Effects of Ground Motion Spatial Variations and Random Site Conditions on Seismic Responses of Bridge Structures

Transcription

1 Effects of Ground Motion Spatial Variations and Random Site Conditions on Seismic Responses of Bridge Structures by Kaiming BI BEng, MEng This thesis is presented for the degree of Doctor of Philosophy of Structural Engineering School of Civil and Resource Engineering May 0

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3 DECLARATION FOR THESIS CONTAINING PUBLISHED WORK AND/OR WORK PREPARED FOR PUBLICATION This thesis contains published wor and/or wor prepared for publication, which has been co-authored. The bibliographical details of the wor and where it appears in the thesis are outlined below. Bi K, Hao H, Ren W. Response of a frame structure on a canyon site to spatially varying ground motions. Structural Engineering and Mechanics 00; 36: -7. Chapter The estimated percentage contribution of the candidate is 50%. Bi K, Hao H, Chouw N. Required separation distance between decs and at abutments of a bridge crossing a canyon site to avoid seismic pounding. Earthquae Engineering and Structural Dynamics 00; 393: Chapter 3 The estimated percentage contribution of the candidate is 60%. Bi K, Hao H, Chouw N. Influence of ground motion spatial variation, site condition and SSI on the required separation distances of bridge structures to avoid seismic pounding. Earthquae Engineering and Structural Dynamics, published online. Chapter 4 The estimated percentage contribution of the candidate is 60%. Bi K, Hao H. Modelling and simulation of spatially varying earthquae ground motions at a canyon site with multiple soil layers. Probabilistic Engineering Mechanics, under review. Chapter 5 The estimated percentage contribution of the candidate is 70%. Bi K, Hao H. Influence of irregular topography and random soil properties on the coherency loss of spatial seismic ground motions. Earthquae Engineering and Structural Dynamics, published online. Chapter 6 The estimated percentage contribution of the candidate is 80%. Bi K, Hao H, Chouw N. 3D FEM analysis of pounding response of bridge structures at a canyon site to spatially varying ground motions. Earthquae Engineering and Structural Dynamics, under review. Chapter 7 The estimated percentage contribution of the candidate is 70%. Kaiming Bi Print Name Signature Date Hong Hao Print Name Signature Date Nawawi Chouw Print Name Signature Date Weixin Ren 0/03/0 Print Name Signature Date

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5 Abstract School of Civil and Resource Engineering Abstract The research carried out in this thesis concentrates on the modelling of spatial variation of seismic ground motions, and its effect on bridge structural responses. This effort brings together various aspects regarding the modelling of seismic ground motion spatial variations caused by incoherence effect, wave passage effect and local site effect, bridge structure modelling with soil-structure interaction SSI effect, and dynamic response modelling of pounding between different components of adjacent bridge structures. Previous studies on structural responses to spatial ground motions usually assumed homogeneous flat site conditions. It is thus reasonable to assume that the ground motion power spectral densities at various locations of the site are the same. The only variations between spatial ground motions are the loss of coherency and time delay. For a structure located on a canyon site or site of varying conditions, local site effect will amplify and filter the incoming waves and thus further alter the ground motion spatial variations. In the first part of this thesis Chapters -4, a stochastic method is adopted and further developed to study the seismic responses of bridge structures located on a canyon site. In this approach, the spatially varying ground motions are modelled in two steps. Firstly, the base roc motions are assumed to have the same intensity and are modelled with a filtered Tajimi- Kanai power spectral density function and an empirical spatial ground motion coherency loss function. Then, power spectral density function of ground motion on surface of the canyon site is derived by considering the site amplification effect based on the onedimensional seismic wave propagation theory. The structural responses are formulated in the frequency domain, and mean pea responses are estimated by the standard random vibration method. The dynamic, quasi-static and total responses of a frame structure Chapter and the minimum separation distances between an abutment and the adjacent bridge dec and between two adjacent bridge decs required in the modular expansion joint MEJ design to preclude pounding during strong ground motion shaing are studied Chapter 3. The influence of SSI is also examined in Chapter 4 by modelling the soil surrounding the pile foundation as frequency-dependent springs and dashpots in the horizontal and rotational directions. i

6 Abstract School of Civil and Resource Engineering A method is proposed to simulate the spatially varying earthquae ground motion time histories at a canyon site with different soil conditions. This method taes into consideration the local site effect on ground motion amplification and spatial variations. The base roc motions are modelled by a filtered Tajimi-Kanai power spectral density function or a stochastic ground motion attenuation model, and the spatial variations of seismic waves on the base roc are depicted by a coherency loss function. The power spectral density functions on the ground surfaces are derived by considering seismic wave propagations through the local site by assuming the base roc motions consisting of outof-plane SH wave and in-plane combined P and SV waves with an incident angle to the site. The spectral representation method is used to simulate the multi-component spatially varying earthquae ground motions. It is proven that the simulated spatial ground motion time histories are compatible with the respective target power spectral density or design response spectrum at each location individually, and the model coherency loss function between any two of them. This method can be used to simulate spatial ground motions on a non-uniform site with explicit consideration of the influences of the specific site conditions. The simulated time histories can be used as inputs to multiple supports of longspan structures on non-uniform sites in engineering practice. Based on the proposed simulation technique, the influences of irregular topography and random soil properties on coherency loss of spatial seismic ground motions are evaluated. In the analysis, random soil properties are assumed to follow normal distributions and are modelled by the one-dimensional random fields in the vertical directions. For each realization of the random soil properties, spatially varying ground motion time histories are generated and the mean coherency loss functions are derived. Numerical studies show that coherency function directly relates to the spectral ratio of transfer functions of the two local sites, and the influence of randomly varying soil properties at a canyon site on coherency functions of spatial surface ground motions cannot be neglected. A detailed 3D finite element analysis of pounding responses between different components of a two-span simply-supported bridge structure on a canyon site to spatially varying ground motions are performed. The multi-component spatially varying ground motions are stochastically simulated as inputs and the numerical studies are carried out by using the transient dynamic finite element code LS-DYNA. Results indicate that the torsional response of bridge structures induces eccentric poundings between the adjacent bridge structures. Traditionally used SDOF model or D finite element model of bridge structure ii

7 School of Civil and Resource Engineering Abstract could not capture the torsional response induced eccentric poundings, therefore might lead to inaccurate pounding response predictions. The detailed 3D finite element model is needed for a more reliable prediction of earthquae-induced pounding responses between adjacent structures. iii

8 Table of Contents School of Civil and Resource Engineering Table of Contents ABSTRACT...I TABLE OF CONTENTS...IV ACKNOWLEDGEMENTS... VIII THESIS ORGANIZATION AND CANDIDATE CONTRIBUTION...IX PUBLICATIONS ARISING FROM THIS THESIS... XII LIST OF FIGURES... XIV LIST OF TABLES... XVIII CHAPTER...- INTRODUCTION BACKGROUND RESEARCH GOALS OUTLINE REFERENCES CHAPTER... - RESPONSE OF A FRAME STRUCTURE ON A CANYON SITE TO SPATIALLY VARYING GROUND MOTIONS INTRODUCTION BRIDGE AND SPATIAL GROUND MOTION MODEL BRIDGE MODEL BASE ROCK MOTION SITE AMPLIFICATION STRUCTURAL RESPONSE EQUATION FORMULATION MAXIMUM RESPONSE CALCULATION NUMERICAL RESULTS AND DISCUSSIONS EFFECT OF SOIL DEPTH EFFECT OF SOIL PROPERTIES EFFECT OF COHERENCY LOSS CONCLUSIONS...- iv

9 School of Civil and Resource Engineering Table of Contents.7 REFERENCES...- CHAPTER REQUIRED SEPARATION DISTANCE BETWEEN DECKS AND AT ABUTMENTS OF A BRIDGE CROSSING A CANYON SITE TO AVOID SEISMIC POUNDING INTRODUCTION BRIDGE MODEL SPATIAL GROUND MOTION MODEL BASE ROCK MOTION SITE AMPLIFICATION STRUCTURAL RESPONSES NUMERICAL RESULTS AND DISCUSSIONS EFFECT OF GROUND MOTION SPATIAL VARIATIONS EFFECT OF THE BRIDGE GIRDER FREQUENCY EFFECT OF THE LOCAL SOIL SITE CONDITIONS CONCLUSIONS APPENDIX APPENDIX A: MEAN PEAK RESPONSE CALCULATION APPENDIX B: CHARACTERISTIC MATRICES REFERENCES CHAPTER INFLUENCE OF GROUND MOTION SPATIAL VARIATION, SITE CONDITION AND SSI ON THE REQUIRED SEPARATION DISTANCES OF BRIDGE STRUCTURES TO AVOID SEISMIC POUNDING INTRODUCTION BRIDGE-SOIL SYSTEM METHOD OF ANALYSIS DYNAMIC SOIL STIFFNESS STRUCTURAL RESPONSE FORMULATION NUMERICAL EXAMPLE INFLUENCE OF SITE EFFECT AND SSI INFLUENCE OF GROUND MOTION SPATIAL VARIATION AND SSI CONCLUSIONS APPENDIX APPENDIX A: ELEMENT FOR [ i ] Z AND [ Z g i] APPENDIX B: PSDS OF THE REQUIRED SEPARATION DISTANCES REFERENCES...4- CHAPTER v

10 Table of Contents School of Civil and Resource Engineering MODELLING AND SIMULATION OF SPATIALLY VARYING EARTHQUAKE GROUND MOTIONS AT A CANYON SITE WITH MULTIPLE SOIL LAYERS INTRODUCTION WAVE PROPAGATION THEORY AND SITE AMPLIFICATION EFFECT GROUND MOTION SIMULATION NUMERICAL EXAMPLES AMPLIFICATION SPECTRA EXAMPLE -PSD COMPATIBLE GROUND MOTION SIMULATION EXAMPLE -RESPONSE SPECTRUM COMPATIBLE GROUND MOTION SIMULATION CONCLUSIONS REFERENCES CHAPTER INFLUENCE OF IRREGULAR TOPOGRAPHY AND RANDOM SOIL PROPERTIES ON COHERENCY LOSS OF SPATIAL SEISMIC GROUND MOTIONS INTRODUCTION THEORETICAL BASIS ESTIMATION OF COHERENCY FUNCTION ONE-DIMENSIONAL WAVE PROPAGATION THEORY GROUND MOTION GENERATION RANDOM FIELD THEORY MONTE-CARLO SIMULATION NUMERICAL EXAMPLE INFLUENCE OF IRREGULAR TOPOGRAPHY INFLUENCE OF RANDOM SOIL PROPERTIES INFLUENCE OF RANDOM VARIATION OF EACH SOIL PARAMETER CONCLUSIONS REFERENCES CHAPTER D FEM ANALYSIS OF POUNDING RESPONSE OF BRIDGE STRUCTURES AT A CANYON SITE TO SPATIALLY VARYING GROUND MOTIONS INTRODUCTION METHOD VALIDATION BRIDGE MODEL SPATIALLY VARYING GROUND MOTIONS NUMERICAL EXAMPLE LONGITUDINAL RESPONSE TRANSVERSE AND VERTICAL RESPONSES TORSIONAL RESPONSE RESULTANT POUNDING FORCE vi

11 School of Civil and Resource Engineering Table of Contents STRESS DISTRIBUTIONS CONCLUSIONS REFERENCES CHAPTER CONCLUDING REMARKS MAIN FINDINGS RECOMMENDATIONS FOR FUTURE WORK vii

12 Acnowledgements School of Civil and Resource Engineering Acnowledgements I would lie to express my deep and sincere gratitude to my supervisor, Winthrop Professor Hong Hao, who supported me persistently during the period of this research. Prof. Hao was always there to listen and give advice, which enabled my research wor to move forward continuously. Many of the ideas in this thesis would not have taen shape without his incisive thining and insightful suggestions. What I learned from him will benefit me greatly in the rest of my life. Many thans go to Associate Professor Nawawi Chouw from the University of Aucland in New Zealand for his invaluable suggestions, critical and insightful reviews of some of the papers involved in this thesis. I would also lie to than Professor Weixin Ren, from Central South University in China, who introduced me to Professor Hao, so that I have the opportunity to pursue my study in UWA. I am indebted to the staff and postgraduate students from School of Civil and Resource Engineering and Centre for Offshore Foundation Systems COFS for their friendship and diverse help during my study in UWA. I would lie to acnowledge the International Postgraduate Research Scholarship IPRS for providing the financial support to me to pursue this study. At last, I wish to express my sincere thans to my parents, my brothers and sisters, for their constant love and inspiration. Without their support, I could not have done it. viii

13 Thesis Organization and Candidate Contribution School of Civil and Resource Engineering Thesis Organization and Candidate Contribution In accordance with the University of Western Australia s regulations regarding Research Higher Degrees, this thesis is presented as a series of papers that have been published, accepted for publication or submitted for publication but not yet accepted. The contributions of the candidate for the papers comprising Chapters ~7 are hereby set forth. Paper This paper is presented in Chapter, first-authored by the candidate, co-authored by Winthrop Professor Hong Hao and Professor Weixin Ren, and published as Bi K, Hao H, Ren W. Response of a frame structure on a canyon site to spatially varying ground motions. Structural Engineering and Mechanics 00; 36: -7. The candidate developed a program to study the combined ground motion spatial variation and local site amplification effect on the seismic responses of a frame structure located on a canyon site. Under the supervision of Winthrop Professor Hong Hao and Professor Weixin Ren, the candidate overviewed relevant literature, carried out parametrical studies, interpreted the results and wrote the paper. Paper This paper is presented in Chapter 3, first-authored by the candidate, co-authored by Winthrop Professor Hong Hao and Associate Professor Nawawi Chouw, and published as Bi K, Hao H, Chouw N. Required separation distance between decs and at abutments of a bridge crossing a canyon site to avoid seismic pounding. Earthquae Engineering and Structural Dynamics 00; 393: The candidate developed a program to study the combined ground motion spatial variation and local site amplification effect on the required separation distances between abutments ix

14 School of Civil and Resource Engineering Thesis Organization and Candidate Contribution and bridge decs and between two adjacent bridge decs of a two-span simply-supported bridge structure crossing a canyon site to avoid seismic pounding. Under the supervision of Winthrop Professor Hong Hao and Associate Professor Nawawi Chouw, the candidate overviewed relevant literature, carried out parametrical studies, interpreted the results and wrote the paper. Paper 3 This paper is presented in Chapter 4, first-authored by the candidate, co-authored by Winthrop Professor Hong Hao and Associate Professor Nawawi Chouw, and has been published as Bi K, Hao H, Chouw N. Influence of ground motion spatial variation, site condition and SSI on the required separation distances of bridge structures to avoid seismic pounding. Earthquae Engineering and Structural Dynamics, published online. This paper is an extension of Paper Chapter 3. The candidate incorporated soilstructure interaction effect SSI into the program developed in Paper. Under the supervision of Winthrop Professor Hong Hao and Associate Professor Nawawi Chouw, the candidate conducted a series of analysis, highlighted SSI effect and local site conditions on the required separation distances to avoid seismic pounding of the bridge structure investigated in Paper, and wrote the paper. Paper 4 This paper is presented in Chapter 5, first-authored by the candidate, co-authored by Winthrop Professor Hong Hao, and has been submitted as Bi K, Hao H. Modelling and simulation of spatially varying earthquae ground motions at a canyon site with multiple soil layers. Probabilistic Engineering Mechanics, under review. Under the supervision of Winthrop Professor Hong Hao, the candidate incorporated local site effect of multiple soil layers into the traditional spatially varying seismic ground motion simulation technique, developed a program to simulate the multi-component spatially varying seismic motions on the ground surface of a canyon site, and wrote the paper. Paper 5 This paper is presented in Chapter 6, first-authored by the candidate, co-authored by Winthrop Professor Hong Hao, and has been published as x

15 School of Civil and Resource Engineering Thesis Organization and Candidate Contribution Bi K, Hao H. Influence of irregular topography and random soil properties on the coherency loss of spatial seismic ground motions. Earthquae Engineering and Structural Dynamics, published online. Based on the program developed in Paper 4 Chapter 5, the candidate studied the influence of irregular topography and random soil properties on the lagged coherency loss function of spatial seismic ground motions. Under the supervision of Winthrop Professor Hong Hao, the candidate overviewed relevant literature, carried out a parametrical study and wrote the paper. Paper 6 This paper is presented in Chapter 7, first-authored by the candidate, co-authored by Winthrop Professor Hong Hao and Associate Professor Nawawi Chouw, and has been submitted as Bi K, Hao H, Chouw N. 3D FEM analysis of pounding response of bridge structures at a canyon site to spatially varying ground motions. Earthquae Engineering and Structural Dynamics, under review. The candidate simulated the multi-component spatially varying ground motions at the supports of a two-span simply-supported bridge structure located at a canyon site based on the program developed in Paper 4 Chapter 5, established the detail 3D finite element model of the bridge, and investigated the pounding responses of the bridge structure by using the transient dynamic finite element code LS-DYNA. Under the supervision of Winthrop Professor Hong Hao and Associate Professor Nawawi Chouw, the candidate overviewed the relevant literature, carried out a parametrical study, highlighted the effect of torsional response induced eccentric poundings, and wrote the paper. I certify that, except where specific reference is made in the text to the wor of others, the contents of this thesis are original and have not been submitted to any other university. Signature: Kaiming Bi May 0. xi

16 Publications Arising From This Thesis School of Civil and Resource Engineering Publications Arising From This Thesis Journal papers. Bi K, Hao H, Ren W. Response of a frame structure on a canyon site to spatially varying ground motions. Structural Engineering and Mechanics 00; 36: -7.. Bi K, Hao H, Chouw N. Required separation distance between decs and at abutments of a bridge crossing a canyon site to avoid seismic pounding. Earthquae Engineering and Structural Dynamics 00; 393: Bi K, Hao H, Chouw N. Influence of ground motion spatial variation, site condition and SSI on the required separation distances of bridge structures to avoid seismic pounding. Earthquae Engineering and Structural Dynamics, published online. 4. Bi K, Hao H. Modelling and simulation of spatially varying earthquae ground motions at a canyon site with multiple soil layers. Probabilistic Engineering Mechanics, under review. 5. Bi K, Hao H. Influence of irregular topography and random soil properties on the coherency loss of spatial seismic ground motions. Earthquae Engineering and Structural Dynamics, published online. 6. Bi K, Hao H, Chouw N. 3D FEM analysis of pounding response of bridge structures at a canyon site to spatially varying ground motions. Earthquae Engineering and Structural Dynamics, under review. 7. Liang JZ, Hao H, Wang Y, Bi K. Design earthquae ground motion prediction for Perth metropolitan area with microtremor measurements for site characterization. Journal of Earthquae Engineering 009; 37: Bai F, Hao H, Bi K, Li H. Seismic response analysis of transmission tower-line system on heterogeneous sites to multi-component spatial ground motions. Advances in Structural Engineering, in print. xii

17 School of Civil and Resource Engineering Publications Arising From This Thesis Conferences papers. Bi K, Hao H, Chouw N. Stochastic analysis of the required separation distance to avoid seismic pounding between adjacent bridge decs. The 4 th World Conference on Earthquae Engineering, Beijing, China, 008; Bi K, Hao H. Seismic response analysis of a bridge frame at a canyons site in Western Australia. The 4 th World Conference on Earthquae Engineering, Beijing, China, 008; Bi K, Hao H. Simulation of spatially varying ground motions with non-uniform intensities and frequency content. Australian Earthquae Engineering Society 008 Conference, Ballarat, Australia, 008; Paper No Liang JZ, Hao H, Wang Y, Bi K. Site characterization evaluation in Perth metropolitan area using microtremor array method. Proceedings of the 0 th International Symposium on Structural Engineering for Young Experts, Changsha, China, 008; Bi K, Hao H, Chouw N. Dynamic SSI effect on the required separation distances of bridge structures to avoid seismic pounding. Australian Earthquae Engineering Society 009 Conference, Newcastle, Australia, 009; Paper No Bi K, Hao H. Analysis of influence of an irregular site with uncertain soil properties on spatial seismic ground motion coherency. Australian Earthquae Engineering Society 009 Conference, Newcastle, Australia, 009; Paper No Hao H, Bi K, Chouw N. Combined ground motion spatial variation and local site amplification effect on bridge structure responses. 6 th International Conference on Urban Earthquae Engineering, Toyo, Japan, 009; Bi K, Hao H. Pounding response of adjacent bridge structures on a canyon site to spatially varying ground motions. Australian Earthquae Engineering Society 00 Conference, Perth, Australia, 00; Paper No. 9. Bi K, Hao H, Zhang C. Analysis of coupled axial-torsional pounding response of adjacent bridge structures. The th International Symposium on Structural Engineering, Guangzhou, China, 00; xiii

18 List of Figures School of Civil and Resource Engineering List of Figures Figure -. Schematic view of a bridge frame crossing a canyon site Figure -. Filtered ground motion power spectral density function on the base roc Figure -3. Different coherency loss functions...- Figure -4. Site transfer functions for different soil depths...-4 Figure -5. Power spectral densities of ground motions on site of different depths...-4 Figure -6. Phase difference caused by seismic wave propagation...-4 Figure -7. Normalized dynamic responses for different soil depths...-6 Figure -8. Normalized total responses for different soil depths...-6 Figure -9. Dynamic, quasi-static and total responses with h A = 0m and h B = 30m...-7 Figure -0. Soil site transfer function for different soil properties...-7 Figure -. Power spectral densities of ground motions at sites...-8 Figure -. Phase difference owing to seismic wave propagation...-8 Figure -3. Normalized dynamic responses for different soil properties...-8 Figure -4. Normalized total responses for different soil properties...-9 Figure -5. Dynamic, quasi-static and total responses medium soil at support B...-9 Figure -6. Normalized dynamic responses for different coherency losses...- Figure -7. Normalized total responses for different coherency losses...- Figure 3-. a Schematic view of a bridge crossing a canyon site; b structural model Figure 3-. Filtered ground motion power spectral density function at base roc Figure 3-3. Effect of ground motion spatial variation on the required separation distance a Δ, b Δ, c 3 Δ Figure 3-4. Left span frequency response function with respect to the frequency ratios.3-6 Figure 3-5. Effect of vibration frequency on the required separation distance Figure 3-6. Effect of soil depth on the required separation distance Figure 3-7. Effect of soil properties on the required separation distance...3- Figure 3-8. Ground motion power spectral density functions with...3- xiv

19 List of Figures School of Civil and Resource Engineering Figure 4-. a Schematic view of a girder bridge crossing a canyon site Figure 4-. Frequency-dependent dynamic stiffness and damping coefficients of the pile group ab horizontal direction and cd rotational direction Figure 4-3. Influence of site effect and SSI on the required separation distances Figure 4-4. Site effect on ground motion spatial variations: a transfer function, Figure 4-5. Contribution of SSI to the required separation distances with different soil conditions f =. 0 Hz a Δ 3, b Δ and c Δ Figure 4-6. Contribution of SSI to the required separation distances with different soil conditions f =. 0 Hz a Δ 3, b Δ and c Δ Figure 4-7. Influence of ground motion characteristics and SSI on the required separation distances f =. 0 Hz a Δ 3, b Δ and c Δ Figure 4-8. Contribution of SSI to the required separation distances with different coherency loss functions f =. 0 Hz a Δ 3, b Δ and c Δ Figure 4-9. Contribution of SSI to the required separation distances with different coherency loss functions f =. 0 Hz a Δ 3, b Δ and c Δ Figure 5-. A canyon site with multiple soil layers not to scale Figure 5-. Amplification spectra of site 3, a horizontal out-of-plane motion; Figure 5-3. Generated base roc motions in the horizontal directions Figure 5-4. Comparison of power spectral density of the generated base roc acceleration with model power spectral density Figure 5-5. Comparison of coherency loss between the generated base roc accelerations with model coherency loss function Figure 5-6. Generated horizontal out-of-plane motions on ground surface Figure 5-7. Comparison of power spectral density of the generated horizontal out-of-plane acceleration on ground surface with the respective theoretical power spectral density Figure 5-8. Comparison of the coherency loss functions between base roc motions Figure 5-9. Generated horizontal in-plane motions on ground surface Figure 5-0. Comparison of power spectral density of the generated horizontal in-plane acceleration on ground surface with the respective theoretical power spectral density Figure 5-. Generated vertical in-plane motions on ground surface Figure 5-. Comparison of power spectral density of the generated vertical in-plane acceleration on ground surface with the respective theoretical power spectral density Figure 5-3. Generated time histories according to the specified design response spectra..5-3 xv

20 School of Civil and Resource Engineering List of Figures Figure 5-4. Comparison of the generated acceleration and the target response spectra.5-4 Figure 5-5. Comparison of coherency loss between the generated time histories with the model coherency loss function Figure 6-. Schematic view of a layered canyon site Figure 6-. A four-layer canyon site with deterministic soil properties not to scale...6- Figure 6-3. Simulated acceleration time histories Figure 6-4. Mean values and standard deviations of the lagged coherency of the horizontal out-of-plane motion at 0.,.0, 5.0 and 9.0Hz Figure 6-5. Comparison of the mean lagged coherency on the base roc from 600 simulations with the target model Figure 6-6. Comparison of the mean lagged coherency between the surface motions j, with that of the incident motion on the base roc Figure 6-7. Standard deviations of the lagged coherency on the ground surface Figure 6-8. Modulus of the site amplification spectral ratio of two local sites Figure 6-9. Amplitudes of the site amplification spectra of two local sites Figure 6-0. Influence of uncertain soil properties on Figure 6-. Influence of uncertain soil properties on the Figure 6-. Influence of uncertain soil properties on the Figure 6-3. Influence of each random soil property on the...6- Figure 6-4. Influence of each random soil property on the...6- Figure 6-5. Influence of each random soil property on the...6- Figure 7-. A typical pounding damage between bridge decs in Chi-Chi earthquae Figure 7-. Different models not to scale: a lumped mass model from []; Figure 7-3. Structural responses based on different models: a relative displacement and7-8 Figure 7-4. a Elevation view of the bridge, b Cross-section of the bridge girder, Figure 7-5. Finite element mesh of the bridge and the nodal points for response recordings...7- Figure 7-6. First four vibration frequencies and mode shapes of the bridge...7- Figure 7-7. Simulated acceleration time histories with soft soil condition and intermediately correlated coherency loss Figure 7-8. Simulated displacement time histories with soft soil condition and intermediately correlated coherency loss Figure 7-9. Comparison of PSDs between the generated horizontal in-plane motions on ground surface with the respective theoretical model value xvi

21 School of Civil and Resource Engineering List of Figures Figure 7-0. Multi-component spatially varying inputs at different supports of the bridge.7-6 Figure 7-. Influence of pounding effect on the longitudinal displacement response Figure 7-. Influence of soil conditions on the longitudinal displacement response Figure 7-3. Influence of coherency loss on the longitudinal displacement response Figure 7-4. Influence of pounding effect on the transverse displacement response Figure 7-5. Influence of soil conditions on the transverse displacement response Figure 7-6. Influence of coherency loss on the transverse displacement response Figure 7-7. Influence of pounding effect on the vertical displacement response Figure 7-8. Influence of soil conditions on the vertical displacement response Figure 7-9. Influence of coherency loss on the vertical displacement response Figure 7-0. Longitudinal displacements of different nodes to case ground motion Figure 7-. Influence of soil conditions on the resultant pounding forces Figure 7-. Influence of coherency loss on the resultant pounding forces Figure 7-3. Stress distributions in the longitudinal direction at left gap of different cases at the time when pea resultant pounding force occur a Case at t=6.7s, b Case 3 at t=7.63s, c Case 4 at t=7.96s and d Case 5 at t=8.04s unit: Pa xvii

22 List of Tables School of Civil and Resource Engineering List of Tables Table -. Parameters for coherency loss functions... - Table -. Parameters of base roc and different types of soil... - Table 3-. Parameters for coherency loss functions Table 3-. Parameters for local site conditions Table 4-. Parameters for local site conditions Table 5-. First two vibration frequencies of the sites Table 7-. Parameters for local site conditions Table 7-. Different cases studied Table 7-3. Mean pea displacements in the longitudinal direction m Table 7-4. Mean pea displacements in the transverse direction m Table 7-5. Mean pea displacements in the vertical direction m Table 7-6. Mean pea rotational angle degree xviii

23 Chapter School of Civil and Resource Engineering Chapter Introduction. Bacground The term spatial variation of seismic ground motions denotes the differences in the amplitude and phase of seismic motions recorded over extended areas. The spatial variation of seismic ground motions can result from the relative surface fault-motion for sites located on either side of a causative fault, solid liquefaction, landslides, and from the general transmission of the waves from the source through the different earth strata to the ground surface []. This thesis concentrates on the latter cause for the spatial variation of surface ground motions. The spatial variation of seismic ground motions has an important effect on the response of large dimensional structures, such as pipelines, dams and bridges. Because these structures extended over long distances parallel to the ground, their supports undergo different motions during an earthquae. Since 960 s, pioneering studies analyzed the influence of the spatial variation of the motions on the above-ground and buried structures. At that time, the different motions at the structures supports were attributed to the wave passage effect, i.e., it was considered that the ground motions propagate with a constant velocity on the ground surface without any change in their shape. The spatial variation of the motions was then described by the deterministic time delay required for the wave forms to reach the further-away supports of the structures. In these early studies, it was recognized that wave passage effect influence the responses of large dimensional structures significantly. After the installation of the dense seismography arrays in the late 970 s to early 980 s, the modelling of spatial variation of the seismic ground motions and its effect on the responses of various structural systems attracted extensive research interest. The array, which has provided an abundance of data for small and large magnitude events that have been -

24 School of Civil and Resource Engineering Chapter extensively studied by engineers and seismologists, is the SMART- array, located in Lotung, Taiwan. The spatial variability studies based on these array data provided valuable information on the physical causes underlying the variations over extended areas and the means for its modelling. It is generally recognized that four distinct phenomena give rise to the spatial variability of earthquae-induced ground motions []: incoherence effect due to scattering in the heterogeneous medium of the ground, as well as due to the superpositioning of waves arriving from an extended source; wave passage effect results from the different arrival times of waves at separation stations; 3 local site effect owing to the spatially varying local soil profiles and the manner in which they influence the amplitude and frequency content of the base roc motion underneath each station as it propagates upward; and 4 attenuation effect results from gradual decay of wave amplitudes with distance due to geometric spreading and energy dissipation in the ground media. For most of the engineering structures, ground motion attenuation over the distance comparable to the dimension of the structure is usually not significant []. This study thus concentrates on the influence of the first three factors on the ground motion spatial variation and bridge structural responses. These dense arrays usually located on the flat-lying alluvial sites, and the recorded ground motions were usually regarded as homogeneous, stationary and ergodic random field. The stochastic characteristics of the spatially varying ground motions can be described by the auto-power spectral density function, cross-power spectral density function and coherency loss function. The auto-power spectral densities of the motions are estimated from the analysis of the data recorded at each station and are commonly referred as point estimates of the motions. Once the power spectra of the motions at the stations of interest have been evaluated, a parametric form is fitted to the estimates, generally through a regression scheme. The most commonly used parametric forms of the auto-power spectral density function are the Tajimi-Kanai power spectrum model [3]. However, this model is inadequate to describe the ground displacement, as it yields infinite power for the displacement as the frequency approaches zero. To correct this, Clough-Penzien suggested introducing a second filter to modify it, which is nown as the filtered Tajimi-Kanai Power spectrum model [4]. Many stochastic ground motion models [5-7] have also been proposed by considering the rupture mechanism of the fault and the path effect for transmission of waves through the media from the fault to the ground surface. The joint characteristics of the time histories at two discrete locations on the ground surface can be depicted by the cross-power spectral density function and coherency loss function. By processing the recorded ground motions at these dense arrays, many empirical [8-] and semi-empirical -

