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Response of pile foundations to simulated earthquake loading : experimental and analytical results volume… Gohl, W. Blair 1991

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RESPONSE OF PILE FOUNDATIONS T O SIMULATED LOADING: EXPERIMENTAL AND VOLUME  ANALYTICAL  EARTHQUAKE RESULTS  II  By W. BLAIR GOHL B . E n g . (Civil) M c G i l l University  1976  M . E n g . (Civil) 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  T H E REQUIREMENTS  DOCTOR OF  FOR T H E D E G R E E OF  PHILOSOPHY  in T H E F A C U L T Y OF G R A D U A T E CIVIL  STUDIES  ENGINEERING  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  COLUMBIA  J u l y 1991 © W . B L A I R . G O H L , 1991  OF  RESPONSE OF PILE FOUNDATIONS TO SIMULATED LOADING: EXPERIMENTAL AND  ANALYTICAL  EARTHQUAKE RESULTS  VOLUME I By W. BLAIR G O H L B . Eng. (Civil) M c G i l l University  1976  M . Eng. (Civil) 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  T H E REQUIREMENTS  DOCTOR OF  FOR T H E D E G R E E  OF  PHILOSOPHY  in T H E FACULTY OF G R A D U A T E CIVIL  STUDIES  ENGINEERING  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  COLUMBIA  J u l y 1991 © W . B L A I R G O H L , 1991  OF  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. copying  of this thesis for scholarly  department  or  by  his or  I further agree that permission for extensive  purposes  her representatives.  may be granted It  by the head of my  is understood  that  publication of this thesis for financial gain shall not be allowed without permission.  Department of  Civil Engineering  The University of British Columbia Vancouver, Canada  Date  DE-6 (2/88)  ,luly ??, 1991  copying  or  my written  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 prepared 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 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. A n 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 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 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  iv  considered i n the numerical discretization of the problem has also been used.  Interac-  tion between the soil and vibrating ground is accounted for using a K e l v i n - V o i g h t model which includes non-linear W i n k l e r 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 i n d 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 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.  v  Table of Contents  Abstract  ii  List of Tables  xii  List of Figures  xv  Acknowledgement 1  Statement of Research  1  1.1  B e h a v i o u r a l Aspects of Single Pile Response to C y c l i c L a t e r a l L o a d i n g  ] .2  Observations of Full Scale Pile Response D u r i n g Earthquake Loading  1.3  E x p e r i m e n t a l Observations of Pile G r o u p Interaction  1.4  N u m e r i c a l Modelling of Single Pile Response to Earthquake L o a d i n g . . .  1.5  Numerical Modelling of Pile G r o u p Behaviour to Static and D y n a m i c  1.6 2  xxxvi  .  1  . .  7 9 15  Loading  25  1.5.1  L o w Frequency, Quasi-Static L o a d i n g  25  1.5.2  Higher Frequency D y n a m i c L o a d i n g  28  Scope of Study  30  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 M o d e l Layout  49  vi  3  4  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  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  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  4.4  Base Motion Excitation of Single Piles  110  4.3.1  Low-level Sinusoidal Shaking  113  4.3.2  Random Earthquake Excitation  115  Soil-Pile Interaction  129  4.4.1  129  Introduction vi i  4.5  5  4.4.2  Earthquake Excitation  144  4.4.3  Low Level Sinusoidal Shaking  157  4.4.4  Near Field Hysteretic Damping  169  Equivalent Visco-Elastic Soil Resistance  172  4.5.1  176  Computed Lateral Winkler Stiffness and Material Damping . . . .  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  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  vi i i  5.5  5.6  5.7 6  Soil-Pile Interaction  348  5.5.1  Hysteretic Damping  356  5.5.2  Non-Linear Modelling of P - Y Hysteresis Loops  358  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  Cyclic Axial Load Behaviour of Model Piles  393  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  6.3  . . . .  