soil stiffness. The real component of lateral soil stiffness kfl(= p/y), gives the in phase soil resistance p occurring in response to relative deflections between the pile and the free field soil, y. Their solution predicts that the lateral soil stiffness varies with depth z. To simplify the use of their method, they have suggested averaging the relationship over the total length of the pile to produce a depth independent relationship between kh and Enf. This procedure results in average Winkler stiffnesses which have been found to adequately model the dynamic pile response. The relationship between kh and Enf is expressed using a simple proportionality as, h = SEnf (4.14) where 8 is a proportionality constant that depends on Kr and H/r0. Kagawa and Kraft's three dimensional solutions have been developed for homogeneous elastic media. Since the near field moduli vary non-uniformly with depth, Kagawa and Kraft compute an average modulus Enj over the pile length to evaluate the factor Kr. Knowing Kr and H/r0, the proportionality constant 8 may be estimated. The average Chapter 4. Centrifuge Test Results 178 < < EC U 0 U / i & o ? l , n Z ' e n L/2r0 = 67 (Yoshidl O 9 7 , 7 ) , | , Yoshinaka(1972) 1 0 2 1 0 \" 1 0 6 1 0 ' 8 LOCAL PILE FLEXIBILITY. Kf Figure 4.48: Proposed relationship between 6 and Kr for various values of H/2r0 at zero frequency and comparison with other researcher's relationships, (after Kagawa and Kraft, 1980a) modulus is computed using the following equation: Enf = &b*?*L (4.15) where and y are depth dependent and is estimated iteratively using equation 4.14 for a trial value of 8. The relationship between 8 and Kr for an H/2r0 of 20, which corresponds approx-imately to the aspect ratio used in the centrifuge tests, is shown in Figure 4.48. The static (zero frequency) relationship is shown since Kagawa and Kraft have shown that the frequency dependence of 8 is not strong provided the frequency of excitation does not match the fundamental frequencies of the far field soil. Pronounced minima in the computed values of 8 occur at resonance of the free field since the relative deflections between the pile and the free field soil are large. These minima become less pronounced as soil damping increases. To evaluate the approximate range of Kr appropriate to the centrifuge tests, the Chapter 4. Centrifuge Test Results 179 Test No. Sand Density Base Motion h(z) (kPa) Kr 17 Loose Sinusoidal 5600 z 0.64 x 104 14 Very Dense Earthquake 4900 z 1.17 x 104 41 Very Dense Sinusoidal 20,000 z1'2 2.03 x 103 Table 4.5: Computed Relative Soil-Pile Stiffnesses, K r , for Low Level Shaking Winkler moduli distributions, kh(z), described in section 4.3 were used to evaluate Enf. The lateral pile deflection profiles y(z) computed using the linear or square root of depth Winkler models were used to evaluate K T . The computed K T values are given in Table 4.5 for a trial 6 value of 1.9. The K r values are seen to range between 2.03 x 103 for centrifuge test 41 to 1.17 x 104 for centrifuge test 14, giving a range in 6 values of 1.8 to 2.0. An average value of 1.9 has been selected for further computations. In practise, kh(z) and y(z) would not be known prior to computing the dynamic pile response for a specific earthquake input and would be estimated iteratively as a function of an effective modulus around the pile and trial values of 6. If a non-linear analysis of the pile response was being carried out, kh would be varied for each time step. If kh. was assumed to remain constant during shaking, then a suitable average value over the duration of shaking would be chosen. For each iteration, the relative pile deflection profile y(z) would be computed and used to estimate average strains and effective moduli. From the estimated modulus variation along the pile, the average modulus and K T value would be computed and a new estimate of 6 made. This would be compared to original estimates and the above steps repeated until convergence between the initial and final estimate of 8 was achieved. An assumption implicit in the use of elastic theory to model soil- pile interaction is that the soil on the back face of the pile is capable of sustaining tension. The assumption Chapter 4. Centrifuge Test Results 180 that a cohesionless soil can sustain tensile stresses is clearly unreasonable and corrections are normally made to account for this effect (Poulos and Davis, 1980). The influence of soil resistance on the back face of the pile was subsequently investigated using the finite element computer code FEADAM84 developed by Duncan et al (1984). A plane strain analysis of a disc (pile) being translated laterally under static conditions was carried out. The finite element code uses isoparametric quadrilateral elements with incompatible displacement modes developed by Wilson et al (1971) and incorporates a non-linear, incremental elastic stress-strain law based on the well known hyperbolic model. The latter describes the soil's stress-strain response using tangent values of Young's modulus and bulk modulus which are stress level dependent. The finite element model used in the calculations is shown in Figure 4.49. The lat-eral boundaries of the finite element mesh were set 12 pile radii from the centre of the disc which had a diameter equal to that of the prototype pile (0.57 m). The location of the lateral boundary provided a match between elastic lateral stiffnesses computed by Baguelin et al (1977), assuming plane strain translation of a rigid disc in a homogeneous elastic medium, and the three dimensional elastic solution of Kagawa and Kraft. The dis-tance to the outer rigid boundary beyond which the soil is not affected by pile movement also agrees with centrifuge and shake table results which show that interaction between piles in a group for both low and high level shaking extends to approximately 12 pile radii from the centre of any pile in the group. Following establishment of the appropri-ate mesh size, lateral loading was applied incrementally to the disc. During each load increment, each element was checked to determine if it was in a state of primary loading, unloading/reloading, tensile failure or shear failure. Tensile failure occurred when the minor principal stress o~3 was found to be negative, as occurred in the active region of the soil behind the translating disc. If an element was in tension, the tangent moduli were assigned small values to minimize the effects of the soil on the back face of the \\ Chapter 4. Centrifuge Test Results 181 o 1 I—I—I—t—i—I SCALE IN METERS Figure 4.49: Plane strain finite element model of a rigid translating disc. disc. If the stress state of an element was such that it exceeded 95 percent of its shear strength computed using the Mohr-Coulomb criteria, the tangent Young's moduli were also assigned very low values. The hyperbolic stress-strain parameters used in the analysis are given in Table 4.6 for both loose and dense sands based on parameters suggested by Byrne et al (1987). Initial vertical and horizontal stresses in the sand have been computed at the 4 pile diameter depth using the sand densities and Ko values given in the table. The computed lateral load - deflection relationship of the disc for each sand density considered is shown in Figure 4.50. To check the influence of the soil on the active face of the disc, initial moduli used in the hyperbolic model calculations were reduced by a factor of 100 in those elements behind the pile (see Figure 4.49). Computed lateral load - deflection Chapter 4. Centrifuge Test Results 182 Parameters Loose Sand Dense Sand Ke 350 1000 ne 0.5 0.5 Kb 200 600 rib 0.25 0.25 R} 0.9 0.65 4f 35° 45° Acf>' 0° 6° K0 0.4 0.6 7 d (kN/m 3) 14.7 16.7 Table 4.6: Hyperbolic Stress-Strain Parameters Used in Finite Element Analysis relationships for the disc were again computed and are shown in Figure 4.50. The revised curves are not significantly different from those computed including the influence of a no-tension soil behind the disc. The comparison shows that the majority of lateral soil restraint is due to passive soil resistance at the front of the disc. The secant lateral stiffnesses (kh = p/y) have been computed from the lateral load - deflection plots given in Figure 4.50. The stiffnesses are shown plotted versus lateral deflection normalized with respect to the pile diameter (y/d) in Figure 4.51. The results are shown with and without the influence of the no-tension soil on the active face of the disc, i.e. the 'no gap' and 'with gap' cases, respectively, shown in the figure. The analysis predicts that the active soil region affects the lateral stiffness more strongly for smaller deflections than when the disc undergoes larger deflections. This is believed due to the influence of shear stresses on the disc. For smaller deflections, the shear stiffness of the sand is higher leading to larger shear stresses on the disc whose horizontal components contribute to the combined lateral resistance of the soil. For larger deflections, the shear stiffness is considerably reduced so that shear stresses drop off. The lateral resistance then approaches the situation where the influence of the soil on the rear half of the disc Chapter 4. Centrifuge Test Results 183 ;(Q) 100-Dimensionless Defl. Y/d (b) <0 D \"5 200 150 100-0.000 0.025 Dimensionless Defl. Y/d 0.050 Figure 4.50: Computed lateral load - deflection relationships for plane strain translation of a rigid disc in a no-tension soil (a) loose sand (b) dense sand Chapter 4. Centrifuge Test Results 184 may reasonably be neglected. The finite element results suggest that the elastic Winkler stiffnesses computed using the Kagawa and Kraft model should be multiplied by 0.5 for large pile displacements which accounts for the inability of soil on the active face of the pile to sustain radial tension stresses and the reduction in the shear stress component of soil resistance on the pile. For smaller deflections, the reduction is not as pronounced. The suggested variation of 8 versus dimensionless pile displacement y/d has been derived from the finite element results and is shown in Figure 4.52. This variation has been used to compute kh(= 6 Enf) as a function of dimensionless pile deflection y/b. The finite element computations have also been used to estimate zones of influence around the pile to permit computation of average strains and moduli in the near field soil. The secant Winkler stiffnesses kh were computed over a range of disc deflections and used to compute the effective secant Young's modulus from the relationships kh — 8 Enf. The average stress-strain response of the near field soil was then expressed in terms of a non-linear hyperbola using the relationship given by Duncan et al (1980) as, ^ - F ,ult where Ei is the initial Young's modulus used in the hyperbolic stress-strain model and 1\\ is the average major principal strain in the near field soil. The ultimate deviatoric stress, <7d,uit is given by 1 2<73sincy o-d,uit = r—77 (4.17) Rjl— sin '/2)), a'v is the effective overburden pressure at the depth of interest, d is the pile diameter and 0 is an arbitrary proportionality constant which depends on depth and the effective friction angle of the soil '. Barton has suggested that 8 is in the range of 1 at shallow depths to values as high as Kp at great depth based on lateral pile load tests carried out using the Cambridge centrifuge. Broms (1964) has suggested 3 ~ 3 while Brinch-Hansen (1961) and Reese, et al (1974) give 3 as a function of depth, confining stress, and effective friction angle. During unloading or reloading, use of the Masing criteria in equation 4.20 gives the following relationship: Knowing the point (yr,pr) at the start of an unload or reload cycle, the p-y hysteresis loop is easily calculated using equation 4.26. Pult = 8Kp(r'vd (4.25) (4.26) Chapter 4. Centrifuge Test Results 198 Sensitivity analyses carried out using the above equations have shown that for a prescribed value of Dmax, yuit and puu the computed p-y hysteresis loops are insensitive to the value of kh. Therefore, kh has been assumed related to the low strain Young's modulus Emax based on the Kagawa and Kraft analysis described in section 4.4, or kh = 8Emax. Here Emax has been computed from the measured low strain shear moduli Gmax m the foundation sand using a Poisson's ratio of 0.2 and 8 has been assigned a value of 1.9. The secant slope of the computed p-y hysteresis loops have been found to be sensitive to the values of puit (or 0) and yuit assumed. In general, reducing the value of 0 or increasing yuit leads to a natter p-y loop. Since it is not possible from the experimental p-y data to separate the effects of the two parameters, yuit has been arbitrarily set equal to 5 percent of the diameter of the pile and the parameter 0 varied until a match of the experimental and computed p-y hysteresis loops was achieved. From a practical point of view, the above sensitivity analyses have demonstrated that the large deflection response of the pile is dominated by the values of puit and yu\\t. Computed p-y hysteresis loops are shown plotted for various depths for tests 12 and 15 in Figures 4.58 and 4.59, respectively. The parameters used in their derivation are given in Table 4.8 and are seen to give an excellent match to the experimental p-y loops measured at peak pile deflection. The value of Dmax used in the calculations is based on average values inferred from the test measurements while the proportionality constant 0 varies from 1.75 to 3.5. These values are in the range recommended by Barton and Broms. The larger 0 values have been used near the soil surface and gradually decrease with depth. This may suggest that puit has a greater dependence on three dimensional out of plane effects near the soil surface while at greater depths the lateral resistance approaches a plane strain passive resistance. The calculations demonstrate that the use of a Masing model in conjunction with the Ramberg-Osgood backbone curve can be successfully used to simulate the experimental p-y curves. Chapter 4. Centrifuge Test Results 199 Depth = 1.71 m (3 pile diam.) 200 150 100 -100 -ISO -200-Depth = 3 Pile Dfam. * Time = 11.72-12.78 s«c • Masing Loop -25 -20 -15 -10 -5 0 5 10 15 Pile Displacement Y (mm) (d) Depth = 2.85 m (5 pile diam.) 4 © 3 vi 100' 75-50 25-0 -25 -50--75 -100-Oapth = 5 Pile Diam. * Time = 11.72-12.78 sec - Masing Loop -25 -20 -15 -10 -5 0 Pile Displacement Y (mm) 10 15 20 25 (e) Depth = 3.99 m (7 pile diam.) a. 30 20 10 0' -10 -20 -30 Depth = 7 Pile Dlom. • Time = 11.72-12.78 sec Masing Loop -10 - 8 - 6 - 4 - 2 0 2 4 6 8 10 Pile Displacement Y (mm) Figure 4.58: Computed Masing loops versus measured p-y hysteresis loops - test 12 (a) z/d = 1 (b) z/d = 2 (c) z/d = 3 (d) z/d = 5 (e) z/d = 7 Chapter 4. Centrifuge Test Results 200 Depth = 1.71 m (3 pile diam.) 125-100-75-•< 50 -2 25-a. 0- -50-i l_ 0- -75--100--125-Oaplh = 3 Pita Dfom. / * 1 = 10.82-11.58 sec Masing Loop (d) Depth = 2.85 m (5 pile diam.) -20 -10 0 10 Pile Displacement Y (mm) 125 100 75 50 25 0 -25 -50 -75 -100 -125 Depth = 5 Pile Diam. f i i * \\ - 10.82-11.58 sec * Masing Loop -20 -10 0 10 20 Pile Displacement Y (mm) (e) -50 Depth = 3.99 m (7 pile diam) Depth = 7 Pile Dtam. :/ • 1 = 10.82-11.58 sec Masing Loop -3 -2 -1 Pile Displacement Y (mm) Figure 4.59: Computed Masing loops versus measured p-y hysteresis loops - test 15 (a) z/d = 1 (b) z/d = 2 (c) z/d = 3 (d) z/d = 5 (e) z/d = 7 Chapter 4. Centrifuge Test Results 201 Test No. z d Dmax (%) ' (deg.) e0 Ko 0 12 1 18.0 35 0.78 0.4 3.5 2 20.0 35 0.78 0.4 2.8 3 20.0 35 0.78 0.4 2.3 » 5 25.0 35 0.78 0.4 1.8 n 7 2.0 35 0.78 0.4 1.8 15 1 12.0 45 0.57 0.6 2.8 » 2 12.0 45 0.57 0.6 2.3 n 3 12.0 45 0.57 0.6 2.1 V 5 12.0 45 0.57 0.6 1.8 V 7 2.0 45 0.57 0.6 1.8 z/d = dimensionless depth. eo = consolidated void ratio of sand. Ko = earth pressure at rest. Table 4.8: Cyclic p-y Curves - Masing Loop Parameters 4.7 Base Mot ion Excitation of Pile Groups 4.7.1 Introduction Centrifuge tests have been carried out on two pile groups to examine pile to pile interac-tion at stress levels representative of full scale conditions. An evaluation of the capability of static pile group interaction factors proposed by Randolph and Poulos (1982) to pre-dict pile group displacements has also been made. Based on pile group displacements computed using the Randolph-Poulos interaction factors, empirically based modifications to their mathematical form have been suggested. Tests on four pile groups were next carried out to help determine whether interaction factors established from the 2-pile tests could be superimposed to give a reasonable estimate of deflections and stiffness of the four pile group. The prediction of the pile group stiffness, taking into account interaction between piles in the group, is necessary in the prediction of the dynamic response of the Chapter 4. Centrifuge Test Results 202 group. The tests were carried out using dense sand foundations prepared by vibrating an originally loose sand using high frequency vibration. Average void ratios of 0.57 were achieved in the sand. Following preparation of the sand foundation, the model piles were pushed 195 ± 5 mm into the sand so that their tips rested on the base of the sand container. The distance between the soil surface and the centre of gravity of mass was 24 ± 2 mm for the two pile tests and 36 ± 1 mm for the four pile tests. The two pile groups were oriented inline, offline or at approximately 45 degrees to the direction of shaking using centre to centre spacing ratios (s/d) ratios of up to 6. The four pile group was placed in a square (2 x 2) configuration using an s/d ratio of 2. Following installation of the model piles and checking of all instrumentation, the cen-trifuge was brought up to test speed which corresponded to a 60 g centrifugal acceleration at the mid-height of the sand foundation. Shear wave velocity measurements were then made. With the low strain stiffness properties of the foundation soil determined, an input motion was applied to the base of the model. Input base, free field and pile cap accelerations were recorded during the tests as well as bending moments along both piles. Changes in dynamic axial load were also measured for one pile in the group. All data are presented at prototype scale using a centrifuge scale factor of 60. The two pile groups were subjected to low level shaking using sinusoidal base mo-tions with peak steady state accelerations of approximately 0.04 g and a predominant input frequency of 0.5 Hz. The forcing frequency selected for the tests is less than the estimated first mode natural frequencies of the pile groups (see section 4.7.2). This mini-mizes dynamic resonance effects and permits interaction between piles to be more clearly discerned. The four pile groups were subjected to sinusoidal and random earthquake base motion. Chapter 4. Centrifuge Test Results 203 (a) o.io -0,10 I I I • I I r , , I I | i . i i | ' 1 1 I 1 1 1 ' 0 5 10 15 20 25 30 35 40 Time (sec) (b) 0.10-1 -0.10 I i i i c ! i i i i i i i i i i i i i i i i i • • i i i • i i 0 5 10 15 20 25 30 35 40 Time (sec) Figure 4.60: Typical input base and free field surface motions - centrifuge test 39 (a) base accelerations (b) free field accelerations 4.7.2 Low Level Shaking - Two Pile Groups Typical sinusoidal base and free field surface accelerations measured during low level shaking of the two pile groups are shown in Figure 4.60a and b. Their characteristics are very similar to those described in section 4.3.1 for single pile test 41, thereby permitting a comparison of single pile and pile group test results. Pile cap accelerations and displacements are shown in Figure 4.61 for an inline shaking test carried out using a spacing ratio s/d = 4. The pile cap accelerations and Chapter 4. Centrifuge Test Results 204 displacements have been processed using a 10 Hz digital cutoff filter to minimize spurious high frequency electronic noise inherent in centrifuge testing. Fourier spectra computed from the accelerations show that frequencies below 10 Hz dominate the pile cap response. Following the initial transient phase of input excitation, the pile cap accelerations and displacements achieve a steady state response after about 6 cycles of shaking. The pile cap accelerations (and displacements) have a slight lack of symmetry, despite the input base motions being reasonably symmetric, for reasons described in section 4.3.1. Maximum acceleration amplitudes are seen to occur in the direction of the first loading pulse. The fundamental natural frequencies of the inline and offline groups have been in-ferred from a comparison of Fourier spectra computed from the pile cap accelerations with corresponding free field response spectra. Maximum amplification of the pile cap accelerations occurs when frequencies contained within the free field accelerations match the natural frequencies of the pile group. This is illustrated in Figure 4.62 where pile cap and free field response spectra are compared for an inline shaking test (s/d = 4). The fundamental frequency of the group is estimated to be 3.6 Hz. Similar comparisons have been made for other pile spacings and directions of shaking from which the funda-mental frequencies of the pile groups have been estimated (Table 4.9). The fundamental frequencies of the offline groups are reasonably constant in the 1.6 to 1.8 Hz range sug-gesting there is very little interaction between the piles. The fundamental frequency of a pile group subjected to inline shaking increases with increasing pile spacing, varying between 3.1 and 3.8 Hz over the range of pile spacings examined. This occurs since the counteracting moment caused by the axial reactions of the piles against the pile cap increases with increasing pile separation, thereby increasing the rotational stiffness of the group. The fundamental frequencies of the groups are substantially higher than the dominant input frequency of 0.5 Hz used in the tests and minimizes dynamic resonance effects. Chapter 4. Centrifuge Test Results 205 Figure 4.61: Pile cap response - inline shaking test 39 (s/d = 4) (a) pile cap accelerations (b) pile cap displacements in direction of shaking Chapter 4. Centrifuge Test Results 206 (a) Frequency (Hz) (b) Frequency (Hz) Figure 4.62: Comparison of Fourier spectra - inline test 39 (s/d = 4) (a) pile cap accel-erations (b) free field accelerations Chapter 4. Centrifuge Test Results 207 Shaking Direction s/d A (Hz) Offline 2 ~1.8 a 4 ~1.8 u 6 ~1.6 Inline 2 ~3.1 a 4 ~3.6 u 6 ~3.8 / i = estimated fundamental frequency bf pile group Table 4.9: Pile Group Fundamental Natural Frequencies Bending moment distributions measured along the piles with centre to centre spacings of two are shown in Figure 4.63 for both inline and offline loading. Peak bending moments averaged over the steady state region of response have been plotted for both directions of shaking. For the offline test, the bending moment distribution shows the same sign of bending moment along the pile and is similar to that for a single free headed pile. This shows that the pile cap provides no restraint during offline loading. Both piles in the group have approximately the same bending moment distribution for both directions of loading. This suggests that the piles may be regarded as being equally loaded and that there is very little interaction between the piles. Bending moments along the pile are larger for one direction of shaking than the other, illustrating the lack of symmetry in pile response due to soil non-linearity. Similar behaviour was seen for larger pile spacings. During inline loading, the bending moment changes sign indicating the restraint of the pile cap against rotation. This restraint is augmented by the fact that the pile tips rested on the rigid base of the soil container and rotations of the pile cap were strongly resisted by axial compressive forces in the piles. The bending moment distributions in the piles are sufficiently different near the soil surface to suggest significant interaction Chapter 4. Centrifuge Test Results 208 G.S. } I: P i l e 1 (+) o P i l e 2 (+) • P i l e 1 (-) -100 -50 0 50 Bending Moment (kN—m) (b) in O -JO -20 -10 0 10 20 30 Bending Moment (kN-m) Figure 4.63: Bending moment vs. depth in a two pile group for s/d = 2 (a) offline shaking (b) inline shaking between the piles. The slopes of the bending moment distributions indicate that the shear load in each pile at the soil surface varies by about 20 percent from the average in the group. When the shaking direction reverses, the loading on one pile in the group (pile no. 2) is reduced as it moves into a softened soil zone created by the adjacent pile. Similar shadowing effects have been observed during pile group tests on the shake table (see section 5.6). Bending moment distributions for inline shaking using a centre to centre pile spacing of 6 pile diameters are shown in Figure 4.64. The bending moments do not vary signif-icantly between piles for each direction of shaking, suggesting that the piles are equally Chapter 4. Centrifuge Test Results 209 13 Q. j — _0> CL > o < o c o in O G.S. B I / / I ' * \\ Pile 1 (+) o Pile 2 (+) a Pile 1 (—) • Pile 2 (-) 1 1 1 1 1 ' 1 ' ' -50 -25 0 25 Bending Moment (kN-m) 50 Figure 4.64: Bending moment vs. depth in a two pile group - inline shaking - s/d = 6 loaded and that there is a low degree of interaction between them. Test measurements for the 2-pile group tests carried out on the centrifuge are summarized in Table 4.10 where peak average values of pile cap acceleration, pile cap displacement, input base and free field acceleration are given. Data are also given for tests where the axis connecting the centres of the two piles was at an angle 8 ~ 45° to the direction of shaking. In the latter case, pile cap accelerations were recorded in a direction perpendicular to the longitudinal axis of the group and lie at an angle 8 to the direction of shaking. Average displacement and acceleration quantities for each test have been computed as peak to peak values in the steady state region of response divided by two. Peak input base accelerations averaged between 0.040 and 0.045 g with the exception Chapter 4. Centrifuge Test Results 210 Shaking Direction Xff A x (g) (g) (g) (mm) Offline 2 0.045 0.060 0.043 5:60 (5.23) « 4 0.042 0.054 0.041 4.75 a 6 0.041 0.053 0.039 4.90 (5.02) Inline 2 0.027 0.043 0.040 2.50 (3.89) a 4 0.044 0.055 0.061 2.50 (2.39) a 6 0.045 0.060 0.063 2.05 (1.91) 0 = 45° 2 0.043 0.056 0.032 4.10 (4.00) 0 = 42.5° 4 0.040 0.060 0.029 3.60 (3.78) 0 = 41° 6 0.042 0.052 0.035 3.50 xi, = average peak input base acceleration (unfiltered). Xff = average peak free field acceleration (unfiltered). Xcap = average peak acceleration of the pile cap (10 Hz filtered data). A x = pile cap displacement recorded at the top of the pile cap/mass assembly in the direction of shaking. The figure in parenthesis refers to pile cap displacements scaled linearly to a common peak base acceleration of 0.042 g. 0 = angle between longitudinal axis of pile group and the direction of shaking. Table 4.10: Pile Group Test Data Chapter 4. Centrifuge Test Results 211 of one inline shaking test carried out using a pile spacing ratio s/d = 2. Lower shaking intensities were accidentally used in this test due to unforeseen changes in the calibration of the span control on the Caltech shaker. For input base accelerations of about 0.04 g, free field accelerations were amplified to values between 0.05 and 0.06 g. For the offline tests, pile cap accelerations were relatively constant at about 0.04 g and were not significantly amplified relative to the input base motions. For the inline tests with an input base acceleration of 0.04 g, pile cap accelerations were approximately 0.06 g and did not vary appreciably with changes in pile spacing. This is is believed due to the constraint imposed on the pile cap by the piles reacting against the rigid base of the sand container since with a compliant base (i.e. where the tips of the piles rest in the sand) it would be expected that pile cap accelerations would increase as pile spacings decrease. The pile cap accelerations were also observed to have a higher frequency content than accelerations recorded during offline shaking. These high frequency acceleration components have resulted in pile cap accelerations larger than those recorded during offline shaking. Pile cap displacements measured parallel and perpendicular to the direction of shak-ing are shown in Figure 4.65 for a test with the piles oriented at 3 ~ 45° to the direction of shaking. A pile spacing ratio s/d = 2 was used in this test although similar behaviour was observed for other pile spacings. The strong components of motion in both of these directions is clearly evident and shows that biaxial bending of the piles in the group oc-curs. Since the purpose of these tests was to determine the significance of pile interaction for shaking at an angle 3 to the longitudinal (inline) axis of the group it was necessary to measure loads and pile group displacements in the direction of shaking. The instru-mented piles were therefore oriented so that the bending strain gauges were inline with the axis of shaking. Average pile cap displacements are plotted against the pile spacing ratio, s/d, for ratios between 2 and 6 for different shaking directions in Figure 4.66. The measured Chapter 4. Centrifuge Test Results 212 (a) Time (sec) (b) 10 -io I i i I 0 5 10 15 20 25 30 35 40 Time (sec) Figure 4.65: P i le cap displacements (a) parallel and (b) perpendicular to the direction of shaking - s/d = 2 (B = 45°) Chapter 4. Centrifuge Test Results 213 I 5 c . Qj 4 E o X l < 0) o c o Soil Surface - 2 5 Pile Top -X. \" i s X Theory / Offline s/d = 4 A P i l e 1 - E x p ' t x P i l e 2 - E x p ' t T h e o r y 0 2 S 5 0 7 5 Bending Moment (kN-m) (C) JE, io i— • 9 > o < o c D 8H 5H 3-t Soil Surface - 2 5 V P i l e T °p X Theory T / */ / Offline s/d = 6 A P i l e 1 - E x p ' t X P i l e 2 - E x p ' t T h e o r y 0 2 5 5 0 7 ! Bending Moment (kN-m) 1 0 0 , Figure 4.67: Comparison of predicted and measured bending moments using a Winkler model for offline shaking (a) s/d = 2 (b) s/d = 4 (c) s/d = 6 Chapter 4. Centrifuge Test Results 217 Shaking Direction Vo Mo a A ; A X (kN) (kN-m) (kN/m 5 / 2 ) (mm) (mm) Offline 2 23.5 45.0 18,000 5.58 5.60 u 4 22.0 39.0 20,000 4.74 4.75 u 6 21.0 43.0 20,000 4.97 4.90 Inline 2 21.0 0.0 13,000 2.49 2.50 u 4 33.0 -18.0 14,000 2.46 2.50 u 6 34.0 -17.0 20,000 2.02 2.05 0 = 45° 2 17.5 20.5 9,500 4.09 4.10 0 = 42.5° 4 13.0 22.5 10,500 3.58 3.60 3 = 41° 6 27.0 15.0 20,000 3.48 3.50 Vo = average shear force per pile at the soil surface Mo = average bending moment per pile at the soil surface a = Winkler modulus proportionality constant A * = computed deflection at top of structural mass Ax = measured average deflection at top of structural mass Table 4.11: Average Forces and Deflections of Centrifuged Pile Groups - Low Level Shaking loading. For larger spacings (s/d = 6) the Winkler modulus distribution is the same as used in the single pile test and shows there is little interaction between the piles. Pile cap deflections in the direction of shaking computed using the above moduli and average loads in the group are given in Table 4.11. The latter are compared with measured deflections and are seen to agree closely. In summary, the centrifuge test data indicate that there is very little interaction between piles for offline shaking and that during inline shaking interaction between piles extends to distances of up to 6 pile diameters from the centre of a pile. Interaction between piles is also seen to be significant using departure angles of 45 degrees and is somewhat stronger than observed for inline shaking, possibly due to the effects of biaxial shaking of the piles. Chapter 4. Centrifuge Test Results 218 10 J . . Q . r— bt 0) XI < S 7 c o •I 6H 3-t ^ X ~ - . a Pile Top Soil Surface \\ I J Theory Inline s/d = 2 / A Pil« 1 - Exp't / x Pil« 2 - Exp't Theory Soli Surface i (b) -50 -25 0 25 50 Bending Moment (kN-m) 12-a. r -'cZ > o XI < c Q 10 9-a Pile 1 - Exp't x Pile 2 - Exp't Theory Soil Surface \\ Pile Top Soil Surface V '. Theory Inline s/d = 4 / -75 -50 -25 0 25 50 Bending Moment (kN-m) 75 (O 10 a. i— Q_ a o 8 XI < S 7 c o \"in Q 6-Pile Top s Soil Surface \\ \\ j Theory Inline ^ ' s/d = 6 / A Pile 1 - Exp't ' x pile 2 - Exp't Theory Soil Surface -75 -50 -25 0 .25 50 75 Bending Moment (kN-m) Figure 4.68: Comparison of predicted and measured bending moments using a Winkler model for inline shaking (a) s/d = 2 (b) s/d = 4 (c) s/d = 6 Chapter 4. Centrifuge Test Results 219 (a) ea-rn 9 § 8 < J3 Q \\ Pile Top ^ Soil Surface - J Theory -J $ = 45° y s/d=2. / c Pile 1 - Exp't f x Pile 2 - Exp'l Theory -50 (b) 50 £, io _ 9 _0J c£ o XI < CD (J c o 8-34 \\ x P i l e T ° P * \\ Soil Surface I Z< Theory / 0 = 42.5° J s/d = 4 / a Pile 1 - Exp't f x Pile 2 - Exp't Theory -50 50 Bending Moment (kN—m) Bending Moment (kN-m) * Pile Top \\ -\\ ^ Soil Surface - i - ^ Theory -/ 0 = 4 1 ° J s/d = 6 I i j A Pile 1 - Exp'l x Pile 2 - Exp'l Theory -50 0 50 100 Bending Moment (kN-m) Figure 4.69: Comparison of predicted and measured bending moments using a Winkler model - 0 ~ 45° (a) s/d = 2 (b) s/d = 4 (c) s/d = 6 Chapter 4. Centrifuge Test Results 220 4.7.3 Pile Group Interaction Analysis Pile group interaction is currently assessed using an equivalent elastic representation of the soil medium. To account for soil non-linearity single pile flexibilities are determined for a particular level of loading. Interaction coefficients derived from two or three dimen-sional elastic continuum analyses are then applied to the single pile flexibilities to account for the influence of one pile on adjacent piles. Since the interaction coefficients are derived using elastic theory they cannot account for the effects of soil non-linearity, including the effects of the inability of cohesionless soils to sustain tension, soil-pile separation (gap-ping) and slip between the soil and the pile. Considering the potential limitations of this approach, it is important to validate the use of these coefficients. The static analysis procedure proposed by Randolph and Poulos has been used since it takes into account three dimensional interaction effects and has been developed for both homogenous and inhomogeneous elastic soil conditions. Two-pile elastic interaction coefficients have also been proposed by Kaynia and Kausel (1982) for dynamic loading conditions. The impor-tance of the frequency of excitation, u, on interaction coefficients has been determined through consideration of the dimensionless frequency of excitation, a 0 , defined by Kaynia and Kausel as «o = ^ r (4.27) where Vs is an average shear wave velocity selected to be compatible with strains in the near field soil. It may be computed from the equation Gc — pV32 where Gc is the effective shear modulus over the critical length of the pile defined subsequently. The effective shear modulus has been estimated in terms of the Winkler modulus distribution used to describe the single pile response to low level sinusoidal excitation (centrifuge test 41), giving a dimensionless frequency of the order of 0.01. Kaynia and Kausel suggest that when ao < 0.1 static pile interaction coefficients may be applied. Chapter 4. Centrifuge Test Results 221 Neglecting interaction between piles, deflections and rotations at the soil surface (UQ, OQ) of a single pile are given as, where fuv is the lateral displacement u at the soil surface under the action of a unit horizontal force v and fum is the lateral displacement under the action of a unit moment m. The flexibility coefficients for pile head rotation, f$v and f$m, are similarly defined. The single pile flexibility coefficients are derived in Appendix D for the two cases of the lateral Winkler stiffness proportional to the square root of depth and linearly proportional to depth. The influence of pile interaction results in deflections and rotations at the soil surface that are increased relative to those computed neglecting interaction . This may be taken into account using the elastic interaction factors rjij given by Randolph and Poulos and computed for various modes of pile interaction. The interaction factors are a function of pile spacing, s, and orientation angle 3. The latter defines the angle in the horizontal plane between the direction of loading and a line connecting the centres of any two piles such that for inline shaking, 3 = 0, and for offline shaking 3 = w/2. The interaction factors between any two piles for the various modes of interaction are given by Randolph and Poulos as, fuvV0 + fumM0 (4.28) (4.29) V V 0.7r7 (4.30) (4.31) V a V V (vuvf (4.33) (4.32) Chapter 4. Centrifuge Test Results 222 In order to avoid problems at very close spacings (where s —+ 0 would imply cv —» oo), a transformation is introduced such that, where an interaction factor is calculated to be greater than 1/3, then the value is replaced by This has the effect of smoothly transforming the hyperbola into a parabola for values of a greater than 1/3. It is also noted that equation 4.34 implies a —• 1 as s —• 0. The interaction factors are seen to depend on the relative stiffness between the soil and the pile and the shear modulus distribution G(z) assumed in the analysis. The latter is expressed by the factor pc which has a value of 1.0 for homogeneous soil conditions, a value of 0.5 where G(z) increases proportionally to the square root of depth, and a value of 0.25 when the modulus increases linearly with depth. The relative soil-pile stiffness is expressed by the ratio E*/G*C where E* = AEI/TTTQ, EI is the flexural rigidity of the pile, ro is the pile radius and G* is the shear modulus averaged over the critical length of the pile, lc, times the factor (1 + v). Here v is the estimated Poisson's ratio of the soil and has been assumed equal to 0.35 in the calculations. The critical length of the pile is given as, For low level shaking, two modulus distributions were used to assess the sensitivity of the computed results to this factor. In the first approach, low strain shear moduli derived from the shear wave velocity measurements were used, resulting in moduli proportional to the square root of depth. In the second approach, effective shear moduli were estimated from Winkler stiffness distributions used in the derivation of single pile flexibility factors. These are considered to represent strain equivalent moduli in the near field soil. The shear modulus distribution G(z) was related to the lateral Winkler stiffness kh(z) using (4.35) Chapter 4. Centrifuge Test Results 223 the following equation: kh(z) ~ KG(Z) (4.36) where K is a proportionality constant which is dependent on pile length, relative soil-pile stiffness, frequency of vibration and depth z. Based on elastic studies of soil-pile interaction carried out by Kagawa Kraft (1980) described in section 4.4, K was set equal to a value of 5.3. From the previous Winkler analysis, kh = OLZ1!2. G(Z) is then given approximately as, G(z) ~ ^— (4.37) Using the above modulus distributions and the known structural properties of the piles, interaction factors have been estimated for various centre to centre pile spacings and orientations with respect to the direction of shaking. The deflection at the soil surface U Q t- of the ith pile in a group subjected to its own moment and shear loading ( M , - , Vi) plus the influence of loads from N adjacent piles (Mj, Vj) is given as, 4 4.84 (4.73) 4.73 4.75 6 5.04 (4.97) 4.97 4.90 Inline 2 2.39 (2.92) 1.92 2.50 4 2.17 (2.50) 1.86 2.50 JJ 6 2.22 (2.08) 2.02 2.05 £ = 45° 2 3.23 (4.03) 2.93 4.10 8 = 42.5° 4 2.85 (3.10) 2.74 3.60 8 = 41° 6 3.61 (3.50) 3.48 3.50 A I | S = pile cap deflection with group interaction using a square root of depth variation in shear modulus (Randolph and Poulos model). Figure in paranthesis refers to pile cap deflection computed using modified interaction factors. Aj; i S = pile cap deflection with no group interaction using single pile flexibility factors. A x = measured deflection at the top of the structural mass in the direction of shaking. Table 4.12: Pile Group Interaction Analysis - Centrifuge Tests -Low Level Shaking better estimate of group response. For shaking with 8 ~ 45°, the calculations indicate an underestimate of group deflections by approximately 21 percent for spacing ratios of 2 and 4. The greater underestimates of group deflection in the latter case result from the fact that an elastic analysis predicts less interaction between piles for 8 ~ 45° relative to the case of inline loading. Group deflections for a spacing ratio of six are slightly overpredicted by approximately 3 percent. Barton (1982) has also observed that group displacements measured during inline shaking for pile spacing ratios of up to 4 are underpredicted by up to about 20 percent using the above elastic interaction factors. For larger pile spacings and for offline shaking over a range of pile separations, Barton's data also show that elastic theory overestimates pile group displacements. These discrepancies are attributed to the effects of local soil yielding which alters the displacement field around a pile from that predicted using elastic Chapter 4. Centrifuge Test Results 226 theory. The latter tests were carried out on the Cambridge geotechnical centrifuge using two, three and six pile groups subjected to cyclic pile head loading. The test data was used to estimate pile group interaction coefficients over a range of pile spacings and angular orientations. Barton also carried out a static, laterally loaded pile analysis incorporating a no tension, elastic-plastic model of the soil's stress-strain behaviour. The analysis shows that the displacement field around a laterally loaded pile decays more rapidly than is indicated using an elastic model of soil response. Use of the Randolph-Poulos elastic interaction factors implies that displacements decay inversely proportionally to distance from the pile, i.e., as 1/r. Barton's non-linear analysis shows that displacements decay approximately proportionally to 1/r 2 for spacing ratios s/b greater than 4. There are other full scale and model test data to support the present experimental findings. Tominaga et al (1983) have summarized a number of full scale, static lateral load tests of single piles embedded in loose to medium dense sandy soils. Horizontal ground surface displacements were reported in the direction of loading and along a line 45 degrees from the direction of loading at various distances from the test piles. The tests showed that ground surface deflections were negligible at distances of about 6 pile diameters from the centre of the loaded pile over a range of pile deflections. Interaction between piles may therefore be neglected beyond this point. Ground surface deflections in the direction of loading were also seen to be similar to those recorded along the 45 degree line, showing that interaction between piles for 8 ~ 45° was as strong as for the inline case. Static and cyclic lateral loading of model pile groups in sand under 1 g conditions have been carried out by several researchers and their results summarized by O'Neill (1983). These data suggest that interaction extends up to distances of up to 6 pile diameters between piles for inline loading and up to 2.5 pile diameters for offline loading. Frequency sweep tests employing low amplitude harmonic loading of a group of 102 model piles have been described by E l Sharnouby and Novak (1984). The piles were Chapter 4. Centrifuge Test Results 227 spaced using an s/d ratio of 3 and rigidly connected to a pile cap. Pile cap excitation in the horizontal and vertical directions was provided by a mechanical oscillator. It was found that to predict the measured dynamic amplification curve for horizontal vibration the amount of interaction predicted using the static elastic model of Poulos had to be substantially reduced. It was also found that group damping was increased due to the interaction effect. Recently, Ochoa and O'Neill (1989) have published two pile interaction coefficients derived from cyclic pile head loading of full scale test piles in medium dense saturated sand. Their published experimental coefficients, while subject to certain inaccuracies in view of the measurement techniques employed, were then compared with Randolph and Poulos' theoretical interaction factors. The coefficients of Ochoa and O'Neill have tended to validate the latter. However, inline interaction between any two piles was found to be dependent on whether the loaded pile was in front of or behind the unloaded pile, due to the inability of cohesionless soil to sustain tension. Interaction factors were also found to depend on the load cycle and amplitude of loading considered with interaction increasing with the amplitude of the load and decreasing with the number of load cycles. Interaction factors have also been estimated from the present test data using the following procedures. Average pile group deflections at the soil surface were first derived from the Winkler models previously described that provided a match of measured pile cap deflections and average bending moments along the piles. Using the average shear and moment loads per pile given in Table 4.11 and the computed pile group deflections at the soil surface, equation 4.39 was used to derive a relationship between T]uv and num. While in theory this equation can be written for each of the two piles in the group, individual pile deflection and shear force at the soil surface could not be defined to sufficient accuracy to permit the accurate solution of the two flexibility equations. It was therefore considered preferable to work in terms of an average shear, moment Chapter 4. Centrifuge Test Results 228 and deflection per pile which could be checked against other measurements such as pile cap deflection and acceleration. Because of the above experimental limitations, it was necessary to assume a relationship between rjuv and 77 u m . The approximate relationship given by Randolph and Poulos was therefore used given as TJUV = y/num. The computed TJUV factors are plotted versus pile spacing ratio (s/d) for various 0 angles in Figure 4.70a-c. Interaction factors derived from other researcher's experimen-tal work involving pile group tests in sand are also plotted. The data points are then compared against the theoretical relationship for rjuv given by Randolph and Poulos. The experimental data show considerable scatter. As pointed out by Ochoa and O'Neill this is due to differences in sand density, load level and numbers of cycles of shaking over which the factors were derived, and inherent experimental error. The experimental data suggest that elastic theories overpredict pile to pile interaction at large pile spacings and generally underpredict interaction at small spacings. To account for the above experimental observations, it is suggested that the static interaction factor rju^ proposed by Randolph and Poulos be modified as follows: for 5 > 8r 0 . Here Fp is a plastic yield correction factor which has been set arbitrarily to a value of 2.0 and F(0) is a correction factor to account for the variation of pile interaction with angle 3. The function F(0) has the form, The relationships between if** and the interaction factors 77\"\", 77 u m , r)6v, and rj6m are assumed to have the same relationships given by Randolph and Poulos. The corrected (4.40) for 2r 0 < s < 8r 0 and, (4.41) (4.42) Chapter 4. Centrifuge Test Results 229 ( a ) 0.7 0.6 > 3 < 0.5-i . O ,8 0 4 H O 0.3 O D 0.2 0.1 0.0 \\ X H A \\ \\ \\ \\ A \\ \\ \\ •V. T r (b) 2 3 4 5 6 7 Pile Spacing Ratio (s/d) 0.7-0.6-> 3 < 0.5-_o t) 0.4-,D U . c q 0.3-\"o o i_ 0.2-0.1-0.0-2 3 4 5 6 7 Pile Spacing Ratio (s/d) 0.7-0.6-> 3 < 0.5-U 0.4H c .2 °-3\" \"o o 0.2-0.1-o.o-2 3 4 5 6 7 Pile Spacing Ratio (s/d) A Centrifuge (Gohl) x Centrifuge (Barton) • Full Scale (Ochoa & O'Neill) B Full Scale (Tominaga et al) Theory (Randolph & Poulos) Revised Relation Figure 4.70: Experimental versus Randolph and Poulos' interaction factor TJUV (a) 0 = 0 degrees (b) 0 = 45 degrees (c) 0 = 90 degrees Chapter 4. Centrifuge Test Results 230 7/uf factor accounts for the observed rapid attenuation of pile interaction for large pile spacings and sets interaction between piles to zero for offline shaking. The plastic yield correction factor Fp results in increased pile deflection for close pile spacings in accordance with the experimental observations. The modified F(B) function results in greater pile interaction for 3 > 0 than is predicted using the Randolph and Poulos elastic interaction model, where F(3) = 1 + cos2 3. Using the above corrected interaction factors, pile group deflections have been recom-puted and are compared with measured pile cap deflections in Table 4.12. The computed deflections are, in general, closer to the measured deflections relative to the deflections computed using the Randolph and Poulos interaction factors. The above approach, while largely empirical, can be tuned to the test conditions. A practical approach to assessing pile interaction under full scale field conditions is to measure ground surface deflections around a static or cyclic laterally loaded pile. Curves of horizontal ground surface deflection, Au, normalized with respect to horizontal pile deflection at the soil surface, uo, can then be constructed versus distance from the pile and orientation angle 3. These curves are equivalent to interaction factor rjuv assuming pure shear loading of the pile at the soil surface and neglecting the reduction in interaction that may occur due to the stiffening effect of adjacent piles. Relationships between TJUV, spacing ratio s/d, and angle 3 can then be derived and the remaining interaction factors estimated from the Randolph-Poulos relationships. 4.7.4 Base M o t i o n Excitation of a 2 x 2 Pile Group Tests on a four pile group subjected to sinusoidal and random earthquake base motions have been carried out to provide a data base against which procedures used to analyse the dynamic response of pile groups to base motion excitation can be checked. The modified interaction factors described in the previous section have also been used to estimate the Chapter 4. Centrifuge Test Results 231 pile group deflections and compared to measured values as a check on their accuracy. Provided computed group deflections are in reasonable accord with those measured, the interaction factors can be used to estimate the translational and rotational stiffnesses of the group. The four pile group was arranged in a square 2 x 2 configuration using a centre to centre pile spacing ratio (s/d) of 2. Following preparation of the dense sand foundations and the checking of all instrumentation, the centrifuge was brought up to the required test speed and subjected to a sequence of base motions. The first test sequence consisted of two tests in which low level sinusoidal base motions were applied to the model piles. The second of these tests (test 43) employed a shaking intensity similar to that used in the two-pile tests, and will be described in detail. After the first test series, the centrifuge was stopped and the sand redensified under 1 g conditions using high frequency vibration. The centrifuge was then brought up to test speed and a series of random earthquake base motions applied. The first two tests used low level shaking in which peak base motions did not exceed 5 percent of gravity. The third test (test 46) is described since shaking intensities with peak base accelerations of up to 0.14 g were larger than applied during the first two tests. It is of interest to check whether interaction factors derived from low level shaking tests using sinusoidal base motion could be applied to tests using stronger earthquake shaking. (a) Sinusoidal Base Motion Sinusoidal base motions used during shaking of the four pile group are shown in Figure 4.71a. The input motions are seen to be symmetric and are dominated by a 0.5 Hz frequency component. Following cessation of the initial transient base motions in which accelerations of up to 0.06 g occurred, the average in-put acceleration was 0.038 g. Free field accelerations measured at the soil surface were amplified through the soil and averaged 0.052 g (Figure 4.71b). Pile cap accelerations Chapter 4. Centrifuge Test Results 232 measured in the direction of shaking during the steady state region of response were ap-proximately 0.050 g (Figure 4.71c). Displacements measured at the top of a cylindrical mass attached to the pile cap are shown in Figure 4.71d. The displacements are seen to have a slight lack of symmetry and an average peak amplitude of 5.2 mm. Bending moment time histories at three different depths along pile no. 1 are shown in Figure 4.72a-c. Bending moments recorded by strain gauge no. 3 are 180 degrees out of phase with strain gauge no. 1. This illustrates the restraint of the pile cap which causes a reversal in curvature of the pile. Bending moments decay to zero at depth. The time history of axial load variation in pile no. 1 is shown in Figure 4.72d. Dynamic axial load changes in the pile result from the pile reacting against the pile cap as it attempts to rotate during shaking. The axial load changes are intensified since the piles rest on a rigid base. Peak bending moments along the piles for both directions of shaking are shown in Figure 4.73. The bending moment in both piles are reasonably similar near the soil surface with an average value close to zero. Greater differences in bending moment between piles exist around the location of maximum bending. When the shaking direction reverses, the maximum bending moment in the group changes from one pile to the other as a result of pile shadowing effects. Average bending moments along each pile, computed as their peak to peak value di-vided by two, have been found to be nearly identical between piles. By fitting a piecewise cubic spline to the measured average bending moments and numerically integrating the computed spline the pile cap rotation, 6t, at the point of fixity between the pile and cap was computed. Since the cap and structural masses are rigidly connected, the top of pile deflection at the underside of the structural mass has been estimated from the measured deflection at the top of the structural mass using the equation yt = Ax — Ah sin 0t (Table 4.13). Chapter 4. Centrifuge Test Results (a) 0.10-1 233 (b) (c) (d) cn 0.05 -0.10-f-0 O.tO T-15 20 25 Time (sec) 10 15 20 25 30 35 40 Figure 4.71: 2x2 pile group response to low level sinusoidal shaking - test 43 (a) input base accelerations (b) free field accelerations (c) pile cap accelerations (d) displacements at top of mass Chapter 4. Centrifuge Test Results 234 (a) (b) (c) (d) 15 20 25 Time (sec) Figure 4.72: 2 x 2 group response for low level sinusoidal shaking - test 43 (a) strain gauge 1 (b) strain gauge 3 (c) strain gauge 7 (d) dynamic axial load Chapter 4. Centrifuge Test Results 235 0 25 Bending Moment (kN—m) Figure 4.73: Bending moment vs. depth for both directions of shaking in a four pile group - low level sinusoidal shaking Chapter 4. Centrifuge Test Results 236 Input Excitation Vt (mm) et (rads) Vt (mm) vV (mm) Sinusoidal^1) 3.53 0.0066 3.64 2.96 Earthquake^2) 42.0 0.0071 38.7 33.5 (1) V0 = 27 kN and M0 = 0 kN-m in the calculations. (2) Vo = 130 kN and M0 = 70 kN-m in the calculations. y* — computed pile group deflection using modified interaction factors. y** = computed pile group deflection using Randolph and Poulos interaction factors. Table 4.13: Measured and Computed Deflections in a 2 x2 Pile Group The average shear force and bending moment per pile at the soil surface have been deduced from measured bending moments and pile cap accelerations, and are given in the table. Assuming each pile in the group is equally loaded, and using the single pile flexibilities and modified interaction factors described in section 4.7.3, pile cap deflec-tions, y t, have been computed by adding up the deflections from a pile under its own loading plus the loads from the surrounding piles in the group. Pile cap deflections have also been computed using the elastic interaction factors of Randolph and Poulos and are compared in Table 4.13. Computed deflections using the modified interaction fac-tors exceed the measured deflections by about 3 percent. Computed deflections using the Randolph-Poulos interaction factors are 16 percent less than those measured. These differences are not large and are acceptable for most pile response calculations. The cal-culations confirm the tendency of the Randolph - Poulos interaction factors to slightly underestimate pile group deflections for small pile separations. (b) Earthquake Base Motion Input base motions used during simulated earthquake shaking of the four pile group during test 46 and their computed Fourier spectra are shown in Figures 4.74a and 4.75a, respectively. Peak accelerations of up to 0.14 g were applied Chapter 4. Centrifuge Test Results 237 to the base of the foundation and were dominated by frequencies in the 0 to 5 Hz range. The measured free field surface accelerations and their corresponding spectra are also shown (Figure 4.74b and 4.75b). The free field accelerations were strongly amplified through the sand to values of up to 0.26 g. A comparison of the input base and free field spectra shows that the strongest amplification occurred at a frequency of about 3 Hz which corresponds to the fundamental frequency of the sand foundation. This value is similar to that inferred from centrifuge test 15 carried out using the same input base motions (see section 4.2). Pile cap accelerations of up to 0.24 g were measured during the test and are shown in Figure 4.74c. Fourier spectra computed from the pile cap accelerations (Figure 4.75c) shows strong amplification of the free field motions around a frequency of 2.6 Hz. This corresponds to the fundamental frequency of the group for the level of shaking employed. When compared to the value of 3.1 Hz measured during low level inline shaking of a two pile group with a centre to centre pile spacing ratio of two, it is clear that there has been a reduction of the lateral stiffness of the pile group under the stronger earthquake shaking. Displacements measured at the top of the structural mass are shown in Figure 4.74d. Maximum displacements of up to 60 mm occurred which are an order of magnitude larger than occurred during the previous tests using sinusoidal base motion. Bending moments recorded at three different depths along pile no. 1 are shown in Figure 4.76a-c. The bending moments are in phase with the pile cap accelerations showing that the flexural behaviour of the piles are dominated by pile cap inertia forces. Dynamic axial load changes in pile no. 1 are also shown in the figure. The bending moment distribution along the piles is shown at a time when pile cap deflections were a maximum (t ~ ll.Osec) and has a form similar to that described in the case of low level sinusoidal shaking (Figure 4.77). Bending moments are nearly identical between the two piles near the soil surface although significant differences exist around the point Figure 4.74: 2 x 2 group response during earthquake shaking - test 46 (a) base accelera-tions (b) free field accelerations (c) pile cap accelerations (d) top of mass displacements Chapter 4. Centrifuge Test Results 239 (a) 25 Frequency (Hz) (b) Frequency (Hz) 10 (c) Est. f. Est. f, 2 4 6 Frequency (Hz) 8 10 Figure 4.75: Computed Fourier spectra during earthquake shaking of a four pile group (a) base accelerations (b) free field accelerations (c) pile cap accelerations Chapter 4. Centrifuge Test Results 240 of maximum bending. The slopes of the bending moment distributions suggest that the shear force per pile near the soil surface varies by up to about 20 percent from the average. By integrating the average bending moment distribution measured along the piles for t ~ 11.0 sec the pile cap rotation has been computed. With this, and the measured deflection at the top of the structural mass the deflection at the pile cap level has been estimated (Table 4.13). This is compared to estimates of group deflection computed using the modified and Randolph-Poulos interaction factors described previously. Sin-gle pile flexibilities used in these calculations have been derived from single pile test 15 subjected to the same earthquake shaking used in the present pile group test (see sec-tion 4.3.2). The single pile flexibilities have been computed using a linear with depth Winkler modulus distribution characterized by kh(z) = nhz with rih set to a value of 4800 k N / m 3 . Pile group deflections computed using the modified interaction factors are less than the measured deflections by 8 percent. Using the Randolph-Poulos interaction factors, computed deflections are 20 percent less than those measured. 