25 School of Civil and Resource Engineering Chapter [3-4] spatial ground motion coherency loss function models have been proposed. These coherency functions usually consist of two parts, the modulus or called lagged coherency, which measures the similarity of the seismic motions between the two stations, and the phase, which describes the wave passage effect. It is generally found that the lagged coherency decreases smoothly as a function of station separation and wave frequency. These proposed ground motion spatial variation models can be applied directly to the stochastic analysis of the linear elastic responses of relatively simple structural models. Previous stochastic studies of ground motion spatial variation effects on the structural responses include the analysis of a simply-supported beam [5], continuous beams [6, 7], an arch with multiple horizontal input [8], an arch with multiple simultaneous horizontal and vertical excitations [9], a symmetric building structure [0], an asymmetric building structure [], and a cable-stayed bridge []. It should be noted that all these studies were based on the ground motion spatial variation models by analyzing the data recorded from the relatively flat-lying sites, the influence of local site effect was not considered. In reality, seismic waves will be amplified and filtered when propagating through a local soil site. The amplifications occur at various vibration modes of the site. Therefore, the energy of surface motions will concentrate at a few frequencies. The power spectral density function of the surface motion then may have multiple peas. These phenomena are not considered in these traditional models. The combined influences of ground motion spatial variation and local site effect on the structural responses needs to be studied. Seismic ground motion spatial variations may result in pounding or even collapse of adjacent bridge decs owing to the large out-of-phase responses. In fact, poundings between an abutment and bridge dec or between two adjacent bridge decs were observed in almost all the major earthquaes [3-7]. Many methods were adopted to reduce the negative effect of pounding. The most direct way to avoid pounding is to provide adequate separation distance between adjacent structures. For bridge structures with conventional expansion joints, a complete avoidance of pounding between bridge decs during strong earthquaes is often impossible, since the separation gap of an expansion joint is usually a few centimetres to ensure a smooth traffic flow. Recently, a modular expansion joint MEJ system has been developed, and used in some new bridges. The system allows a large relative movement between the bridge girders without comprising the bridge s serviceability and functionality. Using a MEJ, it is possible to mae the gap sufficiently large to cope with the expected closing girder movement, and consequently completely preclude pounding between adjacent girders. However, up to now studies of the suitability of such a -3

26 School of Civil and Resource Engineering Chapter system for mitigating adjacent bridge girder pounding responses and its influence on other bridge response quantities under earthquae loading are limited. Chouw and Hao too two independent bridge frames as an example, discussed the influences of SSI and non-uniform ground motions on the separation distance between two adjoined girders connected by a MEJ [8] and then introduced a new design philosophy for a MEJ [9]. In these studies the ground motion spatial variations and soil-structure interaction SSI were included, the influence of local site effect, however, was neglected. In the first part of this thesis Chapters -4, the combined influences of ground motion spatial variation and local site effect on a frame structure Chapter and on a two-span simply-supported bridge Chapters 3-4 structure are extensively studied based on a stochastic method. The abutment and the adjacent bridge dec and/or the two adjacent bridge decs of these structures are connected by a MEJ. The bridge structure is simplified as a multi-degree-of-freedom MDOF system. The structural responses are stochastically formulated in the frequency domain and the mean pea responses are calculated. In particular, Chapter investigates the dynamic, quasi-static and total responses of the frame structure to various cases of spatially varying ground motions. Chapters 3 and 4 study the required separation distances that MEJs must provide to avoid seismic pounding during strong earthquaes, with Chapter 3 presenting the influence of ground motion spatial variation and local site effects and Chapter 4 highlighting the SSI effect. As mentioned above, the stochastic analysis of the structural responses is usually applied to relatively simple structural models and for linear response of the structures owing to its complexity. For complex structural systems and for nonlinear seismic response analysis, only the deterministic solution can be evaluated with sufficient accuracy. In this case, the generation of artificial seismic ground motions is required. An extensive list of publications addressing the topic of simulations of random processes and fields has appeared in the literature [, 30-3]. Most these studies [, 30-3] assumed the power spectral densities for various locations are the same, the amplification and filtering effect of local site effect was neglected. The only study considered different power spectral densities of different locations was reported by Deodatis [3]. This method is based on a spectral representation algorithm to generate sample functions of a non-stationary, multivariate stochastic process with evolutionary power spectrum. The considered varying spectral densities are filtered white noise functions with different central frequency and damping ratio. This method can only approximately represent local site effect on ground motions, since local soil conditions will amplify and filter the incoming waves at various vibration modes of the site as -4

27 School of Civil and Resource Engineering Chapter mentioned above. This phenomenon, however, cannot be considered by this method [3] since only one pea corresponding to the fundamental vibration mode of the site is considered. Moreover, trying to establish an analytical expression for a realistic ground motion evolutionary power spectrum related to the local site conditions is quite difficult since very limited information is available on the spectral characteristics of propagating seismic waves [33]. To incorporate local site effect into the simulation technique is a quite challenging problem in engineering practice. Chapter 5 presents a method to model and simulate spatially varying earthquae ground motion time histories at sites with nonuniform conditions. This approach directly relates site amplification effect with local soil conditions, and can capture the multiple vibration modes of local site, is believed more realistically simulating the multi-component spatially varying motions on surface of a canyon site. Contrasting to the observations on the flat-lying sites, some researchers [34-35] investigated the lagged coherency loss function between the sites with different conditions, they found that the lagged coherency does not show a strong dependence on station separation distance and wave frequency, and the incoherency is generally higher than that on the flatlying sites. These observations suggest that the spatial coherency function measured on flat-lying sedimentary sites may not provide a good description of spatial ground motion coherencies on sites with irregular topography. However, at the present, only very limited recorded spatial ground motion data on sites of different conditions are available. They are not sufficient to determine the general spatial incoherence characteristics of ground motions and derive empirical relations to model spatial ground motion variations at a site with varying site conditions. Moreover, all the previous studies on coherency loss functions assumed the site characteristics are fully deterministic and homogeneous. However, in reality, there always exist spatial variations of soil properties and uncertainties in defining the properties of soils. This results from the natural heterogeneity or variability of soils, the limited availability of information about internal conditions and sometimes the measurement errors. These uncertainties associated with system parameters are also liely to have influence on the lagged coherency loss function [36-37]. Theoretical or analytical analysis in this field is also limited and is in demand. Chapter 6 evaluates the influences of local site irregular topography and random soil properties on the coherency function between spatial surface motions based on the approach proposed in Chapter 5. For bridge structures with conventional expansion joints, a complete avoidance of pounding between bridge decs during strong earthquaes is often impossible as discussed -5

28 School of Civil and Resource Engineering Chapter above. Pounding is an extreme complex phenomenon involving damage due to plastic deformation at contact points, local cracing or crushing, fracturing due to impact, and friction, etc. To simplify the analysis, many researchers modelled a bridge girder as a lumped mass [38-43], some other researchers modelled the bridge girders as beam-column elements [44-45]. Based on theses simplified models, only point to point pounding in D, usually in the axial direction of the structures, can be considered. In reality, pounding could occur along the entire surfaces of the adjacent structures. Moreover, it was observed from previous earthquaes that most poundings actually occurred at corners of adjacent bridge decs. This is because torsional responses of the adjacent decs induced by spatially varying transverse ground motions at multiple bridge supports resulted in eccentric poundings. To more realistically model the pounding phenomenon between adjacent bridge structures, a detailed 3D finite element analysis is necessary. Moreover, ground motion spatial variation, besides bridge structural vibration properties, is a source of pounding responses in strong earthquaes. Owing to the difficulty in modelling ground motion spatial variation, many studies assumed uniform excitations [38, 40-4] or assumed variation was caused by wave passage effect only [39, 44], only a few studies considered combined wave passage effect and coherency loss effect in analyzing relative displacement responses of adjacent bridge structures [4-43, 45]. Study of the combined influences of ground motion spatial variation and local site effects on earthquae-induced pounding responses of adjacent bridge structures have not been reported. Chapter 7 investigates the pounding responses between the abutment and the adjacent bridge dec and between two adjacent bridge decs of a two-span simply-supported bridge located on a canyon site based on a detailed 3D finite element model. The influences of local soil conditions and ground motion spatial variations on the pounding responses are investigated in detail. The influence of torsional response induced eccentric pounding is highlighted.. Research goals This study was undertaen with the aims of:. Investigating the combined influences of ground motion spatial variation and local site effect on the responses of a frame structure.. A comprehensive study of ground motion spatial variation, local site effect and SSI on the required separation distances that MEJs must provide to avoid seismic pounding. 3. Proposing a method to model and simulate spatially varying earthquae ground motions on a canyon site with different soil conditions. -6

29 School of Civil and Resource Engineering Chapter 4. Evaluating the influence of irregular topography and random soil properties on coherency loss function of spatial seismic ground motions. 5. Studying torsional response induced eccentric poundings between adjacent bridge structures during strong earthquaes..3 Outline This thesis comprises eight chapters. The seven chapters following this introductory chapter are arranged as follows: Chapters ~4 formulate the structural responses in frequency domain based on the stochastic method. In particular, Chapter investigates the combined ground motion spatial variation and local site effect on the response of a frame structure located on a canyon site. Chapter 3 and 4 study the required separation distances MEJs must provide to avoid seismic pounding during strong earthquaes, with Chapter 3 presenting the influence of ground motion spatial variation and local site effects and Chapter 4 highlighting the SSI effect. Chapters 5~7 study ground motion spatial variations and structural responses in time domain. Chapter 5 proposes a method to simulate spatially varying ground motions of a site with varying soil conditions. Chapter 6 investigates the influence of local site effect and random soil properties on coherency loss of spatial seismic ground motions. Chapter 7 studies the pounding responses of a two-span simply-supported bridge structure located on a canyon site based on a detailed 3D FE model. Finally, Chapter 8 summarizes the main outcomes of this research, along with suggestions for future studies..4 References. Zerva A, Zervas V. Spatial variation of seismic ground motions: an overview. Applied Mechanics Reviews 00; 563: Der Kiureghian A. A coherency model for spatially varying ground motions. Earthquae Engineering and Structural Dynamics 996; 5: Tajimi H. A statistical method of determining the maximum response of a building structure during an earthquae. Proc. of nd World Conference on Earthquae Engineering, Toyo, Japan, 960;

30 School of Civil and Resource Engineering Chapter 4. Clough RW, Penzien J. Dynamics of Structures. New Yor: McGraw Hill; 993. Joyner WB, Boore DM. Measurement, characterization and prediction of strong ground motion. Earthquae Engineering and Structure Dynamics II-Recent Advances in Ground Motion Evaluation Proc GSP 0, Par City, Utah, 988; Joyner WB, Boore DM. Measurement, characterization and prediction of strong ground motion. Earthquae Engineering and Structure Dynamics II-Recent Advances in Ground Motion Evaluation Proc GSP 0, Par City, Utah, 988; Atinson GM, Boore DM. Evaluation of models for earthquae source spectra in Eastern North America. Bulletin of the Seismological Society of America 998; 884: Hao H, Gaull BA. Estimation of strong seismic ground motion for engineering use in Perth Western Australia. Soil Dynamics and Earthquae Engineering 009; 95: Loh CH. Analysis of the spatial variation of seismic waves and ground movement from SMART- data. Earthquae Engineering and Structural Dynamics 985; 35: Harichandran RS, Vanmarce EH. Stochastic variation of earthquae ground motion in space and time. Journal of Engineering Mechanics 986; : Loh CH, Yeh YT. Spatial variation and stochastic modelling of seismic differential ground movement. Earthquae Engineering and Structural Dynamics 988; 64: Hao H, Oliveira CS, Penzien J. Multiple-station ground motion processing and simulation based on SMART- array data. Nuclear Engineering and Design 989; 3: Harichandran RS. Estimating the spatial variation of earthquae ground motion from dense array recordings. Structural Safety 99; 0: Luco JE, Wong HL. Response of a rigid foundation to a spatially random ground motion. Earthquae Engineering and Structural Dynamics 986; 46: Somerville PG, McLaren JP, Saiia CK, Helmberger DV. Site-specific estimation of spatial incoherence of strong ground motion. Earthquae Engineering and Structural Dynamics II-Recent Advances in Ground Motion Evaluation, ASCE Geotechnical Special Publication No. 0, 988; Harichandran RS, Wang W. Response of simple beam to spatially varying earthquae excitation. Journal of Engineering Mechanics 988; 49:

31 School of Civil and Resource Engineering Chapter 6. Harichandran RS, Wang W. Response of indeterminate two span beam to spatially varying seismic excitation. Earthquae Engineering and Structural Dynamics 990; 9: Zerva A. Response of multi-span beams to spatially incoherent seismic ground motion. Earthquae Engineering and Structural Dynamics 990; 9: Hao H. Arch responses to correlated multiple excitations. Earthquae Engineering and Structural Dynamics 993; 5: Hao H. Ground-motion spatial variation effects on circular arch responses. Journal of Engineering Mechanics 994; 0: Hao H, Duan XN. Multiple excitation effects on response of symmetric buildings. Engineering Structures 996; 89: Hao H, Duan XN. Seismic response of asymmetric structures to multiple ground motions. Journal of Structural Engineering 995; : Soylu K, Dumanoglu AA. Spatial variability effects of ground motions on cablestayed bridges. Soil Dynamics and Earthquae Engineering 004; 43: Hall FJ, editor. Northridge earthquae, January 7, 994. Earthquae Engineering Research Institute, Preliminary reconnaissance report, EERI-94-0; Kawashima K, Unjoh S. Impact of Hanshin/Awaji earthquae on seismic design and seismic strengthening of highway bridges. Structural Engineering/Earthquae Engineering JSCE 996; 3: Earthquae Engineering Research Institute. Chi-Chi, Taiwan, Earthquae Reconnaissance Report. Report No.0-0, EERI, Oaland, California Elnashai AS, Kim SJ, Yun GJ, Sidarta D. The Yogyaarta earthquae in May 7, 006. Mid-America Earthquae Centre. Report No. 07-0, 57, Lin CJ, Hung H, Liu Y, Chai J. Reconnaissance report of 05 China Wenchuan earthquae on bridges. The 4 th World Conference on Earthquae Engineering. Beijing, China, 008; S Chouw N, Hao H. Significance of SSI and non-uniform near-fault ground motions in bridge response II: Effect on response with modular expansion joint. Engineering Structures 008; 30: Chouw N, Hao H. Seismic design of bridge structures with allowance for large relative girder movements to avoid pounding. New Zealand Society for Earthquae Engineering Conference. Wairaei, New Zealand 008; Paper No: Shinozua M. Monte Carlo solution of structural dynamics. Computers & Structures 97; :

32 School of Civil and Resource Engineering Chapter 3. Conte JP, Pister KS, Mahin SA. Nonstationary ARMA modelling of seismic ground motions. Soil Dynamics and Earthquae Engineering 99; : Deodatis G. Non-stationary stochastic vector processes: seismic ground motion applications. Probabilistic Engineering Mechanics 996; 3: Shinozua M, Deodatis G. Stochastic process models for earthquae ground motion. Probabilistic Engineering Mechanics 988; 33: Somerville PG, McLaren JP, Sen MK, Helmberger DV. The influence of site conditions on the spatial incoherence of ground motions. Structural Safety 99; 0: Liao S, Zerva A, Stephenson WR. Seismic spatial coherency at a site with irregular subsurface topography. Proceedings of Sessions of Geo-Denver, Geotechnical Special Publication No. 70, 007; Zerva A, Harada T. Effect of surface layer stochasticity on seismic ground motion coherence and strain estimations. Soil Dynamics and Earthquae Engineering 997; 6: Liao S, Li J. A stochastic approach to site-response component in seismic ground motion coherency model. Soil Dynamics and Earthquae Engineering 00; : Malhotra PK. Dynamics of seismic pounding at expansion joints of concrete bridges. Journal of Engineering Mechanics 998; 47: Janowsi R, Wilde K, Fujino Y. Pounding of superstructure segments in isolated elevated bridge during earthquaes. Earthquae Engineering and Structural Dynamics 998; 7: Ruangrassamee A, Kawashima K. Relative displacement response spectra with pounding effect. Earthquae Engineering and Structural Dynamics 00; 300: DesRoches R, Muthuumar S. Effect of pounding and restrainers on seismic response of multi-frame bridges. Journal of Structural Engineering ASCE 00; 87: Chouw N, Hao H. Study of SSI and non-uniform ground motion effects on pounding between bridge girders. Soil Dynamics and Earthquae Engineering 005; 50: Chouw N, Hao H. Significance of SSI and non-uniform near-fault ground motions in bridge response I: Effect on response with conventional expansion joint. Engineering Structures 008; 30:

33 School of Civil and Resource Engineering Chapter 44. Janowsi R, Wilde K, Fujino Y. Reduction of pounding effects in elevated bridges during earthquaes. Earthquae Engineering and Structural Dynamics 000; 9: Chouw N, Hao H, Su H. Multi-sided pounding response of bridge structures with non-linear bearings to spatially varying ground excitation. Advances in Structural Engineering 006; 9:

34 Chapter School of Civil and Resource Engineering -

35 Chapter School of Civil and Resource Engineering Chapter Response of a Frame Structure on a Canyon Site to Spatially Varying Ground Motions By: Kaiming Bi, Hong Hao and Weixin Ren Abstract: This paper studies the effects of spatially varying ground motions on the responses of a bridge frame located on a canyon site. Compared to the spatial ground motions on a uniform flat site, which is the usual assumptions in the analysis of spatial ground motion variation effects on structures, the spatial ground motions at different locations on surface of a canyon site have different intensities owing to local site amplifications, besides the loss of coherency and phase difference. In the proposed approach, the spatial ground motions are modelled in two steps. Firstly, the base roc motions are assumed to have the same intensity and are modelled with a filtered Tajimi- Kanai power spectral density function and an empirical spatial ground motion coherency loss function. Then, power spectral density function of ground motion on surface of the canyon site is derived by considering the site amplification effect based on the one dimensional seismic wave propagation theory. Dynamic, quasi-static and total responses of the model structure to various cases of spatially varying ground motions are estimated. For comparison, responses to uniform ground motion, to spatial ground motions without considering local site effects, to spatial ground motions without considering coherency loss or phase shift are also calculated. Discussions on the ground motion spatial variation and local soil site amplification effects on structural responses are made. In particular, the effects of neglecting the site amplifications in the analysis as adopted in most studies of spatial ground motion effect on structural responses are highlighted. Keywords: site amplification effect; ground motion spatial variation; dynamic responses; quasi-static responses; total responses. -

36 Chapter. Introduction School of Civil and Resource Engineering Earthquae ground motions at multiple supports of large dimensional structures inevitably vary owing to seismic wave propagation effects. Many researchers have investigated seismic ground motion spatial variations. Most of these studies are based on processing the recorded ground motions at dense seismographic arrays, such as the SMART- array. Many empirical spatial ground motion coherency loss functions have been derived [-5]. In all those studies the site under consideration is assumed to be uniform and homogeneous. Therefore the ground motion power spectral densities at various locations of the site under consideration are assumed to be the same. In other words, the only variations in spatial ground motions are loss of coherency and a phase shift owing to seismic wave propagation. However, this assumption will lead to inaccurate ground motion representation when a site has varying conditions such as a canyon site as shown in Figure -. At a canyon site, the spatial ground motions at base roc can still be assumed to have the same power spectral density, but on ground surface at points A and B the ground motion power spectral densities will be very different owing to seismic wave propagation through different wave paths that cause different site amplifications. Uniform ground motion power spectral density assumption in such a situation may lead to erroneous estimation of structural responses. Some researchers have tried to model the effect of local site conditions on earthquae ground motion spatial variations. Der Kiureghian et al. [6] proposed a transfer function that implicitly modelled the site effect on seismic wave propagation. In the model, the ground motion power spectral density function was represented by a site-dependent transfer function and a white noise spectrum. Typical site-dependent parameters, i.e., the central frequency and damping ratio for three generic site conditions, namely, firm, medium and soft site were proposed. The advantage of this model is that it is straightforward to use. The drawbac is it can only approximately represent the local site effects on ground motions. For example, it is well nown that seismic wave will be amplified and filtered when propagating through a layered soil site. The amplifications occur at various vibration modes of the site. Therefore, the energy of surface motions will concentrate at a few frequencies. The power spectral density function of the surface motion then may have multiple peas. This phenomenon, however, cannot be considered in Der Kiureghian s model since only one pea corresponding to the fundamental vibration mode of the site can be involved. In a recent study [7], derivations of earthquae ground motion spatial variation on a site with uneven surface and different geological properties were presented. -

37 School of Civil and Resource Engineering Chapter In the latter study, spatial base roc motion was modelled by a Tajimi-Kanai power spectral density function [8] together with an empirical coherency loss function [4]. Power spectral density functions of the surface motions were derived based on the one dimensional seismic wave propagation theory. Compared to the model by Der Kiureghian et al. [6], the latter study by Hao and Chouw [7] modelled the base roc motion by the Tajimi-Kanai power spectral density function instead of a white noise, and the seismic wave propagation and specific site amplification effects were explicitly represented in terms of the site conditions such as the soil depth and properties. The multiple vibration modes of local site can be easily considered. Therefore the latter model gives more realistic prediction of local site effects on seismic ground motions besides explicitly relating the site conditions to ground motion model. Previous studies of ground motion spatial variation effects on structural responses include stochastic response analysis of a simply supported beam [9], continuous beams [0, ], an arch with multiple horizontal input [], an arch with multiple simultaneous horizontal and vertical excitations [3], a symmetric building structure [4], an asymmetric building structure [5], and a cable-stayed bridge [6]. Most of these studies assumed linear elastic responses. Many researchers have also performed time history analysis of structural responses to spatially varying ground motions. In these studies, both linear elastic, nonlinear inelastic responses, pounding responses, soil-structure interaction effects were considered. The spatial ground motion time histories were obtained either by considering the wave passage effect only [7], or stochastically simulated to be compatible to a selected empirical coherency loss function [8-]. In most of these studies, the site was assumed to be homogeneous and flat, local site effect was not considered. Using the model developed by Der Kiureghian et al. [6], Zembaty and Rutenburg [3] derived the displacement and shear force response spectra with consideration of ground motion spatial variation and site effects. They concluded that site effects modified the overall behaviour of the multi-supported structure significantly. Dumanogluid and Soylu [4] also used this model and analysed responses of a long span structure to spatially varying ground motions with site effect. It was concluded that although it was difficult to draw general conclusions because of the limited analyses performed, it was clear that ground motion spatial variation and site effects significantly affect the structural responses; considering different site effects at multiple supports generated larger structural responses; the more significant was the difference between the site conditions at the multiple supports, the larger was the structural responses. Another study that used this model to -3

38 School of Civil and Resource Engineering Chapter consider the site effects and ground motion spatial variation was reported by Ates et al. [5]. Similar conclusions were drawn, i.e., site effects significantly affect structural responses. Sextos et al. [0, ] discussed the importance of considering ground motion spatial variations, site effect, soil-structure interaction and nonlinear inelastic responses in bridge response analysis and design. They also outlined the possible numerical approaches for bridge response analysis. In the present study, the spatial ground motion model with site effect derived by Hao and Chouw [7] is used to analyse the responses of a bridge frame on a canyon site. Stochastic method is used to perform parametric analysis in this study. Dynamic, quasi-static and total structural responses are calculated. The influences of site conditions and ground motion spatial variations on structural responses are highlighted. Structural responses to uniform ground motion, to spatial ground motion without considering coherency loss or phase shift and to spatial ground motion without considering the site effect are calculated and compared. Discussions on the ground motion spatial variation and site effect in terms of the site properties on structural responses are made.. Bridge and spatial ground motion model.. Bridge model Figure - illustrates the schematic view of a model bridge frame on a canyon site, in which A and B are the two supports on ground surface, the corresponding points at base roc are ' A and ' B. ρ, j v, j ξ and j h j are the density, shear wave velocity, damping ratio and depth of the soil under support j, respectively, where j represents A or B. The corresponding parameters on the base roc are ρ R, v R and ξ R. The dec of the bridge frame is idealized as a rigid beam supported by two piers. It should be noted that only one bridge frame is modelled in the present study, the adjacent bridge structures are neglected. This simplification implies no pounding between adjacent bridge structures is considered. This is a rational assumption since with the new development of modular expansion joint MEJ, which allows a large joint movement and at the same time without impending the smoothness of traffic flow, completely precluding seismic pounding between adjacent bridge structures is possible [6]. This means that each bridge frame can vibrate independently during an earthquae without pounding between adjacent structures. In the numerical analysis, without losing generality, the viscous damping ratio of the structure is assumed to be 5% in the present study. -4

39 Chapter d School of Civil and Resource Engineering B B A A h A Soil A ρ A, v A, ξ A Soil B ρ B, vb, ξ B h B ' A Base ρ, v roc, ξ R R R ' B Figure -. Schematic view of a bridge frame crossing a canyon site.. Base roc motion Assume the amplitudes of the power spectral densities at different locations on the base roc are the same and in the form of the filtered Tajimi-Kanai power spectral density function 4 4 = g + 4ξ gg S g H P S0 = Γ - + ξ + 4ξ f f f g g g in which H P is a high pass filter [7], S 0 is the Tajimi-Kanai power spectral density function [8], g and ξ g are the central frequency and damping ratio of the Tajimi- Kanai power spectral density function, Γ is a scaling factor depending on the ground motion intensity, and f and ξ f are the central frequency and damping ratio of the high pass filter. In this study, it is assumed that f f = f / π = 0. 5Hz, ξ = 0. 6, f 3 = / π = 5. Hz, ξ = 0. 6 and Γ = 0.0m / s. These values correspond to a pea g g 0 ground acceleration PGA 0.5g with duration spectral density of the base roc motion. g T = 0s [8]. Figure - shows the power f -5

40 Chapter School of Civil and Resource Engineering Figure -. Filtered ground motion power spectral density function on the base roc Ground motion spatial variation at the base roc is modelled with a coherency loss function [8] γ id / v app βd α d / π id / v i i e e e e app ' ' = γ ' ' = - A B A B in which πa / + b / π + c 0.34rad / s 6.83rad / s α = -3 0.a + 0b + c > 6.83rad / s where a, b, c and β are constants, d is the distance between the two supports, v app is the apparent wave propagation velocity. The cross power spectral density function of the motion at points ' A and ' B on the base roc is thus S ' ' i = S g γ ' ' i -4 A B A B It should be noted that the above coherency function was obtained by processing recorded spatial ground motions on ground surface. Here it is used to model spatial variations of ground motion at base roc. This is because no information about ground motion spatial variations at the base roc is available. It is believed that seismic wave propagation through -6

41 School of Civil and Resource Engineering Chapter a heterogeneous soil site will change ground motion spatial variations. The present assumption may lead to some inaccurate estimation of coherency loss between spatial base roc motions. Further research into the influence of local site conditions on spatial ground motion coherency loss is deemed necessary...3 Site amplification Using seismic wave propagation theory presented by Ai and Richards [9], Safa [30] derived the transfer function for shear wave propagation in a horizontal layer as iτ j iξ j iτ iξ U j i + rj iξ j exp = = H j i U i + r iξ exp ' j j j j j j = A or B -5 where U j i and U j i is the Fourier transform of the motion u j t and u j ' t on the ' ground surface and at the base roc, respectively. ξ = / Q is the damping ratio j 4 accounting for energy dissipation owing to seismic wave propagation, and Q is the quality factor; τ = h / v is the wave propagation time from point j j j ' j to j, and r j is the reflection coefficient for up-going waves ρ RvR ρ jv j rj = j = A ρ v + ρ v R R j j or B -6 In engineering application, usually the outcrop motion on the roc surface is available, instead of the base roc motion. The parameters defined above corresponding to the Tajimi-Kanai power spectral density function in Equation - also correspond to the outcrop motion on hard roc. Therefore, the constant in Equation -5, which is a measure of free surface reflection, in the transfer function is dropped. Then it has iτ j iξ j iτ iξ + rj iξ j exp H j i = j = A or B -7 + r iξ exp j j j j The auto and cross power spectral density function at point j and between points A and B are S = H j S AB i i = H j A S g i H * B -7 i S j = A or B ' ' A B i -8

42 School of Civil and Resource Engineering Chapter in which the superscript * represents complex conjugate. The coherency loss between ground motions at points A and B is γ AB i = = γ S H AB i = S S A ' ' A B B i exp [ i θ θ + d / v ] A A * i H B i γ ' ' i A B = exp H i H i A B B app [ i θ θ ] A B γ ' ' A B i -9 * where Im H A i H B i θ A θ B = tan * Re H i H i is the phase difference of motions at points A A B and B owing to wave propagation at the site. This derivation indicates that the wave propagation through a homogeneous site has no effect on coherency loss γ i, but it changes the phase delay between the spatial ground motion at base roc and on ground surface, and changes the ground motion intensity. It should be noted that this derivation is based on assumption that site condition is homogeneous, and ground motion is stationary. In real case, a soil site will not be homogeneous. The soil properties may vary randomly in space. Moreover, ground motion is not stationary. All these will cause coherency loss in spatial ground motions. However, study of the influence of local site conditions on spatial ground motion coherency loss is beyond the scope of the present paper. It should also be noted that the transfer function expressed in Equation -7 is derived for the case with only one soil layer. If multiple soil layers are under consideration, it can be straightforwardly extended based on the seismic wave propagation theory as discussed by Wolf [3]. A ' B'.3 Structural response equation formulation The purpose of this paper is to investigate the ground motion spatial variation and site effect on responses of multi-supported structures, the soil-structure interaction effect is thus ignored. Without losing generality, a 3-DOF mathematical model, with one for the bridge dec and two for the support movements, is used in the present study. Effectively such structural model represents only a single dynamic mode of vibration with two additional inematic degrees of freedom representing the spatial excitations in the longitudinal direction. The dynamic equilibrium equation can be written as -8