414  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  7.3.2  476  Centrifuge Test - Four Pile Group Subjected to Earthquake Excitation  7.4 8  483  Summary  486  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 B. l  Strain Gauges  547 547  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 C. l  Introduction  554 554  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  E  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  Calculation of Soil Resistance - Lateral Pile Displacement Curves  580  E. l  580  Methodology  E. 2 Comparison of Method With Cubic Spline Differentiation F  570  583  Single Pile Response in a Winkler Medium to Base Motion Excitation588 F. l  Equations of Motion  588  F.2  Free Vibration Response  590 Xi  F. 3 Forced Vibration Response  592  G Uncoupled Solution for a Pile - Structural Mass System 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 H. l  I  J  596  Equations of Motion  604 604  H. 2 Solution of Equations of Motion  609  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  Uncoupled Dynamic Analysis of a Pile Group  627  J.l  Finite Element Discretization  628  J.2  Dynamic Solution Methodology  633  xii  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)  . . .  3.1  Centrifuge Scaling Relations  3.2  Summary of Model Pile and Pile Cap Structural Properties Used in Cen-  79  trifuge Tests) 3.3  85  Summary of Pile and Pile Cap Structural Properties - Centrifuge Tests on Two Pile Group  3.4  59  87  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, K , 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-  4.8  T  alent Elastic Winkler Moduli - L A T P I L E Analysis  194  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  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  290  327  5.8  Single Pile Test Characteristics'for Strong Shaking on the Shake Table  5.9  Winkler Model Predictions of Single Pile Deflections - Shake Table Tests  . 334 351  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)  6.3  408  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 Effects - P G D Y N A  7.3  475  Computed Vs. Measured Pile Group Response for Strong Sinusoidal Shaking - Four Pile Group on the Shake Table  7.4  483  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  C y c l i c lateral load - displacement characteristics of soil in soft clay (a) zone of unconfined response (b) confined response (after B e a et a l , 1980)  2.1  3  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 table A - com puted Fourier spectra, (c) Shake table B - measured table accelerations, (d) Shake table B - computed Fourier spectra  2.2  37  T y p i c a l sinusoidal input base motions recorded using a sampling rate of 303 H z per channel - shake table A : (a) table accelerations - moderate intensity shaking (b) table accelerations - high int. ensity shaking (c) Fourier spectrum - moderate intensity shaking (d) Fourier spectrum - high intensity shaking  2.3  39  T y p i c a l earthquake input base motions recorded using a sampling rate of 303 H z per channel - shake table A (a) measured table accelerations (b) Fourier spectrum  2.4  40  G r a d a t i o n curve for C-109 O t t a w a sand used in shake table tests and comparison w i t h Toyoura sand (after Tatsuoka and F u k u s h i m a , 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 Ottawa sand (a) m e d i u m dense sand (D  r  (D 2.7  T  = 30 percent) (c) dense sand (D  T  = 50 percent) (b) loose sand  = 90 p ercent)  Single pile used i n shake table tests showing instrumentation layout  xvi  45 . . .  53  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 ggradients 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 triaxial tests on Nevada 120 sand  93  4.1  Source and receiver bender element voltage outputs  4.2  Lumped mass mechanical model used to describe bender element response  105  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  4.5  Ill  Measured accelerations - centrifuge test 17 (a) input base accelerations (b) free field surface accelerations (c) pile cap accelerations xvii  116  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 ) 4.16 Measured displacements parallel to shaking direction - centrifuge test 12  128 129  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 ) 4.21 Measured displacements parallel to shaking direction - centrifuge test 14  xvi i i  133 134  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 ) 4.26 Measured displacements parallel to shaking direction - centrifuge test 15  138 139  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 A P I 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 A P I 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 A P I 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 K for various values of H/2r r  0  frequency and comparison with other researcher 's relationships,  at zero (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 I and lateral pile e  deflection  188  4.54 Results of elastic analysis - test 17 (a) effective shear strains (b) effective shear moduli (c) Winkler moduli  .191  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 4.57 Construction of unloading and reloading curves based on the Masing rule xxi  193 195  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 accelerations (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 -q (a) 3 = 0 uv  degrees (b) 3 = 45 degrees (c) 3 = 90 degrees  xx i i  22 9  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 accelerations (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 def