4.7.5 Summary The above calculations suggest that the use of an interaction factor approach can lead to a reasonable estimate of pile group deflections provided one can characterize the single pile response for a particular level of shaking. The latter have been determined from experimental measurements using simple Winkler models of lateral soil response. In practice, the single pile response must be estimated for a particular level of shaking prior to the actual earthquake. This may be done using equivalent visco-elastic models of soil response based on strain compatible estimates of near field moduli, as discussed in section 4.4. Alternatively, non-linear cyclic p-y curves may be used as input into the analysis, characterized using an appropriate backbone p-y curve and Masing loops to Chapter 4. Centrifuge Test Results 241 Figure 4.76: 2 x 2 group response to earthquake base motion -(b) strain gauge 3 (c) strain gauge 7 (d) dynamic axial load test 46 (a) strain gauge 1 Chapter 4. Centrifuge Test Results 242 f— SI OJ > o -Q < OJ O c _o Q 12-11-10 9 8 7-6 5 4-3-2 1 0 G.S. o,' x/ II ! 1 j 1 x Pile 1 / / • Pile 2 i -200 0 200 Bending Moment (kN-m) 400 Figure 4.77: Bending moment vs. depth at peak pile cap deflection during earthquake shaking of a four pile group Chapter 4. Centrifuge Test Results 243 model load-unload behaviour (section 4.5). The a priori prediction of single pile response is still difficult and will continue to be supplemented by full scale load tests and other in situ (e.g. pressuremeter) tests. It has been shown that the use of the elastic interaction factors proposed by Randolph and Poulos leads to an underestimate by up to about 20 percent of the measured pile group deflections when the piles are closely spaced (s/d ratios less than 4). Elastic interaction theory overestimates interaction for larger pile spacings. This check has been made for both sinusoidal and earthquake shaking for low to moderately strong shaking. To compensate for the limited pile interaction observed during offline shaking and during inline shaking for pile spacing ratios greater than 6, modifications have been proposed to the above elastic interaction factors. These modified factors result in an increased group deflection for close pile spacings where plastic yield around the piles influences the pile interaction and reduces the extent of interaction for larger pile spacings. The use of these modified factors leads to an improved estimate of group deflections and therefore the combined lateral stiffness of the pile group. Chapter 5 Shake Table Test Results 5.1 Introduction Single piles and pile groups were subjected to sinusoidal and random earthquake exci-tation using the U B C shake tables. The testing was carried out primarily topermit a comparison of centrifuge and shake table test data since it not clear to what extent the low stress levels present in the shake table tests relative to full scale stress conditions will affect the model pile response. The shake table tests also provide a data base against which numerical predictions of pile response to seismic excitation can be checked. Similar kinds of tests were carried out on the shake table as were performed on the centrifuge. Single piles were embedded in both loose and dense sand foundations and the natural frequencies of the pile and free field determined over a range of shaking intensities. The dependence of the measured natural frequencies on test type, duration and shaking intensity was examined. A series of base motion excitation tests were then carried out. Selected tests were compared against the predictions of an uncoupled dynamic response model where the free standing portion of the pile and structural mass (the superstructure) is uncoupled from the embedded pile foundation. In this model, the superstructure is linked to the foundation by a series of springs and equivalent viscous dashpots and measured surface accelerations applied to the free field end of the foundation compliances. The numerical model assumes that the free field motions do not vary significantly over the sand surface due to the influence of the walls of the sand container. Therefore, a 244 Chapter 5. Shake Table Test Results 245 series of tests were also carried out to examine the spatial variation of the free field surface motions to check the validity of this assumption. Data were also compared with centrifuge test data to determine stress level dependent differences in pile foundation response and characteristics of soil-pile interaction for various intensities of shaking. The pile groups consisted of two and four pile groups and were embedded in dense sand. The piles were subjected to sinusoidal base motions using both low level and strong shaking. The strong shaking data obtained on the shake table supplement the low level shaking data obtained from the centrifuge. The pile group tests were compared with data from single pile tests excitation levels to assess group interaction effects and to determine whether these were similar to group behaviour observed on the centrifuge. A limited number of 4-pile group tests were also carried out using strong sinusoidal shaking to determine whether elastic interaction factors determined from the previous two pile tests could be superimposed to describe the displacement response of the larger group. 5.2 Elastic Wave Velocities Wave velocity measurements using piezoceramic bender elements were made prior to each series of shake table tests using the procedures described in section 2.7. The measured velocities were then used to estimate low strain shear moduli in the sand. Typical signals produced by three adjacent receivers are shown in Figure 5.1. The bender source was caused to bend first in one direction by applying a voltage of positive polarity to the element. The voltage polarity was then reversed causing the element to bend in the opposite direction. The receivers responded with a positive voltage change followed by a negative voltage change in response to the positive and negative shear wave pulses, respectively (see Figure 5.1). The voltage traces show two distinct wave arrivals; one corresponding to an apparent Chapter 5. Shake Table Test Results 246 P-wave and the second to a larger amplitude S-wave. It is noted that both the P and S-wave signals show reversal. The existence of a P-wave arrival has been attributed to variations in the vertical alignment of receivers relative to the source and \"near field\" effects, as discussed in section 2.7. The latter provides a theoretical explanation for the existence of shear disturbances propagating at both the P and S wave velocity, and shows that reversal of the shear disturbance propagating at the P-wave velocity is possible. The arrival times of the P and S-waves have been determined from a visual inspection of the recorded voltage signals. The distance between the tips of two adjacent bender elements divided by the transit time of the P or S wave between them gives the average elastic wave velocity over the depth interval in question. At the start of the shake table investigation, the existence of a wave travelling at the P-wave velocity seemed somewhat surprising since it was expected that a bender shear source would produce mainly shear waves. To confirm that the initial wave arrivals could be used to determine P-wave velocities, a series of measurements were carried out in vibrated dense sand having an average void ratio of 0.52. Two different wave sources were used. The first was a bender shear source. The second was a plate which was struck vertically by a hammer to generate a strong P-wave component. Receivers were buried in the sand beneath the sources and their amplified outputs used to estimate P-wave arrival times. P-wave velocities computed from these independent measurements were compared to check the consistancy of test results. The data were also compared against P-wave velocity measurements quoted in the literature for dry sand. Voltage outputs from a bender shear source and receiver having a tip to tip separation distance of 240 mm are shown in Figure 5.2. A voltage of positive polarity was first applied to the source element. The amplified signal from the receiver shows the arrival of an apparent P-wave followed by the arrival of a large amplitude S-wave. A voltage of negative polarity was then applied to the bender source. As shown in Figure 5.2, the Chapter 5. Shake Table Test Results 247 Figure 5.1: Bender element voltage outputs recorded on the shake table (a) receiver R- l (b) receiver R-2 (c) receiver R-3 Chapter 5. Shake Table Test Results 248 (a) so -50 H 1 1 1 ' 1 ' • 1 ' 1 ' ' ' ' 0 1 2 3 Time (msec) Start of Shear *3 20-Q_ \"S O Wave Pulse / n -Time (msec) Figure 5.2: Bender element voltage outputs - confirmatory test in dense vibrated sand (a) source (b) receiver receiver responds wi th a large amplitude voltage output of opposite polarity. The receiver output shows that the P and S-wave undergoes reversal in response to the reversal in direction of shear excitation. A P-wave arrival time of 1.17 msec was determined from the receiver voltage traces, giving an average P-wave velocity of 205 m/sec over the depth interval in question. A n accelerometer wi th its sensitive axis aligned in the vertical direction was buried in the sand at the same depth as the bender receiver. A short section of hollow, rectangular steel channel was embedded 11 m m below the surface of the sand and the channel l ightly struck i n the vertical , downward direction using a hammer. W h e n the hammer makes contact wi th the steel channel, an electrical circuit is completed, generating a voltage pulse to signal the start of wave propagation. A t the instant the P-wave arrives at the Chapter 5. Shake Table Test Results 249 750 - i 500- Start of Vertical Hammer Impact > 250- P -500--750 0 2 3 Time (msec) 4 5 Figure 5.3: Receiver response during P-wave velocity measurements in dense sand using a hammer plate source and an accelerometer receiver. accelerometer, it responds to the mechanical vibration with an electrical signal. P-wave reversal was next carried out by tapping vertically upwards on the steel channel. Since the channel was buried 11 mm below the soil surface, the upward tapping causes unloading of the sand and a reversal in phase of the P-waves. The accelerometer outputs resulting from the hammer impacts are shown in Figure 5.3. The figure shows that a P-wave arrives at the receiver 1.29 msec after the impact and that it undergoes reversal in response to the upward and downward tapping of the plate source. The distance between the source and accelerometer was 246 mm, giving an average P-wave velocity over the depth interval of 191 m/sec. The P-wave velocities computed using the two separate procedures agree to within 7 percent of each other. P-wave velocity measurements have also been made in dry sand by Stokoe et al (1985). The sand samples used in these tests have a similar D50 as the C-109 Ottawa sand and were prepared using air pluviation, achieving an average void ratio of 0.63. Data are presented showing measured P-wave velocities for wave propagation in Chapter 5. Shake Table Test Results 250 the vertical direction versus isotropic confining stress level. Extrapolating the data to the average confining stress existing half-way between the source and receiver, and correcting the data for differences in void ratio using relationships given by Hardin and Drnevich (1972) gives a P-wave velocity of 239 m/sec. This value is in reasonable agreement with the above measurements and shows that the initial wave arrivals recorded by the bender element can be used to estimate P-wave velocities. Incremental wave velocities have been determined in the loose sand from the measured wave arrival times prior to shaking of the sand. Their distribution versus depth is shown plotted in Figure 5.4. The shear wave velocity, Vs, may be related to the maximum shear modulus, Gmax, using Gmax = pVs2 and the equation proposed by Hardin and Black (1968) for estimating Gmax (see section 4.1). Using e0 = 0.72 and R~0 = 0.4 in the preceding formula, shear wave velocities in the loose sand have been computed versus depth. Figure 5.4 shows that the measured shear wave velocities and those computed using the Hardin and Black equation are in very good agreement. The data also demonstrate that the maximum low strain shear modulus of the sand varies as the square root of the depth. Similar data are presented in Figure 5.5 for vibrated dense sand. The measured shear wave velocities show greater scatter than was observed in the loose sand, which may be due to variation in confining stress level in the dense vibrated sand. The data points lie within a scatter band of about 15 m/sec. The data may be approximated by Vs = Az0-25, where A = constant = 187.0 m ° - 7 5 / s e c , z is the depth below the soil surface in metres and V„ is the shear wave velocity in m/sec. The measured wave velocities are, on average, 17 percent larger than computed from the Hardin and Black equation, using values of e0 = 0.53 and R~0 = 0.6 in the computations. The latter K0 value is believed to be representative of a dense vibrated sand but may be increased to higher values during cyclic densification (Youd and Craven, 1975). Discrepancies between measured and computed shear wave velocities are believed due to an inadequate Chapter 5. Shake Table Test Results 251 100 CD o O 200 M— i _ 13 CO 300 H 400 H to _o < D C D m a> o c D 300-400 Q 500 X X i X X - Interpolated X 600- i I • ' ' • I' ' i ' • 100 200 300 400 P—Wave Velocity (m/sec) 500 Figure 5.6: Compression wave velocities in loose sand foundations with no cyclic pre-strain on the shake table. strain Poisson's ratio are computed to be 0.25 and 0.18 for the loose and dense sand, respectively. Chapter 5. Shake Table Test Results 254 ^ 100-E OJ CJ O 200-300 H 400 v> 5 500-600-X l A Shear plate X Bender Source Interpolated 100 200 300 400 P—Wave Velocity (m/sec) 500 Figure 5.7: Compression wave velocities in vibrated dense sand foundations on the shake table. Chapter 5. Shake Table Test Results 255 5.3 Natural Frequency Tests 5.3.1 Introduction In the present section, the results of tests carried out to determine the fundamental natural frequencies of the model pile(s) and of the sand foundation are described. A knowledge of the natural frequencies of the soil-pile system is necessary to determine whether a resonance condition could develop during shaking. Resonance results in signif-icant pile flexure and occurs if a significant component of the input base motion contains frequencies matching the fundamental frequency of the pile(s). While higher mode vi-bration contributes to the overall pile response, theoretical analyses and examination of the experimental data have shown that first mode vibration dominates. Therefore, experiments to define the fundamental frequencies of the pile(s) have been emphasized. The natural frequencies of the sand foundation have also been determined since if the predominant frequencies of the base motion correspond to the natural frequencies of the sand foundation large displacement gradients can occur over the depth of the layer, causing additional pile bending. As noted by Coe et al (1985), resonance of the sand foundation inside the soil container increases the importance of wave reflections from the container boundaries. Unless energy absorbing boundaries are provided, these reflections can cause additional loading on the pile foundations. Coe states that provided the natural frequencies of the sand foundation and piles are well separated, and the system is driven at input frequencies which do not cause resonance of the sand, wave reflections are unlikely to seriously affect the experimental results. The natural frequencies of the pile (or pile group) and those of the free field are in general not identical due to the interaction that occurs between the near field soil and the pile during lateral vibration. This interaction is especially pronounced if the pile supports a structural mass or if the pile is sufficiently stiff that the pile does not move Chapter 5. Shake Table Test Results 256 in concert with the free field soil during the earthquake loading (Flores- Berrones and Whitman, 1982). The natural frequency of the pile (or pile group) will be determined by the structural properties of the pile, the pile head mass, and the lateral stiffness of the near field soil. The lateral soil stiffness is controlled by the insitu moduli which are influenced by the density and mean effective confining stress operative in the near field soil prior to the start of cyclic lateral loading. During cyclic loading, a complex strain field is set up around the pile which further alters the insitu moduli. During the first few cycles of lateral loading, the shear strains in the near field cause dilation and reduction in shear stiffness of the soil. The shear stiffness is reduced with increasing shear strain and amplitude of pile vibration, at a rate dependent on confining stresses operative in the soil. At low confining stress, dilation of the sand will be more pronounced leading to more significant shear stiffness reduction. The mechanism of stiffness reduction or \"strain softening\" during one half of a vibration cycle is due to the strain dependent modulus reduction that occurs in the soil at the front of the pile as the pile pushes into it, and the tensile yielding and resultant softening of the soil at the back of the pile as the pile moves away from the soil. The process reverses itself during the next half cycle of shaking. Cyclic loading at constant amplitude may also lead to a slight stiffening of the soil response due to soil compaction under the repetitive loading. In summary, since the insitu shear moduli which control the lateral stiffness of the soil adjacent to the pile (the near field) are strain level dependent, the natural frequencies of the pile will depend on the amplitudes of the pile vibration at which they are measured. The number of cycles of vibration will also influence lateral soil stiffness since this controls densification processes in dry sand. The natural frequencies of the sand foundation in the free field are similarly dependent on excitation amplitude and number of load cycles. Chapter 5. Shake Table Test Results 257 5.3.2 Test Procedures The natural frequency tests were carried out on a single pile using both loose and dense sand foundations. The method of preparation of the loose sand foundation has been described in section 2.6. Dense sand foundations were prepared using two methods of sand densification. In the first method (method A), a loose sand foundation was prepared and the pile pushed into the soil by hand. Following a series of tests in loose sand, the sand layer was densified using high frequency vibration with the pile restrained from movement during the vibration process. In the second method (method B), the sand was placed in layers and each layer densified using high frequency (25 to 30 Hz) vibration. In this way, care was taken to ensure the sand contained no loose zones. A relative density of approximately 100 percent was achieved. The pile was then pushed into the soil by hand, assisted by high frequency vibration. Kagawa and Kraft (1981b) have suggested that the response of a laterally loaded pile will be determined by the manner in which the loading is applied. This is so because the lateral soil stiffness depends on the stress and strain field created in the near field which in turn depend on the method of loading. For example, during earthquake loading the stress-strain response of the near field soil is dependent on the free field and lateral pile motions. For the case of cyclic pile head loading, the soil acts as a passive resistor and its stress-strain response is due only to the influence of the pile head forces. To determine whether the method of loading significantly influences the measured natural frequencies of a single pile, two types of tests were carried out. The first is referred to as a ringdown test and the second as a hammer impact test. During these tests, the pile was subjected to five cycles or less of low amplitude shaking. In the hammer impact test the base of the sand container was struck by a hammer in a horizontal direction. The impulsive force generates body waves which propagate Chapter 5. Shake Table Test Results 258 (a) c o ~o 1 _ JD o o CL CO Q - 0 . 1 --0 .2 2000 2500 Time (msec) 3000 (0 E E I c <1> E o 0) c 13 C OJ CD -25 2000 2500 Time (msec) 3000 Figure 5.13: Ringdown test R - L l in loose sand (a) pile head displacements (LVDT 2), (b) pile head displacements (LVDT 1), (c) bending moments (strain gauge 3) Chapter 5. Shake Table Test Results 272 with the measured deflection and bending moment. The parameter cv describes the variation with depth of the lateral subgrade stiffness. It has been found that using a Winkler model where the lateral stiffness, k^, is assumed to vary proportionally to the square root of depth (see Appendix D) provides a good description of bending moment variation along the pile for low amplitudes of pile vibration. This assumption has been confirmed by the ringdown test carried out during test series II. Using the above procedure the displacement of the pile at the soil surface at the peak of the first cycle of ringdown was computed to be 0.12 mm using a Winkler subgrade reaction parameter a = 0.045 N / m m 5 ' 2 . The pile deflection at the soil surface is equal to 1.9 percent of the model pile diameter. The peak bending moment measured during the first cycle of vibration was 60 N-mm which equals 1 percent of the pile yield moment. Although pile vibration amplitudes were relatively small during the ringdown test, the deflections were sufficient to induce significant strain softening and modulus reduction in the near field soil, based on the analysis described subsequently. By measuring the time interval between two successive displacement (or bending moment) peaks the natural frequency of the pile has been determined to be 12.0 Hz. Following the tests R - L l and R-L2, ringdown tests R-L3 and R-L4 were carried out using test method B. Pile head accelerations measured at the top of the mass during test R-L3 are shown in Figure 5.14a. Peak accelerations of up to 0.5 g occurred during the first cycle of ringdown and reduced to an approximately zero level after three cycles. The fundamental natural frequency was determined to be 10.5 Hz by measuring the time interval between two successive acceleration peaks in the 0.3 to 0.5 g range. The natural frequency was found to be 11.1 Hz for accelerations less than 0.3 g. A slight increase in natural frequency with decreasing vibration amplitude was observed. Applying the log decrement method to the free vibration phase of the pile head acceleration over the last three cycles of ringdown, a fraction of critical damping of 0.14 is computed. Chapter 5. Shake Table Test Results 273 Displacements of the pile head mass measured using L V D T 1 and 2 are shown in Figure 5.14b and c. Measured bending moments at the location of strain gauge no. 3 are also shown in Figure 5.14d. The initial offset of the LVDT's and strain gauge no. 3 represents the initial displaced position of the pile prior to ringdown. At the start of free vibration, the pile displacement at the soil surface has been computed to be 0.92 mm using the analysis procedure described above with an a value of 0.025 N / m m 5 / 2 . Bending moments at the location of strain gauge 3 prior to ringdown are seen to be about 330 N-mm. Displacements and pile bending moments are clearly larger than those that occurred during test R - L l . From this, one can infer that larger strains were experienced by the near field soil during test R-L3 which has resulted in a softened soil response and a lower measured natural frequency. The equivalent Winkler stiffness used in the above analysis is also seen to be less than the value used in test R - L l , demonstrating the dependence of this parameter on the amplitudes of pile vibration. Pile head accelerations measured during test R-L4 are shown in Figure 5.15a. Accel-eration amplitudes were 0.15 g or less and were damped out after 5 cycles. The measured natural frequency averaged over five cycles of ringdown has been determined to be 11.9 Hz. The fraction of critical damping computed using the log decrement method is equal to 0.09. Displacements and bending moments recorded during the test are given in Figure 5.15b to d. The initial displacement of the pile at the soil surface prior to ringdown has been computed to be 0.28 mm or 4.3 percent of the pile diameter. The initial value of bending moment is seen to be approximately 100 N-mm. The smaller vibration ampli-tudes recorded during this test have resulted in a natural frequency slightly larger than measured in test R-L3. The change in stick-up of the pile head mass was 1.5 mm over the course of the four ringdown tests during test series I. Using the single pile model discussed in section 6.2.5, it is estimated that this would cause a change of at most 0.2 Hz in the measured Chapter 5. Shake Table Test Results 274 (a) 2600 2800 TLme (mLLLLsec) (b) 2600 2800 TLme (mLLLLsec) (0 2600 2800 TLme (mLLLLsec) (d) i ° ° -•(QO H . — i — i — . — . — i — i — i — . — . — . — i . — ' — • — i — • — ' — ' — ' — 2200 2400 2600 2800 3000 3200 Ti,nie (mLLLLsec) Figure 5.14: Ringdown test R - L 3 in loose sand (a) pile head acceleration, (b) pile head displacement ( L V D T 1), (c) pile head displacement ( L V D T 2), (d) bending moment (strain gauge 3) Chapter 5. Shake Table Test Results 275 natural frequency. The changes in natural frequencies discussed above are due mainly to differences in the amplitude of pile vibration and the amount of strain softening in the near field soil. Pile head accelerations recorded at the centre of gravity of the structural mass during ringdown test R-L5 in test series II are shown in Figure 5.16a. Peak accelerations of up to 0.34 g occurred during the first cycle of ringdown and reduced to an approximately zero level after three cycles. By measuring the time between the first two acceleration peaks in the range of 0.14 to 0.34 g the fundamental frequency of the pile has been determined to be 10.0 Hz. The natural frequency was found to be 11.1 Hz for accelerations less than 0.14 g. These results are similar to those measured during test R-L3 and suggest that similar strain levels occurred in the near field soils for both tests. Displacements of the pile head mass as recorded by L V D T 1 and 2 have been used to compute the deflection and slope at the top of the pile. From measured bending moments along the pile recorded prior to ringdown the moment and shear force at the soil surface has been deduced. With these four boundary conditions, the deflection of the pile at the soil surface prior to ringdown has been computed to be 1.24 mm using the static beam equation and is similar to the value of 0.92 mm derived from test R-L3. Peak bending moments measured along the pile at the peak of the first cycle of ringdown are shown in Figure 5.16b. The form of the bending moment distribution is typical of a free headed pile vibrating in its first mode. The bending moment distribution is seen to be matched closely using a square root of depth Winkler model, incorporating an a value of 0.030 N / m m 5 / 2 . Pile deflections computed using the above Winkler model are shown in Figure 5.16c and are in good agreement with pile top deflections deduced from the L V D T measurements. It is of interest to compare the Winkler stiffness distribution described above with the low strain stiffness properties of the sand. Using the procedures described in section 4.4, Chapter 5. Shake Table Test Results 276 (a) CD C o o 0.2 0.1-D 0.0 -0.2 \\ / \\ / Vy 2200 2400 2600 2800 Time (msec) 3000 (b) 0.50 (c) (d) £ c CO E V o D a CO b -0.50 0.50 0.25 0.00 -0.25-'-0.50 E E I z c . o E o D) c TJ C CO CO 2200 2400 2600 2800 Time (msec) 2200 2400 2600 2800 Time (msec) 2200 2400 2600 2800 Time (msec) 3000 3000 3000 Figure 5.15: Ringdown test R-L4 in loose sand (a) pile head acceleration, (b) pile head displacement (LVDT 1), (c) pile head displacement (LVDT 2), (d) bending moment (strain gauge 3) Chapter 5. Shake Table Test Results 277 the relative soil-pile stiffness factor Kr defined by Kagawa and Kraft (1981b) has been computed based on computed pile deflections and the above Winkler modulus distribu-tion. The calculations give Kr = 0.3 x 106 and an average, equivalent elastic Winkler stiffness for the case of pile head loading of kh — 2.8Gnf where Gnf is the equivalent near field shear modulus at depth z. A nearly identical relationship between the Winkler stiff-ness and near field shear modulus is given by Nogami and Novak (1977). Using the latter relationship and assuming kh = cvz1/2, the equivalent elastic shear modulus distribution is given approximately as Gnf ~ 0.007Z1/2, where G„/ has units of N / m m 2 and z is the depth below the soil surface in mm. This may be compared to the low strain, modulus distribution derived from the bender element measurements given as Gmax = 0.9121/2 where Gmax has units of N/mm 2 . The average effective modulus distribution is about 0.7 percent of that computed assuming low strain excitation of the near field soil. This suggests that significant strain softening of the soil has occurred. (c) Hammer Impact Test Pile head accelerations recorded during hammer impact test (test HI-LI) in test series I are shown in Figure 5.17a. Peak accelerations of 1.0 g occurred during the initial transient phase of the excitation and reduced to levels of 0.1 g or less after four load cycles. The free field acceleration measured after the impact is shown in Figure 5.17b for comparison. This shows that at t ~ 1500 msec the free field excitation has essentially ceased and the pile is vibrating freely. The fundamental natural frequency was determined to be 15.0 Hz by measuring the time interval between successive acceleration peaks in the range of 0.1 to 0.35 g. The fraction of critical damping was determined to be 0.13 by applying the log decrement method over the last two cycles of free vibration. Displacements of the pile head mass given by L V D T 2 (Figure 5.17c) indicate that after an initial displacement pulse of 0.15 mm amplitudes were less than 0.02 mm (0.3 Chapter 5. Shake Table Test Results 278 (a) c o ._> H-) o L 250 o < Soil Si / urface\" '/ \"7\" ^/^From 8.M. Integration & Experimental Computed Deflections 1 I ' i — • i 1 I 1 i 1 O.t 0.0 0.1 0.2 0.3 0.4 0.5 0.6 Lateral Deflections (mm) Figure 5.16: Ringdown test R-L5 in loose sand (a) pile head acceleration (b) peak bending moment distribution in the first cycle of ringdown (c) pile deflections in the first cycle of ringdown Chapter 5. Shake Table Test Results 279 percent of the pile diameter) which is within the range of accuracy of the L V D T . The pile bending moment (Figure 5.17d) recorded by strain gauge 3 shows a similar small response after the initial transient free field vibration has died out. The slight compaction of the near field soil during the ringdown and hammer impact tests in test series I, as indicated by the 2.5 mm change in stick-up of the pile head mass (Figure 5.10), suggests that a stiffening of the near field soil has taken place. This, combined with the smaller amplitudes of pile vibration relative to the previous ringdown tests, would explain the measured increase in natural frequency. The effect of increasing soil stiffness has apparently counteracted the softening effect of increased pile head stick-up, resulting in an overall increase in measured natural frequency. (d) Frequency Sweep Tests The natural frequency measured using the hammer impact test may be compared to that determined using a frequency sweep. The frequency sweep tests were carried out using stronger shaking levels than employed in the hammer impact test to examine the influence of higher vibration amplitude and number of cycles of shaking on the measured natural frequency of the pile. Table 5.2 compares the effective intensity of pile head acceleration, given by the RMS value, with corresponding RMS values from the input base and free field accelerations. Peak values of pile head displacement (LVDT no. 2) and bending moment (strain gauge no. 3) are also shown. Figure 5.18 shows the Fourier amplitudes of the pile head accelerations-normalized with respect to the free field surface accelerations. This shows that for the level of shaking used in the frequency sweep the fundamental frequency of the pile is approximately 10 Hz. This may be compared to the value of 15 Hz measured during the previous hammer impact test which shows that the stronger shaking used in the frequency sweep has caused strain softening of the near field soil and a reduction in measured natural frequency. At resonance, pile head accelerations are about 3.0 times those of the free field accelerations Chapter 5. Shake Table Test Results (a) 1 280 (b) (c) c o o v o y < 1300 1400 1500 1600 1700 Time (msec) 0.3 : 0.2-O) c Q 0.1-i \"5 0.0-L . V <0 -0.1-y y < -0.2--0.3- 1 1 1 1 1 1 1 I 1 1 I 1 1 1 1 I 1 1300 1400 1500 1600 1700 Time (msec) 1300 1400 1500 1600 1700 Time (msec) 1800 1900 1800 1900 1800 1900 (d) £ £ I c a> E o o> c TJ c co 40-1 20 -20--40 1300 1400 1500 1600 1700 Time (msec) 1800 1900 Figure 5.17: Hammer impact test H - L l in loose sand (a) pile head acceleration, (b) foun-dation surface acceleration, (c) pile head displacement (LVDT 2), (d) bending moment (strain gauge 3) Chapter 5. Shake Table Test Results 281 Input Freq. Base RMS F . F . RMS R H . RMS Max. Defl. Max. B . M . (Hz) (g) (g) (g) (mm) (N-mm) 5 0.089 0.171 0.185 0.70 190 7.5 0.080 0.173 0.205 0.72 212 10 0.081 0.163 0.276 1.89 363 12 0.071 0.164 0.315 1.10 297 15 0.112 0.243 0.389 0.72 268 20 0.098 0.212 0.254 0.29 163 30 0.136 0.422 0.280 0.20 145 40 0.063 0.278 0.170 0.05 64 50 0.093 0.399 0.193 ~0.02 67 Base RMS = RMS value of input base acceleration F . F . RMS = RMS value of free field surface acceleration P.H. RMS = RMS value of pile head acceleration Max. defl. = maximum deflection recorded by L V D T 2 Max. B . M . = maximum bending moment recorded by strain gauge no. 3 Table 5.2: Frequency Sweep Test Data - Pile Response in Loose Sand (or about 3.5 times the input base accelerations). At frequencies approaching the natural frequency of the sand foundation, pile accelerations are less than those in the free field. These results are in agreement with what one would expect from resonance curves derived from single degree of freedom vibration theory. Peak bending moments measured by strain gauge no. 3 and normalized with respect to the base acceleration RMS value are shown in Figure 5.18b. The significant increase in bending moment at the resonant frequency of the pile is clearly evident. Pile head deflections also increase significantly as the fundamental frequency of the pile is approached (Table 5.2). The relatively broad peak of the pile acceleration amplification curve (Figure 5.18a) suggests a highly damped system response. By modelling the portion of the pile above the soil surface as a single degree of freedom system, subject to base motion excitation due to the free field surface motion, one can estimate the damping of the system from equation 5.4. Using an amplification ratio of 3.0 gives a fraction of critical damping Chapter 5. Shake Table Test Results 282 of 0.18. This value is higher than the values of 0.09 to 0.14 deduced from the previous ringdown and hammer impact tests and suggests that as the strength of shaking increases frictional damping in the near field soil increases. Figure 5.10 shows the change in stick-up that occurred between the end of the hammer impact test and the frequency sweep test at the resonant frequency of 10 Hz was 5 mm. It is estimated based on analyses described in section 6.2 that this would cause a 0.5 Hz reduction in natural frequency from the value of 15 Hz measured during the hammer impact test. The additional reduction in natural frequency to a value of 10 Hz is due to strain softening of the near field soil. (e) Summary The previous tests have shown the influence of amplitude and number of cycles of vibration, and the effect of previous cyclic strain history on the measured natural frequencies of the pile. During an initial ringdown test, the fundamental fre-quency of the pile was seen to be approximately 12 Hz. Increasing the amplitudes of pile vibration caused a softening of the near field soil and resulted in a slight decrease in natural frequency to about 10.5 Hz. This effect is typical of a non-linear system and has been observed by other researchers in the field of pile dynamics (Scott et al, 1982). Effective Winkler stiffness distributions have been deduced from bending moments mea-sured during ringdown test R-L5 and when compared to stiffness estimates made using low strain shear moduli suggests that significant strain softening has occured in the near field soil. Effective lateral soil stiffnesses have also been deduced from other ringdown test measurements. The effective soil stiffness is seen to be strongly dependent on the amplitudes of pile vibration. During the hammer impact test, the pile was subjected to a limited number of small amplitude load cycles. Since the test followed a series of four ringdown tests, the previ-ous cyclic straining caused densification of the near field soil. This combined with the Chapter 5. Shake Table Test Results 283 (a) 4-D DC 3-OJ ~o \"cl 2 £ Natural Frequency = 10 Hz r-r-' 10 Frequency Sweep Test Data Base Accel. RMS = 0.063 to 0.136 G 20 Frequency (Hz) 30 40 50 Figure 5.18: Frequency sweep test in loose sand: (a) normalized Fourier spectra of pile head accelerations; (b) normalized peak bending moment at strain gauge 3 Chapter 5. Shake Table Test Results 284 lower intensity of shaking resulted in a stiffer near field response which counteracted the softening effects of increased pile head stick-up. The fundamental frequency measured during the test was shown to be 15 Hz which is larger than the value of 12 Hz measured at comparable pile head acceleration levels during the ringdown tests. The influence of the method of pile excitation on measured frequency characteristics would appear to be minor based on the ringdown and hammer impact tests carried out in the present study. This is a reasonable conclusion since the hammer impact caused a limited number of cycles of loading of the foundation soil whose motions were rapidly damped out. Little global change to the free field soil properties would therefore be expected. This is confirmed by the settlement plate readings which indicated that small surface settlements resulted from the hammer impact. Following the cessation of the free field motions, the pile vibrated freely in response to the motions of the pile head mass. This is analogous to the previous ringdown tests and one would therefore expect similar test results for similar previous load histories and pile vibration levels. For stronger sustained ground motion, especially in saturated sands where pore pressures build up around the pile due to the combined influence of pile and free field vibration, such conclusions may not be valid. The stronger shaking used during the frequency sweep compared to that used during the hammer impact test resulted in a softened near field soil response and an increase in pile head stick-up. The change in stick-up can account for a 0.5 Hz reduction in natural frequency from the value of 15 Hz measured during the previous hammer impact test. The additional reduction in natural frequency to a value of 10 Hz is due to strain softening of the near field soil. Fractions of critical damping of 9 to 14 percent have been estimated from the hammer impact and ringdown tests, using single degree of freedom vibration theory. The stronger shaking used in the frequency sweep caused the fraction of critical damping to increase Chapter 5. Shake Table Test Results 285 to a value of about 18 percent. This is attributed to the larger strain excitation of the near field soil, resulting in greater frictional energy losses. It is of interest to compare the above damping values with values quoted in the literature based on full scale or centrifuge test results. Prevost and Abdel-Ghaffar (1982) have carried out cyclic pile head loading tests on single piles in loose dry sand using a geotechnical centrifuge. Damping values derived from their data were in the range of 2 to 3 percent for pile cap acceleration levels at prototype scale in excess of 0.15 g. Scott et al (1982) have carried out analogous cyclic pile head loading tests under full scale conditions for single piles in saturated medium dense sand. The piles were shaken over a range of shaking intensities achieving pile cap accelerations of up to 0.17 g. Typical damping values ranged from 4 to 6 percent of critical. The damping values measured at full scale stress levels are clearly lower than obtained on the shake table. This suggests that the sand dilated strongly under the low stress conditions operative on the shake table, resulting in greater softening and hysteretic energy losses in the sand. Measurements of the fundamental natural frequency of the loose sand in the free field show that this lies in the 40 to 50 Hz range depending on the amplitude and number of cycles of shaking. The natural frequencies of the pile and free field are well separated. Therefore, by using base motion input whose frequency content is close to the fundamental frequency of the pile, sand layer resonance is not anticipated and the influence of free field displacements and boundary wave reflections on pile response is minimized. The reduction in natural frequency measured during the large number of cycles of shaking in the frequency sweep test relative to that measured in the hammer impact test at comparable acceleration levels is not large. The measured fundamental frequency is well predicted using the one dimensional wave propagation analysis SHAKE which has shown that effective shear strains and modulus reduction are small for the intensity of Chapter 5. Shake Table Test Results 286 shaking examined during the frequency sweep. While the S H A K E analysis was capable of predicting the measured fundamental frequency of the sand foundation, it was neces-sary to adjust the frictional damping in the sand to match the measured free field surface amplification. The effective damping ratios were found to be approximately 13 percent, considerably higher than the values of 1 to 2 percent computed using the modified equa-tion of Hardin and Drnevich. This again suggests that frictional damping is greater under the low stress conditions operative on the shake table. 5.3.4 Single Pile Tests in Dense Sand Hammer impact, ringdown and frequency sweep tests were also used to determine the fundamental frequencies of a single pile embedded in dense sand. The fundamental frequency of the sand foundation was determined from the free field surface accelerations measured during the hammer impact and frequency sweep tests. Hammer impact (test HI-Dl) and ringdown tests (test R-Dl) were carried out during an initial test series (series I) using a dense sand foundation prepared by vibrating an originally loose sand foundation. The foundation sand had a thickness of 600 mm and an average void ratio after densification of 0.57, corresponding to a relative density of 78 percent. Pile head accelerations and displacements, and free field accelerations were measured during the tests. Additional ringdown (test R-D2) and hammer impact tests (test HI-D2) were carried out during a second test series (series II) in which bending moments along the pile were also measured to permit estimates of the effective soil stiffness to be made. The foundation was prepared as described above, achieving an average void ratio in the sand of 0.53 or a relative density of 90 percent. The ringdown tests were carried out by displacing the pile head mass laterally and then releasing it from its initial displaced position, setting it into free vibration. The distance between the centre of gravity of the pile head mass and the soil surface was Chapter 5. Shake Table Test Results 287 maintained constant at a value of 47 mm ± 0 . 5 mm during the tests. Frequency sweeps using sinusoidal input base motion with peak average accelerations of 0.2 g were also carried out during two separate test series (series III and IV). In test series III, the sand foundation was prepared by placing the sand in layers and densifying each layer individually. The pile was then pushed into the sand, assisted by high frequency vibration. After sand placement, the foundation had a thickness of 605 mm and an average void ratio of 0.50, corresponding to a relative density of 100 percent. During test series IV, an originally loose sand foundation was densified with the pile in place. After densification, the foundation had a thickness of 609 mm and an average jvoid ratio of 0.53, corresponding to a relative density of 90 percent. The frequency sweep data were compared to determine whether the method of foundation densification significantly affected the measured natural frequencies of the pile. Test data were also compared with ringdown and hammer impact test results to determine the effect of the intensity of shaking on the measured natural frequencies. Settlement plate readings indicated that foundation surface settlements were negligi-ble during test series I to IV. (a) Free Field Response Surface accelerations in the sand foundation in the free field recorded during hammer impact test HI-Dl are shown in Figure 5.19a. The maximum acceleration was 1.1 g and was rapidly damped to near zero after four cycles. Applying the log decrement method over two cycles of ringdown in the 0.1 to 0.6 g range gives a fraction of critical damping of 0.14. The fundamental natural frequency determined by measuring the time interval over the first 3 acceleration peaks was estimated to be 56 Hz. The Fourier spectrum of the accelerations (Figure 5.19b) shows a predominant fundamental frequency in the range of 55 to 60 Hz. The presence of higher modes is also indicated by the spectrum and may indicate the influence of wave reflections from the Chapter 5. Shake Table Test Results 288 (a) C q ~o i_ CP o u < 1.5 1 -1 - 1 . 5 (\\. r I 400 450 500 550 Time (millfsec) 600 (b) r 100 150 200 Frequency (Hz) 250 Figure 5.19: Hammer impact test HI-Dl in dense sand (a) Foundation surface accelera-tions (b) the Fourier spectrum boundary of the sand container. Corresponding data from hammer impact test HI-D2 are shown in Figure 5.20. Measured accelerations (Figure 5.20a did not exceed 0.45 g and show the presence of higher frequency acceleration components. By measuring the time interval between the first two acceleration peaks in the range of 0.20 to 0.30 g gives a natural frequency of about 68 Hz. The Fourier spectrum (Figure 5.20b) shows a fundamental frequency in the range of 60 to 70 Hz. The presence of higher frequency modes is also indicated. Test data obtained during the frequency sweep (test series III) are shown in Table 5.3. The data show that surface accelerations were strongly amplified relative to the input base motions with this amplification being most pronounced at an input frequency of about 60 Hz. Chapter 5. Shake Table Test Results 289 Figure 5.20: Hammer impact test HI-D2 in dense sand (a) Foundation surface accelera-tions (b) the Fourier spectrum Chapter 5. Shake Table Test Results 290 Input Frequency Base RMS xff Free Field RMS (Hz) (g) (g) (g) (g) 5 0.20 0.081 0.25 0.131 10 0.20 0.082 0.37 0.176 12 0.24 0.081 0.44 0.210 13.5 0.17 0.070 0.37 0.164 15 0.17 0.065 0.34 0.145 20 0.17 0.066 0.44 0.163 25 0.18 0.079 0.45 0.203 30 0.20 0.075 0.45 0.200 40 0.18 0.071 0.54 0.223 50 0.22 0.099 1.0 0.441 60 0.24 0.109 1.1 0.472 70 0.22 0.088 0.85 0.341 Xb = average peak input base acceleration Base RMS = RMS value of input base acceleration iff = average peak free field acceleration at the soil surface Free field RMS = RMS value of free field surface acceleration Table 5.3: Frequency Sweep Test Data - Free Field Response in Dense Sand Chapter 5. Shake Table Test Results 291 Frequency (Hz) Figure 5.21: Frequency sweep test in dense sand: normalized Fourier spectra of free field accelerations and comparison with S H A K E output The Fourier amplitudes of the surface accelerations normalized with respect to the input base accelerations have been plotted versus the applied forcing frequency in Figure 5.21. The data show a broad amplification peak in the 60 Hz range which is close to the natural frequency measured using the hammer impact test at comparable surface acceleration levels. The reduction in measured natural frequency relative to that mea-sured during the hammer impact test at comparable acceleration levels was not large, indicative that significant strain softening of the foundation sand did not occur. The computer program S H A K E was again used to determine whether one dimensional wave propagation theory could be used to predict the measured fundamental frequen-cies of the sand foundation. Following the procedures described in section 5.3.3, the frequency sweep tests were simulated by inputting variable frequency sinusoidal base motions having peak accelerations of 0.2 g to the rigid base of the sand. The computed Chapter 5. Shake Table Test Results 292 surface accelerations, normalized with respect to the peak base acceleration, are plotted versus frequency in Figure 5.21. The analysis predicts that the fundamental frequency of the sand is 59.5 Hz which is in good agreement with the experimental results. Using the low strain shear modulus distribution gives a fundamental frequency of 60.4 Hz. The SHAKE analysis suggests that strain softening of the sand has been small and predicts maximum shear strains at resonance in the range of 0.00006 to 0.00011. Using the modified Hardin and Drnevich equation (equation 4.12), SHAKE predicts that damping ratios are small and equal to the minimum damping ratio of 1 percent specified for the analysis. The small damping specified has resulted in an overestimate of maximum surface response at resonance (Figure 5.21). Using a back-figured damping ratio of 10 percent gives a good match of the dynamic free field amplification which again suggests that damping in the sand is higher at the low stress levels operative on the shake table. While one dimensional wave propagation theory provides an acceptable prediction of the fundamental frequency of the foundation sand, it is not clear to what extent the soil container boundaries will affect the measured surface accelerations. It is well known that wave propagation is influenced by the dimensions of the container and the compliance of the boundaries (Aoki, 1969). Furthermore, the presence of a semi-rigid boundary can lead to reflected waves in the soil model which causes spatial inhomogeneity in the measured surface accelerations. This is important when using an uncoupled pile response analysis where the free field accelerations are applied to the free field end of foundation compliances linking the superstructure to the embedded portions of the pile. If the free field accelerations vary significantly across the surface of the soil container, this complicates selection of the free field accelerations to be used in the analysis. A series of tests were therefore carried out to determine whether the location of the accelerometer on the surface of the dense sand foundation significantly influences the Chapter 5. Shake Table Test Results 293 measured acceleration response. A sinuosoidal input base motion with a predominant frequency of 5 Hz but containing higher frequency overtones was applied to the base of the sand container. A few tests were also carried out using a 60 Hz sinusoidal input. The base motion had peak average accelerations of about 0.2 g and RMS values of 0.086 to 0.095 g. The surface acceleration response has been measured at a variety of locations in the container and the Fourier amplitudes of the surface accelerations normalized with respect to the input base accelerations have been plotted versus the most significant frequency components contained in the input motion. The data points obtained were then connected to form an average response envelope. For each measurement location, the response envelopes obtained are shown in Fig-ure 5.22. Additional data points obtained from the 60 Hz tests are also plotted. The response curves show the fundamental frequency of the sand to be approximately 60 Hz. The amount of surface amplification is shown to be dependent on measurement loca-tion only at frequencies approaching the fundamental frequency of the sand layer when boundary reflections cause non-uniform surface response. This is in agreement with the experimental results of Coe et al (1985) who state that boundary reflections are mainly important around resonance of the foundation layer. At frequencies corresponding to the natural frequencies of the model pile (e.g., less than about 24 Hz based on the model pile tests described subsequently), the foundation surface response is unaffected by mea-surement location. This is important since the pile will be most affected by foundation ground motions containing frequency components close to the fundamental frequency of the pile. From the test results, it can be concluded that for the soil-pile system examined the low frequency free field accelerations are not significantly affected by the container boundary and do not vary spatially. Chapter 5. Shake Table Test Results 294 Figure 5.22: Effect of location in sand container on measured free field surface response Chapter 5. Shake Table Test Results 295 (b) Hammer Impact Tests Pile head accelerations recorded at the top of the pile head mass during test HI-Dl are shown in Figure 5.23a. Peak accelerations of 1.7 g occurred during the initial transient phase of the excitation and reduced to levels of about 0.2 g after 6 to 7 load cycles. The fundamental natural frequency was determined to be 22.9 Hz by measuring the average time interval between successive acceleration peaks in the range 0.2 g to 0.6 g. The natural frequency was found to be 24.0 Hz on the average for accelerations less than 0.2 g. Fractions of critical damping of 5.5 percent were computed by applying the log decrement method to the free vibration phase of the pile head acceleration over several acceleration cycles in the 0.1 g to 0.6 g range. The Fourier spectrum (Figure 5.23b) shows a fundamental frequency of about 24 Hz. The secondary peak in the spectrum at a frequency of about 20 Hz reflects the change in fundamental frequency with differences in the amplitudes of motion of the pile. Displacements of the pile head mass given by LVDT no. 1 (Figure 5.23c), located 4.8 mm below the top of the pile head mass, indicate that during free vibration after an initial displacement pulse of 0.21 mm displacement amplitudes were less than 0.03 mm (0.5 percent of the pile diameter). Data obtained from hammer impact test HI-D2 are presented in Figure 5.24a to c. Accelerations recorded at the centre of gravity of the pile head mass (Figure 5.24a) show that after cessation of the free field motion at t ~ 2300 msec the pile vibrated freely at a fundamental frequency of 24.5 Hz. This value is similar to that obtained from hammer impact test HI-Dl indicating about the same amount of strain softening occurred in the near field soil in both tests. Fractions of critical damping of 6.0 percent have been computed during free vibration of the pile for accelerations of less than 0.15 g. Displacements recorded by LVDT 1 located 6.3 mm below the top of the pile head mass (Figure 5.24b) show that following a displacement pulse of 0.21 mm displacement amplitudes were small and less than 0.02 mm during free vibration of the pile. Chapter 5. Shake Table Test Results 296 (a) o> c o a i_ a> o o < 400 600 800 Time (millisec) 1000 (b) 50 75 Frequency (Hz) 100 (c) E E —' 0.15-c a> E 0.00-CD O o - 0 . 