43 School of Civil and Resource Engineering Chapter -9 = B A t B B A A B A B A B A t B A t u u v u u v c u u v m & & & && && && -0 where m is the lumped mass of the bridge dec, v t is the total displacement response, and u A and u B are the ground displacement at support A and B respectively, A and B are the stiffness of the two columns. The total response consists of dynamic response and quasistatic response qs t v v v + = - The quasi-static response can be derived as [ ] [ ] B B A A B A B A B A B A B A qs u u u u u u v ϕ ϕ ϕ ϕ + = = + = - in which / and / B A B B B A A A + = + = ϕ ϕ. The dynamic response can be obtained by solving the dynamic equilibrium equation B B A A qs B A u u m mv v cv v m & & && && & & & ϕ ϕ + = = Transfer Equation -3 into frequency domain, the dynamic response can be obtained by ] [ ϕ ϕ ξ i u i u i H i v i H i v i i v B B A A s qs s qs && && && & & + = = + = -4 in which m B A / 0 + = is the circular natural vibration frequency of the structure, 0 ξ is the damping ratio, and i H s is the transfer function of the structure. The power spectral density function of dynamic, quasi-static and total response can then be derived as [ ] { } [ ] { } [ ] { } Re Re Re Re 4 ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ i S S S i H S S S i S S S S i S S S i H S AB B A B B A A s v v v AB B A B B A A v AB B A B B A A s v qs t qs = + + = + + = -5

44 Chapter School of Civil and Resource Engineering in which Re denotes the real part of a complex number. In this study, the uniform ground motion is assumed to be the same as u A. Under uniform ground motion excitation, Equation -5 reduces to S t vu = S vu S vu = H i S qs = S v 4 A u + S qs Re vu s S A [ H i S ] s A -6.4 Maximum response calculation Standard random vibration method [3] is used to calculate the mean pea displacement, it is briefly described in the following. For a zero mean stationary process xt with nown power spectral density function S, its m th order spectral moment is defined as λ m c m S d -7 0 where c is a high cut-off frequency. The zero mean cross rate v and shape factor of the power spectral density function δ, can be obtained by π λ λ v = -8 0 λ δ = -9 λ λ 0 the mean pea response can then be calculated by x max = lnve T + σ -0 lnv T e -0

45 Chapter School of Civil and Resource Engineering where T is the duration of the stationary process, σ = λ is the standard deviation of the 0 process, and max.,δ T 0 δ < v e T =.63δ 0.38 vt 0. δ < vt δ 0.69 In the present study, the high cut-off frequency is taen as 5Hz since it covers the predominant vibration modes of most engineering structures and the dominant earthquae ground motion frequencies..5 Numerical results and discussions The effects of ground motion spatial variations and site conditions on structural responses are investigated in detail in the present study. Dynamic, quasi-static and total responses of the structure in Figure - under different ground motions and site conditions are calculated. The phase shift effect of spatial ground motion owing to seismic wave propagation depends on a dimensionless parameter f 0 t d [, 9,, ], in which f 0 is the structural vibration frequency, and t d is the time lag between ground motions at two points separated by d. In the previous studies without considering the site effect and with a flat ground surface assumption, t d =d/v app, in which v app is the apparent wave propagation velocity corresponding to the spatial motions at the site. In this study, as discussed above, vertical wave propagation is assumed in the local site, then the time lag between motions at points A and B on ground surface can be estimated as t d =d/v app +τ B -τ A, in which τ B and τ A, as defined above, are time required for wave to propagate from B to B and A to A, respectively. The spatial ground motion phase shift effect is investigated by varying the vibration frequency f 0 of the structure. The constants of coherency loss function in Equation - are obtained by processing recorded motions during Event 45 at the SMART- array [8]. It should be noted that this coherency loss function represents highly correlated ground motions. For comparison, two modified coherency loss functions are also used in the study, which represent intermediately and wealy correlated ground motions, respectively. Figure -3 shows different coherency loss functions corresponding to parameters given in Table -. For spatial ground motion without coherency loss, γ i A ' B' = in Equation -. -

46 School of Civil and Resource Engineering Chapter Table -. Parameters for coherency loss functions Coherency loss β a b c 4 highlyevent intermediately wealy Figure -3. Different coherency loss functions The main parameters for base roc and soil conditions can be combined together to form a single coefficient defined as roc/soil impedance ratio [33] I ρ v R R R S = - / ρsvs This impedance coefficient reflects the differences between base roc and soil conditions. Without losing generality, three types of soils are studied in the paper. The corresponding parameters for the soil layer and the base roc are given in Table -. Table -. Parameters of base roc and different types of soil 3 Type ρ g / m v m / s ξ R S Base roc / Firm soil Medium soil Soft soil I / -

47 Chapter.5. Effect of soil depth School of Civil and Resource Engineering The effects of soil depth on structural responses are investigated first. Six different soil depths are discussed, i.e., the bridge frame locates on a flat site with h = h = m, A B 0 ha = hb = 30m, ha = hb = 50m or locates on a canyon site with ha = 0 m hb = 30m, ha = 0 m hb = 50m and ha = 30 m hb = 50m. To preclude the influence of other parameters, the soil under both site A and B are assumed to be firm soil with I 5, and the ground motions are assumed to be intermediately correlated. R / S = As shown in Figure -4, different soil depths lead to different transfer functions. The peas occur at the corresponding vibration modes of the sites. Tae h=50m as a example, the resonant frequencies of the soil layer are f = v / 4h, =,3,5,..., where v s and h is the shear wave velocity and depth of the soil layer respectively, obvious peas can be obtained s at f=.5, 6.75 and.5hz with v s = 450m / s and h=50m. The deeper is the soil, the more flexible is the site, and the lower is the fundamental vibration frequency. The transfer function directly alters the ground motion power spectral density function on ground surface as compared to that at the base roc, as shown in Figure -5. Motions on ground surface have a narrower band, but higher pea, as compared to that at the base roc, indicating the effect of site filtering and amplification on base roc motion. If the ground surface is flat, the time lag between motions at A and B are the same as those at the base roc τ B -τ A =0 because soil properties are assumed to be the same at the two wave paths. In this case, wave propagation through the site will not cause further phase difference. However, if a canyon site is assumed, the time for wave propagating from base roc to ground surface is different τ B -τ A 0, which results in an additional phase difference between motions at A and B, as compared to those at the base roc, as shown in Figure - 6. Dynamic, quasi-static and total responses with varying structural vibration frequencies are calculated, and normalized by the corresponding responses to uniform excitation, which is defined as the motion at Point A, as discussed above. Figures -7 and -8 show the normalized dynamic responses and total responses with respect to the dimensionless parameter, f 0 t d, respectively. This parameter measures the relation between phase shift or time lag of spatial ground motions at points A and B and the fundamental vibration mode with frequency f 0. When a flat site is considered, f 0 t d =f 0 d/v app, and the multiple ground -3

48 School of Civil and Resource Engineering Chapter excitations and the structural vibration mode are in-phase if f 0 =.0,.0L, whereas they t d are out-of-phase if f 0 t d = 0.5,.5 L for the special case [8, 34]. Figure -4. Site transfer functions for different soil depths Figure -5. Power spectral densities of ground motions on site of different depths Figure -6. Phase difference caused by seismic wave propagation through sites of different depths -4

49 Chapter School of Civil and Resource Engineering As shown in Figure -7, if the site is flat, non-uniform ground motion always reduces the dynamic responses as compared to the uniform ground motion. The normalized dynamic responses reach their minimum value at f 0 t d = 0.5,. 5 and maximum value at f 0 t d =.0,. 0 because of the out-of-phase and in-phase ground motion inputs. This observation is the same as those reported in many previous studies [8, 34]. If a canyon site is assumed with Point A on base roc and Point B on soil surface, the maximum responses, however, do not occur at f 0 t d =. 0. This is because of the dominance of site amplification effect on ground motions and resonant responses. The maximum response occurs when the structure is resonant with the soil site. For example, when h A = 0m and h B = 30m, the first pea occurs at f 0 t d =0.65, or f 0 =3.75Hz because t d =d/v app +τ B =d/v app +h B /v B = sec. The second pea can be observed when f 0 =.5Hz. As shown in Figures -4 and -5, the resonant frequencies of the site with soil depth 30m are f = v / 4h = 3.75, =,3,5,... If. s h A = 0m and h B = 50m, the first pea occurs at f 0 t d =0.475, or f 0 =.5Hz because t d =0.sec. Again as shown in Figures -4 and -5,.5 Hz is the fundamental vibration frequency of the soil site with depth 50m. The following peas can also be observed when resonance occurs. If both point A and B locate on soil surface with h A = 30m and h B = 50m, the spatial ground motion wave passage effect dominates the site effect on dynamic structural responses, i.e., the minimum values occur around f 0 t d = 0.5,. 5, and the maximum values around f 0 t d =.0,. 0. This is because, although site A and B have different fundamental vibration modes and different pea values in their respective power spectral density function as shown in Figure -5, the mean pea responses to ground motion at site A and B are similar to each other because they depend on the spectral moments as defined above. Therefore, normalization removes the site amplification effects, which leaves the wave passage effects to govern the normalized dynamic response in this case. It can also be noted that the normalized dynamic responses are always smaller than.0 when wave passage effect dominates, indicating the spatial ground motion phase shift always results in a reduction in dynamic structural responses. Similar observation has also been obtained in previous studies [8, 34]. When the vibration frequency of the structure coincides with the fundamental frequency of the soil layer, however, the normalized pea dynamic responses can be larger than.0, indicating the significance of site amplifications on ground motions and hence on structural responses. These observations indicate the importance of considering both the site and the ground motion spatial variation effects in structural response analysis. -5

50 Chapter School of Civil and Resource Engineering Quasi-static responses are independent of the fundamental vibration frequency of the structure Equations -5 and -6. The normalized quasi-static responses are therefore constant for each case with respect to f 0 t d. The normalized total responses are given in Figure -8. As shown, when the dimensionless parameter f 0 t d is less than.5, the normalized total responses are similar to the normalized dynamic responses, indicating dynamic response dominates the total response. When f 0 t d increases, however, the normalized responses approach to a constant, equal to the quasi-static response. Neither spatial ground motion wave passage effect, nor the site amplification effect is prominent. This is because increasing f 0 t d implies the structure becomes stiffer, as f 0 is increased in this study. The dynamic response is smaller when structure is stiffer. At large f 0 t d, quasi-static response dominates the total response, as shown in Figure -9. This observation indicates the importance of quasi-static responses for stiff structures. Figure -7. Normalized dynamic responses for different soil depths Figure -8. Normalized total responses for different soil depths -6

51 Chapter School of Civil and Resource Engineering Figure -9. Dynamic, quasi-static and total responses with h A = 0m and h B = 30m.5. Effect of soil properties To study the effect of soil properties on ground motion spatial variation and hence on structural responses, different soil types shown in Table - are considered. The soil under point A is assumed to be firm soil I 5 and unchanged in all the cases, while soil R / S = under support B varies from firm soil I 5 to soft soil I 30. The soil depths are assumed to be R / S = R / S = h A = 30m and h B = 50m, and the ground motions are intermediately correlated. Figure -0 shows the transfer function at support B for different cases. Figure - shows the corresponding power spectral density function of motion on ground surface at Point B. For comparison purpose, the power spectral density function of motion at Point A is also shown in these two figures. Figure - shows the phase differences between motions at Point A and B. The normalized dynamic responses and total responses are shown in Figures -3 and -4, respectively. Figure -0. Soil site transfer function for different soil properties -7

52 Chapter School of Civil and Resource Engineering Figure -. Power spectral densities of ground motions at sites with different soil properties Figure -. Phase difference owing to seismic wave propagation through sites with different soil properties Figure -3. Normalized dynamic responses for different soil properties -8

53 Chapter School of Civil and Resource Engineering Figure -4. Normalized total responses for different soil properties Figure -5. Dynamic, quasi-static and total responses medium soil at support B Figure -0 clearly shows again the site effects. As shown, pea value of the transfer function increases, while the frequency band becomes narrower with the decrease of the site stiffness. This directly affects the ground motions on ground surface, resulting in substantial spatial variations between ground motions at Points A and B. Soft soil I 30 and medium soil I 0 significantly amplifies the ground motions at its R / S = R / S = resonant frequencies, firm soil I 5 also amplifies ground motions, but at higher R / S = frequencies and with a less extent. As a result, the ground motion power spectral densities at ground surface are very different as shown in Figure -. Soil properties also affect the seismic wave propagation velocity and hence the phase difference between motions at Point A and B. Figure - shows the phase differences between motions at A and B owing to wave propagation from base roc to ground surface. It shows that the phase -9

54 School of Civil and Resource Engineering Chapter differences vary rapidly with respect to frequency. The softer is the soil, the more drastic variation is the phase difference because the wave velocity is slower. Again, as shown in Figure -3, when the vibration frequency of the structure is low, site effect dominates the dynamic responses. When the soil properties of site A and B are different from each other significantly, i.e., the maximum responses occur when the structure resonates with the soil site. For example, when site B is the medium soil, the first pea occurs at f 0 t d =0.3, or f 0 =.5Hz because t d =0. sec. As shown in Figures -0 and -, the fundamental vibration frequency for the medium site is.5 Hz. When site B is a soft soil site, the first pea occurs at f 0 t d =0.67, or f 0 =0.5Hz because t d = sec, and the second pea at f 0 t d =0.8, corresponding to the second mode of the site B. Subsequent peas of these two cases are associated with the in-phase excitations and the minimum values are associated with the out-of-phase effect. This is because when the structure becomes stiffer, the dynamic response and hence the site resonance effect becomes less significant as compared with the ground motion spatial variation effect. As also can be seen in Figure -3, soft soil amplification effect results in larger dynamic responses, normalized dynamic responses are usually larger than.0 when the responses are dominated by the site effect, and the results are always less than.0 when spatial ground motion phase shift effect governs the dynamic responses. Total responses shown in Figure -4 follow the similar pattern as that discussed above, i.e., the normalized total responses are similar to the normalized dynamic responses when f 0 t d is less than.5. However, if the structure is stiff, the dynamic responses are small and the total responses are dominated by the quasi-static responses, as shown in Figure -5 medium soil at support B..5.3 Effect of coherency loss To investigate the influence of ground motion spatial variation, different coherency losses are considered in the paper as shown in Figure -3, i.e., highly, intermediately and wealy correlated coherency loss functions. Moreover, two special cases, i.e., intermediate coherency loss without considering phase shift cos =. 0, and no coherency loss γ i, are also considered. All the results are normalized by the corresponding ' ' = A B t d uniform excitation. For these cases, the canyon site with h A = 30m and h B = 50m is considered, and medium soil I 0 are assumed at both sites A and B. Figure -6 R / S = -0

55 School of Civil and Resource Engineering Chapter shows the normalized dynamic responses and Figure -7 shows the normalized total responses. Figure -6. Normalized dynamic responses for different coherency losses Figure -7. Normalized total responses for different coherency losses As shown in Figure -6, site effect governs the dynamic responses when f 0 t d <0.5, i.e., pea response occurs at the resonant frequency with f 0 =0. Hz and f 0 t d =0.3. However, if f 0 t d >0.5, spatial ground motion wave passage effect dominates the dynamic responses, i.e., the normalized dynamic responses reach their minimum value at f 0 t d = 0.5,. 5 and their maximum value at f 0 t d =.0,. 0 because of the out-of-phase and in-phase ground motion excitations. The more correlated are the ground motions, the more pronounced are the inphase and out-of-phase effects. This means that the influence of site effect is more significant when the structure is relatively flexible, while spatial ground motion wave passage effect dominates the dynamic responses when the structure is stiff. If multiple ground motion phase shift is not considered, normalized pea response occurs at vibration modes of the soil site, no in-phase or out-of-phase effects are present. For total responses -

56 School of Civil and Resource Engineering Chapter as shown in Figure -7, similar observations can be drawn, i.e., dynamic response dominates total response when f 0 t d <.0, and quasi-static response is more significant when the structure is stiff..6 Conclusions This paper studies the combined effects of ground motion spatial variation and local site conditions on the responses of a bridge frame located on a canyon site. Dynamic, quasistatic and total responses of the model structure to various cases of spatially varying ground motions are investigated. Following conclusions can be drawn:. Wave propagation through multiple sites with different site conditions causes further variations of spatial ground motions. Depending on the soil conditions along each wave path, spatial ground motions at different locations on surface of a canyon site have different power spectral densities and more pronounced phase shift as compared to those on the base roc.. Local site conditions significantly affect spatial surface ground motions, and hence the structural responses. The pea dynamic responses occur when the structure resonates with the site, and when the spatial ground motion and structural vibration mode are in-phase. The minimum dynamic responses occur when the spatial ground motion and structural vibration modes are out-of-phase. 3. Dynamic response governs the total response when the structure is flexible, while quasi-static response dominates it when the structure is stiff. 4. Different site conditions at two structural supports causes more significant spatial variations of ground motions, and hence larger structural responses. 5. Spatial ground motion coherency loss has a relatively less significant effect on structural responses when the structure is flexible and the total response is governed by the dynamic response. However, coherency loss effect is prominent, especially when the structure is stiff. 6. Uniform site assumption leads to underestimation of spatial variations of ground motions on a canyon site, and therefore underestimation of structural responses..7 References. Bolt BA, Loh CH, Penzien J, Tsai YB, Yeah YT. Preliminary report on the SMART- strong motion array in Taiwan. Report No. UCB/EERC-8-3, University of California at Bereley, Bereley, CA, 98. -

57 School of Civil and Resource Engineering Chapter. Harichandran RS, Vanmarce EH. Stochastic variation of earthquae ground motion in space and time. Journal of Engineering Mechanics 986; : Loh CH, Yeah YT. Spatial variation and stochastic modelling of seismic differential ground movement. Earthquae Engineering and Structural Dynamics 988; 64: Hao H, Oliveira CS, Penzien J. Multiple station ground motion processing and simulation based on SMART- data. Nuclear Engineering and Design 989; 3: Abrahamson NA, Schneider JF, Stepp JC. Empirical spatial coherency functions for application to soil-structure interaction analyses. Earthquae Spectra 99; 7: Der Kiureghian A. A coherency model for spatially varying ground motions. Earthquae Engineering and Structural Dynamics 996; 5: Hao H, Chouw N. Modeling of earthquae ground motion spatial variation on uneven sites with varying soil conditions. The 9th International Symposium on Structural Engineering for Young Experts, Fuzhou & Xiamen, Tajimi H. A statistical method of determining the maximum response of a building structure during a earthquae. Proceedings of nd World Conference on Earthquae Engineering, Toyo, Japan 960; Harichandran RS, Wang W. Response of simple beam to spatially varying earthquae excitation. Journal of Engineering Mechanics 988; 49: Harichandran RS, Wang W. Response of indeterminate two span beam to spatially varying seismic excitation. Earthquae Engineering and Structural Dynamics 990; 9: Zerva A. Response of multi-span beams to spatially incoherent seismic ground motion. Earthquae Engineering and Structural Dynamics 990; 9: Hao H. Arch responses to correlated multiple excitations. Earthquae Engineering and Structural Dynamics 993; 5: Hao H. Ground-motion spatial variation effects on circular arch responses. Journal of Engineering Mechanics 994; 0: Hao H, Duan XN. Multiple excitation effects on response of symmetric buildings. Engineering Structures 996; 89: Hao H, Duan XN. Seismic response of asymmetric structures to multiple ground motions. Journal of Structural Engineering 995; :

58 School of Civil and Resource Engineering Chapter 6. Dumanoglu AA, Soylu K. Response of cable-stayed bridge to spatially varying seismic excitation. 5th International Conference on Structure Dynamics, Munich, Germany, 00; Janowsi R, Wilde K, Fujino Y. Reduction of pounding effects in elevated bridges during earthquaes. Earthquae Engineering and Structural Dynamics 000; 9: Hao H. Effects of spatial variation of ground motions on large multiply-supported structures. Report No. UCB/EERC-89-06, University of California at Bereley, Bereley, Monti G, Nuti C, Pinto E. Nonlinear response of bridges to spatially varying ground motion. Journal of Structural Engineering 996; : Sextos AG, Kappos AJ, Patilais KD. Inelastic dynamic analysis of RC bridges accounting for spatial variability of ground motion, site effects and soil-structure interaction phenomena. Part : Methodology and analytical tools. Earthquae Engineering and Structural Dynamics 003; 34: Sextos AG, Kappos AJ, Patilais KD. Inelastic dynamic analysis of RC bridges accounting for spatial variability of ground motion, site effects and soil-structure interaction phenomena. Part : Parametric study. Earthquae Engineering and Structural Dynamics 003; 34: Chouw N, Hao H. Study of SSI and non-uniform ground motion effects on pounding between bridge girders. Soil Dynamics and Earthquae Engineering 005; 50: Zembaty Z, Rutenburg A. Spatial response spectra and site amplification effect. Engineering Structures 00; 4: Dumanoglu AA, Soylu K. A stochastic analysis of long span structures subjected to spatially varying ground motions including the site-response effect. Engineering Structures 003; 50: Ates S, Dumanoglu AA, Bayratar A. Stochastic response of seismically isolated highway bridges with friction pendulum systems to spatially varying earthquae ground motions. Engineering Structures 005; 73: Chouw N, Hao H. Significance of SSI and non-uniform near-fault ground motions in bridge response II: Effect on response with modular expansion joint. Engineering Structures 008; 30: Ruiz P, Penzien J. Probabilistic study of the behaviour of structures during earthquaes. Report No. UCB/EERC-69-03, University of California at Bereley, Bereley,

59 School of Civil and Resource Engineering Chapter 8. Hao H. A parametric study of the required seating length for bridge decs during earthquae, Earthquae Engineering and Structural Dynamics 998; 7: Ai K, Richards PG. Quantitative seismology theory and methods. WH Freeman and Company, San Francisco, Safa E. Discrete-time analysis of seismic site amplification. Journal of Engineering Mechanics 995; 7: Wolf JP. Dynamic soil-structure interaction. New Jersey: Prentice-Hall; Der Kiureghian A. Structural response to stationary excitation, Journal of Engineering Mechanics 980; 06: Roesset JM. Soil amplification of earthquaes. Numerical methods in geotechnical engineering, McGraw-Hill, New Yor, Hao H, Zhang S. Spatial ground motion effect on relative displacement of adjacent building structures, Earthquae Engineering and Structural Dynamics 999; 84:

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61 Chapter 3 School of Civil and Resource Engineering Chapter 3 Required separation distance between decs and at abutments of a bridge crossing a canyon site to avoid seismic pounding By: Kaiming Bi, Hong Hao and Nawawi Chouw Abstract: Major earthquaes in the past indicated that pounding between bridge decs may result in significant structural damage or even girder unseating. With conventional expansion joints it is impossible to completely avoid seismic pounding between bridge decs, because the gap size at expansion joints is usually not big enough in order to ensure smooth traffic flow. With a new development of modular expansion joint MEJ, which allows a large joint movement and at the same time without impeding the smoothness of traffic flow, completely precluding pounding between adjacent bridge decs becomes possible. This paper investigates the minimum total gap that a MEJ must have to avoid pounding at the abutments and between bridge decs. The considered spatial ground excitations are modelled by a filtered Tajimi-Kanai power spectral density function and an empirical coherency loss function. Site amplification effect is included by a transfer function derived from the one dimensional wave propagation theory. Stochastic response equations of the adjacent bridge decs are formulated. The effects of ground motion spatial variations, dynamic characteristics of the bridge and the depth and stiffness of local soil on the required separation distance are analysed. Soil-structure interaction effect is not included in this study. The bridge response behaviour is assumed to be linear elastic. Keywords: required separation distance; MEJ; spatial variation; site effect; dynamic characteristic; stochastic method 3. Introduction For large-dimensional structures, such as long-span bridges, earthquae ground motions at different supports are inevitably not the same owing to seismic wave propagation and local site conditions. Such ground motion spatial variations may result in pounding or even 3-

62 School of Civil and Resource Engineering Chapter 3 collapse of adjacent bridge decs owing to the out-of-phase responses. Poundings between an abutment and bridge dec or between two adjacent bridge decs were observed in almost all the major earthquaes, e.g., the 989 Loma Prieta earthquae and the 994 Northridge earthquae [], the 995 Hyogo-Ken Nanbu earthquae [], the 999 Chi-Chi Taiwan earthquae [3], the 006 Yogyaarta earthquae [4], and more recently the 008 Wenchuan earthquae [5]. More and more earthquae engineers have realized the importance of pounding between adjacent structures. Many methods were adopted to reduce the negative effect of pounding. The most direct way to avoid pounding is to provide adequate separation distance between adjacent structures. Most previous studies on structural pounding have been focused on adjacent buildings. Jeng et al. [6] estimated the building separation required to avoid pounding by using spectral difference method. Kasai et al. [7] defined vibration phase, and proposed a simplified rule to predict the inelastic vibration phase. Penzien [8] proposed a formula for evaluating the required separation distances of two buildings, based on the procedure of equivalent linearization of non-linear hysteric behaviour. Lin [9] proposed a theoretical solution based on random vibration method to predict the statistics of separation distance of adjacent buildings. Hao and Zhang [0] investigated the effect of the spatially varying ground motions on the relative displacement of adjacent buildings. Seismic design codes such as UBC [], Australian Earthquae Loading Codes [] and Chinese Seismic Design Code [3] also specify the required separation distances between buildings. For bridge structures with conventional expansion joints, a complete avoidance of pounding between bridge decs during strong earthquaes is often impossible. This is because the separation gap of an expansion joint is usually only a few centimetres to ensure a smooth traffic flow. Many researchers therefore focused on damaging effects of pounding and strategies to mitigate pounding between bridge decs. Ruangrassamee and Kawashima [4] studied the relative displacement spectra of two SDOF systems with pounding effect. DesRoches and Muthuumars [5] investigated pounding effect on the global response of a multiple-frame bridge. Owing to the difficulty in modelling the spatial variation of ground motions, both studies assumed uniform ground motions. Janowsi et al. [6] and Zhu et al. [7] studied the pounding effect of an elevated bridge caused by wave passage effect. Hao and Chouw [8] and Zanardo et al. [9] considered the pounding effect of simply supported segmental bridges, and they confirmed that spatial ground variations can have strong influences. Janowsi et al. [0] studied several approaches for reducing the 3-

63 School of Civil and Resource Engineering Chapter 3 damaging effects of collisions. To mitigate pounding effect some bridge design codes such as AASHTO [], CALTRANS [] and JRA [3] recommended an adjustment of the vibration properties of adjacent bridge decs so that they have the same or similar fundamental periods. A recent study by Chouw and Hao [4] found that even if the adjacent bridge decs have exactly the same fundamental period, a few centimetre gap size of a conventional expansion joint is not sufficient to completely preclude poundings because of ground motion spatial variations and local site conditions. With the new development of a Modular Expansion Joint MEJ, which allows large relative movement in the joint, completely precluding pounding between bridge decs becomes possible [5]. However, only very limited information on the required separation distances to avoid seismic pounding between adjacent bridge structures is available. Hao [6] analysed various parameters that influence the required seating length to prevent bridge dec unseating; Chouw and Hao [5] studied the influence of soil-structure interaction SSI effect on the required separation distance of two adjacent bridge frames connected by an MEJ; Bi et al. [7] studied site effect on the required separation distance between two adjacent bridge decs. In all these studies, the two adjacent bridge decs were independent and modelled as uncoupled systems, the bearing that connects bridge pier and dec was not considered, and the multiple bridge piers were assumed resting on a flat site with the same ground motion intensity. In this paper, a more realistic bridge model and site conditions are considered in studying the required separation distance between two adjacent bridge decs and between the bridge dec and adjacent abutment. The bridge model is illustrated in Figure 3-. Bearings that connect bridge decs to pier or abutments are included in the model. Each bridge dec is modelled as a rigid beam with lumped mass supported on isolation bearings, and the two adjacent bridge decs are coupled with each other through the pier which is considered as a linear elastic reinforced concrete column. The bearings located on the pier and the two abutments provide with their horizontal flexibility and damping the desired isolation of the bridge girders. The abutments and pier are supported on ground of different elevations. Site amplification effect is considered in the study by a transfer function derived from the one dimensional wave propagation theory. Spatial ground motions are modelled by a filtered Tajimi-Kanai power spectral density function and an empirical coherency loss function. Stochastic method is used in the study to calculate the required separation distances between abutment and dec and between two decs. This paper is a continuation 3-3

64 School of Civil and Resource Engineering Chapter 3 of the previous wor [6, 7]. The primary differences between the present wor and previous wors include: A more realistic bridge model with two adjacent decs coupled on top of the pier instead of two independent bridge girders, and bearings that connect bridge decs to the pier and abutments are considered in this study; The required separation distance between abutments and adjacent bridge dec is also investigated in addition to that between bridge decs; 3 A canyon site with site amplification effect is considered in this paper instead of a uniform flat ground surface with the same ground motion intensity at all the bridge supports; 4 More cases with different bridge girder frequencies and site conditions are considered in this paper. However, it should be noted that the effect of soil-structure interaction is not included, and only linear elastic responses are considered. 3. Bridge model Figure 3- a illustrates the schematic view of a typical bridge crossing a canyon site. Two decs with length d and d are supported by four isolation bearings which are connected to two abutments and one elastic pier. Points A, B and C are the three bridge support locations on the ground surface, the corresponding points at base roc are ' A, ' B and h j is the depth of the soil layer under the jth support, where j represents A, B or C. ρ R, v R and ξ R represent density, shear wave velocity and damping ratio of the base roc, respectively; the corresponding parameters of soil layer are ρ j, v j and ξ j. ' C. A MEJ is installed between the bridge decs and at the two abutments. A MEJ consists of two edge beams and several centre beams, and seals to cover the gaps between the beams and to ensure the watertightness of the joint. Since the seals move with the gap freely almost without any resistance, a MEJ allows a large movement gap equivalent to the sum of a number of small gaps between the beams. Therefore using a MEJ is possible to provide sufficient closing movement between bridge decs to preclude pounding during strong ground shaings. More detailed information regarding the MEJ can be found in [5]. The gap required for a MEJ to avoid pounding between bridge decs and at abutments in terms of the ground motion properties, site conditions and bridge conditions needs to be determined. To simplify the analysis, following hypothesises are made: The bridge decs are assumed to be rigid with lumped mass m and m ; The two bearings located on the abutment and the pier for each dec have the same dynamic characteristics with stiffness 3-4

65 Chapter 3 b and damping cb for the left span, b and cb School of Civil and Resource Engineering for the right span; 3 The pier is modelled as an elastic column with a lumped mass at pier top, the corresponding stiffness and damping are p and c p, respectively; 4 Spatially varying ground motions are considered at different supports; 5 Soil-structure interaction is not considered in the present paper. Based on the above assumptions, a 6 DOF model of the bridge with one DOF for each rigid dec, one for pier, and three for spatial support movements as shown in Figure 3- b, is used in the present study. Δ d Δ d Δ 3 A h A ' A b Soil A h B ' B B Soil B Base roc a Soil m m b b b C C h C ' C A 4 c b c c b b m 3 3 p c p c b 6 C B b 5 Figure 3-. a Schematic view of a bridge crossing a canyon site; b structural model 3.3 Spatial ground motion model 3.3. Base roc motion Assuming ground motion intensities at A, B and C on the base roc are the same, and the coherency loss is measured by an empirical coherency loss function. Its power spectral density is modelled by a filtered Tajimi-Kanai power spectral density function as 4 + 4ξ g / g S g = H P S0 = Γ 3- + ξ / + 4ξ / f f f g g g 3-5

