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Investigation of variation of motions between free field and foundation in seismic soil-structure interaction… Pandey, Bishnu Hari 2013

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INVESTIGATION OF VARIATION OF MOTIONS BETWEEN FREE FIELD AND FOUNDATION IN SEISMIC SOIL-STRUCTURE INTERACTION OF STRUCTURES WITH RIGID SHALLOW FOUNDATION   by BISHNU HARI PANDEY  M.E., The University of Tokyo, 2002  A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF  DOCTOR OF PHILOSHOPHY in THE FACULTY OF GRADUATE AND POSTDOCTORAL STUDIES (Civil Engineering)  THE UNIVERSITY OF BRITISH COLUMBIA (Vancouver)  September 2013  ? Bishnu Hari Pandey, 2013 ii  Abstract  Soil-foundation-structure interaction (SFSI) in buildings during earthquakes is characterized by several aspects including the variation between the free-field and foundation motions. Past procedures for analysing the effect of the foundation on the free-field input motions are all based on the assumptions that the foundation slabs always reduce the motion. Recent guidelines, standard and codes including FEMA-440 and ASCE/SEI41-06 also recognize that foundations with interconnected grade beams or concrete slab will always reduce the free-field motions. It implies that SFSI is beneficial and can be conservatively neglected in regular building design practice. A large number of instrumented buildings that have experienced a numbers of earthquakes in the past provide an opportunity to investigate the SFSI effects and evaluate the methods of estimating the foundation motion based on field data.  In this research study, investigation is carried out on the records of past earthquakes from sites of instrumented buildings over a wide range foundation configurations and site conditions in California. Analysis of records among 26 buildings that have shallow rigid foundation shows that foundation motions are reduced in two-third cases and amplified in one-third cases. The estimations of variation of motion by the ASCE procedure are not in good agreement, even for the cases of reduction of foundation-base motion. It was obvious that the amplification of the motion cannot be captured by the procedure.  Time history simulations of soil-building system have been carried out for varied parameters in 2D continuum finite element models using computer program ABAQUS. The results from iii  simulations confirm that motion may amplify at the foundation depending on period of the building, soil deposit and the predominant period of input motion.   This thesis develops a simple mass?spring?dashpot-based system of soil?foundation?structure interaction, called the 3DOF SFS model, for calculating the foundation-base motion that accounts for dynamic interaction between soil, foundation and building.  The 3DOF SFS model was verified for a building-structure system with varied parameters and for different input motions using results from ABAQUS. Both amplification and reduction cases of foundation motion compared with free field were predicted by the model.    iv  Preface  This thesis is a product of a research project on Soil-structure Interaction in Performance Based Design of Bridges carried out at the University of British Columbia funded by Natural Science Research Council (NSERC).  I am responsible in identification and design of research as well as in all the analysis and interpretation of the results presented in this thesis.   Some parts of the research presented in Chapter 3 have been published in following publications.  ? C.E Ventura, [B.H Pandey] and WDL Finn (2011) Comparison of foundation and free-field motions of instrumented buildings during earthquakes. Conference Proceedings on Experimental Vibration Analysis for Civil Engineering Structures. I drafted the sections 2 and 3 of the paper.  ? W.D.L. Finn, [B.H. Pandey] and C.E. Ventura (2011) Modeling soil-foundation-structure interaction. The Structural Analysis of Tall and Special Building. Volume 20, Issue Supplement S1, 47-62. I prepared the manuscript for section 5 of this paper.  ? [B.H. Pandey], C.E. Ventura and W.D.L. Finn (2012) Vibration Characteristics of Foundation and Free-Field Motions of Instrumented Buildings during Earthquakes. Chapter 43, Conference Proceedings of the Society for Experimental Mechanics Series, Springer publishing. I wrote most of the manuscript. ? [B.H. Pandey], WDL Finn, and C.E. Ventura (2012) Modification of Free-field Motion by Soil-foundation-structure Interaction for Shallow Foundations. Paper 1575, Proceedings of world conference on earthquake engineering. I wrote most of the manuscript. v  Table of Contents  Abstract .......................................................................................................................................... ii?Preface ........................................................................................................................................... iv?Table of Contents ...........................................................................................................................v?List of Tables ..................................................................................................................................x?List of Figures ............................................................................................................................... xi?Acknowledgements .................................................................................................................. xxiii?Dedication ...................................................................................................................................xxv?Chapter  1: INTRODUCTION .....................................................................................................1?Chapter  2: STATE OF PRACTICE IN ESTIMATING FOUNDATION MOTION.............9?2.1? Context ....................................................................................................................... 9?2.2? Free-field and Foundation Motions ......................................................................... 10?2.3? The Soil?Structure Interaction and Foundation Motion .......................................... 11?2.4? Early Studies on the Variation of Motions .............................................................. 13?2.5? Averaging of Motion over Transit Time ................................................................. 14?2.6? Tau-effect Transfer Function ................................................................................... 17?2.7? Variation of Motion due to Spatial Incoherency ..................................................... 20?2.7.1? Calibration of spatial incoherency from instrumented building records ................. 22?2.8? FEMA/ASCE Provisions of Base Slab Averaging for Shallow Foundations ......... 23?2.9? Summary of State of Practice and Research Direction ............................................ 24?Chapter  3: VARIATION OF SEISMIC MOTION FROM FREE FIELD TO FOUNDATION IN INSTRUMENTED BUILDINGS ........................................26?vi  3.1? Introduction ............................................................................................................. 26?3.2? Data Collection from the Records of the Instrumented Buildings .......................... 28?3.3? Comparison of Time Histories and Response Spectra ............................................ 42?3.4? Spectral Analyses of Recorded Data ....................................................................... 44?3.5? Variation of Motion from Free Field to Foundation ................................................ 48?3.5.1? Free field to foundation base in buildings with basement ....................................... 48?3.5.2? Free field to foundation base in buildings with surface foundation ........................ 49?3.6? Example Cases of Reduction of Motion at Foundation Base .................................. 51?3.6.1? Hollywood Storage Building ................................................................................... 51?3.6.2? Newport Beach Hospital Building ........................................................................... 54?3.6.3? Los Angeles Fire Command and Control Centre .................................................... 56?3.7? Example Cases of Amplification of Motion at the Foundation Base ...................... 57?3.7.1? El Centro Imperial County Service Building .......................................................... 57?3.7.2? San Bernardino Office Building .............................................................................. 61?3.8? Comparison of Observed Data with Wave Passage Models ................................... 62?3.8.1? Discussion ................................................................................................................ 67?3.9? Comparison of Observed Data with Spatial Incoherency Models .......................... 68?3.9.1? Discussion ................................................................................................................ 72?3.10? Comparison of Observed Data with FEMA-440/ASCE41-06 Provisions .............. 77?3.11? Discussion of the Procedures Available to Estimate the Motion at the Foundation Base ......................................................................................................................... 79?3.12? Patterns in the Observed Motions of the Foundation Base...................................... 80?3.12.1? Different responses for different earthquakes.......................................................... 81?vii  3.12.2? Different responses to the same earthquake in two directions ................................ 82?3.13? Summary and Concluding Remarks ........................................................................ 83?Chapter  4: CONTINUUM MODEL TO STUDY THE VARIATION OF MOTION FROM FREE FIELD TO FOUNDATION .......................................................................86?4.1? Introduction ............................................................................................................. 86?4.2? Continuum Model of a Soil?Structure System ........................................................ 87?4.3? Input Motions .......................................................................................................... 90?4.4? Parametric Study ...................................................................................................... 95?4.4.1? Foundation width ..................................................................................................... 97?4.4.1.1? Discussion ......................................................................................................... 99?4.4.2? Storey height of building ......................................................................................... 99?4.4.2.1? Constant storey stiffness ................................................................................. 101?4.4.2.2? Constant modal mass and period .................................................................... 103?4.4.2.3? Discussion ....................................................................................................... 104?4.4.3? Soil shear wave velocity ........................................................................................ 105?4.4.3.1? Foundation motion de-amplified in Vs =200 m/s and 300 m/s ....................... 108?4.4.3.2? Foundation motion de-amplified for increased Vs .......................................... 108?4.4.3.3? Foundation motion amplified for increased Vs ............................................... 109?4.4.3.4? Discussion ....................................................................................................... 110?4.4.4? Soil damping .......................................................................................................... 110?4.4.5? Mass of the building .............................................................................................. 112?4.4.5.1? Increasing and decreasing trend of peak spectral motion at foundation ......... 116?4.4.6? Depth of soil deposit .............................................................................................. 118?viii  4.4.6.1? Effect of deposit depth to the variation of motion at foundation under different building mass .................................................................................................. 121?4.5? Patterns in the Results ............................................................................................ 127?4.5.1? Amplification of foundation motion in a system under different earthquakes ...... 127?4.5.2? Reduction of foundation motion in a system under different earthquakes ............ 131?4.5.3? Amplification and reduction of foundation motion in the same system under different earthquakes ............................................................................................. 134?4.6? Effects of Period in the Variation of Motion ......................................................... 136?4.6.1? Fixed-base period of building ................................................................................ 137?4.6.1.1? Increasing and decreasing trends of foundation motion with building period 137?4.6.1.2? Spectral value of motion at foundation at building period ............................. 139?4.6.2? Period of the soil?structure system, Tsys and predominant period of earthquake input, Tp ................................................................................................................. 141?4.6.3? Effect of the building period on the system period ............................................... 148?4.6.4? Identification of period of peak variation from Tb and Tsys ................................... 150?4.7? Conclusions ........................................................................................................... 153?Chapter  5: SOIL?FOUNDATION?STRUCTURE (SFS) MODEL WITH LUMPED PARAMETERS ....................................................................................................155?5.1? Introduction ........................................................................................................... 155?5.2? Soil?Foundation?Structure 3-Degree of Freedom (SFS 3DOF) Lumped Model . 156?5.3? Lumped Parameters of Soil Deposit with Foundation ........................................... 157?5.3.1? Participating mass of soil- foundation ................................................................... 157?5.3.2? Effect of foundation width ..................................................................................... 159?ix  5.3.3? Estimation of static stiffness of soil-foundation slab system ................................ 162?5.3.4? Generalized stiffness of foundation-soil system .................................................... 164?5.3.5? Mass of deposit ...................................................................................................... 165?5.3.6? Properties of soil deposit only ............................................................................... 166?5.3.7? Properties of foundation ........................................................................................ 171?5.3.8? Properties of building ............................................................................................ 172?5.4? Implementation of SFS 3DOF Model ................................................................... 172?5.4.1? Response at the foundation base and free field ..................................................... 175?5.4.1.1? Case of reduction of motion at the foundation base ....................................... 176?5.4.1.2? Case of amplification of motion at foundation base ....................................... 180?5.4.2? Analysis steps using the SFS 3DOF model ........................................................... 183?Chapter  6: CONCLUSIONS AND FUTURE WORK ..........................................................185?6.1? Future Work ........................................................................................................... 186?References  ................................................................................................................................187?Appendices  ................................................................................................................................191?Appendix A ESTIMATION SYSTEM PERIOD OF SOIL-STRUCTURE SYSTEM .............. 191?A.1? Governing equation of shear beam with appended spring?mass........................... 191?A.2? Frequency equation for the system ........................................................................ 195?Appendix B MATLAB CODE FOR SFS-3DOF ........................................................................ 196? x  List of Tables  Table 3-1 Selected instrumented buildings with records at basement level and free field .. 34?Table 3-2 Selected instrumented buildings with records at basement level and ground floor .................................................................................................................................... 35?Table 3-3 Selected instrumented buildings with records at foundation base at ground surface level and free field ....................................................................................... 37?Table 4-1 Selected set of input motion from earthquake events other than the 1994 Northridge Earthquake ........................................................................................... 91?Table 4-2 Set of input motions from the 1994 Northridge Earthquake event selected for analysis ...................................................................................................................... 94?Table 4-3 Dynamic properties of building configurations with constant storey stiffness .. 102?Table 4-4 Dynamic properties of building configurations with constant modal mass and period ...................................................................................................................... 103? xi  List of Figures  Figure 2-1 Change in the response spectra value due to increased damping and period lengthening (NEHRP, 1997) .................................................................................. 12?Figure 2-2 Base translations arising from travelling seismic waves....................................... 15?Figure 2-3 Comparison of the result of the time averaging technique with response spectra of motion recorded at the Hollywood Storage Building basement and PE Lot (free field) in the San Fernando Earthquake, 1971 (east-west direction) ......... 17?Figure 2-4 Tau-factor as a function of frequency and apparent wave velocity (Clough and Penzin, 1995) .......................................................................................................... 19?Figure 2-5 Application of the tau-factor averaging method in the Northridge Earthquake record for a foundation width 80 m and Vs =340 m/s ........................................ 20?Figure 2-6 Transfer functions for lateral components of foundation input motion for rectangular foundations subjected to vertically incident incoherent waves (Veletsos et al., 1997) ............................................................................................. 22?Figure 2-7- Ratio of response spectra for base slab averaging, RRSbsa (ASCE/SEI 41-06, 2007) ........................................................................................................................ 24?Figure 3-1 Sensor layouts at foundation and free field in the building site of a Piedmont three-storey building from the CSMIP database (Naeim, 2005) ....................... 32?Figure 3-2 Comparison of motion at free field and foundation base in time histories for the Los Angeles seven-storey  UCLA Math-Science Building in the 1994 Northridge Earthquake in the north-south direction ........................................ 42?xii  Figure 3-3 Response spectra of acceleration and  velocity histories at free field and foundation base of the Los Angeles seven-storey UCLA Math-Science Building in the 1994 Northridge Earthquake in the north-south direction ..................... 44?Figure 3-4 Spectral acceleration of motions at basement and free field of the Pomona two-storey commercial building in the Upland 1990 Earthquake in the north-south direction .................................................................................................................. 46?Figure 3-5 Comparison of power spectral density function and transfer function of records at free field and basement of the Pomona two-storey commercial building in the Upland 1990 Earthquake in the north-south direction ................................ 47?Figure 3-6 Spectral velocity and displacement of motions at basement and free field of the Pomona two-storey commercial building in the Upland 1990 Earthquake in the north-south direction ............................................................................................. 48?Figure 3-7 Comparison of motions at free field with foundation base at the basement level for (a) peak ground acceleration (PGA) and (b) peak spectral acceleration (Sa) ................................................................................................................................. 49?Figure 3-8 Comparison of motions at free field with foundation base ground surface level for (a) peak ground acceleration (PGA) and (b) peak spectral acceleration (Sa) ................................................................................................................................. 50?Figure 3-9 Hollywood Storage Building configuration and instrument layout (after CSMIP, 2005) ........................................................................................................................ 51?Figure 3-10 Spectral motions at the foundation base compared with free field at the Hollywood Storage Building in the 1994 Northridge Earthquake in (a) east-west direction and (b) north-south direction ...................................................... 52?xiii  Figure 3-11 PSD at the foundation base compared with free field at the Hollywood Storage Building in the 1994 Northridge Earthquake in (a) east-west direction and (b) north-south direction ............................................................................................. 53?Figure 3-12 Transfer function between the motion at free field and foundation base at the Hollywood Storage Building in the 1994 Northridge Earthquake in (a) east-west direction and (b) north-south direction ...................................................... 54?Figure 3-13 Newport Beach 11-storey Hospital Building configuration and instrument layout (after CSMIP, 2005) ................................................................................... 55?Figure 3-14 Spectral motions at the foundation base compared with free field at the Newport Beach Hospital Building in the 1994 Northridge Earthquake in (a) east-west direction and (b) north-south direction .............................................. 55?Figure 3-15 Los Angeles two-storey Fire Command and Control Centre configurations and instrument layout (after CSMIP, 2005) ............................................................... 56?Figure 3-16 Spectral motions at the foundation base compared with free field at the Los Angeles Fire Command and Control Centre  in the 1994 Northridge Earthquake in (a) east-west direction and (b) north-south direction ............... 57?Figure 3-17 El Centro Imperial County Service Building configuration and instrument layout (building layout after CSMIP (2005) and photograph after Pardoen et al. (1983) ................................................................................................................. 58?Figure 3-18 Spectral motions at the foundation base compared with free field at the El Centro Imperial County Service Building in the 1979 Imperial Valley Earthquake in (a) east-west direction and (b) north-south direction ............... 59?xiv  Figure 3-19 PSD at the foundation base compared with free field at the El Centro Imperial County Service Building in the 1979 Imperial Valley Earthquake in (a) east-west direction and (b) north-south direction ...................................................... 60?Figure 3-20 Coherency between the motion at free field and foundation base at the El Centro Imperial County Service Building in the 1979 Imperial Valley Earthquake in (a) east-west direction and (b) north-south direction ............... 60?Figure 3-21 San Bernardino Office Building configuration and instrument layout (after CSMIP, 2005) ......................................................................................................... 61?Figure 3-22 Spectral motions at the foundation base compared with free field at the San Bernardino Office Building in the 1992 Lander Earthquake in the east-west direction .................................................................................................................. 62?Figure 3-23 Comparison of the wave passage model with observed spectral motions at the foundation base at the Hollywood Storage Building in the 1994 Northridge Earthquake in the north-south direction ............................................................. 63?Figure 3-24 Transfer function (tau-factor) applied to modify the free-field motion as per Clough and Penzin (1995) for the Hollywood Storage Building in the north-south direction ........................................................................................................ 64?Figure 3-25 Comparison of the wave passage model with observed spectral motions at the foundation base of the Newport Beach Hospital Building in the 1994 Northridge Earthquake in the north-south direction ........................................ 65?Figure 3-26 Comparison of wave passage model with observed spectral motions at the foundation base at the El Centro Imperial County Service Building in the 1989 Imperial Valley Earthquake in the north-south direction ................................. 66?xv  Figure 3-27 Comparison of the wave passage model with observed spectral motions at the foundation base at the San Bernardino Office Building in the 1992 Lander Earthquake in the east-west direction ................................................................. 67?Figure 3-28 Comparison of the spatial incoherency model with observed spectral motions at the foundation base at the Hollywood Storage Building in the 1994 Northridge Earthquake in (a) east-west direction and (b) north-south direction ................................................................................................................................. 70?Figure 3-29 Comparison of spatial incoherency model with observed spectral motions at the foundation base at the Los Angeles 15-storey Office Building in the 1994 Northridge Earthquake in the north-south direction ........................................ 71?Figure 3-30 Comparison of the spatial incoherency model with observed spectral motions at the foundation base at the El Centro Imperial County Service Building in the 1989 Imperial Valley Earthquake in the north-south direction ................. 72?Figure 3-31 Transmissibility function between free-field and foundation motions for vertically incoherent waves (after Kim and Stewart, 2003) ............................... 73?Figure 3-32 Transfer function and estimation of incoherence parameter, ka, for foundation motion at the Sylmar Hospital Building in the Whittier Earthquake in the longitudinal direction ............................................................................................ 75?Figure 3-33 Transfer function and estimation of incoherence parameter ka for foundation motion at Sylmar hospital building from the Whittier Earthquake in longitudinal direction  ((Kim and Stewart, 2003) ............................................... 75?Figure 3-34 Spatial incoherency parameter verses soil shear wave velocity for sites (Kim and Stewart, 2003) ................................................................................................. 76?xvi  Figure 3-35 Comparison of foundation-base motion applying FEMA-440/ASCE41-06provisions with observed data at the Los Angeles 15-storey Office Building in the 1994 Northridge Earthquake in the east-west direction.......................... 78?Figure 3-36 Comparison of foundation-base motion applying FEMA-440/ASCE 41-06 provisions with observed data at the El Centro Imperial County Service Building in the 1989 Imperial Valley Earthquake in the east-west direction .. 79?Figure 3-37 Comparison of free-field and foundation-base motions at the Eureka 5-storey Residential Building in (a) the 2010 Ferndale Earthquake and (b) the 2008 Trinidad Earthquake in the north-south direction ............................................ 81?Figure 3-38 Comparison of free-field and foundation-base motions at the Fortuna Supermarket Building in (a) the 2010 Ferndale Earthquake and (b) the 1992 Petrolia Earthquake in the north south direction .............................................. 82?Figure 3-39 Comparison of free-field and foundation-base motions at the San Bernardino 3-storey Office Building in the 1992 Landers Earthquake in (a)the east-west direction and (b) the north-south direction ......................................................... 83?Figure 4-1 Features of continuum model used for the analysis .............................................. 88?Figure 4-2 Acceleration time histories of selected ground motions for analysis ................... 92?Figure 4-3 Response spectra of selected ground motion for analysis ..................................... 93?Figure 4-4 Selected records from the 1994 Northridge Earthquake event for analysis (a) acceleration time histories, (b) response spectra ................................................ 94?Figure 4-5 White noise input used in the analysis: (a) frequency content (b) response spectra ..................................................................................................................... 95?xvii  Figure 4-6 Effect of slab width - response spectra of response motion at the foundation slabs and free field for input of the NGA#30 Earthquake record ..................... 97?Figure 4-7 Effect of slab width - response spectra of response motion at the foundation slabs and free field for the input of White noise ................................................. 98?Figure 4-8 Building and mass configurations with constant storey stiffness ...................... 100?Figure 4-9 Building and mass configurations with constant first modal mass and fundamental period ............................................................................................. 100?Figure 4-10 Responses at the foundation base of buildings with different numbers of storeys and same typical storey stiffness to the excitation of the NGA#37 record .................................................................................................................... 102?Figure 4-11  Responses at the foundation base of buildings with different numbers of storeys and same modal mass and building period to the excitation of the NGA#37 record .................................................................................................... 104?Figure 4-12 Variation of motion from free field to foundation base under excitation of the NGA#37 record .................................................................................................... 106?Figure 4-13 Variation of motion from free field to foundation base under excitation of the white-noise motion ............................................................................................... 107?Figure 4-14 Reduction of foundation-base motion from free-field motion for Vs = 200 m/s and Vs = 300 m/s (mb =120 ? 103 kg, H = 15 m) ................................................ 108?Figure 4-15 Amplification and reduction of foundation-base motion from free-field motion for Vs = 200 m/s and Vs = 300 m/s (mb = 100 ? 103 kg, H = 20 m) ................... 109?Figure 4-16 Reduction and amplification of foundation-base motion from free-field motion for Vs = 200 m/s and Vs = 300 m/s (mb = 250 ? 103 kg, H = 30 m) ................... 110?xviii  Figure 4-17 Effect of soil damping in variation of motion from free field to foundation under seismic excitation of (a) the NGA#37 record, ( b) the NGA#30 record 111?Figure 4-18 Change in spectral acceleration of free fields and foundation bases with change in the mass of the building for a system with a 15 m deep deposit with Vs = 200 m/s under the NGA#122 excitation .................................................................... 113?Figure 4-19 Change in spectral acceleration of free fields and foundation bases with change in building period for a system with a 30 m deep deposit with Vs = 300 m/s under the NGA#30 excitation ............................................................................. 115?Figure 4-20 Effect of building mass on the foundation motion for a 20 m deposit with Vs = 200 m/s under the NGA#30 excitation ............................................................... 