1 5 -C L V) o 400 600 800 Time (millisec) 1000 Figure 5.23: Hammer impact test HI-Dl (a) Pile head accelerations (b) Fourier spectrum of pile head accelerations (c) pile head displacements - L V D T 1 Chapter 5. Shake Table Test Results 297 Bending moments measured along the pile at the peak of the first cycle of free vi-bration, after cessation of the free field ground motion, are shown in Figure 5.24c. The bending moments are small and their distribution is typical of a free headed pile vibrating in its first mode. The maximum bending moment recorded by strain gauge no. 2 is about 5 N-mm and is equal to 0.1 percent of the pile yield moment. The measured bending moments are also compared with those predicted using the square root of depth Winkler model described in Appendix C. The proportionality constant ct used in the calculations was equal to 2.5 N / m m 5 / 2 and provides a reasonable match of the measured bending moment distribution. It is noted that this Winkler modulus distribution is substantially larger than used for the ringdown tests in loose sand due to the higher density of the sand and the lower amplitudes of pile vibration in hammer impact test HI-D2. (c) Ringdown Tests Pile head accelerations measured at the top of the pile head mass during ringdown test R - D l are shown in Figure 5.25a. Pile head accelerations of up to 0.3 g occurred during the first cycle of ringdown and reduced to levels of less than 0.05 g after four cycles. The fundamental frequency was determined to be 13.0 Hz in 0.1 to 0.3 g range and increased to a value of about 24 Hz for accelerations less than 0.1 g. The dependence of the measured natural frequency on the displacement amplitude of the pile is readily seen. Fractions of critical damping of 5.2 percent were computed by applying the log decrement method over several acceleration cycles. The Fourier spectrum (Figure 5.25b) shows frequency peaks at about 24 and 13 Hz, confirming the natural frequencies deduced from the pile head accelerations. Pile head displacements measured 4.8 mm from the top of the pile head mass (Figure 5.25c) show that after an initial displacement of 0.5 mm, displacements of the pile head reduced to levels of 0.02 mm or less after two cycles of ringdown. During ringdown test R-D2, pile head accelerations of up to 0.40 g were measured Chapter 5. Shake Table Test Results 298 (a) 2000 2100 2200 2300 2100 2500 2600 2700 2800 2900 3000 Ti_me (msec) (b) 2000 2100 2200 2300 2100 2S00 2600 2700 2800 2900 3000 TLme (msec) (c) 400 a. 350-» 300i o XI < a> o c D 250 Q 200 Experiment \\ A I •I Theory Soil Surface 0 5 10 Bending Moment (N-mm) Figure 5.24: Hammer impact test HI-D2 (a) Pile head accelerations (b) pile head dis-placements - L V D T 1 (c) Bending moment distribution along the pile at the start of free vibration Chapter 5. Shake Table Test Results (a) 299 01 CD O L CO > CO o o a: 500 TLme (ml11 usee 1 1000 (b) 100 150 Frequency (Hz! 200 250 (c) 0.05 0.01 — 0.03 e ° 0.02 0.01 0.00 c CO e CO o O -0.01 ) S\" -0.02 Q -0.03 -0.04 -0.05 500 TLme (mLLLLsec) 1000 Figure 5.25: Ringdown test R - D l (a) pile head accelerations (b) Fourier spectrum of pile head accelerations (c) pile head displacement - L V D T 1 Chapter 5. Shake Table Test Results 300 and reduced to levels of about 0.05 g in three cycles (Figure 5.26a). The fundamental frequency of the pile has been determined to be 12.7 Hz in the 0.1 to 0.4 g range and increased slightly to a value of 13.7 Hz in the 0.05 to 0.1 g range. These values are similar to natural frequencies recorded during the initial phases of ringdown in test R-D l although a marked increase in natural frequency with decreasing amplitude of pile vibration was not observed during test R-D2. Fractions of critical damping of 0.10 were determined over three cycles of ringdown. Displacements located 6.3 mm from the top of the pile head mass were about 1.35 mm (Figure 5.26b) and are larger than observed during test R - D l . The larger pile displacements are believed responsible for the larger damping values observed during test R-D2. Bending moments measured along the pile at the peak of the first cycle of ringdown are shown in Figure 5.26c and are compared to those predicted using the square root of depth Winkler model. An a value of 0.15 N / m m 5 / 2 was used in the calculations, and equals 6 percent of the effective lateral stiffness used to describe the bending moment distribution in hammer impact test HI-D2 (see Figure 5.24). This again illustrates the dependence of the equivalent elastic modulus on vibration amplitudes of the pile and suggests that significantly greater strain softening has occurred in the near field soil during the test. (d) Frequency Sweep Tests Frequency sweep data obtained during test series III are given in Table 5.4. The table compares the effective intensity of pile head acceleration, given by the RMS value, with corresponding RMS values from the input base and free field accelerations. Peak values of bending moment (strain gauge 3) are also shown. Similar pile response data recorded during test series IV are given in Table 5.5. Tables 5.4 and 5.5 show pile head accelerations were strongly amplified relative to the input base motions. The Fourier amplitudes of the pile head accelerations normalized Chapter 5. Shake Table Test Results 301 Jl w 3500 3600 3700 3800 3900 1000 1100 1200 1300 1100 1500 TL .me ( m s e c ) (b) (c) 3500 3600 3700 3600 3900 4000 4100 Tcrne (msec) 4200 4300 4400 4500 400 350 IT CL 300 « 250 O X l < CD o 200 c _o (0 150-100 Soil Surface Theory Oo, A / Experiment / -50 0 50 100 150 200 250 Bending Moment (N-mm) Figure 5.26: Ringdown test R-D2 (a) pile head accelerations (b) pile head displacements - L V D T 1 (c) peak bending moment distribution during first cycle of ringdown Chapter 5. Shake Table Test Results Input Freq. Base RMS F . F . RMS R H . RMS Max. B . M . (Hz) (g) (g) (g) (N-mm) 5 0.081 0.131 0.254 327 10 0.082 0.176 0.222 230 12 0.081 0.210 0.608 > 727 13.5 0.070 0.194 0.436 505 15 0.065 0.145 0.389 403 20 0.066 0.163 0.188 141 25 0.079 0.203 0.169 95 30 0.075 0.200 0.194 69 40 0.071 0.223 0.151 76 50 0.099 0.441 0.288 98 60 0.109 0.472 0.237 53 70 0.088 0.341 0.189 43 Base RMS = RMS value of input base acceleration F . F . RMS = RMS value of free field surface acceleration P.H. RMS = RMS value of pile head acceleration Max. B . M . = maximum bending moment recorded by strain gauge no. 3 Table 5.4: Frequency Sweep Test Series III - Pile Response in Dense Sand Input Freq. Base RMS P.H. RMS Max. B . M . (Hz) (g) (g) (N-mm) 5 0.090 0.265 277 10 0.084 0.180 189 12 0.089 0.290 359 13.5 0.086 0.923 819 15 0.075 0.691 711 17 0.069 0.414 374 20 0.071 0.305 188 Base RMS = RMS value of input base acceleration P.H. RMS = RMS value of pile head acceleration Max. B . M . = maximum bending moment recorded by strain gauge no. 3 Table 5.5: Frequency Sweep Test Series IV - Pile Response in Dense Sand Chapter 5. Shake Table Test Results 303 with respect to the input base accelerations are plotted versus input frequency in Figure 5.27a. This shows the fundamental frequency of the pile to be in the range of 12 to 13.5 Hz. The experimental results appear to be slightly dependent on the method of pile installation. It appears that by installing the pile after foundation densification (test series III), some dilation and loosening of the sand in the near field has occurred. This has resulted in a lower measured natural frequency relative to the case where the sand was densified with the pile in place (test series IV). The Fourier amplitudes of the pile head accelerations normalized with respect to the free field surface accelerations are shown for test series III in Figure 5.27b. The peak amplification ratio at pile resonance is 6.8. Using equation 5.4, this gives an average damping ratio of 0.07 which is lower than observed during similar tests in loose sand. Peak bending moments measured by strain gauge no. 3 and normalized with respect to the base acceleration RMS value are shown in Figure 5.28 for the two test series. The increase in bending moment as the resonant frequency of the pile is approached is clearly evident. The fundamental frequency obtained from the frequency sweeps is approximately one half that measured in the hammer impact and ringdown tests during low amplitude pile vibration. The frequency sweep test results reflect the much greater strain softening caused by the high steady state response amplitudes and larger number of cycles of shak-ing. The percentage reduction in natural frequency is much larger than that measured during the frequency sweep carried out using a loose sand foundation. This implies strain softening and modulus reduction in the near field soil was more pronounced for the dense sand tests. The large dilation of dense sand at the low confining stresses operative on the shake table is believed responsible for the pronounced stiffness reduction. Chapter 5. Shake Table Test Results 304 (a) 15 o =5 10 rr \"a Natural Frequency = 13 Hz 10 Legend A Fdn, Prep. Method A x Fdn. Prep. Method B Frequency Sweep Test Data Base Accel. RMS = 0.065 to 0.109 G 20 30 40 Frequency (Hz) 50 60. 70 (b) 10 o cn oj £ < Natural Frequency = 13 Hz Legend A Fdn. Prep. Method A Frequency Sweep Test Data Base Accel. RMS = 0.065 to 0.109 G •n - r 10 20 30 40 50 60 Frequency (Hz) 70 Figure 5.27: Fourier spectra from frequency sweep test in dense sand (a) pile accelerations normalized with respect to base accelerations - test series III and IV (b) pile accelerations normalized with respect to free field accelerations - test series III Chapter 5. Shake Table Test Results 305 ^ 12000 - 1 1 1 1 1 1 — 1 1 I ' 1 1 1 1 ' 1 ' 1 I ' 1 1 1 1 • ' • • I 1 ' 1 1 1 1 • ' 1 I 1 1 1 • • 1 ' 1 1 I— 1 1 1 1 ! | -* J 0 10 20 30 40 50 60 70 Frequency (Hz) Figure 5.28: Frequency sweep test in dense sand - normalized peak bending moments (e) Summary From the hammer impact tests, the fundamental frequency of the pile measured during small amplitude vibration was seen to be approximately 24 Hz. Increas-ing the amplitude of vibration caused a softening of the near field soil and resulted in a reduction in measured natural frequency. The extent of this reduction depends on the amplitude and number of cycles of vibration. For ringdown tests where relatively large pile displacements occurred the fundamental frequency of the pile reduced to a value of about 13 Hz. Effective lateral soil stiffnesses derived from measured bending moments along the pile shows that the stiffness of the near field soil reduced substantially relative to conditions operative during the hammer impact test. The moderately strong shaking used during the frequency sweep tests compared to that used during the hammer impact tests resulted in a reduction in fundamental fre-quency to a value of about 12 Hz. The measured natural frequency was seen to be slightly dependent on the method of pile installation. The degree of softening during the frequency sweep was more pronounced than observed during comparable tests in loose Chapter 5. Shake Table Test Results 306 sand. This suggests that the dense sand dilated strongly under the low confining stresses operative on the shake table, resulting in significant stiffness reductions. Fractions of critical damping of about 5 percent have been estimated from the hammer impact and ringdown tests during low amplitude vibration, using single degree of freedom vibration theory. During larger amplitude shaking, the fraction of critical damping has been shown to increase to values between 7 and 10 percent. Thes damping values are less than observed during comparable tests in loose sand. Measurements of the fundamental natural frequency of the dense sand in the free field show that this lies in the 60 to 70 Hz range depending on the amplitude and number of cycles of shaking. The fundamental frequency measured during the frequency sweep was well predicted using the computer program SHAKE although in order to predict the free field amplification at the soil surface it was necessary to use damping values considerably higher than anticipated using the equations of Hardin and Drnevich. The reduction in natural frequency observed during the frequency sweep relative to the hammer impact tests at comparable acceleration levels was not large. This suggests that shear strain development in the foundation sand was small which was confirmed by SHAKE. The experimental results also show that the free field accelerations vary spatially only around resonance of the sand layer. Container boundary effects which lead to wave reflections from the sides of the container are therefore unimportant at frequencies well removed from the fundamental frequency of the sand. 5.3.5 N a t u r a l Frequency Tests - 2 P i l e Groups in Dense Sand Ringdown and frequency sweep tests were carried out on two pile groups placed at dif-ferent centre to centre spacings ranging between 2 and 6 pile diameters. The piles were oriented so that a line connecting their centres was either parallel (inline) or perpendic-ular (offline) to the direction of shaking. The piles were pushed into dense vibrated sand Chapter 5. Shake Table Test Results 307 prepared by placing the sand in layers and vibrating each layer individually (densification method B). Following foundation densification the foundation sand had a thickness of 610 mm and an average void ratio after densification of 0.51, corresponding to a relative density of 97 percent. The distance between the centre of gravity of the pile head mass and the soil surface was maintained constant at a value of 67 mm ± 0 . 5 mm during the tests. For each pile group configuration, a ringdown test was carried out by displacing the pile cap 0.5 mm, as measured by L V D T 1 located 4 to 5 mm above the base of the cap, and then releasing the pile cap so that it vibrated freely. Following the ringdown tests, a frequency sweep (test series A) was carried out using shake table B employing peak base accelerations of 0.08 g. At least 20 cycles of shaking were applied for each input frequency. Following test series A, the sand was redensified using high frequency vibration and a second frequency sweep (test series B) performed using peak base accelerations of 0.45 g. The peak base accelerations used during the frequency sweep tests were chosen to match those used in subsequent forced vibration testing in which bending moment distributions along the piles were more carefully studied. The natural frequency tests were carried out to determine: (a) the dynamic amplification and damping characteristics of the pile groups, (b) the dependence of these characteristics on pile spacing and orientation, and (c) the influence of shaking intensity on the natural frequencies of the pile groups. (a) Ringdown Tests Test data from inline and offline ringdown tests PGR1 and PGR6, carried out using a centre to centre pile spacing ratio (s/d) of two, are described in detail to illustrate the influence of shaking direction on the free vibration behaviour of two pile groups. Test data for other pile spacings are summarized in Table 5.6. Accelerations recorded at the centre of gravity of the structural mass during inline test PGR1 are shown in Figure 5.29a. The natural frequency of the group has been Chapter 5. Shake Table Test Results 308 deterrnined by measuring the time over two to three acceleration cycles. Accelerations of up to 0.16 g occurred during ringdown and reduced to zero after three cycles. The fundamental frequency has been determined to be 14.7 Hz in the 0.10 to 0.15 g range, increasing to a value of about 19.5 Hz at lower acceleration levels. Fractions of critical damping of 0.14 were determined by applying the log decrement method over two cycles of ringdown for accelerations less than 0.15 g. Corresponding acceleration data are shown for offline test PGR6 in Figure 5.30a. Pile cap accelerations of up to 0.35 g occurred during the test and reduced to zero after seven cycles of ringdown. The maximum accelerations recorded are clearly larger than measured during test PGR1 due to the absence of pile cap restraint. From the pile cap accelerations shown in Figure 5.30a, the fundamental frequency of the offline group has been determined to be 11.9 Hz for accelerations in the range of 0.20 to 0.35 g, increasing slightly to a value of 12.1 Hz at lower acceleration levels. Fractions of critical damping of 0.05 were determined over 4 cycles of ringdowns in the range of 0.10 to 0.35 g. The effect of pile cap restraint on the dynamic equilibrium of a two pile group sub-jected to inline shaking can be seen by constructing a free body diagram of forces acting on the free standing portion of the group (Figure 5.31a). The equation of moment equilibrium of the free standing portion of the group is given as, W0xcg + Icg0cg - 2APaX = 2M0 (5.5) The corresponding free body diagram for a two pile group subjected to offline shaking is shown in Figure 5.31b. The equation of moment equilibrium for the offline group is, W0xcg + I c g e c g = 2M0 • (5.6) where W0 is the total weight of the pile cap and structural mass, Icg is the mass moment of inertia with respect to the centre of gravity of the structural mass for the relevant Chapter 5. Shake Table Test Results (a) 0.2-, 309 cn O . H 0.0 -0.1 -0.2 3500 3700 3900 TLme 4100 nsec) 4300 4500 (b) 400-£ 300 Q_ a> 250 O XI < CO o u CE 0.00 -0.25--0.50 A A A A A . 1 1500 4600 1700 1800 1900 5000 5100 5200 5300 5400 5500 TLme (msec) (b) .2 bl o 250 O XI < o O 200-Pile Top-^ 7 V-Soil Surface y /) 0.50-C 0.25 0 -0.25 -0.50 -0.75 - | -1 1000 2000 3000 4000 5000 TLme (msec) 6000 7000 Figure 5.33: Typical input base motions at 10 Hz used in pile group frequency tests (a) low level shaking (b) strong shaking Chapter 5. Shake Table Test Results 317 shaking is shown in Figure 5.34b. The fundamental frequency of the group for low level, offline shaking is seen to be about 12 Hz and is identical to average values recorded during the previous ringdown tests. Pile cap accelerations of up to 0.25 g occurred at resonance of the groups. These acceleration levels are similar to the peak accelerations recorded during the offline ring-down tests and suggests that similar amounts of strain softening occurred-in the near field soil during the tests. The dynamic amplification curve for the group is seen to be reasonably independent of pile spacing. Identical behaviour was discerned from the acceleration and bending moment measurements. From this it can be inferred that pile interaction effects are relatively small for offline shaking. The data for stronger shaking (Figure 5.34b) show that the fundamental frequency of the group is in the range of 8 to 8.5 Hz. The reduction in fundamental frequency with increasing excitation amplitude is clearly evident. The dynamic amplification curve is constant over a range of pile spacings, and again suggests that pile interaction effects are small. Pile cap displacements for inline shaking over a range of pile spacings are shown plot-ted versus input frequency in Figure 5.35. The data for low level shaking (Figure 5.35a) shows that as pile spacings increase the fundamental frequency of the group increases, varying from about 15 Hz for a spacing ratio of two to 21 Hz for a spacing ratio of six. The measured fundamental frequencies are close to those measured during the ringdown tests for similar excitation intensities. For stronger shaking intensities, the resonant fre-quencies are shifted to substantially lower levels by the increased strain softening of the near field soil. Dynamic analyses described in chapter 7 indicate that these frequency shifts are due in large part to reductions in axial tension and compression stiffness of the piles, as the sand around the pile shaft progressively dilates and loosens leading to a marked reduction in pile shaft friction. The dependence of the natural frequency of the Chapter 5. Shake Table Test Results 318 Figure 5.34: Frequency sweep tests for a two pile group subjected to offline shaking (a) low level shaking (b) strong shaking Chapter 5. Shake Table Test Results 319 inline group on differences in pile spacing has been seen during the ringdown tests and can be anticipated from the fact that increasing the pile spacing increases the rotational stiffness of the group and reduces the amount of pile interaction. This results in an increased natural frequency with increased pile spacing. Fractions of critical damping have been estimated from the resonant response curves based on the half-power bandwidth method (Clough and Penzien, 1975). For offline, low level shaking damping values ranged between 5 and 7 percent. For stronger shaking these increased slightly to values between 7 and 8 percent. For inline, low level shaking damping values between 9 and 10 percent were inferred from the test data and increased slightly to values as high as 12 percent for stronger shaking. The above damping values are very similar to those inferred from ringdown test measurements. No discernable dependence on pile spacing could be determined from the frequency sweep data since the procedure used to estimate damping ratios is very sensitive to slight inaccuracies in the shape of the amplification curve. Pile group displacements for both low level and high level shaking show a strong frequency dependence as would be predicted from linear structural dynamics theory. Since the fundamental frequencies of the group are shifted with changes in pile separation distance, the dynamic amplification curves are also shifted. This obscures the influence of pile interaction on the dynamic displacements for a particular input frequency. At frequencies substantially less than the resonant frequency of the group, where quasi-static pile group behaviour might be expected to apply, group displacements for close pile spacings exceed those for wider spacings. This is more clearly shown in Figure 5.35a for low level shaking where displacements are plotted to a larger scale. For stronger shaking pile cap displacements for a spacing ratio of two are slightly larger than for wider pile spacings. In both cases, their is a tendency for pile cap displacements to merge to a common value past a spacing ratio of about six. The above data suggest that strong pile Chapter 5. Shake Table Test Results 320 to pile interaction exists for inline shaking. At higher frequencies group displacements for larger pile separation distances exceed those for closer spacings, due to the shifting of the resonant response curves. If one visually aligns the dynamic amplification curves shown in Figure 5.35a and b so that their displacement peaks fall at the same resonant frequency then it can be seen that pile cap displacements are larger for closer pile spacings than for wider pile spacings over a wide range of frequencies. (c) Summary Ringdown and frequency sweep tests carried out in which the piles were subjected to offline shaking show little difference in measured fundamental frequency or dynamic displacement response with changes in pile spacing. This suggests that pile to pile interaction effects are small for this direction of loading. Similar conclusions have been reached by Ting and Scott (1984) based on centrifuge tests carried out on two pile groups in sand subjected to cyclic pile head loading. Reductions in measured fundamental frequency were observed with increasing shaking intensity in the present investigation. Ringdown tests carried out in which the piles were shaken in an inline direction shows that the effective lateral stiffness of the sand was significantly less than for the case of offline shaking, suggesting strong interaction between the piles. Frequency sweep and ringdown tests show that the fundamental frequency of the inline groups increases with increasing pile spacing. This is due to the effects of pile cap restraint, resulting in increased rotational stiffness of the group, and reduced pile group interaction with increasing pile separation. Measured fundamental frequencies for inline shaking also reduce as the intensity of shaking increases. This appears largely due to changes in axial tension and compression stiffness of the piles (see chapter 7), as the soil around the pile shaft loosens during cyclic loading. Measured dynamic amplification curves show that pile cap displacements depend on the input excitation frequency. Correcting for the shift in the dynamic amplification Chapter 5. Shake Table Test Results 321 (Q) 0.30 0.00 I i i i i i i i i i i i i i i i i i i i i i i i i i i i i i 0 5 10 15 20 25 30 Frequency (Hz) (b) 6 0 _ i i i i | i i i i | i i i i | i i i i | i i i i 0 5 10 15 20 25 Frequency (Hz) Figure 5.35: Frequency sweep tests for a two pile group subjected to inline shaking (a) low level shaking (b) strong shaking Chapter 5. Shake Table Test Results 322 curves for each pile group spacing, the data show that group displacements depend on pile spacing for centre to centre pile separations of up to about six pile diameters. Beyond this, pile interaction appears to be small. Chapter 5. Shake Table Test Results 323 5.4 Base Mot ion Excitation of Single Piles Following completion of the natural frequency tests, a series of tests were carried out in which a single pile was subjected to sinusoidal and random earthquake base motion using a range of shaking intensites. Input frequencies in the range of 4 to 12 Hz were selected for the sinusoidal shaking tests. The tests were carried out using both loose and dense sand foundations and elastic wave velocity measurements made prior to each test to determine low strain shear moduli in the foundation sand. The model pile was instrumented to measure bending moment distributions along the pile, and pile head accelerations and displacements. Input base and free field accelerations at the sand surface were also measured. 5.4.1 Free Fie ld Response The natural frequency tests described previously have shown that significant amplifica-tion of the input base motions occurs through the foundation sand at or near its funda-mental frequency. Tests were carried out to provide data illustrating in greater detail the amplification of the free field motions through the sand which were then used to check the results of free field response analyses. Two tests were carried out using loose and dense sand foundations. In the first test, a loose sand foundation was prepared by plac-ing the sand in layers. After soil placement, the sand had an average void ratio of 0.72 and a total thickness of 610 mm. Accelerometers were placed at the base, mid-height and surface of the sand during foundation preparation. The loose sand was then shaken using sinusoidal base motions having a peak acceleration of 0.33 g, an RMS value of 0.12 g and a dominant frequency of 10 Hz. A dense sand foundation was next prepared by vibrating the loose sand on the shake table. Sand was added to the soil container to obtain a final layer thickness of 610 mm and an average void ratio of 0.56. The dense Chapter 5. Shake Table Test Results 324 sand was then shaken using the same input base motions as used in the loose sand test. The accelerometer outputs were recorded during each test and data stored on disc for future processing. The computed Fourier amplitudes of the input base motions are shown for the loose and dense sand tests in Figures 5.36a and 5.37a, respectively. It may be seen that the base motions contain a number of higher frequency overtones which occur in addition to the dominant 10 Hz frequency component. In particular, there is a 50 Hz component which is in close proximity to the fundamental frequency of the foundation sand. Com-puted Fourier amplitudes of the accelerations recorded at the mid-height and-surface of the sand are shown in Figures 5.36b-c and 5.37b-c. These show that the free field motions are significantly amplified as they pass through the sand. The 50 Hz component has the largest amplification since this frequency component is in close proximity to the resonant frequencies of the sand which lie in the 40 to 60 Hz range. 5.4.2 Single Pile Flexural Response - Shake Table vs. Centrifuge Results Due to the low stress conditions operative in the pile foundation model, it was not clear whether the pattern of the pile's flexural response would be similar to that observed under full scale stress conditions present on the centrifuge. A series of shake table tests were therefore carried out using moderately strong shaking for comparison with centrifuge tests having similar prototype shaking intensities. The characteristics of the shake table and centrifuge tests used for this comparison are given in Table 5.7. Pile response in both loose and dense sand foundations was examined. The measured flexural response of the model piles has been plotted in terms of dimen-sionless bending moment (M/M0) versus dimensionless depth below the soil surface (z/d) where M0 is the value of bending moment at the soil surface obtained by interpolating between measured bending moment data points. The bending moment at the soil surface Chapter 5. Shake Table Test Results 325 150 -i 3 \"a £ < (a) FOURIER AMPLITUDES - INPUT BASE MOTION LOOSE SAND TESTS 40 60 Frequency (Hz) 100 (b) FOURIER AMPLITUDES - ACCEL. AT MID-DEPTH LOOSE SAND TESTS ISO-i q) 100-3 a £ < 50 i 20 40 60 Frequency (Hz) 80 100 (c) FOURIER AMPLITUDES - ACCEL. AT SOIL SURFACE LOOSE SAND TESTS 40 60 Frequency (Hz) 100 Figure 5.36: Fourier amplitudes of free field accelerations - loose sand tests (a) base (b) mid-height (c) soil surface Chapter 5. Shake Table Test Results 326 (a) FOURIER AMPLITUDES - INPUT BASE MOTION DENSE SAND TESTS 40 60 Frequency (Hz) (b) FOURIER AMPLITUDES - A C C E L AT MID-DEPTH DENSE SAND TESTS 40 60 Frequency (Hz) 100 (c) FOURIER AMPLITUDES - A C C E L AT SOIL SURFACE DENSE SAND TESTS 300-1 2 5 0 -<» 2 0 0 -XI 3 Frequency (Hz) Figure 5.37: Fourier amplitudes of free field accelerations - dense sand tests (a) base (b) mid-height (c) soil surface Chapter 5. Shake Table Test Results 327 Test No. Sand Density Base Motion Xph (g) (g) S.T. 15 Loose Sinusoidal 0.12(2) 0.31(2) (e0 = 0.72) (20 cycles at 4 Hz) S.T. 22 Loose Earthquake 0.19W 0.40^) (eo = 0.72) (5.5 sec duration) S.T. 26 Loose Sinusoidal 0.10(2) 0.20(2) (eo = 0.72) (30 cycles at 10 Hz) C . T . 12 Loose Earthquake 0.160W 0.180(x) (e0 = 0.77) (30 sec duration) S.T. 13 Dense Sinusoidal 0.10(2) 0.30(2) (e0 = 0.58) (18 cycles at 4 Hz) S.T. 18 Dense Sinusoidal 0.12(2) 0.32(2) (e0 = 0.53) (17 cycles at 4 Hz) S.T. 24 Dense Earthquake 0.20(1) 0.45(1> (eo = 0.54) (5.5 sec duration) C . T . 14 Dense Earthquake 0.05W 0.10W (eo = 0.58) (30 sec duration) C . T . 15 Dense Earthquake 0.14W 0.29W (eo = 0.57) (30 sec duration) (1) Peak acceleration measured during test. (2) Averaged over steady state portion of the acceleration response. C . T . = centrifuge test. S.T. = shake table test. Table 5.7: Single Pile Test Characteristics for Moderate Shaking - Shake Table vs. Cen-trifuge Chapter 5. Shake Table Test Results 328 has been chosen as a convenient normalizing parameter to account for differences in pile cap acceleration, pile head stick-up and structural mass. The results of the comparison are shown plotted in Figures 5.38 and 5.39 where all centrifuge data have been expressed at prototype scale using a centrifuge scale factor of 60. The shake table data indicate that pile bending extends to larger dimensionless depths under the low stress conditions operative in the sand. Maximum dimensionless bending moments were similar during the centrifuge and shake table tests with bending moments decreasing to zero at depth. Under the full scale stress conditions achieved in the cen-trifuge, maximum bending moments occurred in the range of 2.6 to 4.4 pile diameters below the soil surface. The depth of maximum bending was observed to increase with decreasing sand density as the lateral stiffness of the soil decreased. Similar behaviour was observed during shake table testing, with maximum bending moments occurring in the range of 7 to 10 pile diameters for the shaking intensities examined. The elongated bending moment distributions observed in the shake table tests may be anticipated if one considers Matlock and Reese's (1961) definition of characteristic length valid for a pile in a Winkler medium whose stiffness, kh(z), increases linearly with depth as = n^z. The characteristic length factor T is given as, •Ell0'20 (5-7) rih J where the maximum bending moment occurs at the 1.5T depth according to the Matlock and Reese analysis. Considering that the centrifuge test represents to a reasonable ap-proximation full scale behaviour, the ratio of the pile flexural rigidity at prototype scale used in the centrifuge tests to the pile flexural rigidity on the shake table is 3.71 x 107. If one assumes that the rate of increase in lateral Winkler stiffness described by the nh factor is the same between model and prototype and using the above ratio of prototype Chapter 5. Shake Table Test Results 329 Dimensionless B.M. M/MO Figure 5.38: Comparison of pile flexural response observed on the centrifuge and the shake table - moderate level shaking in loose sand Chapter 5. Shake Table Test Results 330 2S 1 • • • • gl - 0 . 5 0 0.5 1 1.5 2 Dimensionless B.M. M/MO Figure 5.39: Comparison of pile flexural response observed on the centrifuge and the shake table - moderate level shaking in dense sand Chapter 5. Shake Table Test Results 331 to shake table flexural rigidities, equation 5.7 gives T.t = — (5.8) 32.6 .... v ; Since the maximum bending moment occurs at the 1.5 T depth which according to the centrifuge data occurs on average at the 3.5 pile diameter depth (= 2000 mm), then Tp = 1330 mm and Tst = 40.8 mm. Therefore, the maximum bending moment is predicted to occur at the 61 mm or 9.6 pile diameter depth which is in reasonable agreement with the shake table test data. The available shaking table data suggests that equivalent rih values derived from measured bending moment distributions along the pile are in fact larger than values derived from centrifuge tests for similar shaking intensities. For example, the peak bending moment distribution measured during shake table test 15 shown plotted in Figure 5.40 is matched reasonably well using a linear versus depth Winkler model where rih = 0.010 N/mm 3 . This value may be compared to the equivalent rih value of 0.0025 N / m m 3 derived from centrifuge test 12 at peak pile deflection (see Table 4.4). Assuming that the shake table nn values are four times those derived from the centrifuge test results gives T.t = (5.9) and a predicted depth of maximum bending equal to 7.3 model pile diameters. The above calculations demonstrate that the bending moment distributions measured on the shake table are in good agreement with what one would infer from laterally loaded pile theory. Further data to lend support to the observation that lateral soil reactions under the low stress regime present in the shake table model are higher than what one would infer from full scale tests is presented in section 5.5. It is not considered essential that the shake table tests simulate full scale behaviour since they were carried out to provide data at model scale as a check against the predic-tions of numerical models used to predict dynamic pile response. It is necessary, however, Chapter 5. Shake Table Test Results 332 600 550-E J , rt 500 OJ > o < OJ o c _D 450 400-350-Soil Surface 300 - 5 0 y V i l e Top / \\ J A Exp't. Theory-k=0.010*z 50 150 Bending Moment (N-mm) 250 Figure 5.40: Theoretical versus measured bending moment distribution - shake table test 15 Chapter 5. Shake Table Test Results 333 that characteristics of the lateral soil resistance be obtained from the test data to use in these analyses. Since many dynamic pile analyses model the soil-pile interaction using linear or non- linear Winkle r springs, it was hoped that the flexural response of the pile measured on the shake table could be successfully simulated using Wink le r models of soil response. Therefore, a number of shake table tests have been examined i n detail to provide information concerning the nature of the lateral soil response during shaking. Particular attention has been paid to those tests carried out using strong shaking to complement the centrifuge test data which were obtained using lower shaking intensities. 5.4.3 S h a k e T a b l e Tes t R e s u l t s Three shake table tests are described in the present section to illustrate typical aspects of pile vibration during strong shaking. Bending moment and pile deflection data obtained during the tests were then used to derive cyclic p-y curves necessary to characterize the soil-pile interaction (section 5.5). Shake table tests 23 and 25 were carried out using random earthquake excitation having peak accelerations of 0.69 g and R M S values of 0.15 g. Test no. 23 was carried out using a loose sand foundation. The test followed an ini t ia l test (test no. 22) using low level earthquake excitation having a peak acceleration of 0.20 g and an R M S value of 0.04 g. Settlements in the loose sand during the in i t ia l shaking amounted to approximately 1.0 m m and resulted in an average void ratio change of 0.003. A n identical test series (test nos. 24 and 25) was carried out following test no. 23 using a dense sand foundation. This was prepared by vibrating the originally loose sand wi th the pile fixed in place during the vibration process. Settlements in the dense sand were negligible during shaking. A third test (test no. 14) was also carried out in dense sand and the pile subjected to approximately 28 cycles of a sine wave wi th a peak average acceleration of 0.60 g at a frequency of 10 Hz . Pertinent test characteristics are given in Table 5.8. Chapter 5. Shake Table Test Results 334 Test No. Sand Density eg (mm) Base Motion (g) (g) 25 Dense 44.4 Earthquake 0.69^ 1.90(3) (eo = 0.54) (5.5 sec duration) 14 Dense 47.7 Sinusoidal 0.60(2) 3.5(2) (eo = 0.57) (28 cycles at 10 Hz) 23 Loose 48.4 Earthquake 0.69(3) 1.0(3) (e0 = 0.72) (5.5 sec duration) (1) Distance of centre of gravity of pile head mass above ground surface. (2) Averaged over steady state portion of the acceleration response. (3) Peak acceleration measured during test. Table 5.8: Single Pile Test Characteristics for Strong Shaking on the Shake Table The sinusoidal and earthquake acceleration inputs at the base of the model for tests 14 and 25 are shown in Figure 5.41a-b. The earthquake input used in test 23 was virtually identical to that used in test 25. Fourier spectra computed from the input base motions are shown in Figure 5.42a-b. The spectra show that the earthquake input is dominated by frequencies in the 0 to 20 Hz range which overlap the measured fundamental frequency of the pile and therefore result in more pronounced flexural response. Higher frequency components up to about 60 Hz are also present. The sinusoidal input motion used in test 14 is seen to be reasonably symmetric and is dominated by a 10 Hz component. Minor higher frequency overtones are also present. The input frequency of 10 Hz selected was believed to be close to the natural frequency of the pile for the level of shaking employed. Free field surface accelerations recorded during the tests 14, 23 and 25 are presented in Figure 5.43a-c. Their corresponding Fourier spectra are shown in Figure 5.44a-c. The data show that the input base motions were amplified through the sand with the strongest amplification occurring around the fundamental frequency of the sand in the Chapter 5. Shake Table Test Results Figure 5.41: Input base accelerations (a) test 14 (b) test 25 Chapter 5. Shake Table Test Results 336 (a) 600-500-j 400 - X> 3 •Q. 300-E < 200 : 100-o : 50 Frequency (Hz) 100 150 (b) 100 CD CL 60 80 Frequency (Hzl HO Figure 5.42: Computed Fourier spectra of input base accelerations (a) test 14 (b) test 25 Chapter 5. Shake Table Test Results 337 40 to 60 Hz range. As a result of this amplification, the free field surface accelerations contain high frequency components not seen to the same degree in the measured pile accelerations. Time histories of accelerations measured on the pile head mass are shown in Figure 5.45a-c. It is noted that the pile head accelerometer used in test 14 was placed on top of the structural mass while in tests 23 and 25 the accelerometer was mounted at the centre of gravity of mass. The data show that pile head accelerations were amplified relative to the free field surface accelerations and do not contain the high frequency components contained within the free field motions. This is shown more clearly by Fourier spectra computed from the measured accelerations (Figure 5.46a-c). From tests 23 and 25 it can be seen that the most significant amplification occurred at frequencies of 10 and 12 Hz for the loose and dense sand tests, respectively. These correspond to the fundamental frequency of the pile for the level of shaking employed. Time histories of pile bending moment at various points along the pile during test 14 are shown in Figure 5.47a- c. Figure 5.48 shows the peak bending moment distributions along the pile for two different times at a peak base motion amplitude of 0.6 g. Bending moments are seen to increase to a maximum near strain gauge 4, located approximately 13 pile diameters below the soil surface, and then decrease to approximately zero at greater depths. This indicates that the pile may be considered long in the sense that the lower parts of the pile do not influence the pile head response to inertia loads applied at the pile head. The time history of bending moment variation at any one depth has the same fre-quency content as the pile head acceleration which shows that the pile response is dom-inated by structural inertia forces. Peak bending moments occurring in the pile also decrease with number of load cycles (Figure 5.47). This indicates that strain hardening of the soil adjacent to the pile is occurring due to compaction of the sand under the Chapter 5. Shake Table Test Results (a) c s CO o o rr 2000 3000 TLme 4000 5000 6000 [mLLLLseconds) 8000 (c) c o ._> H-> . o c CO _ J CD O O 1000 2000 3000 4000 5000 6000 TLme (mLLLLseconds) 7000 8000 Figure 5.43: Free field surface accelerations (a) test 14 (b) test 25 (c) test 23 Chapter 5. Shake Table Test Results 339 (a) 600 50 100 Frequency (Hz) (b) 100 60 80 100 Frequency (Hz! (c) ioo-i 75- ~> o 0-L CO _ ) - 0 . 5 -CO o o -1 -cn - 1 . 5 -3000 4.000 5000 6000 TLme (mLLLLseconds) 8000 Figure 5.45: Pile head mass accelerations (a) test 14 (b) test 25 (c) test 23 Chapter 5. Shake Table Test Results 341 (a) 1500 1200 H •§ 900 H a. E 600-300-IX 50 100 Frequency (Hz) 150 (b ) 200 150 .-> 100 CL e cc 40 60 80 Frequency (Hz) (O 60 80 100 Frequency (Hz) 140 Figure 5.46: Computed Fourier spectra of pile head mass accelerations (a) test 14 (b) test 25 (c) test 23 Chapter 5. Shake Table Test Results 342 repetitive steady state loading. The spatial variation of bending moments along the pile (Figure 5.48) shows that all points along the pile experience the same sign of bending moment at any instant in time with the exception of strain gauge 7 which shows a small moment of opposite sign. Laterally loaded pile models derived from static beam theory have been previously shown to provide a good prediction of the pile's flexural response. These static models also indicate small negative bending moments at large depths. Since static beam theory can often be used to approximate first mode beam flexure (Clough and Penzien, 1975) this suggests that the pile is vibrating in its first mode. Bending moment distributions derived at peak pile deflection during tests 23 and 25 are shown in Figures 5.49 and 5.50, respectively. These show the same general behaviour indicated by test 14 with maximum bending moments occurring at approximately the 15 and 7 pile diameter depths, respectively. Comparing the depth of maximum bending observed,in tests 14 and 25, it can be seen that the stronger intensity of sustained shaking used in test 14 relative to that used in test 25 has resulted in a more softened soil response and an increase in the depth to maximum bending. Bending moment distributions measured during tests 23 and 25 show that as the sand density decreases, the depth of maximum bending increases. The time histories of pile head displacement recorded by an L V D T located near the base of the pile head mass (LVDT 3) are shown in Figure 5.51a-c for tests 23, 25 and 14, respectively. For tests 23 and 14, the maximum recorded displacement is approximately one pile diameter while for test 25 the maximum displacement is about 0.5 pile diameters. These large deflections suggests that significant soil nonlinearity developed during the lateral pile motion. During sinusoidal shaking in test 14, the recorded displacements were observed to have a slight lack of symmetry despite the input base motions being reasonably symmetric. At the end of shaking, small residual displacements were observed for all tests indicative of the cyclic shakedown effects described in section 4.3. Chapter 5. Shake Table Test Results (a) 3oo-250-i 200-f 1 5 0 i 100-i m -200--250 -i -300 4-c—i—i—•—r—i—i i i | i i — — i — i — i — i — i — | — ' — ' I 1 1 1 1 I '—|—'— 1 I ' ' 1 — 0 1000 2000 3000 4000 5000 6000 7000 8000 Time (millisec) g> -100-j \\ -150-CO m -200-^ -250 H -300 4~'—1 ' ' I — 1 — 1 — T — 1 — 1 — 1 1 I '—' ' ' I '—' ' ' I '—'—1-r-1—1—1—T -1—1—1— r-0 1000 2000 3000 4000 5000 6000 7000 8000 Time (millisec) Figure 5.47: Bending moment time histories - shake table test 14 (a) strain gauge 1 strain gauge 4 (c) strain gauge 6 Chapter 5. Shake Table Test Results 344 600-V Pile Top Soil burface ^ \\ 550-500-450-I. 400 H / A Exp't. Theory-k=0.007*z 350-300--500 500 1500 2500 Bending Moment (N—mm) Figure 5.48: Peak bending moment distribution - shake table test 14 Chapter 5. Shake Table Test Results 345 600 H Soil Surface 500 H oj 400-0J > o < OJ (J c _D GO Q 300 H 200-100-^ Pile Top X A Exp'l. Theory—Variable k -500 0 500 1000 Bending Moment (N—mm) 1500 Figure 5.49: Peak bending moment distribution - shake table test 23 Chapter 5. Shake Table Test Results 346 600 Soil Surfcice E E p\" 500 OJ > O Time 1=1672-1751 msec A.P.I. Curve Pile Displacement Y (mm) Figure 5.54: Cyclic p-y curves - shake table test 25 (a) z/d = 1 (b) z/d = 3 (c) z/d = 5 (d) z/d = 10 (e) z/d = 15 Chapter 5. Shake Table Test Results 354 .5 Winkler Modulus KH (N/mm**2) Figure 5.55: Computed secant lateral stiffnesses from p-y curves during low and peak amplitude shaking - shake table test 25 Chapter 5. Shake Table Test Results 355 2.2, increased rates of dilation at these low stress levels could result in larger mobilized friction angles although available data suggests that these effects are small. No direct measurements of peak friction angle at stress levels representative of those on the shake table exist for Ottawa sand, however. Cyclic p-y curves computed for shake table test 23 are shown in Figure 5.56 up to the 25 pile diameter depth for a range of shaking intensities. The p-y curves have been computed over a much larger depth range since the zone of pile bending extends to greater depths in loose sand. The data also indicate nonlinear, hysteretic behaviour indicative of a highly damped system. Approximately linear soil response is observed past the 25 pile diameter depth. For initial low amplitude shaking (cycle A from time t = 3213 to 3322 msec) the secant lateral stiffness is larger than for higher amplitude shaking (cycle B from time t = 4057 to 4209 msec), as expected. To check whether significant densification and stiffening of the lateral soil response occurred under constant amplitude shaking, p-y curves were also computed for a cycle (cycle C from time t = 5116 to 5245 msec) having similar displacement amplitudes as occurred during cycle A. The p-y curves have been computed for cycle A and C at the one pile diameter depth and are shown in Figure 5.57. The p-y curves are not sufficiently different to suggest that significant cyclic densification has occurred. Secant lateral stiffnesses derived from the p-y hysteresis loops for cycles A and B are shown plotted versus depth in Figure 5.58. These indicate the range in effective lateral stiffness between the two cycles of shaking. Using the stiffness distribution computed dur-ing maximum pile displacement, L A T P I L E has been used to compute bending moments along the pile and pile head deflections. The computed bending moment distribution is shown in Figure 5.49 which is in close agreement with the measured bending moment distribution. Computed pile deflections at the location of L V D T 3 are 20 percent less Chapter 5. Shake Table Test Results 356 than those measured (Table 5.9). This is surprising considering the close match of mea-sured bending moments along the pile and may suggest a completely rigid connection between the head of the pile and structural mass was not obtained during the test. Backbone p-y curves computed using the A.P.I, procedures are also shown plotted in Figure 5.56. The curves have been computed assuming a friction angle of 30 degrees derived from triaxial test measurements and an value of 0.00675 N / m m 3 recommended by the A.P.I, for dry loose sand. The A.P.I, curves are again seen to underestimate the lateral soil reactions. 5.5.1 Hysteretic Damping The areas enclosed within the p-y hysteresis loops have been computed to determine hysteretic damping ratios over a range of pile deflections using procedures described in section 4.3.4. The computed damping ratios have been used to estimate equivalent viscous dashpot coefficients along the pile which were then used in equivalent linear analyses of the dynamic pile response (see chapter 6). Data from a shake table test (test no. 30) carried out using low level shaking and a dense sand foundation have also been included to provide damping data for relatively low amplitudes of pile vibration. The test was carried out using sinusoidal shaking at 10 Hz with a peak steady state amplitude of 0.06 g. Bending moment and pile deflection data collected during the test were used to develop cyclic p-y curves at various depths during steady state shaking. The computed p-y curves are plotted in Figure 5.59 up to the 15 pile diameter depth. The p-y hysteresis loops indicate an increase in lateral soil stiffness with depth and were used to compute damping ratios for the small pile deflection levels achieved in the test. The computed damping ratios, D, are plotted versus dimensionless pile deflection, y/d, for tests in loose (Figure 5.60) and dense sand (Figure 5.61). Although there is some scatter in the data, there is a trend for damping to increase with increasing pile Chapter 5. Shake Table Test Results 357 (e) E 1 0.2 £ -0.1-Depth = 25 Pile Diam. i t o Time t=3213-3322 msec A.P.I. Curve Pile Displacement Y (mm) Figure 5.56: Cyclic p-y curves - shake table test 23 (a) z/d = 1 (b) z/d = 5 (c) z/d = 10 (d) z/d = 15 (e) z /d = 25 Chapter 5. Shake Table Test Results 358 0.050 Pile Displacement Y (mm) Figure 5.57: Cyclic p-y curves at the 1 pile diameter depth for constant amplitude shaking - shake table test 23 deflection. When extrapolated to small pile deflections (y/d ~ 0), damping ratios are in good agreement with those inferred from a SHAKE analysis of free field response during a series of frequency sweep tests (see section 5.3). The damping ratios derived from the shake table tests are considerably larger than those inferred from the centrifuge tests over a broad range of pile deflections. This behaviour is attributed to the fact that under the low stress levels present in the sand on the shake table sand particles have a greater freedom to dilate, roll and slide. This would be expected to induce greater frictional energy loss during shear. 5.5.2 Non-Linear Modelling of P-Y Hysteresis Loops The non-linear analysis of pile response to earthquake shaking requires as input a series of backbone p-y curves (see section 6.3). P-y curves were therefore estimated for shake table test 23 to facilitate a dynamic analysis of the pile response. The backbone p-y curve was described using the Ramberg-Osgood equation (see equation 4.20). The ultimate Chapter 5. Shake Table Test Results 359 0 0.0 0.1 0.2 0.3 0.4 Winkler Modulus KH (N/mm**2) Figure 5.58: Computed secant lateral stiffnesses from p-y curves during low and peak amplitude shaking - shake table test 23 Chapter 5. Shake Table Test Results 360 £ -0.04 Depth = 1 Pile D t o m . * Time t=5831-5930 msec -0.10 -0.05 0.00 0.05 0.10 Pile Displacement Y (mm) (b) E 1 Depth = 3 Pile Diam. / '0 y • Time 1=5831-5930 m»c • Time 1=2016-2115 msec -0.10 -0.05 0.00 0.05 0.10 Pile Displacement Y (mm) £ -0.04-Depth = 5 Pile Diam. A 7 / • Time 1=5831-5930 ms«c -0.10 -0.05 0.00 0.05 0.10 Pile Displacement Y (mm) (d) Depth = 10 Pile Diam. j 1 1 . Tim. 1=5831-5930 msec -0.15 -0.10 -0.05 0.00 0.05 0.10 Pile Displacement Y (mm) (e) 0.08 Depth = 15 Pile Diam. X * Time t=5fl31-5930 msec -0.025 0.000 0.025 Pile Displacement Y (mm) Figure 5.59: Cyclic p-y curves - shake table test 30 (low level shaking) (a) z/d = 1 (b) z/d = 3 (c) z /d = 5 (d) z/d = 10 (e) z/d = 15 Chapter 5. Shake Table Test Results 361 75 A Test 2 3 X SHAKE A n a l y s i s A 0.0 0.1 0.2 0.3 0.4 Dimensionless Defl. y/b 0.5 Figure 5.60: Frictional damping ratios, D, versus dimensionless pile deflection y/d in loose sand (shake table test 23) Chapter 5. Shake Table Test Results 362 75 EJ A Test 25 X Test U • Test 30 B SHAKE Analysis X A A X A X X 0.00 0.05 0.10 0.15 Dimensionless Defl. y/b 0.20 Figure 5.61: Frictional damping ratios, D, versus dimensionless pile deflection y/d in dense sand (shake table tests 14, 25 and 30) Chapter 5. Shake Table Test Results 363 z d Dmax ft Vult (mm) 1 0.50 11.5 0.4 3 0.50 10.0 3.2 5 0.50 8.0 3.3 7 0.3 8.0 2.0 10 0.25 8.0 2.0 15 0.20 8.0 2.0 20 0.13 8.0 1.0 25 0.13 8.0 2.5 z/d — dimensionless depth. Table 5.10: Backbone P-y Curve Parameters for Shake Table Test 23 lateral resistance parameter, 6, and pile deflection at failure, yuit, were inferred from the experimental p-y hysteresis loops. Similarly, the damping ratios, Dmax, were determined from the areas within the hysteresis loops. The values of the latter three parameters used in the analysis are given for various depths in Table 5.10. The initial slope of the curve was defined as kh = SEmax with 6 set to a value of 1.4 based on calculations using the elastic model of Kagawa and Kraft (1980a). The low strain shear modulus, Gmax, was determined from the bender element measurements in the foundation sand and related to Emax using a Poisson's ratio of 0.2. A peak friction angle, ', of 30 degrees was used in all calculations. The above parameters were then used to calculate the shape of the backbone p-y curve using the procedures described in section 4.6. To check whether the backbone p-y curves could be used to reproduce the p-y hystere-sis loops derived from the test measurements, a series of Masing loops were constructed at various depths. To reproduce the rather broad shape of the p-y hysteresis loops observed at shallow depths, the mathematical form of the Masing hysteresis loops was modified Chapter 5. Shake Table Test Results 364 from that given in equation 4.19 to the more general form, P - P r K (5.10) where a K value of 2.5 was found to give the best match of the experimental p-y hysteresis loops up to the 10 pile diameter depth. Below this depth a K value of 2.0 was used which describes pure Masing behaviour. Computed hysteresis loops are shown for various depths up to the 15 pile diameter depth in Figure 5.62 and are compared with the p-y data derived from the experimental measurements during the shaking cycle when peak pile deflection occurred. The math-ematical model is capable of matching the observed hysteretic behaviour and indicates that the general shape of the backbone p-y curves are reasonable. Chapter 5. Shake Table Test Results 365 (a) -0.050-Depth = 1 Pile D iam. 0 J * T ime t = 4 0 5 7 - 4 2 0 9 n Mas inq Loop nsec - 5 - 4 - 3 - 2 - 1 0 1 2 3 4 Pile Displacement Y (mm) (b) Pile Displacement Y (mm) (C) ° - 0 .0 o_ £ az 100 : 50 : J La JL 20 10 60 80 Frequency (Hz) 1 i 1 ' 100 120 HO (b) 300 ^ 250^ CO 200 : T 3 •J ._> 150 -J CL £ rr 100 -. 50 : o : 20 40 60 80 Frequency (Hz) 4 ^ 100 120 HO (c) 300-250 : CO 200-XI 150-CL £ az 100-5 0: o : 40 60 80 Frequency (Hz) 100 120 HO Figure 5.64: Computed Fourier spectra - high level, inline shaking (s/d=2) (a) base acceleration (b) free field acceleration (c) acceleration at the e.g. of mass Chapter 5. Shake Table Test Results 371 moment distributions are different between the piles and show that the shear force in each pile varies by up to 25 percent from the average in the group. This non-uniform load sharing diminishes with increasing pile separation. Bending moment distributions for inline shaking of piles at a larger separation distance (s/d = 8) are shown in Figure 5.66. Bending moments do not vary significantly between piles, showing that the piles are equally loaded and that there is a low degree of interaction between them. Test measurements for strong shaking are summarized in Table 5.11 where peak average values of pile cap acceleration, pile cap displacement (LVDT 1), input base and free field acceleration are given. Peak input base accelerations for the tests varied between 0.43 and 0.47 g. Free field surface accelerations were amplified to values between 0.68 and 0.72 g. For the offline tests, pile cap accelerations were approximately the same as those in the free field, varying between 0.68 and 0.75 g. For the inline tests, pile cap accelerations were less than measured during offline shaking due to the restraint of the pile cap, and varied between 0.45 and 0.65 g. There is a tendency for the maximum pile cap acceleration to decrease with increasing pile spacing, as the rotational stiffness of the group increases. The exception to this occurred for a test carried out using a centre to centre pile spacing ratio s/d = 6. Pile cap accelerations were significantly higher than for other spacings, which was confirmed by shear forces in the group deduced from measured bending moments. The reason for this can be seen from examination of the Fourier spectra computed from the measured pile cap accelerations (Figure\" 5.67). The spectra are compared with those computed from pile cap accelerations recorded during an inline test using a spacing ratio s/d = 4. The spectra for the input base accelerations, which did not vary significantly between tests, are also shown. The spectra from the test with s/d = 6 contains a significant 15 Hz frequency component which matches the fundamental frequency of the group for the level of shaking employed (see section 5.2.5). This component, which is present in the input base motion, was not amplified Chapter 5. Shake Table Test Results (a) o < Soil Sur face A Pile I (-0 X Pile 2 (+) • Pile I ( - ) _ H Pile 2 ( - ) - 8 0 0 - 6 0 0 - 4 0 0 - 2 0 0 0 2 0 0 4 0 0 6 0 0 8 0 0 Bending Moment (N-mm) (b) Soil S u r f a c e c Pile 1 f » Pile 2 (•) P i \" 1 (-) P \" « 2 (-) - 4 0 0 - 2 0 0 0 2 0 0 4 0 0 Bending Moment (N-mm) Figure 5.65: Bending moment vs. depth in a two pile group - high level shaking (s/d 2) (a) offline shaking (b) inline shaking Chapter 5. Shake Table Test Results 373 400 Figure 5.66: Bending moment vs. depth in a two pile group -(s/d = 8) high level, inline shaking Chapter 5. Shake Table Test Results 374 Shaking Direction s/d Xff XCg A x (g) (g) (g) (mm) Offline 2 0.43 0.68 0.68 1.09 (1.14) 4 0.46 0.72 0.75 1.16 (1.13)\" « 6 0.45 0.70 0.72 1.11 (1.11) Inline 2 0.46 0.66 0.58 0.72 it 4 0.46 0.71 0.57 0.62 u 6 0.47 0.72 0.65 0.56 a 8 0.46 0.71 0.45 0.38 Xb = average peak input base acceleration Xff = average peak free field acceleration at the soil surface xcg = average peak acceleration at the centre of gravity of the pile cap/structural mass assembly A i = pile cap displacement (LVDT 1) recorded 4 mm above the base of the pile cap (figure in parentheses refers to corrected value scaled to 0.45 g for offline shaking) Table 5.11: Pile Group Test Data - High Level Shaking to as great an extent during the test using a spacing ratio of 4 since the fundamental frequency of the group in the latter case is about 10 Hz. The differences in the degree of amplification of the 15 Hz component are responsible for the significant differences in pile cap acceleration. Average pile cap displacements are plotted against the pile spacing ratio, s/d, for ratios between 2 and 8 for both inline and offline shaking in Figure 5.68. In the case of offline shaking, the measured displacements were all scaled to a peak base acceleration of 0.45 g to account for slight differences in input acceleration level. Scaling was not applied to the inline test data since input base accelerations did not vary substantially between tests. For offline shaking, pile cap displacements are relatively constant over a range of spacings, varying by ± 0 . 0 2 mm from the average. This suggests there is little interaction between the piles. During inline shaking, measured displacements gradually decrease Chapter 5. Shake Table Test Results (a) 375 10 60 Frequency (Hz) HO (b) 20 10 60 80 100 Frequency (Hz) (C) 300-250-CD 200-^ XI D 150 -CL e 100 : 5 0 ~ o : J 4 i fiL :Js Ifr rrJ**i4V 250-o < 150-Soil Surface / / Pile Top Offline s/d= 6 a Pile 1 - Exp't x Pile 2 - Exp't Theory 0 200 400 600 800 Bending Moment (N-mm) Figure 5.70: Comparison of predicted and measured bending moments using a Winkler model for high level, offline shaking (a) s/d = 2 (b) s/d = 4(c)s/d=6 Chapter 5. Shake Table Test Results 380 0.016 and 0.020 N/mm 3 . Lower values were used to describe the flexural behaviour of the pile group for spacing ratios of 2 and 4. This is indicative of the greater amount of pile interaction that occurs for these close pile spacings. For larger spacings (s/d = 6 and 8) the value is the same as used in the single pile test and suggests there is little interaction between the piles. Pile cap deflections computed using the above modulus parameters are within 0.01 to 0.02 mm of measured deflections. The above calculations suggest there is little interaction between piles for offline shaking. During inline shaking, the data suggests interaction between piles extends to distances of up to 6 pile diameters from the centre of a pile. It is of interest to compare measured group deflections with predictions using the single pile flexibilities previously established and elastic interaction factors proposed by Randolph and Poulos. Provided the computed pile group deflections are reasonably accurate, this gives greater confidence in using the method to compute the combined stiffness of larger groups. 