66 Chapter 3 School of Civil and Resource Engineering in which H P is a high pass filter [8], which is applied to filter out energy at zero and very low frequencies to correct the singularity in ground velocity and displacement power spectral density functions. S is the Tajimi-Kanai power spectral density function [9], 0 and g ξ are the central frequency and damping ratio of the Tajimi-Kanai power spectral g density function, respectively. f and ξ f are the central frequency and damping ratio of the high pass filter, respectively. Without losing generality, in this study, it is assumed that f = / π = 0.5 Hz, ξ f = 0. 6, f = / π = 5. 0 Hz and ξ g =0.6. Γ is a scaling factor f f g g depending on the ground motion intensity, assuming a ground acceleration of duration T=0 s and pea value PGA 0.5g, Γ = 0. 0 m /s 3 is estimated in this study according to the standard random vibration approach given in Appendix A. Figure 3- shows the power 4 spectral density of the base roc acceleration and displacement S = /. d S g Ground motions at two distant bridge foundations can vary significantly from each other, because the propagating seismic waves will not arrive at these locations at the same time, and the geological medium in the wave path can affect the characteristics of the propagating waves. In the numerical simulation of spatially varying ground motions usually empirical coherency loss functions are applied. In the present paper, the coherency loss function at points j and n where j, n represents A, B or C was derived from the SMART- array data by Hao et al. [30] and is modelled in the following form γ j' n' i id j ' n' / vapp βd j ' n' α d j ' n' / π id j ' n' / vapp = γ j' n' i e = e e e 3- in which πa / + b / π + c 0.34rad / s 6.83rad / s α = a + 0b + c > 6.83rad / s where a,b, c and β are constants, d is the distance between points j and n, v = 000 ' j ' n app m/s is used in the present paper, which is the apparent wave propagation velocity. It should be noted that the above empirical coherency loss function was derived from the recorded strong motions on ground surface at the SMART- array. It may not be suitable to model ground motion spatial variations on the base roc. How the base roc motion varies spatially, however, is not nown. In this study, the soil site is modelled as a 3-6

67 School of Civil and Resource Engineering Chapter 3 homogeneous medium. It only affects the ground motion intensity and spatial ground motion phase delay. The cross power spectral density function of the motion at points j and n on the base roc is thus S ' ' i = S γ ' ' i 3-4 j n g j n 3.3. Site amplification Even if the motion intensities, i.e., the power spectral density functions, at different locations of the base roc are identical, the surface motions would be different due to the variation in the filtering and amplification effects of the soil layer at the bridge supports. The effect of site amplification can be represented by a frequency-dependent transfer function. In the present study, the transfer function of ground motion due to wave propagation from base roc j to ground surface j is based on the one dimensional wave propagation assumption, and is given in the following form [3] H S j iτ j iξ j + rj iξ j e i = 3-5 iτ j iξ j + r iξ e j j where τ j = h j / v j is the wave propagation time from point j to j, and r j is the reflection coefficient for up-going waves r ρ v ρ v R R j j j = 3-6 ρrvr + ρ jv j The power spectral density function at point j is thus j S j S = H i S 3-7 g and the cross power spectral density function between j and n is S jn S S ' ' j n j n * i = H i H i S i 3-8 where the superscript * represents complex conjugate. 3-7

68 Chapter 3 School of Civil and Resource Engineering a b Figure 3-. Filtered ground motion power spectral density function at base roc a acceleration, b displacement 3.4 Structural responses With the hypotheses mentioned above, the dynamic equilibrium equation of the system shown in Figure 3- can be written as M 0 ss 0 && y Css + 0 && yg 0 0 y& K + 0 y& g K ss T sb K K sb bb y 0 = yg where [ M ss ] is the diagonal lumped mass matrix, [ C ss ] is the viscous damping matrix and [ K ss ] is the stiffness matrix corresponding to the structure degrees of freedom. K ] is the coupling stiffness matrix between the structure degrees of freedom and the support degrees 3-8 [ sb

69 Chapter 3 School of Civil and Resource Engineering of freedom, [ K bb ] is that corresponding to the support movements, [ y] T = { y, y y } are the T total displacements vector of the structure and [ ] { y y y } g g, g, g 3, y = are the ground displacements vector at the bridge supports, and in which the superscript T denotes a matrix transpose. Corresponding characteristic matrices are given in Appendix B. 3 The total structural response equation can be derived from Equation 3-9 as ][& y ] + [ C ][ y& ] + [ K ][ y] = [ K ][ y ] 3-0 [ M ss ss ss sb g Equation 3-0 can be decoupled into its modal vibration equation as T ϕ [ Ksb] q& + ξ q& + q = [ yg ] 3- T ϕ [ M ] ϕ ss where ϕ is the th vibration mode shape of the structure, q is the th modal response, and ξ are the corresponding circular frequency and viscous damping ratio, respectively. The th modal response in the frequency domain can be obtained from Equation 3- as r q i = H i ψ y i 3- j= j gj in which r is the total number of supports, and H = iξ i is the th mode frequency response function. ψ j j ϕ [ Ksb] = 3-4 ϕ M ] ϕ T T [ ss is the quasi-static participation coefficient for the th mode corresponding to a movement j at support j, [ K ] is a vector in coupled stiffness matrix K ] corresponding to support j. sb 3-9 [ sb

70 School of Civil and Resource Engineering Chapter The structural response of the ith degree of freedom is t q t y l i i = = ϕ 3-5 where l is the number of modes considered in the calculation, and i ϕ is the th mode shape value corresponding to the ith degree of freedom. For the system shown in Figure 3-, the relative displacement between the adjacent bridge decs is t q t q t y t y t l l = = = = Δ ϕ ϕ 3-6 The power spectral density function of Δ can then be derived as: = = = = Δ = r j jn r n l l s ns s s s j i S i H i H S * 4 ψ ϕ ϕ ψ ϕ ϕ 3-7 where i S jn is the cross power spectral density function given in Equation 3-8. Similarly, the relative displacement between the abutment and the dec is 3 3 t y t y t t y t y t g g = Δ = Δ 3-8 The power spectral density functions of Δ and 3 Δ can be formulated in the following form = = = = = = = Δ r j l j j r j jn r n l l s ns s s j i S i H i S i H i H S 4 * 4 Re ψ ϕ ψ ϕ ψ ϕ 3-9

71 Chapter 3 S Δ3 = 4 Re 4 r r l l ϕ H i ψ j j= n= = s= r l j= = ϕ H i ψ S j 3 j School of Civil and Resource Engineering * ϕs H s i ψ ns S i jn i 3-0 where Re denotes the real part of a complex quantity. The mean pea responses can be obtained based on Equations 3-7, 3-9 and 3-0 by using the standard random vibration method in Appendix A. 3.5 Numerical results and discussions Numerical calculations are performed on the relative displacement between the abutment and the bridge dec, and between the two adjacent bridge decs of the bridge model shown in Figure 3- subjected to spatially varying ground motions at a canyon site. It should be noted that in the case of strong earthquae non-linear behaviour of piers, bearings and foundations might occur, which will strongly affect the structural responses. However, the current wor mainly concentrates on the effect of ground motion spatial variation and site amplification. Therefore, only linear elastic response is considered to avoid further complicating the discussions. For two independent structures, the frequency ratio is usually used to measure the vibration properties of the two adjacent structures, and it is found that the frequency ratio has a great influence on the required separation distance [0, 4-7]. In the present study, though the two adjacent bridge decs are coupled with each other on the top of the pier, the uncoupled frequency ratio f / f of the two spans is still used to approximately quantify the frequency difference of the two decs. This is because the uncoupled vibration frequency of each span is easy to be determined and has a straightforward physical meaning. In this paper, the numerical results are presented with respect to the dimensionless parameter f / f, where / / π f = and f = / / π, are the b m uncoupled frequency in Hz of the left and right spans, respectively. b m To simplify the analysis, the cross sections of the two decs of the bridge model shown in 4 Figure 3- are assumed to be the same, with mass per unit length. 0 g/m, the lengths of the left and right spans are assumed to be d =d =00 m, so the masses of the two decs 6 5 are m = m =. 0 g. The lumped mass at the top of the pier is m 3 = 0 g. The actual bearing stiffness of a bridge depends on many factors such as the dec dimension and 3-

72 Chapter 3 School of Civil and Resource Engineering weight, bearing types and dimensions, etc. In practice, most commonly used bearings have 6 7 stiffness in the range of 0 N/m to 6 0 N/m. For parametrical study, the bearing 6 stiffness of the left span is assumed to be = 6 0 N/m, which corresponds to the uncoupled frequency of the left span f = 0. 5 Hz. The bearing stiffness of the right span 5 8 varies from 0 N/m to 6 0 N/m in the present paper to obtain different frequency ratios f / f. The stiffness of the pier used in the study is 8 = 0 N/m. The damping coefficient of the right span varies with the changing stiffness to maintain the damping ratio unchanged for the system in the calculation. The damping ratio of 5% is used for bearings and the pier in the study. It should be noted that the assumption of 5% damping might underestimate that of the bearings. Since increase damping will reduce the structural response, this assumption might lead to a conservative estimation of the required gaps to avoid pounding. b p 3.5. Effect of ground motion spatial variations To investigate the influence of spatially varying ground motions on the required separation distance, highly, intermediately and wealy correlated ground motions are considered. The parameters are given in Table 3-. For comparison, spatial ground motions with intermediate coherency loss without considering phase shift cos d / =. 0, spatial ground motions without considering coherency loss γ i. 0, wave passage effect ' ' = A B only and uniform ground motion γ i. 0 are also considered. To preclude the ' ' = A B effect of site amplification, the analysis in this section assumes that the bridge is located on the base roc, i.e. h h = h = 0 m. Previous papers [4-7] considered the required A = B C separation distance between adjacent bridge decs Δ in the present paper, no paper regarding the required separation distance between the abutment and the bridge dec Δ 3 and Δ has been reported. For discussion purpose, the sequences of the required separation distances discussed in the paper are Δ, Δ and then 3 Δ. The effects of ground motion spatial variations on the mean minimum required separation distance to avoid seismic pounding are shown in Figure 3-3. The corresponding standard deviations, which are not shown here, are rather small as compared to the mean pea responses. Therefore, only the mean pea responses will be presented and discussed hereafter. v app 3-

73 Chapter 3 Coherency loss School of Civil and Resource Engineering Table 3-. Parameters for coherency loss functions β a b c Highly Intermediately Wealy As shown in Figure 3-3a, with an assumption of uniform excitation, the relative displacement between the two adjacent bridge decs Δ is relatively small when the fundamental frequencies of the adjacent structures are similar, and is zero when f = f. This is because the vibration modes of the two adjacent spans are exactly the same when the frequencies coincide with each other [6]. Therefore, there is no relative displacement between them. These results correspond well with the recommendations of the current design regulations to adjusting the frequencies of the adjacent bridge spans to close to each other in order to preclude pounding. The ground motion spatial variation effect is most significant when f / f is close to unity, wealy correlated ground motions cause larger relative displacement than highly correlated ground motions. The ground motion spatial variation effect is, however, not so pronounced if the vibration frequencies of the two spans differ significantly. In these situations the out-of-phase vibration of the two spans owing to their different frequencies contributes most to the relative displacement of adjacent bridge decs. When f f. 5, the results are almost constant with the increase / > of the frequency ratio. This is because when the structure is relatively stiff as compared to the ground excitation frequency, the dynamic response of the right span is small. The displacement response of the right span is caused primarily by the quasi-static response associated with the non-uniform ground displacement at the multiple bridge supports, and this quasi-static response is independent of the structural frequency, and is a constant once the ground displacement is defined. As shown in Figure 3-3a, there is one obvious pea occurring at f f This is because at this frequency ratio, the first modal vibration / = frequency of the coupled system is 0.5 Hz, which coincides with the predominant frequency of ground displacement as shown in Figure 3-b. The above observations also indicate that adjusting the frequencies of adjacent bridge decs alone is not sufficient to preclude pounding because ground motion spatial variation also induces relative displacement of adjacent decs. 3-3

74 Chapter 3 School of Civil and Resource Engineering a b c Figure 3-3. Effect of ground motion spatial variation on the required separation distance a Δ, b Δ, c 3 Δ Very few researchers studied the required separation distance between bridge dec and abutment to avoid pounding although pounding damages between bridge dec and 3-4

75 Chapter 3 School of Civil and Resource Engineering abutment have been observed in many earthquaes in the past. In this study, the relative displacement between decs and abutments are calculated. The abutment is assumed to be rigid and has the same displacement of the respective ground motion. For the separation distance between the right bridge span and abutment Δ as shown in Figure 3-3b, one 3 obvious pea can be observed when the modal frequency of the coupled system coincides with the predominant frequency of the base roc ground displacement as mentioned above. The effect of ground motion spatial variation is not prominent. Uniform ground motion assumption gives a good estimation of the relative displacement. This observation indicates that the phase shift effect with the assumption of apparent wave velocity vapp = 000 m/s, and the influence of coherency loss, are not significant in this considered bridge example. As for the relative displacement between the left abutment and the dec, although the stiffness of the left span remains unchanged, Δ in Figure 3-3c is not a constant and varies with the change of f because of the coupling through the centre pier. As can be seen in Figure 3-3c, when the frequency ratio is slightly smaller than, the required separation distance is small. However, when the frequency ratio is slightly larger than unity, maximum separation distance is required. This is because, as shown in Figure 3-4, when the uncoupled vibration frequencies of the left span and right span differ from each other, changing the vibration frequency of the right span has little influence on that of the left span. However, the coupled vibration frequency of the left span fluctuates suddenly with the change of the vibration frequency of the right span when f /f is close to unity. It is interesting to observe that the spatially varying ground motions have positive effects on Δ, i.e., wealy correlated ground motions result in a smaller required separation distance and the largest required separation distance corresponds to the uniform ground excitation case. It should be noted that, the changing of the stiffness of the right span has no influence on the required separation distance of the left span if the system is uncoupled through the pier, and Δ will be a constant. 3-5

76 Chapter 3 School of Civil and Resource Engineering Figure 3-4. Left span frequency response function with respect to the frequency ratios 3.5. Effect of the bridge girder frequency 6 The bearing stiffness of the left span studied above is b = 6 0 N/m, which maes the left dec rather flexible f = 0. 5 Hz. To cover a larger range of possible cases, two other 7 bearing stiffness for the left span, i.e., =.4 0 N/m, and b N/m, are also considered. The corresponding frequencies of the left span are f =.0 Hz and.0 Hz, respectively, which represent intermediate and stiff isolated bridge dec for longitudinal movements. Figure 3-5 shows the results corresponding to the intermediately correlated spatial ground motions. As shown in Figures 3-5a and b, the largest separation distance is required when the modal vibration frequency of the coupled system coincides with the base roc ground displacement as previously discussed. The recommendation of the current design regulations to adjust the adjacent spans to have similar vibration frequencies can be applied when both of them are relatively flexible Figure 3-5 a, f = 0. 5 Hz and.0 Hz. When one of the spans or both of them are relatively stiff, this recommendation does not necessarily produce the minimum separation distance. In fact, the required separation 3-6 distance almost reaches a constant when f f 0. 5 if f. 0 Hz as shown in Figure 3- / > 5a. It is observed again that having the same vibration frequencies of the adjacent spans does not completely rule out the relative displacement because of the ground motion spatial variations. For Δ, the coupling effect can still be observed when the uncoupled vibration frequencies of the two adjacent spans are close to each other, but it is less pronounced with the increasing of the left span frequency as shown in Figure 3-5c. As expected, the higher is the uncoupled frequency of the left span, the smaller separation =

77 School of Civil and Resource Engineering Chapter 3 distance is required. These observations indicate that the relative displacement depends not only on the frequency ratio of the adjacent spans, but also on the absolute uncoupled frequency of the bridge. a b c Figure 3-5. Effect of vibration frequency on the required separation distance a Δ, b Δ, c 3 Δ 3-7

78 Chapter Effect of the local soil site conditions School of Civil and Resource Engineering Local soil site conditions have great influences on the structural responses because of the site filtering and amplification effect on the ground motions. To study the influence of local site effect, three different types of soils are considered, i.e., firm, medium and soft soil. Table 3- gives the corresponding parameters of site conditions. Figure 3-6 shows the required separation distances Δ, Δ, and Δ corresponding to the intermediately correlated 3 ground motions and medium soil with different soil depth. Figure 3-7 shows the results corresponding to different local soil conditions. In these cases, the ground motion is also assumed to be intermediately correlated. The canyon site is considered, with ha = hc = 50m, hb = 30 m. The soil under the pier is assumed to be firm soil, while soil under the two abutments varies from firm soil to soft soil, which is represented by FFF, MFM and SFS, respectively for the three site conditions considered, in which F represents firm, M medium and S soft soil conditions under each support. Table 3-. Parameters for local site conditions Type Density g/m3 Shear wave velocity m/s Damping ratio Base roc Firm soil Medium soil Soft soil Based on the discussion above, one obvious pea occurs when the structural frequency coincides with the central frequency of the ground displacement if the bridge locates on the base roc. When one or all the bridge supports are located on the soil site, additional peas can be observed as shown in Figure 3-6a, b and Figure 3-7a, b. One example is shown in Figure 3-6a and b, when h h = 50 A = C 3-8 m h B varies from 0 to 50 m, another obvious pea occurs at f f 3. This is because when f f 3, the corresponding bearing / = / = 7 stiffness of the right span is = N/m. It is found that with this stiffness the b second modal vibration frequency of the coupled system is.4 Hz, which coincides with the predominant frequency of the ground motion on the 50m soil site as can be seen in Figure 3-8a, indicating resonance occurs at this frequency ratio. Similar conclusions can be obtained when the soil depth is 30 m, in this case the site vibration frequency is.4 Hz. Another example is shown in Figure 3-7a and b, when soft soil SFS is considered, the second pea appears when f f which corresponds to the second modal frequency of / = the coupled system at 0.95 Hz. As can be seen from Figure 3-8b, 0.95Hz is the

79 Chapter 3 School of Civil and Resource Engineering predominant frequency of the soft site. Similar conclusions can be obtained when firm or medium site is considered. These observations indicate that larger separation distance is required when the bridge resonates with local site. Comparing Figure 3-6a with Figure 3-6b, Figure 3-7a with Figure 3-7b, the results show that the local soil site conditions have a more significant effect on Δ 3 than Δ, especially when f f. / >. This is because Δ 3 depends on the absolute response of the structure, while Δ depends on the relative response of the bridge girders. The softer site results in larger absolute structural response Δ 3, which slightly increases the relative displacement Δ. As for Δ, fluctuations can be seen when f and f are close to each other because of the coupling effect as previously discussed. This example also indicates the importance of local site effect on the required separation distance. For a bridge structure locates on base roc directly e.g., [6], only one pea can be observed, for a bridge locates on a canyon site, more peas can be obtained corresponding to different vibration modes of the local site. It should be noted that all the above results are based on the assumption of a 5% structural damping ratio, different damping ratio results in different numerical results. Reference [6] concluded that the required separation distance decreases with the increasing damping ratio. The present paper focuses on the total gap that a MEJ must have in order to avoid pounding, and pounding effect is not considered in the present paper, the effect of damping ratio will be the same as that in the previous paper. Moreover, the above results indicate that, as expected, the largest relative displacement is generated when the bridge structure resonates with ground motions. As dominant ground motion frequency is highly dependent on the site vibration frequencies, a site investigation to determine the site vibration frequency is recommended to design the bridge structure to avoid resonance. 3-9

80 Chapter 3 School of Civil and Resource Engineering a b c Figure 3-6. Effect of soil depth on the required separation distance a Δ, b Δ, c 3 Δ 3-0

81 Chapter 3 School of Civil and Resource Engineering a b c Figure 3-7. Effect of soil properties on the required separation distance a Δ, b Δ, c 3 Δ 3-

82 Chapter 3 School of Civil and Resource Engineering a b Figure 3-8. Ground motion power spectral density functions with a different soil depth, b different soil properties 3.6 Conclusions For bridge structures with conventional expansion joints, completely precluding pounding between bridge decs during strong earthquae excitations is often not possible because the separation gap in a conventional expansion joint is usually only a few centimetres due to serviceability consideration for smooth traffic flow. With the new development of the Modular Expansion Joint, which allows for large relative movements in the joint, completely precluding pounding between adjacent bridge spans becomes possible. This paper investigates the minimum separation distance required to avoid seismic pounding of two adjacent bridge decs coupled on the top of the pier through isolation bearings. The influence of spatial variation of ground excitations, vibration characteristics of the bridge structure and local soil conditions on the separation distances between the adjacent bridge decs, between the abutment and the bridge dec are considered. Following conclusions can be obtained based on the numerical results: 3-

83 School of Civil and Resource Engineering Chapter 3. The required separation distance increases when bridge girders resonate with the local site, or when modal frequency of the bridge coincides with the central frequency of ground displacement. Site conditions influence the separation distance significantly. In general, the deeper and the softer is the local site, the larger is the required separation distance. Effect of spatially varying ground motions can not be neglected when the adjacent bridge decs have similar vibration frequencies. Less correlated ground motions require a larger separation distance to avoid pounding between bridge decs. The coupling effect is significant on the required separation distance between the dec and abutment when the uncoupled frequency ratio of the adjacent spans is close to unity.. A consideration of the frequency ratio of the adjacent spans alone is not enough to determine the required separation distance. The absolute frequency of the bridge also strongly affects the responses. A flexible bridge requires a larger separation distance. The recommendation of current design regulations to mae adjacent spans have similar vibration frequency can be applied when both of them are relatively flexible. When one of the spans or both of them are relatively stiff, this recommendation does not necessarily give the minimum required separation distance. This regulation underestimates the smallest separation distance required between the bridge girder and adjacent abutment. 3. The required MEJ total gap depends on the dynamic properties of the participating adjacent structures and the dynamic behaviour of the supporting subsoil not considered in this wor and the spatially varying ground excitations. Sufficient total gap of a MEJ should be provided in the bridge design to preclude possible poundings during strong earthquaes. 3.7 Appendix 3.7. Appendix A: Mean pea response calculation Standard random vibration method is used to calculate the mean pea displacement, it is briefly described in the following [3]. For a zero mean stationary process xt with nown power spectral density function S, its m th order spectral moment is defined as λ m c m S d A

84 Chapter 3 School of Civil and Resource Engineering where c is a high cut-off frequency. The zero mean cross rate v and shape factor of the power spectral density function,δ, can be obtained by π λ λ v = A3-0 λ δ = A3-3 λ λ 0 the mean pea response can then be calculated by x max = lnv e T + σ A3-4 lnv T e where T is the duration of the stationary process, σ = λ is the standard deviation of the 0 process, and T v e max.,δt 0.45 =.63δ 0.38 vt vt 0 δ < δ < 0.69 δ 0.69 A3-5 In the present study, the high cut-off frequency is taen as c = 5Hz since it covers the predominant vibration modes of most engineering structures and the dominant earthquae ground motion frequencies Appendix B: Characteristic matrices For the bridge model shown in Figure 3-, the mass matrix is m 0 0 [ M = ss ] 0 m 0 B3-0 0 m 3 3-4

85 Chapter 3 School of Civil and Resource Engineering where m, m and m 3 is the lumped mass of the two bridge decs and the pier, respectively. The stiffness matrices can be formulated as b 0 b [ Kss] = 0 b b B3- b b b + b + p b 0 0 K sb ] = 0 0 B3-3 b 0 0 p [ where b and b are the bearing stiffness of the left and right spans, respectively, and p the corresponding stiffness of the pier. 3.8 References. Yashinsy M, Karshenas MJ. Fundamentals of seismic protection for bridges. Earthquae Engineering Research Institute, Kawashima K, Unjoh S. Impact of Hanshin/Awaji earthquae on seismic design and seismic strengthening of highway bridges. Structural Engineering/Earthquae Engineering JSCE 996; 3: Earthquae Engineering Research Institute. Chi-Chi, Taiwan, Earthquae Reconnaissance Report. Report No.0-0, EERI, Oaland, California Elnashai AS, Kim SJ, Yun GJ, Sidarta D. The Yogyaarta earthquae in May 7, 006. Mid-America Earthquae Centre. Report No. 07-0, 57; Lin CJ, Hung H, Liu Y, Chai J. Reconnaissance report of 05 China Wenchuan earthquae on bridges. The 4th world conference on earthquae engineering, Beijing, China, 008; S Jeng V, Kasai K, Maison BF. A spectral difference method to estimate building separations to avoid pounding. Earthquae Spectra 99; 8: Kasai K, Jagiasi AR, Jeng V. Inelastic vibration phase theory for seismic pounding mitigation. Journal of Structural Engineering 996; 0:

86 School of Civil and Resource Engineering Chapter 3 8. Penzien J. Evaluation of building separation distance required to prevent pounding during strong earthquaes. Earthquae Engineering and Structural Dynamics 997; 68: Lin JH. Separation distance to avoid seismic pounding of adjacent buildings. Earthquae Engineering and Structure Dynamics 997; 63: Hao H, Zhang SR. Spatial ground motion effect on relative displacement of adjacent building structures. Earthquae Engineering and Structural Dynamics 999; 84: Uniform Building Code UBC. International Building Officials. Whittier, California; AS 70.4 SAA Earthquae Loading codes. Stands Association of Australia Seismic design code for building and structures-gb-89. Chinese Academy of Building Research, Beijing, Ruangrassamee A, Kawashima K. Relative displacement response spectra with pounding effect. Earthquae Engineering and Structural Dynamics 00; 300: DesRoches R, Muthuumar S. Effect of pounding and restrainers on seismic response of multi-frame bridges. Journal of Structural Engineering, ASCE 00; 87: Janowsi R, Wilde K, Fujino Y. Pounding of superstructure segments in isolated elevated bridge during earthquaes. Earthquae Engineering and Structural Dynamics 998; 75: Zhu P, Abe M, Fujino Y. Modelling of three-dimensional non-linear seismic performance of elevated bridges with emphasis on pounding of girders. Earthquae Engineering and Structural Dynamics 00; 3: Hao H, Chouw N. Response of a RC bridge in WA to simulated spatially varying seismic ground motions. Australian Journal of Structural Engineering 008; 8: Zanardo G, Hao H, Modena C. Seismic response of multi-span simply supported bridges to spatially varying earthquae ground motion. Earthquae Engineering and Structural Dynamics 00; 36: Janowsi R, Wilde K, Fujino Y. Reduction of pounding in elevated bridges during earthquaes. Earthquae Engineering and Structural Dynamics 000; 9: LRFD bridge design specifications and commentary. American Association of State Highway and Transportation Officials AASHTO,

87 School of Civil and Resource Engineering Chapter 3. Caltrans Seismic Design Criteria Version l.. Department of Transportation. Sacramento, California, Design specifications for highway bridges Part V: Seismic Design. Japan Road Association JRA, Toyo, Japan, Chouw N, Hao H. Significance of SSI and non-uniform near-fault ground motions in bridge response I: Effect on response with conventional expansion joint. Engineering Structures 008; 30: Chouw N, Hao H. Significance of SSI and non-uniform near-fault ground motions in bridge response II: Effect on response with modular expansion joint. Engineering Structures 008; 30: Hao H. A parametric study of the required seating length for bridge decs during earthquae. Earthquae Engineering and Structural Dynamics 998; 7: Bi K, Hao H, Chouw N. Stochastic analysis of the required separation distance to avoid seismic pounding of adjacent bridge decs. The 4th world conference on earthquae engineering, Beijing, China, 008; Ruiz P, Penzien J. Probabilistic study of the behaviour of structures during earthquaes. Report No. UCB/EERC-69-03, University of California at Bereley Tajimi H. A statistical method of determining the maximum response of a building structure during an earthquae. Proceedings of nd World Conference on Earthquae Engineering, Toyo, 960; Hao H, Oliveira CS, Penzien J. Multiple-station ground motion processing and simulation based on SMART- array data. Nuclear Engineering and Design 989; 3: Hao H, Chouw N. Modeling of earthquae ground motion spatial variation on uneven sites with varying soil conditions. The 9th International Symposium on Structural Engineering for Young Experts, Fuzhou, China, 006; Der Kiureghian A. Structural response to stationary excitation. Journal of Engineering Mechanics 980; 066:

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89 Chapter 4 School of Civil and Resource Engineering Chapter 4 Influence of ground motion spatial variation, site condition and SSI on the required separation distances of bridge structures to avoid seismic pounding By: Kaiming Bi, Hong Hao and Nawawi Chouw Abstract: It is commonly understood that earthquae ground excitations at multiplesupports of large dimensional structures are not the same. These ground motion spatial variations may significantly influence the structural responses. Similarly, the interaction between the foundation and the surrounding soil during earthquae shaing also affects dynamic response of the structure. Most previous studies of ground motion spatial variation effects on structural responses neglected soil-structure interaction SSI effect. This paper studies the combined effects of ground motion spatial variation, local site amplification and SSI on bridge responses, and estimates the required separation distances that modular expansion joints MEJs must provide to avoid seismic pounding. It is an extension of a previous study [], in which combined ground motion spatial variation and local site amplification effects on bridge responses were investigated. The present paper focuses on the simultaneous effect of SSI and ground motion spatial variation on structural responses. The soil surrounding the pile foundation is modelled by frequency-dependent springs and dashpots in the horizontal and rotational directions. The pea structural responses are estimated by using the standard random vibration method. The minimum total gap between two adjacent bridge decs or between bridge dec and adjacent abutment to prevent seismic pounding is estimated. Numerical results show that SSI significantly affects the structural responses, and cannot be neglected. Keywords: required separation distance; SSI; site effect; ground motion spatial variation; MEJ 4-