117?Figure 4-21 Effect of building mass on the foundation motion for a 30 m deposit with Vs = 200 m/s under the NGA#37 excitation ............................................................... 118?Figure 4-22 Effect of soil deposit depth on the response spectra of the motion at foundation base for a system with building mb = 175 ? 103 kg under excitation of the NGA#122 record .................................................................................................. 120?Figure 4-23 Variation of spectral values from free field to foundation base as depth of deposit increases under excitation of the NGA#37 record ............................... 123?Figure 4-24 Variation of spectral values from free field to foundation base as depth of deposit increases under excitation of the NGA#122 record ............................. 124?Figure 4-25 Variation of spectral values from free field to foundation base as depth of deposit increases under excitation of white noise ............................................. 126?Figure 4-26 Motions at free field and foundation in the system of a building with Tb = 0.4 s resting on a 25 m deep deposit of soil with Vs = 200 m/s excited under the xix  NGA#30, NGA#37, NGA#122, NGA#953 and NGA#993 records and white-noise motion .......................................................................................................... 129?Figure 4-27 Motions at free field and foundation in the system of a building with Tb = 0.3 s resting on a 20 m deep deposit of soil with Vs = 200 m/s excited under the NGA#30, NGA#37, NGA#97, NGA#953 and NGA#993 records and white-noise motion ................................................................................................................... 130?Figure 4-28 Motions  at free field and foundation in the system of a building with, Tb = 0.3 s resting on a 20 m deep deposit of soil with Vs = 300 m/s under excitation of the NGA#30, NGA#37, NGA#122, NGA#953, NGA#993 records and white-noise motion ................................................................................................................... 132?Figure 4-29 Motions  at free field and foundation in the system of a building with, Tb = 0.37 s resting on a 15 m deep deposit of soil with Vs = 200 m/s under excitation of the NGA#30, NGA#97, NGA#106, NGA#122, , NGA#993 records and white-noise motion .......................................................................................................... 133?Figure 4-30 comparisons of motions in the system of a building with Tb = 0.27 s resting on a 15 m deep deposit of soil with Vs = 200 m/s, where foundation motions are (a) reduced under the NGA#30 and NGA#953 records (b) amplified under the NGA#97 and NGA#989 records ......................................................................... 135?Figure 4-31 Change in response spectra of foundation motion with fixed-base period of structure in a soil?structure system with a 25 m deep deposit of soil Vs = 200 m/s for input of the NGA#953 motion ............................................................... 138?xx  Figure 4-32 Change in response spectra of foundation motion with fixed-base period of structure in a soil?structure system with a 25 m deep deposit of soil Vs = 200 m/s for input of the NGA#30 motion ................................................................. 139?Figure 4-33 Variation of foundation response spectra from free field at building period Tb for different systems under five earthquake inputs ......................................... 141?Figure 4-34 Change in foundation spectra with system period for the NGA#953 input .... 143?Figure 4-35 Change in spectra of motion at foundation base with system period for the NGA#30 input ...................................................................................................... 145?Figure 4-36 Variation of foundation response spectra from free field at system period, Tsys, for different systems under five earthquake inputs ......................................... 146?Figure 4-37 Cases of amplification in foundation response spectra in three earthquakes where amplification ceases to occur at the period larger than Tp ................... 148?Figure 4-38 Relation between building period and system period in different deposits .... 149?Figure 4-39 Mode shape of a system in fundamental mode: (a) system with a 20 m deposit, (b) system with a 75 m deposit ............................................................................ 150?Figure 4-40 Controlling period for variation of motion from free field to foundation for earthquake input of the NGA#30 record for systems with (a) 15 m deposit, (b) 50 m deposit .......................................................................................................... 152?Figure 5-1 Soil?foundation?structure (3DOF SFS) lumped-mass model ............................ 157?Figure 5-2 First frequency mode of soil deposit with foundations of 20 and 50 m width for different deposit depths ....................................................................................... 160?Figure 5-3 Static stiffness of the soil?foundation system obtained from pushover analysis in ABAQUS ............................................................................................................... 161?xxi  Figure 5-4 Comparison of static stiffness of the soil?foundation system obtained from ABAQUS, Equation 5.4 (Gazetas, 1991), and a modified formula (Equation 5.5) ......................................................................................................................... 163?Figure 5-5 Cantilever shear beam model of free-field deposit .............................................. 166?Figure 5-6 Ratio of deposit to foundation area for an equivalent uniform shear beam model for the case of foundation width, W = 20 m ...................................................... 170?Figure 5-7 Acceleration response in time history at the foundation level for the NGA#30 record in the soil?structure system with deposit H = 20 m and building Tb = 0.3 s .............................................................................................................................. 174?Figure 5-8 Time history acceleration response at the foundation level for the NGA97 record in a deposit with H = 25 m and building Tb = 0.33 s ........................................ 174?Figure 5-9 Time history acceleration response and comparison of response spectra at the foundation level for the NGA#122 record in a deposit with H = 20 m and building Tb = 0.3 s and free field ........................................................................ 176?Figure 5-10 Time history acceleration response and comparison of response spectra at the foundation level for the NGA#30 record in a deposit with H = 20 m and building Tb = 0.4 s ................................................................................................ 177?Figure 5-11 Time history acceleration response and comparison of response spectra at the foundation level for the NGA#122 record in a deposit with H = 20 m and building Tb = 0.4 s ................................................................................................ 178?Figure 5-12 Time history acceleration response and comparison of response spectra at the foundation level for the  NGA#122 record in a deposit with H = 15 m and building Tb = 0.4 s ................................................................................................ 179?xxii  Figure 5-13 Time history acceleration response and comparison of response spectra at the foundation level for the NGA#97 record in a deposit with H = 25 m and building Tb = 0.4 s ............................................................................................... 180?Figure 5-14 Time history acceleration response and comparison of response spectra at the foundation level for the NGA#953 record in a deposit with H = 15 m and building Tb = 0.4 s ................................................................................................ 181?Figure 5-15 Time history acceleration response and comparison of response spectra at the foundation level for the NGA#37 record  in a deposit with H = 30 m and building Tb = 0.43 s ............................................................................................. 182?Figure A-1 Model of soil?structure system by shear beam with appending spring?mass . 192?  xxiii  Acknowledgements  I am deeply grateful to my supervisor Dr. Carlos Ventura. He has not only provided me academic guidance and research direction for this study but provided me immeasurable support throughout my time at UBC. I feel greatly fortunate to get such a mentor who can easily translate complex concepts of structural dynamics to simple rules of science. I have benefited a lot from his exceptional knowledge and experience in earthquake engineering.  Whenever I was puzzled with research problems, he had simple solution around it by just a little twist.   This work would not have been accomplished without academic, financial and moral support he provided.    I would like to express deepest gratitude to my co-supervisor Dr. Liam Finn. His deep insight and interest in the subject of my thesis and constant encouragement to me are the driving forces to move ahead in the research and get the results.  His passions to the new findings taught me to love research. He never bothered spending hours discussing the problem until a logical way forward comes to sight.  Without his guidance and advice, this thesis would not come out with the results presented.   I would also like to thank supervisory committee members Dr. Donald Anderson and Dr. Mahdi Taiebat.  Dr. Anderson provided valuable suggestions in the thesis. His thorough check of concepts and every equations, units and numbers provided an additional confidence in the research results. Dr. Taiebat provided practical advice and guidance in numerical simulation part of the research.   xxiv  The research is partially funded by Natural Sciences and Engineering Research Council of Canada (NSERC) through Soil-structure interaction project and by BC Ministry of Education (MOE) through BC school retrofit project. The supports are greatly acknowledged.   I have been quite fortunate while at UBC to be with nice people around me. I dragged my friends Jose Centeno, Sheri Molnar, Yavus Kaya and Jason Dowling at Earthquake Engineering Research Facility several times to my research inviting their comments and positions in the research problem. I appreciate their inputs, supports, kindness and friendship.  I would also like especially thank to my good friend Juan Carlos, who provided valuable advice in the early phase of the research.   I would like to devote special thanks to my mother who I left behind at home to pursue the study. Her understanding and moral support always reminded me to work hard being so far from her.     Finally, I would like to thank my family for their support and love. I owe my dear wife Indira a depth of gratitude for her patience, understanding and sacrifices during the past years and continuous support and encouragements, which enabled me to confront challenges that came to the road of PhD study.  To my lovely daughter, Ashvina, who gave us a different perspective of life, I can now assure that I will make up for the time you missed to play with me. This work is dedicated to my late father LaxmiNarayaran and late brother TulsiRam   who had a dream to see me as an engineer.        xxv  Dedication         To LaxmiNarayan and TulsiRam1  Chapter  1:  INTRODUCTION Current code design practice adopts a method to estimate motion at foundation of building with rigid surface foundation that considers foundation slabs always reduce the motion as an effect of seismic soil-structure interaction (SSI). It implies that SSI is beneficial and can be conservatively neglected in regular design. This thesis examines the cases of instrumented buildings where motions are recorded at foundation and free fields in past earthquakes.  From analysis of the records, it is found that motions at the foundation base are reduced in some cases and amplified in other cases. It is also found that motion at foundation of same building is reduced in one earthquake event and amplified in other event.  A continuum model of soil deposit together with a building having generic properties is analyzed varying the system?s parameters to identify the reasons of variation of foundation motion.  From the parametric study, it is found that motion at the foundation could be higher or lower than free-field motion based on combinations of period of building, soil deposit and input motion as well as mass of building relative to deposit depth.    The main contribution of the research study is to identify the problem of current code practice of estimating foundation motion in buildings with shallow surface foundation and to present explanation of why foundation motion varies from one case to another in earthquake loading. An 2  alternative simple model of soil- structure system is also developed that can explain the phenomena.    Introduction  The seismic response of a structure depends on the dynamic characteristics of the structural system and the motion it experiences. If soil-structure interaction (SSI) effects are not considered in structural analysis, motion obtained in the field with no structure on it, known as ?free field motion?, is used as the input motion. To incorporate SSI effects, the input motion needs to be modified.  This is particularly important if the building rests in other than hard rock and has a rigid mat foundation or individual footings are tied together to form an integral unit.  In such a case, motion at the foundation base is not only affected by averaging of wave motion but also by the dynamic interaction between soil deposit and structure.   Approaches to Account SSI Effects There are two approaches to consideration of the seismic soil?structure interaction effects in the structural analysis. The first is the direct modeling approach, where soil, foundation, and structure are incorporated in the same model and analysis is carried out in a single step. This approach can handle all the interactions between model components, and the response of the structure is expected to be representative of the free field. This approach is implemented in a continuum model with the finite element method or the boundary element method. In this approach, there is no need to derive the foundation motion explicitly for the analysis, as the model takes care of the response of each component of the system.  3   However, the computational resource required to develop the model and run the analysis is so large that it is not practical to implement this approach in routine analysis and design practice. It is mostly used for a complicated system and for verification of results obtained from a simple model.   The second approach, known as ?sub-structure approach? divides the analysis model into two components by sub-structuring and to carry out the analysis in stages. First, the soil and massless structure will be considered in an analysis model to find the foundation input motion (FIM). This motion is considered to be different from free-field motion because of the presence of foundation. The foundation is supposed to take care of wave scattering and averaging of the motion over its base area. These effects are regarded as the ?kinematic interaction effects?. In the second stage, analysis of a superstructure with its mass on it is carried out with FIM applied at the foundation base. Further interactions between structure and soil are supposed to take place because of the inertia force generated from structural mass. The effects of such interactions are regarded as ?inertial interaction effects?.    In the sub-structure approach, deriving the proper foundation motion is an important step, as it will serve as the excitation motion for the structure.   In some current design codes, provisions are included to consider SSI effects, which are based on sub-structure approach. In the guidelines and standards like FEMA-440 (2005) and ASCE/SEI 41-06 (ASCE, 2007), provisions are included to estimate the foundation motion for shallow 4  integrated surface foundations in terms of modified design spectra for the response spectra analysis. The modification to the design site spectra is given by ratios of response spectra (RRS), which are always less than 1.0. This implies that the foundation-base motion will always be less than free-field motion suggesting that SSI is beneficial and can be conservatively neglected in regular building design practice.  Basis of Current Methods of Estimating Foundation Motion in ASCE/SEI 41-06  The code provisions in ASCE/SEI41-06 in regards to estimating motion at foundation of building with shallow rigid foundation is based on work of Kim and Stewart (2003), who used instrumented records of several buildings in California, USA, to calibrate a model proposed by Veletsos and Prasad (1997) to account for averaging of spatially variant motion by rigid massless slab over the area. The implementation of the code provisions may result in inaccurate demand on the structure. The reason is as follows: The model proposed by Veletsos and Prasad (1997) can only account averaging of wave motion by the base slab and not dynamic interaction between structure and soil deposit. However, the instrumented records Kim and Stewart (2003) used for calibration of the model are from buildings that must also have dynamic interaction with underlying soil due to the effect of mass of structure. The model, which considers only averaging of wave motion and would always result in reduction of motion at foundation, was calibrated with the data that have effect of both averaging and dynamic interaction.     5  Motivation and Objectives of the Study  A wealth of seismic records obtained from the building sites that have instrumentation at the building foundation as well as at nearby free fields has provided opportunity to evaluate the soil?structure interaction effects objectively. The actual motion at foundation base can be obtained directly from the records. This makes possible to examine the code provisions of estimating foundation motion against the field data.  The provisions in the code that suggest reduction of motion at the foundation base of a rigid shallow foundation compared with free-field motion need to be reviewed against the field data, as the basis the code provision do not seem compatible with the original model of averaging wave motion. In order to get a better understanding of the variation of motion from free field to foundation, a continuum model can be developed where parametric study could explain the cause of such variation. This study utilizes the information obtained from instrumented records and the capability of continuum model for better understanding of the effects.   Goal of this study is to provide understanding of variation of motion from free field to foundation base of buildings with shallow rigid foundation. Following are the major objectives of study to achieve the goal:   ? To revisit the basis of provisions of ASCE and other methods of estimating foundation motion based on the records from instrumented buildings ;  ? To identify the key factors of  and their relations to the variation of motion from free field to foundation base in the seismic loading environment; and   6  ? To propose a simple alternative model that explains the phenomena of variation of motion.   Following are major tasks carried out in this study to meet the objectives: - Conduct literature review on variations of motion from free field to foundation due to seismic soil- structure-interaction. - Collect and analyze the seismic motions recorded at instrumented building sites. This also includes comparison of results with the estimations using ASCE provisions and other methods - Identify the patterns in the results of analysis of recorded motions in terms of variations of motions. - Develop a continuum model of soil deposit with building and analyze the system varying parameters. - Analyze the results to establish the relation between properties of soil-structure system and the variation of motion from free field to foundation.  - Develop an alternative simple physical model to explain the phenomena of variation of motion.  Organization of the Thesis  This thesis is organized as follows: Chapter 2 provides a literature review on the available procedures and models of estimating the motion at foundation from free-field motion. The theoretical basis of models based on wave 7  passage effects and spatial incoherency effects are presented. The chapter also describes provisions in ASCE/SEI41-06 for estimating foundation motion. The need for a critical review of methods and procedure is discussed.  Chapter 3 describes the detailed investigation carried out on the seismic records obtained at the foundation base and free field of instrumented building sites from past earthquakes. Procedures and criteria for data collection are explained, followed by comparative analysis of the motions at foundation base and free fields. Trends and patterns observed from the results of analysis of these motions are discussed. The variations of observed motions between free field and foundation base are compared with available methods and procedures for estimating foundation motion. As it is found that proposed methods and code provisions do not provide reliable estimation of foundation motion, particularly, in the cases when the motion at foundation base is higher than at free field, approach to be taken for further investigation is discussed.   Chapter 4 describes development and results of analysis from a continuum model of soil deposit which also includes a building on top of it. The details of the model developed in the computer program ABAQUS and parameters used for the analysis are given. The effects of foundation width, number of storeys of building, mass of building, soil shear wave velocity, damping of soil and depth of soil deposit in a soil-structure system to the variation of motion from free filed to foundation are discussed. After the synthesis of the results of analysis, patterns are identified and causes of amplification of foundation motion in one case and reduction in another case are presented. The most important factors controlling the variation of motion are discussed.   8  Chapter 5 describes development of a simple alternative model that can explain the phenomena of variation of motion from free field to foundation. Detail steps of development of lumped mass model of soil structure system are presented.  Derivations of parameters of components of the soil?foundation?structure system are discussed in detail. The results of this simple model are verified with the results from ABAQUS run.   Chapter 6 summarizes the conclusions of this research and the major contributions made to the topic. Major findings from the analysis of field data and parametric study carried out in a continuum model are presented in line with goals and objectives of the study.  It also suggests future research work in the field of SSI analysis of buildings with shallow rigid foundations and variation of motions in other modes of vibration.   Appendix A presents formulation of system period of soil-structure system based on shear beam model of soil deposit that has appended mass-spring-dashpot representing building.   Appendix B presents the MATLAB algorithm of analysis of soil-structure system with the SFS 3DOF model.    9  Chapter  2:  STATE OF PRACTICE IN ESTIMATING FOUNDATION MOTION 2.1 Context In the seismic analysis of building structures, it is common practice to apply the earthquake motion estimated for the site to the base of the structure directly. The practice is based on the assumption that the foundation base is fixed and is subjected to the motion that comes at the free-field ground surface. This assumption is valid only in the cases when the building sits on hard rock or very stiff soil and incoming seismic waves hit all the points of the foundation base at the same time. Otherwise, the assumption is not valid, and analysis results may become inaccurate.  For the flexible soil condition, which is the most prevalent case, there are two major ways to address the problem. First, a complete model is prepared that includes both the structure and the soil around it, and analysis is carried out to find the member forces and displacement in the structure. Second, the motion that the foundation actually experiences is estimated, and analysis of the structure is conducted for this motion as an input.   The first approach, usually taken using a continuum model, has a capability of analysing any complex cases including nonlinearity in soil and structure. However, it is computationally 10  expensive and is not feasible in daily design practice. Also, it may still have a problem in defining the extent of soil media that is unbounded in the horizontal direction, if not in the vertical direction, too. The second approach is straightforward in concept and is practical for design office practice once the actual input motion is defined. However, the following question arises: how reliable are the available methods proposed to estimate the foundation motion when compared to field data?   2.2 Free-field and Foundation Motions The seismic motion at the ground surface that is free from any influence of structural vibration is referred to as ?free-field motion?. This motion is influenced by types of seismic waves and direction of propagation (Luco and Wong, 1978) and soil conditions (Idriss and Seed, 1968) when motion at the base rock propagates to the ground surface. When a structure is built on top of this ground, the motion at the same point will deviate from the free-field condition, as the structure imposes constraints and the pattern of the wave motion changes. The dynamic effects from the structure and soil deposit are combined. The effects are reflected in the motion of the foundation, which serves as the continuity point between the two. Hence, the effect of the soil deposit on the base rock motion and the effect of the foundation on the wave pattern cannot be simply added to find the motion at the foundation. Rather, the structure also affects the response of the soil, based on the dynamic properties in relation to the characteristics of input motion. The foundation motion is the result of all these effects coming together.   11  2.3 The Soil?Structure Interaction and Foundation Motion   Classically, the problem of soil?structure interaction is dealt with by defining its effects in two categories: (i) kinematic interaction and (ii) inertial interaction. The kinematic interaction effects are considered as the effects of the geometry of the massless foundation on the seismic waves through averaging. The effect is induced in the wave field where the individual peaks of the spatially variant wave are attenuated by the averaging effect of rigid slab motion. The wave passage effect on the inclined or horizontal wave is also included in the kinematic interaction. For the foundation with embedment, this effect also includes variation of ground motion with depth and scattering of waves at the corner of the foundation. The analysis carried out for such effects will result in reduction of original motion. The motion thus obtained by applying the kinematic interaction is regarded as the effective ?Foundation Input Motion (FIM)?.   Then, the inertial interaction effect is simulated by applying the FIM to the foundation base by replacing the soil by springs and dashpots. The inertial effect usually results in period lengthening owing to added compliant soil properties and increased damping included to represent radiation. The increased period and damping will further reduce the response of the structure. The resulting reduction from the inertial effect can be incorporated in the input motion by modifying the response spectra, as shown in the Figure 2.1 (adopted from NEHRP, 97)   This treatment of soil?structure interaction will effectively reduce the foundation motion. This also implies that the SSI effect is beneficial, in general in reducing base shear, and one can 12  neglect it as a conservative simplification of a complex problem. This leads to the use of free-field motion as the foundation motion for fixed-base analysis.    Figure 2-1 Change in the response spectra value due to increased damping and period lengthening (NEHRP, 1997)  The reliability of estimation of motion at the foundation level, including the suggestion that it be reduced from free-field motion using the approach described above, needs to be closely reviewed against the field evidence. In this chapter, different methods used to estimate the foundation motion, including current code practice, will be explained.   13   2.4 Early Studies on the Variation of Motions  Osawa et al (1974) presented a case of Earthquake Research Institute (ERI) six-storey reinforced concrete building in Tokyo where 29 earthquake records were obtained at various levels of structure and at ground surface 60 m away from building edge.  The observed records showed that variation of motion from free field to foundation depends on the period of structure. They reported that foundation motions were attenuated in the neighborhoods of fixed-base fundamental period of structure. Hradilek et al (1974) also reported that foundation motions were attenuated around fixed-base natural frequencies of building based on field observations in the South Pacific Building in San Francisco by Borcherdt (1970) during a nuclear blast in Nevada and analysis of seismic records in the Los Angeles  Hollywood Storage Building sites during the 1952 Kent  Earthquake.   Later, Minami and Sakurai (1977) also concluded that fundamental period of building structure relative to period of deposit is an important factor in the response of structure including that at the foundation.  Their conclusion was based on analysis of model soil-structure system for the motions of 1940 El Centro Earthquake and the 1952 Taft Earthquake records.   Seed and Idriss (1974) mentioned that response of structure may increase depending on proximity of underlying rock to the foundation based on a finite element analysis of an embedded structure including surrounding soil. The responses quantities obtained in the structure (including at foundation level) were related to free field motion. It shows the effect of deposit 14  depth to the variation of motion. Hasiba and Whitman (1968) and Whitman (1969), also studied the effect of depth of soil deposit to the interaction between structure and soil mass. They concluded that SSI effect is pronounced when the depth of soil deposit is in the range between 0.5R and 4R, where R is the foundation radius.    Pavlyk (1976) showed a theoretical model for an approximate transfer function between free field and foundation motion based on SDOF characterization of soil-structure system. Although the model is not capable to accurately quantify the variation, it shows the effect of stiffness of building to the variation of motion from free field to foundation.  2.5 Averaging of Motion over Transit Time  Newmark et al. (1977) and Morgan et al. (1983) proposed a method to estimate the foundation motion from free-field motion using a simple numerical averaging procedure. The technique is based on a concept that the averaging of motion over a time delay in excitation to parts of the foundation caused by the horizontally propagating waves that impinge first on one side of the building foundation and then on the other side of it gives modified motion at the foundation slab. The averaging is done over the transit interval along the acceleration time history.   Figure 2.2 provides a sketch employed in the calculation of averaging of motion for a given building foundation. The foundation is subjected to shear wave motion of the earthquake. In this case, the transit time, trt , is taken as the time required for a wave to travel along the length of the foundation. This is equal to the length of the foundation divided by the effective wave velocity, 15  which is the velocity of wave motion realized at the ground surface in horizontal direction. At one point, the wave hits the left edge of the foundation with its acceleration (corresponding to time point A of original acceleration history), whereas the right edge receives another acceleration (corresponding to time point B of original acceleration history). The foundation motion at this point of time would be an average acceleration over the time period trt . The average acceleration is computed from velocities corresponding to the acceleration at the beginning (point A) and at the end (point B). As time progresses, the acceleration time history simply slides along the foundation, and new foundation motions are effective.   Figure 2-2 Base translations arising from travelling seismic waves  If the velocity of the motion at time point A in the time history is Au?  and the velocity at time point B is Bu? , the foundation motion at the instant is given by  ? ?BAtruutu ???? ??1 .          (2.1) 16  For a rectangular foundation, it is suggested that same transit interval can be employed in both directions using the geometric mean dimension. In that case, the transit time is calculated by  str Vbat ??            (2.2) where a and b are plan dimensions of the foundation and Vs is the speed of the horizontally propagating shear wave.  Morgan et al. (1983) mentioned the observation that buildings with large foundations have less intensity of motion than do small structures in an earthquake. The primary observation point was the recording at the Hollywood Storage Building in the San Fernando Earthquake, where the motion at the foundation base was reduced by a factor of 2 or 2.5 in the short period range in the response spectra. Figure 2.3 illustrates the deviation of motion from free field to foundation of the Hollywood Storage Building in the 1971 San Fernando Earthquake in the east-west direction. There is a huge reduction of spectral motion up to a period of 0.4 s. The figure also shows the estimated motion at the foundation using this technique. The result is comparable with the recorded foundation motion.   The method is simple in implementation and gives a systematic way of deriving foundation from free field. It does not require an assessment of the frequencies included in the earthquake motion. Because of this, the method has been in use for a long time.  17   Figure 2-3 Comparison of the result of the time averaging technique with response spectra of motion recorded at the Hollywood Storage Building basement and PE Lot (free field) in the San Fernando Earthquake, 1971 (east-west direction)  2.6 Tau-effect Transfer Function  The deviation from free field to foundation base in a rigid foundation is described as a kinematic effect in translation motion in terms of the ?tau-effect? by Clough and Penzin (1995) in their book Dynamics of Structures. It is based on the concept that when the earthquake motions vary significantly within the area of a foundation, they will be constrained by the rigid foundation mat. If the rigid foundation has dimensions greater than the apparent wave length in the frequency range of interest, the motion at the base of the foundation will be the average of the free-field motion over the foundation area.   18  For one-dimensional horizontal wave propagation, they proposed that the modified translational motion is obtained by Fourier transforming the free-field acceleration, multiplying the amplitude by a transfer function, and inverse Fourier transforming the product to get the modified acceleration. The ratio of the amplitude of motion at free field to that at the foundation base is given as a function of frequency and wave length.  The amplitude of the transfer function is given as  ? ???? cos121 ??           (2.3) where ? = ????? ???? , Va is apparent wave velocity, ?? is frequency, D is foundation dimension in the direction of wave propagation, and ? is the wave length. The values of ? decrease from unity at ? = 0 to zero at ? = 2?, as shown in Figure 2.4 (adopted from Clough and Penzin, 1995). This means that if the base dimension of the foundation is very small compared with the wavelength of the ground motion, the tau-effect is negligible and the slab motions will be essentially the same as the free-field motions. On the other hand, if the base dimension of the foundation is fairly large compared with the wavelength of the ground motion, the tau-effect is significant, and the base motion could be much smaller than the free-field ground motion. 19   Figure 2-4 Tau-factor as a function of frequency and apparent wave velocity (Clough and Penzin, 1995)  As the transfer function is always less than unity for any frequency greater than zero, the resulting base motion computed from this method is always less than free-field motion. Also, the variation of motion is monotonically decreased as the period increases in the response spectra curve for a given foundation dimension and given soil site.   Figure 2.5 shows an application of the method for an 80 m wide foundation to the free-field earthquake record of the 1994 Northridge Earthquake in response spectra (5% damping). Soil shear wave velocity Vs is 340 m/s in this example case. To apply this technique, the original time history is made longer adding zeros at the end to make the number total data points in the power of 2. It is transformed to Fourier spectra, which is then multiplied by ? factor given by Equation 2.3. Inverse Fourier transformation is carried out for the resulting Fourier spectra to get new time history. The response spectra of the new time history is presented in Figure 2.5 (dotted line). It is 20  noted that the method gives high reduction of motion at foundation at the short period. The reduction of motion gradually decreases to have the same spectral acceleration at the long period.   Figure 2-5 Application of the tau-factor averaging method in the Northridge Earthquake record for a foundation width 80 m and Vs =340 m/s  2.7 Variation of Motion due to Spatial Incoherency Veletsos and Prasad (1989) and Veletsos et al. (1997) proposed a method to estimate the deviation of motion from free field to foundation caused by other factors than wave passage effects. It is considered that the free-field motion does have spatial variability that, when impinged to the rigid foundation, is averaged over its area, and the resulting motion over the 21  foundation slab deviates from the original free-field motion. Incoherency in the ground motion is assumed to be a result of possible different paths of individual wave trains emanating from different points of an extended source or different timing of impinging to the ground surface. It is claimed that the variability exists even for the vertically propagating shear waves.   The method computes the transfer function, which relates motion at the free field and the foundation base, by accounting for the spatial variation of the wave and assuming foundation?soil contact. The transfer function amplitude is presented in terms of ratio of spectral density of free-field motion to the motion at the foundation base for harmonically excited massless foundations. Figure 2.6 illustrates the amplitude of the transfer function for vertically incident incoherent waves. In the figure, ?x and ?y are spatial incoherency in the x and y directions, be is the effective width of the foundation, and Sll and Sgg are power spectral densities of estimated foundation motion and original free-field motion, respectively. The value of spatial incoherency terms ?x and ?y are not specified in the paper by Veletsos et al. (1997). However, it is shown that the transfer function for translational motion is always less than unity for any value of incoherency. This implies that foundation motion will always be less than free-field motion.   As the model is based on concepts that a free-field wave has variability due to a number of random factors that are not readily quantifiable, the applicability of the model is not viable for problems in the field. Further, since this model considers only the geometrical constraints imposed by the foundation on the wave, the contribution of dynamic interaction between foundation and soil will not be reflected in the estimated motion at the foundation base.   22   Figure 2-6 Transfer functions for lateral components of foundation input motion for rectangular foundations subjected to vertically incident incoherent waves (Veletsos et al., 1997)  2.7.1 Calibration of spatial incoherency from instrumented building records Kim and Stewart (2003) calibrated incoherency parameters in the model developed by Veletsos et al. (1997) using seismic records of instrumented buildings with shallow foundations. The calibrated incoherency parameters are then correlated with soil shear wave velocity of the site. This results in a simple procedure of estimating transfer function between motions at free field and foundation base from shear wave velocity and foundation dimensions. The transfer function is then applied to modify the frequency content of the free-field motion to derive the base motion.   23  2.8 FEMA/ASCE Provisions of Base Slab Averaging for Shallow Foundations FEMA-440 (2005) and ASCE/SEI 41-06 (ASCE, 2007) have provided reduction factors for spectral values due to the action of the foundation slab, as shown in Figure 2.7, for slab foundations with shallow embedment. The foundation input motion for subsequent building analysis and design is obtained by applying the period-dependent reduction factor to the code spectrum. Based on charts of reduction values developed for effective width ranging from 65 ft to 330 ft, Equation (2.4) is provided to calculate the reduction in the response spectra for base slab averaging (RRSbsa). 2.1100,1411 ???????? TbRRS ebsa    ? the value for T = 0.2 s    (2.4) where be = ?(ab) is effective foundation size in feet, a and b are longitudinal and transverse dimensions of the footprint of building foundation in feet, and T is the fundamental period of the building in seconds.   The basis of the FEMA/ASCE provision is the base slab averaging effect, where foundations get reduced motion compared with free-field motion, based on work by Kim and Stewart (2003). The frequency-dependent transfer function from the Veletsos model is derived using the incoherency parameter estimated by Kim and Stewart for different effective widths of foundation. Reduction factors for response spectra are obtained from the transfer function for corresponding periods. This method, too, cannot capture the amplification of the motion at the foundation base. 24    Figure 2-7- Ratio of response spectra for base slab averaging, RRSbsa (ASCE/SEI 41-06, 2007)  2.9 Summary of State of Practice and Research Direction Procedures for calculating the variation of motion as described above are based on the effects of wave passage or spatial incoherency, which always produce the reduction of motion at the foundation level. They support the general understandings that SSI is beneficial and can be conservatively neglected in regular building design practice. However, it was reported that an office building in El Centro, California, sustained major damage during the 1979 Imperial Valley Earthquake despite the fact that the level of shaking in the nearby free field was only about 0.15g. The instrumented record shows that the motion was highly amplified (Poland et al., 2000). 0 0.2 0.4 0.6 0.8 1 1.2Period (s)0.40.50.60.70.80.91Foundation/free-field RRS from base slab averaging (RRSbsa)Simplified Modelbe = 65 ftbe = 130 ftbe = 200 ftbe = 330 ft 25  They also reported that some other buildings experienced amplifications in other earthquakes. Currently available methodologies to estimate the foundation motion do not explain this phenomenon.  In view of these observations, a closer look at individual significant cases with respect to the ASCE provisions and other methods to estimate the variation of motion will be made in this study. That will provide insight into how different the results from actual cases are and what contributes to the discrepancies, if any. This will help to establish a clear picture of the mechanics of interaction that would justify the variation of motion. The case of amplification is more important because analysis using the current procedure could significantly underestimate the demand on the structure.  26  Chapter  3:  VARIATION OF SEISMIC MOTION FROM FREE FIELD TO FOUNDATION IN INSTRUMENTED BUILDINGS 3.1 Introduction  The variation in motion from free field to the foundation base of buildings with rigid shallow foundations is attributed to the soil?foundation?structure interaction. The presence of a structure on top of a deposit of soil modifies the motion of the deposit from what it would otherwise experience should the structure not be present. The modification implies that (a) response of the soil deposit will be different from the results of site response analysis and (b) input motion for the structure sitting on top of the deposit is different from the motion at the free field. Hence, the input motion to the structure at its foundation required for structural analysis cannot be derived simply from site response analysis of a deposit to the base rock motion.  Motions that  the foundation experiences during earthquake shaking is important for design practice, as structural analysis is usually carried out with a model that has a fixed foundation where input motion is applied. In code-based design practice, this motion is in the form of design 27  spectra that constitute the seismicity of the region. The code design spectra modified to take account of local soil conditions represent the free-field motion. In a conventional design practice where seismic soil?structure interaction (SSI) is not taken into account, the spectrum, modified for the local soil conditions, is applied at the base of the foundation. In seismic analysis and design that incorporated SSI effects, the variation of motion from the free field to foundation is also accounted for in the modification of design spectra. For time history analysis, the input motion at the foundation should be the motion that the foundation actually experiences, rather than the motion that the free field experiences.  Instrumented buildings can provide the actual motions experienced at the foundations during earthquakes. Many of the buildings that are instrumented also have a free-field station nearby. This offers a unique opportunity to compare motions at the free field and the building foundation. This will help to develop an understanding of the extent of variation of these motions and what parameters could affect this variation.  There is a large database of seismic motions recorded at the instrumented buildings in California, USA, during past earthquakes. The California Seismic Motion Instrumentation Program (CSMIP) database has one of the largest of such databases of recorded motions at building sites in California. In this study, recorded motions from the CSMIP database are investigated from the point of view of variation of motion from free field to the foundation base of buildings. The results of this investigation are presented in this chapter.  28  First, the procedure and selection criteria of the instrumented buildings in the CSMIP database for this study are described. This is followed by observations about the time history records and their response spectra for some of these buildings. The results of spectral analysis of free-field motions and motions at the foundation are presented for representative buildings. The variations in the motions of those buildings, in terms of response spectra, are discussed in more detail. The cases of motion reduction, amplification, and no significant variation from free field to foundation are examined thoroughly.  In the next step, the recorded foundation motions are compared with the motion estimated for the foundation base from the free field through the use of different techniques proposed by various researchers. The results, after the application of base-slab-averaging techniques, are compared with field data through the use of estimations accounting for spatial incoherency and the method of FEMA/ASCE to estimate the motion of a rigid foundation base. These comparisons provide a justification for further research that is needed to explain the variations observed in the motions. Cases providing some insight into the contributing factors for the variations in the motions are discussed.  3.2 Data Collection from the Records of the Instrumented Buildings The variation of earthquake motions recorded at the free field of a deposit and recorded at the foundation base of structures is evident in the records of the instrumented structures that have experienced one or more earthquakes in the past. In California, USA, a few buildings were instrumented with seismic instrumentation as early as the 1950s. After the 1971 San Fernando 29  Earthquake, CSMIP was established, and more buildings were instrumented to obtain strong motion records. In many of these buildings, earthquake monitoring devices were installed both within the building and at nearby free-field sites. Such records, where data are available both at the foundation base of the building and at a nearby free-field site, allow comparisons of motions. Recorded earthquake motions at instrumented buildings are collected from the CSMIP Instrumented Building Response and 3D Visualization System (Naiem et al., 2005) and the Centre for Engineering Strong Motion Data (CESMD, 2010). There are more than 100 buildings instrumented under the CSMIP, and records of only those buildings that meet the following criteria were selected for this study: 1. The building has a surface or lightly embedded foundation with mat slab or interconnected beams at the grade level so that the foundation can be regarded as a rigid unit. 2. The building has sensors located at a minimum of two of the following locations: free field, foundation base, and ground floor. 3. Earthquake records are available for comparison between sensors at a minimum of two levels and for at least one significant earthquake event. 4. Earthquake records at sensors at different locations have the same north-south and east-west directional components. 5. The free-field station should be at a distance of less than 500 m from the instrumented building of interest. However, it should not be so close to the structure that its motion is affected by the structural response.  30  A total of 44 buildings from the CSMIP building database were selected to study the variation of the motions from the free field to the foundation base during past earthquakes. From 32 different earthquakes, 137 pairs of records have been investigated in this study. Some buildings among those selected do not have corresponding free-field records in the CSMIP database. Free-field records for those buildings are taken from other ground motion databases, including the US Geological Survey Database (USGS, 2010) and the PEER Strong Motion Database (PEER, 2010), ensuring that free-field motions are recorded at a distance of less than 500 m from the instrumented buildings.  Based on the availability of records and building configuration data, the selected buildings are grouped into the following three categories: i) Buildings with records available at foundation base, at basement level, and at free field (BF) ii) Buildings with records available at foundation base, at basement level, and at ground floor level (BG) iii) Buildings with records available at foundation base, at ground surface level, and free field (GF)  Selections of channels in a building station are made to ensure that they provide records of the same earthquake in the same direction as the channel at the free-field station for the purpose of the comparative analysis. Records in both east-west and north-south directions are included from each station, if available. If there is more than one channel available in a single direction at a building foundation base or at ground floor, the channel that is closest to the instrument at free 31  field is chosen. Only records from channels that are installed directly at the base slab are considered to represent the foundation motion for the study. In many instances, sensors at the free field do not have the same orientation as the building reference axis. In such a case, free-field data cannot be directly compared with the building data, since the sensors are not aligned in the same directions. This incompatibility is addressed by transforming the free-field axis to the building reference axis. If ? is the angle between the building reference axis and free-field true axis and AccEW and AccNS are original accelerations recorded at the free field, the modified accelerations that correspond with acceleration records at the foundation base are ???????????? ????????NSEWNSEWAccAccAccAcc????cossinsincos .       (3.1)  An example case is shown in Figure 3.1, which depicts the layout of instrumentation at foundation level and free field for the Piedmont three-storey building. In this case, channel #8 and channel #9 are chosen for the acceleration records of foundation motions in north-south and east-west directions, respectively. The sensors at the free-field station CSMIP 58790 are oriented at 45? to the building axis. Acceleration data recorded at the free field are modified for ?? 45?  using Equation 3.1 to compare with accelerations recorded at the foundation base. 32   Figure 3-1 Sensor layouts at foundation and free field in the building site of a Piedmont three-storey building from the CSMIP database (Naeim, 2005)  Seismic signal processing software, SeismoSignal (SeismoSoft, 2010), is used for necessary baseline corrections to the records taken for the study. In order to eliminate noise in the original signal, frequency filtering is also carried out as necessary. A Butterworth fourth order infinite-impulse-response (IIR) filter is used for band pass filtering for a frequency range of 0.1 to 25 Hz. Table 3.1 lists buildings selected for the study, where seismic records are available at the basement of the building to compare with the records at the nearby free field. Eight buildings have 33 records at the foundation base to pair with free-field records. Peak ground acceleration (PGA) at the free field and peak acceleration (PA) at the foundation as well as their peak spectral acceleration values for 5% damping are presented in the table. 33  Table 3.2 lists buildings where seismic records are available at the basement and ground floor of the building but not at the nearby free field. These sites are selected to investigate how motion is changed from the basement foundation to the ground floor in the buildings. This will provide insight on how the practices of assuming fixed-base conditions at the ground level and applying code motions stand against the field data for lightly embedded buildings.  Table 3.3 lists the buildings with rigid surface foundations and details of records at their foundation base and free field. There are a total of 26 buildings selected, mostly low-rise commercial buildings, schools, and hospitals in this category.  Records at these building sites have low to medium intensity earthquake accelerations. No earthquake was reported to cause significant damage to these buildings except to the El Centro County Service Building in the 1979 Imperial Valley Earthquake. The free-field record at the site of the El Centro County Service Building in the earthquake, however, shows that the intensity of ground motion was medium, with PGA = 0.15g. It can be conveniently assumed that the records at the foundations of these buildings reflect the elastic response of the soil?structure system to the earthquake motions. 34  Table 3-1 Selected instrumented buildings with records at basement level and free field No. Building name Recording station EQ event* Free field  Foundation base at basement level Remarks  Building  Free field PGA (Gal) Peak Sa at ? = 5% (Gal) PA (Gal) Peak Sa at ? = 5% (Gal) EW  NS EW  NS EW NS EW NS 1 Rancho Cucamonga, 4-storey Justice Centre  CSMIP- 23497 CSMIP - 23497 LD 65.5 107.4 267.7 420.3 67.3 98.9 189.6 363.9 Free-field station is 100 m from southeast corner of the building. NR 38.3 63.2 172.9 188.2 37.4 36.3 111.7 106.5 PS 15.7 7.9 69.6 55.9 12.7 13.4 45.5 45.2 UL 214.2 248.2 806.6 805.5 104.6 119.5 394.9 502.2 WT 34.5 36.8 136.7 114.8 25.4 20.5 94.3 56.0 2 Pomona, 2-storey commercial building CSMIP- 23511 CSMIP-23525 UL 187.1 168.5 580.9 617.4 116.6 124.1 296.1 530.9 Free-field station is 87 m from southeast corner of the building. WT 43.0 52.2 170.8 182.7 42.0 47.8 138.0 150.6 3 Los Angeles, 7-storey University hospital CSMIP- 24605 CSMIP- 24605 LD 40.1 48.9 101.5 174.9 24.7 38.9 78.0 108.7 Free-field station is 104 m from southeast corner of building. NR 222.4 477.9 705.0 1428.9 160.0 379.0 512.5 1183.44 Santa Cruz, 5-storey government office building CSMIP-48733 CSMIP-48733 GL 27.4 33.5 92.3 106.8 14.6 21.1 47.4 79.6 Free-field station is located at 105 m from southeast corner of building. 5 Richmond, 3-storey government office building CSMIP- 58503 CSMIP- 58503 LP 132.4 109.4 329.6 434.0 97.6 128.6 250.9 382.7 Free-field station is 200 m from southeast corner of building. 6 Los Angeles, 14-storey Hollywood Storage Building CSMIP - 24236  CSMIP- 24303 NR 234.1 352.9 879.9 1366.1 191.5 277.3 594.4 851.0 Basement at excavated edge and ground floor at free end of same level  WT _ 200.3 _ 655.0 _ 114.4 _ 320.2 35  No. Building name Recording station EQ event* Free field  Foundation base at basement level Remarks  Building  Free field PGA (Gal) Peak Sa at ? = 5% (Gal) PA (Gal) Peak Sa at ? = 5% (Gal) EW  NS EW  NS EW NS EW NS 7 Los Angeles, 15-storey government office building CSMIP-24569  CSMIP-24611  LD 30.9 30.6 91.7 94.3 30.7 29.3 96.4 100.2  NR 127.5 197.4 449.1 522.5 129.3 198.2 393.7 634.1 8 Seal Beach, 8-storey office building  CSMIP- 14578  CSMIP- 14578  LD 43.9 40.8 145.0 128.1 45.6 37.3 142.2 120.4 Free-field channels are 139 m from southeast corner NR 77.6 65.8 356.6 241.1 79.0 55.6 338.1 183.7 *LD: Lander Earthquake (1992); NR: Northridge Earthquake (1994); PS: Palm Spring Earthquake (1986); UL: Upland Earthquake (1990); WT: Whittier Earthquake (1987); GL: Gilroy Earthquake (2002); LP: Loma Prieta Earthquake (1989).  Table 3-2 Selected instrumented buildings with records at basement level and ground floor No. Building  EQ event* Foundation base at basement level  Ground floor level Remarks  PGA (Gal) Peak Sa at ? = 5% (Gal) PA (Gal) Peak Sa at ? = 5% (Gal) Name Station EW NS EW NS EW NS EW NS 1 Sherman Oaks, 13-storey commercial building CSMIP- 24322  CW 16.5 13.7 425.9 344.4 54.5 25.8 543.5 670.7 Basement channels are in second sub-level  LD 27.0 42.0 49.8 43.2 32.3 41.8 103.0 72.6 NR 193.6 518.4 89.6 130.6 379.6 750.8 92.3 144.9 WT 142.1 103.9 779.2 1327.9 170.6 245.7 1189.9 2514.52 Los Angeles, 3-storey commercial building CSMIP- 24332 NR 328.4 311.6 1050.6 1042.0 423.8 318.4 1468.1 1347.2 Basement channels are in Garage level B (height to ground floor is 6.9 m) WT 51.8 50.5 175.8 198.8 78.4 61.7 255.4 272.6 36  No. Building  EQ event* Foundation base at basement level  Ground floor level Remarks  PGA (Gal) Peak Sa at ? = 5% (Gal) PA (Gal) Peak Sa at ? = 5% (Gal) Name Station EW NS EW NS EW NS EW NS 3 Pasadena, 9-storey commercial building CSMIP-24571  LD _ 42.0 _ 143.0 _ 45.7 _ 151.3 Channel at ground floor slab ceiling is taken for GF NR _ 42.0   143.0 _ 45.7   151.3 SM _ 231.2   1120.6 _ 245.1   1180.84 Los Angeles, 52- storey office building  CSMIP- 24602  BB 31.8 26.6 64.0 86.3 41.9 33.2 88.5 116.2 Basement record at E level 17.4 m down from ground LD 32.6 35.0 106.3 85.5 93.5 87.9 126.9 174.7 NR 111.2 139.0 436.4 336.2 125.7 162.3 399.3 448.1 SM 67.2 83.4 212.0 211.2 96.6 123.2 308.6 363.6 5 Los Angeles, 54- storey office building CSMIP-24629 NR 95.3 133.0 296.0 263.1 104.8 166.6 333.7 323.9 Basement record at P4 level 14 m down from ground 6 Los Angeles, 19- storey office building  CSMIP- 24643  NR 309.2 197.8 816.3 584.3 513.3 267.3 1376.3 1376.0Basement record at D level 9.6 m down from ground 7 South San Francisco, 4-storey office building  CSMIP- 58261 LP 157.3 _ 658.1 _ 190.6 _ 665.6 _ Basement resting on piles MH 26.8 _ 130.5 _ 35.4 _ 146.0 _ 8 Hayward, 6-storey office building CSMIP-58462  AR 8.2 _ 14.3 _ 7.2 _ 13.0 _   LP 105.4 91.9 297.0 271.9 104.3 92.7 318.2 286.9 9 Berkeley, 2-storey hospital building CSMIP-58496 LP 114.4 97.7 360.1 283.3 111.4 115.7 379.7 306.5   10 San Francisco, 47-storey office building CSMIP- 58532 LP 157.2 104.7 460.2 435.2 99.7 113.3 503.0 463.2   *CW: Charts Worth Earthquake (2007); LD: Lander Earthquake (1992); NR: Northridge Earthquake (1994); WT: Whittier Earthquake (1987); SM: Sierra Madre Earthquake (1991); BB: Big Bear Earthquake (1992); MH: Morgan Hill Earthquake 1984; AR: Alum Rock Earthquake (2007); LP: Loma Prieta Earthquake (1989). 37  Table 3-3 Selected instrumented buildings with records at foundation base at ground surface level and free field No. Building name Recording station EQ event* Free field  Foundation base at ground surface  Remarks  Building  Free field PGA (Gal) Peak Sa at ? = 5% (Gal) PA (Gal) Peak Sa at ? = 5% (Gal) EW  NS EW  NS EW NS EW NS 1 El Centro, Imperial County services building CSMIP- 01260 CSMIP- 01335-  IP 238.62 234.5 509.31 739.42 342.06 294.79 840.99 1155.44Free-field station is 105 m from northeast corner of building 2 Newport Beach, 11-storey hospital CSMIP- 13589 CSMIP-13610  NR 101.59 82.7 413.11 315.34 75.43 52.11 288.36 204.01 Free-field station is 105 m from northeast corner of building 3 Long Beach, 7-storey office building  CSMIP-14323 CSMIP-14395- WT 65.02 55.01 231.48 149.27 65.58 44.44 263.77 140.44 Free-field station is 60 m from southwest corner of building 4 San Bernardino, 9-storey commercial building  CSMIP-23515  CSMIP-23522- LD 90.78 83.86 262.63 307.49 76.95 80.58 244.35 235.48 Free-field station ( at San Bernardino 2nd and Arrowhead) is about 400 m from building 5 San Bernardino, 3-storey office building CSMIP-23516  CSMIP-23542 LD 76.29 85.26 303.32 295.06  79,93 127.49 490.51 254.05 Free-field station ( at San Bernardino E and Hospital) is about 300 m from building CH 43.01 50.22 122.97 130.82 29.48 40.05 95.76 66.27 SB 81.1 52.09 176.55 106.7 81.22 78.03 281.32 144.94 IW 14.06 10.95 50.86 42.9 6.84 6.89 19.59 15.33 6 Los Angeles, 7-storey UCLA MathSci. Building  CSMIP- 24231  CGS- 24688- NR 247.77 431.96 847.21 1623.70 226.84 280.05 837.96 863.05 Free-field station is on UCLA ground 38  No. Building name Recording station EQ event* Free field  Foundation base at ground surface  Remarks  Building  Free field PGA (Gal) Peak Sa at ? = 5% (Gal) PA (Gal) Peak Sa at ? = 5% (Gal) EW  NS EW  NS EW NS EW NS 7 Sylmar, 6-storey county hospital CSMIP- 24514 CSMIP-24763 NR 581.61 804.86 1337.67 2651.39 444.34 701.22 2103.25 1338.08 Free-field station is about 140 m from northeast corner WT 51.54 55.5 203.44 197.66 54.38 52.29 197.74 200.84 8 Lancaster, 3-storey office building CSMIP-24517  CGS-24526 WT 59.22 61.42 104.08 220.43 62.2 50.37 153.72 180.87 Free-field station is at Lancaster 15 and J (75 m from building) 9 Los Angeles, 2-storey Fire Command and Control Centre CSMIP-24580 CSMIP-24592 LD 52.17 55.28 172.69 170.05 50.97 52.02 189.58 189.56 Free-field station is about 30 m from northwest corner NR 257.98 331.27 968.68 1024.03 209.94 166.78 701.45 707.41 SM 113.4 96.59 381.95 266.42 74.08 75.97 282.39 180.93 10 Park Field, 1-storey school building  CSMIP- 36531  CGS-36138-  PF4 270.72 297.48 972.55 718.66 186.43 194.11 555.06 578.24 Free-field station is at Park Field fault zone 12 at a distance of 250 m from building  PF7 16.27 13.98 48.42 49.02 11.54 14.03 51.68 59.00 11 Templeton, 1-storey hospital CSMIP-36695  CGS-36712 PF5 10.35 18.57 41.02 45.12 9.66 10.75 24.88 44.07 Free-field station is at a distance of 60 m from building  LN 20.8 20.94 50.07 65.75 18.11 17.63 51.07 54.89 12 King City, 2-storey hospital  CSMIP- 47231  CSMIP-47232 PF4 47.44 39.44 124.08 106.99 28.45 30.72 85.98 84.44 Free-field station is at a distance of 280 m from northeast corner of building  PF5 7.86 6.57 18.93 14.52 2.57 3.63 7.05 11.13 39  No. Building name Recording station EQ event* Free field  Foundation base at ground surface  Remarks  Building  Free field PGA (Gal) Peak Sa at ? = 5% (Gal) PA (Gal) Peak Sa at ? = 5% (Gal) EW  NS EW  NS EW NS EW NS 12 Hollister, 1-storey warehouse  CSMIP -47391  CSMIP-47524 LP 179.2 370.89 681.81 1293.44 226.89 368.41 675.59 1233.86 Free-field station is at a distance of 120 m from southeast corner of building  13 San Jose, 3-storey office building  CSMIP-57562 CSMIP- 57563  AR 35.24 46.04 220.43 999.57 17.53 24.93 553.59 469.40 Free-field station is about 155 m from East Tip of the building corner LP   286.65   125.48   154.92   70.705314 Piedmont, 3-storey school building CSMIP- 58334 CSMIP-58790 LY 19.12 13.72 54.66 62.76 4.53 10.66 12.76 31.62   15 Eureka, 5-storey building CSMIP-89494 CSMIP-89509 FDJ 139.91 235.55 432.99 623.46 100.21 143.3 435.77 513.63   TD 28.7   61.53   28.3   59.06   FDF 11.87 11.34 35.22 30.75 5.7 11.23 22.56 40.03 16 La Jolla, 2-storey University Hospital  CSMIP- 03233 CSMIP- 03234 CX 39.26 37.43 124.3 99.34 23.33 21.82 87.59 74.51   SC 30.34 20.04 67.1 74.8 26.6 22.34 80.83 82.03 17 San Bernardino, 1-storey commercial building CSMIP- 23622  CSMIP- 23522 LD 90.59 88.46 262.51 318.62 63.02 69.14 258.33 296.15  Free-field station is at a distance of 500 m from building  18 Fortuna, 1-storey supermarket building  CSMIP-89473 CSMIP -89486 PL 111.1 82.55 309.53 304.41 81.26 155.83 400.83 356.84  Free-field station is at a distance of105 m from northwest corner of the building  FDJ 132.23 135.18 448.25 533.97 120.6 113.84 466.5 415.29 PLA 169.27 175.12 650.37 664.43 175.93 147.41 567.78 657.18 40  No. Building name Recording station EQ event* Free field  Foundation base at ground surface  Remarks  Building  Free field PGA (Gal) Peak Sa at ? = 5% (Gal) PA (Gal) Peak Sa at ? = 5% (Gal) EW  NS EW  NS EW NS EW NS 19 Los Angeles, 4-storey government office building  CSMIP-14766 CSMIP-14767 IW 28.93 42.57 71.45 69.79 21.16 50.17 71.7 93.38  Free-field station is at a distance of 120 m from building corner 20 El Centro, 1-storey hospital building CSMIP-01699 USGS- 01711 CX 50.22 45.95 106.88 122.88 17.55 16.26 48.22 56.62  Free-field station is at a distance of 100 m at El Centro Imperial Street and Ross 21 Crescent City, 1-storey hospital building CSMIP -99261 CSMIP-99269 CS 9.68 10.25 40.43 39.28 11.78 11 45.46 49.56  Free-field station is at a Hwy101 and Washington street FDJ 18.84 20.49 70.28 56.01 18.27 16.69 72.82 66.24 22 San Francisco, 15-storey hospital building  CSMIP- 58257 CSMIP -58306 AM 6.47 6.72 12.21 12.59 3.36 4.85 6.24 6.72   23 Redwood City, 16-storey office building  CSMIP- 58615 CSMIP-58619 AR 16.48 13.29 65.74 58.06 10.92 5.88 45.13 18.52   GL 31.83 35.28 157.72 123.24 12.78 12.24 56.02 57.18 24 Fremont, 2-storey City Library building CSMIP- 57784 CSMIP- 57311 AR 18.76 26.98 72.42 79.14 30.38 35.34 102.2 105.38  Free-field station is at a distance of 300 m from building site 25 Gilroy, 2-storey hospital building CSMIP-57200 CSMIP- 57203 AR 16.46 18.64 54.36 50.67 16.19 14.36 51.86 56.63  Free-field station is at a distance of 95 m from southwest corner of the building  SMT 10.98 11.79 29.9 38.01 11.38 9.48 32.52 43.04 41  No. Building name Recording station EQ event* Free field  Foundation base at ground surface  Remarks  Building  Free field PGA (Gal) Peak Sa at ? = 5% (Gal) PA (Gal) Peak Sa at ? = 5% (Gal) EW  NS EW  NS EW NS EW NS 26 Salinas, 3-storey county hospital building  CSMIP-47796  CGS-47762 SJB 11.04 6.63 31.51 25.62 11.66 3.26 33.43 8.72 Free-field station is about 100 m from building edge SS 15.75 13.06 54.16 52.5 14.96 12.47 59.56 59.27 GL 13.94 9.31 45.22 39.69 11.02 16.85 36.14 29.51 PF4 15.75 13.06 54.16 52.5 14.96 12.47 59.56 59.27 *IP: Imperial Valley Earthquake (1989); NR: Northridge Earthquake (1994); WT: Whittier Earthquake (1987); LD: Lander Earthquake (1992); CH: Chatsworth Earthquake 2007; SB: San Bernardino Earthquake (2009); IW: Inglewood Earthquake (2009); SM: Sierra Madre Earthquake (1991); PF4: Park Field Earthquake (2004); PF5: Park Field Earthquake (2005); PF7: Park Field Earthquake (2007); LN: Lake Nacimiento Earthquake (2009); LP: Loma Prieta Earthquake (1989); AR: Alum Rock Earthquake (2007); LY: Lafayette earthquake (2007); FDJ: Ferndale Earthquake (January 2010); FDF: Ferndale Earthquake (February 2010); TD: Trinidad Earthquake (2008);CX: Calexico Earthquake (2010); SC: San Clemente Earthquake (2004); PL: Petrolia Earthquake (1992); PLA: Petrolia Aftershock (1992); CS: Crescent City Earthquake ( 2005) ; AM: Alamo Earthquake (2008); GL: Gilroy Earthquake (2002); SMT: San Martin Earthquake (2006); SJB: San Juan Bautista Earthquake (2004); SS: San Simeon Earthquake (2003). 