5.6.3 Pile Group Interaction Analysis The methodology used to compute the elastic interaction factors proposed by Randolph and Poulos and the displacement response of a pile group has been described in section 4.6.3. An identical methodology has been followed to describe the response of pile groups subjected to strong sinusoidal shaking on the shake table. Two modulus distributions were used to assess the sensitivity of the computed inter-action factors to this factor. In the first approach, low strain shear moduli derived from the shear wave velocity measurements were used, resulting in moduli proportional to the square root of depth. In the second approach, effective shear moduli were estimated from Winkler stiffness distributions used in the derivation of single pile flexibility factors. The shear modulus distribution G(z) was related to the lateral Winkler stiffness kh(z) using equation 4.35 where K is a proportionality constant which is dependent on pile length, Chapter 5. Shake Table Test Results 381 ^ Pile Top X A 1 Surface' \\ A Inline - (' s/d = 2 : i A Pile 1 - Exp'l ; X Pile 2 - Exp'l 1 Theory 1 (b) 0 100 200 300 400 Bending Moment (N-mm) o si < T Pile Top Soil Suijface 7 / -A: Inline s/d = 4 A Pile 1 - Exp'l X Pile 2 - Exp'l Theory 0 100 200 300 Bending Moment (N-mm) Pile Top-/ <$oil Surface - ) A - ' / / Inline s/d = 6 \\ / Pile 1 - Exp't X — . 1 1 ] r-Pile 2 - Exp't Theory 0 100 200 300 Bending Moment (N-mm) (d) < Pile Top^ - Soil Surface / Inline s/d = 8 A Pile 1 - Exp't x Pile 2 - Exp't Theory 0 100 200 300 Bending Moment (N-mm) Figure 5.71: Comparison of predicted and measured bending moments using a Winkler model for inline, high level shaking (a) s/d = 2 (b) s/d = 4 (c) s/d = 6 (d) s/d = 8 Chapter 5. Shake Table Test Results 382 relative soil-pile stiffness, frequency of vibration and depth z. Based on the Kagawa and Kraft model K was set equal to a value of 3.6. Using a linear versus depth Winkler analysis to describe the single pile response, kh = rihZ. The modulus distribution G(z) is then given as, G(z) ~ ^ (5.11) K Using the above modulus distributions and the known structural properties of the piles, interaction factors have been computed for various centre to centre pile spacings and orientations with respect to the direction of shaking. Considering interaction between two piles and making the simplification that the piles are equally loaded (M\\ =-M2 = Mo and Vi = V2 = VQ) the average deflection of the group at the soil surface has been computed using equation 4.38. Single pile flexibilities developed from test no. 34 have been used in the calculations using a Winkler modulus parameter nh = 0.020 N/mm3. Average pile group rotations were computed in an analogous manner. With estimates of deflection and rotation at the soil surface, and known average loads in the group, pile cap deflections were determined from a structural analysis of the free standing part of the group. Computed deflections at the location of LVDT 1 taking into account group interaction are shown in Table 5.13 for strong shaking. Computed deflections are shown for the different modulus assumptions adopted and compared to the measured deflections. The computations for offline shaking show good agreement between computed and measured pile cap displacement. The effect of modulus amplitude for a particular modu-lus variation with depth does not substantially affect the calculations. Calculations have also been carried out neglecting pile interaction. These show that pile interaction effects are small for offline shaking and result in less than a 5 percent difference in computed de-flections relative to those measured. For all practical purposes, interaction during offline Chapter 5. Shake Table Test Results 383 Shaking Direction s/d A ( l ) A ( 2 ) A i (mm) (mm) (mm) Offline 2 1.13 1.11 1.09 r> 4 1.19 1.20 1.16 •n 6 1.07 1.08 1.11 Inline 2 0.78 0.75 0.72 4 0.62 0.61 0.62 5) 6 0.60 0.59 0.56 8 0.42 0.41 0.38 (1) = pile cap deflection with group interaction - low strain shear modulus (2) = pile cap deflection with group interaction - shear modulus estimated from Winkler analysis A x = measured pile cap deflection at the location of L V D T 1 Table 5.13: Pile Group Interaction Analysis - Strong Shaking shaking may be neglected. For inline shaking computed displacements for small pile separations (s/d = 2) are 0.03 to 0.05 mm larger than measured displacements. These differences are less than 10 percent of measured displacements and are considered acceptable. For larger pile sepa-rations, computed displacements are within 5 to 10 percent of measured displacements. For centre to centre pile spacing ratios of 6 and 8 (see Table 5.13) there is a tendency to overpredict the group displacements. The model therefore predicts that interaction between piles extends to larger distances than actually occurs. Pile cap displacements neglecting interaction have been computed for these larger pile spacings and are found to be within 0.01 mm of measured displacements. This suggests that interaction effects between piles for inline shaking extends within a zone of at most six pile diameters from any pile. The shake table data indicate that the Randolph-Poulos method of estimating in-teraction between piles gives an acceptable (within 10 percent) prediction of pile group Chapter 5. Shake Table Test Results 384 displacements. The key factor in the accuracy of the analysis is the determination of the single pile flexibilities. The interaction factors proposed tend to overestimate inline interaction at larger pile spacings and for this reason it seems reasonable \"to cut off pile interaction beyond a centre to centre pile spacing of 6 pile diameters. Interaction between piles for offline shaking over the range of pile spacings examined appears to be small and can reasonably be neglected. Similar conclusions have been inferred from centrifuge and full scale test data (see section 4.6.3). Interaction factors, nuv, have also been estimated from the test data using the pro-cedures described in section 4.6.3. The computed factors are plotted versus pile spacing ratio for both inline and offline shaking in Figure 5.72. Interaction factors derived from centrifuge and full scale field tests (see Figure 4.71) have also been plotted for compari-son. For inline shaking, interaction factors computed from the shake table data are seen to be lower than factors computed from centrifuge tests for close pile spacings (s/d = 2). The available data do, however, show considerable scatter for close pile spacings. For larger spacings (s/d = 4 and 6) during inline shaking and for a broad range of spacings during offline shaking, the centrifuge and shake table data indicate similar amounts of interaction. 5.6.4 Base Motion Excitation of 2 x 2 Pile Group Sinusoidal base motions used during strong shaking of the four pile (2 x 2) group spaced at an s/d ratio of three and subjected to shaking about its longitudinal (x-x) axis are shown in Figure 5.73a. The average input base acceleration, computed as the peak to peak value divided by two, was 0.47 g. Time histories of free field acceleration, pile cap displacement near the base of the pile cap and near the top of the mass (LVDT 1 and 2, respectively), and pile cap accelerations at the centre of gravity of mass are shown in Figure 5.73b-e. Peak average accelerations of 0.68 g occurred in the free field. Average Chapter 5. Shake Table Test Results 385 A Centrifuge (Gohl) x Shake Table (Gohl) • Centrifuge (Barton) 8 Full Scale (Ochoa & O'Neill) ffi Full Scale (Tominaga et al) Theory (Randolph & Poulos) 2 3 4 5 6 7 Pile Spacing Ratio (s/b) 2 3 4 5 6 7 Pile Spacing Ratio (s/b) A Centrifuge (Gohl) x Shake Table (Gohl) a Centrifuge (Barton) Theory (Randolph & Poulos) Figure 5.72: Shake table interaction factor, nvv, and comparison with other experimental and theoretical (Randolph- Poulos) results (a) inline shaking (b) offline shaking Chapter 5. Shake Table Test Results 386 accelerations at the centre of gravity of the pile cap mass were approximately 0.67 g. Displacements near the top of the structural mass are greater than those measured near the base of the cap from which a peak angular rotation of the pile cap/mass assembly of 0.0106 rads (0.61 degrees) has been computed. Fourier spectra computed from the input base, free field and pile cap acceleration time histories are shown in Figure 5.74. It can be observed that higher frequency components in the 50 to 70 Hz range contained in the input base motion were significantly amplified at the soil surface in the free field. These frequencies overlap the fundamental frequency of the sand layer which accounts for their amplification. These high frequency components axe not contained to any significant degree in the pile cap accelerations since the fundamental frequency of the group is well removed from these higher excitation frequencies. Bending moment time histories at three different depths along pile no. 1 are shown in Figure 5.75a-c. At any instant in time, bending moments have the same sign and show that the pile is dominated by first mode vibration. At the end of shaking, small residual bending moments remain along the pile as a result of the pile displacing slightly from its original position. Residual pile cap displacements were also observed (Figure 5.73). The time history of axial load variation in pile no. 1 is shown in Figure 5.75d. The dynamic axial loads contain higher frequency components not seen in the bending moment time histories. Average bending moments along the piles computed as their peak to peak value di-vided by two are shown in Figure 5.76. The bending moment in both piles are reasonably similar near the soil surface with an average value of 225 N-mm. The slopes of the bend-ing moment distribution in this region are somewhat different and show that the shear force per pile varies by about 18 percent from the average shear load of 5.6 N. The average Chapter 5. Shake Table Test Results 387 Figure 5.73: 2 x 2 pile group response - high level shaking (s/d = 3) (a) base acceler-ation (b) free field surface acceleration (c) acceleration at the e.g. of mass (d) pile cap displacement (LVDT 1) (e) pile mass displacement (LVDT 2) Chapter 5. Shake Table Test Results 388 (a) 400 -sec) 300-i a> itude 200-AmpI 100-40 60 Frequency (Hz) 100 (b) 400--sec) 300-i U) itude 200-AmpI 100-60 Frequency (Hz) 100 (c) 400 -sec 300-cn CD x> 3 200-AmpI 100 : o- IAI»X. , i — i ,. i i , , \\ i., , 0 20 40 60 80 100 Frequency (Hz) Figure 5.74: Computed Fourier spectra (a) base acceleration (b) free field acceleration (c) acceleration at the e.g. of mass Chapter 5. Shake Table Test Results 389 (O) ^ 300-r E 200-0) => - joo l , , , , , , 0 1000 2000 3000 4000 5000 6000 7000 Time (msec) (b) Time (msec) (C) 300-1 Time (msec) z ' x 1000 2000 3000 4000 5000 6000 7000 Time (msec) Figure 5.75: 2 x 2 pile group response - bending moment and axial load time histories (a) strain gauge 1 (b) strain gauge 3 (c) strain gauge 5 (d) dynamic axial load Chapter 5. Shake Table Test Results 390 bending moment distribution is again well described using a linear versus depth Win-kler model where has a value of 0.013 N/mm 3 . This value is substantially less than the value of 0.020 N / m m 3 used to describe the single pile response and is indicative of significant interaction between the piles in the group. With the known average moment and shear load per pile, the Winkler model was used to compute average deflections and rotations at the soil surface. A structural analysis of the free standing portion of the group was then carried out which yielded deflections and rotations of the rigid pile cap. These were linearly extrapolated to give a displacement of 0.97 mm at the location of L V D T 1 which may be compared with an average displacement of 0.98 mm measured by L V D T 1. The close agreement of the computed and measured pile cap deflections confirms the reasonableness of the inferred modulus distribution. Using the above average loads, unit pile flexibilities derived from single pile test no. 34, and interaction factors computed using the Randolph-Poulos methodology, pile displacements and rotations at the soil surface have been computed considering the effects of loading from all piles in the group. A structural analysis of the free standing portion of the group was then carried out which gave a displacement of 0.89 mm at the location of L V D T 1. This is approximately 9 percent less than the measured displacement which is considered acceptable. Using the revised interaction factors described in section 4.6.3, a pile group deflection of 0.95 mm was computed at the location of L V D T 1. This represents a slight improvement on predictions made using the Randolph-Poulos approach. 5.6.5 Summary Two pile groups tests subjected to strong sinusoidal shaking using the U B C shaking table and embedded in dense vibrated sand have shown that interaction between piles is significant during inline shaking for centre to centre pile spacings of up to six pile diameters. During offline shaking, pile to pile interaction can for all practical purposes Chapter 5. Shake Table Test Results 391 Soil Surface V Pile Top X As XA A Pile 1 - Exp't x Pile 2 - Exp't Theory -200 0 200 400 Bending Moment (N-mm) 600 Figure 5.76: Average bending moment vs. depth in the four pile group and comparison with Winkler model Chapter 5. Shake Table Test Results 392 be ignored. These observations confirm conclusions arrived at from centrifuge testing carried out using low to moderately strong shaking. Analysis of the shake table data shows that interaction factors estimated using the Randolph-Poulos methodology gives reasonable estimates of pile group deflection but tends to overpredict interaction for large pile spacings during inline shaking and for a range of spacings during offline shaking. Analysis of the four pile group data suggests that provided one can accurately estimate the single pile flexibilities, the use of a superposition approach incorporating pile to pile interaction factors can lead to acceptable predictions of group response. Interaction effects observed on the shake table and centrifuge have been obtained from tests in dense dry sand where theories of elastic interaction can be most expected to apply. In situations where the sand is saturated and pore pressure rise is significant around the piles, it is expected that interaction between piles will be reduced from that predicted using elastic theory. Similarly, if gaps develop between the pile and the soil as occurs most commonly during cyclic loading of piles in clay, pile interaction will be correspondingly attenuated (Bogard and Matlock, 1983). Interaction between piles results in non-linear soil response which can only be approximately estimated using linear elastic theory. The degree of interaction in sand during cyclic loading will depend on the sand density, lateral load level, the number of cycles of shaking, and the location of one pile with respect to another in the group. Due to the highly complex nature of the soil response it is not surprising that considerable scatter has been observed in interaction factors inferred from inline shaking tests for close pile spacings. From a practical design point of view, the fact that interaction is significant over a limited range of pile spacings can be used to refine currently used elastic theories of pile to pile interaction. Chapter 5. Shake Table Test Results 393 5.7 Cycl ic A x i a l Load Behaviour of M o d e l Piles The dynamic response of pile groups depends on both the lateral and axial stiffness of the piles, appropriately modified for the effects of group interaction. The lateral stiffnesses can be inferred from the measured flexural behaviour of the piles during the model tests. The axial stiffnesses play an important role in determining the rotational stiffness of the group but cannot be determined directly from the measured lateral response of the model piles. Cyclic axial push-pull tests have therefore been carried out on a single pile pushed into vibrated dense sand, having the same embedment as used in the pile group tests. The model pile was located in the centre of a cylindrical container whose walls were located 12 pile diameters from the centre of the pile. The pile tip was located 4 pile diameters above the base of the container. This was considered a large enough distance to minimize the influence of the base on the pile response since prediction of axial pile response is commonly made using soil properties located at most 4 pile diameters below the pile tip (Schmertmann, 1978). The container was mounted in a triaxial testing frame and the head of the pile screwed into a load cell using an appropriate mechanical adaptor. The load cell reacted against a fixed reference beam while the base of the sand container, which was attached to the triaxial frame loading pedestal, was cycled vertically at a constant rate of displacement. This loading procedure moves the sand foundation past the fixed head pile and is approx-imately equivalent to displacing the pile head cyclically, since for small axial loads and relatively short pile lengths pile tip displacements do not vary significantly from those at the top due to elastic shortening of the pile. A displacement rate of 2.0 mm/min was adopted for the tests. Cyclic axial loading of the pile was carried out using a constant displacement amplitude of ± 0 . 3 0 0 mm. These amplitudes exceed the maximum vertical Chapter 5. Shake Table Test Results 394 displacements the piles were subjected to during strong inline shaking by a factor of about three. Cyclic axial load at the head of the pile and displacement of the base of the sand container were measured during the test over 30 cycles of loading. The load-displacement response over a number of different load cycles is shown in Figure 5.77. After about 10 cycles of loading, the axial load- deformation (P — A) behaviour of the pile may be approximated as bi-linear for displacements of up to about 0.15 mm. This is the range of displacement of most interest during cyclic loading of the model pile groups. Beyond this displacement, the axial stiffness gradually increases. During loading and unloading up to a maximum change in displacement of about 0.05 mm, the tangent axial stiffness equals approximately 600 N/mm. For larger displacements, the tangent axial stiffness reduces dramatically to a value between 10 and 20 N/mm. This pronounced reduction in stiffness is due to the reduction in sheax stiffness of the sand as it fails in shear during axial displacement of the pile and is augmented by the significant dilation which occurs under the low stress conditions operative in the test. The reduction in axial stiffness evidenced by the above test has a significant impact on the natural frequencies and dynamic response of the group, and will be discussed further in chapter 7. Chapter 5. Shake Table Test Results 395 Axial Displacement (mm) Figure 5.77: Cyclic axial loading test on a single model pile in dense sand Chapter 6 Single Pile Response to Earthquake Excitation 6.1 Introduction Rigorous analysis of the response of a single pile subjected to earthquake excitation is based on an approach in which the foundation and superstructure are analysed as a com-bined system. Interaction between the pile and the near field soil is accounted for using linear or non-linear lateral compliances (springs and equivalent viscous dashpots) placed along the length of the pile. These compliances model the forces acting on the pile due to relative movement between the pile and the free field ground motions. One particu-larly attractive method of analysis, because of its ability to simulate non-linear soil-pile interaction, is embodied in the computer program SPASM8 (Matlock et al, 1978a,b). In this program, the time history of free field displacements computed at various depths throughout the soil profile is applied to the free field end of the lateral compliances. Soil-pile interaction is simulated using a non-linear Winkler idealization of the lateral soil resistance, incorporating cyclic p-y curves. This procedure automatically accounts for hysteretic damping in the near field soil while radiation damping is approximately accounted for using equivalent viscous dashpots. In absence of experimental data de-scribing the p-y curves, methods to construct backbone p-y curves for sand have been summarized by Murchison and O'Neill (1984) and include procedures recommended by the American Petroleum Institute. Unloading p-y response is described assuming a mod-ified form of Masing behaviour (see section 6.3), and may incorporate cyclic degradation 396 Chapter 6. Single Pile Response to Earthquake Excitation 397 of the lateral soil resistance and/or gapping between the soil and the pile. Alternatively, an equivalent linear representation of the lateral soil stiffness may be used in which the lateral soil resistance is assumed to remain constant during shaking. Hysteretic and radiation damping is simulated using equivalent viscous dashpots. SPASM8 represents the superstructure in a rather approximate manner using a se-ries of one dimensional beam elements. Therefore, structural designers often prefer to simulate the dynamic response of the superstructure using a more detailed plane or 3 dimensional frame analysis. The superstructure is uncoupled from the pile foundation using a series of linear or non-linear pile head compliances. These are complex valued functions representing the in phase and out of phase (damping) forces at the pile head induced by the relative displacement between the ground and the piles during shaking. Using an uncoupled analysis, the free field surface motions are applied to the ends of the foundation compliances. This assumes that the free field motions equal those at the pile cap level and neglects the fact that differences could exist between the free field and pile cap motions due to differences in stiffness between the soil and the piles (Gazetas, 1984a). Gazetas has shown that the effects of kinematic interaction cause the pile cap motions to be less than those in the free field, especially for higher input frequencies. To check the validity of the above coupled and uncoupled analysis procedures, pre-dictions of single pile response measured during shake table and centrifuge testing were made. Details of the modelling procedures and results of the analyses follow. Chapter 6. Single Pile Response to Earthquake Excitation 398 6.2 Uncoupled Non-Linear Analysis In the present section, an uncoupled, non-linear analysis is described which was used to predict the dynamic response of the model piles. Pile head compliances were computed to represent the embedded portion of the pile and were derived for the case of cyclic pile head loading, neglecting the effects of the free field ground motion on the equations of motion of the pile (see chapter 1). This is a simplification commonly made and appears to be a reasonable approximation based on the experimental results and calculations using a closed form analytic solution described subsequently. 6.2.1 Pile Head Stiffnesses The pile head stiffnesses used in the analyses represent the in phase (real) component of pile head compliance, K^, and equal the equivalent elastic force (or moment) at the pile head in the ith direction acting in phase with a unit amplitude harmonic displacement in the jth direction. Since the lateral soil stiffness along a pile is non-linear and dependent on the amplitudes of pile vibration, pile head stiffnesses were selected that were compatible with an appropriate level of pile head displacement at a given instant in time. The pile head stiffnesses used in the analyses are defined as: Kuu = pile head shear force due to unit lateral displacement (u = 1.0) while pile head rotation 0 = 0 R~uj> = pile head moment due to unit lateral displacement (u = 1.0) while pile head rotation 0 = 0 = pile head moment due to unit pile head rotation (0 = 1.0) while pile head displacement u = 0 Computation of the pile head stiffness relies on a determination of the lateral soil stiffness along the pile, kh(z). This was derived using two different procedures. In the Chapter 6. Single Pile Response to Earthquake Excitation 399 first, a lateral Winkler stiffness distribution, kh(z), proportional to depth or to the square root of depth was selected that gave the best match of measured bending moments and deflections along the pile at various times during shaking. From the Winkler stiffness distribution, the static pile head stiffnesses were derived (see Appendix D) and their variation with the level of pile head deflection determined. Using the computer program PILAY developed by Novak and Aboul- Ella (1978) the dependence of these compliances on frequency was also determined. A shear modulus distribution was selected having either a square root or linear with depth distribution that provided a match of the static compliances. By varying the frequency of pile head excitation, the pile head stiffnesses were found to be insensitive to frequency in the range of interest, suggesting that static pile head stiffnesses may be used in the calculations. The above matching procedure is considered to be the most accurate method of determining the effective pile head stiffness. Since the latter procedure is based on a back analysis of the test data, it is of interest to estimate the lateral soil stiffness distribution using a soil-pile interaction model that might reasonably be used prior to an earthquake. Non-linear backbone p-y curves were therefore computed at various depths along the pile using the recommendations of the American Petroleum Institute (1979). While cyclic p-y curves generated from the centrifuge test data have shown that use of the A.P.I, procedures overestimates lateral soil stiffness, this analysis was carried out to demonstrate the sensitivity of computed pile response to inaccuracies in foundation compliance. A range of pile head shear, Vo, and moment, M 0 , loads were applied to the pile at the soil surface to cover the anticipated range of structural inertia loads applied to the pile for the intensity of shaking considered. For each load level, the computer program L A T P I L E was used to compute the deflected shape of the pile and the secant Winkler stiffness along the pile compatible with the pile deflection level. The L A T P I L E model yields a Winkler stiffness distribution that is non-uniform with depth. Therefore, the pile head stiffnesses K{j were computed using as Chapter 6. Single Pile Response to Earthquake Excitation 400 input into L A T P I L E the non-uniform distribution of secant Winkler modulus, fc^z). Pile head shear and moment were then applied that resulted in unit pile head deflection and zero rotation (fixed head case). The shear load required for this condition is equivalent to Kuu while the moment load equals K^u. For the case of a free headed pile, a shear load was applied that resulted in unit pile head deflection, giving the lateral stiffness for a pinned head pile, Kun. It may be shown that the following relationship exists (Novak, 1979): K*+ = KuK \\ K (6-1) Equation 6.1 was then used to compute the rotational stiffness K^. Since the pile head stiffnesses are dependent on the amplitudes of pile vibration, this was taken into account by expressing the pile head stiffnesses, i f a s a function of the pile head deflection at the soil surface, yo. It was found that if one plots the logarithm of the pile head stiffness K{j versus the absolute value of y0, an approximately linear relationship results. This has the form, log Kij = \\og(Kij)o - m\\y0\\ (6.2) where (if,j)o is the reference stiffness at zero pile head deflection. During each time step in the dynamic solution adopted (see Appendix H), the above equation was used to compute the pile head stiffnesses based on deflection levels computed for the preceding time step. 6.2.2 Pile Head Damping The imaginary component of pile head compliance, K™, represents the integrated effect of damping forces along the pile which consist of frictional (hysteretic) and radiative en-ergy losses. For an equivalent viscously damped system, the viscous dashpot coefficients dj are related to as C,j = K^/LO. The pile head damping forces and moments Chapter 6. Single Pile Response to Earthquake Excitation 401 are determined using a unit pile deflection profile, u(z), which is compatible with the boundary conditions used to estimate the pile head stiffnesses, K{j. The unit deflection profile u(z) should not be confused with the real deflected shape of the pile y(z) occurring during shaking. The velocity amplitude, tt(z), of the pile is then u>u(z), where u; is the frequency of excitation. The damping forces (/<*) per unit length along the pile due to the unit amplitude pile head motions are represented using velocity proportional damping where fd(z) = ceq(z)u(z). The equivalent viscous dashpots ceq vary with depth z along the pile since the hysteretic component of damping varies with pile deflection and strains in the near field. Following the approach of Gazetas and Dobry (1984), we decompose ceq into the sum of its radiation and hysteretic damping components, or ceq = c? + c^. Expressions for Cr have been given by Gazetas and Dobry. These have been found to be in good agreement with dashpot coefficients derived by Novak et al (1978) considering plane strain vibration of a rigid disc in a viscoelastic medium, and by Roesset and Angelides (1980) who used an efficient three dimensional finite element formulation to derive average dashpot coefficients along the pile. The expressions proposed by Gazetas and Dobry are frequency and depth dependent. For depths greater than 2.5 times the pile diameter the radiation dashpot coefficient is given as in which a 0 is the dimensionless frequency factor given by a 0 = u)r0/Va, to is the frequency of excitation, r 0 is the pile radius and is the free field shear wave velocity. For shallower depths, the following approximate relationship has been suggested by Gazetas and Dobry, Alternatively, Kagawa and Kraft (1981a) and Berger et al (1977) have proposed using Lysmer and Richart's (1966) analogue to estimate a frequency independent coefficient c r (6.3) (6.4) Chapter 6. Single Pile Response to Earthquake Excitation 402 where cr = 4r0p(Vp + Vs) and Vp is the compressional wave velocity in the free field. This provides dashpot coefficients in reasonable agreement with equation 6.3 for dimensionless frequencies of the order of 0.5. For lower dimensionless frequencies, the use of Lysmer and Richart's dashpot coefficient leads to radiation damping coefficients less than those predicted by Dobry and Gazetas. The hysteretic viscous dashpot, c^ , is defined from equation 4.13 in terms of D, kh and u. Both D and kh vary with depth and are dependent on the lateral pile deflection occurring during shaking, y(z), which in turn is proportional to the average maximum shear strain, 7 m a I , in the near field soil. The latter may be estimated using the procedures described in Appendix C for a given pile deflection profile or alternatively based on experimental relationships derived between D and y/d. For a pile deflection profile, u(z), compatible with the pile head boundary conditions u(0) = 1.0 and 0(0) = 0.0, the resultant damping force, Fd is given as The corresponding pile head moment, Af^, due to damping forces along the pile is given as The equivalent viscous dashpot coefficients are then Cuu = Fd/tou(0) and = Md/uju(0) where u(0) = 1.0. Similarly, = Md/utl>(0) and C u t / , = Fd/u;0(O) for the deflection boundary conditions 0(0) = 1.0 and u(Q) = 0.0. For a given centrifuge or shake table test, an average value of u> was chosen that corresponded to the dominant frequency contained within the test. Equivalent viscous dashpots, C t J , were also computed over the appropriate frequency range using the computer program PILAY for comparison with the above procedures using the same shear wave velocity distribution and unit pile deflection profile. The (6.5) (6.6) Chapter 6. Single Pile Response to Earthquake Excitation 403 theory used to compute the real and imaginary components of pile head compliance is described by Novak and Aboul-Ella (1978). The procedure is based on a finite element discretization of the pile, incorporating complex valued, plane strain soil reactions given by Novak et al (1978). The lateral soil response is represented using a Winkler model since soil response at one depth is assumed to be independent of response at another. The PILAY computer code has a built in frequency cut-off which calculates the viscoelastic soil reactions using ao = 0.3 if dimensionless frequencies prescribed for the analysis are less than this value. This accounts for the fact that use of plane strain soil reactions lead to zero lateral stiffness under static (zero frequency) conditions. It was found that using the radiation damping parameters proposed by Dobry and Gazetas for a dimensionless frequency ao = 0.3 gave pile head damping coefficients C,j very similar to those computed using PILAY for a given u(z) profile. Therefore, PILAY was used to estimate damping parameters for the dynamic response calculations discussed subsequently. The integration approach described above is, however, conceptually simpler than the theory used in PILAY and demonstrates clearly the concept of a pile head viscous dashpot coefficient. The damping analysis requires as input the shear wave velocity distribution in the soil. To estimate radiation damping in the free field, shear moduli, G7/, compatible with effective strains occurring during shaking were computed from a free field response analysis (SHAKE). The shear moduli were then used to compute effective shear wave velocities, V ,^ from the equation Gfj = pVa2. The shear wave velocities were then used to compute the dashpot coefficients due to radiation damping in the free field, (C,j) r . Hysteretic damping is mainly due to frictional energy losses in the near field, which are generally higher than those in the far field due to the high shear strains that occur in the soil adjacent to the pile. Therefore, material damping ratios, D, and shear moduli, Gnj, representative of the effective shear strains in the near field were used in the PI-L A Y analysis, giving viscous dashpot parameters representing the combined influence of Chapter 6. Single Pile Response to Earthquake Excitation 404 radiation and hysteretic damping. The analysis was then repeated, neglecting hysteretic damping, giving the dashpot coefficients due to radiation damping alone. The difference between the latter two sets of coefficients represents the damping due to frictional energy losses in the near field, (C,j)/,. The hysteretic damping coefficients were then added to those due to radiation damping in the far field, giving the combined viscous dashpot coefficient C,j . 6.2.3 C o u p l e d V e r s u s U n c o u p l e d A n a l y t i c a l S o l u t i o n The foundation compliances used in the uncoupled analysis are derived for the case of cyclic pile head loading so that the effects of the free field ground motions on the pile response at the soil surface are neglected. The significance of this on the computed superstructure response is illustrated using a closed form analytic solution for both a coupled and uncoupled analysis. In the coupled analysis, the base motions are harmonic of the form ub = u0etwt and are applied at the pile tip. The pile is surrounded by a homogeneous Winkler medium with the soil reaction, p, given by p = k3(y — it) where ks is the Winkler subgrade modulus, y is the relative lateral pile deflection with respect to the moving base and u is the relative free field displacement. The undamped pile-superstructure response is then solved assuming that the pile vibrates in its first mode. A similar approach has been used by Flores-Berrone and Whitman (1982) who have neglected the stick-up of the pile and the structural mass in their analytic solution. The analysis is described in Appendix F and leads to a closed form solution for the natural frequencies and forced vibration response of the soil-pile system. The results of an analysis using this coupled solution are summarized in Table 6.2 using the soil and structural properties given in Table 6.1. The structural properties are identical to those used in the centrifuge tests and are quoted at prototype scale. The Winkler modulus used in the analysis is representative of a loose dry sand. The analysis Chapter 6. Single Pile Response to Earthquake Excitation 405 -50 0 50 100 150 Bending Moment (kN-m) Figure 6.1: Computed bending moment distribution - coupled versus uncoupled analytic solution shows that the second natural frequency of the pile lies above the frequencies contained in the earthquake base motions used in the centrifuge. This suggests that second mode contributions to vibration response will not be significant. Experimental evidence also shows that the pile vibrates substantially in its first mode. Peak bending moments and pile deflections relative to the input base displacements are plotted versus depth in Figures 6.1 and 6.2, respectively. The free field displacements are also shown in Figure 6.2 which demonstrate that the pile displacements exceed the free field displacements near the soil surface due to the influence of the pile head inertia forces. Chapter 6. Single Pile Response to Earthquake Excitation 406 0 10 20 Relative Displacement (mm) Figure 6.2: Computed pile and free field displacements relative to base motion - coupled analytic solution Chapter 6. Single Pile Response to Earthquake Excitation 407 a) Pile properties • Pile flexural rigidity (EI) = 172614 kN-m 2 • Mass moment of inertia of structural mass (Icg) = 53.11 kN-sec2-m • Pile head mass (mo) = 53.2 kN-sec 2/m • Pile mass per unit length (ra) = 0.003657 kN- sec 2/m 2 • Embedded depth of pile (H) = 11.90 m • Stick-up between soil surface and underside of mass (Z) = 0.90 m • Distance between C G . and underside of mass (Az) = 0.99 m b) Soil and Base Motion Properties • Lateral subgrade modulus (k3) = 13600 k N / m 2 • Peak base acceleration = 0.10 g • Input earthquake forcing frequency = 3.14 rad/sec Table 6.1: Pile and Soil Properties Used in Test Case The coupled analysis yields the bending moment and shear force acting on the pile at the soil surface. The bending moment distribution along the pile due to this shear and moment loading, neglecting the influence of the free field ground motions, has been computed using a constant with depth Winkler modulus ka (Scott, 1981) to be compatible with the assumptions of the dynamic solution. The bending moment distribution is compared to that computed using the coupled dynamic analysis in Figure 6.1. Agreement between the two approaches is quite good, especially near the soil surface where structural inertia effects dominate. This suggests that provided one can predict pile shear forces and bending moments at the soil surface using an appropriate dynamic analysis, one can obtain a reasonable estimate of maximum flexural stresses in the pile near the soil surface using a simplified static solution. It also suggests that the flexural response of the model pile is dominated by structural inertia forces rather than by the free field ground motions. Chapter 6. Single Pile Response to Earthquake Excitation 408 • First mode natural frequency = 10.90 rad/sec • Second mode natural frequency = 59.73 rad/sec • Peak absolute pile acceleration at soil surface = 0.114 g • Peak absolute pile acceleration at underside of mass = 0.117 g • Peak bending moment at soil surface = 117.9 kN-m • Peak bending moment at underside of mass = 62.7 kN-m • Peak shear force at soil surface = 61.3 kN • Peak shear force at underside of mass = 61.3 kN Table 6.2: Superstructure Response to Harmonic Base Motion - Coupled Analysis (Test Case) An uncoupled analysis of the above test case has also been carried out. Details of the solution are shown in Appendix G which gives the natural frequencies and undamped forced vibration response of the superstructure. Vibration of the superstructure occurs in response to the free field surface motions which are linked to the base of the superstructure via the foundation compliances. The forced vibration solution also assumes that the superstructure vibrates in its first mode. The derivation of the dynamic foundation spring stiffnesses for a homogeneous Winkler foundation (constant k^ versus depth) is shown in Appendix G. These are derived for the case of cyclic pile head loading analogous to the Novak and Aboul-Ella procedure. The dynamic spring stiffnesses have been found to be independent of frequency in the low frequency range for the soil and structural properties being considered. Using the pile and soil properties given in Table 6.1,the foundation spring stiffnesses have been derived. The natural frequencies and forced vibration response of the su-perstructure occurring in response to the ground motions used in the previous coupled analysis have then been computed (see Table 6.3) and may be compared to the results of Chapter 6. Single Pile Response to Earthquake Excitation 409 Foundation spring constants: • kuu = 36,290 kN/m • = 48,448 kN-m/m • kw = 129,270 kN-m/rad Results: • First mode natural frequency = 10.90 rad/sec • Second mode natural frequency = 59.75 rad/sec • Peak absolute pile acceleration at soil surface = 0.114 g • Peak absolute pile acceleration at underside of mass = 0.117 g • Peak bending moment at soil surface = 117.0 kN-m • Peak bending moment at underside of mass = 62.2 kN-m • Peak shear force at soil surface = 61.0 kN • Peak shear force at underside of mass = 60.9 kN Table 6.3: Superstructure Response to Harmonic Base Motion - Uncoupled Analysis (Test Case) the coupled analysis. The comparison demonstrates that the uncoupled procedure gives results in excellent agreement with the more rigorous coupled approach and suggests that foundation compliances derived for the case of cyclic pile head loading may be used in the uncoupled analysis without significant loss of accuracy. The present investigation has focused on pile response in dry sand where significant free field deformations do not develop. The model piles also support a relatively heavy structural mass so that structural inertia forces dominate the pile response. In cases where one has a deep deposit of soft clay or liquefiable sand and the pile supports a relatively light superstructure, the free field ground motions are expected to play a larger role in the pile response. Chapter 6. Single Pile Response to Earthquake Excitation 410 6.2.4 Uncoupled Equations of Mot ion - Single Pile To incorporate the effects of foundation damping using the pile head viscous dashpots previously described, it has proved convenient to discretize the free standing portion of the pile above the soil surface using a 4 degree of freedom beam element whose stiffness matrix is derived from a cubic polynomial displacement field. This element has been found to predict natural frequencies of the uncoupled pile-mass system in good agreement with predictions using a displacement field derived from the analytic solution described in the previous uncoupled analysis. The mass properties of the beam element are based on a consistent mass formulation. The beam element is linked to the foundation via pile head compliances consisting of springs and equivalent viscous dashpots. A rigid structural mass having rotational and translational mass properties to represent the superstructure has also been linked to the beam element. In the case of the shaking table tests, the influence of the L V D T spring forces on the stiffness of the soil-pile system has been included. The model is described fully in Appendix H leading to generalized dynamic equations of motion of the form: [M]{x} + [C]{x} + [K]{x} = {P(t)} (6.7) where [M], [C] and [K] are the mass, damping and stiffness matrices of the 4 degree of freedom system, respectively, {P{t)} represents the load vector which depends on the input free field surface motions, and {x} the nodal degrees of freedom. The above matrix equation is then solved using a step by step integration procedure described in Appendix H incorporating deflection dependent, secant foundation stiffnesses. The damping matrix is assumed constant during shaking. It is noted that this solution procedure implicitly accounts for all vibration modes of the system and no assumptions as to the form of the damping matrix are assumed, unlike solutions using modal superposition. Foundation deflections at the soil surface used in the computation of the foundation Chapter 6. Single Pile Response to Earthquake Excitation 411 stiffnesses if,j were computed from the following equation, 2/o = V0fuv + M0fum (6.8) where Vo and M0 are the shear force and moment at the soil surface and fuv and fum are foundation flexibility coefficients for shear and moment loading, respectively, computed for a particular time step. A similar equation may be written for pile rotation, 0o, at the soil surface. Knowing t/o, 0o, and the shear and bending moment in the free standing portion of the pile above the soil surface, deflections and rotations at the underside of the pile head mass were computed using a structural analysis. Deflections at the location of the displacement sensor on the structural mass were then computed assuming a rigid connection between the pile and the mass. The above procedure assumes that pile head deflections are dominated by structural inertia forces which the experimental data indicates is a valid assumption. Shear forces and bending moments in the pile were computed at each time step from the computed nodal deflections and rotations using the equations of beam flexure. Since translational accelerations were computed at the top of the pile, it was necessary to extrapolate these to the particular location of the accelerometer used in a test taking into account rotational accelerations of the mass. This was done using equations of rotational kinematics. In the case of the centrifuge tests, the accelerometer was located as shown in Figure 3.4. The acceleration at this edge location, xe, was then computed as xe = x t - 0t (i?! sin 0-Ro)- (6t)2 Rt cos B (6.9) where RQ is the vertical distance between the top of the pile and the centre of gravity ( C G . ) of the structural mass, Ri is the radial distance between the C G . and the ac-celerometer and 0 is the angle from the horizontal between the line connecting the the C G . and the accelerometer. The rotational velocities 0t and accelerations 0t at the top Chapter 6. Single Pile Response to Earthquake Excitation 412 of the pile were determined by expressing the computed time history of pile top rotations 8t using a Fourier series of N harmonics selected to have a high enough frequency range to cover the frequencies contained within the pile head motions. Rotational velocities and accelerations were then obtained by appropriate differentiation of each term in the series. In the case of the shake table tests, the pile head accelerometer was located either at the C G . of mass or on top of the mass. The acceleration at either of these locations was then computed as, xph = xt + Ho0t (6.10) where H0 is the vertical distance between the underside of the structural mass and the location of the accelerometer. 6.2.5 Prediction of Ringdown Test Results Use of the uncoupled foundation model is first illustrated in an analysis of two ringdown tests described in section 5.3. One test (test R-L5) was carried out using a loose sand foundation while in the second test (test R-D2) the model pile was embedded in dense sand. During each test, the pile was displaced laterally a known amount and then released, setting the pile into damped free vibration. From measured bending moments along the pile and pile head deflections, the initial displacement and slope of the pile at the soil surface and at the underside of the pile head mass were determined. The latter correspond to the four degrees of freedom used in the dynamic analysis and represent the initial conditions of the problem. Setting the load vector P(t) to zero, equation 6.7 was solved, incorporating pile head stiffnesses that were varied with the level of pile deflection. The pile head stiffnesses were determined from a back analysis of bending moment distributions measured during the ringdown tests over a range of pile deflections. A square root of depth Winkler model, where k^ = az1/2, was used in the analysis. The Chapter 6. Single Pile Response to Earthquake Excitation 413 modulus parameter a was selected by trial and error until a reasonable match of measured bending moments and pile head deflections was achieved. With this modulus distribution, the pile head stiffnesses were computed for a given pile deflection, yo. The above analysis was repeated for a range of pile deflections and equation 6.2 fitted to the data. Pile head damping coefficients CV, were computed using PILAY for the frequency range of interest during ringdown using low strain shear wave velocities measured in the free field. Hysteretic damping ratios D used to compute near field hysteretic damping coefficients were estimated based on pile deflection levels at the peak of the first cycle of ringdown according to damping ratios derived from experimental p-y curves (see section 5.5.1). The stiffness and damping parameters of the soil- pile system used in the analysis of the ringdown tests are given in Table 6.4. Structural properties of the model pile used in the analysis not previously described during description of the experimental set-up are also included. Measured and computed displacements at the location of L V D T 2 and bending moments at the soil surface are plotted versus time for tests R-L5 and R-D2 in Figures 6.3 and 6.4, respectively. The computed time histories of displacement and bending moment are seen to be in reasonable agreement with measured quantities, demonstrating that the natural frequen-cies and hence the stiffness and mass properties of the soil-pile system are approximately correct. It is noted that the non-linear foundation stiffnesses incorporated in the model correctly simulate the observed reduction in fundamental period of the soil-pile system as vibration amplitudes decrease. The computed time histories predict little dynamic response after 2 to 3 cycles of ringdown, as was observed, and suggests that estimates of system damping are reasonable. The results demonstrate that use of an uncoupled dynamic analysis can lead to acceptable predictions of pile response under the test con-ditions examined. The model was next applied to the prediction of pile response to Chapter 6. Single Pile Response to Earthquake Excitation 414 Foundation compliance parameters (test R-L5): • Kuu:m = 0.86 N/mm 2 , (Kuu)0 = 37.6 N/mm • Ku+ : m = 0.765 N/mm-rad, (Ku^,)0 = 1946.0 N/rad • : m = 0.674 N, (if^)o = 176246.0 N-mm/rad • D = 0.15 (constant with depth) Foundation compliance parameters (test R-D2): • Kuu : m = 1.37 N/mm 2 , ( A ^ o = 100.0 N/mm • ^ : m = 0.628 N/mm-rad, (# t t t >) 0 = 2828.0 N/rad • i f ^ : m = 0.0 N, (K^)0 = 125000.0 N-mm/rad • D = 0.15 (constant with depth) Structural Properties: • Dist. between soil surface and pile top (test R-L5) = 15.2 mm • Dist. between soil surface and pile top (test R-D2) = 20.8 mm • Dist. between pile top and L V D T 2 (test R-L5) = 21.3 mm • Dist. between pile top and L V D T 2 (test R-D2) = 18.8 mm Table 6.4: Ringdown Analysis Parameters - Tests R-L5 and R-D2 simulated earthquake shaking. 6.2.6 P red i c t ion of P i l e Response to Free F i e ld G r o u n d Mo t i ons Computations carried out using the uncoupled foundation model to predict pile response to the measured free field ground motions are described subsequently and compared with measured response to assess the accuracy of the model. Four tests covering a range of shaking intensities and sand densities were selected to ensure that the model would be tested under conditions where approximately linear to highly non-linear soil response occurred. Two tests were conducted on the Caltech centrifuge using low to moderately Chapter 6. Single Pile Response to Earthquake Excitation (a) 2-E E c E o _ D D _ in 415 100 150 200 Time (msec) 250 300 (b) 300 150 200 Time (msec) 300 Figure 6.3: Computed versus measured ringdown time histories - test R- L5 (a) displace-ments at L V D T 2 (b) bending moments at the soil surface Chapter 6. Single Pile Response to Earthquake Excitation (a) 416 ( b ) 400 100 150 200 Time (msec) 250 100 150 200 Time (msec) 300 300 Figure 6.4: Computed versus measured ringdown time histories - test R- D2 (a) displace-ments at L V D T 2 (b) bending moments at the soil surface Chapter 6. Single Pile Response to Earthquake Excitation 417 strong shaking. Centrifuge test 41 was carried out using low level sinusoidal shaking in dense sand while test 12 used moderately strong earthquake excitation of piles in loose sand. Two tests carried out using the U B C shaking table (table A) were also selected for analysis. Tests 14 and 23 were carried out using dense and loose sand foundations, respectively, and subjected to very strong shaking. In test 14, sinusoidal base motions were used while in test 23 the pile was subjected to random earthquake excitation. Static pile head stiffnesses were derived for each test from a back analysis of bending moment distributions measured along the pile at various stages during shaking. Laterally loaded pile models based on Winkler subgrade stiffnesses having linear or square root of depth variations were found to adequately describe the measured bending moment dis-tributions. The stiffnesses were again found to vary in a non-linear manner with pile deflection and were fitted to a semi-logarithmic equation given by equation 6.1. A typi-cal distribution used to describe foundation stiffnesses for centrifuge test 12 is shown in Figure 6.5. The experimentally derived pile head stiffnesses are also compared against those predicted using the computer program L A T P I L E and backbone p-y curves recom-mended by the American Petroleum Institute for cyclic loading in loose sand (see section 6.2.1). The predicted increase in pile head stiffness using the A.P.I, recommendations is readily apparent. Radiation damping coefficients (C,j) r were computed using as input into PILAY shear wave velocity distributions computed from free field response analyses carried out using S H A K E . The analysis is carried out using equivalent elastic soil properties which are estimated iteratively according to a specified level of effective strain (Schnabel et al, 1972). Low strain shear moduli, Gmax, determined from bender element measurements were used as basic input. The variation of secant shear modulus in the free field, G/f, normalized with respect to Gmax versus shear strain, 7, was estimated using equations 4.9 to 4.11. Peak friction angles derived from triaxial test measurements and K0 values Chapter 6. Single Pile Response to Earthquake Excitation 418 CO to OJ c C/) T> O OJ X _0J 100-10-KPP (kN-m/rad) KUP (kN/rad) KUU (kN/m) 15 20 Pile Defl. Y0 (mm) — r -25 30 KPP - Exp't. KUP Exp't. KUU Exp't. KPP - API KUP r API KUU - API Figure 6.5: Experimental variation of the logarithm of pile head stiffness K{j versus pile deflection y0 and comparison with A.P.I, recommendations - centrifuge test 12 Chapter 6. Single Pile Response to Earthquake Excitation 419 Shear Strain (%) Figure 6.6: Normalized shear modulus-shear strain attenuation relationships used in SHAKE analysis of centrifuge tests equal to 0.4 and 0.6 for loose and dense sand, respectively, were used in the calculations. The Gff/Gmax versus 7 relationships used for the analysis of the centrifuge tests are shown in Figure 6.6. Similar relationships have been computed for the low stress levels present on the shake table (see Figure 2.6). Damping ratios, D, used in SHAKE have been computed at various shear strain levels using equation 4.12. As a preliminary check on the ability of the one dimensional SHAKE analysis to simulate measured free field response, a back analysis of the free field test described in section 5.4.1 was carried out. During the test, a 0.61 m thick loose sand foundation was subjected to sinusoidal base motion having a peak acceleration of 0.33 g and a dominant 10 Hz frequency component (Figure 6.7a-b). Using a D m t n value of 13 percent, the Fourier spectra of free field accelerations computed at the mid-depth and top of the sand Chapter 6. Single Pile Response to Earthquake Excitation 420 layer are shown in Figure 6.7c-d. Frequencies of up to 60 Hz have been considered in the analysis. The computed spectra are compared with Fourier spectra computed from accelerations measured at the mid-depth and top of the sand. The computations clearly show the amplification of the ground motions through the sand. Computed Fourier am-plitudes are in reasonable agreement with those measured, especially at the soil surface, and demonstrate that S H A K E provides a reasonable simulation of the free field response. An interesting result of the S H A K E analysis is that peak shear strains are computed to be small and in the range of 0.004 to 0.013 percent. Strain compatible shear moduli have been computed at effective strain levels equal to 65 percent of maximum shear strains computed during shaking. Since effective strains are small, effective shear moduli are computed to be within 3 percent of low strain moduli Gmax- It would be expected that a one dimensional analysis, which ignores the constraining effect of the boundaries of the soil container, would provide an upper bound estimate of shear strains in the sand. Therefore, an analysis taking into account the presence of the container boundaries would likely predict somewhat smaller strain levels. The Fourier spectra of free field surface accelerations computed using S H A K E for shake table tests 14 and 23 are presented in Figures 6.8 and 6.9 and are compared against those measured. For test 23 carried out in loose sand, a Dmin value of 13 percent has been adopted for the computations and frequencies of up to 60 Hz have been considered in the analysis. For test 14 carried out in dense sand, a Dmin value of 10 percent was used and frequencies of up to 100 Hz have been considered. Peak accelerations of 0.75 and 0.85 g were computed for tests 14 and 23, respectively. These may be compared against measured values of 0.80 and 0.70 g, respectively. The computed frequency content and amplitudes of the free field surface motions are sufficiently close to measured quantities to permit accurate estimates of effective shear stiffnesses in the free field. The S H A K E analyses again predict that peak shear strains are low and effective shear moduli within Chapter 6. Single Pile Response to Earthquake Excitation 421 (b) 1500 3500 5500 Time (msec) 7500 30 40 Frequency (Hz) 50 60 10 30 40 Frequency (Hz) (d) 60 150 o \"5. 