90 Chapter 4 4. Introduction School of Civil and Resource Engineering Observations from past strong earthquaes revealed that for large-dimensional structures such as long span bridges, pipelines and communication transmission systems, ground motion at one foundation may significantly differ from that at another. There are many reasons that may result in the variability of seismic ground motions, e.g., wave passage effect results from finite velocity of travelling waves; loss of coherency due to multiple reflections, refractions and super-positioning of the incident seismic waves; site effect owing to the differences of local soil conditions; additionally to the above, seismic motion is further modified by the soil surrounding the foundation, which is nown as soil-structure interaction SSI effect. Seismic ground motion variations may result in pounding or even collapse of adjacent bridge decs owing to the out-of-phase responses. In fact, poundings between an abutment and bridge dec or between two adjacent bridge decs were observed in almost all the major earthquaes, e.g. the 994 Northridge earthquae [], the 995 Hyogo-Ken Nanbu earthquae [3], the 999 Chi-Chi Taiwan earthquae [4], the 006 Yogyaarta earthquae [5], and more recently the 008 Wenchuan earthquae [6]. Notwithstanding the extensive research carried out over the last 30 years, very limited studies involve a comprehensive consideration of the coupling effects of ground motion spatial variation, site amplification and SSI due to the complexity of these problems. In the early stage, the sweeping assumptions that the foundations are fixed and the ground motions at all locations are the same prevail. These assumptions are often used in analysis of pounding responses between adjacent structures. For example, by assuming uniform ground motion input, Ruangrassamee and Kawashima [7] calculated the relative displacement spectra of two SDOF systems with pounding effect; DesRoches and Muthuumar [8] investigated pounding effect on the global response of a multiple-frame bridge. After installation of the SMART- array in Lotung, Taiwan, more and more researchers realized the importance of variation of ground motions and some researchers studied one factor or two factors of ground motion spatial variations on structural responses. Janowsi et al. [9] and Zhu et al. [0] studied the pounding effect of an elevated bridge caused by spatial ground motions with only wave passage effect. Taing combined wave passage effect and coherency loss effect into consideration, Hao [] investigated the required seating length to prevent bridge dec unseating; Zanardo et al. [] carried out a parametric study of the pounding phenomenon of multi-span simply supported bridges. To model the combined effects of ground motion spatial variation and multi-site amplification, Der Kiureghian [3] proposed a transfer function that implicitly 4-

91 School of Civil and Resource Engineering Chapter 4 models the site effect on spatial seismic ground motions. Using this model, Dumanogluid and Soylu [4] analysed the stochastic responses of a cable-stayed bridge to spatially varying ground motions with site effect; Ates et al. [5] investigated the effects of spatially varying ground motions and site amplifications on the responses of a highway bridge isolated with friction pendulum systems. In all these studies of ground motion spatial variation effects on structural responses, SSI effects are however neglected. On the other hand, Wolf [6] presented a uniform approach in the frequency domain to analyse the freefield response of local site with multiple soil layers and SSI effect on structural responses; Spyraos and Vlassis [7] assessed the effect of SSI on the response of seismically isolated bridge piers; Maris et al. [8] presented an integrated procedure to analyse the problem of soil-pile-foundation-superstructure interaction and investigated the effect of SSI on the Painter Street Bridge in California. These studies concentrate on SSI effect, ground motion spatial variation are not considered. Chouw and Hao [9, 0] studied the influence of SSI and non-uniform ground motions on pounding between bridge girders, but neglected the site amplification effect on spatial ground motions in their study. Shrihande and Gupta [] proposed a stochastic approach for the linear analysis of suspension bridges subjected to earthquae excitations with consideration of ground motion spatial variation and SSI. The studies with the broadest scope nown to the authors are that by Sextos et al. [, 3], who implemented ground motion spatial variation, site effect and SSI into a computer code ASING, and studied the inelastic dynamic responses of RC bridges in time domain. To preclude pounding effect, the most straightforward approach is to provide sufficient separation distances between adjacent structures. For bridge structures with conventional expansion joints, a complete avoidance of pounding between bridge decs during strong earthquaes is often impossible. This is because the separation gap of an expansion joint is usually only a few centimetres to ensure a smooth traffic flow. With the new development of modular expansion joint MEJ, which allows large relative movement in the joint, completely precluding pounding between bridge decs becomes possible [4]. Though the MEJ systems have already been used in many new bridges, very limited information on the required separation distance that a MEJ should provide to preclude seismic pounding is available. Chouw and Hao too two independent bridge frames as an example, discussed the influences of SSI and non-uniform ground motions on the separation distance between two adjoined girders connected by a MEJ [4] and then introduced a new design philosophy for a MEJ [5]; In a recent study [], the authors combined ground motion spatial variation effect with site effect, studied the minimum total gap that a MEJ must provide to avoid seismic pounding at the abutments and between bridge decs. It should 4-3

92 School of Civil and Resource Engineering Chapter 4 be noted that these studies either neglected site effect [4, 5] or SSI []. To the best nowledge of the authors, the comprehensive consideration of the coupling of ground motion spatial variation, site effect and SSI on the required separation distances that MEJs must provide to preclude seismic pounding has not been reported. The aim of this paper is to study the combined effects of ground motion spatial variation, site amplification and SSI on relative responses of adjacent bridge structures. It is an extension of a previous wor [] in which SSI effect is not considered. The present wor therefore focuses on the SSI effect on bridge structural responses. Random vibration method is adopted in the study. Spatial ground motions on the base roc are assumed to have the same intensity, which are modelled by a filtered Tajimi-Kanai power spectral density function. The wave passage effect and coherency loss effect of the spatial ground motions on the base roc are modelled by an empirical coherency loss function. Site amplification effect is included by a transfer function derived from the one dimensional wave propagation theory. SSI effect is modelled by using the substructure approach. The soil surrounding the pile foundation is modelled by equivalent frequency-dependent horizontal and rotational spring-dashpot systems. With linear elastic response assumption, the bridge responses are formulated and solved in the frequency domain. The power spectral density functions of the relative displacement responses between adjacent bridge decs and between bridge dec and abutment are derived and their mean pea responses are estimated. The minimum total gaps between abutment and bridge dec and two adjacent bridge decs connected by MEJs to avoid seismic poundings are then determined. The numerical results obtained in this study can be used as references in designing the total gap of MEJs. 4. Bridge-soil system Figure 4- a illustrates the schematic view of a typical girder bridge crossing a canyon site. The superstructure of the bridge is adopted from []. The length and total mass for each dec are d =d =00m and m =m =. 0 6 g, respectively. The two bearings for each dec have the same dynamic properties with an effective stiffness b and an equivalent viscous damping cb for the left span, and b and c b for the right span. The concrete pier with a height of L=0m is modelled as an elastic column with a lumped mass m 3 = 0 5 g at pier top, the lateral stiffness of the pier is p =0 8 N/m. To simplify the analysis, a constant damping ratio of 5% is used for bearings and the pier. Different from Reference [], where all the foundations are assumed rigidly fixed to the ground surface, the pier in 4-4

93 School of Civil and Resource Engineering Chapter 4 the present study is founded on a rigid cap which is supported by a pile group with the diameter of each pile d=0.6m and axis to axis distance between two adjacent piles s=3m. The length of the pile is l=m. To preclude seismic pounding between adjacent bridge structures, an MEJ is installed at the pier and at the two abutments respectively. The three bridge support locations on the ground surface are denoted as point, and 3 as shown in Figure 4-a, the corresponding points at base roc are ', ' and 3'. The soil depth of the three sites is assumed to be 50, 30 and 50m respectively. This paper focuses on the relative responses in the horizontal direction, and also owing to the fact that the vertical stiffness of bridge structure is usually substantially larger than that in the horizontal direction, the structure is assumed to be rigid in the vertical direction in this analysis. Moreover, because the abutment of a bridge is usually very stiff as compared to the pile foundation, SSI between foundation and abutment is less significant as compared to that between pile and the surrounding soil, and is often neglected in the analysis [6]. The SSI effects between the abutments and the supporting soils are also neglected in this study. Therefore only the dynamic interaction effect between the pile foundation and the surrounding soil is considered. The soil surrounding the pile foundation at site 3 is modelled as springs and dashpots with the frequency-dependent coefficients h, c h in the horizontal direction and r, c r in the rotational direction. Only viscous damping, which is developed through the energy emanating from the foundation in the soil medium, is considered. The corresponding values of these coefficients are related to the pile and soil conditions, which will be discussed in Section For simplicity, the rigid cap supporting the pier is assumed massless. With the above assumptions, the bridge can be modelled as a five-degree-of-freedom system as shown in Figure 4-b: the dynamic displacements u and u of the bridge dec movement relative to the free field motion u g and u g ; the horizontal displacement u 0 of the pile foundation at site 3 relative to the free field motion u g3 ; the rotational response φ of the pier at the foundation level and the dynamic response u 3 at the pier top. 4-5

94 Chapter 4 School of Civil and Resource Engineering Δ d Δ 3 d Δ 3 Site Site 3 Site ' Base ' 3 roc ' d s a b m m b b b c b c c b b m 3 p c p c b c h h r c r ug u ug3 u0 Lφ u3 u u g m m 3 m φ L b Figure 4-. a Schematic view of a girder bridge crossing a canyon site and b structural model 4-6

95 Chapter Method of analysis School of Civil and Resource Engineering In the present paper, the structural responses are calculated in the frequency domain. Freefield ground motions are used as input in the analyses. In general, the ground motions at the supports are different from the free-field seismic motion. These differences are caused by the scattered wave fields, which generate between soil and structure interface. However, for motions that are not rich in high frequencies, the scattered fields are wea. The support motions therefore can be approximately considered equal to the free-filed motions [8, 7]. Similar to Reference [], the free-field spatial ground motions in the present study are derived from the base roc motions together with the transfer function of local site. To avoid repetition, they are not presented here. The detailed derivation of the spatial ground motion power spectral density function and the ground motion parameters can be found in []. Substructure method is used to analyse SSI effect. The dynamic impedances of the foundations are defined in Section The mean pea responses of the system are estimated based on random vibration method after the power spectral densities of the relative displacements are derived in Section Dynamic soil stiffness As mentioned above, the soil-abutment interaction is ignored at sites and, the only SSI considered in the paper is the pile group embedded in a uniform stratum at site 3. The dynamic stiffness of a pile group K G can be calculated using the dynamic stiffness of S ' a single pile K in conjunction with dynamic interaction factors α. This method can be used with confidence for pile groups not having a large number of piles, say less than 50 [8]. The dynamic stiffness and damping of a single pile can be described in terms of complex stiffness K S S S = + ic 4- where superscript S represents the values for a single pile, is circular frequency, S and S c are the stiffness and equivalent viscous damping of the pile. The corresponding expressions for S and S c can be readily obtained for use from the previous studies [8, 9], and the coefficients suggested by Gazetas [9] are used in the present paper. 4-7

96 Chapter 4 School of Civil and Resource Engineering In geotechnical practice, when the response of a pile group is of interest, such pile-soil-pile ' interaction effects are often assessed through the use of an interaction factor α. For two ' identical piles, the frequency-dependent dynamic interaction factor α is defined as w = = w ' ' qp α α 4- qq where w qp is the dynamic displacement of pile q caused by pile p and w qq is the displacement of pile q under own dynamic load. Gazetas et al. [30] presented the dynamic interaction factors for floating pile groups in graphs; Dobry and Gazetas [3] developed a simple analytical solution for computing the dynamic impedances of pile groups due to pile-soil-pile interaction. The simple solution proposed by Dobry and Gazetas [3] is adopted herein. The frequency-dependent dynamic stiffness and damping coefficient of the pile group then can be estimated as G = Re, Im K / h K h c = 4-3a h G h in the horizontal direction, and G = Re, Im K / r K r c = 4-3b r G r in the rotational direction, where G K h and G K r are the complex stiffness of the pile group in the horizontal and rotational direction, respectively. Im and Re denote the real and imaginary part of the pile group impedances Structural response formulation With the hypotheses mentioned above, the total displacements of bridge decs and the pier are u t ug + u t t =, u = ug + u, u3 = ug3 + u0 + Lφ + u

97 School of Civil and Resource Engineering Chapter The dynamic equilibrium equations of the idealized model in Figure 4-b can be expressed in the matrix form as follows: + + = / 0 0 / g g g b b b b b b b b b b b b b b b b g g g b b b b b b b b b b b b b b b b g g g o r b b b b b b b b b b h b b b b b b b b b b p b b b b b b b b b b b b o r b b b b b b b b b b h b b b b b b b b b b p b b b b b b b b b b b b o u u u u u u c c c c c c c c c c c c c c c c u u u m m m m m L u u u u L L u u u u L c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c L u u u u m m m m m m m m m m m & & & && && && & & & & & && && && && && φ φ φ 4-5 We define the circular frequencies of the structure and the subsoil system in the following form m b =, m b =, 3 3 m m m p + + = 4-6a 3 m m m h h + + =, 3 L m m m r r + + = 4-6b the corresponding viscous damping ratio of the bridge dec, pier and the subsoil system are expressed as b b c ξ =, b b c ξ =, p p c 3 3 ξ = 4-7a h h h h c ξ =, r r r r c ξ = 4-7b With the aid of Equations 4-6 and 4-7, and by assuming the mass ratios

98 Chapter 4 m = m α, 3 m = m 3 School of Civil and Resource Engineering β 4-8 Equation 4-5 can be expressed in the frequency domain as { u i } [ Z i] { u i } [ Z i] = 4-9 g g where { u i } { u i u i u i u i Lφ i } T 3 0 = 4-0a and { u i } { u i u i u i } T g g g g3 = 4-0b are the dynamic response vector and the input ground motion vector, respectively. [ Z i ] and [ Z g i] are the impedance matrices of the system, which are in the following form z [ Z i] = z 5 i M i L L z z 5 55 i M i 4-a [ Z g z i] = z g g 5 i M i L L z z g3 g 53 i M i 4-b The elements of these two matrixes are given in Appendix A. The dynamic response of the bridge structure can then be calculated by { u i } = [ Z i] [ Z i] { u i } [ H i] { u i } g g = g 4- For the bridge model shown in Figure 4-, the minimum separation distance a MEJ must provide to preclude pounding equals the relative displacement of the bridge system, which can be expressed in the frequency domain as 4-0

99 Chapter 4 School of Civil and Resource Engineering t t Δ i = u i, Δ i = u i, Δ i = u i u i The power spectral density functions of Δ, Δ and Δ 3 thus can be derived, and the corresponding expressions are given in Appendix B. It should be noted that Δ and Δ, i.e., the relative displacement between bridge dec and abutments, only depend on the respective dynamic response of the bridge dec, whereas the relative displacement response between two adjacent decs, Δ 3, depend on both bridge dec dynamic response and spatial ground displacements. After the derivation of the power spectral density functions of the required separation distances, the mean pea responses can be estimated based on the standard random vibration method [3]. In the present study, the high cut-off frequency is assumed to be 5 Hz since it covers the predominant vibration modes of most engineering structures and the dominant earthquae ground motion frequencies. 4.4 Numerical example This section carries out parametric studies of the effects of ground motion spatial variation, site effect and SSI on the minimum separation gaps that the MEJs between bridge decs or at abutments of the bridge shown in Figure 4- must provide. Three types of soil, i.e., firm, medium and soft soil, are considered. Table 4- gives the corresponding parameters of soil and base roc. Figure 4- shows the frequency-dependent dynamic stiffness and damping coefficients of the pile group at site 3 corresponding to different soil properties. Because this study concentrates on analysing SSI effect and the influences of varying site conditions on bridge responses have been extensively studied in Reference [], soil conditions at the three sites are assumed to be the same in each case in the present study. To study the ground motion spatial variation effect, highly, intermediately and wealy correlated ground motions are studied. The corresponding parameters can be found in []. Table 4-. Parameters for local site conditions Type Density g/m 3 Shear wave velocity m/s Damping ratio Poisson s ratio Base roc Firm soil Medium soil Soft soil

100 Chapter 4 School of Civil and Resource Engineering For two independent structures, the frequency ratio is usually used to measure the vibration properties of the two adjacent structures, and it is found that the frequency ratio has a great influence on the required separation distances between the two structures [,, 4, 5]. In the present study, though the two adjacent bridge decs are coupled with each other on the top of the pier, the uncoupled frequency ratio f / f of the two spans is still used to approximately quantify the frequency difference of the two decs. This is because the uncoupled vibration frequency of each span is easy to calculate and has a straightforward physical meaning. In this paper, the numerical results are presented with respect to the dimensionless parameter f / f. The stiffness of the left span is assumed to be 7 a constant with =.4 0 N/m, which gives an uncoupled frequency b f = b / m / π.0 Hz. The bearing stiffness of the right span varies from = N/m to.5 0 N/m to obtain different frequency ratios f / f. As mentioned above, a constant damping ratio of 5% is used for bearings and the pier, thus the damping coefficient of the right span also varies with the stiffness to eep the damping ratio unchanged in the calculations. Figure 4-. Frequency-dependent dynamic stiffness and damping coefficients of the pile group ab horizontal direction and cd rotational direction 4.4. Influence of site effect and SSI 4-

101 School of Civil and Resource Engineering Chapter 4 Figure 4-3 shows the influences of site effect and SSI on the total relative response. The ground motions are assumed to be intermediately correlated and the apparent wave velocity is v = 000 m/s. Results corresponding to soft, medium and firm soils with SSI app bold lines or without SSI thin lines are compared, and the effect of SSI is examined. Figure 4-3. Influence of site effect and SSI on the required separation distances a Δ, b 3 Δ and c Δ As shown in Figure 4-3, for Δ 3 and Δ the largest required separation distance occurs at f / f = 0.8, or when the uncoupled vibration frequency of the right span is 0.8 Hz. This is because 0.8 Hz coincides with the central frequency of ground displacement as shown in Figure 4-4c. This significantly increases the displacement response of the right span and hence the relative displacement response Δ 3 and Δ. When soft soil bold solid line is considered, additional amplification in Δ 3 and especially in Δ occurs at f / f = because the bridge system resonates with local site. At this frequency ratio the vibration frequency of the right span is 0.75 Hz, which coincides with the predominant frequency of the soft soil site with a depth of 50m and shear wave velocity 50 m/s. This again amplifies the dynamic response of the right span and hence increases the corresponding relative displacement. For medium soil bold dash line, one obvious pea occurs at f / f. 0 in Δ owing to the same reason. However, this pea is not observed in Δ 3, this is because Δ 3 measures the relative displacement between the two adjacent bridge decs, the two spans tend to vibrate in phase when their frequencies are close to each other, therefore the relative displacement is the smallest although both adjacent span displacements are large owing to resonance. When the site is relatively stiff dash dotted line, no obvious additional pea can be observed, because the fundamental 4-3

102 School of Civil and Resource Engineering Chapter 4 vibration frequency of the site is relatively high compared to the soft and medium site. In this situation, the dynamic response of the right span is small even at resonance with the site. The response of the stiff right span is primarily determined by the quasi-static response associated with the non-uniform ground displacement at the multiple bridge supports, and the quasi-static response is independent of the structural frequency, and is a constant once the ground displacement is defined. Similar observation was also made in a previous study []. As for the relative displacement Δ between the left abutment and the dec, although the stiffness of the left span remains unchanged, Δ in Figure 4-3c is not a constant and varies with the change of f because of the coupling through the pier. Δ experiences a sudden significant variation when f / f is close to unity. This is because changing f has only insignificant influence on the vibration frequency of the left span when f differs pronouncedly from f. However, when f is close to f, or f / f is close to unity, changing the vibration properties of the right span has a significant effect on the actual vibration frequency of the left span through coupling at the pier. This results in a sudden change in the response of the left span. Moreover, when right span resonates with the soil site, it also more significantly affects the responses of the left span through coupling. Therefore corresponding pea responses also occur at respective frequency ratios when the right span resonates with the site. However, the pea response at f f 0. 8 is not / = observed in Δ. This is because f is a constant in the simulations, and this pea is associated primarily with the ground displacement, which has insignificant effect on the dynamic responses of the left span through the dynamic coupling at the pier. The same phenomenon was also observed in the previous study []. Larger Δ in Figure 4-3b is usually obtained when the site is soft. However, for Δ 3 and Δ, medium soil bold dash line might give larger responses than the soft soil condition. This is because the frequency of the left span is.0 Hz, which coincides with fundamental frequency of the medium soil site with a depth of 50 m and shear wave velocity 00 m/s. These observations indicate that the effect of local site conditions should not be neglected because the structure might resonate with local site to therefore generate larger structural responses, and hence larger separation distances will be required. 4-4

103 School of Civil and Resource Engineering Chapter 4 Compared with the results without SSI effect in Figure 4-3, SSI only slightly changes the frequency content of the bridge, i.e., the peas appear at almost the same frequency ratio. This is because the power spectral densities of the required separation distances can be expressed as the product of power spectral density of motions on the ground surface S i and the frequency response function of the structure H i as shown in Appendix B. Local site amplifies certain frequencies significantly at various vibration modes of the site, which results in the energy of the surface motion concentrates at a few frequencies. Tae soft site for example, at the site fundamental frequency of 0.75 Hz the soft soil amplifies ground motions on the base roc 4 times as shown in Figure 4-4a, which alters the ground motions on ground surface significantly from the base roc motion as can be seen in Figure 4-4b. The influence of site effect is more significant than that of the frequency response function. That is why larger relative displacement occurs at the same frequency ratio of the bridge spans with or without considering the SSI effect because this frequency ratio corresponds to the resonance of either one bridge span with the site. Figure 4-4. Site effect on ground motion spatial variations: a transfer function, b PSD of surface acceleration and c PSD of surface displacement To observe the contribution of SSI more clearly, the required separation distances with consideration of SSI are subtracted by those without SSI effect, and the results are shown in Figure 4-5. As expected the influence of SSI is significant especially for soft and medium soil site condition. The required separation distances will be underestimated when SSI effect is ignored since most of the values shown in Figure 4-5 are larger than zero. As shown in Figure 4-5b, when soft or medium soil is considered, the influence of SSI on Δ increases and becomes most pronounced when the structure resonates with local site. The contribution of SSI on relative displacement response is nearly 0. m for soft soil when f / f is around As previously discussed, the right span resonates with local site at this frequency ratio. When f f 0. 75, the influence of SSI on the total responses / > 4-5

104 Chapter 4 School of Civil and Resource Engineering decreases and becomes almost a constant when the right span is stiff enough, indicating quasi-static response dominates the total response. It is generally true that SSI effect is more obvious on a soft soil site than on a medium soil site. When a firm site is considered the influence of SSI can be neglected. For Δ 3, similar observations of different SSI effect can be obtained. It should be noted that no obvious pea can be observed when resonance occurs for medium soil, owing to the two spans tend to vibrate in phase as discussed above. For Δ, it is observed again that SSI effect on soft site is more prominent than on firm site. Figure 4-5. Contribution of SSI to the required separation distances with different soil conditions f =. 0 Hz a Δ 3, b Δ and c Δ Figure 4-6. Contribution of SSI to the required separation distances with different soil conditions f =. 0 Hz a Δ 3, b Δ and c Δ Reference [] concluded that considering only the frequency ratio of the adjacent spans is not enough to determine the required separation distances. The absolute frequency of the bridge also strongly affects the responses. The same fact is expected in this study when SSI effect is considered. To cover a wide range of possible cases, a relatively stiff left span with f =.0 Hz is additionally investigated. Figure 4-6 shows the contribution of SSI to the total responses. For Δ 3 and Δ, similar conclusions can be obtained as in Figure 4-5, i.e., 4-6

105 Chapter 4 School of Civil and Resource Engineering the contribution of SSI of a soft soil site is generally larger than that of a firm soil site and is most significant when resonance occurs. For Δ, however, the influence of site conditions is less prominent as compared to the case with f =. 0 Hz, the influence of SSI can actually be neglected. This is because the left span is relatively rigid, which reduces the SSI effect. Moreover, the right spans are only wealy coupled through the bearings at the pier cap, and the coupling effect is less pronounced when the left span bearings are stiff Influence of ground motion spatial variation and SSI To study the influences of ground motion spatial variation and SSI effect on the required separation distances, highly, intermediately and wealy correlated ground motions are considered. The soils under the three foundations are assumed to be medium soil. The apparent wave velocity is v = 000 m/s. As shown in Figure 4-7, whether or not SSI is app considered, the influence pattern of ground motion spatial variation on the required separation distances does not change too much. Tae Δ 3 in Figure 4-7a for example, when SSI is not considered, ground motion spatial variation effect is most significant when f / f is close to unity, where wealy correlated ground motions result in larger required separation distance. The ground motion spatial variation effect is, however, not so pronounced if the vibration frequencies of the two spans differ significantly. In these situations, the out-of-phase vibration of the two spans owing to their different frequencies contributes most to the relative displacement of the adjacent bridge decs. When SSI is included, similar results can be observed. Similar observations can be made on Δ and Δ. These observations are also similar to those reported in [] where SSI is not considered. As shown, the influence of changing spatial ground motions from highly correlated to wealy correlated has insignificant effect on the relative displacement responses. This is because the local site effects dominate the structural responses, as discussed above. Under a uniform site condition at the multiple bridge supports, the effect of cross correlation between spatial ground motions will be more prominent as observed in many previous studies e.g., []. 4-7

106 Chapter 4 School of Civil and Resource Engineering Figure 4-7. Influence of ground motion characteristics and SSI on the required separation distances f =. 0 Hz a Δ 3, b Δ and c Δ To see the influence of SSI more clearly, Figures 4-8 and 4-9 show the subtracted displacements between the results with and without consideration of SSI effect for soft f =. 0Hz and stiff f =. 0 Hz left span bearings, respectively. It is obvious in Figure 4-9 that the effect of SSI is most significant at f / f 0. 5, or f =. 0. As discussed above, natural vibration frequency of the 50 m medium site is about.0 Hz. At this frequency ratio, the right span resonates with the local site, which results in large responses. Large right span response also affects the response of the left span through the coupling at the pier, but at a less scale. Figure 4-8, however, does not show the same response phenomenon. For Δ in Figure 4-8b, similar observation can be obtained when right span resonates with the site at f / f. 0 or f =. 0. For Δ 3 and Δ in Figure 4-8a and c, however, no such peas can be observed. This is because both the left and right spans resonate with the site at f f. 0. Since the two adjacent spans tend to vibrate in-phase / at this vibration frequency, the relative displacement response Δ 3 is the smallest, and the effect of SSI is also insignificant. As for Δ, it shows a fluctuation when f / f is close to unity as discussed in the previous section. It is interesting to observe that with an increase in the spatial variability of the ground motion, the required separation distance becomes less sensitive to the dynamic interaction, i.e., SSI effect becomes more pronounced when the spatial ground motions have higher correlations, whereas the SSI effect is less prominent if the spatial ground motion is less correlated. These results are in agreement with that of Shrihande and Gupta [], where they investigated the influences of ground motion spatial variation and SSI on the bending moment at the mid-point of the centre span of a suspension bridge based on the stochastic approach in the frequency domain. These observations are, however, not fully consistent 4-8

107 Chapter 4 School of Civil and Resource Engineering with those in [4], in which it was concluded that SSI effect is more prominent only when the spatial ground motions are highly correlated in the range of f / f between 0.55 and.0. The later study [4] was carried out in the time and Laplace domain by using 0 sets of stochastically simulated time histories as inputs, which may give biased results because of the limited number of simulations. The results obtained in this study demonstrate the importance of considering SSI effect, especially when the spatial ground motions are highly correlated. Figure 4-8. Contribution of SSI to the required separation distances with different coherency loss functions f =. 0 Hz a Δ 3, b Δ and c Δ Figure 4-9. Contribution of SSI to the required separation distances with different coherency loss functions f =. 0 Hz a Δ 3, b Δ and c Δ 4.5 Conclusions Based on the fixed base assumption, Reference [] investigated the combined effect of ground motion spatial variation and site effect on the required separation distances that modular expansion joints MEJs must provide to prevent seismic pounding. This paper is an extension of Reference [] by including the SSI effect using the substructure method. The soil surrounding the pile foundation is modelled by equivalent frequency-dependent horizontal and rotational spring-dashpot systems. The combined effect of site condition 4-9

108 School of Civil and Resource Engineering Chapter 4 and SSI, combined effect of ground motion spatial variation and SSI are investigated, and the SSI effect is highlighted. Following conclusions are obtained:. The influence of SSI on the required separation distances is significant. Larger separation distances to avoid seismic pounding are usually required when SSI is considered.. SSI effect can not be neglected especially when the structures are founded on soft site. The contribution of SSI is relatively small when firm site is considered. 3. SSI effect is most evident when the structure resonates with local site. 4. SSI effect on the required separation distances is more prominent when the spatial ground motions are highly correlated. Otherwise ground motion spatial variation effect is more pronounced. Local site conditions are always important and should not be neglected for an accurate structural response analysis. 4.6 Appendix 4.6. Appendix A: Element for [ i ] z i + iξ z = + i = 0 3 i iξ 4 i = iξ 5 i = iξ Z and [ i] z = + / A4- z + z + / / Z g z i = 0 z i = + + iξ z 3 i = + iξ / A4- z 4 i = + iξ / z i = + i / 5 ξ z 3 i = α + iξ / z 3 i = β + i ξ / z 33 i = + α + iξ / + β + i ξ / + + α + β 3 + i3ξ 3 z 34 i = + α + iξ / + β + i ξ / A4-3 z i = + α + i ξ / + β + i / 35 ξ 4-0

109 Chapter 4 z 4 i = α + iξ / z i = β + i / 4 ξ 43 i + α + iξ / + β i ξ School of Civil and Resource Engineering z = + / A4-4 z i = + α + i ξ / + β + i ξ / + + α + β + i ξ 44 h h h 45 i = + α + iξ / + β i ξ / z + z 5 i = α + iξ / z i = β + i / 5 ξ 53 i + α + iξ / + β i ξ 54 i = + α + iξ / + β i ξ z = + / A4-5 z + z i = + α + i ξ / + β + i ξ / + + α + β + i ξ 55 r r r / z g i = + iξ / i 0 z g = z g 3 i = + iξ / A4-6 i z g = 0 z g i = + i / A4-7 ξ z g 3 i = + iξ / z g 3 i = α + iξ / z g i = β + i / A4-8 3 ξ z g 33 i = α + iξ / β + iξ / + z g 4 i = α + iξ / z g i = β + i / A4-9 4 ξ z g 43 i = α + iξ / β + iξ / + z g 5 i = α + iξ / z g i = β + i / A4-0 5 ξ z g 53 i = α + iξ / β + iξ / + 4-

110 School of Civil and Resource Engineering Chapter Appendix B: PSDs of the required separation distances [ ] [ ] [ ] Re Re Re i S i H i H i S i H i H i S i H i H i S i H i S i H i S i H S Δ = B4- [ ] [ ] [ ] Re Re Re i S i H i H i S i H i H i S i H i H i S i H i S i H S i H S Δ = B4- [ ][ ] { } [ ][ ] { } [ ][ ] { } Re Re Re i S i H i H i H i H i S i H i H i H i H i S i H i H i H i H i S i H i H i S i H i H i S i H i H S Δ = B4-3 where i S j 3, =,, j is the cross power spectral density function between points j and on the ground surface, which can be obtained from Reference []. i H j is the frequency response function, and can be determined by Equation References. Bi K, Hao H, Chouw N. Required separation distance between decs and at abutments of a bridge crossing a canyon site to avoid seismic pounding. Earthquae Engineering and Structural Dynamics 00; 393: Hall FJ, editor. Northridge earthquae, January 7, 994. Earthquae Engineering Research Institute, Preliminary reconnaissance report, EERI-94-0; 994.