42  3.3 Comparison of Time Histories and Response Spectra For each building site, the foundation-base and free-field motions are compared in time histories and response spectra of acceleration, velocity, and displacement. Figure 3.2 shows a case of time history comparisons of motions from records at the building sites of the Los Angeles seven-storey UCLA Math-Science Building in the 1994 Northridge Earthquake in the north-south direction. Comparison in the acceleration time history records does not offer a clear picture except that it depicts reduced motion at the foundation compared with free field. This is because there are many condensed sharp peaks in the acceleration history.    Figure 3-2 Comparison of motion at free field and foundation base in time histories for the Los Angeles seven-storey  UCLA Math-Science Building in the 1994 Northridge Earthquake in the north-south direction 43  Comparison in the velocity time history shows that foundation motion follows the trend of free-field motion in cycles with long periods. Displacement time histories of the motions show that foundation motion is not reduced throughout. Comparisons in time histories of both velocity and displacement, however, do not offer period-dependent change in the foundation motion from free field.  This suggests that the comparison of motions should be better examined in the frequency or period domain; this would provide the period-dependent variation of the motion from free field to foundation. Comparison of motions in the response spectra is an obvious choice, as this will show the differences among the maximum response for each motion for a structure with a specific period. The presentation of comparison of free-field and foundation-base motions in response spectra also provides a direct relation to design spectra specified in the codes.  The same sets of records at the Los Angeles seven-storey UCLA Math-Science building are presented in their response spectra in Figure 3.3. It is clear from the comparison of acceleration response spectra that motion at the foundation base is reduced from that at free field in the period range up to 0.3 s. The maximum reduction of motion is more than 50%.  The comparison of motions recorded at free field and foundation base will be presented in response spectra in most of the cases in this chapter. The presented spectral acceleration, spectral velocity, and spectral displacement of records at free field and foundation base are developed for damping of ? = 5%. 44   Figure 3-3 Response spectra of acceleration and  velocity histories at free field and foundation base of the Los Angeles seven-storey UCLA Math-Science Building in the 1994 Northridge Earthquake in the north-south direction  3.4 Spectral Analyses of Recorded Data Some records are also investigated for their spectral characteristics, since they provide important information about the motions in terms of signal strength, their variation from free field to foundation base, and correlation in the frequency domain. The following spectral characteristics are considered in the study: a) Power spectral density function  45  Power Spectral Density (PSD) shows the strength of signal for the records in the frequency domain. In order to develop PSD, the acceleration time history records are further modified with zero padding at the end to make the length of a record equal to a number in the next power of 2. The PSDs are estimated by Welch?s averaged modified accelerogram method with a Hamming window of one fourth the length of the accelerogram used with 2% overlap. The outcomes of PSDs are smoothened by a moving average with a span of three data points. b) Coherency  The records at free field and foundation from the same earthquake events are checked for their coherence by means of magnitude-squared coherence given by )()( 22yyxxxyxy SSS??            (3.2)  where Sxy is the cross spectrum of two records measured and Sxx and Syy are their PSD functions. A high value of coherence at a given frequency, with maximum of 1, indicates that the two records are highly correlated at that frequency. The length of window, overlaps, moving average section, and length of fast Fourier transform (FFT) points are similar to those used for estimating PSD functions. c) Transfer function  The change in acceleration signal from free field to foundation base is represented by the transfer function between two acceleration records. The transfer function, Tf, is given by yyxyf SST ? ,           (3.3) 46  where Syy is the PSD of the free-field record and Sxy is cross-spectral density of records at free field and foundation base.  3.4.1 Example case of a Pomona two-storey commercial building Figure 3.4 presents response spectra of acceleration recorded at foundation base and free field in the Upland Earthquake in a Pomona two-storey Commercial Building in the north-south direction. The record shows reduction in spectral acceleration at basement compared with free field in a short period range up to 0.4 s. There are peaks in the response spectra of free-field motion at periods of 0.4, 0.25, and 0.13 s. The motion at the foundation has significantly lower spectral values at the latter two periods. However, the comparisons of velocity and displacement spectra do not show significant variation between the two motions.  Figure 3-4 Spectral acceleration of motions at basement and free field of the Pomona two-storey commercial building in the Upland 1990 Earthquake in the north-south direction The PSD functions of the motion help in understanding this difference in observations. Three major peaks in the acceleration spectra of free-field motion correspond to the peaks in the PSD 47  function at frequencies 2.5, 4.0, and 7.5 Hz, as shown in Figure 3.5. It shows that the first peak at 2.5 Hz has significant signal strength corresponding to energy associated with the motion at that frequency. However, the other two peaks are comparatively low in PSD, implying that the motion does not carry significant energy associated with those periods. This suggests that the first two peaks in the response spectra of the free field are not of high significance and so the difference between spectral motions at free field and foundation base at corresponding periods is. This phenomenon is reflected in the velocity and displacement spectra in Figure 3.6, where the difference between spectral values of motions at free field and foundation is not significant. The transfer function shows reduction in motion at frequencies near 4.6 and 7.5 Hz that simply correspond to the highest difference between spectra of motions at foundation and free field.  Figure 3-5 Comparison of power spectral density function and transfer function of records at free field and basement of the Pomona two-storey commercial building in the Upland 1990 Earthquake in the north-south direction 48    Figure 3-6 Spectral velocity and displacement of motions at basement and free field of the Pomona two-storey commercial building in the Upland 1990 Earthquake in the north-south direction  3.5 Variation of Motion from Free Field to Foundation 3.5.1 Free field to foundation base in buildings with basement Figure 3.7 shows the comparison of motion at the free field with the motion at the foundation base of the buildings with basement floors. Figure 3.7(a) illustrates the comparisons of peak acceleration values of time histories, and Figure 3.7(b) illustrates the comparison of peak values of spectral acceleration at 5% damping. Both graphs show that motions at the foundation base are less than corresponding motions at the free field. This is clearly due to the effect of embedment, which is well established with the concept that the seismic motion diminishes with the depth from the ground surface. Apart from the embedment effect, some other interactions might create the variation, but their effects may not be strong enough to be distinct. 49   (a)        (b) Figure 3-7 Comparison of motions at free field with foundation base at the basement level for (a) peak ground acceleration (PGA) and (b) peak spectral acceleration (Sa)     3.5.2  Free field to foundation base in buildings with surface foundation Figure 3.8 shows the comparison of motion recorded at the free field with foundation-base motion of the buildings that have surface foundations. Figure 3.8(a) illustrates the comparisons of peak acceleration values of time histories, and Figure 3.8(b) illustrates the comparison of peak values of spectral acceleration at 5% damping. Unlike in the case of buildings with embedment, no clear trend of reduction or amplification is observed in the motion at the foundation base compared with that at free field. There are more cases of higher (PGA) values at the free field compared with the foundation base. However, the cases of amplification observed in peak spectral acceleration at the foundation base are comparable with cases of reductions. The cases of amplification either in peak acceleration or peak spectral acceleration are one third of the total of all pairs of motions. 50      (a)      (b) Figure 3-8 Comparison of motions at free field with foundation base ground surface level for (a) peak ground acceleration (PGA) and (b) peak spectral acceleration (Sa)  This raises a concern that design practice using the fixed-base analysis of structure using the code motion may not always be safe. As discussed earlier, the motion prescribed in the code represents the motions at the free-field site at the location of a new or existing structure for reference site class.  The modification to reference code motion is carried out with the local site condition taken into consideration. The resulting motion is expected to be at the free-field ground surface of the site. If this motion is further modified to the foundation of the structure, the analysis carried out based on free-field motion will be inaccurate, and design may not be safe if the motion is amplified in the actual condition. In this case, the practice that considers reduction of foundation motion due to base slab averaging as per the provisions in ASCE/SEI41-06 and ASCE7-10 will result in unsafe design. The observation of amplification of motion in the building sites in the actual earthquake is contrary to the views of building codes that suggest otherwise. 51   The cases of reduction and amplification of motion at foundation base from free field are examined in further detail in the following sections to get a better understanding of what contributes to those variations. Comparison will be made with available methods of predicting variation and code provisions.  3.6 Example Cases of Reduction of Motion at Foundation Base 3.6.1 Hollywood Storage Building The Hollywood Storage Building is a 14-storey reinforced concrete frame structure supported by concrete footings on piles. The building has a partial basement. It is a box-type warehouse building with few windows. It has concrete flat slabs and exterior concrete shear wall as a lateral load resisting system. The plan dimensions of the building are 66 m in the east-west direction and 15.5 m in the north-south direction. Figure 3.9 shows the configuration of the building and the instrument layout.  Figure 3-9 Hollywood Storage Building configuration and instrument layout (after CSMIP, 2005) 52   This building experienced several earthquakes in the past. A detail account of earthquake histories of the building can be found in a research report by Trifnunac et al. (2001). In most of the cases, the building foundation shows lower response compared with the free-field station located at a nearby parking lot.  Figure 3.10 shows the response spectra of the records at free field and foundation base in the 1994 Northridge Earthquake in east-west and north-south directions. The foundation-base motions are reduced in the short period range up to 0.25 s in both directions. The foundation motions are almost similar in the longer period.  (a)      (b) Figure 3-10 Spectral motions at the foundation base compared with free field at the Hollywood Storage Building in the 1994 Northridge Earthquake in (a) east-west direction and (b) north-south direction  Figure 3.11 shows comparison of the motions in power spectral density function. This allows examination of the energy of the signal in the frequency range where variation of motion is 53  prominent. The PSD of free-field motion is significantly higher in the range of 4.0 to 7.0 Hz, which corresponds to a period less than 0.25 s. Figure 3.11(a) shows that there is a little hump of foundation-base motion exceeding free field motion at around 3.8 Hz, which is also reflected in the response spectra. However, the signal does not have strength to create the significant variation.      (a)     (b) Figure 3-11 PSD at the foundation base compared with free field at the Hollywood Storage Building in the 1994 Northridge Earthquake in (a) east-west direction and (b) north-south direction     Figure 3.12 illustrates transfer function, Tf of the motions from free field to foundation base. It is observed that Tf is reduced significantly in the frequency range above 4.0 Hz in the east-west direction. However, the Tf is approximately 1.0 in the low frequencies in the north-south direction; this suggests that there is no significant variation of motion in that range.  54   (a)        (b) Figure 3-12 Transfer function between the motion at free field and foundation base at the Hollywood Storage Building in the 1994 Northridge Earthquake in (a) east-west direction and (b) north-south direction     It is noted from Figures 3.10 and 3.11 that comparison of motions at free field and foundation base in response spectra can capture the variation of motions reasonably. In consideration of this, discussion will be focused on comparisons of response spectra for the rest of the cases.  3.6.2 Newport Beach Hospital Building The Newport Beach Hospital Building is an 11-storey rectangular reinforced concrete building structure with concrete footings at its foundation. It has plan dimensions of 47 and 23.5 m in building length and width, respectively. The footing is continuous around the perimeter wall. The building has a flat slab floor supported by beams and columns in the interior and shear walls on the perimeter. Figure 3.13 shows the configuration of the building and the instrument layout. 55   Figure 3-13 Newport Beach 11-storey Hospital Building configuration and instrument layout (after CSMIP, 2005) Figure 3.14 shows the comparison of motions at free field and foundation base in their response spectra at the Newport Beach Hospital Building in the 1994 Northridge Earthquake in east-west and north-south directions. Foundation motions are significantly reduced in the short period range. There is an amplification of foundation motion at a period of approximately 0.9 s in the north-south direction, but the increment is not significant.   (a)       (b) Figure 3-14 Spectral motions at the foundation base compared with free field at the Newport Beach Hospital Building in the 1994 Northridge Earthquake in (a) east-west direction and (b) north-south direction 56  3.6.3 Los Angeles Fire Command and Control Centre The Los Angeles Fire Command and Control Centre is a steel frame building with floors of concrete-filled steel decking. The two-storey building has plan dimensions of 55 m by 25.6 m. The building has concrete spread footings founded on hard soil. The building has base-isolated rubber bearings under all columns. Figure 3.15 shows the configuration of the building and the instrument layout. The set of recording channels placed below the isolator system are used in this analysis to compare with free-field records.  Figure 3-15 Los Angeles two-storey Fire Command and Control Centre configurations and instrument layout (after CSMIP, 2005)  Figure 3.16 shows comparison of foundation-base motion of the Los Angeles Fire Command and Control Centre building with free-field motion in the 1994 Northridge Earthquake in east-west and north-south directions. The peak of base motion is significantly lower than that of the free-field site in both directions. The motion at the building base is slightly amplified in the period range after 0.4 s in the east-west direction, which is not evident in the north south direction. 57   (a)     (b) Figure 3-16 Spectral motions at the foundation base compared with free field at the Los Angeles Fire Command and Control Centre  in the 1994 Northridge Earthquake in (a) east-west direction and (b) north-south direction      3.7 Example Cases of Amplification of Motion at the Foundation Base 3.7.1 El Centro Imperial County Service Building The Imperial County Service Building was built in 1968 and was demolished in 1980, after the building sustained severe damage in the 1979 Imperial Valley Earthquake (Pardoen et al., 1983). It was a reinforced concrete building with a shear wall in the longitudinal direction and frame in the transverse direction. It was a rectangular building with plan dimensions of 41 m by 26 m. The building was founded on piles that were interconnected with reinforced concrete link beams at pile caps. The soil underneath the building was alluvium sand. Figure 3.17 shows the configuration of the building and the instrument layout. 58    Photograph Figure 3-17 El Centro Imperial County Service Building configuration and instrument layout (building layout after CSMIP (2005) and photograph after Pardoen et al. (1983)  Figure 3.18 shows comparison of foundation-base motion of the building with free-field motion in the 1979 Imperial Valley Earthquake in east-west and north-south directions. The motion at the foundation base was significantly higher than that at free field in the short period range in both directions. At some periods, the amplification was about 100% different from the free-field record value. While the free-field motions in east-west and north-south directions are comparable, the foundation-base motion in the north-south direction is more amplified in the period range up to 0.5 s.  59  The amplification of foundation motion in this building was also reported in one of the CSMIP data utilization reports (Poland et al., 2000).   (a)       (b) Figure 3-18 Spectral motions at the foundation base compared with free field at the El Centro Imperial County Service Building in the 1979 Imperial Valley Earthquake in (a) east-west direction and (b) north-south direction     The PSD function of the records at free field and foundation base of the building also show consistent amplification in the entire frequency range (Figure 3.19). The PSD of foundation motion in the north-south direction in the range of 2 to 5 Hz is significantly higher than the free-field motion. The amplifications of motion, however, do not correspond to noticeable incoherency of two motions at the frequencies where they happen, as illustrated in Figure 3.20. 60      (a)       (b) Figure 3-19 PSD at the foundation base compared with free field at the El Centro Imperial County Service Building in the 1979 Imperial Valley Earthquake in (a) east-west direction and (b) north-south direction     (a)      (b) Figure 3-20 Coherency between the motion at free field and foundation base at the El Centro Imperial County Service Building in the 1979 Imperial Valley Earthquake in (a) east-west direction and (b) north-south direction       61  3.7.2 San Bernardino Office Building The San Bernardino Office Building is a three-storey steel moment frame building with truss joist plywood floors with thick concrete overlay. The steel moment frame serves as the lateral load resting system for the building. The building is rectangular in plan, 37 m by 41 m. The foundation has spread footings interconnected with grade beams. Figure 3.21 shows the configuration of the building and the instrument layout.  Figure 3-21 San Bernardino Office Building configuration and instrument layout (after CSMIP, 2005)  Figure 3.22 shows the comparison of motions at free field and foundation base in the 1992 Lander Earthquake in the east-west direction. The foundation motion was significantly amplified in the range 0.6?1.8 s. 62    Figure 3-22 Spectral motions at the foundation base compared with free field at the San Bernardino Office Building in the 1992 Lander Earthquake in the east-west direction  3.8 Comparison of Observed Data with Wave Passage Models Newmark et al. (1977) proposed the numerical averaging procedure to produce modified motion over the foundation slab from free field based on the concept that the averaging of motion happens over a time delay in excitation of parts of the foundation; this is caused by the horizontally propagating waves that impinge first on one side of the building foundation and then move to the other side. The averaging is done for the transit interval which is moved along the acceleration time history. 63  Similarly, the wave passage effect in modifying the translatory motion of the rigid slab is quantified by Clough and Penzin (1995) in terms of the ?tau-effect?. This method uses a technique of modifying the Fourier amplitude of the original motion by a ?tau-factor? and conducting inverse Fourier transformation to get the modified acceleration.  Figure 3.23 shows the comparison of estimations of foundation motion made by applying both of these techniques on the data for the Hollywood Storage Building in the Northridge Earthquake in the north-south direction. The averaging technique proposed by Newmark et al. (1977) provides a good estimate in the short period range. However, it overestimates the motion in the medium period range. The tau-factor technique is not in good agreement in the short period range, as it overestimates the foundation motion. In the long period range, motions at free field and foundation base do not differ significantly, and this technique corresponds well with observed data, since the tau-factor is close to unit in lower frequency (longer period).   Figure 3-23 Comparison of the wave passage model with observed spectral motions at the foundation base at the Hollywood Storage Building in the 1994 Northridge Earthquake in the north-south direction 64   Figure 3.24 shows the tau-factor transfer function for the building in the north-south direction. It is clear that there is no reduction in amplitude up to a frequency of 2 Hz (period longer than 0.5 s).  Figure 3-24 Transfer function (tau-factor) applied to modify the free-field motion as per Clough and Penzin (1995) for the Hollywood Storage Building in the north-south direction  Figure 3.25 shows the comparison of estimated motions of foundation base with recorded motion at the Newport Beach 11-storey Hospital Building in the 1994 Northridge Earthquake in the north-south direction. The record shows significant reduction of motion at the foundation from that at the free field. Both methods of averaging techniques using wave passage effects do not capture the level of reduction. For the periods longer than 0.6 s, foundation motion is slightly higher than free-field motion. 65   Figure 3-25 Comparison of the wave passage model with observed spectral motions at the foundation base of the Newport Beach Hospital Building in the 1994 Northridge Earthquake in the north-south direction  Figure 3.26 shows a case of amplification of motion at the foundation base. The figure compares the motions of foundation base estimated from wave passage techniques with recorded motion at the El Centro Imperial County Service Building in the 1989 Imperial Valley Earthquake in the north-south direction. The estimated motions are not close to the recorded motion in this case. The significant motion observed at the period of approximately 0.4 s was not comparable with the estimations. The wave passage averaging technique by Newmark et al. (1977) gives some amplification in the long period of response spectra; this was, however, overestimated.  66   Figure 3-26 Comparison of wave passage model with observed spectral motions at the foundation base at the El Centro Imperial County Service Building in the 1989 Imperial Valley Earthquake in the north-south direction  Another case of amplification is presented in Figure 3.27. The figure shows comparison of the motion of the foundation base estimated from wave passage techniques with recorded motion at the San Bernardino 3-storey Office Building in the 1992 Lander Earthquake in the east-west direction. Both models grossly underestimate the foundation-base motion.  67   Figure 3-27 Comparison of the wave passage model with observed spectral motions at the foundation base at the San Bernardino Office Building in the 1992 Lander Earthquake in the east-west direction  3.8.1 Discussion It is observed from the field data that wave passage models cannot capture the variation of motion in all cases. The main reason is that the basis of these methods is simply to account for the wave passage effect for the horizontally travelling wave, which may not be prevalent in all cases.  In the method proposed by Newmark et al. (1977), the transit interval was proposed based on the calibration of a limited number of earthquake records. Since the method is targeted to create a smoothening function for the peaks that appear in the high-frequency range, it will not be able to 68  estimate the variation of motion from dynamic interaction of building and soil system. The authors also noted that the total effect in deviation of motion could be from both wave passage and classical soil?structure interaction. Also, for the same reason, the method may not be applicable to the far-field motions that do not contain a high-frequency signal.  The tau-effect method also takes wave passage effects into account only for estimating the motion of the foundation and is strictly derived for the horizontally propagating wave. As the situation in real earthquake cases is likely to be different, the method may not be reliably applicable. Also, since the transfer function given by the tau-factor is always less than unity, the modified motion to represent the foundation will always be less than the original motion. Observations in the earthquake show that the foundation motion may not always be less than free-field motion. Actually, the base motions in many cases are higher than free-field motion. Even in the cases where motion is reduced at the foundation, it is not that the highest reduction occurs at the shortest period, as it would be in the tau-effect method.  3.9 Comparison of Observed Data with Spatial Incoherency Models Veletsos and Prasad (1989) and Veletsos et al. (1997) proposed a model to estimate the foundation-base motion of a rigid slab based on the concept of averaging of motion experienced by the rigid slab at individual points over the area. The concept is that the variation of motions within an area due to spatial incoherency will be diminished by the action of the rigid slab and the resulting motion will be different from what the motion would be without the foundation. 69  Averaging the  motion will result in lower maxima if the wavelength is smaller than foundation dimension in the direction of wave travel.  The model gives a transfer function amplitude between free-field and foundation motions that depends on several parameters, including dimensions of foundation, soil shear wave velocity, and incoherency of the ground motion. Figure 3.28 illustrates the transfer function as a function of the non-dimensional frequency parameter. The transfer function is given by ? ?? ? ???????? ??????????? ??????242441)2(241)2(222yeyerfyxexerfxSSHyxgguu ????    (3.4) where sa Vakx ??            (3.5a) and sa Vbky ??            (3.5b) where a and b are half of the foundation dimensions, ? is the circular frequency, Vs is the shear wave velocity, and ka is an incoherency parameter. For a vertically incident wave, the transfer function amplitude is a function of a0 defined as saeVkra ??0            (3.6) where abre ?  is the effective width of the foundation. The method to estimate the ground incoherency parameter, ak , was not described in the original model by Veletsos et al. (1997). The incoherency parameter used in the model was later 70  calibrated by Kim and Stewart (2003) based on data of seismic records in instrumented buildings with shallow foundations. This allows estimation of the transfer function that can be applied to free-field ground motion to get the foundation-base motion.  The spatial incoherency model is applied to estimate motion at the foundation base, and the results are compared with the observed recorded motion at the foundation base in instrumented buildings in past earthquakes. Figure 3.28 shows an example case of comparison of motion estimated by this procedure with the observed record of the Hollywood Storage Building in the 1994 Northridge Earthquake in east-west and north-south directions. The model gives a reasonable estimate of foundation-base motion in the north-south direction but overestimates the motion in the east-west direction in most parts of the response spectra.   (a)      (b) Figure 3-28 Comparison of the spatial incoherency model with observed spectral motions at the foundation base at the Hollywood Storage Building in the 1994 Northridge Earthquake in (a) east-west direction and (b) north-south direction     71   Figure 3.29 shows the comparison of estimated motion with recorded motion at the Los Angeles 15-storey Office Building in the 1994 Northridge Earthquake in the north-south direction. They do not match for the major part of the response spectra. The estimated motion was significantly lower than the actual motion observed in the short period, and it overestimated the motion at a period longer than 0.4 s.  Figure 3-29 Comparison of spatial incoherency model with observed spectral motions at the foundation base at the Los Angeles 15-storey Office Building in the 1994 Northridge Earthquake in the north-south direction  In the case of the El Centro Imperial County Service Building, the model failed to provide a reasonable prediction of foundation motion in the short period range, as shown in Figure 3.30. As foundation motion is significantly amplified, the model missed capturing the phenomena. It is reported that the El Centro Imperial County Service Building sustained major damage, requiring 72  demolition, during the Imperial Valley Earthquake despite a level of shaking in the nearby free field of only about 0.15g (Pardoen et al., 1983).  