50 E < o-f 0 A- 1 Exp't, Computed 20 30 40 Frequency (Hz) 50 60 Figure 6.7: S H A K E analysis of free field test 10 (a,b) input base accelerations and computed Fourier spectra (c,d) computed versus measured Fourier spectra at foundation mid- depth and sand surface, respectively Chapter 6. Single Pile Response to Earthquake Excitation 422 6 percent or less of Gmax. Free field response analyses have also been carried out for centrifuge tests 41 and 12. The analyses have been carried out at prototype scale and have considered frequencies of up to 5 Hz. Large strain damping values, Dmax, of 20 and 25 percent have been adopted for tests 41 and 12, respectively. Low strain damping values of 1 percent have also been assumed for each test. Free field surface accelerations computed using S H A K E for centrifuge test 41 are again expressed in terms of their Fourier amplitudes in Figure 6.10 and are compared against measured quantities. The computed spectra for test 41 are in close agreement with those measured for frequencies of up to 3.5 Hz. Discrepancies exist for larger frequencies. However, the amplitude of the dominant 0.5 Hz frequency component is well predicted, resulting in peak acceleration amplitudes of 0.047 g during steady state shaking. This agrees closely with measured values of 0.05 g. Effective shear moduli ranging from 66 to 93 percent of Gmax were computed over the height of the sand layer. Fourier spectra derived from surface accelerations computed using S H A K E for cen-trifuge test 12 are presented in Figure 6.11 and are compared against measured spectra. The dominant frequencies contained within the measured surface motions are seen to be present in the computed motions although discrepancies in amplitude do exist at se-lect frequencies. To check that the effective stiffness properties of the sand used in the S H A K E analysis were sufficiently accurate, the natural frequency of the sand has been computed using strain compatible shear stiffnesses. The fundamental frequency has been found to be 2.7 Hz which is close agreement with a value of 2.75 Hz inferred from the experimental data (section 4.3). The computed and recorded time histories of free field surface acceleration are shown in Figure 6.12. The measured and recorded accelera-tions are seen to be in reasonable agreement which suggests that the effective stiffness and damping properties of the sand used in the computations are acceptable. Effective Chapter 6. Single Pile Response to Earthquake Excitation 423 Q_ E < 400 300 o Frequency (Hz) Figure 6.9: Fourier spectra of free field surface accelerations computed using S H A K E versus measured spectra - shake table test 23 (a) measured spectra (b) computed spectra Chapter 6. Single Pile Response to Earthquake Excitation 425 40 Figure 6.10: Fourier spectra of free field surface accelerations computed using S H A K E versus measured spectra - centrifuge test 41 Chapter 6. Single Pile Response to Earthquake Excitation 426 Figure 6.11: Fourier spectra of free field surface accelerations computed using S H A K E versus measured spectra - centrifuge test 12 (a) measured spectra (b) computed spectra Chapter 6. Single Pile Response to Earthquake Excitation 427 - 0 . 2 -— 0 . 3 I ' i ' ' | ' ' ' ' i 1 ' 1 ' l ' 1 ' ' I ' ' ' ' l ' ' ' ' 0 5 10 15 20 25 30 Time (sec) Figure 6.12: Free field surface accelerations computed using SHAKE versus measured accelerations - centrifuge test 12 (a) measured accelerations (b) computed accelerations Chapter 6. Single Pile Response to Earthquake Excitation 428 using S H A K E , where G/j < Gmax, were used. A constant with depth Poisson's ratio of 0.35 was used to compute P-wave velocities which is believed appropriate for larger strain excitation of the free field soil. Pile head damping coefficients (C,j) r were then computed at the dominant frequency of excitation appropriate to the centrifuge or shake table test. In the case of earthquake excitation, this corresponds to the measured excitation frequency where maximum pile response was recorded during the test. In normal design prior to an earthquake event, it would appear logical to select the fundamental frequency of the pile as the frequency of interest, provided the earthquake motions are expected to contain significant frequency components around the fundamental frequency of the pile. Since PILAY uses a frequency cut-off (a0 = 0.3) below which damping coefficients are assumed independent of frequency, and dimensionless frequencies used in the centrifuge and shake table tests are less than this cut-off value, the computed damping coefficients were found to be independent of frequency within the range of interest. Hysteretic damping coefficients (C,j)/, were estimated using the following procedures. An effective near field shear modulus distribution, Gnf(z), was selected for use in PILAY that would result in pile head stiffnesses K~ij equal to those inferred from the test mea-surements using either a linear or square root of depth Winkler modulus distribution. The Gnf distribution was selected to have a depth variation compatible with the Win-kler model adopted. Since system damping was assumed to be constant during shaking it was necessary to select a reference deflection level at which to compute the reference K~ij values. This was arbitrarily selected equal to the deflection at which maximum pile response occurred. Effective shear wave velocities were then computed from the appro-priate modulus distribution and used as input into PILAY. Unit pile deflection profiles u(z) were then computed consistent with the lateral stiffness distribution selected. Chapter 6. Single Pile Response to Earthquake Excitation 429 Neglecting near field hysteretic damping, a fictitious set of radiation damping param-eters were computed using the input near field moduli and a large strain Poisson's ratio value of 0.4. The analysis was then repeated using a prescribed variation of material damping D(z) along the pile, resulting in a new set of damping coefficients including the effects of hysteretic damping. The difference between the latter two sets of damping coefficients equals the component due to near field hysteretic damping. This was added to the radiation damping component to give the combined damping of the system. Selection of an appropriate material damping distribution D(z) represents the largest uncertainty in the analysis. Since lateral pile deflections and average effective strains around the pile decrease with depth, D(z) was selected to have a linearly decreasing distribution versus depth, ranging from a maximum value at the pile head to a minimum value below a point where pile bending is small. In the case of centrifuge test 12, hysteretic damping was varied from a maximum value of 25 percent to a value of zero below the 10 pile diameter depth. For centrifuge test 41 where the pile was subjected to low level shaking, the experimental data suggests that the near field soil response exhibited very little hysteretic behaviour. Therefore, hysteretic damping was neglected for this test. For shake table tests 14 and 23 which were carried out using very strong shaking, a maximum damping ratio of 35 percent was adopted and was decreased to a minimum value of 10 percent below the 20 pile diameter depth. The rotational damping coefficient was found to be more sensitive to the material damping distribution adopted than were the coefficients Cuu and Cu1/,. This may be anticipated from equation 6.6 since evaluation of moment quantities involves terms fdzdz rather than terms fddz. Using the above procedures, foundation stiffness and damping coefficients have been derived for each test and are given in Tables 6.5 and 6.6. The hysteretic and radiation damping components of the damping coefficients have been separated to indicate the relative contributions of each component. Structural stiffness and mass properties for the Chapter 6. Single Pile Response to Earthquake Excitation 430 model piles used in the uncoupled response analyses have been described in chapters 2 and 3. Additional structural parameters not given previously, including the free standing lengths of pile above the soil surface and the distance of displacement transducers above the top of the pile, are also presented in the table. Using the recorded free field surface motions as input into the uncoupled dynamic analysis, the time history of pile response has been computed for each of the four tests. Absolute accelerations and displacements at the measurement point location and shear force and bending moment at the soil surface have been computed for comparison with measured quantities. The results of the computations are plotted in Figures 6.13 to 6.17. Model predictions are seen to be in excellent agreement with measured pile response for centrifuge tests 41 and 12. Centrifuge test 12 is considered to provide a more stringest check of the accuracy of the analysis procedure since the test involves random earthquake excitation containing a wide spectrum of frequencies in the 0 to 5 Hz range. Using the pile head stiffnesses derived using the A.P.I, p-y curves, pile response during centrifuge test 12 was also computed. Dynamic pile response was similarly well predicted (Figure 6.14a) although peak pile deflections are seen to be lower than measured. This suggests that the pile response is relatively insensitive, for this particular example, to errors in foundation stiffness. To check the accuracy of the stiffness and mass properties of the soil-pile system the fundamental frequencies of the pile were checked using pile head stiffnesses computed at peak pile deflection during each test. For centrifuge test 41, the fundamental frequency has been computed to be 1.51 Hz which may be compared with the value of about 1.8 Hz estimated from the experimental measurements (Table 4.2). The slight discrepancy between measured and computed fundamental frequency is not serious, however, since the frequency of excitation of 0.5 Hz used in the test is well removed from the resonant frequency. Therefore, predictions of dynamic pile response Chapter 6. Single Pile Response to Earthquake Excitation Foundation compliance parameters (C.T. 41): • Kuu : m = 0.0142 kN/m per mm, (Kuu)0 = 63,095 kN/m • Ku+ : m = 0.032 kN/rad per mm, {KU^,)Q = 79,430 kN/rad • : m — 0.014 kN-m/rad per mm, (K^)0 = 158,500 kN-m/rad • (C««)/i = 0 kN-sec/m • (C««)r = 717 kN-sec/m • (Cu^)/i = 0 kN-sec/rad • (Cuj,)r = 358 kN-sec/rad • (Ctw)/i = 0 kN-m-sec/rad • (CtW-)r = 262 kN-m-sec/rad Foundation compliance parameters (C.T. 12): • Kuv:m = 0.0067 kN/m per mm, ( i f u u ) 0 = 22,387 kN/m • if u V , : m = 0.0036 kN/rad per mm, {Ku+)0 = 39,810 kN/rad • : m = 0.0016 kN-m/rad per mm, (K^)Q = 123,000 kN-m/rad • (Cuv.)h = 164 kN-sec/m • (CUu)r = 637 kN-sec/m • {Cu*)h = 179 kN-sec/rad • (CW)r = 379 kN-sec/rad • (CW)/i = 241 kN-m-sec/rad • (CW)r = 332 kN-m-sec/rad Structural Properties: • Dist. between soil surface and pile top (C.T. 41) = 0.90 m • Dist. between soil surface and pile top (C.T. 12) = 0.99 m • Dist. between pile top and displ. transducer (C.T. 41 and 12) Table 6.5: Uncoupled Analysis Parameters - Centrifuge Tests Chapter 6. Single Pile Response to Earthquake Excitation Foundation compliance parameters (S.T. 14): • Kuu : m = 0.067 N/mm per mm, (KUU)Q = 43.6 N/mm • Kuj, : m = 0.047 N/rad per mm, (Ku^,)0 = 1,995 N/rad • : m = 0.0 N-mm/rad per mm, (K^)Q — 113,000 N-mm/rad • (Cuu)h = 0.010 N-sec/mm • (Cuu)r = 0.074 N-sec/mm • (CW)fc = 0.40 N-sec/rad • (Cw-)r = 0.71 N-sec/rad • (Cw,)h = 20.5 N-mm-sec/rad • {Cw>)r = 9.8 N-mm-sec/rad Foundation compliance parameters (S.T. 23): • Kuu : m = 0.109 N/mm per mm, (Kuu)o = 17.4 N/mm • Kuj, : m = 0.077 N/rad per mm, (ifu^,)o = 1,122 N/rad • : m = 0.035 N-mm/rad per mm, (K^)0 — 104,713 N-mm/rad • (Cuu)h = 0.007 N-sec/mm • (Cm)r = 0.059 N-sec/mm • (C„^)k = 0.33 N-sec/rad • (CW)r = 0.68 N-sec/rad • (PiH>)h = 21-5 N-mm-sec/rad • (Cw)r = 11.1 N-mm-sec/rad Structural Properties: • Dist. between soil surface and pile top (S.T. 14) = 22.3 mm • Dist. between soil surface and pile top (S.T. 23) = 23.0 mm • Dist. between pile top and displ. transducer (S.T. 14) = 15.9 mm • Dist. between pile top and displ. transducer (S.T. 23) = 22.8 mm Table 6.6: Uncoupled Analysis Parameters - Shake Table Tests Chapter 6. Single Pile Response to Earthquake Excitation 433 are unlikely to be seriously affected. For centrifuge test 12, the computed fundamental frequency is 0.97 Hz which agrees closely with the estimated value of 1 Hz inferred from test measurements. Using the pile head stiffnesses derived from the A.P.I, p-y curves for a maximum base shear of 100 kN, the fundamental frequency is predicted to be 1.2 Hz. The slight change in natural frequency has not significantly affected the computed pile response since the earthquake ground motions contain a spectrum of frequencies surrounding and including the fundamental frequency of the pile. In cases where the earthquake ground motions are dominated by a frequency where resonance of the pile occurs, an imperfect prediction of the fundamental frequency of the pile would result in more serious discrepancies between computed and measured pile response. In general, it may be stated that a satisfactory match of computed and measured natural frequencies is necessary in order to make high quality predictions of dynamic pile response. Shake table test 14 provides an opportunity to assess the accuracy of damping pa-rameters used in the analysis. The test was carried out using a frequency of excitation of 10 Hz which is close to the measured fundamental frequency of the pile for the intensity of shaking used. In such circumstances, the computed pile response is very sensitive to the value of damping used in the analysis. Using the foundation stiffnesses computed at peak pile deflection, the fundamental frequency of the pile has been computed to be 10.0 Hz which implies that resonance of the pile will occur during shaking. Using the damping parameters given in Table 6.6 it has been found that pile response was underpredicted. Peak bending moments and shear forces at the soil surface during steady state shaking of about 14 N and 750 N-mm were predicted which may be compared to measured quantities of 20 N and 1000 N-mm, respectively. This shows that damping in the system is over-predicted. It is not clear whether this is an inherent weakness in the theoretical damping model used or whether the boundaries of the soil container have minimized radiation damping in the system. It was found necessary to reduce the damping coefficients given Chapter 6. Single Pile Response to Earthquake Excitation 434 in Table 6.6 by 25 percent to predict the peak steady state amplitudes of pile response measured during the test. Computed amplitudes of pile response were somewhat larger than measured quantities during the ramp-up phase of excitation used in the test. Pile response was matched during steady state shaking and during the ramp-down phase of excitation using the modified damping coefficients. Computed pile response for shake table test 23 using random earthquake shaking is also seen to be in excellent agreement with measured response (Figure 6.17). The analysis suggests that where the ground motions are not dominated by frequencies which match the fundamental frequency of the pile, potential inaccuracies in damping or stiffness coefficients used in the analysis do not significantly affect the computed results. The comparisons show that an uncoupled analysis can give high quality predictions of pile response provided reasonable estimates of foundation stiffness and damping are made. The accuracy of the computed foundation compliances is particularly important where the free field ground motions have a dominant frequency component which matches the fundamental frequency of the pile. An uncoupled analysis is a computationally more efficient procedure than using a fully coupled solution. However, an accurate determina-tion of free field surface motions prior to an actual earthquake event is necessary in order to obtain meaningful estimates of pile response. Predictions of pile response using foundation stiffnesses derived from the A.P.I, p-y curves were seen to be very good for the one test case examined. The sensitivity of the computed pile response to the form of the p-y curves used is examined in greater detail subsequently using a fully coupled analysis of pile response. Chapter 6. Single Pile Response to Earthquake Excitation 435 Figure 6.13: Computed pile response using exp't. pile head stiffnesses versus measured pile response - C .T . 41 (a) pile head accelerations (b) pile head displacements (c) shear force at soil surface (d) bending moment at soil surface Chapter 6. Single Pile Response to Earthquake Excitation 436 Figure 6.14: Computed pile response using exp't. pile head stiffnesses versus measured pile response - C . T . 12 (a) pile head accelerations (b) pile head displacements (c) shear force at soil surface (d) bending moment at soil surface Chapter 6. Single Pile Response to Earthquake Excitation 437 Figure 6.15: Computed pile response using A.P.I, pile head stiffnesses versus measured pile response - C . T . 12 (a) pile head accelerations (b) pile head displacements (c) shear force at soil surface (d) bending moment at soil surface Chapter 6. Single Pile Response to Earthquake Excitation 438 (a) (b) (c) (d) < - 2 . 5 -3000 5000 Time (msec) 3000 5000 Time (msec) 3000 5000 Time (msec) E E l z o 2 Time (msec) Figure 6.16: Computed pile response using exp't. pile head stiffnesses versus measured pile response - S.T. 14 (a) pile head accelerations (b) pile head displacements (c) shear force at soil surface (d) bending moment at soil surface Chapter 6. Single Pile Response to Earthquake Excitation (a) 439 (b) (c) (d) 600 -t 400 200 0 c » E o cn -200-c 3000 5000 Time (msec) 3000 5000 Time (msec) 3000 5000 Time (msec) nMIfJlkllii Exp't. Computed * *M ^pi(1||fl| 3000 5000 Time (msec) Figure 6.17: Computed pile response using exp't. pile head stiffnesses versus measured pile response - S.T. 23 (a) pile head accelerations (b) pile head displacements (c) shear force at soil surface (d) bending moment at soil surface Chapter 6. Single Pile Response to Earthquake Excitation 440 6.3 Coupled Dynamic Pile Analysis 6.3.1 Introduction One aspect of the seismic design of a pile supported structure that may be important is the effect of coupling of the pile to the supporting foundation since the effects of seismic ground motion on bending along the pile are taken into account. The coupled solution used in the present study was developed by Matlock et al (1978a) and is embodied in the commercially available computer program SPASM8 (Single Pile Analysis with Support Motion). The mainframe version of the program originally received was downloaded to an IBM P C - A T computer and adjustments made to various array sizes to permit specification of larger amounts of free field motion input. A restart facility present in the original program was also used, where necessary, to permit analysis in stages where the length of the shaking time history was in excess of program capabilities. The revised program was found to require approximately 580 Kbytes of R A M for execution and generally required 20 to 30 minutes of run time to complete a problem involving 1500 time steps and about 100 beam elements. The predictive capabilities of SPASM8 have not been extensively checked against full scale test measurements since there are limited data available on fully instrumented pile foundations during earthquake shaking. For this reason, the program has been used to model selected pile shaking tests carried out on the Caltech centrifuge and the U B C shake table. Three centrifuge tests have been considered; one involving low level sinu-soidal excitation (centrifuge test 41) and two using moderately strong earthquake shaking (centrifuge tests 12 and 15) carried out using loose and dense sand foundations. Cen-trifuge test 41 was examined to determine whether linear elastic models of soil response could be used to simulate soil-pile interaction during low level shaking. Tests 12 and Chapter 6. Single Pile Response to Earthquake Excitation 441 15 provided a check of the program's capabilities to simulate pile response when sig-nificant soil non-linearity occurred in the near field soil response. A back analysis of shake table test 23 was also carried out to determine whether shake table test data could be used to provide a data base against which dynamic pile analysis could be checked. The soil-pile interaction during shaking was simulated using cyclic p-y curves derived from the experimental measurements and from procedures recommended by the Ameri-can Petroleum Institute (1979). In this way, the sensitivity of computed pile response to potential inaccuracies in the p-y curves used was examined. 6.3.2 Methodology Using SPASM8, the pile is modelled as a linearly elastic beam column incorporating the flexural rigidity of the pile, EI, and the effects of spatially varying axial load, P(z), on pile bending (P — A effects) during lateral seismic loading. Simplified superstructure effects can be simulated by increased stiffness along the pile member within the structural system and by rotational constraints at appropriate joints. The structural model used in the SPASM8 analysis is shown in Figure 6.18. Interaction between the near field soil and the pile during shaking is represented using non-linear lateral springs ka placed along the length of the pile. Equivalent viscous dashpots are also placed in parallel with the near field springs to simulate the radiation of P and S waves away from the pile (Figure 6.18). Time varying free field displacements, which are varied along the length of the pile, are applied to the free field end of the spring-dashpot assemblage and represent the input seismic excitation applied to the pile. The input ground displacements have been computed using the free field response analysis, S H A K E . The S H A K E analysis has also been used to compute equivalent elastic shear moduli Gjj from which shear wave velocities have been computed. Radiation damping coefficients were then computed from the shear wave velocity distribution using equations Chapter 6. Single Pile Response to Earthquake Excitation 442 (a) (b) RIGID CYLINDRICAL MASS E O cr) W T S NODAL STATION NOOAL STATION j INTERACTION-E L E M E N T 0.57mm DIA.— S T E E L PILE ^ U F F < t ) | J g ^ O F F < < > i — STATION 13 SOIL SURFACE INPUT F R E E FIELD > DISPLACEMENTS AT E A C H STATION. STATION 97 RIGID CYLINDRICAL MASS z NODAL STATION i — j U F F < \" i 1^ NODAL STATION j JJ|R)_^ U F F ° ' i INTERACTION— ELEMENT 6.35mm DIA. -ALUMINUM PILE — STATION 4 SOIL SURFACE INPUT FREE FIELD DISPLACEMENTS AT EACH STATION. O 20mm S C A L E NONUNfAR p - 7 ntSPONSE / v 2 P , | V p ' V , | 8 COKNCCICO TO I R H - H U O SOU O l S P l A C E M t K T . T| ABSOLUTE RCKRCNCE AXIS INTERACTION ELEMENT Figure 6.18: Structural model used in SPASM8 analysis (a) centrifuge tests (b) shake table tests Chapter 6. Single Pile Response to Earthquake Excitation 443 6.3 and 6.4. The dynamic pile analysis assumes that the radiation damping coefficients remain constant during shaking. The spring stiffnesses k3 for a particular pile deflection level are defined from non-linear p-y curves. A backbone p-y curve f(y) is defined by the program user in the (+p,+y) and (-p,-y) quadrants. Unloading p-y response at a peak loading point (pmaxi Dmax) during shaking is simulated by following a path in p-y space that mimics the initial back-bone response specified using a function f(y — ymax)- This differs from pure Masing response described in section 4.6 where an unloading function of the form f(y — y m ax / 2 ) is used. The unloading criteria used in SPASM result in a smaller estimate of-near field hysteretic damping than using the Masing criteria. The p-y curves during initial loading are represented by a parallel series of sub-elements consisting of springs and frictional Coulomb sliders (Matlock et al, 1978a,b). The groups of sub-elements at each point along the pile are assumed to be unaffected by each other under deformation, thereby simulating a Winkler foundation. As lateral pile deflections increase, the spring sub-elements progressively detach at specific deflection levels, resulting in a gradual decrease in lateral soil stiffness as soil resistance builds. At a deflection level corresponding to the ultimate lateral soil resistance, all springs have detached leaving only an ultimate soil resistance equal to the combined sum of resistances from the Coulomb sliders. During initial unloading, all sub-elements are re-activated and gradually detach with further unloading. The frictional resistances represented by the slider sub-elements are assigned a negative value during unloading, resulting in a gradually decreasing soil resistance as unloading proceeds. The near field soil model can also be used to simulate the hardening of soil resis-tance with number of cycles of shaking, which was observed during shaking tests in loose dry sand. The Coulomb resistances are increased by a percentage of the ultimate lat-eral resistance during each complete load reversal up to a specified maximum value. To Chapter 6. Single Pile Response to Earthquake Excitation 444 simulate the formation of gaps between the pile and the soil, which occurs during shak-ing of piles in saturated sands (Barton, 1982) and cohesive soils (Bogard and Matlock, 1983), a number of the mechanical sub- elements may also be specified as gap elements. During unloading or reloading when a p-y hysteresis loop crosses the p = 0 axis, the sub-elements designated as gap elements are nullified. This results in increasing (or de-creasing) deflection along the p = 0 axis, thereby simulating the presence of a gap. Once computed deflections are large enough such that these exceed the deflections at which the remaining sub-elements are activated, the p-y loop follows the path defined by the sub-element spring stiffnesses and Coulomb resistances. Since the experimental p-y data do not indicate the formation of gaps during pile shaking, this feature of the analysis was not employed. The governing differential equation of motion solved using SPASM8 is given as myr + Ely™ + P(z)y\" + kayr + Cryr = kaur + c>ur - mub (6-H) where yT is the pile deflection relative to the moving base, ub is the input base acceleration, uT are the free field ground displacements relative to the moving base, ka is the lateral stiffness of the near field soil, and Cr are the radiation damping coefficients per unit length of pile. It is noted that the above time dependent variables also vary along the length of the pile. From the right hand side of equation 6.11, it can be seen that the pile is driven by forces due to the relative free field ground motions ur and added inertia forces due to the input base accelerations ub. The above equation is solved using an implicit (Crank-Nicolson) finite difference procedure which marches through time at time step Ar (Matlock et al, 1978a). For the particular step by step integration procedure used, the lateral soil stiffness ka represents a tangent value of stiffness at a particular deflection point along the p-y hysteresis loop. During each time step, a number of iterations are performed to ensure compatibility between computed relative deflections between the Chapter 6. Single Pile Response to Earthquake Excitation 445 pile and the free field (yr — ur) and the tangent soil stiffness ks defined from the p-y hysteresis loop. The steps used in each analysis are described as follows: 1. Carry out a free field response analysis (SHAKE) using the measured time history of base accelerations as input. 2. Compute the time history of shear strains and equivalent elastic moduli compatible with effective strain levels for each sub-layer used in S H A K E . 3. For each time step used in the S H A K E analysis, integrate the computed shear strain distribution to obtain relative free field displacements ur(z, i) which are used directly as input into SPASM8. 4. From the measured time history of input base accelerations, compute pile iner-tia forces m« 0 for each pile increment (station). This represents a time varying, distributed lateral load applied to the pile. 5. Compute appropriate backbone p-y curves and radiation damping coefficients at various depths along the pile. Note that 'p' represents the soil resistance in units of force for each pile station, and not force per unit length used in most p-y curve specifications. Similarly, the damping coefficients computed using the equations of Gazetas and Dobry or other suitable estimates are multiplied by the beam incre-ment length to give units of force over velocity per beam station. 6. Compute all mass and flexural stiffness properties of the model pile and structural mass. 7. Select an appropriate beam increment length and time step interval to be used in the analysis that will ensure convergence of the solution for each time step within Chapter 6. Single Pile Response to Earthquake Excitation 446 a reasonable number of iterations. It is noted that specification of too large a time step can lead to spurious oscillations in the dynamic solution. A time step interval equal to twice the sampling interval used in the centrifuge experiments (0.0192 sec at prototype scale) and a 0.15 m (6 in) beam increment length was found to give acceptable results. A time step equal to 0.0033 sec was used to model shake table test 23, which equals the sampling interval used in the test. A beam increment length of 12.7 mm (0.5 in) was also used. For each analysis, the time history of bending moment and shear force at the soil surface, and displacement at the top of the pile head mass relative to the moving base of the soil container were computed for comparison with experimental measurements. Results of the free field response and SPASM8 analyses are described subsequently. 6.3.3 Results of Analysis The time histories of free field surface acceleration computed using S H A K E are shown for centrifuge tests 41 and 15 and shake table test 23 in Figures 6.19a, 6.20a and 6.21a, respectively. Similar data have been presented previously for centrifuge test 12 in Figure 6.12. The computed accelerations may be compared against measured surface accelerations shown in Figures 6.19b, 6.20b and 6.21b, respectively. Agreement be-tween measured and computed acceleration is seen to be acceptable. Computed free field displacements relative to the input base motion versus time are shown at the mid-depth and surface of the sand foundation for each test in Figures 6.22 to 6.25. The increasing free field displacement with distance above the base of the sand is readily apparent as is the decreasing amplitudes of ground displacement with decreasing intensities of base motion and increasing sand density. A series of preliminary analyses were next carried out for centrifuge test 12 to examine Chapter 6. Single Pile Response to Earthquake Excitation 447 Figure 6.19: Free field surface accelerations computed using S H A K E versus measured accelerations - centrifuge test 41 (a) computed (b) measured Chapter 6. Single Pile Response to Earthquake Excitation 448 Figure 6.20: Free field surface accelerations computed using S H A K E versus measured accelerations - centrifuge test 15 (a) computed (b) measured Chapter 6. Single Pile Response to Earthquake Excitation 449 Figure 6.21: Free field surface accelerations computed using S H A K E versus measured accelerations - shake table test 23 (a) computed (b) measured Chapter 6. Single Pile Response to Earthquake Excitation 450 Figure 6.22: Relative free field displacements computed using S H A K E -41 (a) mid-depth of sand layer (b) top of sand layer centrifuge test Chapter 6. Single Pile Response to Earthquake Excitation 451 (a) 10 5 -51 - i o I I | • . . . | • . . . | 0 5 10 15 20 25 30 Time (sec) (b) lo-i 1 l 1 1 1 i • i i i i , 0 5 10 15 20 25 30 Time (sec) Figure 6.23: Relative free field displacements computed using S H A K E - centrifuge test 15 (a) mid-depth of sand layer (b) top of sand layer Chapter 6. Single Pile Response to Earthquake Excitation 452 Figure 6.24: Relative free field displacements computed using S H A K E - centrifuge test 12 (a) mid-depth of sand layer (b) top of sand layer Chapter 6. Single Pile Response to Earthquake Excitation 453 Figure 6.25: Relative free field displacements computed using S H A K E - shake table test 23 (a) mid-depth of sand layer (b) top of sand layer Chapter 6. Single Pile Response to Earthquake Excitation 454 the influence of the radiation damping coefficients used in the dynamic pile analysis. Coefficients were calculated using both the equations of Gazetas and Dobry and the frequency independent equation of Lysmer and Richart (see section 6.2.2). Radiation damping coefficients were computed at different depths based on effective shear wave velocity distributions determined using S H A K E . When using the equations of Gazetas and Dobry, an input frequency of 1 Hz was selected for the calculations corresponding to the fundamental frequency of the pile and the dominant frequency present in the measured pile response. The calculations showed that radiation damping coefficients computed using the Lysmer approach were about one half those computed using the equations of Gazetas and Dobry for the input frequency examined. Using p-y curves derived using the A.P.I, procedures for loose sand, the pile response to the input ground motions was determined for each set of damping parameters used. The computed shear force at the soil surface is shown in Figure 6.26 for each analysis. Differences in computed pile response are seen to be small. It was therefore decided to use the equations of Gazetas and Dobry for future analysis since both sets of damping coefficients produced similar results. Various researchers (Gazetas and Dobry, 1984; Nogami and Novak, 1977) have sug-gested that radiation damping only exists when pile excitation frequencies exceed the fundamental frequency of the free field soil (/ > / n ) . In the present case, this cor-responds to a frequency of about 2.7 Hz which lies above the majority of frequencies contained in the measured pile response. Therefore, an analysis was also carried out in which radiation damping was arbitrarily set to l/100th of previously computed values to simulate a zero radiation damping condition. The computed pile response was found to be markedly in excess of measured response and indicates that radiation damping is important when / < / „ , despite the presence of potentially reflective sand container boundaries. Therefore, radiation damping was included in all further analyses. Chapter 6. Single Pile Response to Earthquake Excitation 455 Figure 6.26: Computed shear force at the soil surface (cent, test 12) using radiation damping coefficients computed using the equations of Gazetas and Dobry (1984) and Lysmer and Richart (1966). Chapter 6. Single Pile Response to Earthquake Excitation 456 Analyses were carried out to determine the influence of the form of the input backbone p-y curves on pile response during all tests examined. P-y curves were computed using both the A.P.I, recommendations for loose and dense dry sand and using the Ramberg-Osgood model described by equation 4.20. The parameters used to describe the backbone p-y curves based on the Ramberg- Osgood model have been given in Table 4.8 for cen-trifuge tests 12 and 15 and Table 5.10 for shake table test 23. The p-y data derived from centrifuge test 41 has shown a near linear, non- hysteretic soil response. Therefore, the p-y curves used in the analysis were modelled assuming linear elastic behaviour using a square root of depth Winkler modulus distribution. P-y curves were computed using each procedure over a range of depths up to the 21 pile diameter depth for the centrifuge tests and up to the 25 pile diameter depth for shake table test 23. Computed shear force and bending moment at the soil surface using both sets of p-y curves are shown for all four tests in Figures 6.3.3 to 6.43 and are compared with measured response. An examination of the figures for centrifuge tests 12 and 41 and shake table test 23 shows that use of the two different sets of backbone p-y curves produces relatively small changes in computed surface response. Similar conclusions have been drawn from the results of the uncoupled dynamic response analysis described in section 6.2. A somewhat greater sensitivity to the type of p-y curve input into the dynamic analysis was seen for centrifuge test 15, but in general the computed surface response was not overly sensitive to the form of the p-y curve input. While the computed pile response at the soil surface was found to be relatively insensitive to the form of the p-y curves used in the analysis, the computed location and value of maximum bending moment at depth along the pile was dependent on the soil stiffness distribution used. The differences in computed maximum bending moment are described subsequently since this is often of most interest in pile design. Chapter 6. Single Pile Response to Earthquake Excitation 457 The frequency content of the computed response is in good agreement with that mea-sured. However, major discrepancies often exist between computed maximum amplitudes of pile response and measured response at peak pile deflection. The computed maximum bending moment and shear force at the soil surface, the maximum bending moment at depth along the pile and the depth of maximum bending are summarized in Table 6.7 for each analysis carried out. The computed results are compared to measured quantities and show that the SPASM8 analysis generally leads to an overly stiff, underprediction of peak pile response. Depths of maximum pile bending are similarly underestimated. This observation is further substantiated by comparing computed pile head displacements at the top of the structural mass with measured quantities for centrifuge test 12 in Figures 6.30 and 6.31. The calculations were carried out using both sets of backbone p-y curves and show that computed displacments are substantially less than measured quantities. Similar results were seen for all tests examined but these results have not been plotted. 6.3.4 Summary The use of a fully coupled dynamic analysis, while theoretically more rigorous than us-ing an uncoupled approach, is computationally more difficult and relies on estimates of input parameters that are difficult to evaluate precisely. Experience gained in using the program suggests that the most important input to the analysis is the time history of free field displacement. It is important to match the computed frequency content and amplitude of ground motions with the earthquake motions anticipated in order to achieve meaningful estimates of pile response. The SPASM8 results show that the com-puted time history of pile response is in good agreement with measured time histories, suggesting that the frequency content of the input free field displacements are reason-able. The fact that computed peak amplitudes of pile response are less than measured may suggest that the S H A K E analysis has underestimated the amplitudes of the input Chapter 6. Single Pile Response to Earthquake Excitation 458 Test No. P-y Curve Type Mo 7 t-'max V0 C . T . 41 A.P.I. 79.6 112.0 1.07 40.0 C . T . 41 Exp't. 79.8 (120.0) 101.9 (172.0) 1.37 (2.0) 40.5 (52.5) C . T . 15 A.P.I. 213.6 369.5 1.68 147.3 C.T . 15 Exp't. 157.1 (270.0) 257.6 (460.0) 1.52 (2.0) 110.9 (150.0) C . T . 12 A.P.I. 235.0 306.2 1.37 115.5 C.T . 12 Exp't. 263.3 (230.0) 332.2 (335.0) 1.07 (2.28) 129.5 (100.0) S.T. 23 A.P.I. 376.7 479.0 38.1 7.3 S.T. 23 Exp't. 368.3 (380.0) 471.0 (908.0) 38.1 (100.0) 7.1 (8.0) Note: All bending moment and shear force quantities for the centrifuge tests are expressed in kN-m and kN units, respectively. All bending moment and shear force quantities for the shake table tests are expressed in N-mm and N units, respectively. Mo = maximum bending moment at the soil surface. Vo = maximum shear force at the soil surface. Mmax = maximum bending moment at depth along the pile. Zmax — depth below the soil surface of maximum bending moment. Figures in parantheses refer to experimentally measured quantities. Table 6.7: Computed Versus Measured Pile Response Chapter 6. Single Pile Response to Earthquake Excitation 459 200 100 v_ O OJ JO CO -100--200-10 15 Time (sec) Figure 6.27: Computed shear force at soil surface using A.P.I, p-y curves and comparison with measured response - centrifuge test 12 I z c OJ CD c c 0J m 400 200-= -200 -400 Time (sec) Figure 6.28: Computed bending moment at soil surface using A.P.I, p-y curves and comparison with measured response - centrifuge test 12 Chapter 6. Single Pile Response to Earthquake Excitation 460 CD O D OJ S CO 200-1 100--100--200 Time (sec) Figure 6.29: Computed shear force at soil surface using experimentally derived p-y curves and comparison with measured response - centrifuge test 12 i c CD CD c c CD CQ 400 200-Fi -200 -400 15 Time (sec) Figure 6.30: Computed bending moment at soil surface using experimentally derived p-y curves and comparison with measured response - centrifuge test 12 Chapter 6. Single Pile Response to Earthquake Excitation 461 Time (sec) Figure 6.31: Computed displacement at the top of the structural mass using A.P.I, p-y curves and comparison with measured response - centrifuge test 12 ground motion. This, in fact, would be expected in using an equivalent linear analysis. However, the magnitude of the ground displacements cannot be evaluated directly from the experimental measurements and must be inferred from a matching of computed free field surface accelerations with those measured. The computed pile response at the soil surface was found to be relatively insensitive to the exact details of the p-y curves and small variations in the magnitude of radiation damping coefficients input into SPASM8. Pile bending response at depth was, however, very dependent on the lateral soil stiffness distribution used. In summary, the uncertainties of input into a fully coupled analysis would appear to outweigh its advantages. The parameters necessary for an uncoupled analysis are easier to evaluate using cyclic pile head loading tests and measured or computed free field surface accelerations, and would appear to give more accurate results than using a Figure 6.32: Computed displacement at the top of the structural mass using experi-mentally derived p-y curves and comparison with measured response - centrifuge test 12 Chapter 6. Single Pile Response to Earthquake Excitation 463 o D CD JC CO 15 20 25 Time (sec) 40 Figure 6.33: Computed shear force at soil surface using A.P.I, p-y curves and comparison with measured response - centrifuge test 41 150 100-Exp't. Computed-Run 1 * * Time (sec) Figure 6.34: Computed bending moment at soil surface using A.P.I, p-y curves and comparison with measured response - centrifuge test 41 Chapter 6. Single Pile Response to Earthquake Excitation 464 CD O OJ - C C O Time (sec) Figure 6.35: Computed shear force at soil surface using experimentally derived p-y curves and comparison with measured response - centrifuge test 41 150 Time (sec) Figure 6.36: Computed bending moment at soil surface using experimentally derived p-y curves and comparison with measured response - centrifuge test 41 Chapter 6. Single Pile Response to Earthquake Excitation 465 Figure 6.37: Computed shear force at soil surface using A.P.I, p-y curves and comparison with measured response - centrifuge test 15 fully coupled approach. One major exception to this philosophy would be if pile bending response is not dominated by structural inertia forces, as is the case in the present study, but instead by sharp transitions in free field ground displacements. The latter could be expected where one region of the ground is softened, for example by a liquefaction failure, and translates significantly relative to a stiffer, unliquefied layer. In such a case, it is considered necessary to properly account for these large free field motions and a fully coupled analysis should be undertaken. Chapter 6. Single Pile Response to Earthquake Excitation 466 c OJ c TD C OJ m 4 0 0 2 0 0 -f i - 2 0 0 - 4 0 0 -15 20 Time (sec) 30 Figure 6.38: Computed bending moment at soil surface using A.P.I, p-y curves and comparison with measured response - centrifuge test 15 CD o o 2 (rad/sec) 0J3 (rad/sec) P G D Y N A - no lumped masses M A C E - no lumped masses 586.2 580.8 588.1 590.7 5666.64 5091.8 P G D Y N A - with lumped masses M A C E - with lumped masses 164.4 157.1 206.9 178.6 297.5 392.7 Table 7.1: Computed Natural Frequencies - P G D Y N A vs. M A C E Solution and foundation spring elements employed are very similar to those computed by M A C E , thereby providing a check on the accuracy of the matrix assemblage routines employed. When the mass properties of the lumped masses attached to the structural frame were included in both analyses, differences in computed first mode frequency of about 5 percent existed. Greater discrepancies existed for the second and third modes of vibration. As a check on the accuracy of the eigenvalue solution employed in P G D Y N A , a check was made that the eigenvalue equation det {[K] — u>2[M}) was satisfied for u = u>i and ui = u>2-This equation was found to be satisfied, demonstrating the accuracy of the eigenvalue solution employed but suggesting that there are slight differences between the form of the lumped mass matrix used in M A C E and P G D Y N A . Agreement between the computed natural frequencies was considered sufficiently close for the first two modes of vibration, suggesting that the results from P G D Y N A were of acceptable accuracy. To check the effects of pile group interaction on computed natural frequencies of the two pile system described above, a sample earthquake record (see Figure 7.2) was used as an input base motion in P G D Y N A . The foundation spring stiffnesses were assumed to have a maximum stiffness (&,j)o at approximately zero pile deflection and to have a semi-logarithmic variation versus pile head deflection, of the form described by equation 6.2. Thus, at a lateral deflection uref the lateral pile head stiffnesses reduce to (fc,j)rey. Chapter 7. Uncoupled Dynamic Solution for a Pile Group 474 LUMPED MASSES - B E A M 1 ELEMENTS UgCO 10 mm —I G L O B A L COORDINATE S Y S T E M - INPUT FREE FIELD SURFACE MOTIONS AT EACH FOUNDATION NODE POINT Kwwf Ku<\" N O N - L I N E A R FOUNDATION SPRINGS (ALSO EQUIVALENT VISCOUS DASHPOTS NOT SHOWN). BegmEtement Properties : (elements 1 and 3) • £7« = EI., = 4.65 x 10« N - m m ' • EA = 1.216 x 10\" N • GJ = 3.87 x 10\" N - m m 1 • GA = 5.06 x 10 s N • ra = 5.0 x 1 0 _ \" N - s e c ! / m m 2 • n = shear r igidity coefficient = 1.334 Note: For beam element 2 which represents a rigid pi le cap, assume flexural, shear and axia l r igidity are 100 times those of beam elements 1 and 3. LumpedMassProperlies : (per mass) • mo = 6.82 x 1 0 \" 1 N - s e c 2 / m m • = / „ = 0.45 N-sec ' -mm • / „ = 0.90 N-sec ' -mm • Zv - 29.2 m m FoundationSpringStif fnesses (i) Linear E last ic Case • k,% = 36.2 N / m m ' • Jt„„ = 20.0 N / m m • ~ 0 N / r a d • k t t = 80,000 N - m m / r a d (ii) Non-L inear Case j • (fc«„)o = 36.2 N / m m , = 12.0 N / m m , u„t = 0.05 m m • (*ww)o = 20.0 N / m m , = 8.0 N / m m , w„, = 0.05 m m • (***)o = 80,000 N - m m / r a d , (***)„/ = 50,000 N - m m / r a d , u„, = 0.05 m m • K* ~ 0 N / r a d EquivalentViscous Dashpots (assumed constant) • c\\,u = 0.1 N -sec /mm (hysteretic plus radiation damping) • Cvj, = 2.5 N-sec / rad (ditto) • = 80.0 N-mm-sec / rad (ditto) • c™ = 1.0 N -sec /mm (ditto) Figure 7.1: Plane frame structural model for two pile group subjected to inline shaking on the shake table Chapter 7. Uncoupled Dynamic Solution for a Pile Group 475 Method of Analysis Time Step No. C J 2 (At = 0.0033 sec) (rad/sec) (rad/sec) P G D Y N A - no interaction 2 164.4 206.8 10 164.3 204.6 484 108.3 163.0 500 141.9 163.5 P G D Y N A - with interaction 2 164.4 206.5 r> 10 164.2 201.7 484 80.1 162.4 V 500 102.5 163.2 Table 7.2: Computed Natural Frequencies With and Without Group Interaction Effects - P G D Y N A Similarly, the axial compression and tension stiffnesses (assumed equal for this exam-ple) have a small deflection stiffness (kw)o and a reference stiffness (kw)Tef at a reference axial deflection of wref. The non-linear pile head stiffness parameters used in this sam-ple analysis are presented in Figure 7.1. Deflection independent damping coefficients representing hysteretic and radiative energy losses were also selected, as shown in the figure. Computed first and second mode natural frequencies are presented in Table 7.2 at various time steps during the dynamic analysis for the cases with and without the effects of pile group interaction. It is noted that the 484th time step represents a point of maximum pile head deflection. The computed natural frequencies are seen to vary during shaking, as expected, with the stiffest response occurring at the start of shaking during, low amplitude vibration and the softest response during larger amplitude vibration. The effects of pile group interaction using the modified interaction factors presented in chapter 4 are seen to result in a significant reduction in computed first mode frequency for a particular time step due to the increased pile head deflection level. The effects of group interaction are generally less pronounced for higher mode vibration. Chapter 7. Uncoupled Dynamic Solution for a Pile Group 476 Time (sec) Figure 7.2: Sample earthquake record used as input into P G D Y N A 7.3 Dynamic Analysis Results 7.3.1 Shake Table Test - Four Pile Group Subjected to Strong Sinusoidal Shaking A dynamic analysis of the response of a four pile group embedded in dense sand and shaken using strong sinusoidal shaking on the U B C shake table (see section 5.6.4) has been carried out using P G D Y N A . Using pile head stiffness and damping parameters derived from single pile test 34 carried out using similar shaking intensities as the four pile test, an attempt has been made to predict the pile group response based on the single pile data. The methodologies used to compute the single pile head stiffnesses have been de-scribed in chapter 6 and are based on fitting the observed bending moment distribution along the pile at a particular instant in time using an appropriate Winkler modulus distri-bution. For single pile test 34, a linear with depth modulus distribution using = 0.020 Chapter 7. Uncoupled Dynamic Solution for a Pile Group 477 N/mm 2 was found to provide a reasonable match of the measured bending moment dis-tribution at peak pile deflection. With this modulus distribution, the static pile head stiffnesses were computed giving kuu = 48 N/mm, ku^, = 2089 N/rad and k^ = 147, 700 N-mm/rad. These static pile head stiffnesses have been found to be reasonably inde-pendent of frequency in the frequency range of interest to the pile group test, based on analyses carried out using the computer program PILAY. The single pile stiffnesses have been found to vary, with pile deflection amplitude; however, since peak pile group re-sponse was of most interest for steady state sinusoidal excitation, a decision was made to use constant stiffness values determined when the single pile deflection achieved a peak amplitude. Dynamic analyses of the free field soil response to strong sinusoidal shaking using the computer program S H A K E have shown that relatively small shear strains are induced in the free field. Thus, low strain shear moduli measured using bender element tech-niques have been input into PILAY to provide estimates of pile head radiation damping coefficients. The radiation damping coefficients for horizontal and vertical modes of vi-bration were computed as cruu = 0.091 N-sec/mm, = 0.91 N-sec/rad, c^ = 12.8 N-mm-sec/rad, and crww = 0.49 N-sec/mm. Hysteretic dashpot coefficients accounting for material damping in the soil around the pile have also been estimated using PILAY and methodologies described in chapter 4. Hysteretic damping ratios at various depths along the pile have been estimated from average strain levels around the pile at peak lateral pile deflection (see Appendix C). Once the effective strain level was computed for a particular depth, and with estimates of the minimum and maximum damping ratio for the sand, the hysteretic damping ratio was computed at various depths along the pile using equation 4.12 for use in the PILAY analysis. The computed distribution of hysteretic damping ratio versus depth, D(z), varied from 18 per cent near the head of the pile to a small strain value of 10 percent Chapter 7. Uncoupled Dynamic Solution for a Pile Group 478 near the pile tip. A shear modulus distribution G(z) was computed for use in the PILAY analysis to provide pile head static stiffnesses identical to those computed using the linear with depth Winkler model. With estimates of G(z) and D(z), the pile head hysteretic damping coefficients were computed. These were found to have values of c£ u = 0.003 N-sec/mm, = 0.08 N-sec/rad, c^, = 2.8 N-mm-sec/rad and c^w -- 0.02 N-sec/mm. These are seen to have values generally less than 10 percent of the corresponding radiation damping coefficients. The variation of axial compression and tension stiffness versus axial pile deflection was found to have a significant effect on the dynamic response of the pile group. This is not surprising since the axial stiffness of each pile in the group, modified appropriately for pile group interaction, influences the rocking stiffness of the group. The single pile axial push-pull data described in chapter 5 (see Figure 5.77) was used to develop estimates of secant axial stiffness versus axial pile movement. The initial axial stiffness of the single pile for both tensional and compressional loading was estimated to be 600 N/mm from the test data. The ultimate compressional and tensional capacity of the pile was estimated to be 40 N and 25 N, respectively, appropriate for 10 to 30 load cycles. It is significant to note that under constant amplitude cycling, there was a continual reduction in ultimate axial capacity. Under several hundred cycles of shaking it might be expected that the axial stiffness of the pile, coupled with the breaking of shaft friction due to vibration of the sand bed, would reduce to substantially lower values. It has been assumed that the axial load (Pax) versus displacement (w) curve may be modelled using a hyberbolic curve given by the following equation: where wuit is the axial deflection at which the ultimate axial capacity in tension or compression is developed. Based on Figure 5.77, wuit has been estimated to have a value Chapter 7. Uncoupled Dynamic Solution for a Pile Group 479 of 0.15 mm. Values of secant axial stiffness versus axial deflection have been derived from equation 7.1 using kw — Pax/w and values of Puit appropriate to the compression or tension loading case. Values of log kwc for compression loading and log kwt for tension loading have been plotted versus axial deflection w on a semi-logarithmic scale, from which a linear equation between log kw and w for both tension and compression loading has been developed. This is identical to the lateral stiffness variation specified by equation 6.2. For purposes of the present analysis, small deflection axial stiffnesses of 600 N/mm have been assumed equal for tension and compression loading. The tensional stiffness was assumed to reduce in a semi-logarithmic fashion to a value of 251 N/mm at an axial uplift of 0.08 mm. Similarly, the compressional stiffness was assumed to have a value of 380 N/mm at an axial deflection of 0.07 mm. Using the structural model, and beam element and lumped mass properties presented in Figure 7.3, the dynamic response of the 4-pile group to the measured free field surface motions (see Figure 5.73) has been computed. The first mode frequency of the group at peak pile deflection has been computed to be 33.2 Hz. It is noted that this is substantially larger than the value of approximately 9 Hz interpolated from the natural frequency curves presented in Figure 5.35 for strong sinusoidal shaking of 2-pile groups shaken in an inline direction. In order to match the measured natural frequency, it was found necessary to reduce the single pile axial stiffnesses specified in the PGDYNA analysis to very low values of the order of 2.5 N/mm. This implies that the several hundred cycles of shaking used during the frequency sweep tests reduced the axial stiffness of piles in the group to values substantially less than implied from Figure 5.35 due to progressive reduction in ultimate axial capacity and breakdown of shaft friction. During preliminary runs of PGDYNA it was found necessary to adjust the single pile damping coefficients used in the analysis to control spurious high frequency amplitudes Chapter 7. Uncoupled Dynamic Solution for a Pile Group 480 computed in the analysis. Radiation damping coefficients were increased 50 percent over and above values computed by the P I L A Y analysis. This implies that either the plane strain model of radiation damping used in the P I L A Y model leads to underestimates of radiation damping, or that the use of frequency independent radiation damping coeffi-cients necessary for time domain analysis cannot effectively damp out the low amplitude, high frequency components of the input ground motion. The analysis was also found to be sensitive to the value of rotational pile head stiffness used. To provide a reasonable match of computed pile cap displacement the single pile head rotational stiffnesses were reduced by 26 percent to a value of 110,000 N-mm/rad. This may indicate that conical depressions formed around each pile in the group during shaking have been augmented by interaction between the closely spaced piles in the group. This would cause lateral soil stiffness reduction near the soil surface, which would be expected to mainly affect the rotational pile head stiffness. This would not be seen to the same degree during the single pile tests. Computed time histories of pile cap acceleration at the centre of gravity of mass of the structural mass, pile cap displacements, shear force and bending moment at the soil sur-face averaged over every pile in the group, and the axial force variation for one pile in the group (the axial force variation for an opposing pile in the group was of opposite phase) are presented in Figure 7.4. The computed results are compared to measured quantities. In general, the computed accelerations, shears and bending moments are seen to over-predict the measured response while the axial loads were significantly underpredicted. The measured axial load variation shows considerable variation in peak amplitude with an average peak to peak amplitude of 16 N. Measured pile cap displacements are seen to have significant residual values at the end of shaking due to plastic shakedown of the sand. Since an equivalent elastic dynamic analysis has been used, residual displacements are not computed. Chapter 7. Uncoupled Dynamic Solution for a Pile Group 481 GLOBAL COORDINATE SYSTEM ® LEGEND BEAM TYPE LUMPED MASSES FOUNDATION SPRINGS AND EQUIVALENT VISCOUS DASHPOTS (3 TRANSLATIONAL, 3 ROTATIONAL, 2 CROSS-TERMS). FREE FIELD GROUND ACCELERATIONS APPLIED AS INPUT MOTIONS TO FOUNDATION NODAL POINTS. SCALE BeamElementProperties : (element 1) • EIXX = EI„ = 4.65 x 106 N-mm2 • EA = 1.216 x 106 N • GJ = 3.87 x 106 N-mm2 • GA=5.06 x 10s N • m = 5.0 x 10~8 N-sec2/mm2 • n = shear rigidity coefficient = 1.334 Note: For beam element 2 which represents a rigid pile cap, assume flexural, shear and axial rigidity are 100 times those of beam element 1. LumpedMassProperties : (per mass) • m0 = 8.15 x IO\"1 N-sec2/mm • hz = 0.60 N-sec2-mm • Ixx = 0.35 N-sec2-mm • 7™ = 0.95 N-sec2-mm • Zcg = 29.2 mm Figure 7.3: Structural model for four pile group subjected to inline shaking on the shake table Chapter 7. Uncoupled Dynamic Solution for a Pile Group 482 . COMPUTED VS. MEASURED AVERAGE BASE SHEAR - PGDYNA , , COMPUTED VS. MEASURED AVERAGE B.M. AT SOIL SURFACE - PGDYNA C) SHAKE TABLE TEST 8C (CU SHAKE TABLE TEST 8C Time (msec) Time (msec) COMPUTED VS. MEASURED DYNAMIC AXIAL LOAD - PGDYNA SHAKE TABLE TEST 8C Time (msec) Figure 7.4: P i le group response computed using P G D Y N A versus measured response for a 4-pile group subjected to inline shaking on the shake table Chapter 7. Uncoupled Dynamic Solution for a Pile Group 483 Pile Group Quantity Computed Measured Accel, at C G of Mass (g) 0.90 0.67 Pile cap displacement (mm) 0.8 0.97 Average pile head shear (N) 7.0 5.5 Average pile head moment (N-mm) 250 200 Pile dynamic axial force (N) 7 16 Table 7.3: Computed Vs. Measured Pile Group Response for Strong Sinusoidal Shaking - Four Pile Group on the Shake Table A summary of peak amplitudes of computed response versus measured average quan-tities (computed as peak to peak quantities divided by two) is presented in Table 7.3. The results suggest that use of an uncoupled dynamic analysis, using the measured free field surface motions as the input motions, leads to a conservative overprediction of the bending response of the pile group. 