111 School of Civil and Resource Engineering Chapter 4 3. Kawashima K, Unjoh S. Impact of Hanshin/Awaji earthquae on seismic design and seismic strengthening of highway bridges. Structural Engineering/Earthquae Engineering JSCE 996; 3: Earthquae Engineering Research Institute. Chi-Chi, Taiwan, Earthquae Reconnaissance Report. Report No.0-0, EERI, Oaland, California Elnashai AS, Kim SJ, Yun GJ, Sidarta D. The Yogyaarta earthquae in May 7, 006. Mid-America Earthquae Centre. Report No. 07-0, 57, Lin CJ, Hung H, Liu Y, Chai J. Reconnaissance report of 05 China Wenchuan earthquae on bridges. The 4 th World Conference on Earthquae Engineering. Beijing, China, 008; S Ruangrassamee A, Kawashima K. Relative displacement response spectra with pounding effect. Earthquae Engineering and Structural Dynamics 00; 300: DesRoches R, Muthuumar S. Effect of pounding and restrainers on seismic response of multi-frame bridges. Journal of Structural Engineering ASCE 00; 87: Janowsi R, Wilde K, Fujino Y. Pounding of superstructure segments in isolated elevated bridge during earthquaes. Earthquae Engineering and Structural Dynamics 998; 75: Zhu P, Abe M, Fujino Y. Modelling of three-dimensional non-linear seismic performance of elevated bridges with emphasis on pounding of girders. Earthquae Engineering and Structural Dynamics 00; 3: Hao H. A parametric study of the required seating length for bridge decs during earthquae. Earthquae Engineering and Structural Dynamics 998; 7: Zanardo G, Hao H, Modena C. Seismic response of multi-span simply supported bridges to spatially varying earthquae ground motion. Earthquae Engineering and Structural Dynamics 00; 36: Der Kiureghian A. A coherency model for spatially varying ground motions. Earthquae Engineering and Structural Dynamics 996; 5: Dumanogluid, AA, Soylu K. A stochastic analysis of long span structures subjected to spatially varying ground motions including the site-response effect. Engineering Structures 003; 50: Ates S, Bayratar A, Dumanogluid, AA. The effect of spatially varying earthquae ground motions on the stochastic response of bridges isolated with friction pendulum systems. Soil Dynamics and Earthquae Engineering 006; 6: Wolf JP. Dynamic soil-structure interaction. Englewood Cliffs, NJ: Prentice Hall;

112 School of Civil and Resource Engineering Chapter 4 7. Spyraos CC, Vlassis AG. Effect of soil-structure interaction on seismically isolated beiges. Journal of Earthquae Engineering 00; 63: Maris N, Badoni D, Delis E, Gazetas G. Prediction of observed bridge response with soil-pile-structure interaction. Journal of Structural Engineering 994; 00: Chouw N, Hao H. Study of SSI and non-uniform ground motion effect on pounding between bridge girders. Soil Dynamics and Earthquae Engineering 005; 5: Chouw N, Hao H. Significance of SSI and non-uniform near-fault ground motions in bridge response I: Effect on response with conventional expansion joint. Engineering Structures 008; 30: Shrihande M, Gupta VK. Dynamic soil-structure interaction effects on the seismic response of suspension bridges. Earthquae Engineering and Structural Dynamics 999; 8: Sextos AG, Pitilais KD, Kappos AJ. Inelastic dynamic analysis of RC bridges accounting for spatial variability of ground motion, site effects and soil-structure interaction phenomena. Part : Methodology and analytical tools. Earthquae Engineering and Structural Dynamics 003; 34: Sextos AG, Pitilais KD, Kappos AJ. Inelastic dynamic analysis of RC bridges accounting for spatial variability of ground motion, site effects and soil-structure interaction phenomena. Part : Parametric study. Earthquae Engineering and Structural Dynamics 003; 34: Chouw N, Hao H. Significance of SSI and non-uniform near-fault ground motions in bridge response II: Effect on response with modular expansion joint. Engineering Structures 008; 30: Chouw N, Hao H. Seismic design of bridge structures with allowance for large relative girder movements to avoid pounding. New Zealand Society for Earthquae Engineering Conference. Wairaei, New Zealand 008; Paper No: Tongaonar NP, Jangid RS. Seismic response of isolated bridges with soil-structure interaction. Soil Dynamics and Earthquae Engineering 003; 3: Fan K, Gazetas G, Kaynis A, Kausel E, Ahmad S. Kinematic seismic response of single piles and pile groups. Journal of Geotechnique Engineering 99; 7: Nova M, Sharnouby BL. Stiffness constants of single piles. Journal of Geotechnical Engineering 983; 097:

113 School of Civil and Resource Engineering Chapter 4 9. Gazetas G. In: Fang HY, editor. Foundation vibrations, foundation engineering handboo, nd edition 99; Gazetas G, Fan K, Kaynia A, Kausel E. Dynamic interaction factors for floating pile groups. Geotechnical Engineering 99; 70: Dobry R, Gazetas G. Simple method for dynamic stiffness and damping of floating pile groups. Geotechinique 988; 384: Der Kiureghian A. Structural response to stationary excitation. Journal of Engineering Mechanics 980; 066:

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115 Chapter 5 School of Civil and Resource Engineering Chapter 5 Modelling and simulation of spatially varying earthquae ground motions at a canyon site with multiple soil layers By: Kaiming Bi and Hong Hao Abstract: In a flat and uniform site, it is reasonable to assume the spatially varying earthquae ground motions at various locations have the same power spectral density or response spectrum. If a canyon site is considered, this assumption is no longer valid because of different local site amplification effect. This paper models and simulates spatially varying ground motions on surface of a canyon site in two steps. In the first step, the base roc motions at different locations are assumed to have the same intensity, and are modelled by a filtered Tajimi-Kanai power spectral density function or other stochastic ground motion attenuation models. The ground motion spatial variation is modelled by an empirical coherency loss function. The power spectral density functions of the surface motions on the canyon site with multiple soil layers are derived based on the deterministic wave propagation theory, assuming the base roc motions consist of out-of-plane SH wave or in-plane combined P and SV waves propagating into the site with an assumed incident angle. In the second step, a stochastic method to generate spatially varying time histories compatible with non-uniform spectral densities and a coherency loss function is developed to generate ground motion time histories on a canyon site. Two numerical examples are presented to demonstrate the proposed method. Each generated ground motion time history is compatible with the derived power spectral density at a particular point on the canyon site or response spectrum corresponding to the respective site conditions, and any two of them are compatible with a model coherency loss function. Keywords: ground motion simulation; spectral representation method; wave propagation theory; power spectral density function; response spectrum 5-

116 Chapter 5 5. Introduction School of Civil and Resource Engineering In the design of structures to resist strong earthquae ground excitations, properly define ground motions is crucial for a reliable analysis of structural responses. Besides ground motion time histories, ground motion response spectrum and power spectral density function are the most commonly used parameters to define earthquae action. For the point structures, owing to the dimensions of such structures are relatively small compared to the wavelengths of the seismic motions, it is reasonable to assume the ground motions over the entire structural base are the same. Many ground motion power spectral density functions have been developed by different researchers, e.g., the Tajimi-Kanai power spectrum model [] and the Clough-Penzien model []. Both of them were proposed by assuming the base roc excitation was a white noise random process, and the surface ground motion was estimated by calculating the responses of a single soil layer to the white noise excitation. Many stochastic ground motion models [3-5] have also been proposed by considering the rupture mechanism of the fault and the path effect for transmission of waves through the media from the fault to the ground surface. Local site effect, which amplifies and filters the incoming seismic waves and hence changes their amplitudes and the frequency contents, has also been intensively studied by many researchers. Wolf [6] presented a uniform approach in the frequency domain to analyze both the free-field response and the soil-structure interaction effect based on the wave propagation theory and finite element approach. Wolf [7] and Safa [8] also presented methods to model the propagating shear waves in layered media in the time domain. For large dimensional structures, such as long span bridges, pipelines, communication transmission systems, the ground motions at different stations during an earthquae are inevitably different, which is nown as the ground motion spatial variation effect. There are many reasons that may result in the spatial variability in seismic ground motions, e.g., the wave passage effect owing to the different arrival times of waves at different locations; the loss of coherency due to seismic waves scattering in the heterogeneous medium of the ground; the site amplification effect owing to different local soil properties. It has been proved that ground motion spatial variations have great influence on the structural responses and in some cases might even govern the structural responses. The ground motion spatial variations are usually modelled by a theoretical/semi-empirical power spectral density function and a coherency loss function. Many ground motion spatial variation models have been proposed especially after the installation of the SMART- array in Lotung, Taiwan. Zerva and Zervas [9] overviewed these models. It should be noted that 5-

117 School of Civil and Resource Engineering Chapter 5 most of these models were proposed based on the seismic data recorded from the relatively flat-lying sites. Taing different soil conditions into consideration, Der Kiureghian [0] proposed a theoretical coherency loss function, in which the ground motion power spectral density function was represented by a site-dependent transfer function and a white noise spectrum. Typical site-dependent parameters, i.e., the central frequency and damping ratio for three generic site conditions, namely, firm, medium and soft site were proposed. The advantage of the model is that it can consider different soil properties at different support locations and it is straightforward to use. The drawbac is that it can only approximately represent the local site effects on ground motions. For example, it is well nown that seismic wave will be amplified and filtered when propagating through a layered soil site. The amplifications occur at various vibration modes of the site. Therefore, the energy of surface motions will concentrate at a few frequencies. The power spectral density function of the surface motion then may have multiple peas. This phenomenon, however, cannot be considered in Der Kiureghian s model since only one pea corresponding to the fundamental vibration mode of the site is modelled. The ground motion power spectral density functions and spatial variation models can be used directly as inputs at multiple supports of structures in spectral analysis of structural responses. This approach, however, is usually applied to relatively simple structural models and for linear response of the structures owing to its complexity. For complex structural systems and for nonlinear seismic response analysis, only the deterministic solution can be evaluated with sufficient accuracy. In this case, the generation of artificial seismic ground motions is required. Many methods are also available to generate artificial spatially correlated time histories at different structural supports. Hao et al. [] presented a method of generating spatially varying time histories at different locations on ground surface based on the assumption that all the spatially varying ground motions have the same intensity, i.e., the same power spectral density or response spectrum. The variation of the spatial ground motions is modelled by an empirical coherency loss function and a phase delay depending on a constant apparent wave propagation velocity. If the considered site is flat with uniform soil properties, the uniform ground motion intensity assumption for spatial ground motions in the site is reasonable. However, for a canyon site or a site with varying soil properties, because local site conditions affect the wave propagation hence the ground motion intensity and frequency contents, the uniform ground motion power spectral density assumption is no longer valid. Deodatis [] developed a method to simulate spatial ground motions with different power spectral densities at different locations. The method is based on a spectral representation algorithm [3, 4] to generate sample functions of a 5-3

118 School of Civil and Resource Engineering Chapter 5 non-stationary, multivariate stochastic process with evolutionary power spectrum. Similar to the Der Kiureghian [0] model, the considered varying spectral densities are filtered white noise functions with different central frequency and damping ratio. This method thus can only approximately represent local site effects on ground motions as discussed above. Moreover, trying to establish an analytical expression for a realistic ground motion evolutionary power spectrum related to the local site conditions is quite difficult since very limited information is available on the spectral characteristics of propagating seismic waves [5]. On the other hand, many studies of site amplifications of seismic waves have also been reported. Taing the site amplification effect into consideration, Hao [6] developed a numerical method to calculate the site amplification effects on ground motion time histories by assuming seismic waves consisting of SH, and combined P and SV waves. Wang and Hao [7] then further extended this method to include the effects of random variation of soil properties on site amplifications of seismic waves. Some computer programs, such as SHAKE [8], EERA [9] and NERA [0], are also available to calculate site responses to incoming seismic waves and hence the ground motion time histories on the ground surface by solving the fundamental dynamic equations of motion in the frequency domain. It should be noted that these approaches only simulate ground motion time histories at one point on ground surface, ground motion spatial variations are not considered. Studies which consider both the ground motion spatial variation effect and the site amplification effect are limited. This paper combines the wave propagation theory [6] and the spectral representation method [3, 4] to derive the power spectral density functions of the spatially varying ground motion on surface of a canyon site with multiple soil layers. The ground motion spatial variations are modelled in two steps: firstly, the spatially varying base roc ground motions are assumed to consist of out-of-plane SH wave or in-plane combined P and SV waves and propagate into the layered soil site with an assumed incident angle. The spatial base roc motions are assumed to have the same intensity and frequency contents and are modelled by the filtered Tajimi-Kanai power spectral density function []. The spatial variation effect is modelled by an empirical coherency loss function []. The surface motions of a canyon site with multiple soil layers are derived based on the deterministic wave propagation theory. The auto power spectral density functions of ground motions at various points on ground surface and the cross power spectral density functions between ground motions at any two points are derived. The spectral representation method is then 5-4

119 School of Civil and Resource Engineering Chapter 5 used to generate spatially varying ground motion time histories compatible to the derived auto and cross power spectral density functions. Compared to the wor by Deodatis [], in this study the power spectral density functions at different locations of a canyon site are derived based on the wave propagation theory, which directly relates the local soil conditions and base roc motion characteristics with the surface ground motions, thus local site effect can be realistically considered. Besides the filtered Tajimi-Kanai power spectral density functions used in this study, other stochastic ground motion attenuation models for different regions can be straightforwardly used to model base roc motion. The current approach also allows for a consideration of different incoming wave types and incident angles to the soil site, which have great influence on the surface motions. For the completeness of the study, the ground motion time histories at a site with different soil properties represented by different response spectra is also considered in the paper, and a numerical example is given. The proposed approach can be used to simulate ground motion time histories at an uneven site with nown non-uniform site conditions. The simulated time histories can be used as inputs to long-span structures with multiple supports resting on site of varying conditions. 5. Wave propagation theory and site amplification effect The one-dimensional D wave propagation theory proposed by Wolf [6] is adopted in the present study to consider the influence of local site effect. It should be noted that the seismic waves are assumed to incident with an angle to the base roc and soil layer interface, and then propagate vertically in the soil layers in the D wave propagation theory, the scattering and diffraction of waves by canyons, which is a D wave propagation problem, are not considered. Further studies are needed to incorporate the scattering and diffraction effect into the simulation technique. For completeness, the D wave propagation theory is briefly introduced here. More detailed information can be found in Reference [6]. For a harmonic excitation with frequency, the dynamic equilibrium equations can be written as e = e or { Ω} = { Ω} c p c s 5-5-5

120 Chapter 5 School of Civil and Resource Engineering where e and { Ω} are the Laplace operator of the volumetric strain amplitude e and rotational-strain-vector { Ω }. c p and c s are the P- and S-wave velocity, respectively. This equation can be solved by using the P- and S-wave trial function. The out-of-plane displacements with the amplitude v is caused by the incident SH wave, while the in-plane displacements with the amplitude u and w in the horizontal and vertical directions depend on the combined P and SV waves. The amplitude v is independent of u and w, hence, the two-dimensional dynamic stiffness matrix of each soil layer for the out-of-plane and in- L L plane motion, [ S ] and [ ], can be formulated independently by analysing the SH SP SV relations of shear stresses and displacements at the boundary of each soil layer. Assembling the matrices of each soil layer and the base roc, the dynamic stiffness of the total system is obtained and denoted by [ S SH ] and [ S P SV ] equation of the site in the frequency domain is thus, respectively. The dynamic equilibrium [ S ]{ u } = { P } or S ]{ u } { P } SH SH SH [ 5- P SV P SV = P SV where { u SH } and { SH } to the incident SH wave, { u } and { } P are the out-of-plane displacements and load vector corresponding P SV P are the in-plane displacements and load P SV vector of the combined P and SV waves. The stiffness matrices [ S SH ] and [ S P SV ] depend on soil properties, incident wave type, incident angle and circular frequency. The dynamic load { P SH } and { } P depend on the base roc properties, incident wave type, P SV incident wave frequency and amplitude. By solving Equation 5- in the frequency domain at every discrete frequency, the relationship of the amplitudes between the base roc and each soil layer can be formed, and the site transfer function [ H ] at each soil layer can be estimated. In the present study, only the motion on the ground surface is of interest. To illustrate Equation 5-, a site consisting of a single homogeneous layer resting on a half-space is used as an example. The input on the base roc is assumed to be SH wave. It can be directly extended to more soil layers or combined P and SV waves. Assembling the L dynamic stiffness matrix of the layer [ S ] and of the half-space [ SH ] SH R S SH, the stiffness matrices S, out-of-plane displacements { u } and load vector { P } can be expressed as [6] SH SH 5-6

121 Chapter 5 [ S SH L * L L t G cos t d ] = L sin t d School of Civil and Resource Engineering L L cos t d + ip sin t d { u } SH = [ v, v ] t b T 5-3 R * R T { P } = [0, it G v ] SH 0 where is the wave number, t is a parameter related to the incident angle, G is the shear modulus, d is the depth of the soil layer, superscript R and L represents base roc and soil layer respectively, v t and v b are the displacement at the top and bottom of the soil layer, T denotes transpose, and i is the unit imaginary number. The above formulation can be easily extended to more soil layers by assembling the proper layer stiffness to the stiffness matrix. More detailed information for a multiple-layer site and for the case with combined P and SV wave can be found in Reference [6]. Substituting Equation 5-3 into Equation 5-, the ratio of surface motion v t to outcropping motion v 0 is vt H = = 5-4 v L i 0 L cos t d + sin t d p in which p R * G L * L = t G / t G is the impedance ratio. Considering linear elastic response only, the auto power spectral density functions of ground motions at various points on ground surface and the cross power spectral density functions between ground motions at any two points can be derived as S = H i * S i = H i H i S γ ij i ii j i g S g ' ' i j d ' ' i j i =,,..., n, i i, j =,,..., n 5-5 where H i i, H j i are the site transfer function at support i and j, respectively; superscript * denotes complex conjugate; S is the ground motion power spectral g 5-7

122 Chapter 5 density at the base roc; γ, i i ' j ' d i ' j ' School of Civil and Resource Engineering is the coherency loss function of spatial ground motions at the base roc, which is related to the distance between location i and j directly underneath the point i and j on ground surface as illustrated in Figure Ground motion simulation Spatial earthquae ground motions on the base roc are assumed as stationary random processes with zero mean values and having the same power spectral density function. This is a reasonable assumption since the distance from the source to the site is usually much larger than the dimension of the structure. The cross power spectral density function of ground motions at n locations in a site can be written as: S S i S n i S i S Sn i S i = 5-6 Sn i Sn i Snn where S and S ij i, i, j =,,..., n are the auto and cross power spectral density ii function respectively, defined in Equation 5-5. The matrix S i is Hermitian and positive definite, it can be decomposed into the multiplication of a complex lower triangular matrix L i and its Hermitian L H i : H S i = L i L i 5-7 The decomposition can be performed by using the Cholesy s method. The lower triangular matrix L i is in the following form: L 0 0 L i L 0 L i = 5-8 Ln i Ln i Lnn and 5-8

123 Chapter 5 Lii = Sii i = i i S Sij i Si i S = Lij i = S S jj i * i i * j i School of Civil and Resource Engineering / i =,,..., n 5-9 j =,,..., i After obtaining L i, the stationary time series u i t, i =,,..., n, can be simulated in the time domain as [] u t i i N = m= n= A im cos[ t + β + ϕ n n im n mn ] n 5-0 where A im β = tan im = 4Δ L im Im[ L Re[ L i, im im i], i] 0 N 0 N 5- are the amplitudes and phase angles of the simulated time histories which ensure the spectrum of the simulated time histories compatible with those given in Equation 5-6; ϕ mn n is the random phase angles uniformly distributed over the range of [ 0,π ], ϕ mn and ϕ should be statistically independent unless m = r and n = s ; rs N represents an upper cut-off frequency beyond which the elements of the cross power spectral density matrix given in Equation 5-6 is assumed to be zero; Δ is the resolution in the frequency domain, and n = nδ is the nth discrete frequency. Directly use Equation 5-0 to generate ground motion is quite time consuming. Ground motions can be generated more efficiently in the frequency domain based on the fast Fourier transform FFT technique. The Fourier transform of u i t is in the following form [] i Ui i n = Bim n[cosαim n + i sinαim n ], n =,,..., N 5- m= 5-9

124 Chapter 5 School of Civil and Resource Engineering where B im n is the amplitude at frequency n, and α im n is the corresponding phase angle, defined by: B im = A / α = β + ϕ im n n im im n n mn n 5-3 The corresponding time series u i t can be obtained by inverse transforming U i into the time domain. i n The time series generated by Equation 5-0 or 5- are stationary processes. In order to obtain the non-stationary time histories, an envelope function ζ t is applied to t, the non-stationary time histories at different locations are obtained by u i fi t = ζ t ui t, i =,,..., n 5-4 It should be noted that if the local site effect is not considered, the cross-power spectral density functions given in Equation 5-5 become Sij i = Sg γ ' ' d ' ', i, i, j =,,..., n 5-5 i j i j This is because H i i = H j i = when the site amplification effect is not considered. In this case the spatial ground motions will have the same power spectral density function S g, the spatial variation is modelled by the coherency loss function only. Then the above approach is the same as that proposed by Hao et al. []. In other words, it is a special case of the present study. In engineering practice, design response spectrum for a given site is more commonly available instead of the ground motion power spectral density function. Therefore it will be very useful to generate ground motion time histories that are compatible to the given design response spectrum. In previous wors of generating spatially varying ground motion time histories [, ], this is achieved by two steps. First the spatially varying ground motion time histories are generated using an arbitrary power spectral density function, and then adjusted through iterations to match the target response spectrum. Usually a few iterations are needed to achieve a reasonably good match [, ]. In this paper, a similar 5-0

125 School of Civil and Resource Engineering Chapter 5 approach is used. However, the ground motion power spectral densities that are related to the target design response spectra are derived first. The time histories are generated to be compatible with these power spectral densities. With this approach, the iterations might not be necessary for the simulated spatially varying time histories to be compatible to the multiple target response spectra. Even if iterations are needed, the response spectra of the simulated time histories converge to the target spectra faster. Therefore the current approach is computationally more efficient. The method proposed in this paper is introduced in the following. For a given acceleration response spectrum RSA, the corresponding power spectral density S can be estimated by [] ξ π S = RSA /ln ln p 5-6 π T where ξ is the damping ratio, T the time duration and p the probability coefficient, usually p []. Using the above approach, the generated time histories usually match well with the multiple f target response spectra. If the response spectra of the generated time histories RSA i do not match satisfactorily the target spectra, iterations need be carried out by adjusting the power spectral density function, which is done by multiplying S by the ratio f [ RSA / RSA i ], and perform the simulation again. This process can be repeated until satisfactory compatibility is achieved. Usually after 3 or 4 iterations, good match can be obtained, as compared to the method by Deodatis [], in which good match can be obtained usually only after more than 0 iterations. 5.4 Numerical examples An alluvium canyon site with multiple soil layers shown in Figure 5- is selected as an example, in which h is the layer depth, G is the shear modulus, ρ density, ξ damping ratio, υ Poisson s ratio and α incident angle. Ground motions on the base roc and on ground surface at three different locations indicated in the figure will be simulated in the study. 5-

126 Chapter 5 School of Civil and Resource Engineering 3 No.3 Silt sand, h=6m, G=0MPa, 3 ρ = 000g / m, ξ = 5%, υ = No. Sandy fill, h=5m, G=30MPa, 3 ρ = 900g / m, ξ = 5%, υ = No. Soft Clay, h=5m, G=0MPa, 3 ρ = 600g / m, ξ = 5%, υ = No.3 Silt sand, h=6m, G=0MPa, 3 ρ = 000g / m, ξ = 5%, υ = No.4 Firm clay, h=7m, G=30MPa, ρ = 600g / m, ξ = 5%, υ = α α α Base roc, G=800MPa, 3 ρ = 300g / m, ξ = 5%, υ = ' Figure 5-. A canyon site with multiple soil layers not to scale 5.4. Amplification spectra The site amplification effect is studied first. For conciseness, only the amplification spectra at site 3 are plotted. Figure 5-a shows the amplification spectra for the horizontal out-ofplane motions when SH wave propagates into the site with different incident angles. Figure 5-b and Figure 5-c show the amplification spectra for the in-plane horizontal and vertical motions with an assumption that the incoming waves consist of combined P and SV waves and the amplitude of the vertical motion is /3 of that of the horizontal component. As shown in Figure 5-, different incoming waves and incident angles significantly affect the site amplification spectra hence the surface motions in each direction. The site amplifies the incident waves at various frequencies corresponding to respective vibration modes of local site. Thus, the motions on the ground surface, which can be obtained by multiplying the power spectral density function of the base roc motion with the site amplification spectra at each discrete frequency, strongly depend on the local site conditions. Moreover, the ground motion power spectral density functions may consist of multiple distinctive peas associated with the multiple modes of the site. The commonly used filtered white noise is not able to represent the ground motion power spectral density with multiple peas. Table 5- gives the first two horizontal and vertical vibration frequencies of the site. 5-

127 Chapter 5 School of Civil and Resource Engineering Figure 5-. Amplification spectra of site 3, a horizontal out-of-plane motion; b horizontal in-plane motion; and c vertical in-plane motion Table 5-. First two vibration frequencies of the sites in the horizontal and vertical directions Site Horizontal Hz Vertical Hz Site Site Site To illustrate the proposed algorithms in the paper, two numerical examples are chosen to simulate spatially varying ground motion time histories at the three locations of the canyon site shown in Figure 5-. In the first example, the site amplification effect is included, the ground motion time histories are simulated to be compatible with the power spectral density functions modelled by Equation 5-5, and a coherency loss function. In the second example, the spatial ground motion time histories are simulated to be compatible with the multiple response spectra associated with the respective site conditions Example -PSD compatible ground motion simulation In this example, the ground motion time histories at different locations of the ground surface shown in Figure 5- are generated. The motion on the base roc is assumed to have the same intensity and frequency contents and is modelled by the filtered Tajimi-Kanai power spectral density function as 4 + 4ξ gg S g = H P S0 = Γ ξ + 4ξ f f f g g g 5-3

128 Chapter 5 School of Civil and Resource Engineering where H P is a high pass filter function [3], which is applied to filter out energy at zero and very low frequencies to correct the singularity in ground velocity and displacement power spectral density functions. S is the Tajimi-Kanai power spectral density function [], g and ξ g are the central frequency and damping ratio of the Tajimi- Kanai power spectral density function, f and ξ f are the central frequency and damping ratio of the high pass filter. In the analysis, the horizontal out-of-plane motion is assumed to consist of SH wave only, while the in-plane horizontal and vertical motion are assumed to be combined P and SV wave. The parameters of the horizontal motion are assumed as 3 g = 0πrad / s, ξ = 0. 6, f = 0. 5π, ξ f = 0. 6 and Γ = m / s. These parameters g correspond to a ground motion time history with duration T=0s and pea ground acceleration PGA 0 0.g and pea ground displacement PGD 0.08m based on the standard random vibration method [4]. The vertical motion on the base roc is also modelled with the same filtered Tajimi-Kanai power spectral density function, but the amplitude is assumed to be /3 of the horizontal component. It should be noted that if a specific site and an earthquae scenario is considered, a stochastic ground motion attenuation model can be easily used to replace the filtered Tajimi-Kanai power spectral density function to represent the specific base roc motion. The Sobczy model [] is selected to describe the coherency loss between the ground motions at points i ' ' and j i j at the base roc: γ ' ' ' ' ' ' ' ' ' ' i j i j i j app i j app i j i = γ i exp id cosα / v = exp βd / v exp id cosα / v 5-8 app in which, γ i ' ' i j is the lagged coherency loss, β is a coefficient which reflects the level of coherency loss, β = is used in the present paper, which represents highly correlated motions; d ' is the distance between the points i ' and i ' j ' j, and d = d 00m is ' ' ' ' = 3 assumed; α is the incident angle of the incoming wave to the site, and is assumed to be 60 ; v app is the apparent wave velocity at the base roc, which is 768m/s according to the base roc property and the specified incident angle. 5-4

129 School of Civil and Resource Engineering Chapter 5 To model the temporal variation of the simulated ground motions, the simulated stationary time histories are multiplied by the Jennings envelope function [5], which has the following form t / t0 0 t t0 ζ t = t < t t n exp[ 0.55 t tn] tn < t T with t0 = s and t n = 0s in this study. In the simulation, the sampling frequency and the upper cut-off frequency are set to be 00 Hz and N = 5Hz, and the time duration is assumed to be T=0s. To improve the computational efficiency, the ground motions are generated in the frequency domain by using the FFT technique as discussed above, and N=048 is used in the paper. The three generated horizontal base roc motions are shown in Figure 5-3a and 5-3b for acceleration and displacement respectively. The PGAs and PGDs of the simulated motions are.7,.,.33m/s and 0.086, , m respectively, which are close to the theoretical PGA of 0.g and PGD of 0.08 m. Figure 5-4 shows the comparisons of the power spectral densities of the generated time histories with the target filtered Tajimi-Kanai spectral density function. It shows that power spectral densities of the simulated motions match well with the target spectrum. Figure 5-5 shows the coherency loss functions between the generated time histories and the Sobczy model, good match can also be observed except for γ '3' in the high frequency range. This, however, is expected because as the distance increases, the cross correlation between the spatial motions or their coherency values decrease rapidly with the frequency. Previous studies e.g., [6]] revealed that the coherency value of about 0.3 to 0.4 is the threshold of cross correlation between two time histories because numerical calculations of coherency function between any two white noise series result in a value of about 0.3 to 0.4. Therefore the calculated coherency loss between two simulated time histories remains at about 0.4 even the model coherency function decreases below this threshold value. 5-5

130 Chapter 5 School of Civil and Resource Engineering Figure 5-3. Generated base roc motions in the horizontal directions a acceleration; and b displacement Figure 5-4. Comparison of power spectral density of the generated base roc acceleration with model power spectral density 5-6

131 Chapter 5 School of Civil and Resource Engineering Figure 5-5. Comparison of coherency loss between the generated base roc accelerations with model coherency loss function Assuming the incoming motions at the base roc consist of SH wave with an incident angle of o α = 60, the horizontal out-of-plane acceleration and displacement time histories on the ground surface are shown in Figure 5-6. It is obvious that the site amplification effects alter the frequency contents and increase the amplitudes of the incoming wave. Different wave paths result in different site amplification effect. For the given example, the PGAs and PGDs on the ground surface reach 4.3, 6.53, 3.3m/s and , , 0.07m at the three different locations as shown in Figure 5-6. As compared with the motions at the base roc, site significantly amplifies the horizontal out-of-plane ground acceleration. This is because the fundamental vibration frequency of site is 4.85Hz as given in Table 5-, which is very close to the central frequency of the filtered Tajimi-Kanai power spectral density function of ground motions at the base roc, so resonance occurs. Site and 3 also amplify the base roc motion, but with a less extent. It is interesting to find that, though the site amplifies the PGA of surface motions, it is not necessarily result in larger PGD. For the given example, the PGDs even decrease 34%, 37% and 5% respectively. This might attribute to the fact that local soil layers also filter the frequency content of the incoming waves. Although PGA of site is the largest, the PGD is the smallest because site is the stiffest among the three sites. On contrary, PGA of site 3 is the smallest, but PGD is the largest because site 3 is the softest. In general, the softer is the site, the larger is the PGD. Figure 5-7 shows the comparisons of the simulated power spectral densities with the theoretical values, good agreements are observed. 5-7