Figure 3-30 Comparison of the spatial incoherency model with observed spectral motions at the foundation base at the El Centro Imperial County Service Building in the 1989 Imperial Valley Earthquake in the north-south direction 3.9.1 Discussion It is observed that the result gives approximation of foundation-base motion in some buildings but the model does not capture the variation in other buildings, particularly where amplifications have been observed. This is because this method also always gives a reduction in the foundation motion compared with free-field motion. The model gives the transfer function as less than unity, which implies the reduction of foundation motion compared with that of free field. Figure 3.31 shows the transfer function for a slab with dimension plan dimensions a and b for a vertically incident wave. 73   Figure 3-31 Transmissibility function between free-field and foundation motions for vertically incoherent waves (after Kim and Stewart, 2003)  It was discussed earlier that this method comes as a result of calibration of the spatial incoherency model with field data obtained from records of instrumented buildings. The incoherency parameter used Veletsos? model as calibrated by Kim and Stewart (2003). While the model was expected to be reliable because data obtained from  instrumented buildings  were used for its calibration, comparisons of estimated motion with field records show otherwise. Therefore, the process of the calibration of the model is reviewed in this study, and the following shortcomings are found in the process: 1. The condition for the calibrations of incoherency parameters is set to have only a vertically propagating incoherent incident wave field. However, there were no conditions that can be applied in selecting a data set that has only the vertically incident wave field. This may lead to results not consistent with the original model by Veletsos et al. (1997), 74  as the calibrated incoherency parameters may reflect not only the spatial incoherency but also wave passage due to inclined wave field.  2. The approach of this procedure is based on the assumption that variation of ground motion between two points can be fully characterized by the spatial incoherency reflected by a single parameter, ka. The values of ka are determined by calibration of observed transfer function between recorded motion at free field and the base of the foundations. However, the observed transfer functions of the sites are not smooth and contain a large number of spurious peaks. It is claimed that the motions are smoothened and filtered and coordinate points of transfer function for calibration are appropriately selected, but there would be the possibility of bias in determining the ka value because the method of filtering and smoothening strongly affects the calibrated value.  3. It is found that the transfer function data are sensitive not only to filtering and smoothening but also to the length of zero padding in the Fourier transform. With the records obtained from instrumented buildings processed for linear base line correction, band pass filtering between 0.1 and 25 Hz with a fourth-order Butterworth filter and zero padding to make the data length of the next 2n number gives the transfer function as shown in Figure 3.32. With selected coherent coordinates as recommended in the paper by Kim and Stewart (2003), the calibrated ka value for the transfer function is found to be 0.074 in contrast to 0.13, as reported (Figure 3.33). The ka value is found to be sensitive to the data points chosen and the data processing. 75   Figure 3-32 Transfer function and estimation of incoherence parameter, ka, for foundation motion at the Sylmar Hospital Building in the Whittier Earthquake in the longitudinal direction   Figure 3-33 Transfer function and estimation of incoherence parameter ka for foundation motion at Sylmar hospital building from the Whittier Earthquake in longitudinal direction  ((Kim and Stewart, 2003)  4. Kim and Stewart used the values of ka obtained from calibration to find their relation with the shear wave velocity of the soil. With linear regression fitting of the data as shown in Figure 3.34, they proposed the following equation: )/(104.7037.0 4 smVsa ??????        (3.7)  76   Figure 3-34 Spatial incoherency parameter verses soil shear wave velocity for sites (Kim and Stewart, 2003) It can be seen from the figure that the deviation of data from the regression line is large, and it appears that the function is not necessarily dependent only on the single parameter Vs. From the conceptual viewpoint, soil deposit with higher shear wave velocity would have larger stiffness and so be able to transmit higher frequency signals that would be suppressed by averaging. So, the dependency ka to Vs is already built into the original transfer function that ka is calibrated to.  When all these points are considered, the reliability of the method cannot be guaranteed to produce the expected result in all cases.  More importantly, this method results only in reduction of motion from free field to foundation, which is not the universal case in earthquake observation. This is because the transfer function proposed by Veletsos et al. (1997), which always has a value less than unity, is adopted in the 77  process of calibration. Although the field data might have cases of amplification, the model provides for only the motions at the foundation.  3.10 Comparison of Observed Data with FEMA-440/ASCE41-06 Provisions FEMa-440 (2005) and ASCE/SEI41-06 (ASCE, 2007)   have provided reduction factors for spectral values due to the action of the foundation slab for a slab foundation with shallow embedment, as discussed in Chapter 2. The basis of the ASCE provision is the kinematic interaction effect where foundations have reduced motion compared with free field motion, based on the spatial incoherency model by Veletsos et al. (1997) and calibration of incoherency by Kim and Stewart (2003). The provisions suggest the reduction of motion at the foundation level of buildings with foundations of concrete slab or interconnected components with grade beam. The provisions are not applicable for buildings located in soft clay and consisting of flexible floor and roof diaphragms.  Figure 3.35 presents the comparison of observed foundation-base motion at the Los Angeles 15-storey office building in the 1994 Northridge Earthquake in the east-west direction with the foundation motion as per FEMA-440/ASCE41-06 provisions. The FEMA-440/ASCE41-06 motion is not close to the observed recorded motion at the foundation. FEMA-440/ASCE41-06 provisions largely underestimate motion in the short period.   78   Figure 3-35 Comparison of foundation-base motion applying FEMA-440/ASCE41-06provisions with observed data at the Los Angeles 15-storey Office Building in the 1994 Northridge Earthquake in the east-west direction  Figure 3.36 presents another case where foundation motion was amplified. Since the ASCE provisions do not give any amplification of motion from free field, it is obvious that it does not work in this case. Analysis of building structure with the motion as given by the provision would grossly underestimate the seismic demand in the structure.  79   Figure 3-36 Comparison of foundation-base motion applying FEMA-440/ASCE 41-06 provisions with observed data at the El Centro Imperial County Service Building in the 1989 Imperial Valley Earthquake in the east-west direction  3.11 Discussion of the Procedures Available to Estimate the Motion at the Foundation Base The FEMA/ASCE approach for calculating motion reduction at the foundation level does not produce a reasonable estimate of motion variation. Other approaches for calculating the variation motion also do not correspond to the observed data. The basis of all methods for quantifying the variation of motion between foundation base and nearby free field as described above is the averaging effect of wave passage or spatial incoherency, which always produces the reduction of 80  motion at the foundation level. However, in one third of the observed cases, motions are amplified in the foundation base. The ASCE approach, which accounts for reduction of motion only at the foundation, needs to be closely reviewed from all aspects, including the basis of the averaging model it used. A clear picture of the mechanics of interaction should be established, which would justify the variation of motion, both with amplification and de-amplification.  3.12 Patterns in the Observed Motions of the Foundation Base The procedures available to estimate the foundation motion do not give a framework that could suggest whether the motion will be reduced or amplified from that of the free field. Actually, one cannot come to any scenario that gives amplification of the motion at the foundation based on the models and parameters these procedures use. This requires the investigation to take a different approach to identify specific conditions that result in reduction and other conditions that result in amplification of the foundation motion.  A few cases of instrumented records shed light on the parameters that may play a role in the variation of motion. In some buildings, the foundation motion is higher than free-field motion in one earthquake and is lower in another earthquake. This suggests that the input motion of an earthquake is an important parameter to consider for the variation of motion; this has not been featured in any available models.  In other buildings, foundation motion is amplified in one direction and reduced in another direction for the same earthquake. For a building that is fairly symmetric in two directions, this 81  can again be attributed to the variation of input motion in two directions. In separate cases, free-field motions in two directions are similar, but motions at the foundation are dissimilar for buildings with different levels of lateral resistance. This implies that different building periods in two directions may result in different foundation motions.  3.12.1 Different responses for different earthquakes Figure 3.37(a) shows the comparison of foundation motion at the Eureka 5-storey Residential Building with corresponding free-field spectra in the 2010 Ferndale Earthquake in the north-south direction. There is significant reduction at foundation motion. However, the same building was subjected to the Trinidad Earthquake in 2008, when amplification of foundation motion was observed (Figure 3.37(b)). This shows that the characteristics of the excitation affect the response at the foundation level. Whereas the predominant period of record was 0.38 s in the Ferndale Earthquake, the Trinidad Earthquake record had its peak at 0.12 s.     (a)      (b) Figure 3-37 Comparison of free-field and foundation-base motions at the Eureka 5-storey Residential Building in (a) the 2010 Ferndale Earthquake and (b) the 2008 Trinidad Earthquake in the north-south direction 82  In the Fortuna Supermarket Building, foundation motion was amplified compared with free field for the input motion from the 2010 Ferndale Earthquake, as shown in Figure 2.38(a). The foundation of the same building had its motion reduced in the 1992 Petrolia Earthquake compared with free field. The Petrolia Earthquake had its predominant period at 0.46 s, but free-field motion at the building site at the Ferndale Earthquake shows that the peak in the response spectra occurred at 0.16 s (Figure 2.38(b)). The spectral shapes of the two motions are very different.  (a)      (b) Figure 3-38 Comparison of free-field and foundation-base motions at the Fortuna Supermarket Building in (a) the 2010 Ferndale Earthquake and (b) the 1992 Petrolia Earthquake in the north south direction      3.12.2 Different responses to the same earthquake in two directions In some buildings, it is observed that the same earthquake can produce different responses at the foundation in longitudinal and transverse directions relative to free field. Figure 3.39(a) shows the comparison of foundation motion at the three-storey San Bernardino Office Building with corresponding free-field spectra in the 1992 Lander Earthquake in the east-west direction. The foundation motion was amplified in the period range 0.5?1.5 s. However, motion was reduced at 83  the foundation over the entire period range in the north-south direction in the same earthquake, as shown in Figure 3.39(b). The foundation plan is rectangular, with comparable dimensions of 135 ft by 120 ft (41.1 m by 36.6 m). The layout of the columns is also symmetrical in two directions. The difference in motion in two directions could be attributed to different responses in those directions at the foundation relative to free field.  (a)       (b)  Figure 3-39 Comparison of free-field and foundation-base motions at the San Bernardino 3-storey Office Building in the 1992 Landers Earthquake in (a)the east-west direction and (b) the north-south direction     In general, symmetrical buildings showed a consistent behaviour in terms of variation of motion from free field to foundation in different directions under an earthquake when the free-field motions in two directions are similar.  3.13 Summary and Concluding Remarks Data from instrumented buildings in California are used to discuss the variation of motion at foundation base from that at free field. Twenty-six of these buildings have rigid shallow surface foundations with a wide range of configurations and site conditions. They have earthquake 84  records at both the foundation level and nearby free field. Analyses of earthquake records obtained from these sites in response spectra clearly show the effect of soil?foundation?structure interaction. Two thirds of the cases of these pairs of records showed significant reductions in spectral values at the foundation level for periods less than approximately 0.5 s, and one third of the records showed amplification in spectral values. Past procedures based on wave passage effect and averaging of spatial variation due to incoherency in analysing the effect of the foundation on the free-field input motions are all based on the assumption that the foundation slabs always reduce the motion. FEMA-440/ASCE41-06 recognizes that foundations with interconnected grade beams or a concrete slab will always reduce the free-field motions except for buildings sitting on soft clay sites and having flexible roof and floor diaphragms. It also presents a formula for calculation of the spectral reduction factor for design. These methods and FEMA-440/ASCE41-06 provisions were applied to the free-field data of records, and the results were then compared with the recorded data at the foundation level. From these comparisons, it was observed that the agreement was generally poor. It was obvious that the methods that include FEMA-440/ASCE41-06 cannot capture the amplification of foundation motion.  A few cases of instrumented records gave indications about the parameters that may play a role in the variation of motion with amplification in some cases and reduction in other cases. In some buildings, the foundation motion is higher than free field in one earthquake and is lower in another earthquake. In other buildings, foundation motion is amplified in one direction and reduced in another direction. These provide a direction for investigation that should consider several parameters, including input motion of the earthquake, its relations with characteristics of 85  soil deposit and the building structure. The variation of motion will be investigated in the next chapters with a model that includes these parameters. 86  Chapter  4:  CONTINUUM MODEL TO STUDY THE VARIATION OF MOTION FROM FREE FIELD TO FOUNDATION 4.1 Introduction Current methods of estimating foundation motion in an earthquake described in Chapter 2 do not explain the amplification of motion at the foundation base relative to free-field motion observed in a number of the instrumented building sites. Details of the observations have been presented in Chapter 3. The explanation of phenomenon that motions at foundation amplify in some cases and reduce in other cases is investigated in this chapter using a continuum model of soil deposit and building.      A continuum model of a system that includes soil deposit and a building represented by a one-dimensional beam with lumped masses at the storey levels is described followed by a detail account of characteristics of input motions selected for the study. The analyses of the soil?structure system are carried out, with the parameters of components being varied to investigate their effects on how the motion at the foundation base differs from free-field motion. Results of analyses are presented to show the effects of foundation width, storey height of the building, soil shear wave velocity, damping of soil, mass of the building, and depth of soil deposit on the 87  variations of motions. System with all combinations of above mentioned parameters of soil deposit and building are analysed separately for a number of input motions.  Representative results that illustrate the effects of parameters are presented.   Both cases of amplification and reduction of motions at foundation base relative to free field do occur in varieties of combinations of parameters. Cases of amplification and reduction of foundation motion take place even in a same system under different earthquake.   Patterns in the results of analysis are studied for the cases of amplification and reduction of motions at the foundation base under different earthquake input motions. The effects of period of components, system, and input motion are investigated and their relations to the variation of motions are established.    4.2 Continuum Model of a Soil?Structure System Figure 4.1 illustrates key features of the simple continuum model of soil deposit with a building structure used in a finite element program. The analysis is carried out in 2D, which made it possible to run several analyses varying parameters to investigate the problem.     The system consists of a building resting on a soil deposit of depth H over a base rock. Each floor of the building is lumped in a single mass point supported by a column of equivalent storey stiffness. In the model, the building has a rigid foundation of a 60 cm thick concrete slab of 88  width W resting on the soil deposit. The soil deposit is extended both sides from the foundation to make it 20 times of foundation width in total.  Figure 4-1 Features of continuum model used for the analysis   The soil deposit is discretized into 2D plain-strain quadrilateral elements. Mesh size of the elements ranges from 1.0 to 2.0 m, which is less than 1/10th of the wavelength for maximum shear wave velocity Vs = 500 m/s considered in the study. The bottom face of the foundation slab is tied to upper elements of the soil deposit and shares the common nodes. Only horizontal translatory motion is prescribed at the foundation. Rocking and uplifting of foundation slab is restrained.  The building is represented by beam elements with assigned stiffness properties. The cantilever columns (bending beams) are assigned in the building model.  Point mass inertias are assigned to the nodes at floor level to represent the lumped mass.  89  The model is implemented in computer program ABAQUS (Simulia), a general finite element program.  The boundary of the soil deposit at the bottom is fixed. Lateral degrees of freedom of elements of deposit are restrained so that shear beam action is created along the deposit height. Infinite boundary elements are placed at the ends of the deposit on both sides to simulate the half-space condition that ensures no waves are reflected back and energy is dissipated at model edges rather than reflected back into the system. Infinite elements are used in unbounded domains in which the region of interest is small in size compared with the entire medium and provide ?quiet? boundaries to the finite element model in dynamic analyses (Simulia). They serve to represent the far-field regions.  The soil in the deposit is homogenous and has elastic properties. It has mass density ?s = 1800 kg/m3 and Poisson?s ratio ? = 0.3. Soil shear wave velocity, Vs, and soil damping, ?s, are varied to investigate the potential effect in the soil?structure interaction. For the concrete, mass density ?c of 2500 kg/m3 and Raleigh damping of 3% are used throughout the analyses. Width of foundation, W, number of building stories, nst, typical floor mass, mst, and depth of deposit, H, are varied in the analyses.  Output results of the time history analysis of the system model are recorded at the bottom of the foundation slab and at the lumped mass points. The response quantities at the contact surface layer are the same for both the foundation and soil deposit.  90  For the free-field response, another set of analyses are carried out, with the foundation and structure removed from the system model. In the deposit-only model, output time history quantities are taken at the same node point at the upper face of the soil deposit. Although any point in the ground surface level away from the structure in the system model could be a candidate point to serve as a free-field reference point, this approach of doing a separate set of analysis for deposit only ensures that there would be absolutely no interference from the structure on the reference free-field point.  4.3 Input Motions A total of nine motions, eight records from different earthquake events including three from the 1994 Northridge Earthquake event and one white-noise motion are used as input motion to run time history analysis. Table 4.1 summarizes major characteristics of the selected earthquake records except those from the Northridge events. All of these earthquake records are obtained from the PEER NGA Database (PEER, 2010). All records have similar peak acceleration values of approximately 0.1g. However, they have different period, as shown in Table 4.1. Figure 4.2 shows acceleration time histories of the records and Figure 4.3 shows response spectra of the motions for 5% critical damping.  The selection set of earthquakes is made after consideration of the variation in frequency content and predominant periods for similar peak ground accelerations in the earthquake records. This variability in the input motion is important to consider in the analysis to get an understanding of the effect of earthquake characteristics on the variation of motion from free field to foundation base. 91  Table 4-1 Selected set of input motion from earthquake events other than the 1994 Northridge Earthquake  The selected records have different predominant period Tp, and mean period, Tm.  Predominant period, Tp is defined as the period which corresponds to the highest spectral acceleration value in the response spectra of the motion. Among the selected earthquake records, NGA#106 has lowest value of Tp = 0.18 s and NGA#122 has highest value of Tp = 0.76 s. Mean period, Tm corresponds to average period in the response spectra curve weighted by corresponding Fourier amplitudes (Rathje et al, 2004). It accounts the distribution of Fourier amplitude coefficients of the record.  The record NGA#106 has shortest mean period Tm = 0.2 s and NGA#37 has a long mean period Tm = 1.23s. Although NGA#37 and NGA#122 records have predominant periods close to each other, their mean periods are significantly different. The long mean period of NGA#37 record represents the sustained spectral acceleration.   Earthquake record ID Event (station) Peak Acceleration PA (g) Predominant Period Tp (s) Mean Period Tm (s) NGA#30 The 1966 Park Field Earthquake  (Cholame - Shandon Array #5) 0.11 0.3 0.42 NGA#37 The 1968 Borrego Mtn Earthquake (LA Hollywood storage free field) 0.112 0.64 1.23 NGA#97 The 1973 Point Mugu Earthquake (Port Hueneme station) 0.112 0.52 0.7 NGA#106 The 1975 Oroville Earthquake (Oroville seismograph station) 0.108 0.18 0.2 NGA#122 The 1976 Friuli Earthquake (Codroipo station) 0.093 0.76 0.73 92   Figure 4-2 Acceleration time histories of selected ground motions for analysis   The NGA#30 earthquake record has a single dominant peak at 0.3 s period and has low spectral acceleration values in period ranges longer than 0.8 s. In contrast, the NGA#37 earthquake record has a predominant period, Tp of 0.64 s. The response spectrum of this record has a second peak at 1.42 s. The NGA#97 record has a smooth spectrum. The NGA#106 has a very sharp short period peak for spectral acceleration, and spectral acceleration drops to sustained low value after the peak in the spectra curve. 93   Figure 4-3 Response spectra of selected ground motion for analysis  A second set of ground motions has three records selected from the Northridge Earthquake event, which hit the number of instrumented buildings described in Chapter 3. The original records are linearly scaled to make them have peak acceleration of approximately 0.1g so that the resulting motion have similar peak horizontal acceleration to other earthquake records. The motions in this set are similar both in peak acceleration and spectral shape. Predominant periods of motions are slightly different. Time history analyses of the model with this set of input motions will serve to confirm the trend in the analysis results obtained using the first set of motions. Table 4.2 lists the details of the ground motions. Time histories and response spectra of the motion ( at 5% critical damping)  are illustrated in Figure 4.4. 94  Table 4-2 Set of input motions from the 1994 Northridge Earthquake event selected for analysis   Figure 4-4 Selected records from the 1994 Northridge Earthquake event for analysis (a) acceleration time histories, (b) response spectra  In addition to earthquake records, a white-noise motion is used in the time history analysis. A random signal is generated with discrete Fourier amplitudes, which is passed through a Butterworth filter with cut-off frequencies f1 = 0.1 Hz and f2 = 15 Hz.  The resulting Fourier Earthquake ID Station Peak Acceleration PA (g) Predominant Period Tp (s) Mean Period Tm (s) NGA#953 Beverly Hills ? 14145 Mulhol 0.103 0.54 0.63 NGA#989 Los Angeles ? Chalon Rd 0.093 0.58 0.628 NGA#993 Los Angeles ?Fletcher Dr 0.120 0.52 0.524 95  spectrum is converted to time history by applying inverse Fourier transformation.   Figure 4.5 shows the white-noise input signal in Fourier spectrum along with its response spectra. The response spectrum of the motion has its peak at 0.12 s.   (a)        (b) Figure 4-5 White noise input used in the analysis: (a) frequency content (b) response spectra  4.4  Parametric Study Following parameters are investigated in relation to their effects to the variation of motions at free field and foundation in a soil-structure system:  ? Foundation width  ? Storey height of building  ? Soil shear wave velocity ? Damping of soil  ? Mass of building ? Depth of soil deposit 96  Results of time history analyses are presented for the motion at the centre of the foundation-soil interface of soil-structure system and motion at the top of the soil deposit when structure is not present. Motions at these points are represented by response spectra of acceleration response history at 5% critical damping. Spectral acceleration is chosen over time history as the former can be directly related to period building, soil deposit and input motion.   In situations where variation of motions between free field and foundation base need to be compared for a number of different soil-structure systems, difference in period dependent spectral values of response spectra of motions at foundation and free field are presented. The variation of spectral acceleration is given by  ))(()()(fieldFreeafieldFreeaFoundationaa TSMaxTSTSS?????         (4.1) where Foundationa TS )(  is spectral acceleration at foundation and Foundationa TS )(  is spectral acceleration at free field. The difference between these quantities is normalized by peak spectral acceleration at free field.  A positive value of aS?  indicates the amplification of motion and a negative value of aS?  indicates reduction of foundation motion compared to free-field motion.   References to the fixed-base building period, Tb, fundamental period of soil deposit, Ts and predominant period of input motion, Tp are made in the presentation of results where they are relevant. Tb is obtained from the mass and stiffness properties of model building and Ts is given by  ss VHT 4?             (4.2) 97  Where H is height of the soil deposit and Vs is the soil shear wave velocity.  A 10% soil damping used in the analyses  unless stated other otherwise.   4.4.1 Foundation width Figure 4.6 illustrates the effect of the width of the foundation slab on horizontal motion at the foundation level for a given earthquake. Motions at the foundation base are compared for the case of 20 m wide and 30 m wide rigid foundations under the NGA#30 earthquake record. These results are for a system with single-story building with roof mass of 100 ?103 kg mass resting on a 32 m deposit of soil with shear wave velocity of 100 m/s.  The building model is 5 m long stick column with stiffness EI = 1.83 ?109 N-m2. The 20 m and 30 m wide foundations have their dead masses 30 and 45 t respectively.       Figure 4-6 Effect of slab width - response spectra of response motion at the foundation slabs and free field for input of the NGA#30 Earthquake record 98    It is observed that the responses at the slab for two different foundations do not differ significantly. The spectral shapes of two cases are similar.   The peak responses in the above cases occur at the short period range.  A case of different input motion and building mass is presented in Figure 4.7, which depicts the effect of foundation width to the variation of motions in longer period range. Input motion in this case is white noise and building mass is 400 ? 103 kg. Other parameters of the system are similar as above.   Comparing spectral accelerations of foundation motion for cases of 20 m and 30 m wide foundations shows that the effect of change in foundation width to the variation is minimal.     Figure 4-7 Effect of slab width - response spectra of response motion at the foundation slabs and free field for the input of White noise 99  4.4.1.1 Discussion Observations mentioned above are representative of several cases with different earthquakes, building configurations and their masses, and soil shear wave velocities. The fact that the shapes of response spectra of motion at foundation for different width remain same implies that period of soil-structure system do not change with change of foundation width.    Gazetas et al. (2013) showed significant effect of foundation width in the structural response in earthquake loading. However, their results are related to the rocking mode of foundation vibration.    4.4.2 Storey height of building The effect of storey height on the variation of motions at free field and foundation base is investigated by changing the storeys of a building. Typical storey height of the building considered is 5 m.  Foundation-base motions are obtained for systems with one-storey, two-storey, three-storey, and five-storey lumped models of buildings and results are compared.  Two configurations are considered in mass and stiffness properties along the building height while varying the number of storeys in the building. In the first set of building configurations, typical storey stiffness is kept same and total mass of the building distributed equally at each floor level is also kept constant while changing the number of storeys. Figure 4.8 shows the configuration of building models in this set.   100   Figure 4-8 Building and mass configurations with constant storey stiffness  In the second set of building configurations, first modal mass and fundamental period of the building are kept constant while varying the number of storeys.   Figure 4-9 Building and mass configurations with constant first modal mass and fundamental period  101   Typical story stiffness has been modified for each building model in this set to keep the modal mass and building period same.  Building models in this set   can be typically represented by a single degree of freedom (SDOF) system with same generalized properties in the first mode.  Figure 4.9 shows the configuration of building models in this set.  4.4.2.1 Constant storey stiffness  Analyses are carried out for the model with a 32 m deep deposit of soil under seismic excitation of the NGA#37 record at the bottom of the deposit. Shear wave velocity of soil in the deposit is 200 m/s. Total building mass m = 200 ? 103 kg. Typical storey stiffness in the building is EI = 1.83 ? 109 N-m2 and storey height is 5 m. Foundation in each building is 20 m wide and 0.6 m thick.   Table 4.3 shows the generalized building mass and fundamental building period for these configurations. With addition of one more storey to a single-storey building, effective period is increased about two times. The effective first period of a five-storey building with the same total dead mass is six times larger than that of a single-storey building. The changes in effective period and mass of building also change the first mode period of the total system, as shown in the last column of Table 4.3.    102  Table 4-3 Dynamic properties of building configurations with constant storey stiffness Building configuration 1st mode building period (s) Generalized mass (kg) First mode period of system (s) One storey 0.42 200 x 103 0.66 Two storey 0.87 108 x 103 0.91 Three storey 1.4 84.2 x 103 1.42 Five storey 2.7 67.7 x 103 2.71   Figures 4.10 shows the comparison of responses at foundation level in terms of acceleration response spectra ( at 5% critical damping)  at the foundation level for buildings with different heights (storeys) to the earthquake excitations of the NGA#37 record. The response for a single-storey building differs significantly from other buildings with more storeys. The foundation motion is amplified at the period of 0.63 s in one-storey. The two-storey building model has the lowest response at the foundation, which is less than that of the free field.  