7.3.2 Centrifuge Test - Four Pile Group Subjected to Earthquake Excitation The experimental results from centrifuge test 46, carried out on a four pile group em-bedded in dense sand and shaken using random earthquake excitation have also been analysed using P G D Y N A . Pile head stiffnesses derived from single pile test 15 have been used in the analysis, since this test was carried out using the same input motions as used in the pile group test. A linear with depth modulus distribution using the parameter nh = 4800 k N / m 3 has been used to characterize single pile response at peak pile deflection (see section 4.3.2), giving kuu = 21,600 kN/m, ku^ = 41,000 kN/rad and k^ = 127,000 kN-m/rad. The p-y curves developed from test 15 indicate that the lateral secant soil modulus kh is reasonably linear over the range of shaking intensities examined in the test. Therefore, the values of kij were assumed to be constant during shaking. These static stiffnesses have been found to be relatively independent of frequency within the Chapter 7. Uncoupled Dynamic Solution for a Pile Group 484 frequency range of interest (0 to 5 Hz) based on analyses carried out using the computer program PILAY. Since the bases of the piles rested on the rigid base of the soil container, the com-pressional axial stiffness of a single pile in the group was computed as kwc = EA/l where I is the embedded depth of the pile. Use of the latter equation implicitly neglects sand friction on the pile shaft which may be a reasonable assumption during vibration of the surrounding sand mass. The computations gave kwc =4.7 x 105 kN/m which was as-sumed to remain constant during shaking. This value of axial stiffness was compared to that computed using PILAY based on a shear modulus distribution that provided a match of the above lateral pile head stiffnesses. An axial stiffness value of 5.1 x 105 kN/m was computed over the frequency range of interest, which is in close agreement with the above simplified approach. A compressional stiffness value of 4.7 x 105 kN/m was therefore adopted for subsequent calculations. For tensional uplift loading on a single pile, the ultimate uplift capacity of the pile Tuit was estimated as, where Ko is the at rest earth pressure coefficient set equal to a value of 0.6, 7^ is the dry density of the soil (= 17.1 kN/m3), 6 is the coefficient of sand-pile friction estimated to have a value of 21 degrees, H is the embedded length of the pile (= 11.7 m) and 5\" is the pile perimeter (=1.79 m). Using equation 7.2 and the above parameters, Tun at prototype scale was computed to have a value of 483 kN. Assuming wuit equals 2 percent of the pile diameter and that k{ equals 4.7 x 10s kN/m, equation 7.1 was used to compute the variation of secant tension stiffness kwt versus axial uplift w. A straight line variation of log kwt vs w was passed through the data points, giving a small deflection stiffness value (kwt) of 4.7 x 105 (7.2) Chapter 7. Uncoupled Dynamic Solution for a Pile Group 485 kN/m, reducing to a value of 0.47 x 10s kN/m at a deflection of 0.010 m. Radiation damping coefficients were computed using PILAY based on shear moduli computed using S H A K E during an analysis of free field soil response for centrifuge test 15. The results of the PILAY analysis gave cruu = 900 kN-sec/m, cru1p = 530 kN-sec/rad, C W — 470 kN-m-sec/rad and crww = 2300 kN-sec/m. Hysteretic damping coefficients were computed using a material damping distribution derived from the experimental p-y curves for centrifuge test 15. The damping ratios used in the PILAY analysis were varied from 10 percent near the head of the pile to a value of zero at depths greater than 4.5 m. Using the procedures described in chapter 4, the hysteretic damping coefficients at a frequency of 2 Hz were computed as c\\u = 80 kN-sec/m, = 80 kN-sec/rad, c^, = 110 kN-m-sec/rad and = 50 kN-sec/m. The hysteretic damping coefficients were also assumed to be constant during shaking. Using the structural model shown in Figure 7.5 and the above values of pile head compliance, the dynamic response of the 4-pile group to the measured free field surface motions (see Figure 4.74) has been computed. The first mode frequency of the group has been computed to be 2.55 Hz. This is in good agreement with a value of 2.6 Hz inferred from a frequency analysis of the experimental pile group response. Computed time histories of pile cap acceleration at the pile cap level (where accelerations were measured during the test), displacements at the top of the structural mass, shear force and bending moment at the soil surface averaged over each pile in the group, and the axial force variation for one pile in the group are presented in Figure 7.6, where they are compared with the measured dynamic response. In general, the computed peak accelerations, bending moments and shear forces ex-ceed measured quantities, suggesting that use of unsealed free field motions as input leads to a conservative prediction of pile group response. Alternatively, the radiation damping parameters computed using PILAY may be too low. The computed axial load variation is Chapter 7. Uncoupled Dynamic Solution for a Pile Group 486 Pile Group Quantity Computed Measured Pile cap accel. (g) 0.35 0.24 Structural mass displacement (mm) 57 60 Peak pile head shear (kN) 230 150 Peak pile head moment (kN-m) 250 70 Peak pile dynamic axial force (kN) 575 770 Table 7.4: Computed Vs. Measured Peak Pile Group Response for Moderate Level Earthquake Shaking - Four Pile Group on the Centrifuge seen to be somewhat less than measured response. The reason for this is unclear but may be due to the fact that since the pile tips rested on the rigid base of the soil container, there was a ringing effect as the pile tips alternately impacted and lifted off the container base. The fact that the computed axial loads are underpredicted has resulted in a sig-nificant overestimate of bending moment at the soil surface, as may be expected from principles of moment equilibrium (see equation 5.5). Computed peak displacements at the top of the structural mass are in reasonable agreement with measured quantities. An examination of the computed time histories of response also shows that the frequency content of measured pile group response is adequately captured. A summary of peak amplitudes of computed response versus measured quantities is presented in Table 7.4. 7.4 Summary Based on the analyses presented, an uncoupled dynamic response analysis would appear to provide conservative predictions of measured pile group response, with a tendency to overpredict response due to the neglect of the effects of kinematic interaction and/or underestimates of damping in the soil-pile system. Pile stiffness and damping parameters derived from single pile tests or from appropriate dynamic analyses, pile to pile interaction factors and the free field surface motions are the sole non- structural inputs required for Chapter 7. Uncoupled Dynamic Solution for a Pile Group 487 GLOBAL COORDINATE SYSTEM LEGEND ® BEAM TYPE LUMPED MASSES 6 UJt) Im FOUNDATION SPRINGS AND EQUIVALENT VISCOUS DASHPOTS ( 3 TRANSLATIONAL, 3 ROTATIONAL, 2 CROSS-TERMS). FREE FIELD GROUND ACCELERATIONS APPLIED AS INPUT MOTIONS TO FOUNDATION NODAL POINTS. 1.14 m SCALE BeamElementProperties : (clement 1) • EI„ = EIZI = 172,614 kN-m2 • EA = 5.51 x 106 kN • GJ = 178,815 N-mm2 • GA = 2.20 x 10s kN • m = 3.657 x IO'3 kN-sec2/m2 • n = shear rigidity coefficient = 1.334 Note: For beam element 2 which represents a rigid pile cap, assume flexural, shear and axial rigidity are 100 times those of beam element 1. LumpedMass Properties : (per mass) • m 0 = 56.15 kN-sec2/m • /« = hx = 160.0 kN-sec2-m • Im = 320.0 kN-sec2-m • Zc = 0.96 m Figure 7.5: Structural model for symmetrical four pile group tested on the Caltech cen-trifuge Chapter 7. Uncoupled Dynamic Solution for a Pile Group 488 (a) (b) E 5 E o Time (sec) M«osur«d Computed Time (sec) (c) (d) (e) Time (sec) Figure 7.6: Pile group response computed using P G D Y N A versus measured response for a symmetric 4-pile group subjected to moderate level, earthquake shaking on the centrifuge Chapter 7. Uncoupled Dynamic Solution for a Pile Group 489 the analysis. The principle advantage of an uncoupled analysis is that the single pile compliances are relatively simple to obtain using either full scale field tests, which is the preferred method, or using appropriate numerical analysis. A key assumption in the analysis is that kinematic interaction effects may be ne-glected. Based on elastic analyses presented by Waas and Haartman (1984), such an approach appears reasonable for small pile groups. For larger pile groups, the effects of kinematic interaction can cause a significant reduction in effective motions transmitted to the pile group. Therefore, the use of unsealed free field motions can be expected to lead to a conservative prediction of large pile group response. The program P G D Y N A , while developed to analyse the response of pile groups to earthquake excitation, is easily adapted to the case of dynamic machine loading. In the latter case, the actual loadings applied to the pile group are reliably known or can be determined. For higher frequency machine loadings, the effects of soil inertia or so-called frequency effects, on the pile head compliances should be accounted for in the frequency range of interest. The use of P G D Y N A to analyse machine foundation loadings represents an interesting topic for future research. Chapter 8 Summary and Suggestions for Future Work 8.1 Introduction A series of model tests on single piles and pile groups containing up to four piles per group and embedded in dry sand have been subjected to simulated earthquake shaking. The purpose of the tests was to examine aspects of soil-pile interaction during single pile tests and to obtain a data base against which commonly used procedures to compute single pile response to earthquake excitation could be checked. Two different numerical models used in predicting single pile response to earthquake excitation have been examined in the present study. The pile group tests were carried out primarily to examine the significance of pile to pile interaction under cyclic loading conditions and to assess the capability of currently used elastic interaction factors to compute pile group deflections. Since simple numerical models are not readily available for computing pile group response to earthquake shaking, taking into account group interaction effects and soil non-linearity, an attempt has been made to develop such a model. A limited verification of the model has been carried out by comparing computed pile group response against measured data. A series of model tests were carried out under small scale (1 g) conditions using shaking tables available at the University of British Columbia (UBC). Another series of shaking tests were carried out using a geotechnical centrifuge available at the California Institute of Technology (Caltech). For the latter tests, the model pile(s) and the pre-pared sand foundations were placed in a soil container which was spun at the end of 490 Chapter 8. Summary and Suggestions for Future Work 491 the centrifuge arm, achieving centrifugal accelerations averaging 60 g at the mid-depth of the model. With the increased gravitational field acting on the model piles and sand foundation, stress conditions in the model are representative of full scale (prototype) conditions using a centrifuge scale factor of 60. An electronically controlled hydraulic actuator attached to the underside of the soil container on the centrifuge supplied the horizontal shaking motions to the model piles. The influence of a superstructure on pile response was represented by attaching a structural mass to the pile(s). The pile- superstructure system was then subjected to base motion excitation using both sinusoidal and random earthquake input. The model piles tested on the U B C shake tables were subjected to both low and high level intensities of shaking while low to moderate intensities of shaking were achieved on the Caltech centrifuge. Prior to each shake table and centrifuge test, the small strain shear modulus distri-bution within the sand foundation was measured using piezoceramic bender elements; one bender element acting as a vibrating surface source and a series of buried bender elements acting as receivers at different depths in the sand. The use of bender elements proved particularly useful in the centrifuge environment since the surface source could be excited remotely while in flight on the centrifuge. Receiver response was amplified on the rotating centrifuge arm and then transmitted via electrical slip rings to a high speed digital storage oscilloscipe for storage and processing. The measurement of low strain shear moduli within the sand foundations has been used in subsequent analysis of free field soil response. Knowledge of the low strain moduli also provides an anchor point to assess the degree of strain softening in near field soil response during pile vibration. Chapter 8. Summary and Suggestions for Future Work 492 8.2 Single Pile Test Results The single pile tests were carried out to elucidate characteristics of single pile response to earthquake ground shaking where significant degradation in soil response due to pore water pressure build-up does not occur. Cyclic 'p-y curves' derived from the test data have allowed one to examine the lateral stiffness and damping characteristics of the near field soil, and the dependence of the lateral soil stiffness and damping on initial sand density and the intensities of shaking. The near field soil stiffness was, in turn, related to shear strains computed in close proximity to the pile using equivalent elastic methods of analysis. Since shear strains vary radially and circumferentially from the pile, a strain averaging procedure was developed in which an effective zone of influence over which strains were averaged was estimated based on finite element results. Observations of pile bending and cyclic p-y response observed under low level stress conditions on the shake table were also compared with that observed on the centrifuge to examine stress level dependent differences in lateral soil response. The dynamic response of selected single pile tests were computed using both cou-pled and uncoupled methods of analysis. In the uncoupled analysis, the below ground stiffness properties of the piles were replaced by a, series of deflection dependent, pile head springs. The latter were derived from back analysis of the test data using static, laterally loaded pile models incorporating Winkler models of soil response. Equivalent viscous dashpots representing radiation damping in the far field were computed using the computer program PILAY. The dashpot parameters were based on free field shear wave velocities derived from the one dimensional, wave propagation analysis S H A K E . The hysteretic damping component of the dashpot coefficients were computed from the test data at peak pile deflection using material damping ratios versus depth derived from the experimental p-y curves. Alternatively, the average shear strain distribution was Chapter 8. Summary and Suggestions for Future. Work 493 computed in the near field for a given lateral pile deflection profile using equivalent elas-tic methods of analyses developed for the present study. Having estimated the effective shear strain and from this the equivalent elastic shear modulus for a given depth, the material damping ratio was computed using the method given by Hardin and Drnevich (1972). With the pile head stiffness and damping properties determined, the free field accelerations measured during the test were applied as input to the free field end of the pile head compliances linking the superstructure to the pile foundation. The effects of kinematic interaction on the effective motions applied to the base of the superstructure were neglected. A fully coupled analysis of single pile response was carried out using the computer program SPASM8. In this analysis, the time history of computed free field displacements were applied along the embedded length of the pile, thereby implicitly accounting for the effects of kinematic interaction. The pile and structural mass were modelled using a series of elastic beam elements. The below ground portions of the pile were linked to the vibrat-ing, free field soil using Kelvin-Voight interaction elements which contained a non-linear Winkler spring and an equivalent viscous dashpot. The latter accounted for radiation damping in the surrounding soil mass and were computed using the equations of Gazetas and Dobry (1984), based on estimates of shear wave velocity in the free field. Tangent lateral spring stiffnesses were derived within the SPASM8 computer program based on input backbone p-y curves, inferred from the experimentally derived p-y curves using a matching procedure. Backbone p-y curves computed using methodologies outlined by the American Petroleum Institute (1979) were also specified to assess the sensitivity of the SPASMS output to the form of the input p-y curve. Unloading p-y response in the SPASM8 program was modelled assuming Masing behaviour. The principal conclusions from the single pile tests and associated analyses are: Chapter 8. Summary and Suggestions for Future Work 494 1. Low strain moduli (Gmax) measured in loose dry sand which has not been subjected to cyclic pre-strain are in excellent agreement with predictions made using an equa-tion relating Gmax to mean confining stress level and initial void ratio proposed by Hardin and Black (1968). Values of Gmax measured in dense vibrated sand may be somewhat higher (approximately 15 percent) than that inferred from the Hardin and Black equation using a K0 value of 0.6, due possibly to the effects of vibration on mean confining stress in the sand. Similar conclusions were obtained from both the shake table (low stress level) and centrifuge (high stress level) tests. 2. Elastic wave arrivals travelling at the compression (P) wave velocity were identified clearly during the shake table tests. The existance of true compression waves from a shear source can be attributed to imperfect geometric alignment of the receiver relative to the source. Alternatively, shear disturbances propagating at the P-wave velocity have been attributed on theoretical grounds to near field effects. A comparison of elastic P and S V wave velocity distributions gives low strain Poisson's ratios in the range of 0.15 to 0.20, which appear reasonable based on the existing literature. 3. The fundamental frequencies of vibration of a single pile increase with increasing sand density owing to the higher shear stiffness of the surrounding soil. For a given initial sand density, the fundamental frequencies reduce appreciably with increasing vibration amplitude, owing to strain softening of the near field soil. Under the low confining stresses existing on the shake table, the degree of softening appears more pronounced for an initially dense sand foundation due to the large amounts of sand dilation that occur under low stress conditions. For constant amplitudes of pile head vibration, the fundamental frequency of vibration was not overly sensitive to the method of pile vibration (ringdown or free field excitation). Chapter 8. Summary and Suggestions for Future Work 495 4. Under shake table test conditions, the fundamental frequencies of the free field sand layer were in the range of 40 (loose sand) to 60 Hz (dense sand). The natural frequency of vibration of the sand was well predicted using S H A K E using low strain shear moduli distributions. This suggests that limited strain softening occurred in the sand during the frequency sweep and hammer impact tests conducted. The peak amplitudes of free field surface acceleration were not well predicted using commonly accepted small strain damping ratios in the range of 2 to 3 percent. When the small strain damping ratios of the loose and dense sand were adjusted upward to within the range of 10 to 14 percent, the peak amplitudes of surface acceleration were well predicted. This may suggest that under the low stress conditions operative on the shake table, the dilative tendencies of the sand promote greater internal energy losses. Larger than expected damping ratios have also been inferred from the cyclic p-y hysteresis loops computed from the single pile tests conducted on the shake table. 5. The fundamental frequencies of vibration of the sand foundation (free field) mea-sured on the shake table are well separated from those of the pile. Indirect mea-surements on the centrifuge using frequency domain analysis of the measured free field and pile cap motions, indicate fundamental frequencies of the free field in the range of 2.5 to 3 Hz, which are somewhat higher than the first mode frequencies of vibration of the single piles tested. Experimental measurements on the shake table show that where the sand is driven at frequencies substantially removed from the fundamental frequency of vibration of the layer, the spatial variation of free field surface acceleration within the sand container due to three dimensional boundary effects is not significant. Thus, when the pile is excited at frequencies away from the fundamental frequency of the sand, it is reasonable to assume spatial homegeneity Chapter 8. Summary and Suggestions for Future Work 496 of the earthquake motions at the soil surface. 6. Free field soil amplification measured at different levels throughout the sand during shake table tests is well predicted using S H A K E , despite the potential three dimen-sional influence of the sand container boundaries. In general, the use of S H A K E to compute the free field surface accelerations and, by inference, the vertical distribu-tion of free field accelerations and displacements, provides results in good agreement with shake table and centrifuge test measurements. 7. Equivalent elastic moduli, inferred from theoretical correlations between lateral Winkler stiffness and elastic shear modulus, indicate that effective near field mod-uli are substantially less than measured small strain moduli in the free field. This result is in agreement with theoretical predictions of strain field development around a single pile based on measured amplitudes of pile vibration. Secant Winkler mod-ulus distributions versus depth derived from cyclic p-y curves also show reasonable agreement with theoretical correlations between effective shear strain in the near field, equivalent elastic shear modulus and secant Winkler modulus based on the theory proposed by Kagawa and Kraft (1981a,b). 8. Cyclic p-y curves have been developed from measured bending moment distribu-tions along the model piles using a technique developed by Ting (1987). The latter technique has been shown in the present study to provide more accurate predictions of soil reaction pressure along the pile than using the more common procedure of fitting a cubic spline distribution through the bending moment data, followed by differentiation of the cubic spline. The computed p-y curves show nearly elastic response with little internal damping for piles tested in dense vibrated sand and subjected to low level shaking on the centrifuge. With increasing intensities of shaking in dense sand and for piles tested in loose sand over the range of shaking Chapter 8. Summary and Suggestions for Future Work 497 intensities examined, greater non- linear and hysteretic response was observed. As lateral pile deflections decrease with increasing depth, linear elastic p-y response gradually develops. Cyclic p-y response observed on the shake table showed larger hysteresis loops than were evident on the centrifuge, indicative of the large energy dissipation in the near field soils. Damping ratios derived from the shake table tests for given normalized pile deflection ratios (y/b) were correspondingly larger than observed on the centrifuge. The shape of the curve passing through the end points of the p-y hysteresis loop (the backbone p-y curve) were poorly predicted using methods suggested for use by the American Petroleum Institute. In general, the API curves were considerably stiffer than those inferred from the centrifuge test results. On the shake table, the opposite was observed. 9. Secant Winkler moduli derived from the slope of the end- points of the p-y hysteresis loops increased with depth under the increasing overburden stress for both the centrifuge and shake table tests. The accuracy of the inferred lateral soil stiffness distribution was checked using a laterally loaded pile model, which models soil response using a series of elastic Winkler springs. Using the above model, pile deflections and bending moments for a given pile head loading were computed and found to be in good agreement with experimental measurements. The distribution of secant Winkler moduli developed from the p-y curves were found to depend on the amplitudes of pile vibration, with lower moduli corresponding to higher amplitudes of pile deflection. The moduli distributions may be approximated using linear or square root of depth distributions, upon which many closed form, analytic solutions for laterally loaded pile response are based. 10. The observed bending moment distributions along the piles are well predicted using simple, static Winkler models of lateral pile response to pile head loading. This Chapter 8. Summary and Suggestions for Future Work 498 implies for the test conditions examined that the pile response to the earthquake shaking is dominated by structural inertia forces caused by the mass of the super-structure, and that the pile vibrates predominantly in its first mode. A corollary of this observation, which is substantiated by theoretical calculations, is that provided one can accurately predict the shear forces and bending moments applied to the pile at the soil surface, one can also accurately predict the below ground, flexural behaviour of the pile. 11. The location of maximum bending in a model pile tested in the 1 g environment of the shake table is much greater relative to its diameter than was observed for similar intensities of shaking on the centrifuge. This fact may be anticipated from dimensionless analysis and critical pile length factors proposed by Matlock and Reese (1960). 12. The dynamic response of a single pile supporting a structural mass is well predicted using an uncoupled dynamic analysis, provided one incorporates the dependence of pile head stiffness at the soil surface on pile deflection level. The good agreement between computed and measured pile response for the test cases analysed substan-tiate the neglect of kinematic interaction. Therefore, one can input the free field surface motions directly into the analysis. Radiation damping parameters used in the analysis, derived using the computer program P I L A Y , have been found, in general, to provide a low estimate of radiative energy losses. Their use would be expected to provide a conservative over-estimate of pile response. 13. Use of a fully coupled analysis incorporated in the computer program SPASM8, while theoretically more rigorous, requires accurate input of the free field displace-ment time histories along the pile. These are seldom known accurately. In the present case, using the computer program S H A K E , free field accelerations have Chapter 8. Summary and Suggestions for Future Work 499 been computed and are in reasonable agreement with measured quantities. The accuracy with which dynamic strains and displacements can be computed using an equivalent elastic model is open to question. Using best estimates of the free field displacement time histories, the SPASM8 results have generally underpredicted pile bending moments, shear forces and displacements. The results have been found rel-atively insensitive to the form of backbone p-y curve used in the input. It is not clear whether the overly stiff pile response predicted using SPASMS is due to an inherent weakness in the program or whether the amplitudes of input ground dis-placement are in error. The latter questions represent a topic for future research. 8.3 Pile Group Test Results Both two and four (2 x 2) pile groups embedded in dense dry sand were tested using low to moderate intensities of shaking on the Caltech centrifuge. Low level sinusoidal shaking was used for the two pile tests, which were oriented inline, offline and at 45 degrees to the direction of shaking. Low intensities of shaking were used to examine whether pile to pile interaction factors derived using elastic theories of soil response could be used to predict the pile group response. Similar tests, using stronger levels of sinusoidal shaking, were carried out on the U B C shake table. The four pile group tests were carried out to determine whether pile interaction factors derived from the two pile tests could be superimposed to yield the displacement of the larger group. Low to moderatel level sinusoidal and earthquake excitation was applied to the four pile groups on the centrifuge. Strong sinusoidal shaking was employed in the four pile tests on the U B C shake table. The results of the pile group tests indicate: 1. The effects of pile to pile interaction for low and strong intensities of shaking are Chapter 8. Summary and Suggestions for Future Work 500 particularly pronounced where a line connecting the centres of the piles are oriented inline and at 45 degrees to the direction of shaking. Interaction effects are small where a line connecting the centres of the two piles is at 90 degrees (offline) to the direction of shaking. 2. Except for very close pile spacings (centre to centre pile spacings less than 3 pile diameters), offline interaction may for all practical purposes be neglected. 3. For inline shaking, interaction effects are stronger than predicted using elastic the-ory for close pile spacings less than about 3 pile diameters. For larger pile spacings, interaction effects die off at a much quicker rate than predicted using elastic the-ory. Beyond a spacing of six pile diameters, the test data indicate that interaction between piles may be neglected. 4. Pile interaction was observed to be as strong for 45 degree shaking as for inline shaking based on the centrifuge test results. This is at odds with elastic theory and may be due to the fact that since the piles were connected by a rigid pile cap, the input sinusoidal excitation caused biaxial shaking of the piles which led to more pronounced stiffness reduction in the near field soil. 5. The use of static elastic interaction factors proposed by Randolph and Poulos (1982) led to a slight (up to 20 percent) under-estimate of pile group deflection for close pile spacings and a slight over-estimate of pile group deflection for larger pile spacings. The increased pile group deflection for close pile separations is attributed to the zones of plastic yield around each pile in the group, which overlap and produce larger pile group deflections than predicted using elastic theory. For larger pile separations, the strain field caused by loading of one pile in the group is attenuated by plastic soil yield more rapidly than predicted using elastic theory. Empirically Chapter 8. Summary and Suggestions for Future Work 501 based modifications to elastic interaction factors proposed by Randolph and Poulos (19S2) have been suggested based on the present study, which produce somewhat better estimates of pile group deflection. 6. A comparison of experimentally derived interaction factors proposed by several researchers for cohesionless soils shows considerable scatter in the data, since inter-action between piles can be expected to be affected by load level, number of cycles of loading, the location of the loaded pile relative to the unloaded pile and the accu-racy of the test measurements. The interaction effects observed during the present study suggest that cyclic loading may attentuate the effects of pile interaction more strongly than observed during static monotonic loading. Based on the interaction factors developed from the present study, a computer model (PGDYNA) has been developed to analyse the dynamic response of a superstructure sup-ported by a group of piles. An uncoupled analysis is considered in which uni-directional, horizontal ground surface motions are applied to the head of every pile in the group, with the pile head compliances being represented by elastic or non-linear springs and equiv-alent viscous dashpots. Either two or three dimensional superstructure response may be considered. At every time step in the analysis, the effects of pile to pile interaction on the real and imaginary components of pile head stiffness are considered, resulting in increased pile group deflections relative to the case where interaction is neglected. The program has been applied to the analysis of 4-pile groups tested on the shake table and centrifuge. The computed pile group response generally exceed measured response. The results indicate that predictions of pile group response are complicated by uncertainties in the degree of radiation damping to be expected in the system and by an inadequate knowledge of what effective input motions should be used in the analysis. The latter are not necessarily the same as the free field surface motions, as indicated by theoretical Chapter 8. Summary and Suggestions for Future Work 502 elastic analyses published by Waas and Haartman (19S4). The effects of kinematic in-teraction appear to be more pronounced for pile groups than for single piles, leading to reductions in the effective input motions to be used in uncoupled pile group analyses. 8.4 Suggestions for Future Work The present study has resulted in the creation of a significant experimental data base relating to the response of single piles and small pile groups to simulated earthquake excitation. Experimental techniques and analysis methods have been developed which it is hoped will be of use to future researchers. The data is limited in that it has been confined to pile response in dry sand where pore water pressure build-up during earthquake shaking has not occurred. The effects of pore pressure build-up and, in the extreme, liquefaction of the free field soils are expected to result in larger free field ground displacements being imposed on the pile(s) and a corresponding decrease in structural inertia forces transmitted to the heads of the piles. Pile to pile interaction effects are also expected to be attenuated significantly. It is therefore recommended that model pile data be obtained where the effects of pore water pressure build-up on free field soil response are taken into account. This would provide an additional data base against which various methods of assessing single pile and pile group response to earthquake excitation can be checked. 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Chang (1982), 'Liquefaction and Permanent Deformation Under Dynamic Conditions - Numerical Solution and Constitutive Relations', from 'Soil Mechanics - Transient and Cyclic Loads', ed. G.N. Pande and O.C. Zienkiewicz, pp. 71-101 532 Appendix A Shake Table Tests - Instrumentation and Data Acquisition A . l Strain Gauges The model piles used in the shake table tests were instrumented with temperature com-pensated, foil type strain gauges. The resistance strain gauge, because of its inherent linearity, very small mass, wide frequency response (from zero to more than 50 kHz) and ease of installation, is ideally suited for use in dynamic testing (Lissner and Perry, 1961). Gauges were mounted on opposite sides of the pile cross-section using procedures recommended by the strain gauge manufacturer. Particular care was taken when passing the electrical lead wires up the inside of the pile to avoid removal of the wire insulation which can result in electrical shorting of the gauges. The outside of the pile was coated with an epoxy based lacquer, known as M-coat, to prevent abrasion of the gauges by the sand during testing. Bending strains were measured using the half bridge configuration shown in Figure A . l . Bending of the beam causes equal and opposite resistance changes in the two active gauges. The bridge configuration is such that voltage output across the two arms of the bridge are of opposite polarity. The net voltage output is given by the difference in voltage change for each arm so that the net output doubles the voltage change across each arm. Tensional or compressional strains due to axial load or temperature change result in equal voltage changes across each arm of the bridge and a net voltage output of zero. 533 Appendix A. Shake Table Tests - Instrumentation and Data Acquisition 534 2 ACTIVE GAUGES MOUNTED ON OPPOSITE SIDES OF PILE Figure A . l : Half bridge configuration to measure pile bending strains It may be shown that the voltage output across the two arms of the bridge, E o u t , is given as: Eout = Ein \" ™ o x (A.l) where E{n is the input excitation voltage, k is the gauge factor, and tmax is the maximum bending strain. Equation A . l shows that the bridge output is directly proportional to the input excitation voltage and that there is a linear relation between bridge output and bending strain (or applied bending moment). Axial bending strains were measured using the bridge configuration shown in Figure A.2. Axial straining of the pile due to dynamic axial loading causes equal resistance changes in the two active gauges which induces equal and opposite voltage drops across each arm of the bridge. The net voltage drop across both arms of the bridge is then given by the addition of the absolute values of the voltage drops for each arm. Bending strains have no effect on bridge output. Although temperature changes are considered to have a minor effect on bridge output for short term dynamic loading, \"dummy\" completion Appendix A. Shake Table Tests - Instrumentation and Data Acquisition 535 E;, in R,, R 3 ARE ACTIVE GAUGES R2 » R 4 A R E C 0 M P L E T I 0 N RESISTORS R Figure A.2: Half bridge configuration to measure pile axial strains resistors were mounted on a piece of aluminum tubing to ensure that temperature changes during the dynamic testing had no effect on bridge output. This assumes negligible temperature gradients between the active and dummy resistors. If this is not the case, then one should use gauges that have internal temperature compensation, as was done during the testing. The relation between output and input voltage is the same as cited in equation A . l , where tmax refers to the axial strain. The bridge voltage outputs are small and require amplification. This was achieved using amplifiers with built in bridge balancing circuitry to allow the strain gauge outputs to be electronically zeroed prior to start of a test. The amplifiers have a negligible attenuation of output for frequencies of up to about 1 kHz which is well above the range of input frequencies used in testing. Dynamic effects are therefore negligible. An input excitation voltage of 5.0 volts and an amplifier gain of 1000 was used during testing. The strain gauges were calibrated using a static loading procedure. Such static cal-ibrations are generally applicable for dynamic strain measurements provided that the Appendix A. Shake Table Tests - Instrumentation and Data Acquisition 536 gauge is properly applied and that the smallest pile curvatures of interest, which con-trol the strain gradients being measured, are large compared with the gauge dimensions (Keast, 1967). The length of the active area of the gauges used is 2 mm which represents 31 percent of the pile diameter. The strains measured therefore reflect average values over the strain gauge dimension. To calibrate the bending strain gauges, the pile was clamped as a cantilever beam and various weights hung from the end of the pile. Graphs of load, or applied bending moment, versus strain gauge output were constructed for each of the seven pairs of strain gauges and showed excellent linearity. The strain gauge pair used to measure axial load was calibrated by applying axial dead load to the model pile. Linearity between strain gauge output and axial load was also observed. Gauges were periodically recalibrated over the course of testing to check whether there had been changes in bridge output due to cyclic fatigue. Changes were found to be minor. A.2 Displacement Transducers (LVDT's) Two linear variable displacement transducers (LVDT's) were used to measure displace-ments and rotations of the structural mass supported by the model pile(s). The LVDT's have full scale displacements of ± 1 2 . 7 mm (±0 .50 in) and use a constant 6.0 volt DC input power supply. A DC-driven L V D T consists of a primary and secondary coil assem-bly and magnetic core which, when displaced along the axis and within the core of the coil assembly, produces a voltage output proportional to the displacement. Resistance to movement of the core is small since there is a small gap between the coil bore and magnetic core. The small mass of the core also results in negligible dynamic inertia. Since the primary coil requires A C excitation, the D C input is converted to A C us-ing a carrier oscillator. The axial core position determines the amount of A C voltage Appendix A. Shake Table Tests - Instrumentation and Data Acquisition 537 induced in the secondary windings which is converted to D C output using a frequency demodulator. These secondary circuits are connected in series opposition so that the resultant output is proportional to the core displacement from electrical center. The internal signal conditioning of the L V D T results in a strong signal output so that no am-plification of output is required. Some deamplification and phase alteration of the L V D T output should be expected when the instrument is measuring high frequency, dynamic displacements. Specifications supplied by the manufacturer of the L V D T indicates that at operating frequencies of up to 30 Hz, attenuation of output is down by less than 3 percent. Since the majority of forced vibration tests on the model piles were carried out using input frequencies of less than 30 Hz, corrections for these effects have not been made. The LVDT's were calibrated statically using a micrometer with 0.025 mm (0.001 in) divisions. The L V D T output, measured using a digital voltmeter, was plotted versus displacement over the expected range of motions of the model pile. Excellent linearity and negligible hysteresis of the L V D T output was observed. The static results were also checked using a dynamic calibration procedure in which the L V D T was used to measure known displacements of the shake table at frequencies varying between 5 and 30 Hz. The L V D T output was measured using an oscilloscope and confirmed that over the above range of frequencies the static and dynamic calibration results were essentially identical. The L V D T core makes contact with the pile head mass via a small plexiglass ball which maintains contact with the pile using a low stiffness spring. Care was taken at the start of every test to keep the spring tension force to a minimum but it was recognized that restraint to pile vibration could occur as a result of L V D T spring forces. Spring stiffnesses were measured by hanging small weights from the ends of the springs and measuring the spring displacements using a scale with 1 mm divisions. Such measurements showed the springs to have a stiffness of approximately 0.04 N/mm, which was constant for a wide Appendix A. Shake Table Tests - Instrumentation and Data Acquisition 538 range of spring displacements. The magnitude of the net spring force acting on the mass was minimized by placing the two LVDT's on opposite sides of the pile. An analysis was also carried out to estimate the natural frequencies of a single pile, incorporating the influence of the L V D T springs (see Appendix H). The analysis has shown that the springs have a negligible effect on the measured natural frequencies of the pile and therefore do not significantly affect its forced vibration response. A.3 Accelerometers Three accelerometers were used during testing; one placed on the base of the sand con-tainer to measure input table motions, one embedded in the sand at the soil surface to measure free field surface accelerations, and another accelerometer mounted on the struc-tural mass to measure pile head accelerations. Each accelerometer consists of a miniature mass placed at the end of a cantilever beam. The accelerations of the mass are sensed by strain gauging the cantilever beam, since strain in the beam is directly proportional to accelerations of the mass. The motion of the mass is damped by surrounding the beam/mass system in a low viscosity oil to prevent damage to the accelerometer due to transient, high acceleration shocks. The natural frequencies of the table accelerometers used on the Earthquake Engineer-ing (table A) and Soil Dynamics (table B) shake tables are 1200 and 800 Hz, respectively. The natural frequencies of the pile head and free field accelerometers are 550 and 400 Hz, respectively. The full scale range of each table accelerometer is ± 5 0 g, and ± 1 0 g for both the pile head and free field accelerometers. Since the accelerometers are damped to about 0.7 times critical, the dynamic response characteristics of the accelerometers are constant up to frequencies equal to 40 percent of the resonant frequency of the instrument (Bradley and Eller, 1961). Therefore, the output of the accelerometers is not affected by Appendix A. Shake Table Tests - Instrumentation and Data Acquisition 539 dynamic resonance effects in the range of frequencies of interest. The weights and dimensions of the accelerometers used to measure free field and pile head accelerations were chosen to minimize the influence of the accelerometer on the dynamic response of the model. The free field accelerometer has a weight of 0.83 N (0.187 lb) and the size of a 25 mm (1 in) cube. The weight of the pile head accelerometer is 0.05 N (0.01 lb) and has the size of a 15 mm (0.59 in) cube. These weights are less than 1 percent of the weight of the object whose accelerations are being measured. Their effect on the dynamic response of the model is considered negligible. Constant input excitation voltages were applied to the accelerometers and their out-puts were amplified using variable gain amplifiers having a frequency bandwidth over which their output is constant of more than 200 Hz. The accelerometers were calibrated using static and dynamic loading procedures. During the static calibration, the sensitive axis of the accelerometer was rotated through various inclinations, 6. The component of the weight of the accelerometer mass in a direction perpendicular to the axis of the can-tilever beam is given as mg sin 6, where m is the mass of the accelerometer. The effective \"acceleration\" acting on the mass then equals g sin 6. Graphs of amplified accelerometer output versus acceleration were found to be linear. During the dynamic calibrations, the free field and pile head accelerometers were subjected to different accelerations using the shake table. Their outputs were measured using an oscilloscope and were compared with the table accelerometer output, which was assumed to accurately measure the table motions. This comparison led to calibration factors in good agreement with the static calibration results. Appendix A. Shake Table Tests - Instrumentation and Data Acquisition 540 A.4 Data Acquisition The majority of data acquisition was carried out using a 16 channel, analogue to digital (A/D) converter with 12 bit resolution. The A / D board was manufactured by Data Translation Inc. The full scale voltage input was set to ± 1 0 volts and the maximum digital sampling rate for all 16 channels was 27.5 kHz. The A / D converter interfaces with an IBM-PC microcomputer using software which controls the rate of data acquisition per channel, At (sec/sample), number of channels of analogue data, M , number of samples per channel, N, and the maximum voltage that can be input into the system. Analogue to digital data conversions are done sequentially, or multiplexed, for each channel sampled and data stored on disc. The multiplexing procedure means that for one complete scan of all M channels there is a slight time shift between consecutive channels equal to At/M. A sampling rate of 3300 ^sec/sample (303 Hz) for each channel was used in the majority of shake table tests, giving a maximum time shift of about 3100 psec between the first and last sample for all 16 channels. This effect has not been corrected for since the time shift is insignificant relative to the duration of testing. Using the above sampling interval and storing 2400 samples per channel, which was limited by available buffer size on the microcomputer, a test duration of 7.92 sec was achieved. Prior to shaking of the pile foundation model, the start of the A / D conversion was signalled using an external voltage trigger. Where fewer channels of data were required, the data acquisition system available in the Earthquake Engineering Laboratory was used. It has a combined data acquisition rate of 1.6 kHz and can sample up to 16 channels of data. Using the above sampling rate of 303 Hz per channel, five channels of data can be sampled without a reduction in rate of data acquisition. The data acquisition system interfaces with software programmed for use on a PDP 11 microcomputer, which allows the user to select appropriate sampling Appendix A. Shake Table Tests - Instrumentation and Data Acquisition 541 parameters. The start of data acquisition is signalled by an external trigger, as above, and test data stored on disc. A .5 Spectral Analysis and Waveform Aliassing The selection of the time interval between data points, At, and the number of samples per channel of data, N, affects the accuracy with which frequency analysis of the digitized data can be carried out. A sampling rate of 303 Hz was selected for the majority of tests to minimize waveform aliassing over the frequency range of 0 to 150 Hz, since preliminary testing had shown that the pile and free field motions were dominated by frequencies in the above range. Use of a higher rate of sampling reduces the total time in which data can be acquired, due to limitations in buffer storage capacity. Waveform aliassing is a problem resulting from the discretization of an originally con-tinuous waveform. With this discretization process, the existence of very high frequencies in the original signal may be misinterpreted if the sampling rate is too slow, or may be indistinguishable from genuine low frequency components. In Figure A.3, it can be seen that digitizing a low frequency signal results in exactly the same set of discrete values as digitizing a high frequency signal using the same sampling parameters. The frequency content of the high frequency signal computed using a spectral analysis would also be in error. It can be shown that provided one uses a sampling rate equal to twice the high-est frequency contained in the analogue signal, known as the Nyquist frequency (fmax), then the original waveform can be interpolated from the digitized data using Shannon's theorem (Taylor, 1983). Response spectra computed from the digitized data are then accurate from zero to / m o x . Spectral analysis of the digitized waveforms was carried out using the discrete Fourier transform package, DFOURT, available in double precision from the U B C Computing Appendix A. Shake Table Tests - Instrumentation and Data Acquisition 542 Figure A.3: Waveform aliassing: (a) sampling a lower frequency waveform at interval AT; (b) sampling a higher frequency waveform at interval At (after Sheriff and Geldart, 1983) Appendix A. Shake Table Tests - Instrumentation and Data Acquisition 543 Centre. The latter computes the complex-valued Fourier amplitudes, C n , for N — 1 dis-tinct harmonics at discrete frequency intervals Af = l/(NAt). Due to the assumption of periodicity in the data over the time interval during which it was acquired, transform val-ues are repeated infinitely outside the returned frequency range from 0 to (N — l)/(NAt). Computed Fourier amplitudes above the Nyquist frequency fmax = 1/2At are rejected. The Fourier cosine and sine coefficients, an and bn, respectively, may be computed directly from the real and imaginary components of Cn using an = Re(Cn/N) and bn = —Im(Cn/N). The absolute amplitude, | C n | , at discrete frequency, / n , is given as \\Cn\\ = yjan + b\\. When the amplitude is plotted versus frequency, the frequency content of the digitized waveform is indicated. A.6 Digital Filtering Reduction in instrument noise which leads to improvement in the resolution of measure-ment was achieved by digitally filtering the recorded data. Digital filtering was found to be especially important for low amplitude signals. Noise levels were highest for the strain gauge and LVDT outputs. Strain gauge y noise was found to be concentrated around a frequency of 60 Hz while noise from the LVDT's was found to have a much broader frequency content. Measurements of pile head acceleration show that the first mode response of the model pile occurs at frequencies of 25 Hz or less, dependent on the amplitudes and duration of shaking, and that first mode vibration dominates the pile response. Therefore, a low pass filter using a cut-off frequency of 50 Hz was applied to the strain gauge and LVDT outputs during low amplitude shaking of the model pile(s). The frequency response characteristics of the pile were first determined from the unfiltered pile head accelerations to ensure that significant frequency components were not removed from the strain gauge and LVDT outputs. Appendix A. Shake Table Tests - Instrumentation and Data Acquisition 544 Digital filtering was carried out by computing the Fourier amplitudes of the recorded digitized data, and then setting the Fourier coefficients to zero above the cutoff frequency. The digital data, expressed in the frequency domain, was then converted to the time do-main using the inverse Fourier transform contained in program DFOURT. The accuracy of this inversion process was checked by computing the Fourier transform of a measured acceleration time history, and then inverting it back to the time domain. No difference could be discerned between the original and inverted signal. The truncation of a Fourier series above a certain cutoff frequency can lead to spurious oscillations in the time history of response due to so-called Gibbs phenomenon (Sheriff and Geldart, 1983). The effect of this can be seen by representing the filtered output, z(t), in terms of a Fourier integral (Clough and Penzien, 1975): 1 f°° z(t) = — / H(oj)c{u)exp(iu>t)duj (A.2) 27T J—oo where H(u>) is the frequency dependant, complex-valued transfer function representing the digital filter, and c(u>) is the Fourier amplitude distribution of the unfiltered signal. The filtered output can also be expressed in the time domain using the convolution integral given by Clough and Penzien as: *(<) = f y{r)h{t - r)dr (A.3) J 0 where y(t) is the original, unfiltered time history and h(t) is the impulse response function, which represents the weighting function commonly used in time domain, least squares filtering (Sheriff and Geldart, 1983). The relationship between h(i) and H(OJ) is given as (Oppenheim and Schafer, 1975): 1 f00 h(t) = — / H(u)exp(iu>t)du (AA) One represents the complex valued, filter transfer function H(u) for a low pass cut-off Appendix A. Shalte Table Tests - Instrumentation and Data Acquisition 545 filter using the equation: H(u) 1 + i, \\to\\ < uc 0, |u>| > uc (A.S) where uc is the cutoff frequency. From equations A.4 and A.5 the impulse response function h(t) is given as: The oscillatory nature of h(t) for both its real and imaginary component is readily seen from the above equation, suggesting that the filtered output z(t) computed from equation A.3 may undergo similar oscillation. To determine the seriousness of these oscillations, a numerical test was carried out using a sine series having the distribution of frequency dependant amplitudes shown in Figure A.4a. The frequency content and distribution of amplitudes shown in the figure is typical of the response observed during shaking of the test pile for low amplitude excitation. The unfiltered input is shown for several cycles and is compared to the filtered output computed using a 50 Hz cutoff filter in Figure A.4b. The figure shows that the filtered signal undergoes minor oscillation for the first half cycle but that these oscillations are rapidly damped to zero after one cycle. This behaviour is in agreement with observations made by Oppenheim and Schafer who state that Gibbs oscillations are less pronounced if the dominant energies of the unfiltered signal lies in a relatively narrow frequency band relative to that of the filter. The performance of the 50 Hz cutoff filter has been found to be satisfactory when applied to signals dominated by lower frequency components. sina>c7; + 0 (A.6) Appendix A. Shake Table Tests - Instrumentation and Data Acquisition 546 SAMPLING PARAMETERS > N = # OF DATA POINTS = 2400 t= TIME INTERVAL BETWEEN DATA POINTS =0.0033 sec Figure A.4: Test case showing the effect of a low pass filter on a sine series (a) spectral input (b) comparison of unfiltered and filtered time series data computed using a 50 Hz cutoff filter Appendix B Centrifuge Tests - Instrumentation and Data Acquisition B . l Strain Gauges Foi l type, resistance strain gauges were applied to the outside of the model piles to measure bending and axial strains, as described previously for the shake table tests. The outside of the pile was coated with M-coat to prevent abrasion of the gauges by the sand during testing. The gauges were wired in a half bridge configuration using a chevron bridge circuit (see Figure B . l ) and lead wires brought through holes in the pile cross section up the inside of the pile. Using a chevron circuit, the Wheatstone bridge used by the bending strain gauges is completed using a common set of completion resistors. The strain gauge signals were amplified prior to transmission from the centrifuge arm using a 16 channel, variable gain amplifier. The electrical configuration of this amplifier is such that one of the output leads from each amplified signal is common to system ground. This minimizes the number of electrical slip rings required to transmit the data from the arm (Figure B . l ) . A 5.0 volt input power supply was used for the strain gauges. The strain gauge amplifier is series connected with a fourth order, But terworth filter whose output is -3 dB down at 5 kHz . A n analogue filter wi th this specification removes frequency components above about 10 k H z and is designed to minimize the high frequency noise associated with centrifuge testing. Frequency components below 3 k H z are not significantly attenuated by the filter. Specifications supplied by the manufacturer of the 547 Appendix B. Centrifuge Tests - Instrumentation and Data Acquisition 548 AMPLIFIED SIGNALS TO HIGH SPEED DATA J AQUISITION COMMON CENTRIFUGE GROUND L E G E N D AMPLIFIER WITH ANALOGUE FILTER 5.0v BENDING STRAIN GAUGE PAIR # I BENDING STRAIN GAUGE PAIR # 2 BENDING STRAIN GAUGE PAIR # n AXIAL STRAIN GAUGE CONFIGURATION* COMMON 350-n. COMPLETION RESISTORS FOR BENDING STRAIN GAUGES BRIDGE EXCITATION O SLIP RING h^n .i ACTIVE GAUGES ON OUTSIDE tiW 1 OF PILE -W- BRIDGE COMPETION RESISTORS * BRIDGE COMPLETION RESISTORS MOUNTED ON PIECE OF STAINLESS STEEL AT SAME TEMPERATURE AS MODEL PILE Figure B . l : Chevron bridge circuit used