132 Chapter 5 School of Civil and Resource Engineering Figure 5-6. Generated horizontal out-of-plane motions on ground surface a acceleration; and b displacement For the coherency loss function between surface motions at a canyon site, the analysis based on the recorded seismic data showed larger variability than that on the flat-lying sites [7, 8]. Further studies revealed that local site effect not only causes phase difference of the coherency function [0], but also affects its modulus [9, 30]. Figure 5-8 shows the comparison of the lagged coherency loss functions of the base roc motions solid line with those of the simulated surface motions dashed line in Figure 5-6. As shown, the coherency loss between the surface motions is smaller than the corresponding base roc motion. These results are consistent with those obtained from recorded surface ground motions [7, 8]. Same results of coherency loss between in-plane surface ground motions, which are not shown here, are also obtained. They indicate wave propagation through nonuniform paths cause further coherency loss between spatial ground motions. More detailed 5-8

133 School of Civil and Resource Engineering Chapter 5 discussions of the influences of local site conditions on coherency loss of surface ground motions can be found in Reference [30]. Figure 5-7. Comparison of power spectral density of the generated horizontal out-of-plane acceleration on ground surface with the respective theoretical power spectral density Figure 5-8. Comparison of the coherency loss functions between base roc motions solid line and those between surface motions dashed line Assuming the incoming motions on the base roc are combined P and SV waves with the P wave incident angle o o 60 and SV wave incident angle 75.4, the horizontal and vertical inplane motions on the ground surface are generated. Figure 5-9 shows the generated horizontal in-plane acceleration and displacement time histories. The comparisons between the theoretically derived power spectral densities and those of the generated time histories are shown in Figure 5-0. As shown, the simulated time histories match the target spectral density functions well. It can also be noticed that the horizontal in-plane motions on the ground surface are similar to the simulated out-of-plane motions based on SH wave assumption. This is expected because both the SH and SV waves have the same characteristics as mentioned above. Figure 5- shows the simulated vertical in-plane ground motion time histories. The comparisons of the spectral density functions of the simulated time histories with the corresponding target functions are shown in Figure 5-. Good agreements are observed again. As expected, the simulated vertical motions are 5-9

134 School of Civil and Resource Engineering Chapter 5 different from the horizontal motions because the vertical motions have rather different spectral density functions from the horizontal motion owing to the different vertical vibration modes from the horizontal vibration modes of the site. As shown in Figure 5-, the PGAs and PGDs of the vertical motions are 3.,.78,.67m/s and 0.093, 0.089, m respectively for the three sites. Site amplifies the base roc motion most because the fundamental vertical vibration frequency of site is 5.5Hz as given in Table 5-, which is close to the central frequency of the filtered Tajimi-Kanai power spectral density function of ground motion at the base roc. Resonance results in the significant site amplification. Figure 5-9. Generated horizontal in-plane motions on ground surface a acceleration; and b displacement 5-0

135 Chapter 5 School of Civil and Resource Engineering Figure 5-0. Comparison of power spectral density of the generated horizontal in-plane acceleration on ground surface with the respective theoretical power spectral density Figure 5-. Generated vertical in-plane motions on ground surface a acceleration; and b displacement 5-

136 Chapter 5 School of Civil and Resource Engineering Figure 5-. Comparison of power spectral density of the generated vertical in-plane acceleration on ground surface with the respective theoretical power spectral density The above results also indicate that surface ground motion energy may concentrate at one or more frequency bands depending on the site vibration modes and the frequency of incident motions to the site. Multi-layered site amplifies seismic wave energy at frequencies around its vibration modes. Wave propagation in the soil site also results in a loss of spatial ground motion coherency. Therefore it is important to model the wave propagation in the local site to reliably predict surface ground motions Example -Response spectrum compatible ground motion simulation In this example, spatially correlated time histories on ground surface are generated to be compatible with the design spectra for different site conditions specified in the New Zealand Earthquae Loading Code [3]. The sub-soil classes at the three sites are assumed to be shallow soil Class C, roc Class B and deep/soft soil Class D, respectively. The pea ground acceleration PGA for the three sites is assumed to be m/s. The corresponding response spectra normalized to PGA of m/s in the New Zealand Earthquae Loading Code for the three sites are plotted in Figure 5-4 dashed line. The Sobczy model [] is again selected to describe the coherency loss between the ground motions at any two locations i and j. The seismic wave apparent velocity is assumed as 000m/s. It should be noted that the Sobczy model for spatial ground motion coherency is suitable for a flat site. As observed above and in a few previous studies [9, 30], this model overestimates coherency of spatial ground motions on surface of a canyon site. However, it is adopted here in this example to model spatial ground motion coherency loss because there is no suitable model available. Moreover, the coherency loss between spatial ground motions on a canyon site is not well understood yet. Some previous studies e.g., [0] also adopted the coherency model for ground motions on flat-laying site to 5-

137 School of Civil and Resource Engineering Chapter 5 model spatial ground motions on non-uniform site. If a proper coherency model was available, it could be easily implemented in the simulation procedure described above. Figure 5-3. Generated time histories according to the specified design response spectra a acceleration; and b displacement The shape function in the form of Equation 5-9 is applied to modulate the simulated stationary time histories. Figure 5-3 shows the generated acceleration and displacement time histories at the three locations on the ground surface after 4 iterations with the damping ratio ξ = and probability coefficient p = The sampling frequency and the upper cut-off frequency are set to be 00 and 5Hz, respectively. The time duration is T=0s. As shown in Figure 5-3, though the PGAs for the three sub-soils are almost the same, about m/s, as defined in the design spectra, the PGDs are very different because of the different frequency contents of ground motions corresponding to the different soil 5-3

138 School of Civil and Resource Engineering Chapter 5 conditions. The PGD of surface motion increases with the decrease in soil stiffness. In this example, the PGD of the three sites reaches , and 0.4m respectively. Figure 5-4 and Figure 5-5 show the response spectra and coherency loss function of the generated time histories and the prescribed models, good matches are observed. Figure 5-4. Comparison of the generated acceleration and the target response spectra Figure 5-5. Comparison of coherency loss between the generated time histories with the model coherency loss function 5.5 Conclusions This paper presents a method to model and simulate spatially varying earthquae ground motion time histories at sites with non-uniform conditions. It taes into consideration of the local site effects on ground motion amplification and spatial variation. The base roc motions can be modelled by using a filtered Tajimi-Kanai power spectral density function or a stochastic ground motion attenuation model. The site specific ground motion power spectral density function is derived by considering seismic wave propagations through the local site by assuming the base roc motions consisting of out-of-plane SH wave and inplane combined P and SV waves with an incident angle to the site. The spectral representation method is used to simulate the spatially varying earthquae ground motions. It is proven that the simulated spatial ground motion time histories are compatible with the respective target power spectral densities or design response spectra individually, and the 5-4

139 School of Civil and Resource Engineering Chapter 5 model coherency loss function between any two of them. This method can be used to simulate spatial ground motions on a non-uniform site with explicit consideration of the influences of the specific site conditions. It leads to a more realistic modelling of spatial ground motions on non-uniform sites as compared to the common assumption of uniform ground motion intensity in most previous studies. The simulated time histories can be used as inputs to multiple supports of long-span structures on non-uniform sites in engineering practice. 5.6 References. Tajimi H. A statistical method of determining the maximum response of a building structure during an earthquae. Proc. of nd World Conference on Earthquae Engineering, Toyo, Japan, 960; Clough RW, Penzien J. Dynamics of Structures. New Yor: McGraw Hill; Joyner WB, Boore DM. Measurement, characterization and prediction of strong ground motion. Earthquae Engineering and Structure Dynamics II-Recent Advances in Ground Motion Evaluation Proc GSP 0, Par City, Utah, 988; Atinson GM, Boore DM. Evaluation of models for earthquae source spectra in Eastern North America. Bulletin of the Seismological Society of America 998; 884: Hao H, Gaull BA. Estimation of strong seismic ground motion for engineering use in Perth Western Australia. Soil Dynamics and Earthquae Engineering 009; 95: Wolf JP. Dynamic Soil-structure Interaction, Englewood Cliffs, NJ: Prentice Hall; Wolf JP. Soil-structure interaction analysis in time domain, Englewood Cliffs, NJ: Prentice Hall; Safa E. Discrete-time analysis of seismic site amplification. Journal of Engineering Mechanics 995; 7: Zerva A, Zervas V. Spatial variation of seismic ground motions: An overview. Applied Mechanics Reviews 00; 563: Der Kiureghian A. A coherency model for spatially varying ground motions. Earthquae Engineering and Structural Dynamics 996; 5: Hao H, Oliveira CS, Penzien J. Multiple-station ground motion processing and simulation based on SMART- Array data. Nuclear Engineering and Design 989; :

140 School of Civil and Resource Engineering Chapter 5. Deodatis G. Non-stationary stochastic vector processes: seismic ground motion applications. Probabilistic Engineering Mechanics 996; 3: Shinozua M. Monte Carlo solution of structural dynamics. Computers and Structures 97; : Shinozua M, Jan CM. Digital simulation of random processes and its applications. Journal of Sound and Vibration 97; 5: Shinozua M, Deodatis G. Stochastic process models for earthquae ground motion. Probabilistic Engineering Mechanics 988; 33: Hao H. Input seismic motions for use in the seismic structural response analysis. The Sixth International Conference on Soil Dynamics and Earthquae Engineering, 993; Wang S, Hao H. Effects of random vibrations of soil properties on site amplification of seismic ground motions. Soil Dynamics and Earthquae Engineering 00; 7: Idriss IM, Sun JI. User s manual for SHAKE9, in User s Manual for SHAKE9, Department of Civil and Environmental Engineering, University of California at Davis, Baedet JP, Ichii K, Lin CH. EERA, a computer program for equivalent linear earthquae site response analysis of layered soil deposits, in EERA, A Computer Program for Equivalent Linear Earthquae Site Response Analysis of Layered Soil Deposits, University of Southern California, Baedet JP, Tobita T. NEAR, a computer program for nonlinear earthquae site response analysis of layered soil deposits, in NEAR, A Computer Program for Equivalent Linear Earthquae Site Response Analysis of Layered Soil Deposit, University of Southern California, 00.. Sobczy K. Stochastic Wave Propagation, Netherlands: Kluwer Academic Publishers; 99.. Kaul MK. Stochastic characterization of earthquae through their response spectrum, Earthquae Engineering and Structural Dynamics 978; 6: Ruiz P, Penzien J. Probabilistic study of the behaviour of structures during earthquaes. Report No. UCB/EERC-69-03, University of California, Bereley, Der Kiureghian A. Structural response to stationary excitation. Journal of Engineering Mechanics 980; 06:

141 School of Civil and Resource Engineering Chapter 5 5. Jennings PC, Housner GW, Tsai NC. Simulated earthquae motions. Report of Earthquae Engineering Research Laboratory, EERL-0, California Institute of Technology, Hao H. Effects of spatial variation of ground motions on large multiply-supported structures. Report No. UCB/EERC-89-06, University of California, Bereley, Somerville PG, McLaren JP, Sen MK, Helmberger DV. The influence of site conditions on the spatial incoherence of ground motions. Structural Safety 99; 0: Liao S, Zerva A, Stephenson WR. Seismic spatial coherency at a site with irregular subsurface topography. Proceedings of Sessions of Geo-Denver, Geotechnical Special Publication 007; pp Lou L, Zerva A. Effects of spatially variable ground motions on the seismic response of s sewed, multi-span, RC highway bridge. Soil Dynamics and Earthquae Engineering 005; 5: Bi K, Hao H. Influences of irregular topography and random soil properties on coherency loss of spatial seismic ground motions. Earthquae Engineering and Structural Dynamics 00, published online. 3. Standards New Zealand. Structural design actions, Part 5: Earthquae actions in New Zealand NZS ,

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143 Chapter 6 School of Civil and Resource Engineering Chapter 6 Influence of irregular topography and random soil properties on coherency loss of spatial seismic ground motions By: Kaiming Bi and Hong Hao Abstract: Coherency functions are used to describe the spatial variation of seismic ground motions at multiple supports of long span structures. Many coherency function models have been proposed based on theoretical derivation or measured spatial ground motion time histories at dense seismographic arrays. Most of them are suitable for modelling spatial ground motions on flat-lying alluvial sites. It has been found that these coherency functions are not appropriate for modelling spatial variations of ground motions at sites with irregular topography []. This paper investigates the influence of layered irregular sites and random soil properties on coherency functions of spatial ground motions on ground surface. Ground motion time histories at different locations on ground surface of the irregular site are generated based on the combined spectral representation method and onedimensional wave propagation theory. Random soil properties, including shear modulus, density and damping ratio of each layer are assumed to follow normal distributions, and are modelled by the independent one-dimensional random fields in the vertical direction. Monte-Carlo simulations are employed to model the effect of random variations of soil properties on the simulated surface ground motion time histories. The coherency function is estimated from the simulated ground motion time histories. Numerical examples are presented to illustrate the proposed method. Numerical results show that coherency function directly relates to the spectral ratio of two local sites, and the influence of randomly varying soil properties at a canyon site on coherency functions of spatial surface ground motions cannot be neglected. Keywords: coherency loss function; irregular topography; random soil properties; Monte- Carlo simulation 6-

144 Chapter 6 6. Introduction School of Civil and Resource Engineering For large dimensional structures, such as long-span bridges, pipelines, communication transmission systems, their supports inevitably undergo different seismic motions during an earthquae owing to the ground motion spatial variation. Past investigations indicate that the effect of the spatial variation of seismic motions on the structural responses cannot be neglected, and can be, in cases, detrimental []. Ground motion spatial variation effect has been extensively studied by many researchers especially after the installation of strong motion arrays e.g. the SMART- array in Lotung, Taiwan. Many empirical [3-7] and semiempirical [8-9] models have been proposed mostly for flat-lying alluvial sites. These coherency functions usually consist of two parts, the modulus or called lagged coherency, which measures the similarity of the seismic motions between the two stations, and the phase, which describes the wave passage effect, i.e., the delay in the arrival of the wave forms at the further away station caused by the propagation of the seismic wave. It is generally found that the lagged coherency decreases smoothly as a function of station separation and wave frequency. To consider local site effect, Der Kiureghian [0] proposed a theoretical model to describe coherency function of motions on the ground surface, in which he assumed that site effect influences the phase of the coherency function only, while it does not affect the lagged coherency. Contrasting to the observations on the flat-lying sites, Somerville et al. [] investigated the coherency function of ground motions on a site located on folded sedimentary rocs the Coalinga anticline, and found that the lagged coherency does not show a strong dependence on station separation and wave frequency, and the incoherency is generally higher than that on the flat-lying sites. They attributed the chaotic behaviour to the wave propagation in a medium having strong lateral heterogeneities in seismic velocity. Liao et al. [], based on the seismic data recorded at the Parway array in Wainuiomata Valley, New Zealand, compared the lagged coherency functions of different station combinations, i.e., four groups with station pairs located on the sediments, one group with one sedimentary station and one roc station. They concluded that the lagged coherency between the sediment and roc stations exhibit large variability and follow no consistent pattern. These observations suggest that the spatial coherency function measured on flat-lying sedimentary sites may not provide a good description of spatial ground motion coherencies on sites with irregular topography. These observations also indicate that the theoretical model proposed by Der Kiureghian [0] might not be able to reliably describe the influence of local site effect on the coherency function. 6-

145 School of Civil and Resource Engineering Chapter 6 Although it was observed that the heterogeneity of site conditions strongly affect the ground motion spatial variations [, ], all the previous studies and theoretical and empirical coherency models mentioned above assumed the site characteristics are fully deterministic and homogeneous. However, in reality, there always exist spatial variations of soil properties and uncertainties in defining the properties of soils. This results from the natural heterogeneity or variability of soils, the limited availability of information about internal conditions and sometimes the measurement errors. These uncertainties associated with system parameters are also liely to have influence on the coherency function. Zerva and Harada [] modelled horizontal soil layers at a site as a -DOF system with random characteristics to study the effect of uncertain soil properties on the coherency function. They pointed out that the spatial coherency of motions on the ground surface is similar to that of the incident motion at the base roc except at the predominant frequency of the layer, where it decreases considerably. The effect of uncertain soil properties should also be incorporated in spatial variation model of ground motions. Their explanation for this phenomenon was that for input motion frequencies close to the mean natural frequency of the oscillators, the response of the systems was affected by the variability in the value of this natural frequency, and resulted in loss of correlation []. However, it should be noted that a -DOF system cannot realistically represent the multiple predominate frequencies that may exist at a site with multiple layers and multiple modes. Liao and Li [3] developed an analytical stochastic method to evaluate the seismic coherency function, in which a numerical approach to compute coherency function is developed by combining the pseudo-excitation method with wave motion finite element simulation techniques. An orthogonal expansion method is introduced to study the effect of uncertain soil properties on the coherency function. The results also demonstrate that the lagged coherency values tend to decrease in the vicinity of the resonant frequencies of the site. This method is, however, difficult to be implemented and sometimes a little arbitrary to select the absorbing boundary conditions, and is difficult to explain why the lagged coherency function varies significantly over relatively short distances owing to the inherent limitations of using finite element method to model wave motion in a unbounded medium [4]. It is obvious that the effects of irregular topography and random soil properties of a site on the coherency function of spatial ground motions cannot be neglected. However, at the present, only very limited recorded spatial ground motion data on sites of different conditions are available. They are not sufficient to determine the general spatial incoherence characteristics of ground motions and derive empirical relations to model spatial ground motion variations at a site with varying site conditions. On the other hand, 6-3

146 School of Civil and Resource Engineering Chapter 6 to the best nowledge of the authors, no more theoretical/analytical analysis in this field can be found except for the studies mentioned above [0,, 3]. The present study investigates the influence of a layered canyon site and randomly varying soil properties on coherency function of spatial ground motions. The site is assumed consisting of horizontally extended multiple soil layers on a half-space base roc. The base roc motions at different locations are assumed to have the same intensity, and are modeled by a filtered Tajimi-Kanai power spectral density function. The spatial variation of ground motions on the base roc is accounted for by an empirical coherency function for spatial ground motions on a flat-lying site. Using the one-dimensional wave propagation theory [5], the power spectral density functions of spatial ground motions at various locations on surface of the canyon site can be derived by assuming the base roc motions consisting of out-of-plane SH wave or in-plane combined P and SV waves propagating into the site with an assumed incident angle. The spatially varying ground motion time histories can then be generated based on the spectral representation method. In order to tae into consideration the random soil properties, Monte-Carlo simulation method is used in the study. The random soil properties considered include the shear modulus, density and damping ratio of each layer, and they are all assumed to have normal distributions in the vertical direction and are modelled as independent one-dimensional random fields [6]. In numerical calculations, for each realization of the random soil properties, spatial ground motion time histories are generated. These time histories are then used to calculate the lagged coherency between any two ground motion time histories. The numerical calculations include the following steps: random generation of soil properties; estimation of ground motion power spectral density functions at various points on the canyon surface; 3 simulations of spatial ground motion time histories; and 4 calculations of coherency functions. These steps are repeated until the estimated mean and standard deviation of the lagged coherency between ground motions at any two points converge. Numerical examples are presented to demonstrate the proposed method and to study the effects of irregular topography and random soil properties on coherency function of spatial ground motions. 6-4

147 Chapter 6 6. Theoretical basis School of Civil and Resource Engineering 6.. Estimation of coherency function Let u j t and u t be the recorded simulated acceleration time histories at locations j and of a site, and the corresponding Fourier transform of the time histories are U and U, respectively. The smoothed auto spectral density function of ground motion at location j or is then j S ii M W mδ U + mδ i = j or n = 6- i n m= M and the cross power spectral density function between motions at stations j and is S j M = W mδ U + mδ U + mδ n 6- m= M j n n Δ is the frequency step, n = nδ is the n-th discrete frequency, and denotes the complex conjugate. where W is the spectral smoothing window, The coherency function of the spatial ground motions can be obtained as [4] S j γ j = 6-3 S S jj The coherency function in Equation 6-3 is generally a complex function and can be written as [ iθ ] γ = γ exp 6-4 j j j in which γ j is the lagged coherency, S j S Im θ j = tan is the phase angle, Re j Im and Re denote the imaginary and real part of a complex number. 6-5

148 School of Civil and Resource Engineering Chapter 6 Based on the analysis above, the coherency function can be readily estimated if the acceleration time histories at each location are available. The simulation of ground motion time histories is based on the one-dimensional wave propagation theory [7] and the spectral representation method. These two parts are briefly introduced in Sections 6.. and 6..3, more detailed information can be found in Reference [5]. 6.. One-dimensional wave propagation theory For a site with horizontally extended multiple soil layers on a half space base roc, the base roc motions can be assumed to consist of out-of-plane SH wave or in-plane combined P and SV waves propagating into a site with an assumed incident angle. For a harmonic excitation with frequency, the dynamic equilibrium equations can be written as [7] e = e or { } { } Ω = Ω c p c s 6-5 where e and { Ω} are the Laplace operator of the volumetric strain amplitude e and rotational-strain-vector { Ω }. c p and c s are the P- and S-wave velocity, respectively. This equation can be solved by using the P- and S-wave trial function. The out-of-plane displacements with the amplitude v is caused by the incident SH wave, while the in-plane displacements with the amplitude u and w in the horizontal and vertical directions depend on the combined P and SV waves. The amplitude v is independent of u and w, hence, the two-dimensional dynamic stiffness matrix of each soil layer for the out-of-plane and in- L L plane motion, [ S ] and [ ], can be formulated independently by analysing the SH SP SV relations of shear stresses and displacements at the boundary of each soil layer. Assembling the matrices of each soil layer and the base roc, the dynamic stiffness of the total system is obtained and denoted by [ S SH ] and ] [ SP SV equation of the site in the frequency domain is thus [7], respectively. The dynamic equilibrium [ S ]{ u } = { P } or S ]{ u } { P } SH SH SH [ 6-6 P SV P SV = P SV where { u SH } and { SH } to the incident SH wave, { u } and { } P are the out-of-plane displacements and load vector corresponding P SV P are the in-plane displacements and load P SV vector of the combined P and SV waves. The stiffness matrices [ S SH ] and [ S P SV ] depend 6-6

149 Chapter 6 School of Civil and Resource Engineering on soil properties, incident wave type, incident angle and circular frequency. The dynamic load { P SH } and { } P depend on the base roc properties, incident wave type, P SV incident wave frequency and amplitude. By solving Equation 6-6 in the frequency domain at every discrete frequency, the relationship of the amplitudes between the base roc and each soil layer can be formed, and the site transfer function [ H ] in the out-of-plane and in-plane directions can be estimated Ground motion generation Consider a canyon site with horizontally extended multiple soil layers resting on an elastic half-space as shown in Figure 6-, in which h m, G m, ρ m, ξ m and υ m is the depth, shear modulus, mass density, damping ratio and Poisson s ratio of layer m. The spatially varying base roc motions are assumed to consist of out-of-plane SH wave or in-plane combined P and SV waves and propagating into the layered soil site with an assumed incident angle as discussed above. The incident motions at different locations on the base roc are assumed to have the same power spectral density, and are modelled by a filtered Tajimi-Kanai [8] power spectral density function. The spatial variation of ground motions at base roc is modelled by an empirical coherency function for spatial ground motions on a flat site. The cross power spectral density functions of surface motions at n locations of the layered site can be written as: S S i S n i S i S Sn i S i = 6-7 Sn i Sn i Snn where S j S = H i * i = H i H i S γ j jj j g S g ' ' j d ' ' j j =,,..., n, i j, =,,..., n 6-8 are the auto and cross power spectral density function respectively. In which S is the ground motion power spectral density on the base roc; γ, i is the coherency j ' ' d j ' ' g function between location ' j and ' on the base roc; H j i, H i are the site transfer 6-7

150 Chapter 6 School of Civil and Resource Engineering function at locations j and on the ground surface, which can be formulated based on onedimensional wave propagation theory discussed in Section 6... j Layer m-:, m h m j Layer :, G, ρ, ξ, υ Base roc: M Layer l: hl, Layer m: hm, G, ρ, ξ, h G, B Figure 6-. Schematic view of a layered canyon site m ρ, ξ, B B m υ B υ m G, ρ, ξ, M l G, m l l υ ρ, ξ, m m l υ m Decomposing the Hermitian, positive definite matrix S i into the multiplication of a complex lower triangular matrix L i and its Hermitian L H i H S i = L i L i 6-9 the stationary time series u j t, j =,,..., n, can be simulated in the time domain directly [6] u t j j N = m= n= A jm cos[ t + β + ϕ n n jm n mn ] n 6-0 where A jm β = tan jm = 4Δ L jm Im[ L Re[ L i, jm jm i], i] 0 N 0 N 6- are the amplitudes and phase angles of the simulated time histories which ensure the spectra of the simulated time histories compatible with those given in Equation 8; ϕ mn n is the random phase angles uniformly distributed over the range of [ 0,π ], ϕ mn and ϕrs should be statistically independent unless m = r and n = s ; N represents an upper cut-off frequency beyond which the elements of the cross power spectral density matrix given in Equation 6-7 is assumed to be zero. 6-8

151 Chapter 6 School of Civil and Resource Engineering The generated time series by Equation 6-0 are stationary processes. In order to obtain the non-stationary time histories, an envelope function ζ t is applied to t. The nonstationary time histories at different locations are then obtained by u j f j t = ζ t u j t, j =,,..., n Random field theory In engineering practice there are always some uncertainties in the soil properties because of the reasons mentioned above. The random field theory [6] is widely used to describe the variability of soil properties. In this theory the random soil property u z is characterized by the mean value u, standard deviation σ u and the correlation distance δ u. σ u measures the intensity of fluctuation or degree to which actual value of u z may deviate from. δ u measures the correlation level or persistence of the property from one point to another in a site, small values of δ u suggest rapid fluctuation about the average, while large values of δ u imply a slowly varying component is superimposed on the average value of u. Consider a one-dimensional random field u z with mean value u z and standard deviation σ, its local average process u Z z of u z over the interval Z centered at z is defined as: u / u z = z Z Z u z' dz' Z z Z / It can be seen that the local average u Z z depends on the specific location of the interval z within the statistically homogeneous soil layer. The mean and variance of u Z z are [6] [ z ] = E[ u z ] E u Z Var = u z [ u z ] = σ λ Z Z u 6-4 where λ Z is a variance reduction function of u z 6-9, which measures the reduction of point variance σ under local average. The variance function λ Z can be derived from u auto-correlation function ρ Δz in the following form u

152 Chapter 6 School of Civil and Resource Engineering Z Δz λ Z = u Δz d Δz Z ρ Z By using the exponential auto-correlation function [9] ρ Δ z = exp Δz / δ 6-6 u u the variance reduction function can be derived as [9] Z / δu [ Z / δ + e ] λ Z = 6-7 u Z / δ u In this study, the shear modulus, density and damping ratio of each soil layer of the site are regarded as random fields, and are assumed to follow normal distributions in the vertical direction. These random fields can be modelled by introducing the mean value, standard deviation and correlation distance of each parameter as mentioned above. Tae shear modulus as an example + COV λ φ G = G + σ G λ Z φ = G Z 6-8 where G and σ are the mean value and standard deviation of shear modulus, λ Z is G the variance reduction function and φ is a normal distributed random process with zero mean and unity variance. COV = σ G / G is the coefficient of variation Monte-Carlo simulation Monte-Carlo simulations have been extensively used in many scientific fields with random parameters. It was found that for the range of variability usually present in soil properties, Monte-Carlo based method, though computationally intensive, might be the simplest and most direct method. Other methods, which are basically expansion based, do not provide accurate results when the coefficients of variation of soil properties are large [0]. In this study, Monte-Carlo simulations are also employed to account for the influence of random soil properties on spatial ground motions. In Monte-Carlo simulations, soil properties are randomly generated according to their distributions. Each set of random soil properties are considered as deterministic in estimating the power spectral densities of ground motions. 6-0

153 School of Civil and Resource Engineering Chapter 6 Then spatial ground motion time histories are simulated according to the procedures described above. 6.3 Numerical example To study the influence of irregular topography and random soil properties on the coherency function between different motions on the ground surface, a four-layer canyon site resting on the base roc is selected as an example as shown in Figure 6-. The mean values of the corresponding soil properties of each soil layer and base roc are also given in the Figure. The motions on the base roc are assumed to have the same intensities and frequency contents and are modelled by the filtered Tajimi-Kanai power spectral density function in the following form: 4 + 4ξ gg S g = H P S0 = Γ ξ + 4ξ f f f g g g where H P is a high pass filter function [], which is applied to filter out energy at zero and very low frequencies to correct the singularity in ground velocity and displacement power spectral density functions. S is the Tajimi-Kanai power spectral density 0 function [8], g and ξ g are the central frequency and damping ratio of the Tajimi-Kanai power spectral density function, f and ξ f are the corresponding central frequency and damping ratio of the high pass filter. Γ is a scaling factor depending on the ground motion intensity. In the analysis, the out-of-plane horizontal motion is assumed to consist of SH wave only, while the in-plane horizontal and vertical motions are assumed to be combined P and SV waves. The parameters for the horizontal motion are assumed as = 0π rad/s, ξ g = 0.6, f = 0. 5π, ξ = 0. 6 and Γ = m /s 3. These parameters correspond to a f ground motion time history with duration T = 0 s and pea ground acceleration PGA 0.g based on the standard random vibration method []. The vertical motion on the base roc is also modelled with the same filtered Tajimi-Kanai power spectral density function, but the amplitude is assumed to be /3 of the horizontal component of PGA 0.g. g 6-