Figure 4-10 Responses at the foundation base of buildings with different numbers of storeys and same typical storey stiffness to the excitation of the NGA#37 record 103   These observations indicate that the configuration of a building (and effective mass) affect the foundation motions. While the building configuration (storey height) affects the effective stiffness of the building, change in distribution of masses alters the effective inertia force to be transmitted to foundation and soil.  These results suggest that it is not only total mass of the building but also distribution of mass along the building affects the soil?structure interaction. 4.4.2.2 Constant modal mass and period   Analyses are also carried out for a model with same deposit under same earthquake input but with different storey stiffness. Storey height is 5 m and foundation in each building is 20 m wide and 0.6 m thick. While changing the number of storeys, the first modal mass of the building and period are set to be 200 x 103 kg and 0.42 s respectively. The storey stiffness and typical floor mass in each building of are listed in Table 4.4. There is a slight deviation in the system period as a result of change in storey height.   Table 4-4 Dynamic properties of building configurations with constant modal mass and period Building configuration Typical storey stiffness EI (N-m2) Typical floor mass (kg) First mode period of system (s) One storey 1.83 ? 109 200 x 103 0.66 Two storey 14.32 ? 109 175.7 x 103 0.67 Three storey 48.0 ? 109 145.85 x 103 0.68 Five storey 220.5 ? 109 104.1 x 103 0.69  Figures 4.11 shows the comparison of responses at foundation level in terms of acceleration response spectra ( at 5% critical damping)  at the foundation level for these buildings models  104  with different heights (storeys) to the earthquake excitations of the NGA#37 record. Foundation responses of all buildings in the set are practically same.  The foundation motions are amplified relative to free-field motion. The peak spectral accelerations occur at the period 0.62 s, which is close to predominant period of earthquake record, Tp for NGA#37.     Figure 4-11  Responses at the foundation base of buildings with different numbers of storeys and same modal mass and building period to the excitation of the NGA#37 record 4.4.2.3 Discussion Results above suggest that first mode parameters of building are important in defining the foundation response when number of storeys is changing. This also confirms that results of analysis of soil-structure system with SDOF building   can be used for a system with more 105  number of storeys as long as modal mass and modal period are same and horizontal motion at foundation level is concerned.    Considering these results, parametric study in the variation of motions between foundation base and free field in the forthcoming section are carried out for SDOF building with modal mass represented by mb and first mode period represented by Tb.     4.4.3 Soil shear wave velocity Analyses of the soil?building system are carried out for different soil shear wave velocities, Vs, of the deposit. Typical results are shown in Figures 4.12 and 4.13, which present the spectral accelerations of motions at free field sand foundation as well as   variation of motion in terms of difference in those spectra for soil shear wave velocity Vs of 200, 300, and 500 m/s. Results are presented for the system with soil deposit depth H = 32 m, foundation slab width, W = 20 m, and a one-storey building with floor mass of 100 x 103 kg.  Figure 4.12 shows the variation of spectral acceleration under excitation of the NGA#37 record, and Figure 4.13 shows the case under excitation of white-noise motion. It is observed that the effect of soil shear wave velocity is significant in the variation. For the deposit with lower soil shear wave velocity Vs, the variation of motion is more pronounced compared with a deposit with higher Vs. For the case of lower Vs, amplification of motion at the foundation base occurs in a longer period range. However, the period range where reduction occurs does not change significantly with soil shear wave velocity.  106  This observation gives important hints that soil property is a significant factor in influencing the period at which amplification occurs.   Figure 4-12 Variation of motion from free field to foundation base under excitation of the NGA#37 record 107   Figure 4-13 Variation of motion from free field to foundation base under excitation of the white-noise motion  A study is carried out to investigate whether the soil shear wave velocity is the sole factor that decides amplification (or reduction) of foundation-base motion compared with free-field motion. This aspect is important in light of ASCE provisions that suggest the reduction of motion at the foundation base with the exception of soft clay.  108  4.4.3.1 Foundation motion de-amplified in Vs =200 m/s and 300 m/s The first case is presented in Figure 4.14 for a system with building mass mb = 120 x 103 kg resting on a 15 m deep deposit with two different soil shear wave velocities, Vs, under the same excitation of the NGA#122 record. The motion at the foundation reduces in both cases. The variation is less in the case of Vs = 300 m/s than that in the case of Vs = 200 m/s.   Figure 4-14 Reduction of foundation-base motion from free-field motion for Vs = 200 m/s and Vs = 300 m/s (mb =120 ? 103 kg, H = 15 m)  4.4.3.2 Foundation motion de-amplified for increased Vs   Figure 4.15 presents the case where foundation motion is amplified for the case of Vs = 200 m/s but reduced for the case of Vs = 300 m/s. This is for a system with building mass mb = 100 x 103 kg resting on a 20 m deep deposit excited by motion of the NGA#953 record. Both amplification and reduction of foundation-base motion are significant near spectral peak regions of the motions. Peak occurs near the predominant period of input earthquake record in in the case of Vs 109  = 200 m/s. In the system with Vs =300 m/s, peak spectral acceleration and maximum variation of motion occur at period of the deposit, Ts.     Figure 4-15 Amplification and reduction of foundation-base motion from free-field motion for Vs = 200 m/s and Vs = 300 m/s (mb = 100 ? 103 kg, H = 20 m)  4.4.3.3 Foundation motion amplified for increased Vs   Figure 4.16 presents a case where motion at the foundation base is amplified compared to free field in larger soil shear wave velocity whereas foundation motion is de- amplified in lower soil shear wave velocity. In a system with building mass mb = 250 x 103 kg resting on a 30 m deep deposit, motion at the foundation base is amplified in the system where Vs = 300 m/s and reduced where Vs = 200 m/s. In both cases peak spectral acceleration occur at the period close to predominant period of earthquake. The input motion for the analyses of both cases is NGA#953. It is the same input to the case presented earlier where foundation motion was de-amplified for a system with deposit having Vs =300 m/s.   110   Figure 4-16 Reduction and amplification of foundation-base motion from free-field motion for Vs = 200 m/s and Vs = 300 m/s (mb = 250 ? 103 kg, H = 30 m)  4.4.3.4 Discussion These results suggest that soil shear wave velocity significantly affects the variation of motion.  In some cases, higher soil shear wave velocity results in de-amplification of foundation motion for a same system. However, under different deposit depth and building mass configuration, higher soil shear wave velocity results in amplification of foundation motion. This suggests that soil shear wave velocity is not the sole factor to determine the type of variation of motion (amplification or de-amplification) but it is an important parameter to look into in combination with mass of building, depth soil deposit and input motion.    4.4.4 Soil damping Damping is one of the important characteristics of a dynamic system. The possible effect of damping in the variation of motion from free field to foundation base in a soil?structure system is investigated by changing the critical damping ratio of the soil deposit. A critical damping of 111  building structure ?b = 5% is adopted in all analyses, as the damping of structure is relatively constant from one type of building to another. However, two different critical damping values ?s = 10% and 20% are considered for the soil deposit, and comparisons of variation of motions at the foundation base from free field is made for each case.  Figures 4.17 (a) and (b) present two cases of soil damping in a system under excitation of the NGA#37 record and the NGA#30 record, respectively. The system has a building with mass mb = 150 ? 103 kg sitting on a 32 m deep deposit with Vs = 300 m/s. The figure shows normalized variation from free field to foundation base. Under both excitations, the variation of motion in the case of ?s = 10% is higher than when ?s = 20%. The pattern of variation and periods of peak are similar in both cases of soil damping for input of both the NGA#30 and NGA#37 records. The peaks of the variation also occur at the same period in both cases of damping. This suggests that the effect of damping is mostly limited to the amplitude of the variation. The higher the damping, the lower will be the variation of motion from free field to foundation base.  (a)      (b) Figure 4-17 Effect of soil damping in variation of motion from free field to foundation under seismic excitation of (a) the NGA#37 record, ( b) the NGA#30 record 112  4.4.5 Mass of the building Results of analyses show that the mass of the building affects the response at the foundation significantly in a building?soil system under seismic excitation. Two cases are presented in Figures 4.18 and 4.19 where mass of the building, mb, is changed from 50 ? 103 kg to 100 ? 103 kg and 150 ? 103 kg. Storey stiffness EI is 1.83 ? 103 N-m2 and foundation width is 20 m for all systems considered.  In the first case, the system has a 15 m deep soil deposit with Vs =200 m/s, critical damping of soil ?s = 10% and it is excited by the NGA#122 record. The motions at the foundation base are presented together with free-field motion in response spectra in Figure 4.18. For the building mass, mb = 50 ? 103 kg, the motion at the foundation-base is amplified compared with free field at the periods of high spectral values. When building mass is changed to mb = 100 ? 103 kg, the spectral acceleration at the foundation base is mostly reduced, except in the post peak region of response spectra. When building mass is further increased to mb = 150 ? 103 kg, the foundation motion is significantly reduced in a wide range of period. The spectral value of foundation-base motion is only about 50% of the free-field motion in the period up to 0.4 s.  This result suggests that the same earthquake can induce significantly different responses at the foundation base of two different buildings in the same site if the masses of the buildings are different. In this particular case, foundation motion is reduced as the mass of building increases. 113   Figure 4-18 Change in spectral acceleration of free fields and foundation bases with change in the mass of the building for a system with a 15 m deep deposit with Vs = 200 m/s under the NGA#122 excitation 114   Figure 4.19 shows the results of another case where increase in building mass causes further amplification of foundation-base motion. The system considered in this analysis has a 30 m deep deposit with Vs = 300 m/s excited by the NGA#30 records. The mass of the building is changed from 50 ? 103 kg to 100 ? 103 kg and 150 ? 103 kg in this case, too. Storey stiffness EI is 1.83 ? 103 N-m2 and foundation width is 20 m for all systems considered.  It is observed that the foundation motion gets higher when the building mass changes from 50 ? 103 to 100 ? 103 kg. The motion at the foundation-base is, however, reduced when the mass of the building is further increased to 150 ? 103 kg. The peak of the spectra at the foundation base and free field are at the same level when mb = 150 ? 103 kg. The effect in variation of motion from free field to foundation base with change in the mass of the building in this case is not as significant as the other case discussed earlier.  Soil deposit and input motion are different in systems of these two cases. This suggests that extent of effect of the building mass to the variation of motion depends on underlying soil deposit and excitation motion.   The role of the mass of the building in the variation of motion needs to be examined more closely, as the results discussed above provide an important insight that both amplification and reduction of foundation motion from free field is realised when the mass of the building is modified but the same deposit and excitation motion are kept. The phenomena that foundation motion is increased as building mass increases up to a point followed by reduction of the motion when the mass is further increased indicates that there should be a critical mass for a given system that separates the increasing and decreasing trends of foundation motion. 115   Figure 4-19 Change in spectral acceleration of free fields and foundation bases with change in building period for a system with a 30 m deep deposit with Vs = 300 m/s under the NGA#30 excitation 116  4.4.5.1 Increasing and decreasing trend of peak spectral motion at foundation Figures 4.20 and 4.21 present the effect of building mass on foundation motion as compared with free-field motion, with a clear trend observed in the results. Figure 4.20 illustrates response at the foundation base for mass of building, mb, of 50 ? 103, 80 ? 103, 100 ? 103, 120 ? 103, 150 ? 103, 175 ? 103, and 200 ? 103 kg in a system with a 20 m deposit of soil with Vs = 200 m/s under excitation of the NGA#30 record. For building mass less than 150 ? 103 kg, peak spectral motion at the foundation is higher than that at the free field. Peak spectral acceleration keeps increasing when mass of the building is increased from mb = 50 ? 103 kg to 80 ? 103 and 100 ? 103 kg. There are, however, some reductions in the motion at out of peak region at short periods.    When mb is further increased, the peak value of spectral acceleration of foundation motion starts decreasing. The peak spectral acceleration in the case of mb = 200 ? 103 kg is 50% less than that when mb = 100 ? 103 kg. For this building-soil system under excitation of NGA#30 motion, mass mb = 120 ? 103 kg separates the increasing and decreasing trends of peak foundation motion in response spectra.    It is also observed that the peak of spectral acceleration of motion at the foundation base occurs at a certain period when amplification increases with increased mb. In the case of the NGA#30 input to the system of a 20 m deep deposit with soil Vs = 200 m/s, all the peaks of amplified motion occur at T = 0.42 s as long as foundation motion increases with increase in building mass. This period is close to the fundamental period of the deposit Ts = 0.4 s.  Shape of spectra curve and the period corresponding to its peak value change when peak spectral motion at foundation decreases with increase in building mass. Spectral shapes of foundation motion for the cases of mb = 150 ? 103, 175 ? 103 and 200? 103 kg deviate from that of rest of the cases. 117   Figure 4-20 Effect of building mass on the foundation motion for a 20 m deposit with Vs = 200 m/s under the NGA#30 excitation  A trend of increasing amplification is observed in the case of the NGA#37 input to the system of a 30 m deposit with Vs = 200 m/s with a building of mass, mb, of 50 ? 103, 80 ? 103, 100 ? 103, 120 ? 103, 150 ? 103, 175 ? 103  and 200 ? 103  kg as shown in Figure 4.21. Building mass is further increased to mb = 250 ? 103, 300 ? 103, 400 ? 103 and 500 ? 103 kg to investigate whether the system follows the same trend. The spectral peak of foundation motion is increased when mb is increased up to 250 ? 103 kg. When the building mass is set to be more than 250 ? 103 kg, the response at the foundation base starts reducing. For mb = 500 ? 103 kg, the peak spectral acceleration value at the foundation is about one third of the peak observed at the case when mb = 200 ? 103 kg (Sa = 2.5g vs. Sa = 0.77g). In the latter case, peak spectral acceleration at foundation base is only half of peak spectral acceleration at free field. 118   In this case of  30 m deep deposit under excitation of NGA#37 motion, peaks of spectral acceleration of motions at the foundation base occur at the period  T = 0.63 s as long as amplification increases with increased mb.  This period is close to pre dominant period of earthquake, Tp = 0.64 s.   Figure 4-21 Effect of building mass on the foundation motion for a 30 m deposit with Vs = 200 m/s under the NGA#37 excitation 4.4.6 Depth of soil deposit In this study, the soil underneath the foundation is assumed to have a certain depth at which it is excited by bedrock motion. Time history analyses are carried out for systems with varying depths of soil deposit for the same set of other parameters to study the effect of depth on the variation of motion at foundation base from free field. The depth of soil deposit H is changed 119  from 10 to 15, 20, 25, 30, 40, 50, 60 m, and 75 m in the soil?structure system. Analyses are repeated for different building mass, soil shear wave velocity, and different excitation inputs.   Figure 4.22 presents a case  for the systems of building with mass mb = 175 ? 103 kg, soil shear wave velocity Vs = 200 m/s, with deposit depths of 10, 15, 20, 30, 50, and 60 m excited by motion of the NGA#122 record. For the system with a 10 m deep deposit, peak spectral motion at foundation is 25% less than free-field motion. Both spectra have similar shapes, and peaks are at the same period. When deposit depth is increased to 15 m and other parameters of system are kept same, the free-field spectral motion is increased and foundation spectral motion is decreased. Still, most of the peaks and valleys in the spectral acceleration curve occur at the same period in both spectra. However, in the system with a 20 m deep deposit, the dominant part of the spectrum of acceleration at foundation is shifted to a longer period (T = 0.5 s). This indicates that the building and deposit interact significantly to the excitation motion and period of resulting motion at foundation-soil interface deviates.  When the deposit depth is increased to 30 m, the period of the peak spectral acceleration is shifted for both foundation motion and free-field motion. The peak of spectra of acceleration at foundation is about 20% higher than the corresponding spectral value of free-field motion. In deeper deposits with H of 50 and 60 m, the motions at free field and foundation base are not significantly different. 120   Figure 4-22 Effect of soil deposit depth on the response spectra of the motion at foundation base for a system with building mb = 175 ? 103 kg under excitation of the NGA#122 record  This phenomenon of change in response at the foundation base relative to that at free field for different depths of deposits is observed for other cases of building, soil shear wave velocity, and 121  input motion. However, the patterns of change in foundation response are not the same for all the cases of earthquake input. The magnitude of variation and the periods at which the significant variations occur differ from one case input motion to other.  4.4.6.1 Effect of deposit depth to the variation of motion at foundation under different building mass Figures 4.23, 4.24, and 4.25 illustrate the spectral acceleration at free field and foundation base as well as their variations in systems of three different buildings to the input motion of the NGA#37 and NGA#122 records and white-noise, respectively.  The variation in spectral acceleration aS? is the difference in spectra normalized by peak spectral acceleration of motion at free field.   The depths of deposit considered are 10, 15, 20, 25, 30, 40 and 75 m. Soil shear wave velocity Vs is 200 m/s for the cases presented here.  In each case of input motion, building mass is varied from mb= 50 ? 103 kg to 100 ? 103 and 150 ? 103 kg. Corresponding fixed-base building periods are Tb = 0.22, 0.3 and 0.37 s respectively.   In the case of the NGA#37 input, the variation of motion is prominent in a system with a deposit of shallow depth when the building is lightweight (mb = 50 103 kg) and period Tb = 0.22 s. Foundation motion is reduced by 40% at the spectral peak that occurs at period T = 0.22 s. The first graph of Figure 4.23 shows that in the system with a 10 m deep deposit, foundation motion is reduced at short periods and amplified at long periods. The cases of 15 and 20 m deep deposits 122  have amplifications of foundation motion. Variation of motion is not significant in other systems with deeper deposits.  123  Figure 4-23 Variation of spectral values from free field to foundation base as depth of deposit increases under excitation of the NGA#37 record As building mass is increased, resulting in an elongation of the period of the building, the variation of motion from free field to foundation is extended to deeper deposits. For the case of building mass mb = 100 ? 103 kg and building period Tb = 0.3 s, the variation of motion is higher in the 15 m deep deposit. For the same building, deposits with intermediate depth (H = 20, 25, and 30 m) also have amplifications in spectral motions. However, deposits with H = 50, 60, and 75 m still have minimal variation of motion from free field to foundation.   When building mass is further increased to make mb = 150 ? 103 kg and Tb = 0.37 s, systems with deeper deposits have more variation in motion from free field to foundation. In this case, variation of motion in a shallow deposit is now reduced. The most dominant variation of motion occurs for deposits with H = 25, 30, and 40 m. A system with a 40 m deep deposit has amplified spectral motion at the foundation throughout the large period range and gets more than 40% increase at peak spectral acceleration. A system with deep deposits with H = 50, 60, and 75 m also has some variation in motion in the foundation from free-field motion.  The trend in variation of motion is that the larger the mass of the building and its period, the bigger the difference between motion at foundation base and free field for a system with deeper deposits. As the depth of deposit increases, the variation of motion will be less in the cases of lightweight buildings. This suggests that mass of building has relation with depth of deposit to produce variation of motion at foundation from free field.    124   Figure 4-24 Variation of spectral values from free field to foundation base as depth of deposit increases under excitation of the NGA#122 record 125  The same systems of building and deposit are studied for other input motions. In the case of input of the NGA#122 record, a significant variation is noticed only for the system with a 10 m deposit for a lightweight building with mb = 50 ?  103 kg and Tb = 0.22 s (Figure 4.24). The foundation motion is reduced by 38% at a period of 0.2 s compared with free-field motion and is increased by 40% at a period of 0.33 s. Foundation motion does not deviate from free-field motion in deeper deposits. As mass of the building is increased, foundations in deeper deposits have more pronounced variation from free field. The motion at foundation in a system with a 40 m deep deposit is amplified at the entire period range in the spectra curve. The only difference in the trend in the case of input of the NGA#122 record compared with the NGA#37 record is that significant variation of motion from free field to foundation occurs at different periods in spectra curves.  The results of the variation of motion in the case of white-noise acceleration input are presented in Figure 4.25. A similar trend is observed as in the other cases of input motions. The variation is noticeable only for the system with shallow deposits for a lightweight building with mb = 50 ? 103 kg and Tb = 0.22 s. In these case,  foundation motion is not significantly different from free-filed motion in deeper deposits. However, as mb is increased and building period Tb becomes larger, motion at the foundation base resting in deeper deposits has a pronounced variation from free-field motion. For a building with mb = 100 ? 103 kg and Tb =0.3 s, the foundation motion is reduced by more than 50% compared with free-field motion at period T = 0.27 s for a 15 m deep deposit. Significant amplification of foundation motion is observed for a building with mb = 200 ? 103 kg and Tb = 0.43 s for systems with 25 and 30 m deep deposits. 126  The periods associated with significant variation of motion from free field to foundation base in the case of white-noise input are different from the cases of the NGA#37 and NGA#122 inputs.  Figure 4-25 Variation of spectral values from free field to foundation base as depth of deposit increases under excitation of white noise 127   The following are the major findings in regards to the effect of change in deposit depth to the variation of motion from free field to foundation base: 1. Depth of soil deposit affects the variation of motion from free field to foundation significantly. 2. A building with less mass and a short period is affected only in shallow depth deposits. The effect is prominent in deeper deposits when the mass of the building is increased and the building period becomes larger.  3. Spectral acceleration of motion at foundation base could be higher than that at free field at some period range and lower in another period range. Reduction and amplification of spectral motion at foundation do not necessarily happen in entire period range.   4. The period range at which significant reduction and amplification of foundation motion in acceleration spectra relative to that at free field occur is different under different excitations for same system of building and deposit. The level of variation of motion also differs with excitation motion.  4.5 Patterns in the Results This section describes the important patterns in the variation of motion relating to dynamic characteristics of building, deposit, and input excitations.  4.5.1 Amplification of foundation motion in a system under different earthquakes In some cases of building-soil system, it is observed that peak spectral acceleration at foundation base is higher than free field for all input motions considered in this study. Figure 4.26 presents 128  spectral accelerations (5% damping)  of motions at foundation base and free field of  a soil- structure system with a building having mb = 175 ?103 kg, Tb = 0.4 s resting on a 25 m deep deposit of soil with Vs =200 m/s for six different input motions including five earthquake records and a white-noise motion. Foundation width is W = 20 m and critical damping of soil 10%.   The amplifications of motion at the foundation base are 10% to 30% of free-field motion at their peaks in response spectra. It is noteworthy that the period that corresponds to largest amplification of the foundation motion is T = 0.52 s for all input motion.  This period is close to period of deposit Ts =0.5 s. The spectral shape of the response spectra of free field and foundation base do not have significant differences. This implies that properties of soil deposit significantly influence the response of the system.   Figure 4.27 presents another case example of amplified motion at foundation compared to free field.  The figure shows spectral accelerations of motions at foundation base and free field of  a soil- structure system with a building having mb = 100 ?103 kg, Tb = 0.3 s resting on a 20 m deep deposit of soil with Vs =200 m/s for six different input motions including five earthquake records and a white-noise motion.   The amplifications of motion at the foundation base are 10% to 30% of free-field motion at their peaks in response spectra. Peak response and largest amplification occur at period of soil deposit, Ts = 0.4 s for all cases except NGA#97 and NGA#953.  Like in earlier case, the spectral shape of the response spectra of free field and foundation base do not have significant differences.   129   Figure 4-26 Motions at free field and foundation in the system of a building with Tb = 0.4 s resting on a 25 m deep deposit of soil with Vs = 200 m/s excited under the NGA#30, NGA#37, NGA#122, NGA#953 and NGA#993 records and white-noise motion 130   Figure 4-27 Motions at free field and foundation in the system of a building with Tb = 0.3 s resting on a 20 m deep deposit of soil with Vs = 200 m/s excited under the NGA#30, NGA#37, NGA#97, NGA#953 and NGA#993 records and white-noise motion 131  4.5.2 Reduction of foundation motion in a system under different earthquakes Cases of amplification of foundation motion compared to free-field motion in a system are presented in section 4.5.1. In some other cases of building-soil system, it is observed that peak spectral acceleration at foundation base is significantly de-amplified  in relation to free field for all input motions considered in this study. Figure 4.28 presents spectral accelerations (5% damping)  of motions at foundation base and free field of  a soil- structure system with a building having mb = 100 ?103 kg, Tb = 0.3 s resting on a 20 m deep deposit of soil with Vs = 300 m/s for six different input motions including five earthquake records (NGA#30, NGA#37, NGA#122, NGA#953, NGA#993) and a white-noise motion. Foundation width is W = 20 m and critical damping of soil 10%.   The peak reductions of motion at the foundation base are 20% to 50% of free-field motion at their peaks in response spectra. In most of cases on inputs, significant peak response and maximum reduction of motion at foundation occur at around period of deposit, Ts = 0.27 s. The spectral shape of the response spectra of free field and foundation base do match each other in these cases too. This suggests that properties of soil deposit have an influence the response of the system. However, presence of building drops the response at the foundation.  The fact that the period of the building (Tb = 0.3 s) is also close to deposit period suggest that interaction between building mass and soil deposit is significant.  This could be related to a two degree of freedom system where responses are controlled by two modes of the system.      132   Figure 4-28 Motions  at free field and foundation in the system of a building with, Tb = 0.3 s resting on a 20 m deep deposit of soil with Vs = 300 m/s under excitation of the NGA#30, NGA#37, NGA#122, NGA#953, NGA#993 records and white-noise motion  Figure 4.29 illustrates another example of reduction of spectral accelerations at foundation base compared to field in a system under different input motions. The system has a building mass mb = 150 ?103 kg, Tb = 0.37 s resting on a 15 m deep deposit of soil with Vs = 200 m/s. It is excited 133  by NGA#30, NGA#97, NGA#106, NGA#122, NGA#993 and a white-noise motion. Reduction of spectral acceleration of foundation motion is significant at period associated with peak spectral acceleration of free field motion. Spectral acceleration of foundation motion is slightly higher at long period out of the peak spectral region.   Figure 4-29 Motions  at free field and foundation in the system of a building with, Tb = 0.37 s resting on a 15 m deep deposit of soil with Vs = 200 m/s under excitation of the NGA#30, NGA#97, NGA#106, NGA#122, , NGA#993 records and white-noise motion 134   4.5.3 Amplification and reduction of foundation motion in the same system under different earthquakes Some soil?structure systems exhibit foundation motions are amplified in some earthquakes and are reduced in others relative to the free-field motions. This suggests that the characteristics of the earthquake motions relative to the system characteristics is an important determinant in whether  motions at foundation slab reduces or amplifies  compared to free-field motions.  Figure 4.30 presents spectral accelerations (5% damping)  of motions at foundation base and free field of  a soil- structure system with a building having mb = 80 ?103 kg, Tb = 0.27 s resting on a 15 m deep deposit of soil with Vs = 200 m/s for different input motions.  Figure 4.30(a) illustrates the cases of motion reduction at the foundation base for the input of the NGA#30 and NGA#953 records. Peak spectral motion at the foundation-base is reduced by 30% in these cases. The spectral shape of the foundation motion is also different from the free-field spectra when the system is excited by the NGA#953 record. This indicates that motion at the base of the foundation is not simply following only the motion of deposit.  Figure 4.30(b) illustrates the cases of amplified motion at the same foundation for the input of the NGA#97 and NGA#989 records. Like in the NGA#953 input, the spectral shape of the foundation motion differs from the free-field spectra when the system is excited by the NGA#989 record.  