in pile strain gauging Appendix B. Centrifuge Tests - Instrumentation and Data Acquisition 549 instrument amplifier show that the attenuation of output is negligible below a frequency of about 5 kHz. Therefore, the analogue filter connected in series with the amplifier controls the frequency content of the output signal. The bending and axial strain gauges were calibrated as described in Appendix A using a static loading procedure. Amplified voltage outputs were routed through the amplifier and data acquisition system used during actual testing. The calibration curves established showed excellent linearity. B.2 Displacement Transducers The pile cap mass was instrumented using a non-contact photovoltaic displacement trans-ducer (PIN-SC/10D) manufactured by United Detector Technology Inc. The light emit-ting diode (L.E.D.) used by the displacement sensor is mounted on the top of the pile head mass, as described in section 3.3. The displacement sensor was mounted on a rigid cross-bar located approximately 25 mm above the L .E .D . and pile head displacements measured relative to the moving base of the centrifuge soil container. The photovoltaic displacement detector is a dual axis position sensor that provides X and Y axis position information in the horizontal plane of a light spot (L.E.D.) on the detector surface. These devices sense the centroid of the light spot and provide continuous amplified analogue output as the light spot moves from null point to the limit of the active area. The displacement transducer has its own amplifier and has a useful frequency range from 0 to approximately 1000 KHz. The device was calibrated by fixing the position detector and moving the L . E . D . in fixed displacement increments in the X and Y directions. Output of the device was read using a digital voltmeter and a calibration curve relating displacement to voltage output of the device obtained. Appendix B. Centrifuge Tests - Instrumentation and Data Acquisition 550 An L V D T was mounted in the hydraulic actuator used to supply the input base motion. The L V D T was calibrated statically by moving the soil container in fixed dis-placement increments and measuring these using a dial gauge. The output of the L V D T was transmitted through the centrifuge slip rings and read using a voltmeter. B.3 Accelerometers Entran miniature accelerometers (model EGA-125F-500D) were used to measure input base, free field and pile head accelerations during testing. The accelerometers employ a fully active Wheatstone bridge consisting of semiconductor strain gauges. The strain gauges are bonded to a simple cantilever beam which is end- loaded with a mass. The strain gauges sense the bending and hence the acceleration of the mass. The accelerometers have a very small size and mass which is important in centrifuge modelling. The accelerometers are 6.8 mm long by 3.7 mm wide by 2.7 mm high and have a weight (minus the leads) of only 0.005 N (0.57 gm). Each accelerometer has a range of ± 5 0 0 g and is damped to 0.7 of critical using a viscous fluid medium. This helps to eliminate resonance and allows a useful frequency range of DC to 1 kHz. The strain gauge bridge used in the accelerometer is powered by a 15 volt D C power supply. The accelerometer outputs were amplified during testing using a gain of 100. The gain of the instrument amplifiers is constant ( ± 1 percent flatness) for frequencies less than 1 kHz. The amplifier contains an analogue filter with the characteristics described previously to minimize the high frequency noise inherent with centrifuge testing. The accelerometers were calibrated by taping them to the base of the centrifuge soil container with the sensitive axes of the accelerometers oriented parallel to the radius of the centrifuge arm. The centrifuge was then spun at various speeds so that the accelerom-eters were subjected to different centrifugal accelerations. The amplified accelerometer Appendix B. Centrifuge Tests - Instrumentation and Data Acquisition 551 outputs were transmitted via the centrifuge slip rings and read using a voltmeter. Cali-bration curves of acceleration versus amplified voltage output were established for each accelerometer and were found to be linear. B.4 Data Acquisit ion Prior to centrifuge spin-up, all instrumentation was checked to ensure that it was in working order and had not been damaged during pile installation. The amplified strain gauge and accelerometer signals were then zeroed electronically using bridge balancing circuitry contained within the instrument amplifiers. The location of the photovoltaic displacement sensor was also adjusted to give optimum initial voltage output so that after centrifuge spin-up and during shaking of the model pile(s) the transducer output did not exceed the range of the data acquisition system. During each single pile test, the amplified instrumentation output was transmitted via electrical slip rings and then fed into a 16 channel analogue to digital (A/D) converter. The A / D converter interfaces with a Zenith micro-computer and data is stored on disc. The A / D converter stores data using 12 bit resolution in the ± 2 volt range and has a combined sampling rate of 100,000 data scans/sec. The sampling rate is divided equally by the number of channels sampled (=16), giving a data sampling interval, At, of 160 /usee between consecutive samples for the same channel. Each channel of data is sampled sequentially giving an offset in time equal to Ar/16 between adjacent channels. This offset is minor relative to the total test duration and consequently has not been corrected for. A total of 5461 data points were collected per channel, giving a total run duration at model scale of 0.87 sec. For the pile group tests, up to 21 channels of data were sampled. This necessitated using a second parallel high speed data acquisition system. The system is manufactured Appendix B. Centrifuge Tests - Instrumentation and Data Acquisition 552 by R-C Electronics of Santa Barbara, Calif. It has 12 bit resolution and can acquire up to 9 channels of data in the ± 1 0 volt range at a combined sampling rate of up to 106 samples/sec. Six channels of data were sampled at 160 fisec per point and data stored on disc. To ensure that signals recorded by the two separate data acquisition systems were matched in time, the input base accelerations were recorded by both systems. B.5 Data Processing The 12 bit A / D converter acquires data in the range of ± 2 volts (or ± 1 0 volts for the second parallel A / D system) and in this range will give a number between 0 and 4095 as follows: -2 (-10) volts corresponds to 0 0 volts corresponds to 2048 2 (10) volts corresponds to 4095 The above relationships were used to obtain digitized voltage data from the binary output of the A / D converter. The digital data stored in each channel was then converted to the required engineering quantity of interest at model scale using the appropriate instrument calibration factor. Finally, a further scale factor was applied as described in section 4.1 to convert from model scale to prototype scale. Time histories of response at prototype scale were plotted using a time interval between each data point of 60 x 0.00016 sec = 0.0096 sec, corresponding to a centrifuge scale factor of 60. Using this sampling interval, which corresponds to a prototype sampling rate of 104 Hz, frequencies of up to 52 Hz (the Nyquist frequency) can be resolved. Since input base motions contained frequencies of at most 10 Hz, this rate of sampling is more than adequate to avoid waveform aliassing. Appendix B. Centrifuge Tests - Instrumentation and Data Acquisition 553 Fourier spectra were then computed from the recorded acceleration time histories to determine the dominant frequencies of response. Low amplitude signals were digitally filtered using a 10 Hz cutoff filter to improve their signal to noise ratio. Appendix C Strain Fields A r o u n d Laterally Loaded Piles C . l Introduction Calculations of dynamic pile response are often made by modelling the interaction be-tween the soil and the pile using lateral soil stiffnesses derived from the theory of elasticity. The determination of these stiffnesses relies on making reasonable estimates of strains in the near field soil since elastic moduli necessary for the computations are strain level dependent. One method of estimating the strains around a pile idealizes the problem as that of plane strain vibration of a rigid disc in an elastic medium. It is anticipated that the assumption of plane strain vibration, while consistent with the use of elastic Winkler models to represent soil-pile interaction, will underestimate strains around the pile since only in plane strains (er, eg, and 7^) are considered. This is demonstrated by comparing the plane strain solution with a solution where displacements are assumed confined to the horizontal plane. Using the plane displacement assumption, there are five strain components and only vertical strains (ez) are neglected. Since computed strains are dependent on the elastic moduli selected for the com-putations and soil is inherently non-linear, equivalent elastic parameters are used that are compatible with average strains around the pile. It is demonstrated that this ap-proach yields reasonable estimates of strains around the pile by comparison with a plane strain finite element solution given by Atukorala et al (1986). In the latter analysis, the hyperbolic model described by Duncan et al (1980) was used to represent the soil's 554 Appendix C. Strain Fields Around Laterally Loaded Piles 555 stress-strain behaviour, thereby providing a basis for estimating strain and stress level dependent moduli. C.2 Navier's Equations of Motion in Three Dimensions Navier's equations of motion for an undamped elastic medium subjected to horizontal earthquake base motion, are given as 2C (A + 2C7)A,r nz,e + 2G{le,z = pu tt - w2pub cos OeT* (C.l) r (A + 2G ) -A , < , -2Gn r , 2 - | - 2Gn 2 i r = pvitt + w 2pub sin 0eiwi (C.2) 2G 2G (A + 2G) A,z — (rUe),r + — £2r,e = pw,tt (C.3) where A = er + e9 + £j = - (ru),r + - v * + (C.4) r r ft* = ^r(W.r-«^) (C5) \"r = \\ ( l WS-V.>) (C-6) = ^ ( ^ - W . r ) (C.7) and u, v, and w are the soil displacements relative to the rigid base motions, ub, in the radial (r), circumferential (9) and vertical (z) directions, respectively. Lame's constant A is defined in terms of the effective shear modulus G and Poisson's ratio v as C.3 Simplifications to the Three Dimensional Equations of Motion The maximum shear strain in the horizontal r — 6 plane is assumed to control the de-gree of shear modulus reduction around the pile. This is determined from the radial Appendix C. Strain Fields Around Laterally Loaded Piles 556 and circumferential strains, er and eg, which a.re determined from displacements u and v. To define these displacements, equations C . l and C.2 are used which express dynamic equil ibrium in the horizontal plane. To% simplify the previous equations one or both of the following approximations are adopted: 1. vertical displacements w are negligible relative to horizontal displacements during lateral vibration of the pile. 2. out of plane displacement curvatures uiZZ and viZZ are small and may be neglected. Assumption (1) has been made by several researchers in the field of pile dynamics ( Taj imi , 1969; Nogami and Novak, 1977; Kagawa and Kraft; 1981b) and is referred to as the plane displacement assumption. Dynamic solutions to the problem of a pile subjected to base motion excitation have been presented by Taj imi and Kagawa and Kraft based on elastic continuum models. Kagawa and Kraft 's analytic solution has been compared wi th three dimensional, elastic finite element solutions in which vertical soil displacements were considered. The comparison indicated that the dynamic lateral load-deflection relationships of piles were not significantly affected by the neglect of vertical soil displacements. Assumption (2) reduces the equations of motion of the system to a plane strain prob-lem. It is of interest to compare dynamic lateral stiffnesses computed using a plane strain model wi th solutions made using the plane displacement assumption for homogeneous elastic media. Kagawa and Kraft (1982) have summarized a variety of these predictions with particular reference to earthquake loading. The lateral soil stiffness at a particular depth, expressed in terms of a multiple of the equivalent elastic Young's modulus of the soil, was seen to vary over the pile length. However, Kagawa and Kraft have derived average lateral soil stiffnesses which adequately reproduce the dynamic response of the soil-pile systems examined. The average lateral resistance was found to be dependent on Appendix C. Strain Fields Around Laterally Loaded Piles 557 the aspect ratio (length to diameter, l/d) of the pile and the relative soil-pile stiffness, Kr, given as K T = EI/ESTQ where EI is the flexural rigidity of the pile, ES is the equivalent elastic modulus of the soil averaged over the length of the pile, and r 0 is the pile radius. Kagawa and Kraft and other researchers have found that the lateral soil stiffness does not have a strong frequency dependence due to soil inertia effects in the low frequency range. The plane strain stiffnesses proposed by Novak et al (1978) were found to be in excellent agreement with the previous computations for long rigid piles (l/d > 60, KR > 106). Similar conclusions have been made by Roesset and Angelides (1980), who used a three dimensional finite element model, and Nogami and Novak (1977) who used an analytic so-lution based on the plane displacement approximation. Roesset, and Nogami and Novak have confined their analyses to the case of cyclic pile head loading. While the available literature suggests that elastic lateral stiffnesses are reasonably well estimated using plane strain solutions it is not clear whether computed strains around the pile due to lateral pile vibration are sensitive to the analytic assumptions adopted. Therefore, a study has been made of the plane displacement and plane strain models of lateral pile vibration. Each model was used to estimate shear strains around the pile and the results compared. C.4 Solution to Navier's Equations of Motion - Plane Displacement Case The solution to equations C . l and C.2 has been outlined in detail by Tajimi (1969) for the case where vertical displacement w is neglected. The displacements u and v may be expressed as, U = Ur + Ujf (C.9) V = Vr + Vff (CIO) Appendix C. Strain Fields Around Laterally Loaded Piles 558 where subscript r refers to the ground displacement induced by relative movement be-tween the pile and the free field (the latter is not influenced by the presence of the pile) and subscript ff refers to the free field ground motion. The latter may be computed using a number of standard analytical or numerical approaches. Since one is interested in computing displacements and strains in the near field soil due to relative movement between the pile and free field, the free field component due to the rigid base motions is not considered. Tajimi's solution for the relative displacements is given as, n=l,3,5,... I n=l,3,5. where + anKQ(cxnr) M M + 0nKo(M + B n K l ^ \" ^ 1 sin knz cos 6eiwt (C. l l ) i y - {u/ugf} UJg = 2H 2H and H is the layer thickness, Vs and Vp are the shear and compression wave velocities of the elastic layer, and u is the forcing frequency of excitation. Provided there is no slip between the pile and the soil at the soil-pile interface (r = ro) and the lateral pile vibration yp relative to the free field motion is co-linear with ur for 6 = 0 then equations C . l l and C.12 give vr = 0 for 6 = 0 and ur = 0 for 6 = TT/2. Assuming a rigid pile cross-section gives, ur(r0,6 = 0) = -vr(r0,6 = TT/2) (C.13) Appendix C. Strain Fields Around Laterally Loaded Piles 559 Equations C . l l to C.13 result in the following relationship between An and Bn Bn = An , V ° ; (C.14) The relative pile deflection yp computed using an appropriate dynamic model or measured during base motion excitation is expressed in terms of the following Fourier series, y p O M ) = £ UnSin knze^ (C.15) n=l ,3,5,... where 1 f^H nirz Un = HJ0 yP(z)™wdz (ai6) It is noted that yp is only defined in the interval from z = 0 (the pile tip) to z = H (the soil surface). Therefore, yp was extended symmetrically over the interval H < z < 2H to evaluate the above integral. Since ur(r = ro, 6 = 0) = yp, from equations C . l l , C.14 and C.15 the coefficient An is given as, An = where Fn = Fn(an,8n,r0) r o [2Jh(r,0) and *(r, 9) given as, u(r,0) + (C.21) r v(r,6) = - $ l ( ? - $ r (C.22) r equations C.19 and C.20 can be written in the form, (A + 2G) (V 2$),r + - (V 2*),* = -P\"2(®,r + (C.23) r r (A + 2G) -(V 2$),e - G (V21'),r = * - * r) (C.24) r r In the above equations, y 2 refers to the Laplacian operator defined in the r — 9 plane with respect to a function / as, V 2 / = /.rr + \"/.r + \\fj» (C.25) Appendix C. Strain Fields Around Laterally Loaded Piles 561 Adding equations C.23 and C.24 leads to two distinct equations in = 0 (C.27) where q = ito^JA+P2G.\" = itoVp, s = iuyj^, i = y/—\\, Vs is the shear wave velocity and Vp is the P- wave velocity. Equations C.26 and C.27 represent two uncoupled eigenvalue problems which are solved separately to yield and $ and hence displacements u and v. The above equations may be solved using a standard separation of variables technique involving variables R(r) and 0(0). This leads to two ordinary differential equations in R and 0 given as, r2R{r)\" + rR(r) - {(ar)2 + m2)R = 0 (C.28) 0\" + m2Q = 0 (C.29) where a = q or s when applied to equations C.26 and C.27, respectively. The solution for R(r) is given as, R(r) = AIm(ar) + BKm(ar) (C.30) where Im and Km axe modified Bessel functions of the first and second kind respectively of order m and A and B are integration constants. For a bounded solution in R as r goes to infinity, A — 0. The solution for Q(9) is, 0(0) = C sm(m9) + D cos{m9) (C.31) The solutions for $ and ^ are therefore, $(r,0) = {B1Km(qr))(C1sm(m9) + D1cos{m9)) (C.32) tf(r,0) = (B2Km(sr))(C2sm(m9) + D2cos(m9)) (C.33) Appendix C. Strain Fields Around Laterally Loaded Piles 562 The integration constants are solved for by using the following boundary conditions which assume a rigid pile cross section and no slip between the soil and the pile: u(r = r o ,0 = O) = l.O (C.34) v(r = r0,6 = 0) = 0.0 (C.35) u(r = r0,e = TT/2) = 0.0 (C.36) v(r = r0, 0 = TT/2) = -1.0 (C.37) Since u is proportional to and v is proportional to <&j, from boundary condition equations C.35 and C.36 one must have C\\ — D2 = 0. Combining the remaining integration constants and assuming with no loss of generality m = 1 , this gives $(r,0) = A K^qr) cos 0 (C.38) tf(r,0) = 5 /^(sr) sin 0 (C.39) From equations C.21 and C.22, the displacements u and v are given as, u(r,0) = A cos 0K[(qr) -\\ Kx(sr) cos 0 r = cos 0[-A{-Kx{qr) + qK0(qr)) + -K^sr)] (C.40) r r v{r,0) = smOl^Mq^ + B^KtW + sKoisr)] (C.41) Using the compatibility condition that u(r = r0,9 = 0) = — v(r = ro,0 = T/2) one can relate constants A and JB as, From equations C.40 to C.42 one can write displacements u(r, 6) and v(r, 0) as,' u(r,6) = A c o s ^ ( r ) (C.43) u(r,0) = ,4 sin 0jF2(r) (C.44) Appendix C. Strain Fields Around Laterally Loaded Piles 563 where functions T\\ and J-2 are defined as, * ( r ) = \" - W ^ ^ - (*M+ ,«,(„)) (C.45) 2 / ^ ( ^ 0 ) 4 - ^ 0 / ^ 0 ) / ^ ( j r ) \\ ^ ( q r ) ^ = 2/r i ( 5r-o) + , r 0 / ro ( 5 r 0 ) + 5 ^ r ) J ~ ( C \" 4 6 ) Since the pile (disc) vibrates at a particular depth with amplitude yp one can solve for constant A as, A = z & r (C.47) The strains in the r — 0 plane are derived from the displacements using the equations, tr = u,r (C.48) u 1 te = - + -vfi (C.49) r r 1 v Ire = -u,e + vtT (C.50) r r The above equations are equally applicable for the plane strain or plane displacement analyses. The maximum shear strain in the horizontal plane is given as 7 m ax = 2/? with R defined by, ^ (^ ) 2 +(T ) ! <™> First order derivatives of Bessel functions necessary in the above calculations have been obtained from Abramowitz and Stegun (1972). Bessel functions have been evaluated numerically using polynomial approximations given in the same reference. C.6 Comparison of Plane Strain Analyt ic Solution W i t h N o n - Linear Finite Element Solution Atukorala et al (1984) have carried out a plane strain finite element analysis of a 0.3 m diameter rigid disc translating laterally in a soft clay. The analysis was performed to Appendix C. Strain Fields Around Laterally Loaded Piles 564 Parameter Value Geq 2900 kPa V 0.495 P 1.6 kN-sec2/m 4 v. 42 m/sec 424 m/sec 6 racl/sec yp 0.003 m Table C . l : Elastic Soil Properties Used in Plane Strain Analytic Solution compute non-linear pressure-displacement (p-y) relationships of the disc, thereby simu-lating the lateral loading behaviour of a long flexible pile. The undrained stress-strain behaviour of the clay was modelled using a non-linear, incremental elastic procedure based on the hyperbolic model. The disc was translated a distance of 0.003 m and strain compatible shear moduli computed at various locations around the disc. An average shear modulus of 2900 kPa was computed within 5 pile radii of the outside edge of the disc. Using the equivalent elastic soil properties given in Table C . l maximum shear strain distributions have been computed from the analytic solution described previously. These strain distributions are shown in Figure C . l (a) to (c) over a range of dimensionless distances (r/r 0 ) and departure angles 9. The angle 9 is shown in the figures. Shear strains computed using the analytic solution agree closely with strains computed using the finite element model, demonstrating that provided one can make a reasonable estimate of the equivalent elastic shear modulus, Geq, and Poisson's ratio, i/, one can compute strain distributions in good agreement with non-linear methods of analysis. To estimate the average shear modulus, Geq, to be used in the above calculations requires that one choose an effective zone of influence for the pile. One can then com-pute average maximum shear strains within this zone and using the equations of Hardin Appendix C. Strain Fields Around Laterally Loaded Piles 565 (a) in 0.001 £ Analytic Solution X Finite Element Solution 2 3 * Dimensionless Distance R/RO (b) Analytic Solution x Finite Element Solution 2 3 4 Dimensionless Distance R/RO (c) in o o .c V) E 3 E Analytic Solution X Finite Element Solution 2 3 4 Dimensionless Distance R/RO Figure C . l : Computed maximum shear strains around a vibrating disc - analytic versus finite element solution (a) 0 = 10 degrees (b) 0 — 40 degrees (c) 0 = 80 degrees Appendix C. Strain Fields Around Laterally Loaded Piles 566 and Drnevich (1972) relate Geq/Gmax to the average shear strain. Matlock (1970) has suggested, based on Skempton's (1951) earlier work on the settlement of spread footings, that the zone of influence of the pile extends 2.5 pile diameters from the edge of the pile. Using this assumption, Matlock has computed the average principal strain, ex, in the zone of influence as, * * iSs where yp is the lateral pile deflection and d is the pile width. An identical concept has been adopted by Kagawa and Kraft (1981a) who relate the maximum average shear strain, 7 m c u . , in the zone of influence to Z\\ using elasticity relationships. Thus 7max = (1 + (C.53) (1 + V)VP Icd where 7£ is an appropriate strain influence factor. Kagawa and Kraft suggest lt be assigned a value between 2 and 3. Based on the above concept, maximum shear strains have been computed using the analytic solution in a zone from 0 < 9 < TT/2 and r 0 < r < Nr0 and averaged over this area using the equation, 1 / - T T/2 rNr0 7max = j j o J 1ma*{r,9)rdrd9 (C.54) where A = \\[(Nr0)2 — (rl)] and N is a factor selected to define the radial extent of the zone of influence. Variations in N from 3 to 7 appear reasonable based on finite element studies described in the main text. The smaller zones of influence correspond to lower amplitude pile deflections. Appendix C. Strain Fields Around Laterally Loaded Piles 567 o rj i_ 3 (/) '5 in _p Q) CQ CL 0) Q -5 5 15 Relative Pile Displacement (mm) Figure C.2: Pile displacement profiles yp(z) used in analysis of strain field around vibrat-ing pile C.7 Comparison of Strain Fields - Plane Displacement versus Plane Strain Solutions Using the pile displacement profile shown in Figure C.2 derived at peak pile deflection from centrifuge test 15, Fourier amplitude coefficients Un have been evaluated considering 10 terms in the Fourier series. This has been found to provide a close matching of measured pile displacements as shown in Figure C.2. Displacements and strains around the pile have been computed 1.0 m below the soil surface using the plane displacement solution and the parameters given in Table C.2. Shear moduli Geq used in the calculations have been selected to be compatible with average shear strains computed using equation C.54 and N = 6. Appendix C. Strain Fields Around Laterally Loaded Piles 568 Parameter Value Geq 2618 kPa V 0.40 P 1.7 kN-sec2/m 4 OJ 3.14 rad/sec VP 0.0194 m ' 0 0.285 m If max 3.8% (PD) Imax 0.7% (PS) Imax 1.9% (M) Table C.2: Parameters Used in Plane Strain/Plane Displacement Analyses Computed maximum shear strains due to relative motions between the pile and the free field are shown versus radial distance from the pile at the 1.0 m depth and at various departure angles 6 in Figure C.3. The strains computed are also compared with those computed using the plane strain (PS) analysis, demonstrating that the plane displacement (PD) analysis leads to higher estimates of shear strain around the pile. The average effective shear strain in the assumed zone of influence (TV = 6) are compared for both methods of analysis in Table C.2 as well as with the approximate method proposed by Matlock. The Matlock (M) procedure gives average strains approximately midway between strains computed using the plane strain and plane displacement analyses. Appendix C. Strain Fields Around Laterally Loaded Piles 569 0 5 10 15 20 Dimensionless Distance R/RO Figure C.3: Computed maximum shear strains around a vibrating disc - plane displace-ment versus plane strain solution (a) 6 = 0 degrees (b) 0 = 40 degrees (c) 9 = 90 degrees Appendix D Static Laterally Loaded Pile Solutions D . l Introduction Closed form solutions for the problem of a single pile subjected to static moment and shear loading of the soil surface have been obtained by numerous researchers. These solutions are adequately summarized by Poulos and Davis (1980) and Scott (1981a). Many of these solutions idealize the soil behaviour using concepts of linear elasticity, although extensions have been made to account approximately for soil non-linearity by assuming that the soil reaches its ultimate lateral resistance above a certain depth (Scott, 1979). Below this depth, the soil is assumed to have linear elastic properties. Laterally loaded pile solutions incorporating soil non-linearity commonly use non-linear Winkler springs to model the soil response, based on the use of lateral soil resistance (p) versus lateral pile deflection (y) curves. Finite difference computer programs such as L A T P I L E (Byrne and Janzen, 1984) are used to solve for the pile response using a series of small load steps and iterations within each step to achieve solution convergence. Linear elastic solutions fall into two categories; those which assume an isotropic elastic continuum, and those which idealize the soil as a Winkler medium. The latter assumes the pile rests on a bed of independent springs and neglects shear coupling between the springs. Despite this theoretical shortcoming, which generally leads to a less stiff solution than is predicted using elastic continuum models, the model is extensively used in practice and has been found to adequately model laterally loaded pile response. 570 Appendix D. Static Laterally Loaded Pile Solutions 571 The use of the above models relies on a determination of modulus parameters which describe the variation of lateral modulus with depth. These parameters are most com-monly derived from the matching of theoretical solutions to experimentally determined deflections or bending moments along a pile during lateral loading (Poulos and Davis, 19S0). Alternatively, insitu tests may be carried out using pressuremeter testing and the results scaled to estimate non-linear 'p-y' curves for use in analysis (Atukorala et al, 1986). In absence of such experimental data, 'p-y' curves may be estimated using a variety of empirically based procedures described by Murchison and O'Neil (1984) in the case of cohesionless soils and by Gazioglu and O'Neil (1984) in the case of cohesive soils. Lateral subgrade moduli compatible with the pile deflections can then be computed at various depths. Attempts have also been made to relate lateral subgrade moduli kh to equivalent elastic moduli G (see chapter 1). Unfortunately, the average shear strains in the soil adjacent to the pile that determine the modulus to be used in the calculations are not known with certainty. Strain levels must be estimated iteratively using relationships be-tween pile deflection and average strain level in the near field soil, using elastic continuum or non-linear finite element solutions. The most rigorous approach to the laterally loaded pile problem involves using three dimensional finite elements incorporating an appropriate constitutive law. This is a highly complex, expensive procedure that is a subject of continued research. The problem of defining an appropriate three dimensional constitutive law to be used in these calculations remains. Linear elastic, static and dynamic finite element analyses have been carried out by several investigators (see chapter 1) to analyze the lateral response of piles to pile head and earthquake loading. Attempts to account for soil non-linearity have also been made by using equivalent elastic parameters that are compatible with average strains in each Appendix D. Static Laterally Loaded Pile Solutions 572 soil element. Slip between the soil and the pile is not accounted for. The equivalent elastic procedures used in calculations of dynamic pile response are similar to those used in the analysis of dynamic free field response, based on total stress approaches (Schnabel et al, 1972). Barton (1982) has extended the above procedures by using a non-linear, elastic-plastic constitutive law which was incorporated in a three dimensional finite element code to model static lateral loading of a single pile. This approach holds considerable promise but remains to be extended to djniamic lateral loading. Since the intent of the present study is to verify analysis procedures that are routinely used in the prediction of dynamic pile and pile group response, Winkler subgrade reaction models have been used to model the interaction between the soil and the pile. It has been shown in several theoretical studies that static lateral soil stiffnesses may be used to model lateral pile response during low frequency dynamic loading since inertia effects caused by vibration of the surrounding soil mass are small (Nogami and Novak, 1977; Kagawa and Kraft, 1980a; Sanchez-Salinero, 19S2). It has also been found in the present study that bending moments and deflections measured during base motion excitation of the model piles are well described using static solutions based on the Winkler approximation. Coefficients of subgrade reaction, which describe the variation of lateral modulus with depth, have been derived from experimentally measured bending moments along the pile. These coefficients have been found to be dependent on initial placement density of the sand and amplitudes of lateral pile deflection. The strain compatible shear moduli inferred from the Winkler subgrade reaction model are substantially less than measured low strain shear moduli. This is due to the high shear strains that occur in the near field soil adjacent to the pile during lateral vibration. The derived moduli have also been compared with lateral stiffnesses estimated using non-linear 'p-y' curves recommended by the American Petroleum Institute (1979). Two distributions of lateral subgrade modulus have been used in the calculations; Appendix D. Static Laterally Loaded Pile Solutions 573 one based on a variation proportional to the square root of depth, and the other linearly proportional to depth. These models are described in detail by Scott (1984) and Matlock and Reese (1960), respectively, and are summarized in the following sections. D.2 Winkler Modulus Proportional to the Square Root of Depth The static equation of beam (pile) flexure subjected to lateral soil pressure along its length p(z) is given as, EI^+p(z) = 0 (D.l) where EI is the flexural rigidity of the pile and y(z) is its lateral deflection as a function of depth coordinate z. Using the Winkler hypothesis, the soil pressure is given as p(z) — khy(z) where kh is a function of z and is assumed to vary proportionally to the square root of depth. Thus, kh = cxzxl2 and a is a lateral modulus parameter. Substituting this definition of p(z) into equation D . l gives £/0 + azx'2y = 0 (D.2) Defining parameter A as A 4 + n = a)EI where n = 1/2 and substituting into equation D.2 leads to the following equation, 0 + X4+nzx/2y = 0 (D.3) The above equation may be simplified by using the dimensionless variables W(Z) and Z defined as, W = Xy Z = Xz (DA) Application of the chain rule of differentiation to equation D.4 gives the following Appendix D. Static Laterally Loaded Pile Solutions 574 relationships, dy _ dW dz dZ Sy_ _ d2W dz2 ~ dZ2 d3y = x2d*W dz3 dZ3 £1 = A 3 — dz4 dZ4 d'y \\ 3 d ' w m ^ ~ x -J^r (D-5) Substituting the above relationships into equation D.3 leads to the following simpli-fied equation, ^ + Z^W = 0 (D.6) Equation D.6 has been solved by Franklin and Scott (1979) for a semi-infinite pile leading to an equation in terms of functions Wi(Z) and W2(Z) and unknown coefficients Cx and c2. Functions W\\ and W2 and their derivatives have been tabulated by Scott (1981). Coefficients Ci and c 2 are evaluated by imposing the pile head boundary condi-tions. Using the sign conventions shown in Figure D . l , the following equilibrium and beam bending relationships hold, dM V = dz d2„ (D.7) dz M 2y E~I ~ Iz2 where V is the shear force and M is the bending moment at a point z along the pile. Appendix D. Static Laterally Loaded Pile Solutions 575 SOIL SURFACE +v V+^ -Y-dz ^ 'A dz ^ z M + l M d z o Z Figure D . l : Pile deflection, rotation, shear force and bending moment positive sign conventions Appendix D. Static Laterally Loaded Pile Solutions 576 Assuming the pile is subjected to moment M0 and shear Vo loading at the soil surface leads to the following boundary conditions at Z = 0, = fy EI dz3 d*W ~ X 1& Mo d2y EI d2W or, Vo \\2EI Mp XEI [ClWi\" + c2W^\"]z=0 [ c ^ ' + c ^ V o (D.9) Using the tabulated derivatives of Wi and W2 evaluated at Z = 0 given by Scott, the following system of equations are obtained from which unknown coefficients c\\ and c2 may be solved for, 0.8923Cl + 0.2904c2 = XEI -0.5846C l + 0.3032c2 = (D.10) XEI Knowing C\\ and c2 one may readily calculate W(Z) or any of its derivatives in terms of the tabulated functions Wi(Z) and W2(Z) and their derivatives for a particular di-mensionless depth Z. Using the relationships given by equations D.4, D.5 and D..7 one can then compute pile deflections, rotations, moments and shear forces along the pile. Appendix D. Static Laterally Loaded Pile Solutions 577 Identical procedures may be followed for other boundary conditions. For example, pile head flexibility coefficients fuv and f$v are derived assuming VQ — 1.0 and Mo = 0.0 at z = 0. This leads to the following simultaneous equations, 0.8923cj + 0.2904c2 = -0.5846c, + 0.3032c2 = VTT ; (D.ll) Solving for c\\ and C2 gives, yo = fuv = \\(ciW1(0) + c2W2(0)) 1 , 0o = fev = CiW{(0) + c2Wf,(0) = -0 .27607 C l - 0.88649c2 (D.12) where fuv is the lateral displacement u at the soil surface resulting from a unit lateral force v and fgv is the corresponding rotation 0. Similarly, fum (= —fgv) and fgm are derived by assuming M0 — 1.0 and V0 — 0.0 at z = 0. Solving equations D.12 for C j and C2 gives the pile head flexibilities for application of a unit moment, y0 — fum and 0Q = fgm. The above flexibilities are derived assuming the pile head is free to translate and rotate under the action of the unit load and moment. The computed flexibilities may then be used to estimate pile defledtions and rotations under combined states of moment and shear loading. Appendix D. Static Laterally Loaded Pile Solutions 578 Static pile head stiffnesses K{j are defined in terms of somewhat different boundary conditions using Novak's notation (Novak, 1974) defined as, Kuu = shear force at the pile head for y0 = 1.0 and 0o = 0.0 KUip = moment at the pile head for y0 = 1.0 and 0o = 0.0 Kj>u — shear force at the pile head for y0 = 0.0 and 60 — 1.0 (= Kutp) K^ = moment at the pile head for y0 = 0.0 and 0o = 1.0 Using the solution for the pile head deflection and rotation at the soil surface and applying the definitions of Kuu and Ku^, given above for yQ = 1.0 and 60 = 0.0 gives the following equations, ci + c 2 = A -0.27607c! - 0.88649C2 = 0.0 (D.13) Solving for ci and c 2 gives, Vo = KUu = A 2 £ / ( 0 . 8 9 2 3 6 c 1 + 0.2904c2) MQ = Ku4> = A £ J ( - 0 . 5 S 4 6 C l +0.3032c2) (D.H) Similarly, using the above definitions for K^u and K^ leads to the following equa-tions, ci + c2 = 0.0 -0.27607cj - 0.8S649c2 = 1.0 (D.15) Appendix D. Static Laterally Loaded Pile Solutions 579 Solving for cx and c 2 gives VQ — K ^ U = X2EI (0.89236d + 0.2904c2) M 0 - Iiecewise cubic spline to the data using procedures described (E.1) 580 Appendix E. Calculation of Soil Resistance - Lateral Pile Displacement Curves 581 in detail by Burden et al (1978). The calculations were carried out using computer pro-gram \"FITPACK\" available in the U B C computer library. The accuracy of the double integration was checked by computing the deflection and slope at the head of the pile. From these, the deflection at the location of the displacement transducer was calculated assuming a rigid connection between the pile and the structural mass. Discrepancies be-tween the measured and computed deflections at the top of the mass inevitably occurred due to the limited number of data points used to define the bending moment distribution and inherent measurement error. In the case of the centrifuge tests, these differences at prototype scale amounted to 0.5 to 1.0 mm which is within the range of accuracy of the displacement transducer. To account for these slight discrepancies, computed deflections along the pile were scaled a constant amount to match the measured deflection. It is noted that pile deflections computed relative to the base are assumed to be approximately the same as deflections relative to the free field ground motions. For reasons discussed in section 4.3, this approximation appears valid since lateral pile response for the test conditions examined is dominated by structural inertia forces. Two different algorithms were used to compute soil reactions p(z) along the pile. The accuracy of each method was assessed by comparing computed p(z) distributions with closed form solutions for a simple test case using static beam theory. The first algorithm has been described by Ting (1987), although complete details of the formulation were not supplied in his paper. The bending moment distribution is assumed to be described by a 7th order polynomial distribution of the form, M(z) = EJ V = z4J2ajZj (E.2) where z is referenced from the tip of the pile. The soil reaction p{z) is related to M(z) using static beam theory by p = cPM/dz2, neglecting pile inertia forces. Estimates of the Appendix E. Calculation of Soil Resistance - Lateral Pile Displacement Curves 582 dynamic pressure contribution due to these inertia forces using a maximum pile accel-eration of 0.25 g and the known mass properties of the pile indicate that the maximum dynamic force contribution per unit length of pile is approximately 0.5 kN/m. This is small relative to the computed soil reactions from static beam theory and consequently has been ignored. The soil reaction p(z) is then given as, P(Z) TF = y • = (j + 4)(j-r3)X>^+2 (E.3) j=o From equation E.2 at each location 2,- of measured bending moment we can write an equation for the error Ei caused by the above polynomial approximation. Thus, fii = !l?-4EV (E.4) j=o We can then form an error function E consisting of the square of the sum of the above individual errors, or E = YIEf. We impose the constraint that at the soil surface z = z\\ the net soil reaction pressure is zero since we have carried out the pile loading tests in a cohesionless sand. A constraint equation g can then be written as, ? ° E ^ 1 E ^ 2 -15zf E * , 1 1 E * , 1 2 E * « 1 3 E * t 1 4 -21z4 -6zl -10z3 -I5zf - 2 l 2 x 5 0 a 0 ai « 2 as / E * 4 < ^ J2z?y\" E * f t f T.zjy'l 0 (E.7) where y\" = Mi/EI is the pile curvature at locations 2,- below the soil surface. At each point in time during a load cycle, equation E.7 is solved. This gives the unknown coefficients a,- and the soil reaction distribution p(z) from which p can be computed for any desired depth. The second algorithm used to compute soil reaction pressures involves taking second derivatives of the cubic spline fitted to the measured bending moment data. The second derivatives were evaluated using the computed program \"FITPACK\". As noted by Ting, slight errors or deviations in the measured bending moments become greatly magnified during double differentiation. This is shown in the following worked example. E.2 Comparison of Method W i t h Cubic Spline Differentiation A cantilever beam of length / (Figure E. l ) is loaded by a moment, M0, and shear force, Vo, at x = 0. The beam is also loaded by a lateral force distribution p(x) given by, '9.7 p(x) = -q sin (E.8) where q is the load amplitude and £ is a factor describing the decay of the lateral force along the length of the beam. Appendix E. Calculation of Soil Resistance - Lateral Pile Displacement Curves 584 SOIL SURFACE \"777- /// p(x) = -qSin e _ X x PILE TIP Figure E . l : Lateral loading of a cantilever beam - test case Appendix E. Calculation of Soil Resistance - Lateral Pile Displacement Curves 585 Applying basic statics the bending moment distribution along the beam is given by, Using the input parameters M0 = 200 kN-m, VQ = 90 kN, / = 12.0 m, q = 65.0 kN/m and £ = 0.15 the bending moment computed along the pile is shown in Figure E.2. This distribution is typical of that measured during the single pile tests. The exact bending moment distribution is assumed to be \"sampled\" at 7 discrete points along the pile within a range of accuracy of ±5 kN-m. The data points used in the analysis are shown in Figure E.2 and do not lie precisely on the exact bending moment distribution. This is intended to show the effect of potential measurement error on computed lateral reaction forces along the pile. Using the sampled data points in the previous two algorithms and assuming a flexural rigidity EI = 172,614 kN-m 2 which represents the rigidity of the prototype pile in the centrifuge tests, lateral pressures along the beam have been computed (Figure E.3). Ting's \"Lagrange multiplier\" method gives an excellent prediction of lateral reaction forces along the beam while the \"cubic spline\" procedure is shown to produce erratic results due to slight errors in the input bending moment data. Based on the above comparison, the Lagrange multiplier algorithm has been used to computed lateral soil reactions from the experimental bending moment data. Experience with the method shows that it is a robust procedure and provides reliable results. M(x) = M0 + V0x + jr F2i^(-icos(F2x) + F2sm(F2x)) + jr | | [e-«* (-£ sm(F2x) - F2 cos(F2x)) + F2] - -+ (E.9) where Fx = £ 2 + (2TT//)2 and F2 = 2TT//. Appendix E. Calculation of Soil Resistance - Lateral Pile Displacement Curves 586 12 H 1 1 1 1 1 1 1 1 1 0 100 200 300 400 500 Bending Moment (kN-m) Figure E.2: Bending moment distribution along laterally loaded beam - test case Appendix E. Calculation of Soil Resistance - Lateral Pile Displacement Curves 587 Figure E.3: Distribution of lateral reaction force p(x) along a laterally loaded beam -test case Appendix F Single Pile Response in a Winkler M e d i u m to Base Mot ion Excitation F . l Equations of Mot ion The undamped equations of motion of a single pile supporting a structural mass above the soil surface (see Figure F. l) are given as follows: „r84y 82y , , S2ua ,„ , ii4 + miF + sV = sU ~ m~w ( } where y is the relative pile displacement with respect to the moving base, u is the lateral free field motion with respect to the base, ug is the time dependent lateral base motion, EI is the pile flexural rigidity, ra is the mass per unit length of the pile, and ks is the constant with depth, Winkler subgrade modulus. The assumption of a constant with depth Winkler modulus is not quantitatively correct since it varies with depth and depends on pile deflection level, the shear stiffness properties of the soil and the relative soil-pile stiffness (Kagawa and Kraft, 1980a,1981a). This assumption does, however, enable a closed form solution to be obtained and illustrates in a qualitative sense some interesting aspects of the seismic behaviour of piles. Equation F . l represents the dynamic behaviour of the pile above the soil surface (region A) while equation F.2 represents the dynamic pile response below the ground (region B). 588 Appendix F. Single Pile Response in a Winkler Medium to Base Motion Excitation5S9 REGION @ R E G I O N ® MASS m 0 ROTATIONAL MOMENT OF INERTIA, I CG Z — ^r'r> SOIL<^ —rrr 7, TTT? k s -WV— -vw—1 -wv—I PILE EI, MASS/LENGTH m Z=H SOIL-PILE INTERACTION GOVERNED BY CONSTANT LATERAL SPRING CONSTANT k. Z = 0 Ug=U 0 e iwt Figure F . l : Single pile model Appendix F. Single Pile Response in a Winkler Medium to Base Motion Excitation590 F.2 Free Vibrat ion Response The free vibration solution to equation F . l , in which the right hand side of the equation is set equal to zero, is given by the following solution (Clough and Penzien, 1975), y(z,t) = ^z)e^ (F.3) where $(z) is a shape function given by $(z) = A' sin az + B' cos az + C' sinh az + D' cosh az (F.4) and a 4 = u>2m/EI. Similarly, the free vibration solution to equation F.2 is given by equation F.3 where = e^(Acos 0z + B sin Bz) + e~0'(C cos Bz + D sin Bz) (F.5) and / , 9 \\ 0 - 2 5 The 8 unknown integration constants A through D and A' through D' are solved for by using the 4 boundary conditions at the pile tip (z — 0) and at the pile head (z = H + z), plus 4 deflection and force compatibility conditions at the soil surface (z = H). Assuming zero moment and shear at the pile tip gives the following two boundary conditions for $(z) in region B: (*a=o = 0 (F.7) (*\"),=o = 0 ( F- 8) where the primes denote differentiation with respect to depth z. Considerations of shear force and moment equilibrium at the pile head, taking into account the inertia and finite size of the structural mass, gives the following additional boundary conditions in region A: EI mz=H+, + \"»ow3 mz=H+-z + mohu2 mz=H+, = 0 (F.9) Appendix F. Single Pile Response in a Winkler Medium to Base Motion Excitation591 -mohW mz=H+J = 0 (F.10) where mo is the mass, Icg is the mass moment of inertia of the structural mass, and h is the distance between the top of the pile and the centre of gravity of the structural mass. Compatibility of deflection, slope, shear force and moment at the soil surface leads to the following relationships between the shape functions $(z) in regions A and B: (HH))A = (*(H))B (F. l l ) (nH))A = ($'(H))B (F.12) ($\"(H))A = (*\"(#))B (F-13) (*m(H))A = (*'\"(H))B (F.14) Using the above boundary and compatibility conditions, and evaluating the various shape functions and their derivatives at the pile tip, soil surface and the pile head, leads to a matrix equation of the form [C] {a} = {0} (F.15) where [C] is an 8 x 8 matrix involving the structural properties of the pile and pile head mass, the Winkler soil stiffness and the frequency parameter u. The vector {a} contains the 8 unknown integration constants. In order for non-trivial solutions of {a} to exist, the determinant of [C] must equal zero. The determinant will be zero only for selected values of w, providing a multiplicity of solutions COJ, j = 1,2,... ,n , which are the eigenvalues or natural frequencies of the soil-pile system. The procedure used to determine these eigenvalues involves varying LOJ incrementally, computing the determinant of [C] for each Uj selected, and finding an eigenvalue whenever det[C] = 0. Appendix F. Single Pile Response in a Winkler Medium to Base Motion Excitation592 F.3 Forced Vibrat ion Response Considering equation F.2, we seek a solution of the form y — yh -\\- yp where yp and y^ are particular and homogeneous solutions, respectively. Following the solution procedure outlined by Flores-Berrones and Whitman (1982), the particular solution is assumed to have a form similar to the solution for the free field ground motion, u(z,t). The latter is obtained from the solution to the shear beam equation for the case of constant shear wave velocity versus depth. The governing undamped shear beam equation is, 82u TriS2u 82u„ Jp - = ( F - 1 6 ) Assuming zero relative displacement between the rigid base and the soil and that shear stresses are zero at the soil surface gives the following boundary conditions: 00,=o = 0 (F.17) ( £ L -0 (R18) The solution to the shear beam equation subject to the above boundary conditions is (Flores-Berrones and Whitman, 1982), u(z, t) = u9(t) [cos 8z -f tan foY sin for — 1] (F.19) where ug(t) = u0etwt, 8 = u>0/Vs, OJ0 is the frequency of input base motion, V^ , is the free field shear wave velocity, and UQ is the amplitude of input base motion. Thus, yp is assumed to be of the form: yp(z, t) = (A + B cos 8z + C sin Sz) eiwot (F.20) Substituting equation F.20 into equation F.2 and collecting similar terms leads to yp(z,t) = u0 {r(cosfor + tan SH sin Sz) - 1} eiwot (F.21) Appendix F. Single Pile Response in a Winkler Medium to Base Motion Excitation593 where T = ks/(EIS4 + ks -miwg). The homogeneous solution is given by the free vibration solution for all n modes of vibration. Thus, yk(z,t) = f2*i(z)ei\"it (F.22) 3=1 where $j(z) is given by equation F.5 for each of j natural vibration modes and ujj is the jth natural frequency of the soil-pile system. It is noted that $j(z) contains 4 unknown integration constants Aj through Dj for each natural vibration mode. The total solution y(z,t) for region B is then given by n V(z, *) = 12 ^ j(z)eiu''t + u0 {r(cos Sz + tan SH sin Sz) - 1} eiwot (F.23) Considering the region of the pile above the soil surface (region A) whose behaviour is governed by equation F . l , the total solution is again given by the sum of a homogeneous (free vibration) and particular (forced) solution. It may be shown that a particular solution that satisfies equation F . l is given by yp(z,t) = -u0eiwot (F.24) where u0 is the displacement amplitude of the base motion, noting that ug = — u>lu0e%wot in equation F . l . The total solution for region A is then given by y(z,t) = f :* i (z)e^* -uoe^ (F.25) 3 = 1 where $j(z) is given by equation F.4 for each of j natural vibration modes, incorporating the 4 unknown integration constants Aj through Dj. The 8 unknown integration constants for each mode may be solved by imposing the 8 boundary and compatibility conditions given in equations F.7 to F.14. Since the available experimental evidence suggests that the pile vibrates substantially in its first mode, i.e. the deflected shape of the pile at any instant in time is similar to that observed Appendix F. Single Pile Response in a Winkler Medium to Base Motion ExcitationbQA under static pile head loading and does not exhibit any 'cross-over' points, the above integration constants are solved only for the first mode response (j=l). An identical approach has been adopted by Flores-Berrones and Whitman and also by Kagawa and Kraft. Assuming zero moment and shear at the pile tip gives &!'(z = 0)] e*'w>' -u0TS3 tan SHeiuJot = 0 &!(z = 0)] eih,>* -u0VS2eiuot = 0 (F.26) (F.27) where the derivatives of $(z) applicable to region B may be evaluated from equation F.5. Imposing compatibility of deflection, rotation, shear and bending moment at the soil surface (z = H) gives sin2<$fT [* = {^(z = H)] A e«'w>* (F.29) sin 2 SHy cos SH t '&j'(z = H)\\ e^' = [&!'(z = H)] eiw>1 Imposing moment and shear force equilibrium at the pile head (z = H + z) leads to (F.31) EI [&?(z = H + z)] a e'\">< = -moco2 = H + z)]A e^* -m0hujl $j(z = H + z) IA (F.32) EI &;{z = H + z)\\ a eiu'* = moto2h (z = H + z))A ^ + &j(z = H + z)] e^' (F.33) Appendix F. Single Pile Response in a Winkler Medium to Base Motion Excitation595 where 70 = Icg + m0h2. It will be noted subsequently that the pile vibrates at frequency OJQ, leading to inertia terms proportional to COQ. Equations F.27 to F.33 lead to a system of simultaneous equations of the form, [C]i=i {*) = W eiuot-iuJlt (F.34) where [C]j = 1 is an 8 x 8 matrix for the first mode whose terms are a function of u>i and C L > 0 , {X} is a vector of unknown modal integration constants Aj-\\ through Dj=i and A'j=l through D'j=1 and {b} is a collection of known 'forcing' terms. Solving equation F.33 for {x}, which are multiplied by the term e*wo*-«wi^ a n ( j substituting these into the total solution y(z,t) for region A or B, it is seen that the pile vibrates at the forcing frequency OJ0 of the base motion. A similar result has been given by Flores-Berrones and Whitman who have not considered the effect of pile head stick-up in their solution. Once the integration constants have been determined, the solution y(z,t) for region A or B is readily obtained, from which moment and shear force distributions along the pile have been derived using elementary beam theory. Appendix G Uncoupled Solution for a Pile - Structural Mass System G . l Equations of Mot ion The undamped equations of motion of a single pile supporting a structural mass above the soil surface and linked to the below ground portion of the soil-pile system by foundation compliances (see Figure G.l) are given as follows: r,^S4y 82y 62ua , EIsT<+mW = -™lF where y is the relative pile displacement with respect to the input free field surface motions, ug is the absolute ground displacement applied to the free field end of the foundation compliance springs, EI is the pile flexural rigidity, and rn is the mass per unit length of the pile. G.2 Free Vibrat ion Response The free vibration solution to equation G . l , in which the right hand side of the equation is set equal to zero, is given by the following solution (Clough and Penzien, 1975), y(z,t) = ^(z)eiut (G.2) where $(z) is a shape function given by $(z) = A' sin az + B' cos az + C sinh az + D' cosh az (G.3) and a4 = u>2m/EI. The 4 unknown integration constants A' through D' are solved for by using the 4 boundary conditions at the soil surface (z — 0) and at the pile head (z = I). 