135   Figure 4-30 comparisons of motions in the system of a building with Tb = 0.27 s resting on a 15 m deep deposit of soil with Vs = 200 m/s, where foundation motions are (a) reduced under the NGA#30 and NGA#953 records (b) amplified under the NGA#97 and NGA#989 records  The free-field spectra in all cases of earthquake inputs have peaks at the same period ( Ts = 0.3s). The peaks of response spectra of motions at foundation do not always occur at this period. This is because effect of building in the system response is associated with characteristics of input motion.    136  This observation of different types of variation for different earthquakes for the same system confirms that amplification or reduction of foundation motion is strongly depend on input motion.  The observations in the recorded data in earthquakes in some instrumented buildings, for instance, Eureka 5-storey residential building and Fortuna supermarket building also show this phenomenon as discussed in chapter 3.   4.6 Effects of Period in the Variation of Motion From the last section, the following parameters are identified that significantly affect the variation of motions from free field to foundation base: i) Characteristics of input excitation ii) Building mass and period iii) Soil shear wave velocity iv) Depth of deposit  All these parameters are associated with the dominant period of components (building and soil deposit), the system of soil?structure, and input motion.   The building parameters storey height, floor mass and their distribution, and storey stiffness are related to the fundamental period of the building. Similarly, the depths of deposit and soil shear wave velocity are related to the fundamental period of soil deposit as evident from equation 4.1.   137  This section will describe how parameters of characteristics of input motion, period of components and system can describe the phenomena of amplification and reduction of motion at the foundation base compared with free field.  4.6.1 Fixed-base period of building 4.6.1.1 Increasing and decreasing trends of foundation motion with building period  Figure 4.31 shows changes in response spectra of motions at foundation base when the building period is increased in a particular soil?structure system. It illustrates the results of the case of a 25 m deep deposit with soil Vs = 200 m/s for the NGA#953 input and building with varying its period, Tb. In a given soil?structure system, effect of building period, Tb, is observed depicting a trend that peak spectral acceleration of foundation motion is increased as Tb is increased up to a particular period. If Tb is further increased, response motion at the foundation decreases. In this case, when Tb is increased from 0.22 to 0.27, 0.3, 0.33, 0.37, and 0.4 s, the peak of the response spectra of foundation motion increases from 2.9g to 3.65g. When Tb is further increased to 0.43 s, the spectral motion decreased. For the system with Tb = 0.48 s, the peak of the spectral motion at foundation is reduced by 30% and becomes less than peak free-field motion. The vertical dotted lines in the graph mark the fixed-base periods of the buildings in the system represented by the same color for their response spectra curve.   This phenomenon is prevalent in all cases of soil deposit and earthquake inputs considered in the study. However, the critical period at which the trend of response at the foundation base reverses 138  varies from one system to another. It is observed that the critical period also varies with earthquake.   Figure 4-31 Change in response spectra of foundation motion with fixed-base period of structure in a soil?structure system with a 25 m deep deposit of soil Vs = 200 m/s for input of the NGA#953 motion  Figure 4.32 presents the results of a time history analysis of the same system for input motion of the NGA#30 record. In this case the critical building period is Tb = 0.33 s.   It is recognised that although period of building has effect to the variation of motion at foundation from free field, it is not the period which increase or decrease of spectral peaks occur at.  139   Figure 4-32 Change in response spectra of foundation motion with fixed-base period of structure in a soil?structure system with a 25 m deep deposit of soil Vs = 200 m/s for input of the NGA#30 motion  4.6.1.2 Spectral value of motion at foundation at building period  It is found that the foundation motion is reduced in most of the cases in the neighbourhood of Tb. Figure 4.33 shows the variation of motion, )( ba TS? , which is difference in response spectra value of motions at foundation and free field at the building period  normalized by peak spectral value of free-field motion. The variation is given by  140  )}({)()()( TSMaxTSTSTSffabffabfnaba??? ??? ,        (4.3) where fnaS ? is spectral acceleration at foundation and ffaS ?  is spectral acceleration at free field.  Analyses are carried out for system with building with period Tb = 0.22, 0.27, 0.3, 0.33, 0.37, 0.4, and 0.43 s and soil deposit with depth H = 10, 15, 20, 25, 30, 40, 50, 60, and 75 m for soil shear wave velocity Vs = 200 m/s and 300 m/s for earthquake inputs of the NGA#30, NGA#37, NGA#97, NGA#106 and NGA#122 records.   Most of cases have reduction of motion (negative value of variation) at the building period Tb. Reduction of foundation motion is up to 60% from peak of free-field motion. The few data points that show amplification (above the zero line) belong to the few cases where response spectra of motion at the foundation base are amplified at the entire period range. These motions are, however, not significant.  This suggests that spectral acceleration of foundation motion is generally lower than spectral acceleration of free-field motion at the fixed-base building period. However, there will be exceptions, as in some cases foundation motion is higher than free-field motion in the entire ranges of periods in the  response spectra curve.  141   Figure 4-33 Variation of foundation response spectra from free field at building period Tb for different systems under five earthquake inputs  4.6.2 Period of the soil?structure system, Tsys and predominant period of earthquake input, Tp From the observations discussed in earlier sections, it is evident that the period of a building and soil deposit relative to each other affect the motion at the foundation. The parameter that best describes the relation between the two is the fundamental period of the soil-structure system, Tsys. The relation between combined system of building and soil deposit to the component periods (fixed base building period, Tb and period of soil deposit, Ts) is developed in Appendix 142  A using discrete model of soil structure system.  The first frequency of the system can be approximated by   ? ? ? ?? ? ?????? ??????? 22222222 41121 sbsbsbsys rr ???????      (4.4) Where sys? is circular frequency of soil-structure system, b? is fist mode frequency of fixed based building, s?  is frequency of soil deposit, r is ratio between mass of building and participating mass of soil deposit in the vibration. A method to estimate the mass ratio, r using discrete model of soil- structure system is also presented in Appendix A.   For the purpose of investigation carried out in this section, system period, Tsys is obtained from Eigen value analysis of system in continuum model in computer program ABAQUS.  Figure 4.34 shows a case of how the system period affects the foundation motion in a system for a given deposit. As the deposit parameters is kept constant, system period is changed by changing building period.  Results shown in Figure 4.34 are taken from the case of input of the NGA#953 record to a system with 30m deep soil deposit having Vs = 300 m/s.  The fundamental period of the system is varied from 0.4 to 0.59 s. The input motion NGA#953 record has its predominant period Tp = 0.54 s.  The first graph in the figure (top left corner) shows that free-field motion and motion at the foundation base are almost the same when the system period Tsys is 0.4 s. There is a small variation of motion in the spectra at period equal to Tsys and at the predominant period of the earthquake, marked by a dotted line at 0.54 s. 143   Figure 4-34 Change in foundation spectra with system period for the NGA#953 input When system period is increased to 0.41 s, the variation is bigger with peak of foundation motion being 10% higher than free-field motion. The amplification is noticeable when Tsys is set to be 144  0.43 s. The foundation motion is amplified in a larger range, and the peak is about 40% higher than corresponding motion at free field. For Tsys = 0.47 s, the peak spectral motion at the foundation is further amplified. When system period is further increased, the peak spectral motion at the foundation-base is now lower but is still higher than the peak of free-field spectral motion. When the system period is matched with the predominant period of earthquake, Tsys = Tp = 0.54 s, the peak values of spectral motions are almost the same.  There is a small shift in the period of peaks. Upon further increase in the system period, the spectral motion at the foundation base is reduced in the peak region. At Tsys = 0.59 s, there is 30% reduction in peak spectral motion from free field to foundation.  This phenomenon is evident when the predominant period of the input motion is relatively longer than the system period under consideration. For input motion with shorter Tp, the system period is likely to be larger than this. In those cases, the motion at the foundation base is lower than free field motion. If the system period is significantly larger than Tp, the reduction is less in the region of peak spectra.  Figure 4.35 present the results of a case with the NGA#30 input to a system of a 15 m deep deposit of soil with Vs = 200 m/s, where Tp = 0.3 s. The fundamental period of the system varies from 0.34 to 0.47 s. In all cases, foundation motion is reduced compared with the motion at free field. When Tsys is increased from 0.34 to 0.39 s, the foundation motion is further decreased in the peak spectral period. However, for the long system period, the peak spectral motion at foundation base is increased, albeit less than free-field motion. There is a small localised hump of increased spectral value in the neighbourhood of the system period. 145   Figure 4-35 Change in spectra of motion at foundation base with system period for the NGA#30 input  Figure 4.36 shows the variation of motion (difference in response spectra) at the system period, Tsys, for earthquake inputs of the NGA#30, NGA#37, NGA#97, NGA#106 and NGA#122 records. The variation is given by  )}({)()()( TSMaxTSTSTSffasysffasysfnasysa??? ??? ,        (4.5) Analyses are carried out for system with building with period Tb = 0.22, 0.27, 0.3, 0.33, 0.37, 0.4, and 0.43 s and soil deposit with depth H = 10, 15, 20, 25, 30, 40, 50, 60, and 75 m for soil shear wave velocity Vs = 200 m/s and 300 m/s.   146   The figure shows that in most of the cases, motion is amplified. There are some data points (inverted triangles) showing some reduction to the input of the NGA#106 record. This is because the predominant period of the NGA#106 record is 0.18 s and the system periods of most of the cases are longer than this. As discussed earlier, the foundation response spectra in those cases may be reduced even in the system period.  Figure 4-36 Variation of foundation response spectra from free field at system period, Tsys, for different systems under five earthquake inputs For a system that has a very short fundamental period, the foundation motion may be amplified in most earthquakes. This could happen in a short period building resting on relatively stiff soil deposit of shallow depth. If the earthquake input has a longer predominant period or has multiple 147  peaks, other systems may also get motion amplified at the foundation. This is because the system period will be shorter than the predominant period of the earthquake.  Figure 4.37 shows %age amplification of foundation motion in response spectra at the period of   peak spectral acceleration of foundation motion. These results are from system configurations where periods of the building, Tb, are 0.22, 0.27, 0.3, 0.33, 0.37, 0.4, and 0.43 s and depths of deposit, H, are 10, 15, 20, 25, 30, 40, 50, 60, and 75 m with soil shear wave Vs = 200 m/s and 300 m/s. Results are presented for only three earthquakes (NGA#953, NGA#37 and NGA#122) for clarity of graph points. The points marked in the graph represent only cases of amplification of foundation motion compared to free field in their peaks in response spectra.  The amplification points are plotted against their corresponding period, T. The predominant period of input earthquake records are marked by dotted lines.   For the NGA#953 record, most of peak spectral acceleration represented by green square box markers occur on or before Tp, NGA#953 = 0.54 s. Similarly, for input of the NGA#37 record, most peak spectral acceleration is at periods less than Tp, NGA#37 = 0.64 s. For the case of the NGA#122 record, which has a longer predominant period of Tp, NGA#122 = 0.76 s, the peak spectral values when foundation motions amplify extend well up to this period.  The results show that amplification of foundation motion at the peak of response spectra is most likely to occur at period   of predominant period of earthquake or shorter period.  This observation corresponds to the characteristic of spectral shape that dominant peak occurs usually 148  at long period associated with fundamental mode of response of a system subjected to the motion.   Figure 4-37 Cases of amplification in foundation response spectra in three earthquakes where amplification ceases to occur at the period larger than Tp  4.6.3 Effect of the building period on the system period In Figure 4.38, the variation of system period, Tsys, with building period, Tb, is illustrated for systems with different soil deposits.  Period of building has been varied for a system with given soil deposit and system period information is obtained from results of Eigen analysis carried out in the soil-structure model.   It is confirmed that with change in building period significantly affects the period of system in shallow deposits. However, as deposit depot is increased, there is less effect of change in 149  building period to the system period. For instance, for system with 40m deep deposit, the system period remain almost same when building period is increased from 0.22 s to 0.43 s.   Figure 4-38 Relation between building period and system period in different deposits  Figure 4.39 presents the response behaviour of systems in their fundamental mode for system with 20 and 75 m deposits. The soil shear wave velocity Vs for both cases is 300m/s and building mass is 150 ?103 kg.  Fixed base building period is 0.37s.   Figure 4.34(a) is the first mode shape of a system with a 20 m deposit where the modal response is limited to below the building and immediate surroundings. This indicates that the response at the foundation and its surrounding is controlled by the building itself. Figure 4.34(b) shows the case of  first mode shape of a system with a 75 m deposit. The entire deposit is activated in this case. Response at 150  the bottom of building (foundation level) follows the motion of the rest. In this case, the interaction between  the deposit and building is lesser compared to earlier case.   Figure 4-39 Mode shape of a system in fundamental mode: (a) system with a 20 m deposit, (b) system with a 75 m deposit  4.6.4 Identification of period of peak variation from Tb and Tsys  Figure 4.40 illustrates how the variation of motion from free field to foundation are related to  building period and system period in systems with two deposits  of different depths.  In Figure 4.40(a), response spectra of motions at foundation base and free field are compared for systems with a 15 m deposit with Vs = 300m/sec and buildings with varying period. The input motion is the NGA#30 motion. As the building period is varied in one system to another, system period system period also changes accordingly. With the change in these period, the pattern of variation of motion from free field to foundation also changes.     151  Free field and foundation motion are almost similar in the case when Tb= 0.22 s and Tsys = 0.32s. When Tb and Tsys are increased to 0.27 and 0.34 s, foundation motion is reduced by 25% at the peak of response spectra and foundation motion is lower than that free field up to the period T = 0.34 s which coincides with Tsys.  Foundation motion is further reduced when Tb and Tsys are increased but the reduction is limited to the period range up to about Tsys.  In the case of Tb = 0.33 s and Tsys = 0.39 s, peak spectral acceleration is reduced but there is change in variation at the period T = 0.39 s. Similar observation is made in next case where reduction of spectral motion is limited up to the period of Tsys when Tb=0.37s and Tsys is 0.43s. In all these cases, spectral motion at foundation is lower at the period of Tb.   For the similar change in building period in a system with 50m deep deposit with  soil shear wave velocity Vs = 200 m/s, the system period do not vary. When building period is changed from Tb= 0.22 s to 0.27, 0.3, 0.33 and 0.37 s, system period remain same around 1.0 s. The variation of motion at foundation and free field is not as significant as the case with deposit depth H = 15m.  However, it is noteworthy that response spectra of foundation motions have trough at the building periods.   It is noted from these observations that system period is related to variation of spectral acceleration from free field to foundation in soil-structure system with shallow deposits. In systems with deeper deposits, the relation is not noticeable. In deep deposit systems, variations of motion occur in the neighbourhood of the building period in response spectra. Foundation motions are reduced though the variations are small and localised.  152   Figure 4-40 Controlling period for variation of motion from free field to foundation for earthquake input of the NGA#30 record for systems with (a) 15 m deposit, (b) 50 m deposit 153   4.7 Conclusions It is shown from simulation that motion at the foundation base can amplify from the free-field motion in some combination parameters of soil structure system and earthquake. It is shown that period of building, soil deposit, and combined soil?structure system as well as the predominant period of input motion are key parameters that affect the motion at the foundation base relative to the free field.   The following major conclusions are drawn from the simulation of soil?structure systems with a wide range of properties by the continuum model in ABAQUS: 1. The higher the soil shear wave velocity of the deposit soil, the lower will be the variation in the ground motion from free field to foundation. 2. When the predominant period of earthquake motion is longer than the period of the soil?foundation?structure system, amplification of foundation motion occurs. In the response spectra, amplification is small and local when system period is far from predominant period of earthquake. The amplification becomes higher for a large period range in response spectra when system period is close to predominant period of earthquake. 3. In the response spectra, spectral acceleration at the foundation is reduced at the fixed- base building period in all cases except when soil-structure system period is close to predominant period of earthquake. 154  4. The variation of motion from free field to foundation is not affected by system period for the case of a deep deposit. In this case, only local reduction of motion can be observed at building period in the response spectra. 155  Chapter  5:  SOIL?FOUNDATION?STRUCTURE (SFS) MODEL WITH LUMPED PARAMETERS 5.1 Introduction The Chapter 4, it was shown that relationship between the periods of soil deposit, structure and input motion are found to be the major parameters that determine whether the motion at the foundation base of building increases or decreases compared with the free-field motion in an earthquake. These findings were developed using finite element analysis in the ABAQUS computer program.   In this chapter a simple lumped mass model is developed which provides a much simpler method of exploring whether slab motion at foundation is amplified or not relative to free-field motion. The model is developed for homogeneous viscous-elastic soil. Only translational motion at the foundation is considered in the development of the model. The model includes building, foundation and soil deposit, each represented by a set of a lumped mass, a dashpot and a spring. This model is developed for 2D model of soil deposit that represents plane strain condition with long foundation footing. This model is taken for the sake of simplicity in the study. It also allows 156  to verify the results with that from ABAQUS model developed in Chapter 4. Similar approach can be taken to develop model for other foundations with plan square or circular plan shapes.   The analytical model is intended to provide an approximate response of the system at the foundation level when the system is excited at the bottom of the soil deposit. Results of time history analysis of the model show good agreement with the ABAQUS continuum model described in Chapter 4. Cases of both amplified and reduced motion at the foundation base in comparison with free-field motion are checked.   5.2 Soil?Foundation?Structure 3-Degree of Freedom (SFS 3DOF) Lumped Model Figure 5.1 shows a lumped model of the soil?foundation?structure system that includes the properties of the individual components- soil-deposit, foundation and building. The system has three degrees of freedom in the translational mode. In this model, foundation has a designated degree of freedom so that response quantities can be obtained from analysis directly at this level.     In the model, Ks, ms, Cs represent stiffness, damping and mass of soil deposit, Kf, mf, Cf represent stiffness, damping and mass of foundation and Kb, mb, Cb  represent stiffness, damping and mass of building respectively. These are generalized properties in their first mode of vibration.    157   Figure 5-1 Soil?foundation?structure (3DOF SFS) lumped-mass model  Following sections describe the development of mass, stiffness and dashpot properties of soil deposit, foundation and building components.  5.3 Lumped Parameters of Soil Deposit with Foundation Lumped properties of soil deposit with foundation are first developed to derive discrete properties of soil deposit and foundation. This approach allows use of past studies on soil-foundation system.   5.3.1  Participating mass of soil- foundation The ratio of building mass to soil mass that is participating in the first mode of vibration of the deposit is not readily known. Despite the fact that the model shown in Figure 5.1 has a definite cross sectional area A of soil column and height H, the actual soil mass that is activated in the vibration together with an overriding building may not necessarily be bound within those 158  dimensions. Previous studies that explored the participating volume of the soil in the vibration (Crockett and Hammond, 1949; Rao and Nagraj, 1960) were based on the concept of bulb pressure. They developed approximate methods to take into account the effect of the participating mass in the determination of system period. The applications of the method are found based on empirical relations developed from the limited forced vibration test. In the 1970s other researchers ruled out the concept of participating mass, citing the lack of experimental evidence to assess the mass of soil (Richard et al,1970; Gazetas,1975) and no further significant research study has been carried out in this field. Since then, research has been focused on developing the dynamic impedance function for the foundation?soil, *K . *K  is a frequency-dependent parameter representing characteristics of foundation and soil by spring and dashpot, respectively. Instead of assigning both mass and springs for soil, the practice has been to assign only springs with an impedance value.  The dynamic impedance function, *K  is defined as ?????? ????????? ??nniKK ????? 21 22* , (5.1) where K is static stiffness, ?  is excitation frequency, n?  is natural frequency, and ? is critical viscous damping (Gazetas, 1983). Although the concept of participating mass is disregarded, Equation 5.1 shows that the impedance function *K is also a function of inertial mass. The termn? , the natural frequency of the soil, is related to mass and static stiffness by mKn ?? . (5.2)  159  Mass m in Equation 5.2, which is embedded in Equation 5.1, must be the mass of soil deposit participating in the vibration of the foundation. Use of the frequency-dependent impedance function in the analysis of foundation vibration requires parameters to be derived from semi-empirical equations. It is necessary to quantify the soil mass that participates in the foundation vibration for the simple model of the soil?structure system adopted in this study.  5.3.2 Effect of foundation width  Eigenvalue analysis is carried out for rigid foundation?soil systems for cases of different foundation width, W, and depth of deposit, H, in ABAQUS to find the first frequency modes. The soil shear wave velocity Vs = 200 m/s , mass density of soil ?s = 1800 kg/m3 and foundation thickness is 0.6 m. Results shown in Figure 5.2 are compared with deposit-only frequencies, which can be obtained from Equation 4.2.  sdp VHT 4?   and  HVsdp 2?? ?     (5.3)   It is observed that estimates of the period from Equation 5.3 match well with the results of simulation in ABAQUS. It suggests that equation for estimating fundamental period of soil deposit can also be used for system of deposit and foundation (with no structure on top of it). This also implies that adding foundation slab to the deposit does not affect the period noticeably.  160   Figure 5-2 First frequency mode of soil deposit with foundations of 20 and 50 m width for different deposit depths  The static stiffness of the foundation?soil system can be determined from static pushover analysis. Stiffness derived from pushover analyses carried out in ABAQUS for deposits with an overriding foundation are presented in Figure 5.3. The deposit depth is increased from 10 to 75 m for three foundation widths of W = 20, 30, and 50 m. The shear modulus of soil, G = 72 MN/m2 and Poisson?s ratio ? = 0.3. In the analysis, the rigid foundation block is pushed horizontally to a specific displacement, and the force exerted in the block is recorded. The stiffness of the system, K, is derived from the relation K = F/?, where F is exerted force and ? is the corresponding displacement at the foundation?soil interface.  161  The results presented in Figure 5.3 show that stiffness increases with foundation-base dimension. The effect is more pronounced in shallow depth deposits. Stiffness decreases as the depth of deposit increases.   Figure 5-3 Static stiffness of the soil?foundation system obtained from pushover analysis in ABAQUS  Observations from the results presented in Figures 5.2 and 5.3 suggest that change in foundation width does not affect the period but modifies the stiffness of combined system of soil and foundation slab. This provides the important insight that any change in the stiffness of the foundation?soil system due to an increase in foundation dimension must be accompanied by a 162  proportional change in participating mass so that the fundamental period of combined system of soil and foundation slab remains the same.   5.3.3 Estimation of static stiffness of soil-foundation slab system Many researchers proposed static stiffness of the foundation?soil system in semi-empirical form based on vibration test data (Richard and Whitman, 1967; Richard et al, 1970; Gazetas, 1975). The formulations were proposed mostly for circular-shaped foundations atop a half-space and hence did not take base rock into account. Later, Gazetas (1991) presented a comprehensive list of spring stiffness and dashpot coefficients for varieties of foundation configurations. He also proposed static stiffness of the soil?foundation system for a deposit resting on base rock. Static stiffness of the strip foundation with width W resting on a soil deposit of depth H is ?????? ??? HWGK 122? . (5.4) The parameters G and ? are soil shear modulus and Poisson ratio, respectively. Figure 5.4 shows that results obtained using Equation 5.4 match with the ABAQUS output for the cases of deep deposits combined with small foundation width, except for the case of a 10 m deep deposit.  Equation 5.4 is modified to account for the effect of base rock at shallow depth. Based on the results from analysis in ABAQUS, it is possible to propose that the stiffness for the foundation?soil system can be approximated by  ???????????????????43122HWGKc ? . (5.5) 163   Figure 5.4 shows the results of the modified equation for estimating the static stiffness of soil?foundation system. The modified equation is able to provide results close to the ABAQUS simulation not only in the deep deposits but also in the shallow deposit cases.  Figure 5-4 Comparison of static stiffness of the soil?foundation system obtained from ABAQUS, Equation 5.4 (Gazetas, 1991), and a modified formula (Equation 5.5) 164   5.3.4 Generalized stiffness of foundation-soil system The stiffness provided by Equation (5.5) corresponds to the static force?displacement relationship of a rigid foundation block sitting on the top of the soil deposit. If the system of soil deposit and foundation block is represented by a single degree of freedom system (SDOF) to carry out dynamic analysis, adjustment needs to be made for this stiffness corresponding to a generalized stiffness.   Chopra (2001) developed equation of motion for a beam with distributed mass and elasticity in terms of generalized mass m~ , generalized stiffness k~  and generalised excitation )(~ tuL g???  as shown in Equation 5.6.   )(~~~ tuLukum g???? ???           (5.6) where gu??  is input motion  and generalized mass, generalized stiffness and generalized excitation for bending beam are given by  ??Hdxzzmm02)]()[(~ ?   ? ???HdxzzEIk02)]()[(~ ?          (5.7) ??HdxzzmL0)]()[(~ ?  Where )(zm is mass per unit length of beam,  )(zEI  is bending stiffness and )(z?  is shape function. The equation 5.7 is SDOF system for ground motion excitation except  L~  replaces the 165  mass of SDOF, m~ . The Equation 5.6 can be normalized by an intensity factor mL ~~~ ??  so that same input motion )(tug??  can be used in the dynamic analysis.    For a homogeneous uniform shear beam with mass density ?, area A and soil shear modulus G, the factor ?~   is given by  ???? HHdzzAdzzA020)]([)(~????          (5.8) The shape function of deposit for its first mode is assumed to be a simple sine function, given by Equation 5.9 ??????? Hzz 2sin)(??  (5.9) where H is the total deposit height. Introducing Equation 5.9 to Equation 5.8, the intensity factor is given by  ?4~ ??             (5.10) Corrected stiffness of foundation-soil system associated with generalized coordinate system is given by   cmc KK ?4?            (5.11) 5.3.5  Mass of deposit Using Equations 5.3, 5.5 and 5.11, and the relation between mass and stiffness, MK?2? , gives the mass of the deposit . 166  ???????????????????4323 1232HWBHmdp ??? . (5.12) Where B is unit plan dimension of foundation normal to width.  5.3.6 Properties of soil deposit only A method to estimate the mass and stiffness properties of a foundation?soil system was discussed earlier. These are lumped parameter properties of a combined system of the soil and the foundation. To develop a model that gives response at the foundation base, the properties of the deposit-alone should be separated from the foundation?soil system.  The equation of motion for the soil deposit without foundation can be derived if it is assumed that the deposit behaves as a cantilever shear beam, as shown in Figure 6.8.  Figure 5-5 Cantilever shear beam model of free-field deposit Carvajal (2011) developed a model for a soil deposit represented by shear beam.  Equation 5.13 represents the equation of motion of the soil deposit for earthquake loading.   167  ? ????????1 )(),(),(),(n tgntHnntHnntHnn uIuKuCuM ????? , (5.13) where zAMHznn ??? ??02)( ;  zGAKHznn ??? ????0)( ;  ?nnn KMC 2? ; zAIHznn ??? ???0)(  and ? is the mode shape function.   If only the first mode is considered and ? ?Hz 2sin ?? ?  is used for the response of the deposit, the mass, dashpot, stiffness, and ground motion intensity parameters can be obtained as 2AHM ?? ; HGAK 82?? ; ??? 2GAC ? ; ??AHI 2? . (5.14)  This is equivalent to the equation of motion for a single degree of freedom system (SDOF) with the parameters of mass, stiffness, damping, and scaling factor for input motion. The parameters, H, ?, ? and G required to solve the equation are available from the properties of the soil deposit. The parameter A , representing the cross-sectional area of the shear beam, is not definite when the structure is not at the top of the deposit, which is unbounded in the horizontal direction. However, it is obvious that one can normalize Equation 5.13 with A , as it appears on both sides of the equation. As a result, the value of parameters defined in Equation 5.14 will take unit value of A . The response at the free field of the deposit to any input motion can be determined by solving the SDOF equation of motion with the parameters defined in Equation 5.14.  168  Taking unit value for area A  will not work when the parameters of the deposit defined in Equation 5.14 are combined with the parameters of the foundation and building to solve the equation of motion for the soil?foundation?structure system. This is because the area A  of the deposit is no longer a common denominator for the combined system. When the components of soil deposit, foundation, and building are combined into a single model, a definite value of cross-sectional area of soil deposit is required. Equating the mass defined in Equation 5.14 and the mass of deposit defined in Equation 5.12 gives the area of deposit ????????????????????????? 433124HWHA ?? . (5.15)  The area given by Equation 5.15 corresponds to a shear beam that has a uniform cross-section area throughout its length. In the case of soil deposit, this is equivalent to having the same intensity of pressure from foundation to base rock. If the force is generated from vibration, the intensity is a function of mass participation. In the soil deposit, the mass participation is not uniform but decays as it goes downwards. In the liquefaction analysis of soil deposit, the Shear Mass Participation Factor, rd, is used to reflect the decay of inertial mass that induces shear stress at a depth from the surface (Cetin and Seed, 2004).   This suggests that the shear beam used in the model should also have a varying cross section based on intensity. Alternatively, a uniform shear beam with the same total intensity (mass participation) can be used so that it would have the same result.  169  The variation of intensity from foundation downwards is assumed to follow a sine function in Equation 5.16. ??????? ??? 2sin , (5.16) where Hz??  is the normalized depth of the point of interest in the deposit. The total intensity of mass participation throughout the depth is given by 212sin102102 ?? ????????? ????? ddI . (5.17)  It suggests that the shear beam with decaying intensity of sine function in one direction will result in only half of the mass of uniform intensity. As the intensity decay function in vertical direction applies in both longitudinal and transverse directions, the total reduction will be one fourth of the beam with uniform intensity.   The equivalent area is given by ????????????????????????? 4331244HWHAeqv ?? . (5.18) For a building with a 20 m wide foundation sitting in a deposit of 20 m width, the area of the deposit will be approximately 10 times that of the foundation area. Figure 6.9 shows an example of variation of ratio of area of deposit to foundation area for an equivalent uniform shear beam model. 170   Figure 5-6 Ratio of deposit to foundation area for an equivalent uniform shear beam model for the case of foundation width, W = 20 m  Mass, stiffness, and dashpot properties of the deposit can be obtained using this area, eqvA , in Equation 5.14. Equations 5.19 and 5.20 will give the mass and stiffness of soil for use in the SFS 3DOF model. ????????????????????????? 43231224HWHM s ??? . (5.19) ???????????????????431232HWGKs ?? . (5.20) 171  5.3.7 Properties of foundation If one considers foundation and underlying soil as a combined system, the total stiffness of it can be regarded as the sum of stiffness of springs of soil and foundation connected in series. The total stiffness is related to individual stiffness by Equation 5.21. fssf KKK111 ?? , (5.21) where sfK  is total stiffness, sK  is soil stiffness, and fK  is foundation stiffness. The sfK  is the stiffness of the soil?foundation system, given by Equation 5.5, and sK  is given by Equation 5.20.  The stiffness of the foundation, fK , is given by Equation 5.22. 11612216 43??????????????????????? HWGK f . (5.22) Mass of the foundation, fM , is the product of volume and mass density of material for the foundation.  Local damping for the foundation DOF is set to be same as structural damping. The force at the foundation spring is the interaction force that is induced at the soil?foundation interface during the dynamic loading. From Equation 5.22, it is obvious that fK  will be increased with soil stiffness. Also, the stiffness depends on the ratio of foundation width to deposit depth. For a very 172  small ratio in the deep deposit, the stiffness of the foundation is least affected by the foundation width, W.   5.3.8 Properties of building Component properties of building can be easily obtained using generalized mass and stiffness properties of building for fixed -based analysis. The procedure to obtain the generalized properties is also discussed in Chapter 4.   5.4  Implementation of SFS 3DOF Model The structural analysis of this model to input motion at the bottom will give the responses at the foundation level and building. The dynamic analysis of this simple system can be carried out in any software such as SAP2000, SeismoStruct etc. All input parameters for the model are already defined in previous sections.  A MATLAB code is developed to solve the problem numerically with the time integration technique. First, the Eigen matrix is solved to obtain modal parameters of the system without using damping. Along with the modes, local viscous type damping is used to develop the damping matrix where the damping ratios of soil and building are ?soil = 10% and ?building = 3%, respectively. Step-by step integration is carried out to solve the equation of motion. The solution procedure involves the Newmark?beta method with ? = 1/2.  173  The response can also be obtained using MATLAB with simple script. In this study, the analysis is carried out in MATLAB, using the algorithm presented in Appendix B.  The results of an example case are presented in Figure 5.7. The system considered for the analysis has a 20 m deep soil deposit with soil shear wave velocity Vs = 200 m/s, mass density ?s = 1800 kg/m3, and Poisson ratio ? =0.3. The building has its fixed-base stiffness Kb = 4.39 x 107 N/m and lumped mass mb = 100 t. The rigid slab foundation has a width of 20 m.   Time history results of the SFS 3DOF model are compared with ABAQUS results for acceleration response at the foundation to the input of NGA#30 earthquake records applied at the base rock. The results from the SFS 3DOF model and ABAQUS match very well for the dominant peaks. The SFS 3DOF model is missing some responses in the time history where high frequency signals occur at time t = ~7 s. This is because the model is based on the first mode of soil deposit and may not capture the response that is contributed by higher modes. Also, there is some discrepancies with the ABAQUS results in peak amplitude in the 9?12 s range of the time history. It is apparent that the damping in the 3DOF system for that frequency is underestimated. However, the overall response of the system is well traced by the proposed model.  A case of a different earthquake and a different deposit is presented in Figure 5.8. Analysis is carried out for a soil?structure system with a 25 m deep deposit and a building mb = 120 t to the input motion of NGA#97 for the continuum model in ABAQUS and for the SFS 3DOF model using MATLAB.  174   Figure 5-7 Acceleration response in time history at the foundation level for the NGA#30 record in the soil?structure system with deposit H = 20 m and building Tb = 0.3 s   Figure 5-8 Time history acceleration response at the foundation level for the NGA97 record in a deposit with H = 25 m and building Tb = 0.33 s 175  The results are very comparable each other in period, phase, and amplitude. The SFS 3DOF model underestimated the amplitude of high frequency cycles appearing between 4.5 and 5.0 s. Also, some cycles have been missed at approximately 4.0 and 8.0 s in the response time history.  5.4.1 Response at the foundation base and free field The following section will provide example cases of motions amplified and reduced at the foundation base in comparison with free-field motion, using results of the SFS 3DOF model in time history and response spectra (5% damping). The results of the model are compared with the results from continuum analysis. It is shown that the SFS 3DOF model can capture the response of the soil?structure system reasonably well.  Figure 5.9 shows a case of a soil?structure system with 20 m deep deposit and building with 0.3 s period subjected to input motion of the NGA#122 earthquake record. In this case, the foundation motion amplified slightly.  The time history response results from the SFS 3DOF model compared reasonably well with the ABAQUS model with some discrepancy in amplitude in some cycles. The response is underestimated in the early stage of time history. However, the response during the significant peak corresponds well with the results from ABAQUS. In the response spectra, the results are well captured with a good estimate of peak Sa value. From the response spectra graph, it is clear that the SFS 3DOF model underestimates the response in very short periods. Nonetheless, the difference is not significant, and the results from the SFS 3DOF model can be considered as reliable. 176   Figure 5-9 Time history acceleration response and comparison of response spectra at the foundation level for the NGA#122 record in a deposit with H = 20 m and building Tb = 0.3 s and free field  5.4.1.1 Case of reduction of motion at the foundation base Figure 5.10 shows a case of significant reduction of foundation motion from free field. The reduction happens as the system period of the soil?structure system is larger than the predominant period of the earthquake (Tp = 0.3 s). The response at the foundation level is dominated by system period Tsys = 0.5 s. The SFS 3DOF model estimated the period accurately 177  and captured this response very well. There is a slight reduction in response at the lower period, which is expected. The reduction of the motion at the foundation level is more than 30%. The effective duration of input earthquake motion in this case was short and rich in particular frequency content, and hence the response spectra are comparatively smooth.  Figure 5-10 Time history acceleration response and comparison of response spectra at the foundation level for the NGA#30 record in a deposit with H = 20 m and building Tb = 0.4 s  The same soil?structure system is subjected to another earthquake NGA#122 record, which has more significant cycles in the time history. Time history and response spectra results are shown in Figure 5.11. This earthquake has predominant period Tp = 0.76 s, which is not in the 178  neighborhoods of the system period. The foundation motion is reduced at the free-field peak period. The foundation motion is heavily reduced at the building period Tb = 0.4 s and increased at the system period Tsys = 0.5 s. The effect of soil?structure interaction is significant in this case. This phenomenon is well traced by the simple SFS 3DOF model. The comparison of response spectra from ABAQUS and SFS 3DOF models shows that the general trend is well captured despite discrepancies at few minor peaks. In the time history response, some peaks in the early stage are underestimated.  Figure 5-11 Time history acceleration response and comparison of response spectra at the foundation level for the NGA#122 record in a deposit with H = 20 m and building Tb = 0.4 s 179   For the same building and same earthquake input, analysis is carried out for a 15 m deep deposit. Results are presented in Figure 5.12. In this case, foundation response is affected significantly by the soil?structure system. The SFS 3DOF model is able to capture the interaction effects. However, the local peaks are not estimated well. In the time history plot, it is clear that the SFS 3DOF model is underestimating the peaks in the range up to t = 12 s and is overestimating afterwards. However, the difference is within 10%.  Figure 5-12 Time history acceleration response and comparison of response spectra at the foundation level for the  NGA#122 record in a deposit with H = 15 m and building Tb = 0.4 s 180  5.4.1.2 Case of amplification of motion at foundation base The results of the SFS 3DOF model are compared with the results from ABAQUS where amplification at the foundation base is observed compared with free field. Figure 5.13 shows results of a case of a soil?structure system with 25 m deep deposit and building with period Tb = 0.4 s subjected to the input motion of NGA#97 earthquake record. The response from ABAQUS shows that there is some modulation with frequency of 0.25 Hz (4.0 s). The foundation motion is amplified in the neighbourhood of system period Tsys = 0.51 s, which is close to the predominant period of earthquake Tp = 0.52 s.   Figure 5-13 Time history acceleration response and comparison of response spectra at the foundation level for the NGA#97 record in a deposit with H = 25 m and building Tb = 0.4 s 181  The SFS 3DOF model captures this response reasonably well in the time history and almost traced the result in the response spectra. The discrepancy in the time history appears in the transition of modulated response. Some short period responses are also underestimated.  A system is analysed with the SFS 3DOF model to a Northridge earthquake record (NGA#953) where huge amplification is obtained. The results are presented in Figure 5.14 in time history and response spectra. The peaks in the time history response from the SFS 3DOF model are slightly lower than the ABAQUS result. It is also reflected in response spectra. However, the error is less than 5%. The amplification at the peak response is about 90%, which is well estimated by the proposed model.  Figure 5-14 Time history acceleration response and comparison of response spectra at the foundation level for the NGA#953 record in a deposit with H = 15 m and building Tb = 0.4 s 182  Results of the response of a soil?structure system with a 30 m deep deposit to an earthquake with long duration of significant cycles are presented in Figure 5.15. There is an increase of motion at the foundation base of approximately 30% compared with free-field motion at a predominant period of earthquake Tp = 0.64 s. The period of the system is Tsys = 0.62 s. Since the system period is close to the predominant period of earthquake, the amplification of motion at the foundation base is observed in a large range of period in the response spectra.  The response from the SFS 3DOF model predicts this amplification well. Small discrepancies appear in the time history response.  Figure 5-15 Time history acceleration response and comparison of response spectra at the foundation level for the NGA#37 record  in a deposit with H = 30 m and building Tb = 0.43 s 183  5.4.2 Analysis steps using the SFS 3DOF model  1. Obtain fixed-base parameters of the building, mb, Kb, and ?b. 2. Obtain mass of the soil that participates in the vibration using Equation 5.19, ????????????????????????? 43231224HWHM s ??? . 3. Obtain stiffness of soil and deposit using Equation 5.20, ???????????????????431232HWGKs ?? . 4. Obtain foundation mass mf from the physical properties and get foundation stiffness from Equation 5.22, 11612216 43??????????????????????? HWGK f . 5. Assign critical damping ratios of building ?b and soil ?s to develop dashpot characteristics of building, bbbb KmC ?2? , foundation, bfff KmC ?2? , and soil, sfsfsfs KmC ?2? . 6. Analyse the SFS 3DOF system with parameters defined in steps 1 to 5 for the input motion and get the response at the foundation?soil degree of freedom. 7. Conduct site response analysis of the free-field soil deposit for the input ground motion using available commercial software and get the free-field motion at the ground surface. 184  8. Compare the results of steps 6 and 7 for the variation of motion from free field to foundation. Comparison can be made in the response spectra using the spectra of each response motion obtained at steps 6 and 7. 185  Chapter  6:  CONCLUSIONS AND FUTURE WORK FEMA-440 and ASCE/SEI41-06 state that slab foundations always reduce the free field motions and therefore result in lower spectral demands in the structures.  Ninety eight  pairs of free field motion recorded during earthquakes were compared and one- third of records show the slab amplified the motions. This implies that FEMA/ASCE recommendations need to be re-examined.   Detail studies were conducted to investigate the conditions under which ground motions are amplified or reduced by the slab. Many of these studies were conducted using a continuum model of the soil and a spring- mass model of the structure. Analyses were conducted using ABAQUS for a wide range of system parameters. Because the ABAQUS analyses are computer time intensive, a simplified model of plane strain foundation was analysed. These studies show that amplification occurs when the system period is close to the dominant period of the input motion. When these two periods are far apart, the motions at the slab are reduced with respect to the field. This analytical conclusion is strongly supported by the cases where same foundation amplifies the motion or reduces the motion in different earthquakes.   186  To allow a more general investigation of the effects of various parameters without excessive computational demands, a three lumped mass model was developed. This model was validated running some of the cases which were analysed using ABAQUS.   6.1 Future Work The basis of FEMA/ASCE recommendation from both data and analysis point of view should be re-examined in light of the fact that it appears in contradiction to the findings from field data and analytical results presented in this thesis.  Future work will be to generalize the plane strain lumped mass model to a 3D model so that the effect of the slab foundation on the free field motion can be estimated for a specific building.  Some aspects that new model might include are:  ? the effect of a non-vertical incident wave on the variation of motion; ? different shapes of foundation slab in deriving the stiffness and mass properties of soil mass in the model; ? springs and masses for rocking and torsional motions of the foundation;  ? nonlinearity of the soil.     ?  187  References ASCE/SEI 41-06. (2007). ?Seismic rehabilitation of existing buildings?. ASCE Standard ASCE/SEI 41-06. American Society of Civil Engineers. Virginia, USA. Borcherdt, R.D. (1970). ?Effects of local geology on ground motion near San Francisco Bay?. Bull.Seism.Soc.Am., Vol.60 (1): 29-62.  Carvajal, J.C. (2011). ?Seismic embankment ?abutment ?structure interaction of integral abutment bridges?. PhD thesis, The University of Brtish Columbia. Canada. CESMD. (2010). ?Centre for Engineering Strong Motion Data?, USA www.strongmotioncenter.org/. Accessed between 2010-2013. Cetin, K. O., and Seed, R. B. (2004). ?Nonlinear shear mass participation factor, rd for cyclic shear stress ratio evaluation.? Soil Dyn. Earthquake Engrg. Vol.24 (2): 103?113. Clough, R.W. and Penzien, J. (1995). ?Dynamics of structures, Third edition?. Computer and Structures Inc. USA. Cook, R., Malkus, D., Plesha, M. and Witt, R. (2001). ?Concepts and applications of finite element analysis?. John Wiley & Sons, Singapore.  Crockett, J. H. A. and Hammond, R. E. R. (1949). ?The Dynamic principles of machine foundations and ground?. Proceedings of the Institution of Mechanical Engineers ,Vol.160: 512-531. CSMIP. (2005). ?CSMIP Instrumented Building Response Analysis and 3-D Visualization System?. California Strong Motion Instrumentation Program, Department of Conservation, CA, USA. FEMA-440. (2005). ?Improvement of Nonlinear Static Seismic Procedures ATC-55?. Federal Emergency Management Agency, Washington D.C.   Gazetas, G. (1975). ?Dynamic Stiffness Functions of Strip and Rectangular Footings on Layered Soil?. S.M. Thesis. Massachusetts Institute of Technology. USA.  188  Gazetas, G. (1983). ?Analysis of machine foundation vibrations: State of the art.? Soil Dyn. & Earthquake Eng. Vol.3 (1): 2-42.  Gazetas, G. (1991) ?Formulas and charts for impedances of surface and embedded foundations.? J.Geotech. Eng., Vol. 117 (9): 1363-1381.  Gazetas, G., Anastasopoulos, I., Adamidis, O. and Kontoroupi, T. (2013). ?Nonlinear rocking stiffness of foundations?. Soil Dyn. & Earthquake Eng. Vol.47 : 83-91. Hasiba, T. and Whitman, R.V. (1968). ?Soil-structure interaction during earthquakes?. Soils and Foundations, Vol. 8 (2): 1-12.  Hradilek, P.J., Carriveau, A.R., Saragoni, G.R. and Duke, C.M. (1974). ?Evidence of soil-structure interaction in earthquakes?. Proceedings of 5th World Conference on Earthquake Engineering, Rome, Italy. Idriss, I.  and Seed, H.B. (1968). ?Seismic response of Horizontal soil layers?. J. Soil. Mech. & Found. Div., ASCE, Vol.94 (SM4): 1003-1031.  Kim, S. and Stewart, J.P. (2003). ?Kinematic soil-structure interaction from strong motion recordings?. J. Geotech . &  Geoenv. Engrg., Vol. 129(4): 323-335. Minami, K. and Sakurai, J. (1977). ?Earthquake response spectra for soil-foundation-building systems?.  Proceedings of 6th World Conference on Earthquake Engineering, New Delhi, India. Morgan, J.R., Hall, W.J. and Newmark, N.M.(1983). ?Seismic response arising from travelling waves?. J. Struct. Engrg., ASCE, Vol. 109(4): 1010-1027. Naeim, F., Lee, H, Hagie, S., Bhatia,H. and Skliros, K. (2005). ?CSMIP-3DV Technical Manual?. John A. Martin & Associates, Inc, Los Angeles, CA. NEHRP. (1997). "NEHRP guidelines for the seismic rehabilitation of buildings, FEMA 273?. Federal Emergency Management Agency, Washington D.C.  Newmark, M., Hall, W. J. and Morgan, J. R. (1977). ?Comparison of building response and free field motion in earthquakes?. Proceedings of 6th World Conference on Earthquake Engineering, New Delhi, India. Osawa, Y., Kitagawa, Y. and Ishida, K. (1974). ?Response analysis of earthquake motions observed in and around a reinforced concrete building including building-subsoil system?. Proceedings of 5th World Conference on Earthquake Engineering, Rome, Italy. Pardoen, G.C., Moss, P.J. and Carr, A.J., (1983). ?Elastic analysis of the County Services Building?. Bull. Seism. Soc. Am., Vol.73 (6): 1903-1916. 189  Pavlyk, V.S. (1976). ? Approximate evaluation of the difference between ground vibrations and those in the foundation of a structure?.  Osnovaniya, Fundamenty i Mekhanika Gruntov, No.5: 20-22.  PEER. (2010). ?PEER strong motion database?, Pacific Earthquake Engineering Research Centre. USA  http://peer.berkeley.edu/smcat/ . Accessed between 2010-2012. Poland, C., Sun, J. and Meija, L. (2000). ?Quantifying the effect of soil-structure interaction for use in building design?. Data Utilization Report CSMIP/00-02. California Department of Conservation, CA, USA. Rao, H.A.B. and Nagraj, C.N. (1960). ?A new method for predicting the natural frequency of foundation-soil systems?. Struct. Engrg. Vol.8(10): 310-316.  Rathje, E.M., Faraj, F., Russell, S. and Bray, J.D. (2004). ?Empirical relationships for frequency content parameters of earthquake ground motions?. Earthquake Spectra: Vol. 20 (1): 119-144. Richard, F.E., J.R. Hall and R.D. Woods. (1970). ?Vibrations of Soils and Foundations?. Englewood Cliffs, NJ, Prentice-Hall Inc. New York.  Richard, F.E. and Whiteman, R.V. (1967). "Comparison of Footing Vibration Tests with Theory." ASCE J. Soil Mech.  Fdn. Engrg., Vol.83(SM6): 143-167. SAP2000. ?Structural analysis program?. Computer and Structures Inc. USA, http://www.csiberkeley.com/sap2000. Accessed between 2009-2013.   Seed, H.B., and Idriss, I.M. (1974). ?Soil-structure interaction of massive embedded structures during earthquakes?. Proceedings of 5th World Conference on Earthquake Engineering, Rome, Italy. SeismoSignal. Earthquake Engineering Software Solution, Italy. http://www.seismosoft.com/. Accessed between 2009-2013.  SeismoSoft. Earthquake Engineering Software Solution, Italy. http://www.seismosoft.com/. Accessed between 2009-2013.  SeismoStruct. Earthquake Engineering Software Solution, Italy. http://www.seismosoft.com/. Accessed between 2009-2013. Simulia. ?ABAQUS: Unified finite element analysis?, USA. http://www.simulai.com. Accessed between 2009-2013. Trifunac, M.D., Hao,T.Y. and Todorovska, M.I. (2001). Response of a 14-story reinforced concrete structure to nine earthquakes: 61 years of observation in the Hollywood Storage 190  Building?. Report CE 01-02, Department of Civil Engineering, University of Southern California, USA.   USGS. (2010). ?National Strong-Motion Project Earthquake Data Sets?. US Geological Survey USA,  http://nsmp.wr.usgs.gov/nsmn_eqdata.html. Accessed between 2010-2012. Veletsos, A.P. and Prasad, A.M. (1989). ?Seismic interaction of structures and soils; stochastic approach?. J. Struct. Engrg., Vol. 115: 935-956. Veletsos, A.P., Prasad, A.M. and Wu, W.H. (1997). ?Transfer functions for rigid rectangular foundations?. J. Earthquake Engrg.  & Struct. Dynamics, Vol. 26: 5-17. Whitman, R.V. (1969). ?Equivalent lumped system for structure founded upon stratum of soil?. Proceedings of 4th World Conference on Earthquake Engineering, Santiago, Chile. Wong, H. L. and Luco, J. E. (1978). ?Tables of Impedance Functions and Input Motions for Rectangular Foundations?. Report CE78-15, Dept. of Civil Engineering, University of Southern California, Los Angeles, California. 191  Appendices Appendix A   ESTIMATION SYSTEM PERIOD OF SOIL-STRUCTURE SYSTEM  A mathematical model of coupled vibration of the soil deposit column and building is developed in this section. The soil deposit is modelled as a pure shear beam fixed at the base rock with a spring?dashpot-mass system appended at the other end of the beam to represent the building. An approximate solution of the system is obtained using the Galerkin method. This method gives an approximate solution in the discrete form to the differential equation of a continuous system (Cook et al., 2001). The fundamental mode of the deposit is used to approximate the shape function in the Galerkin procedure. The model developed gives a frequency equation of the soil-structure system.  A.1 Governing equation of shear beam with appended spring?mass A shear beam with cross-sectional area A and length H representing the  soil deposit is shown in Figure A.1.  The properties of the beam are soil shear modulus, G, and mass density, ?s. Mass and stiffness of the appended SDOF system are mb and Kb, respectively. These are generalized mass and generalized stiffness of a fixed-base building structure for the first mode. The longitudinal coordinate system for both the main system and appended mass?spring starts from the base. The u(z,t) and xH(z,t) are the displacements of the main beam and appended mass, respectively. The displacement of the lumped mass is referenced by the point of its connection to the main beam, which is at the free end of the beam (z = H) in this case.  192   Figure A-1 Model of soil?structure system by shear beam with appending spring?mass  The differential equation of vibration of the shear beam without damping is )},()({),(),( 2222tHutxKztzuGAttzuA Hbs ????????? , (A.1) where As?  is mass per unit length of the beam and GA  is the shear stiffness. In Equation A.1, the first and second terms are related to mass inertia and resisting force by static stiffness, respectively. The right-hand-side term appears as an external force function that accounts for the effect of the SDOF spring?mass appended to the beam. 193   The differential equation of vibration of the mass?spring is )},()({)(22tHutxKdttxdm HbHb ??? . (A.2) The general solution of Equation A.1 can be expressed in the form of tiezUtzu ??? )(),( , (A.3) where )(zU  is the mode shape function of the beam that depends on the space variable only and ?  is the natural frequency of the combined system of beam and SDOF mass?spring.  The solution of Equation A.2 can be expressed as tiH eHXtx ??? )()( . (A.4) The term )(HX of the solution in Equation A.3 is a mode shape function for the mass?spring SDOF system. It does not depend on the coordinate system of the shear beam but is associated with the location where the SDOF is connected with the shear beam.  Introducing Equations A.3 and A.4 into Equation A.2 gives - tibtib eHUHXKeHXm ??? ?? ??? )}()({)( 2 . (A.5) If 2bbb mK ?? , which represents the fixed-base frequency of the building, is introduced into Equation A.5, then the mode shapes of the SDOF spring?mass and shear beam are related by ? ? )(11)( 2 HUHXb???? . (A.6)  194  Introducing Equation  A.6 into Equation A.4 and combining the result with Equation A.1 leads to ? ? ?????? ??????? )()(11)()( 2222 HUHUKzzUdGAzUAbbs ????  and ? ? )(1)()(22222HUmzUAzzUdGAbbs ????? ???? . (A.7)  Equation A.7 is the governing differential equation for a shear beam with an attached SDOF mass?spring. The solution of this equation provides mode displacements along the beam.  The solution to Equation A.7 is obtained using Galerkin?s method. The method requires an approximate function to provide a solution. The Galerkin form of the problem statement would be ? ? 0)()(1)()(022222???????????? dzzUHUmzUAz zUdGAHbbs ????? . (A.8)  When each term of Equation A.8 is integrated over the length of the beam, Galerkin`s solution is obtained in terms of frequency as ? ? bbdpdp mmK 2221 ???? ??? ,  (A.9) Where dzdzdUGAKHdp ? ??????? 02 and dzUAmHdp ??02? . (A.10) 195  Here, )(zU is noted as U  for simplicity. dpK  and dpm are modal stiffness and modal mass of the deposit associated with the mode shape function and are related to each other by the corresponding frequency of the soil deposit, 2dpdpdp mK ?? .  A.2 Frequency equation for the system Equation A.9 can be rearranged to form a quadratic equation on ?2 as 02222224 ?????????? ??? dpbdpbbdpbmm ??????? . (A.11) Solutions of this equation for ?2 give two frequencies of the system. With dpb mmr ? , solutions of the first, ?1, and second, ?2, frequency of the soil?structure system are ? ? ? ?? ? ?????? ??????? 222222221 41121 dpbdpbdpb rr ???????  (A.12-i) and ? ? ? ?? ? ?????? ??????? 222222222 41121 dpbdpbdpb rr ??????? . (A.12-ii)  These two frequencies are regarded as representing the frequency of the two degrees of freedom system, where the beam (soil deposit) and mass?spring (building) are two discrete components of the complete system. Mass and spring properties of the building can be easily obtained using generalized mass and stiffness properties. A procedure to obtain the mass and stiffness of the deposit is explained in Chapter 5. Mass ratio r in the Equation A.12 can be obtained from the mass properties of soil deposit and building.  196  Appendix B   MATLAB CODE FOR SFS-3DOF %%Set the input parameters%%%%%%%%%%%%%%%%%%%%%%%%%%% INPUT ( dt, inp, Vs, roh_s, Wf, H, Zeta_s, myu, Mb, Kb, Zeta_b, roh_c, Af, tf)  SF=(4/pi)^3*4/(2-myu)*(1+(Wf/H)^(3/4)); G= Vs^2 *roh_s; As=SF*Af; Ms = roh_s*As*H/2; Mf= roh_c*Af*tf; Ks= pi()^2 *G*As/(8*H); Kf=((pi^2/4)* (G/(2-myu))*As/H*(1+(Wf/H)^0.75)))/((pi^2/8)*As/H-2/(2-myu)*(1+(Wf/H)^0.75)); VM=[Mb Mf Ms]; Dmp=[Zeta_b zeta_s Zeta_s]; VK=[Kb Kf  Ks] t=0:dt:length(inp)-1*dt; N=3; %%%%%%%%%%%%%%%%%%%% End of setting input parameters%%%% % %Formation of mass and stiffness matrix %%%%%%%%%%%%%%%%%     EM=zeros(N,N); for i=1:N      EM(i,i)=VM(i); end EK=zeros(N,N);   EK(1,1)=VK(1); for i=2:N-1 197        EK(i,i-1)=-VK(i-1); EK(i,i)=VK(i)+VK(i-1); EK(i,i+1)=-VK(i); end        EK(1,2)=-VK(1); EK(N,N-1)=-VK(N-1); EK(N,N)=VK(N-1)+VK(N); %%%%%%%%%%%%%%%% End of mass and stiffness matrix %%%%%%%%% %% Eigen value analysis %%%%%%%%%% [U,W0]=eig(EK,EM); for j=1:N     W(j)=sqrt(W0(j,j));  VW1(j)=j;     for i=1:N       VW2(i,j)=U(i,j);     end     end  %% sort Eigen values from minimum to maximum %%%%%%%%%%%%%%%%% for k=1:N-1     for j=1:N-k       if  W(j) >=W(j+1)         temp=W(j); W(j)=W(j+1);   W(j+1)=temp; temp=VW1(j);     VW1(j)=VW1(j+1); VW1(j+1)=temp;    end   end end   198  %% sort Eigen vectors according to eigen values %% %%%%%%%%%%%%%%% for j=1:N     J1=VW1(j);  for i=1:N      U(i,j)=VW2(i,J1);     end  end  %% matrix of eigen vector multiplied by participation factor %%%%%%%%%%  for j=1:N     Utm1=U(:,j)'*EM*ones(N,1);     Utmu=U(:,j)'*EM*U(:,j); beta=Utm1/Utmu;   U(:,j)=beta*U(:,j);  End  %%% Formation of Damping matrix ( local viscous damping) %%%%%%%%%%% VW1=zeros(N,N);VW2=zeros(N,N);EC=zeros(N,N); for i=2:N      EC(i,1)=EK(i,i)+EK(i-1,i); end for i=1:N         for j=1:N           VW1(i,j)=Dmp(j)*W(i)/W(1);         end  end  for j=1:N 199      s=0; for i=1:N      s=s+VW1(j,i)*VW2(i,j);  end     Dmp(j)=s/EC(j,2); end  EC=EM*U; for i=1:N     VW1(i,i)=2*Dmp(i)*W(i); end    VW2=EC*VW1;VW1=U'*EM*U; for j=1:N    VW1(j,j)=1/VW1(j,j); end    EC=VW2*VW1; for i=1:N      for j=1:N       VW1(j,i)=U(i,j);       end  end  VW2=EC*VW1;  EC=VW2*EM;  %%%%%%%%%%%%% step-by-step integration %%%%%%%%% Bt=1/6; NN=length(inp); 200  A1=(1/2-Bt)*dt^2; A2=Bt*dt^2; A3=1/2*dt; E1=ones(N,1); IEM=EM\eye(N,N); for i=1:N     Dis(i,1)=0;  Vel(i,1)=0; end AM=EM+A3*EC+A2*EK;  for M=2:NN     As=Acc(:,M-1);  Vs=Vel(:,M-1);  Ds=Dis(:,M-1);  AF=-EM*E1*inp(M)-EC*(Vs+A3*As)-...     EK*(Ds+dt*Vs+A1*As);  Acc(:,M)=AM\AF;  Dis(:,M)=Ds+dt*Vs+A1*Acc(:,M);  Vel(:,M)=Vs+A3*(As+Acc(:,M)); end %% Absolute Acceleration %%  for M=1:NN     Acc(:,M)=IEM*(-EC*Vel(:,M)-EK*Dis(:,M)); end %%%%%%End of Computing %%%%%%%%%  

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