596 Appendix G. Uncoupled Solution for a. Pile - Structural Mass System 597 FIXED REFERENCE U 8 - J kuu AAA-MASS m 0 , ROTATIONAL MOMENT OF INERTIA T^g Z= L - EI, m z = o FOUNDATION LEVEL (SOIL SURFACE) Figure G . l : Single pile model - uncoupled analysis Appendix G. Uncoupled Solution for a Pile - Structural Mass System 598 The shear force and moment in the pile at the soil surface must equal the lateral force and restraining moment induced by relative movement between the pile and the free field surface motions. This gives the following two boundary conditions for $ ( 2 ) at the soil surface: - EI&\"(0) = fcuu$(0) - ku*&(0) (G.4) £ I $ \" ( 0 ) = k^'(0) - V $ ( 0 ) (G.5) where the primes denote differentiation with respect to depth z, and kuu, k^, ku^, and k$u represent the real components of the elastic foundation stiffness. It is noted that ku^, equals k^,u for an elastic system according to Betti's law. This, however, is not necessarily true for an inelastic system. Considerations of shear force and moment equilibrium at the pile head, taking into account the inertia and finite size of the structural mass, gives the following additional boundary conditions: EI®\"(1) + m0uj2(l) + m0huj2$'(l) = 0 (G.6) EW{1) - mouj2h$(l) - u2Icg$'(l) - m0h2uj2^'(l) = 0 (G.7) where ra0 is the mass, Icg is the mass moment of inertia with respect to the centre of gravity of the structural mass, and h is the distance between the top of the pile and the centre of gravity of the structural mass. Using the above boundary and compatibility conditions, and evaluating the various shape functions and their derivatives at the soil surface and the pile head, leads to a matrix equation of the form [C] {a} = {0} (G.8) where [C] is a 4 x 4 matrix involving the structural properties of the pile and pile head mass, the pile head stiffnesses kjj and the frequency parameter u. The vector {a} contains Appendix G. Uncoupled Solution for a Pile - Structural Mass System 599 the 4 unknown integration constants. In order for non-trivial solutions of {a} to exist, the determinant of [C] must equal zero. The determinant will be zero only for selected values of CJ , providing a multiplicity of solutions u>j, j' = 1, 2,. .. , n, which are the eigenvalues or natural frequencies of the pile-structural mass system. The procedure used to determine these eigenvalues involves varying Uj incrementally, computing the determinant of [C] for each u>j selected, and finding an eigenvalue whenever det[C] = 0. G.3 F o r c e d V i b r a t i o n R e s p o n s e Considering equation G.l , we seek a solution of the form y = Vh + Vp where yp and yh axe particular and homogeneous solutions, respectively. The homogeneous solution is given by the free vibration solution for all n modes of vibration. Thus, yh(z,t) = ± ^ y u \" t (G.9) i=i where <&j(z) is given by equation G.3 for each of j natural vibration modes and u>j is the jth natural frequency of the soil-pile system. It is noted that $j(z) contains 4 unknown integration constants Aj through Dj for each natural vibration mode.lt may be shown that a particular solution that satisfies equation G.l is given by yp(z,t) = -ugei\"°i (G.10) where ug is the absolute displacement of the free field surface motion, noting that ug = —ujQUgexwot for a harmonic input motion. The total solution is then given by »(z,*) = E* iWe i , ' > £ -« , c f a * 1 (G.ll) The 4 unknown integration constants for each mode may be solved by imposing the 4 boundary conditions given in equations G.4 to G.7. Since the available experimental Appendix G. Uncoupled Solution for a Pile - Structural Mass System 600 evidence suggests that the pile vibrates substantially in its first mode, the above inte-gration constants are solved only for the first mode response (j = 1). The boundary condition at the soil surface is expressed as, - EI&l'MJ\"** - {kuu$(0) - K^'(0)} eiwS + kuuugeiwot = 0 (G.12) El&jWe™* - {^$'(0) - ^$(0)} e«'^ - k^uugeiuot = 0 (G.13) where the derivatives of i and u>0, {%} is a vector of unknown modal integration constants A'^ through Dj=1 and {&} is a collection of known 'forcing' terms. Solving equation G.16 for {x}, which are multiplied by the term e\"\"0'-\"\"1', and substituting these into the total solution y(z,t), it is seen that the pile vibrates at the forcing frequency UJ0 of the base motion. Once the integration constants have been determined, the solution y(z,t) is readily obtained, from which moment and shear force distributions along the pile have been derived using elementary beam theory. Appendix G. Uncoupled Solution for a Pile - Structural Mass System 601 G.4 Pile Head Stiffnesses To compute the dynamic pile head stiffnesses necessary for use in the uncoupled analysis, one solves the governing differential equation for a pile undergoing forced vibration due to harmonic pile head loads. The soil reaction pressure is given by p(z, t) = khy(z)etult where the lateral Winkler modulus kh, is assumed to be constant with depth. This assumption is identical to the one made when solving the coupled soil-pile response to base motion excitation (see Appendix F) , thereby permitting a comparison of the two approaches. The governing differential equation to be solved is E i 2 ; + m j £ + k h y = 0 ( G - 1 7 ) The above equation is a simplification of the differential equation assuming the pile is subjected to base motion excitation and that the soil pressure is due to relative motion between the pile and the free field soil. Pile head compliances are commonly derived using the above simplification, and are inherent in the use of computer programs developed by Novak and his co-workers for use in seismic response analysis. This approach assumes that the stiffness properties of the soil-pile system are dominated by soil reactions generated near the pile head due to the action of structural inertial loads rather than by the action of the free field ground motions. Assuming that the pile is subject to steady state harmonic excitation at the head of the pile gives a solution of the form y{z,i) = $(z)e l u , i, where u> is the frequency of excitation. The pile motion is thereby induced through the boundary conditions of the problem. Substituting this solution into equation G.l7 leads to an ordinary differential equation of the form iv(z} + 43H(z) = 0 (G.18) Appendix G. Uncoupled Solution for a Pile - Structural Mass System 602 where ( 7 —- 2\\ ° - 2 5 The solution to equation G.l8 is t) = [ePz (A cos Sz + B sin Bz) + e~0z (C cos Bz + D sin Bz)] eiwt (G.20) Assuming zero moment and shear force at the pile tip (z = 0) leads to the following two boundary conditions $\"'(0) = 0 (G.21) $\"(0) = 0 (G.22) At the ground surface (z = H) we are interested in deriving the pile head forces and moments occurring in response to unit pile head translations and rotations. Con-sidering unit pile head translation under fixed head conditions, the following boundary conditions are obtained = 1-0 (G.23) = 0.0 , (G.24) From equations G.21 to G.24 which involve the unknown coefficients A through D, the following matrix equation results [C){x} = {b} (G.25) where {b}T =< 0,0,1.0,0 >, [C] is a 4 by 4 matrix involving the structural properties of the pile, frequency ui and the Winkler stiffness parameter kh, and {x} contains the 4 unknown integration constants. Solving equation G.25 for the integration constants, we obtain $ ( 2 ) from which the spring constants kuu and k^u are obtained as follows, kuu = -EW\\H) (G.26) fc*tt = EW{H) (G.27) Appendix G. Uncoupled Solution for a Pile - Structural Mass System 603 Considering unit rotation of the pile head gives the following additional boundary conditions = 0.0 (G.28) = 1.0 (G.29) which changes the vector {b}T to the form < 0,0,0,1.0 >. Solving for the coefficients A through D using equation G.25 and hence <&(z) gives the following spring constants, ku4> = -EI&\"(H) (G.30) le** = EW\\H) (G.31) The frequency dependent pile head stiffnesses have been computed over a range of frequencies of interest to the present investigation. These have been found to be reason-ably independent of frequency, implying that static pile head stiffness values can be used in the uncoupled analysis. Appendix H Finite Element Solution for a Pile - Structural Mass System H . l Equations of Mot ion The uncoupled analytic solution to the problem of a single pile subjected to earthquake shaking described in Appendix G has not considered the effects of foundation damping. To take this into account it has proved convenient to discretize the superstructure using finite elements. A beam element has been used to represent the free standing portion of the pile while a rigid mass having translational and rotational inertia properties has been used to represent the structural mass. The stiffness matrix of the beam element has been derived using a cubic displacement field to represent the deflected shape of the pile, considering rotational and translational (horizontal) degrees of freedom at the endpoints of the beam element. The distributed mass properties of the pile have been represented using a consistent mass matrix. The resulting 4 by 4 stiffness and mass matrix are given in Clough and Penzien (1975). Using principles of virtual work, the virtual work done by the inertia forces acting on the beam-rigid mass system, 8Wj, is given by SWi = (8f)T[mb]{r} + m0x^Sx^ + Icg0cg60cg (H.l) where [mj is the consistent mass matrix for the beam element expressed in terms of the global degrees of freedom, {f } is the global degrees of freedom of the structure (see Figure H.l) expressed in terms of absolute displacements relative to a fixed reference line, xcg and 0cg are the lateral translation and rotation, respectively, at the centre of gravity of the structural mass, and mo and Icg are the mass and rotational mass moment 604 Appendix H. Finite Element Solution for a Pile - Structural Mass System 605 FREE STANDING PILE- MASS SYSTEM ± AZ FIXED REFERENCE STRUCTURAL MASS REPRESENTED AS A LUMPED MASS WITH MASS m 0 AND MASS MOMENT OF INERTIA, I CG E l , m MOVING BASE FOUNDATION COMPLIANCES ( , Cfj- ) Figure H . l : Structural layout assumed for finite element discretization of pile-structural mass system (uncoupled analysis) of inertia of the structural mass defined with respect to its centre of gravity. Referring to Figure H . l , the absolute displacements may be expressed in the following form, ~x~c~g~ = ug(t) + r 3 + r4Az (H.2) where ug(t) is the time variation of the input free field surface motion defined with respect to a fixed reference frame, Az is the distance between the top of the pile and the centre of gravity of the structural mass, is the lateral translation at the top of the beam element relative to the input motion ug(t), and r 4 = 0cg is the rotation of the structural mass. The latter relation assumes a rigid connection between the beam element and the mass, and that rotational free field surface motions are negligable. Using the above Appendix H. Finite Element Solution for a Pile - Structural Mass System 606 relationships, the following equations hold Up + f3 + f4Az (H.3) SxTg = 8r3 + 8r4Az (EA) noting that since ug(t) are the prescribed time variation of the input free field motions, their virtual variation is zero. The relationship between the relative and absolute global degrees of freedom of the system, neglecting rotational free field surface motions, may be written as {r} = {r} + (I) ug(t) (H.5) where (I)T = (1,0,1,0). Substituting equations H.2 to H.5 in H.l gives 6Wr = {6{r})T[mb][{r} + (I)ug] + m0ug8r3 + moAzug8r4 + m0f36r3 + m0Azf38r4 + m0Azf48r3 + (m0Az2 + Icg) r\\8rA (H.6) Equation H.6 may be simplified to a matrix equation of the form m = (8{r})T[M] {f} + {8{r}f{P(t)} (H.7) where [M] is the global mass matrix incorporating the mass properties of the pile and the rigid structural mass and {P(t)} is the load vector which depends on the time dependent free field surface motions. The symmetric global mass matrix is given as 156m/420 22ro//420 54?7z/420 -13ro//420 4m/2/420 13?7i//420 -m/2/140 156m/420 + m o -22m//420 + m0Az . . . m/2/105 + / o [Af] = (H.8) Appendix H. Finite Element Solution for a Pile - Structural Mass System 607 where m = ml, m is the mass per unit length of the beam, / is the free-standing length of the pile above the soil surface and IQ — ICG + mnAz 2. The load vector {P(t)} is defined as {P(t)) \\ (H.9) ml/2 m/ 2/12 ml / 2 + m0 \\ -ml21'12 + m0Az J The virtual work due to the elastic strain energy in the pile and foundation springs, 8WS, is given by £Wa = {&{r})T[h]{r} + fcuurx^i + k^r2Sr2 — k^uriSr2 - ku^r28rx (H.10) where kuu, k^, and k^u{= ku^) are the equivalent elastic pile head stiffnesses. The above equation may be expressed in the form SWS = (6{r})T[K] {r} where [K] is the symmetric global stiffness matrix given by ( H . l l ) (H.12) 12EI/l3 + kuu 6EI/l2-ku^ -12EI/13 6EI/12 AEI/l + kw -6EI/12 2EI/1 12EI/13 -6EI/12 AEI/l where EI is the flexural rigidity of the pile. During shake table testing, additional spring forces acted on the pile head mass due to the presence of the LVDT's. The LVDT's were placed on opposite sides of the pile head mass so their effect on the combined stiffness of the soil-pile system was partially Appendix H. Finite Element Solution for a Pile - Structural Mass System 608 (H.13) self-cancelling. By inspection, it can be shown that the stiffness matrix changes to the following form: l2EI/l3 + kuu 6EI/l2-ku^ -12EI/13 6EI/P AEI/l + k^ -6EI/12 2EI/1 12EI/13 -6EI/l2 + k*mu 4EI/l + k*mm where kmu — ks(ht — hb), k*mm = k3(h2 — h2), ks is the measured L V D T spring force (= 0.041 N/mm), and ht and hb are the respective distances between the L V D T placed near the top and bottom of the structural mass and the top of the pile. Neglecting internal damping of the pile cross-section, the virtual work done by the equivalent viscous damping forces acting on the embedded portion of the pile is given by 8Wd = cuurx8rx + c^r28r2 — c^urx8r2 — cu^r28ri (H.14) where cuu, and c^u(— cu^) axe the pile head equivalent viscous dashpots. The above equation may be expressed in the form 8Wd = (8{r})T[C]{r} where [C] is the symmetric foundation damping matrix given by (H.15) [C] = (H.16) 0 0 0 0 0 The equations of motion of the uncoupled, viscously damped system may therefore be written in the form [M}{r} + [C]{r} + [K}{r} = {P(t)} (H.17) Appendix H. Finite Element Solution for a Pile - Structural Mass System 609 H.2 Solution of Equations of Mot ion Equation H.l7 has been solved using a time step integration procedure based on a constant average acceleration in each time increment At (Clough and Penzien, 1975). The latter procedure provides an unconditionally stable solution. Knowing the initial displacement, velocity and accelerations conditions at time t we can write {r(t + At)} = {r(t)> + Y {*(*)} + f{r(t + A*)} (H.18) At2 At2 {r(t + At)} = {r(t)} + {r(t)At + — { f («)} + ^ -{r(* + At)} (H.19) At time t = t + At, the equation of motion of the system is given as [M]{r(t + At)} + [C]{r(t + At)} + [K]{r(t + At)} = {P{t + At)} (H.20) Substituting equations H.18 and H.19 into equation H.20 gives an equation of the form [K*]{r(t + At)} = {P*(t + At)} (H.21) where [ir] = [M} + {C]f + [K]^- (H.22) and {P*(t + At)} = {P(t + At)} - [C] At2 {r(0} + {r(*)}Ai + — {r(t)} (H.23) We may solve equation H.21 using Gaussian elimination or an equivalent matrix decomposition scheme to give the nodal accelerations at time t + At from which we can solve for the nodal velocities and displacements using equations H.18 and H.19. One then proceeds to the next time step. The time step chosen in the integration procedure Appendix H. Finite Element Solution for a Pile - Structural Mass System 610 is chosen equal to 0.10 times the lowest natural period of interest in the system. Since only the first two modes of the system are of interest in the structure response, a time step equal to O.IOT2 has been chosen where T2 is the natural period of the second mode. To account for the experimental observation that the pile head stiffnesses, R~ij, vary in a non-linear manner with the amplitudes of pile vibration, Kij was expressed in the following functional form: where y0 is the amplitude of pile vibration at the soil surface, {K{j)o is the pile head stiff-ness at zero pile head deflection and m is an experimentally determined slope parameter. For each time step, knowing the pile deflection at the end of the preceding time step, the pile head stiffnesses are readily evaluated using the above expression. The global stiffness matrix [K] is then calculated. To start the time step integration it it necessary to know the initial conditions of the problem. It is assumed that at t = 0 the initial nodal displacements and accelerations are zero. The initial velocities of the system {r(0)} are evaluated based on impulse-momentum considerations, described as follows. From Newton's law at t = 0, log Kij = log(/v t j ) 0 - m\\y0\\ (H.24) (H.25) where {Fo} = {P(0)} - [/v]{r(0)} (H.26) - W O ) } (H.27) From impulse-momentum considerations over time t = 0 to t + At, (H.28) Appendix H. Finite Element Solution for a Pile - Structural Mass System 611 or, (H.29) Equation H.29 is then used to solve for the initial nodal velocities of the system. At each time step At the nodal displacements, velocities and accelerations are solved for. Accelerations at any other point on the rigid structural mass are readily solved for from considerations of rigid body kinematics. Knowing the acceleration at the centre of gravity of the structural mass, xcg, the shear force acting on the pile at the soil surface, V0, is then given as since inertia forces due to accelerations of the pile cross section relative to those of the pile head mass are small. The bending moment at the soil surface, M 0 , may be evaluated using the beam bending equations knowing the nodal displacements and rotations of the pile. Alternatively, MQ may be calculated using the following equation, The rotational accelerations of the pile head mass r4 are evaluated by expressing the pile top rotations r4 in terms of a Fourier series having a sufficient number of terms to adequately capture the significant frequenc}' components of the pile head response. The rotational accelerations are then calculated from the second derivative of each term in the series. This procedure has been found to effectively filter spurious high frequency response in the computed rotational accelerations. V0 = m0xcg (H.30) M0 = m0xcgZcg + Icgr4 (H.31) Appendix I Shake Table Tests - Low Level Shaking of 2-Pile Groups 1.1 Test Data Sinusoidal base motions used during low level shaking of the two pile groups spaced at an s/d ratio of two and subjected to inline shaking are shown in Figure I.la. The input motions are seen to be unsymmetric which is a feature of the shake table B actuator and has resulted in unsymmetric pile group response. Time histories of the free field surface accelerations, pile cap displacement (LVDT 1) measured approximately 4 mm above the base of the pile cap, and pile cap accelerations at the centre of gravity of mass are shown in Figure I.lb-d. The lack of symmetry in the pile group response is clearly evident. Pile cap accelerations are seen to be approximately in phase with the input base and free field accelerations. Pile cap displacements are correspondingly 180 degrees out of phase with the measured pile cap accelerations. Fourier spectra computed from the input base, free field and pile cap acceleration time histories are shown in Figure 1.2. These show that the free field surface motions contain a number of higher frequenc}' components that are significantly amplified around the resonant frequency of the sand layer at about 60 Hz. The pile cap accelerations do not contain these high frequency components, suggesting that the pile group filters out the higher frequency ground motions. This is to be expected since the fundamental frequency of the pile group is much less than the resonant frequency of the sand foundation. Bending moment distributions measured along the piles with centre to centre spacings 612 Appendix I. Shake Table Tests - Low Level Shaking of 2-Pile Groups 613 Figure L I : Pile group response - low level inline shaking (s/d — 2) (a) base accelera-tion (b) free field surface acceleration (c) acceleration at the e.g. of mass (d) pile cap displacement (LVDT 1) Appendix I. Shake Table Tests - Low Level Shaking of 2-Pile Groups 614 (a) 20 \"40 60 80 Frequency (Hz) 100 120 140 (b) 20 40 60 80 100 Frequency (Hz) (C) 100 XI Q_ E cn 40 60 80 100 Frequency (Hz) 140 Figure 1.2: Computed Fourier spectra - low level inline shaking (s/d — 2) (a) base acceleration (b) free field acceleration (c) acceleration at the e.g. of mass Appendix I. Shake Table Tests - Low Level Shaking of 2-Pile Groups 615 of two are shown in Figure 1.3 for both inline and offline loading. Peak bending moments averaged over the steady state region of response have been plotted for both directions of shaking to illustrate changes in loading intensity (if any) on each pile as the shaking direction reverses. For the offline test, the bending moments in both piles are reasonably similar for both directions of loading, suggesting that the piles may be regarded as equally loaded along their length. Identical behaviour was seen for larger pile spacings. During inline loading, the bending moments in the piles have the same general distribution indicating that the piles are vibrating in phase. The bending moments are sufficiently different to indicate that non-uniform soil pressures act on each pile as a result of pile to pile interaction. The slope of the bending moment distributions near the soil surface suggests that the shear load in each pile varies by about 13 percent from the average in the group. It is observed that when the shaking direction reverses, the maximum loading and hence bending moment changes from one pile to the other. During one half cycle of loading the \"front\" pile in the group attracts more load than the trailing \"rear\" pile which has moved into a softened soil zone created by the front pile. During the next half cycle of loading, the above process reverses. This is referred to as the \"shadowing effect\"and has also been observed during centrifuge tests carried out during the present investigation. Bending moment distributions for inline shaking using a centre to centre pile spacing of 6 pile diameters are shown in Figure 1.4. The bending moments do not vary signifi-cantly between piles, suggesting that the piles are equally loaded and that there is a low degree of interaction between them. Test measurements for low level shaking are summarized in Table 1.1 where peak average values of pile cap acceleration, pile cap displacement (LVDT 1), input base and free field acceleration are given. Average displacement and acceleration quantities Appendix I. Shake Table Tests - Low Level Shaking of 2-Pile Groups 616 (a) 400-1 : 1 1 Figure 1.3: Bending moment vs. depth in a two pile group - low level shaking for s/d = 2 (a) offline shaking (b) inline shaking Appendix I. Shake Table Tests - Low Level Shaking of 2-Pile Groups 617 Soil Surfoce X i A A Pile 1 (+) X Pile 2 (+) • Pile 1 (-) B Pile 2 (-) ffi Avg. Curve (+) X Avg^Curye_(-) Soil Surface -100 - 5 0 0 50 Bending Moment (N-mm) 100 Figure 1.4: Bending moment vs. depth in a two pile group - low level, inline shaking -s/d = 6 Appendix I. Shake Table Tests - Low Level Shaking of 2-Pile Groups 618 Shaking Direction s/d Xb Xff xcg A x (g) (g) (g) (mm) Offline 2 0.063 0.082 0.130 0.13 (0.14) « 4 0.064 0.080 0.118 0.11 (0.12) a 6 0.075 0.100 0.125 0.12 (0.11) Inline 2 0.085 0.115 0.152 0.10 (0.11) 4 0.092 0.105 0.122 0.05 (0.05) 6 0.084 0.112 0.105 0.04 (0.04) xb = average peak input base acceleration iff = average peak free field acceleration at the soil surface 'xcg = average peak acceleration at the centre of gravitjr of the pile cap/structural mass assembly Ai = pile cap displacement (LVDT 1) recorded 4 mm above the base of the pile cap (figure in parentheses refers to corrected value scaled to 0.088 or 0.069 g for inline and offline shaking, respectively) Table LI: Pile Group Test Data - Low Level Shaking for each test have been computed as peak to peak values in the steady state region of response divided by two. Peak input base accelerations for the offline tests varied between 0.063 and 0.075 g. Free field surface accelerations were amplified to values between 0.08 and 0.10 g with the higher values corresponding to the higher input acceleration levels. Pile cap accelerations were relatively constant between 0.12 and 0.13 g. For the inline tests, peak base accelerations and free field surface accelerations ranged between 0.084 to 0.092 g and 0.105 to 0.115 g, respectively. Pile cap accelerations varied between 0.15 and 0.10 g and were highest when the piles were closely spaced. This occurs since the rotational stiffness of the group, which is determined to a large extent by the spacing of the piles and the axial reactions of the piles on the cap, is minimized. Average pile cap displacements are plotted against the pile spacing ratio, s/d, for ratios between 2 and 6 for both inline and offline shaking in Figure 1.5. The measured displacements were all scaled to the same average peak base acceleration given in Table Appendix I. Shake Table Tests - Low Level Shaking of 2-Pile Groups 619 1.1, appropriate to whether the group was loaded in an inline or offline direction. This was done to account for slight differences in input acceleration level and is considered acceptable for small variations in the levels of input excitation. For offline shaking using a pile spacing ratio s/d = 2, pile cap displacements are seen to be slightly larger than displacements measured for larger pile separations. Past a spacing ratio of 4, displacements stay relatively constant within a band of ±0.01 mm. This is within the range of accuracy of the L V D T and indicates there is little increase in group deflection due to interaction between the piles. During inline shaking, variations in measured displacement do not exceed 0.01 mm beyond centre to centre spacings of 4 pile diameters, suggesting that interaction effects are small. For smaller pile spacings (s/d = 2) measured displacements exceed those for larger spacings by about 0.05 mm. This is partly due to differences that occur in the average moment loading per pile as centre to centre pile spacings change. For small pile spacings, the average bending moment per pile is significantly greater than for larger pile separations due to differences in the counteracting moment caused by the axial reactions of the piles against the pile cap. It then follows that pile group displacements will be greater for smaller pile separations even if pile interaction effects are unimportant. Using single pile flexibilities derived from an offline test described subsequently, calculations show that interaction effects can account for a 0.02 mm increase in group displacement for the group with a spacing ratio s/d — 2. For larger spacings, group effects account for increases in displacement of 0.01 mm or less. These differences are small due to the low intensities of shaking used in the tests. Appendix I. Shake Table Tests - Low Level Shaking of 2-Pile Groups 620 Figure 1.5: Influence of pile spacing on pile cap displacement for low level shaking. Appendix I. Shake Table Tests - Low Level Shaking of 2-Pile Groups 621 1.2 Effects of Group Interaction on Lateral Soil Stiffness Since the fundamental frequency and dynamic amplification characteristics of the pile group for inline shaking varies with pile separation distance, differences in dynamic pile cap displacement and the dynamic forces acting within the group for a particular input frequency occur with changes in pile spacing. To distinguish this from the effects of pile interaction, the measured bending moments in the group were used to estimate equivalent lateral soil stiffnesses necessary to describe the flexural response of the piles. These were compared with lateral stiffnesses determined from offline test PG4C carried out using a centre to centre pile spacing of 6 pile diameters. At such large pile separations, interaction between piles is considered negligible. Differences in effective lateral stiffness can be used to infer the influences of pile interaction provided shaking intensities are similar. Average bending moments measured along the piles during test PG4C are shown in Figure 1.6 where the data points represent the peak to peak bending moment at a particular depth, divided by two. The measured bending moments are matched closely by a square root of depth Winkler model (see Appendix C) using a modulus parameter a = 0.30 N / m m s / 2 and dynamic loads V0 = 0.95 N and M0 = 70.0 N-mm in the calculations. The latter have been derived from measured pile cap accelerations and bending moments along the pile. Figure I.7a compares the bending moment distribution predicted using the above Winkler model with average bending moments measured during offline shaking of a two pile group using a centre to centre spacing ratio of two. The bending moments plotted for a particular depth have been computed as their peak to peak amplitude during steady state vibration, divided by two. A modulus parameter a = 0.28 N / m m 5 ' 2 was used in the calculations. Using the average loads per pile given in Table 1.2 a good match of the measured bending moments results. The smaller a value relative to that used to describe Appendix I. Shake Table Tests - Low Level Shaking of 2-Pile Groups 622 4 0 0 A Pile 1 - Exp't x Pile 2 - Exp't Interpoloted Theory Soil Surface - 5 0 0 5 0 Bending Moment (N-mm) 100 Figure 1.6: Pile group test P G 4 C - bending moments vs. depth for low level offline shaking (s/d = 6) and comparison to Winkler model Appendix I. Shake Table Tests - Low Level Shaking of 2-Pile Groups 623 Shaking Direction s/d Vo Mo cv At A x (N) (N-mm) (N/mm5/2) (mm) (mm) Offline 2 1.01 75.0 0.28 0.13 0.13 u 4 0.93 68.0 0.30 0.11 0.11 u 6 0.95 70.0 0.30 0.11 0.12 Inline 2 1.17 50.0 0.25 0.11 0.11 u 4 -0.95 0.0 0.25 0.04 0.05 u 6 -0.S5 3.0 0.30 0.03 0.04 Table 1.2: Average Forces and Deflections - Low Level Shaking the single pile test indicates there is a slight amount of interaction between the piles. Similar matching procedures were adopted for an s/d ratio of four (Figure 1.7b) using an a value of 0.30 N /mm 5 ' 2 . This is identical to the value obtained from test PG4C under similar loading intensities and suggests there is little interaction between the piles. As an additional check on the accuracy of the above matching procedures, the Winkler model was used to predict the measured pile cap displacements. Pile deflections and rotations at the soil surface were first computed assuming that shear and moment loads per pile at or above the soil surface do not vary between piles, as may be observed from the test measurements. With known values of moment and shear force at the soil surface and the top of the pile, the lateral deflection yt and rotation 6t at the top of the pile were determined from a structural analysis of the free standing portion of the piles. The deflection at the location of LVDT 1 was then computed as A x = yt — Az sin 0t where Az is the vertical distance between the top of the pile and the location of the LVDT. Computed deflections at the location of LVDT 1 are given in Table 1.2 and are compared with measured deflections, A i . These agree within 0.01 mm or less which is considered satisfactory and within the range of accuracy of the LVDT. Similar calculations have been carried out for the case of inline shaking (see Table Appendix I. Shake Table Tests - Low Level Shaking of 2-Pile Groups 624 (a) 400 350 E 300-a. f-a> SI cu 250-o XI < O c a 200-l/> a 150-100 Soil Surface Interpol Dted Pile Top 7 5 ^ T h e o r y Offline s/d = 4 a Pile 1 - Exp't X Pile 2 - Exp't Interpolated Theory Soil Surface -50 0 50 Bending Moment (N—mm) 100 (b) 350-300 _5> '<£ a> 250-o XI < o a 200 150-100 -50 Pile Top Soil Surface / A Interpolated Theory Offline s/d = 2 a Pile 1 - Exp't x Pile 2 - Exp't Interpolated Theory Soil Surface - 1 — 0 50 100 Bending Moment (N-mm) 150 Figure 1.7: Comparison of predicted and measured bending moments using a Winkler model for low level, offline shaking (a) s/d = 2 (b) s/d — 4 Appendix I. Shake Table Tests - Low Level Shaking of 2-Pile Groups 625 1.2). Average measured bending moments along the piles are shown in Figure I.8a-c for s/d ratios of 2,4 and 6, respectively. Bending moments applied to each pile in the group are less than those measured during offline loading due to the effects of pile cap restraint. The average bending moment per pile at the soil surface is reasonably constant between piles. Variations in shear force occur between the piles for close pile spacings, as noted previously, but for purposes of computing deflections of the rigid pile cap, an average shear force per pile has been used in the calculations. Computed bending moment distributions using the Winkler model are in good agree-ment with those measured (Figure 1.8). The modulus parameter a used ranged between 0.25 and 0.30 N / m m 5 ' 2 . Lower a values were used to describe the average flexural be-haviour of the piles for spacing ratios of two and four and is indicative of the greater amount of pile interaction that occurs at these closer pile spacings. For larger spacings (s/d — 6) the a value is the same as used in test PG4C and suggests there is little interaction between the piles. Pile cap deflections computed using the above modulus parameters are within 0.01 mm of measured deflections which is within the accuracy of the experimental measurements. Appendix I. Shake Table Tests - Low Level Shaking of 2-Pile Groups 626 o XI < Soil Surface T Pile Top Theory Inline s/d = 2 A Pile 1 - Exp't x Pile 2 - Exp't Theory Soil Surfoce — I — 50 (b) X I < Pile Top^ x Soil Surface^ x Interp olated Theory-\"^ ^N^ Inline \\ s/d = 4 A Pile 1 - Exp't X Pile 2 - Exp't Interpolated ! Theory Soil Surface Bending Moment (N-mm) -25 o 25 Bending Moment (N-mm) (C) Pile Top J o 250-O XI < \"fi^ Interpolated Theory \\ Soil Surface 7\" Inline s/d = 6 Pile 1 - Exp't Pile 2 - Exp't Interpolated Theory Soil Surface -25 0 25 Bending Moment (N-mm) Figure 1.8: Comparison of predicted and measured bending moments using a Winkler model for inline, low level shaking (a) s/d = 2 (b) s/d = 4 (c) s/d = 6 Appendix J Uncoupled Dynamic Analysis of a Pile G r o u p The computer program P G D Y N A has been developed to analyse the response of a pile group supporting a superstructure to earthquake input motions applied at the heads of the piles. The superstructure is idealized using a series of beam elements and rigid lumped masses. Either three dimensional or planar two dimensional analyses may be considered. The interaction between the foundation soil and each pile in the group is represented by a series of non-linear springs and equivalent viscous dashpots. The latter account for radiation and hysteretic damping in the soil medium surrounding the piles during shaking. The pile head spring stiffnesses and hysteretic damping coefficients may be varied with the pile head deflection level which is computed taking into account the effects of pile to pile interaction. Interaction effects are incorporated through use of the pile head interaction factors described in chapters 4 and 5. Options exist within the program to use either the modified interaction factors proposed in the present study or the factors suggested by Randolph and Poulos (1982). The effects of pile group interaction on radiation damping are similarly accounted for by writing interaction equations using the appropriate imaginary quantities for single pile flexibility, pile head forces (and moments), and frequency dependent interaction factors defined by Kaynia and Kausel (1982). The methodologies used in the computer program are outlined subsequently. 627 Appendix J. Uncoupled Dynamic Analysis of a Pile Group 628 J.l Finite Element Discretization The structural frame and concentrated masses above the soil surface (the superstructure) have been discretized using three dimensional beam elements and lumped masses. The superstructure is assumed to remain elastic during shaking and internal damping is ne-glected. The beam element stiffness matrix has been described by Coates et al (1972) and incorporates the bending, axial and torsional stiffness of the beam. The effects of shear distortion on the beam element stiffness matrix have also been incorporated using the Timoshenko beam formulation summarized by Cook (1981). However, the effects of shear distortion have been found to have a relativelj' minor effect on computed dynamic response. A consistent mass formulation described by Cook has been used to describe the translational and rotational mass proporties of each beam element. The 12 degree of freedom, symmetric beam stiffness [kb] matrix expressed with respect to the local coordinate system shown in Figure J . l is given as follows: [h] = [hi] M [h2] [k22] where [ku], [ki2], and k22] are submatrices defined as, ( J . l ) EA/l 0 1 2 5 / . 13+I2lgz 0 0 1 2 E / „ f3 + 12/3j, 0 0 0 GJ/l 0 0 < 3 -r l2/ 3 ! , 0 Erv(4l2+12g„) 0 6EIzl l3+\\Ugz 0 0 0 EIz(4P+l2gz) P + 1 2 / ^ (J.2) Appendix J. Uncoupled Dynamic Analysis of a Pile Group 629 -EA/l 0 0 0 0 0 0 -12EI? l3+12lgz 0 0 0 -6EI,l [k22] EA/l 0 13+I21gz 0 0 -\\2EIy l3+121gy 0 6EI,,l l3+12lgy 0 0 0 \\2Ely l3+12lgy 0 0 0 -GJ/l 0 0 0 0 0 GJ/l 0 0 -6EIyl l3+12lgy 0 EIy(2l2-\\2gy) l3 + 12lgy 0 0 0 0 6Ehl P+12/o* 0 0 0 EIz(2l2-12gz) l3+\\2lgz 0 -QEIrl (J.3) /3 + l2/5s, 0 EI,,(4l2+12gy) l3+\\2lgy l3+12lgz 0 0 0 EIz(4l2+l2gz) 13+I2lgz (J.4) where EIZ and EIV are the flexural rigidities with respect to the local z and y axes (see Figure J. l ) , E A is the axial stiffness and GJ is the torsional stiffness of the beam element. The shear rigidity parameter gz is defined as, nEIz 9z AG (J.5) where n equals 1.33 and 1.20 for a circular and rectangular beam cross section, respec-tively, A is the cross sectional area of the beam and G is the shear modulus of the beam. A similar equation for gy may be written with EIZ being replaced by EIy. The beam element mass matrix is given as, [mb] = [mn] [m 1 2 ] [?7z1 2] [m 2 2 ] (J.6) Appendix J. Uncoupled Dynamic Analysis of a Pile Group 6 3 0 where [mn], [m1 2], and m22] are submatrices defined as, [m n] = [m12] = 1/3 0 0 0 0 0 156m/420 0 0 0 22mZ/420 156/7i/420 0 22ml/420 0 1/3 0 0 477z/2/420 0 4m/2/420 0 -13m//420 0 0 0 -3m/2/420 1 / 6 0 0 54?n/420 0 0 0 0 0 0 0 13777//420 0 0 0 0 54m/420 0 0 1/6 13m//420 0 0 0 0 0 -13777//420 0 -3?7i / 2 /420 0 (J .T) (J.8) [m22] 1/3 0 156?77/420 156777/420 0 1/3 (J.9) 0 0 0 0 0 0 0 -22m//420 -22m//420 0 0 0 4m/2/420 0 4m/2/420 where m = ml, m is the mass per unit length of the beam and / is the beam element length. The mass properties of the rigid portions of the superstructure have been represented using a 6 degree of freedom (3 translations, 3 rotations) lumped mass matrix [m;]. The Appendix J. Uncoupled Dynamic Analysis of a Pile Group 631 8 11 Figure J . l : Three dimensional beam element and local nodal numbering system matrix has been derived with respect to the local coordinate system shown in Figure J.2 and is given as, [mi\\ = m0 0 0 0 m 0 0 0 0 m 0 0 0 m 0 A 0 0 0 0 0 0 m0Az c + m0l 0 0 0 0 0 0 m0Az 0 0 0 0 lcg,y 0 Icg,z + mQAz7 (J.10) mnAz 0 where ra0 is the mass of the rigid element, Icg,x, ICg,y a n d Icg,z a r e the centroidal mass moments of inertia with respect to the local x, y and z axes, and Az is the vertical distance from the bottom of the mass to its centre of gravity. The foundation spring matrix [kj] is derived using single pile analysis programs such as PILAY or L A T P I L E taking into account the non-linearity of the soil-pile system and subsequently modified for pile group interaction effects. The matrix is defined with respect to the local coordinate system shown in Figure J.3 and is given as, Appendix J. Uncoupled Dynamic Analysis of a Pile Group 632 Figure J.2: Three dimensional lumped mass element and local nodal numbering system [*/] = 0 0 0 0 -Koz —kugx 0 0 (J. l l ) 0 0 0 0 kw 0 0 0 0 kww kulgx 0 0 k6xW kgx6x 00 0 0 0 k6uev0 0 0 0 0 where i f c u u , kVVj kww represent the translational spring stiffnesses for the local x, y and z axes, respectively, kgxgx,kgygv, kgzgz are the rotational spring stiffnesses for rotations about the subscripted axes indicated, and kwgx, kugz are the cross-coupled spring coefficients relating in plane pile head forces induced by in plane rotations. It is noted that the cross-stiffness term for torsion about the local y axis is neglected. Symmetry of the cross-coupled spring coefficients is also assumed (i.e. kugz = kgzU, etc.) which is strictly valid for purely elastic behaviour. However, for small displacement excursions of the pile head for each time step, it is assumed that equivalent elastic spring coefficients may be Appendix J. Uncoupled Dynamic Analysis of a Pile Group 633 adopted. The foundation damping matrix [c/] is comprised of a series of equivalent viscous dashpot coefficients c,j and has an analogous form to [k/]. It is assumed that each term C{j may be decomposed into a hysteretic damping term, c,-^, and a term due to radiative energy losses, c,j i r, following the approach described in chapter 6. The above stiffness, mass and damping matrices have been expressed with respect to a local coordinate system. Prior to assembly of each element matrix into the stiffness, mass or damping matrix of the combined superstructure-pile foundation system, it is necessary to transform each element matrix into the global coordinate system shown in Figure J.4. This was done using the three dimensional transformation matrix described by Coates (1972), expressed in terms of three orientation angles relating the angular changes that one must rotate the local coordinate system through to align itself with the global coordinate system. Following transformation of each local stiffness, mass or damping matrix to the global coordinate system, each element matrix was selectively added into the combined global stiffness, mass and damping matrix. These matrix addi-tion procedures follow standard finite element approaches outlined by Cook using locator matrices defined for each element. J.2 Dynamic Solution Methodology A time step integration procedure has been adopted to analyse the uncoupled pile group-superstructure response to earthquake excitation. The steps in the analysis are outlined as follows: 1. Assemble all superstructure mass and stiffness matrices, expressed in terms of a three dimensional framework, into a combined global stiffness [Kss] and mass matrix [Af„]. The superstructure mass matrix is comprised of both beam element and Appendix J. Uncoupled Dynamic Analysis of a Pile Group 634 (a) TRANSLATIONAL SPRINGS PILE HEAD-LATERAL SPRING -STIFFNESS, Kww = K33 -AXIAL SPRING STIFFNESS, K w = K g 2 v<2) V\\ Li j e y (5 ) / L_ LATERAL SPRING V STIFFNESS, e ( 6 ) / -u( l ) e,(4) w(3) (b) ROTATIONAL SPRINGS: PILE HEAD Ke j e 2 = K 6 6 I V P I L E H E A D - ^ * ^ V » x ' K « « PILE HEAD-1 -TORSIONAL STIFFNESS, K e y 0 y ' K55 y (c) COUPLED SPRING STIFFNESS: PILE HEAD—s f LATERAL SPRING STIFFNESS K u o =K DUE TO ROTATION 9 Z ° Z L B p X -x ,u PILE HEAD -- LATERAL SPRING STIFFNESS K w f ( = K . DUE TO ROTATION 6 X °* 3 4 7^p/W—\\ •z'w) NOTE' NEGLECT ADDITIONAL AXIAL PILE HEAD STIFFNESS IN LOCAL y DIRECTION DUE TO PILE HEAD ROTATIONS. Figure J.3: Three dimensional foundation spring element and local nodal numbering system Appendix J. Uncoupled Dynamic Analysis of a Pile Group 635 GLOBAL COORDINATE SYSTEM LUMPED MASS y x BEAM ELEMENT—-z PILE PILE HEAD SPRINGS a EQUIVALENT VISCOUS DASHPOTS AT SOIL SURFACE TO REPRESENT EMBEDDED PILE. Figure J.4: Global coordinate system lumped mass contributions. Both [KSs] and [Maa] are assumed to remain constant for the entire duration of shaking. 2. Form the three dimensional mass vector {MI} = [Maa]*{L}, where Li = 1 if global degree of freedom i is aligned with the direction of the input earthquake excitation, and is 0 otherwise. 3. For the planar (two-dimensional) problem, eliminate the out- of-plane global de-grees of freedom and condense [Kaa], [Mss] and {MI}. 4. Form the three dimensional pile head locator matrix with size N x 6 to facilitate addition of each pile head stiffness and damping matrix to the global stiffness and damping matrix, where N is the number of piles in the pile group. Condense this locator matrix to a two dimensional framework, if required. Appendix J. Uncoupled Dynamic Analysis of a Pile Group 636 5. For the initial time step in the dynamic solution, form the single pile head stiffness [kf] and damping matrices [c/] for either the three or two dimensional case, neglect-ing pile group interaction effects. These matrices are formed initially with respect to the local pile head coordinate system. Small displacement values of pile head stiffness and hysteretic damping are used for the initial time step, assuming that k^ and c,-^ /, vary with the level of pile head deflection modified for group effects during subsequent time steps. Values of axial pile head stiffness, kvv, have been assumed to be different in compression in tension. Therefore, the initial estimate at time t=0 of kvv has been set equal to the average of the small displacement values of the tension and compression stiffnesses. The radiation damping coefficients for a single pile used to construct [cy] have been assumed to remain constant during shaking. Coefficients Cij>r have been estimated based on equivalent elastic moduli in the free field, as described in chapter 6. 6. Based on the single pile head stiffness and damping matrices constructed for each pile in the group, transform these to the global coordinate system. Using the foundation locator matrix, add the pile head stiffness matrices for all piles in the group into [Kss]i thereby forming the global stiffness matrix [K]. Similarly, form the global damping matrix [C] from the sum of the contributions from the single pile head damping matrices. 7. From the single pile head stiffness matrix, transform this to the global coordinate system for each pile in the group. The global single pile stiffness is then inverted to form the real (in phase) component of the single pile head flexibility matrix [Fr]. The latter is used subsequently in calculations of pile group interaction carried out with respect to the global coordinate system. Appendix J. Uncoupled Dynamic Analysis of a Pile Group 637 8. From the pile head damping matrix [cj] expressed in terms of the local coordi-nate system for a single pile, compute the imaginary component of single pile head stiffness, [fcym], as [k'™] = u;o[c/] where u0 is the dominant frequency of excitation in the earthquake input motion. For random earthquake excitation containing a multiplicity of input frequencies, it is common in time domain analysis to choose the dominant frequency of excitation equal to the fundamental resonant frequency of the pile group corresponding to the frequency of maximum pile group response. After transforming [kj71] to the global coordinate system using the appropriate ge-ometric transformation matrix, invert the imaginary pile head stiffness to form the imaginary component of single pile head flexibility [jP , m]. The latter is assumed to stay constant during shaking and is used subsequently in group interaction calcula-tions carried out using the imaginary (out of phase) pile head forces and interaction factors defined by Kaynia and Kausel (1982). 9. For the initial time step of the dynamic solution, assume that the single pile head stiffness matrix constructed neglecting group effects, [kj], equals the modified pile head stiffness matrix, [k^], expressed in terms of local pile head coordinates and considering group effects. Inverting [kj] gives a pile head flexibility matrix in the local pile head coordinate system, [fj]. The latter is also assumed for the initial time step to be equal to a local pile head flexibility matrix taking into account group effects, [fj]. It is also assumed for the initial time step that the single pile head damping matrix expressed in terms of the local coordinate system, [c/], equals the pile head damping matrix, [c^], modified for group effects. The methods used to account for the effects of pile group interaction on the coefficients in these matrices are described subsequently. Appendix J. Uncoupled Dynamic Analysis of a Pile Group 638 10. For the initial time step of the dynamic solution, set initial nodal displacements and accelerations to zero. Solve for the initial nodal velocities {x} from the equation, [M„]{x) = {P(t0)}At (J.12) where {P(t0)} = — {-W-0(\"//)o 1 S the initial nodal force vector at time t = 0 derived from the mass vector {MI} and the initial free field surface acceleration (uff)o-11. With the initial conditions established, solve the global equations of motion of the system using a Gaussian elimination procedure (UBC subroutine SLE) for the next time step, which are given as [M33]{x} + [C]{x} + [K]{x} = -{MI}uJf(t) (J.13) using the constant average acceleration technique described in Appendix H and the time history of free field surface motions as input. It is noted that the use of equation J.13 and the method of solution implicitly assumes that secant values of foundation stiffness and damping are added into the global stiffness and damping matrices. 12. Based on the pile head displacements and rotations, computed relative to the input free field motions and expressed in terms of the global coordinate system, trans-form these to the local pile head coordinate system. This gives the generalized displacement vector {<$},• for the ith pile in the group. Compute the real (in phase) components of the pile head interaction forces, {Qr}i for all N piles in the group us-ing the appropriate equations of beam flexure. These interaction forces are stored for subsequent solution output and plotting. The local interaction forces {Qr}i Appendix J. Uncoupled Dynamic Analysis of a Pile Group 639 are then transformed to the global coordinate system and stored for future use in calculations of pile group interaction. 13. Compute the local pile head velocity vector {£},• for the ith pile in the group, using procedures identical to those described above. Compute the imaginary (out of phase) components of the pile head interaction forces, {Q,m}i for each of the N piles in the group using the pile head damping matrix modified for group interaction effects. Thus, {Qtm}i = [cy],- * The local interaction forces {Qtm}i are also transformed to the global coordinate system and stored for future use in calculations of pile group interaction in the imaginary domain. 14. Make a preliminary estimate of the pile head deflections for the ith pile in the group, {Sg}i, taking into account pile group interaction effects and carried out with respect to the local pile head coordinate system. This calculation is based on the pile head forces {Qr}i computed for the present time step and the pile head flexibility matrix modified for group effects, [/'],-, established at the end of the previous time step. Thus, {S9}i ~ [ff]% * {Qr}i- The computed pile group deflections are used subsequently to estimate effective moduli distributions in the near field soil which are then used to calculate pile group interaction coefficients. 15. Based on the average of the pile head deflections in the horizontal plane averaged over all piles in the group, u, assume a relationship between the effective Winkler modulus distribution along the pile, kh(z) and u based on available single pile data. Two modulus distributions are allowed for; one based on a linear versus depth Winkler modulus distribution and the other based on a square root of depth distribution. Thus, it is assumed that either kh(z) — cvz or kh(z) = az 1 ' 2 , where a is related to Ti via a prescribed mathematical relationship derived from single pile test Appendix J. Uncoupled Dynamic Analysis of a Pile Group 640 data. In the current version of the program it is assumed that log a = log a0 — m \\u\\, analogous to equation 6.2. 16. Once the value of a is estimated for the current time step, it is assumed that kh(z) ~ KG(Z) where G(z) is the shear modulus distribution versus depth and K is an input proportionality constant. Once G(z) is prescribed, calculations of static pile group interaction can be made using the modified pile group interaction factors and approaches outlined in chapter 4. Alternatively, an option in the program allows one to use the static (zero frequency) elastic interaction factors for lateral modes of vibration proposed by Randolph and Poulos. It has been found that calculations of pile group interaction factor are not overly sensitive to the amplitudes of shear modulus for a given distribution versus depth. Thus, potential inaccuracies in estimating the effective shear modulus distribution are not of great consequence. 17. Based on the pile head interaction forces (and moments) computed for the present time step and expressed in terms of the global coordinate system for either a three dimensional or two dimensional case, and the single pile head flexibility matrix established in the previous time step and transformed to the global coordinate system, compute pile head nodal displacements (uf,t^,iof) in the horizontal and vertical directions taking into account pile group effects. The method of calculation for lateral displacement modes is described in chapter 4 and equation 4.37. An analogous equation can be written to compute vertical pile head displacements considering axial pile head forces throughout the group, where the static vertical interaction factor (r]w<}) presented by Randolph and Poulos and also by Scott (1981a) has been used in the analysis. The equation for r)V}q is given as, Appendix J. Uncoupled Dynamic Analysis of a Pile Group 641 (J-14) l n r m f l X where s t J is the radial distance between the centre of pile i and pile j , r, max IS the maximum radial distance between piles where vertical interaction is considered important, and r 0 is the pile radius. For purposes of the present analysis, rmax has been set equal to a value of 12 pile radii. It is noted that for lateral vibration in the three dimensional case, only in-plane forces and moments for all j piles in the group are assumed to contribute to in-plane pile head displacements at pile i. 18. While the influence of the frequency of excitation on the real (in phase) components of interaction factors is considered small, allowance has been made in the computer program to multiply the static interaction factors by a frequency correction factor, fx which is assumed to be a function of the dimensionless frequency of excitation ao, the pile to pile spacing ratio ^ , and (for lateral vibration only) the departure angle B (see section 4.6.3). The frequency correction factors have been estimated based on dynamic elastic interaction analyses presented by Kaynia and Kausel (1982). Polynomial expressions have then been developed from the curves presented by Kaynia and Kausel to compute the frequency correction factors for various modes of pile to pile interaction. Thus, the interaction factor n defined for the dominant angular frequency of excitation OJ0 and the particular mode of interaction is computed as where 77(00 = 0, ^,/?) is the static interaction factor corrected appropriately for the effects of soil non-linearity, as discussed in chapter 4. 7?(a0, —, B) -- n(a0 = 0, — , B)fi (J.15) Appendix J. Uncoupled Dynamic Analysis of a Pile Group 642 19. By summing the contributions of all the in phase pile head forces (and moments) to deflections of the ith pile in the group, the real component of pile head deflection accounting for group effects is computed. Knowing the pile head group deflections (uf, if , wf) transformed to the local coordinate system for the iih pile, the pile head stiffness matrix [A^ ],- is modified for the next time step for each of the N piles in the group. Use is made of the semi- logarithmic relationship between the various components of pile head stiffness k,j and the pile head group deflection presented in equation 6.2. An option to iterate a specified number of times within the time step is also available in the program. In this way, the average value of global pile head stiffnesses computed during the present and previous iteration, taking into account group effects, are added into the elastic structural stiffness matrix and the time step repeated to recompute dynamic displacements and forces within the group based on the revised pile head stiffnesses. It has been found for the problems analysed that use of the above iteration procedure leads to convergeance in the pile head stiffness after 3 to 10 iterations, depending on the irregularity of the input motions used. 20. It is noted that depending on the sign of the pile head axial deflection, the axial pile head stiffness kvv is assigned a value appropriate to either tensional or compressional loading. A computer algorithm has been developed such that when there is a reversal in direction of axial movement, a zero value of axial displacement is assigned at the point of reversal and subsequent changes in axial deflection are computed relative to this revised origin. The changes in axial deflection relative to this moving origin are used to compute axial stiffness using equation 6.2. For a subsequent reversal in axial movement, the moving origin is again updated, incremental axial deflections are computed and axial stiffnesses continually revised. Appendix J. Uncoupled Dynamic Analysis of a Pile Group 643 21. After convergeance of the pile head stiffness is obtained within a particular time step, the pile head stiffness matrix for the ith pile in the group is computed. Invert-ing [kf]i gives the pile head flexibility matrix, [fj], expressed in local coordinates and accounting for group interaction effects.Transforming [kj]i to the global coor-dinate system and using the foundation locator matrix, add the components of [k9^ to the global structure stiffness matrix, [Kaa], thereby forming the combined global stiffness matrix [K] for the next time step. 22. Based on the pile head interaction forces computed for a particular iteration within the present time step and the single pile flexibilities established during a previ-ous iteration, compute single pile head deflections relative to the local pile head coordinate system, neglecting group interaction effects. Based on the single pile deflections, revise the single pile head stiffness matrix [fc/]t- for all i piles in the group, thereby accounting for the effects of pile head deflection on single pile stiff-ness. Inverting [kj] gives the revised single pile flexibility matrix, [fj], which is subsequently transformed to the global coordinate system to form the global flexi-bility matrix [Fj] for use during calculations of pile to pile interaction in the next iteration for a particular time step. 23. A similar semi-logarithmic relationship between the pile group deflections com-puted above, taking into account pile to pile interaction, and hysteretic damping coefficients, c,-^ /,, is adopted. With these revised damping coefficients, the pile head damping matrix, expressed with respect to the local coordinate system, [cj'h], is computed. The revised hysteretic damping matrix is subsequently transformed into the global coordinate system so that it can be added into the global system damping matrix, along with the pile head radiation damping components. Appendix J. Uncoupled Dynamic Analysis of a Pile Group 644 24. The effects of group interaction on the imaginary (out of phase) components of pile head deflection may be expressed in an analogous fashion to the real components as, uf = v07/r + £ V o v ; « t f ^ ( J- 1 6) k k _ T/im fim,g . T\\/fim fim,g f J 17\\ — V0,j JuV,j ' 0,k J UTTlyj where VQL™ and MQ™ represent the imaginary components of the pile head forces (and moments) for the jth pile expressed with respect to the global coordinate sys-tem, f™ and are the imaginary components of the single pile head flexibilities for shear and moment loading, respectively, 7 / J ^ ' t m and 7 ? J ™ ' , m are the frequency dependent, imaginary components of pile head interaction factors for shear and moment loading between the jih and kth pile, and fu™f and are the imagi-nary components of the pile head flexibilities for the jth pile accounting for group interaction effects for shear and moment loading, respectively. 25. Assuming that interaction effects due to shear and moment loading may be decou-pled, then equation J.17 gives, fim,g 0,j J uv ' £—ik 0,kJ uv Ijk / T 1 Q \\ Juv,j ~ T^k l J - i b i V0,j A/lim fim I l\\/fr fr um,im rim,g _ nl0,jJum < L^k iu0,kJum'ljk M 1 C n Jurn,j A/fim ' ' mo,j where the group flexibility coefficients are developed with respect to the global coordinate system. Appendix J. Uncoupled Dynamic Analysis of a Pile Group 645 26. Equations similar to equations J.18 and J.19 may be written for the imaginary components of pile head flexibility for other modes of interaction. The following global pile head flexibility matrix is then written for the jth pile in the group, fuvx 0 0 0 0 fumz 0 fwq 0 0 0 0 0 0 fwvz fwmx 0 0 0 0 fll>xVz fil>xmx 0 0 0 0 0 0 frl>ymv 0 fl/>zVx 0 0 0 0 ftl>zmz (J.20) 27. Equation J.20 is computed for every pile in the group, and inverted to form the imaginary component of pile head stiffness, [kj\"1'9], modified for group interaction effects and expressed with respect to the global coordinate system. The equivalent viscous dashpot coefficients representing radiation damping are then computed as (J.21) where [cf9] is expressed with respect to the global coordinate system. 28. Revise the global damping matrix computed during the previous iteration for a particular time step by adding in the pile head hysteretic damping matrices [c9^] and the radiation damping matrices [cf9] modified for group effects for all i piles in the group. 29. Store the combined hysteretic and radiation pile head damping matrices [cy+/l'fl],- for the ith pile in the group, transformed to the local coordinate system and modified for group interaction effects. The combined group damping matrix is used in the next iteration to compute the imaginary components of pile head force and moment. Appendix J. Uncoupled Dynamic Analysis of a Pile Group 646 30. At the end of a specified number of iterations for a particular time step and after convergeance in system stiffness has been achieved, an option exists to compute the natural frequencies of the system. With the revised global stiffness matrix [K] and the constant mass matrix [Afss], compute the natural frequencies for the particular time step by solving the frequency equation det [[K] — OJ2[MSS\\] using the available computer library subroutine PRITZ. 31. Go back to step 11 and repeat the solution process for all time steps. 32. After marching through all time steps, print out solution results including pile head interaction forces, and nodal point accelerations and displacements in the direction of shaking. It is noted that accelerations are expressed in terms of absolute quantities by adding horizontal accelerations computed relative to the moving base to the input base accelerations. Absolute values of horizontal displacement un,,-at the soil surface have been computed using the computed pile head interaction forces and appropriate equations of pile group interaction. These displacements vary slightly between piles. The difference between these absolute values of pile head displacement and displacements computed relative to the moving base, «o,t, is calculated for each pile in the group giving a 'free field' deflection Au 0 , ; . The average of all values of Auo,« is computed and added to relative displacements computed above the soil surface on the superstructure. The sum of the computed relative displacements and the average free field displacement is assumed to give a reasonable estimate of absolute deflection levels. "@en ; edm:hasType "Thesis/Dissertation"@en ; edm:isShownAt "10.14288/1.0050510"@en ; dcterms:language "eng"@en ; ns0:degreeDiscipline "Civil Engineering"@en ; edm:provider "Vancouver : University of British Columbia Library"@en ; dcterms:publisher "University of British Columbia"@en ; dcterms:rights "For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use."@en ; ns0:scholarLevel "Graduate"@en ; dcterms:title "Response of pile foundations to simulated earthquake loading : experimental and analytical results volume I"@en ; dcterms:type "Text"@en ; ns0:identifierURI "http://hdl.handle.net/2429/30882"@en .