154 Chapter 6 School of Civil and Resource Engineering No.4 Sandy fill, h=5m, G=30MPa, 3 ρ = 900g / m, ξ = 5%, υ = j No.3 Soft Clay, h=5m, G=0MPa, 3 ρ = 600g / m, ξ = 5%, υ = No. Silt sand, h=6m, G=0MPa, 3 ρ = 000g / m, ξ = 5%, υ = No. Firm clay, h=m, G=30MPa, 3 ρ = 600g / m, ξ = 5%, υ = j Base roc: G=800MPa, 3 ρ = 300g / m, ξ = 5%, υ = Figure 6-. A four-layer canyon site with deterministic soil properties not to scale The Sobczy model [3] is selected to describe the coherency loss between the ground motions at points j and on the base roc: γ ' ' ' ' ' ' ' ' ' ' j j j app j app j i = γ i exp id cosα / v = exp βd / v exp id cosα / v 6-0 app where β is a coefficient reflecting the level of coherency loss, β = is used in the present paper, which represents intermediately correlated motions; d ' is the distance j ' between the points j and, and d = 00 m is assumed; α is the incident angle of the j ' ' incoming wave to the site, and is assumed to be 60 ; v app is the apparent wave velocity on the base roc, which is 768 m/s according to the base roc property and the specified incident angle. Seismic waves are assumed propagating vertically from the base roc to the ground surface. Tae the canyon site with deterministic soil properties as an example. Assuming the soil properties of each soil layer equal to their mean values as given in Figure 6-, the acceleration time histories on the base roc and the ground surface are simulated based on the procedures presented in Sections 6.. and The sampling frequency and the upper cut-off frequency are set to be 00 Hz and = 0 Hz, respectively. 048 sampling points are used in each set of ground motion time histories. As ϕ mn in Equation 6-0 is a random variable uniformly distributed over the range of [ 0,π ], any realization of a random angle ϕ mn will result in a generation of a set of spatial ground acceleration time histories which are compatible with the spectral density function in Equation 6-8. Figure 6-3 shows one set of the simulated acceleration time histories. 6- N

155 Chapter 6 School of Civil and Resource Engineering Figure 6-3. Simulated acceleration time histories The coherency function between different motions on the ground surface can be estimated after the generation of acceleration time histories. However, it needs to be emphasized that coherency estimates depend strongly on the type of the smoothing window and the amount of smoothing performed on the raw data. Abrahamson et al. [4] noted that the choice of the smoothing window should be directed not only from the statistic properties of the ground motion time histories, but also from the problem for which it is analysing, so that the required resolution is not lost. They suggested an -point Hamming window, if the coherency estimates is to be used in structural analysis, for time windows less than approximately 000 samples and for structural damping coefficient 5% of critical [4]. It should also be noted that if no smoothing is performed on the raw data, the lagged coherency will always be unity for each frequency, and no information about the coherency can be extracted from the data. To obtain the mean lagged coherency functions on the base roc and ground surface, Monte-Carlo simulation method is used as discussed in Section Convergence test needs to be conducted to chec the number of Monte-Carlo simulations required to obtain converged simulation results. Since a larger number of Monte-Carlo simulations is required for the simulation to converge if the random variables under consideration have larger 6-3

156 School of Civil and Resource Engineering Chapter 6 COV, the case with the largest COV considered in this study, i.e., a COV of 60% for shear modulus and damping ratio of each soil layer and 5% for soil density, which will be further discussed in Section 6.3., is used to perform the convergence test. The mean values and standard deviations of the lagged coherency function of the horizontal out-of-plane motion at 0.Hz,.0Hz, 5.0Hz and 9.0Hz are used as the quantity for convergence test. As shown in Figure 6-4, the corresponding values virtually unchanged after 600 simulations, indicating the simulations converged with 600 simulations. Results of the simulated inplane motions, which are not shown, also converge after 600 simulations. Therefore, 600 simulations are performed for each case in the subsequent calculations. Figure 6-5 shows the comparison between the mean lagged coherency functions from the 600 simulated spatial ground motion time histories on the base roc smoothed by the -point Hamming window with the target model. It is evident that very good agreement can be obtained except for the frequencies near zero. In fact, theoretically, coherency should tend to be unity as frequency tends to zero, however, coherency estimates from ground motion time histories, due to smoothing, can rarely reach this value. Figure 6-4. Mean values and standard deviations of the lagged coherency of the horizontal out-of-plane motion at 0.,.0, 5.0 and 9.0Hz Figure 6-5. Comparison of the mean lagged coherency on the base roc from 600 simulations with the target model 6-4

157 Chapter Influence of irregular topography School of Civil and Resource Engineering Assuming all the soil properties are deterministic and equals to their mean values, the influence of irregular topography is studied first. Figure 6-6 shows the mean values of the lagged coherency functions between the spatial ground motions of points j and on the ground surface of the canyon site. For comparison purpose, the lagged coherency between incident motion on the base roc at j and is also plotted. Figure 6-7 shows the corresponding standard deviations. As shown, the standard deviations have a general trend of increasing with frequency, but are relatively small, all less than 0.3. This indicates that the lagged coherency is more difficult to be accurately modelled at high frequencies. Nonetheless, as the standard deviations are relatively small as compared to the mean lagged coherency values, including them will change the lagged coherency value, but not the overall trend. Figure 6-6 shows that the coherency function between surface ground motions differs from that between base roc motions significantly. At all frequencies, the coherency loss functions on the ground surface are smaller than those on the base roc, i.e., the coherency function on the base roc is the upper bound of the coherency of spatial ground motions on the surface of a canyon site. This conclusion is in agreement with that of Lou and Zerva [5], and Liao et al. []. It indicates that wave propagation through a local site even with deterministic site properties further reduces the cross correlation between spatial ground motions on the base roc. As shown, there are many obvious peas and troughs in the coherency function of surface motions. These peas and troughs directly relate to the modulus of the spectral ratio of two local sites, namely H i / H j i, as shown in Figure 6-8. H j i and H i are the transfer functions of site j and respectively. They are the spectral ratio of the surface motion at j or to the corresponding bedroc motion at j or, which can be calculated based on the onedimensional wave propagation theory as discussed in Section 6... Figure 6-9 shows the modulus of the transfer functions at sites j and. It is obvious that site amplifies the motions on the base roc significantly, which maes the energy of surface ground motions concentrate at a few frequencies corresponding to the various vibration modes of the site. This result indicates the importance of considering the multiple modes of a local soil site when estimating the seismic wave propagation and site amplification. The present result is an extension of those obtained with a -DOF model []. With a -DOF model, the influence of the higher vibration modes of the site on site amplification and hence the spatial ground motion coherency cannot be included. Comparing Figure 6-6 with Figure 6-8, it can be noted that when the spectral ratios differ from each other, the spatial ground motions on the ground surface are least correlated with a minimum lagged coherency value. 6-5

158 School of Civil and Resource Engineering Chapter 6 Taing the horizontal out-of-plane motion as an example, four obvious minima can be observed around the frequencies 0.78,.90, 4.0 and 7.0 Hz, which correspond to the four evident peas in the spectral ratio as shown in Figure 6-8a. Similar conclusions can be obtained for the in-plane motions. This is expected because the lagged coherency measures the similarity of the motions at two different locations. If two sites amplify the ground motions to the same extent at certain frequencies, the coherency loss is mainly caused by the incoherence effect and wave passage effect, local site effect has little influence on the lagged coherency. However, if the site amplification spectra are different from each other at certain frequencies, the local site effect on wave propagation is different. Therefore surface ground motions will be different at these frequencies, which results in spatial surface ground motions less correlated. These observations coincide with the recorded data from the Coalinga anticline in California [] and the Wainuiomata Valley in New Zealand []. These observations also indicate that site effect will not only cause phase difference of the coherency function [0], but will also affect its modulus. Liao and Li [3] used the auto-power spectral density of ground motion at one location of the site to identify the lagged coherency function on the ground surface, and concluded that the surface layer irregularity of a site can reduce the lagged coherency function values in the vicinity of the resonant frequencies of the site. To examine their observation, the horizontal out-of-plane motion of site j is used as an example. The fundamental vibration frequency of the site is about.5 Hz as shown in Figure 6-9a. According to Liao and Li s conclusion, the lagged coherency should have a minimum value at this frequency. However, the present results actually display a pea value in the lagged frequency at this frequency as shown in Figure 6-6a. This contradicts with Liao and Li s conclusion. This is because in the present example, both wave paths from j to j and to or both sites amplify the bedroc motion around this frequency, although to a different extent. Therefore wave propagation through the two sites does not significantly reduce the cross correlation of spatial bedroc motions at this frequency. This observation demonstrates that using the amplitude of the power spectral density of ground motion at just one location to assess the influence of wave propagation in an canyon site and hence the coherency function of spatial surface ground motions may not lead to a reliable coherency estimation. The spectral ratio between the two considered sites or two wave paths is a more reliable and appropriate parameter to measure the local site effect on cross correlation of spatial surface ground motions. 6-6

159 Chapter 6 School of Civil and Resource Engineering Figure 6-6. Comparison of the mean lagged coherency between the surface motions j, with that of the incident motion on the base roc Figure 6-7. Standard deviations of the lagged coherency on the ground surface Figure 6-8. Modulus of the site amplification spectral ratio of two local sites Figure 6-9. Amplitudes of the site amplification spectra of two local sites 6-7

160 Chapter Influence of random soil properties School of Civil and Resource Engineering The influence of randomly varying soil properties on the coherency loss functions between the surface motions is studied in this section. Without losing generality, assuming shear modulus, damping ratio and soil density are random fields in all soil layers, and all follow a normal distribution. The mean values of soil properties in every layer are given in Figure 6-. According to a more specific review and summary [6], in most common field measurements, the coefficients of variation COV for the cohesion and undrained strength of clay and sand are in a range of 0% to 00%. The statistical variation of the soil density is, however, relatively small as compared with other soil parameters. Therefore, in the present study, it is assumed that the shear modulus and damping ratio have COV of 0%, 40% and 60% for all soil layers, while the COV of soil density is assumed to be 5% in all the cases. Vanmarce [6] studied the scale of soil fluctuation, and concluded that the correlation distance of various soils vary from 0.6 to 46m. For typical clay, it is about 5 m. The correlation distance of 4 m is used in the present paper. It should be noted that in the present study, only the random fluctuations of soil properties in the vertical direction are considered, those in the horizontal direction are neglected because seismic waves are assumed propagating vertically and modeled with the one-dimensional wave propagation theory. Figure 6-0. Influence of uncertain soil properties on the mean values of lagged coherency functions 6-8

161 Chapter 6 School of Civil and Resource Engineering Figure 6-. Influence of uncertain soil properties on the standard deviations of lagged coherency functions Figure 6-. Influence of uncertain soil properties on the mean spectral ratios of two local sites 6-9

162 Chapter 6 School of Civil and Resource Engineering Figure 6-0 and Figure 6- show the influence of random variations of soil properties on the mean values and standard deviations of the lagged coherencies of spatial surface motions. For comparison purpose, the corresponding values with deterministic soil properties COV=0, and that of the incident motion on the base roc are also plotted. As shown, the influence of random soil properties on the lagged coherency between the motions on the ground surface should not be neglected, especially for in-plane motions. The lagged coherency between the motions on the ground surface is smaller than the incident motion on the base roc as observed above. When the COV of soil properties is 0., the mean lagged coherencies are similar to those obtained by deterministic analysis. Increasing COV of soil properties in general leads to smaller lagged coherencies between the motions on the ground surface, but could result in larger coherency values at certain frequencies where the spectral ratios of the two sites differ from each other significantly as shown in Figure 6-. In this case, larger COV leads to smaller spectral ratios, which results in the relatively larger lagged coherency values. As shown in Figure 6-, larger COV of soil properties results in larger variations of the lagged coherency function on the ground surface, as expected. It should be noted that these observations are based on the simulated data from a canyon site. If a flat site is under consideration, and the randomness of soil properties in the horizontal direction is neglected, the two local sites amplify ground motions on the base roc to the same extent although randomness in the vertical direction is considered. In this case, the spectral ratios of two local sites equal unity, and the coherency function on the ground surface is then the same as that on the base roc incident motion. The random soil properties have no influence on the coherency function on the ground surface in this case. This observation proves again that the influences of local site on surface ground motion spatial variations depend on the similarity of the two wave paths. If the two wave paths are the same, local site will not affect the surface ground motion spatial variations Influence of random variation of each soil parameter To investigate the effect of random variation of each soil parameter on the lagged coherency function between different motions on the ground surface, assuming only one soil parameter, namely either shear modulus, soil density or damping ratio, is random, while the other two parameters are assumed to be deterministic in the calculation. The COVs for shear modulus and damping ratio are assumed to be 40% and the COV for soil density is assumed to be 5%. Figure 6-3 and Figure 6-4 shows the mean values and the corresponding standard deviations of the lagged coherency respectively. The corresponding 6-0

163 School of Civil and Resource Engineering Chapter 6 values with deterministic soil properties, and that between the incident motions on the base roc are plotted again for comparison purpose. As shown, mean values and standard deviations of the lagged coherency obtained by considering only the damping ratio or soil density as random parameter are almost the same as those with deterministic soil property assumption, indicating the influence of random damping ratio and soil density on lagged coherency is insignificant and can be neglected. On the other hand, the influence of the random variations of shear modulus is obvious especially for the horizontal motions. These results can be explained by the spectral ratios of the two local sites as shown in Figure 6-5, in which the influence of random damping ratio and soil density on the spectral ratios is insignificant while the influence of the shear modulus is pronounced. Because the lagged coherency function directly relates to the spectral ratios of two local sites as discussed above, this leads to the observations of lagged coherency functions in Figure 6-3 and Figure 6-4. It should be noted that all the results obtained above are based on the assumption of a correlation distance δ u of 4 m for a typical clay site. In fact, the correlation distance varies in a relatively wide range [6], when larger correlation distance is considered, similar conclusions can be obtained but more prominent variation will be observed. These results are not shown in the current paper owing to the page limit. Figure 6-3. Influence of each random soil property on the mean values of lagged coherency functions 6-

164 Chapter 6 School of Civil and Resource Engineering Figure 6-4. Influence of each random soil property on the standard deviations of lagged coherency functions Figure 6-5. Influence of each random soil property on the mean spectral ratios of two local sites 6-

165 Chapter Conclusions School of Civil and Resource Engineering This paper evaluates the influence of local site irregular topography and random soil properties on the coherency function between spatial surface motions. Following conclusions are drawn:. The coherency function between surface ground motions on a canyon site is different from that between base roc motions. The lagged coherency function on the base roc is the upper bound of that on the ground surface.. For a canyon site, the coherency function of spatial surface ground motions oscillates with frequency. The maximum and minimum coherency values are related to the spectral ratios of two local sites or two wave paths. When the spectral ratios of two local sites differ from each other significantly, the spatial ground motions on the ground surface are least correlated. The coherency function models for motions on a flat-lying site cannot be used to model that of motions on a canyon site. 3. The influence of random soil properties on the lagged coherency function depends on the level of variations of soil properties. In general, the more significant are the random variations of soil properties, the larger is the local site effect on spatial surface ground motion variations. The random variations of soil damping ratio and density have insignificant effect on the lagged coherency as compared to the random variations of shear modulus. It should be noted that the soil nonlinearities also affect the surface motion spatial variations, but are not considered in the present paper. It is suggested to monitor some canyon sites to chec the results obtained in the present paper. Further study is also needed to develop analytical or empirical relation of local site characteristics with ground motion spatial variations for easy use in engineering application. 6.5 References. Somerville PG, McLaren JP, Sen MK, Helmberger DV. The influence of site conditions on the spatial incoherence of ground motions. Structural Safety 99; 0:-3.. Saxena V, Deodatis G, Shinozua M. Effect of spatial variation of earthquae ground motion on the nonlinear dynamic response of highway bridges. Proceeding of th World Conference on Earthquae Engineering, Aucland, New Zealand,

166 School of Civil and Resource Engineering Chapter 6 3. Loh CH. Analysis of the spatial variation of seismic waves and ground movement from SMART- data. Earthquae Engineering and Structural Dynamics 985; 35: Harichandran RS, Vanmarce EH. Stochastic variation of earthquae ground motion in space and time. Journal of Engineering Mechanics 986; : Loh CH, Yeh YT. Spatial variation and stochastic modelling of seismic differential ground movement. Earthquae Engineering and Structural Dynamics 988; 64: Hao H, Oliveira CS, Penzien J. Multiple-station ground motion processing and simulation based on SMART- array data. Nuclear Engineering and Design 989; 3: Harichandran RS. Estimating the spatial variation of earthquae ground motion from dense array recordings. Structural Safety 99; 0: Luco JE, Wong HL. Response of a rigid foundation to a spatially random ground motion. Earthquae Engineering and Structural Dynamics 986; 46: Somerville PG, McLaren JP, Saiia CK, Helmberger DV. Site-specific estimation of spatial incoherence of strong ground motion. Earthquae Engineering and Structural Dynamics II-Recent Advances in Ground Motion Evaluation, ASCE Geotechnical Special Publication No. 0, 988; Der Kiureghian A. A coherency model for spatially varying ground motions. Earthquae Engineering and Structural Dynamics 996; 5: Liao S, Zerva A, Stephenson WR. Seismic spatial coherency at a site with irregular subsurface topography. Proceedings of Sessions of Geo-Denver, Geotechnical Special Publication No. 70, 007; -0.. Zerva A, Harada T. Effect of surface layer stochasticity on seismic ground motion coherence and strain estimations. Soil Dynamics and Earthquae Engineering 997; 6: Liao S, Li J. A stochastic approach to site-response component in seismic ground motion coherency model. Soil Dynamics and Earthquae Engineering 00; : Chen Y, Li J. Effect of random media on coherency function of seismic ground motion. World Earthquae Engineering 007; 33:-6 in Chinese. 5. Bi K, Hao H. Simulation of spatially varying ground motions with non-uniform intensities and frequency content. Australia Earthquae Engineering Society 008 Conference, Ballart, Australia, 008; Paper No

167 School of Civil and Resource Engineering Chapter 6 6. Vanmarce EH. Probabilistic modelling of soil profiles. Journal of the Geotechnical Engineering Division 977; 03: Wolf JP. Dynamic soil-structure interaction. Prentice Hall: Englewood Cliffs, NJ, Tajimi H. A statistical method of determining the maximum response of a building structure during an earthquae. Proceedings of nd World Conference on Earthquae Engineering, Toyo, 960; Vanmarce EH. Random fields: analysis and synthesis. Cambridge: MIT Press, Yeh CH, Rahman MS. Stochastic finite element methods for the seismic response of soils. Internal Journal for Numerical and Analytical Methods in Geomechanics, 998; 0: Ruiz P, Penzien J. Probabilistic study of the behaviour of structures during earthquaes. Report No. UCB/EERC-69-03, University of California at Bereley; Der Kiureghian A. Structural response to stationary excitation. Journal of the Engineering Mechanics Division 980; 066: Sobczy K. Stochastic wave propagation. Netherlands: Kluwer Academic Publishers, Abrahamson NA, Schneider JF, Stepp JC. Empirical spatial coherency functions for applications to soil-structure interaction analysis. Earthquae Spectra 99; 7: Lou L, Zerva A. Effects of spatially variable ground motions on the seismic response of a sewed, multi-span, RC-highway bridge. Soil Dynamics and Earthquae Engineering 005; 5: Baecher GB, Chan M, Ingra TS, Lee T, Nucci LR. Geotechnical reliability of offshore gravity platforms. Report MITSG 80-0, Sea Grant College Program, Cambridge: MIT Press;

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169 Chapter 7 School of Civil and Resource Engineering Chapter 7 3D FEM analysis of pounding response of bridge structures at a canyon site to spatially varying ground motions By: Kaiming Bi, Hong Hao and Nawawi Chouw Abstract: Previous studies of pounding responses of adjacent bridge structures under seismic excitation were usually based on the simplified lumped mass model or beamcolumn element model. Consequently, only point to point pounding in D, usually the axial direction of the structures, could be considered. In reality, pounding could occur along the entire surfaces of the adjacent bridge structures. Moreover, spatially varying transverse ground motions generate torsional responses of bridge decs and these response will cause eccentric poundings. That is why many pounding damage occurred at corners of the adjacent decs as observed in many previous earthquaes. A simplified D model cannot capture torsional response and eccentric poundings. To more realistically investigate pounding between adjacent bridge structures, a two-span simply-supported bridge structure located at a canyon site is established with a detailed 3D finite element model in the present study. Spatially varying ground motions in the longitudinal, transverse and vertical directions at the bridge supports are stochastically simulated as inputs in the analysis. The pounding responses of the bridge structure under multi-component spatially varying ground motions are investigated in detail by using the transient dynamic finite element code LS-DYNA. Numerical results show that the detailed 3D finite element model clearly captures the eccentric poundings of bridge decs, which may induce local damage around the corners of bridge decs. It demonstrates the necessity of detailed 3D modelling for realistic simulation of pounding responses of adjacent bridge decs to earthquae excitations. Keywords: pounding response; eccentric pounding; torsional responses; 3D FEM; local site effect; spatially varying ground motions 7-

170 Chapter 7 7. Introduction School of Civil and Resource Engineering For bridge structures with conventional expansion joints, a complete avoidance of pounding between bridge decs during strong earthquaes is often impossible since the separation gap of an expansion joint is usually a few centimetres to ensure a smooth traffic flow. Therefore, pounding damages of adjacent bridge structures have always been observed in previous major earthquaes. In the 97 San Fernando earthquae, it was found that impacts between bridge decs and abutments were the source of extensive damages to highway bridges with seat type abutments []. In the 989 Loma Prieta earthquae, poundings between the lower roadway and columns supporting the upper-lever dec of the Southern viaduct section at the China Basin in California led to significant damage to the decs and column sides []. Reconnaissance reports from the 995 Kobe earthquae identified pounding as a major cause of fracture of bearing supports, which subsequently led to the unseating of bridge decs [3]. Surveys conducted after the 999 Chi-Chi Taiwan earthquae revealed that 30 bridges suffered some damages due to poundings at the expansion joints [4]. Poundings between adjacent bridge structures were also observed in the more recent 006 Yogyaarta earthquae [5] and 008 Wenchuan earthquae [6]. The most straightforward approach to avoid seismic pounding is to provide sufficient separation distances between adjacent structures. Previous studies on the required separation distances to avoid seismic pounding between adjacent structures mainly focused on buildings. Studies on the adjacent bridge structures are relatively less, probably because with conventional expansion joints it is not possible to provide sufficient separations between bridge decs while not affecting the smooth traffic flow as mentioned above. However, with the recent development of modular expansion joint MEJ in bridge engineering, the separation gap can be sufficiently large, which maes avoiding pounding possible. Hao [7] analysed the effect of various bridge and ground motion parameters on the relative displacement between adjacent bridge decs, and defined the required seating length for bridge decs to prevent unseating. Chouw and Hao [8] studied the influence of soil-structure interaction SSI and ground motion spatial variation effects on the required separation distance of two adjacent bridge frames connected by a MEJ. More recently, Bi et al. investigated local site effect [9] and SSI [0] on the required separation distances between bridge structures crossing a canyon site to avoid seismic pounding. 7-

171 School of Civil and Resource Engineering Chapter 7 Pounding is an extremely complex phenomenon involving damage due to plastic deformation, local cracing or crushing, fracturing due to impact, and friction when two adjacent bridge decs are in contact with each other. To simplify the analysis, many researchers modelled a bridge girder as a lumped mass. For example, Malhotra [] investigated a concrete bridge that experienced significant pounding during California earthquaes with a lumped mass model; Janowsi et al. [] presented an analysis of pounding between superstructure segments of an isolated elevated bridge induced by the seismic wave passage effect; Ruangrassamee and Kawashima [3] calculated the relative displacement spectra of two single-degree-of-freedom SDOF systems with pounding effect; DesRoches and Muthuumar [4] examined the factors affecting the global response of a multiple-frame bridge due to pounding of adjacent frames; Chouw and Hao [5, 6] studied the influence of ground motion spatial variation and SSI on the relative response of two bridge frames. Some other researchers modelled the bridge girders as beam-column elements. For example, Janowsi et al. [7] discretized the superstructure segments and piers as 3D elastic beam-column elements, and investigated several approaches for reducing the negative effects of pounding between superstructure segments of an isolated elevated bridge. Chouw et al. [8] modelled the girders and piers as D beam elements, and studied the effects of multi-sided poundings on structural responses due to spatially varying ground motions. Based on these simplified lumped mass model or beam-column element model, only D point to point pounding, usually in the axial direction of the structures, can be considered. In a real bridge structure under seismic loading, pounding could tae place along the entire surfaces of the adjacent structures. Moreover, it was observed from previous earthquaes that most poundings actually occurred at corners of adjacent bridge decs as shown in Figure 7-. This is because torsional responses of the adjacent decs induced by spatially varying transverse ground motions at multiple bridge supports resulted in eccentric poundings. To more realistically model the pounding phenomenon between adjacent bridge structures, a detailed 3D finite element analysis is necessary. Zanardo et al. [9] modelled the box-section bridge girders as shell elements and piers as beam-column elements, and carried out a parametric study of pounding phenomenon of a multi-span simply-supported bridge with base isolation devices. Julian et al. [0] evaluated the effectiveness of cable restrainers to mitigate earthquae damage through connections between isolated and non-isolated sections of curved steel viaducts using three-dimensional non-linear finite element response analysis. Although 3D FE models of bridge structures were developed in those two studies [9, 0], neither the surface to surface nor eccentric 7-3

172 School of Civil and Resource Engineering Chapter 7 pounding was considered, instead the pounding was simulated by the contact elements which lined the external nodes of adjacent segments together. Zhu et al. [] proposed a 3D contact-friction model to analyse pounding between bridge girders of a three-span steel bridge. This method overcomes the limitation of the previous studies that pre-define the pounding locations, therefore provides a more realistic modelling of pounding responses between bridge decs. The drawbac of the method is that it could not model material non-linearities during contacts. The tas to search contact pairs is also very time consuming and the searching algorithm is complicated. More recently, Janowsi [] analyzed the earthquae-induced pounding between the main building and the stairway tower of the Olive View Hospital based on the non-linear finite element method FEM, and concluded that the use of FEM with a detailed representation of the geometry and the non-linear material behaviour maes the study of earthquae-induced pounding more reliable than using the discrete lumped mass or beam-column element models. To the best nowledge of the authors, a simultaneous study of surface to surface, and torsional response induced eccentric pounding between adjacent bridge structures based on a detailed 3D FEM has not been reported yet. Figure 7-. A typical pounding damage between bridge decs in Chi-Chi earthquae Pounding between adjacent bridge decs occurs because of large relative displacement responses. Ground motion spatial variation, besides differences in vibration properties of adjacent bridge structures, is a source of relative displacement responses. Owing to the difficulty in modelling ground motion spatial variation, many studies assumed uniform excitations [, 3, 4, 0-] or assumed variation was caused by wave passage effect only [, 7]. Only a few studies considered the combined wave passage effect and coherency loss effect in analyzing relative displacement responses of adjacent bridge structures [5, 6, 8, 9]. It should be noted that all these studies mentioned above assumed that the analyzed structures locate on a flat-lying site, the influence of local site effect, which further intensifies ground motion spatial variation at multiple structural supports, are neglected. 7-4

173 School of Civil and Resource Engineering Chapter 7 Studies revealed that local site effect not only causes further phase difference [3, 4], but also affects the coherency loss between spatial ground motions [5]. These differences will significantly affect the structural responses [9, 0, 4]. Consequently, neglecting local soil effect on the spatial ground motion variations at multiple supports of a bridge structure crossing a canyon site may lead to inaccurate estimation of bridge responses. In this study, pounding responses between the abutment and the adjacent bridge dec and between two adjacent bridge decs of a two-span simply-supported bridge located on a canyon site are investigated. A detailed 3D finite element model of the bridge is constructed in ANSYS [6], and then LS-DYNA [7] is employed to calculate the structural responses. To model the local site effect on spatial ground motions, the base roc motions are assumed consisting of out-of-plane and in-plane waves and are modelled by a filtered Tajimi-Kanai power spectral density function and an empirical coherency loss function. Seismic waves then propagate vertically through local soil sites to ground surface. The three-dimensional spatially varying ground motions at different supports of the bridge structure are then stochastically simulated based on the combined spectral representation method and the one dimensional wave propagation theory. The simulated spatial ground motions are used as inputs to calculate structural responses. The influences of pounding effect, local soil condition and ground motion spatial variation effect on the structural responses are investigated in detail. It should be noted that the present study concentrates on modelling the surface to surface pounding and torsional response induced eccentric pounding. The material non-linearities and pounding induced local damage are not considered in the present study, which will be included in the subsequent studies. 7. Method validation A multi-span concrete bridge studied by Malhotra [] is selected to investigate the reliability of various models used in simulating pounding responses. They are a lumped mass SDOF model, a beam-column element model by using the contact element to simulate the pounding effect, and a 3D FE model. In [], Malhotra studied the collinear impact between two concrete rods based on the stereomechanic method, and then applied the procedure to the analysis of pounding effect of a 300m multi-span concrete bridge separated by an intermediate hinge. The bridge was simplified as two uncoupled SDOF systems as shown in Figure 7-a. The length, mass, column stiffness and damping ratio for the short span are L s =00m, m s =. 0 6 g, 7-5

174 School of Civil and Resource Engineering Chapter 7 s =07MN/m and ξ s =0.05, respectively. The corresponding parameters for the long span are L l =00m, m l = g, l =94MN/m and ξ l =0.05. These parameters correspond to the vibration frequencies for the short and long span of f s =.5 and f l =0.996Hz, respectively []. The separation gap between the short and long spans is 5cm. Using the stereomechanic method, the parameters given above are enough. However, for the beam-column element model and 3D finite element model, these parameters are insufficient. Therefore, the following parameters are also used based on the nown properties of the bridge []. They are: Young s modulus of the bridge decs and piers E=35GPa; densityρ=400g/m 3 ; rectangular cross section of the decs.5m with.5m in the transverse direction of the bridge; heights of the bridge piers h = 9m, with cross section m and 0.9.5m for the short and long span, respectively. Figure 7-b and c shows the beam-column element model and the detailed 3D finite element model, respectively. The beam-column model is constructed in ANSYS, and an impact element is used to model the pounding effect. The stiffness p and damping c p of the impact element are two important parameters that need to be determined. Previous investigation suggested a p varying from 0 to 40 times of the lateral stiffness of the stiffer adjacent structures [8]. p is assumed to be 5000 MN / m in the present study as suggested in [6]. The dashpot constant cp determines the energy dissipated during impact. It is determined by relating it to the coefficient of restitution e at pounding as follows []: with c p = ζ m m s l p p 7- ms + ml ln e ζ p = 7- π + ln e In the present study, e=0.46 is used [], which corresponds to a damping ratio of ζ = 0.4. p The 3D finite element model is constructed in ANSYS, but the calculations are carried out by using LS-DYNA. Eight-node solid elements of size 0.m are used for both decs and piers in the model. The treatment of sliding and impact along contact surfaces is an 7-6

175 School of Civil and Resource Engineering Chapter 7 important issue in the modelling. To realistically consider the poundings between entire surfaces of adjacent bridge decs, the contact type CONTACT AUTOMATIC SURFACE TO SURFACE in LS-DYNA is employed. This contact algorithm is used to avoid penetration at the contact interfaces. a b c Figure 7-. Different models not to scale: a lumped mass model from []; b beam-column element model; and c 3D finite element model The bridge is excited in the longitudinal direction only by the first 6.3s of the 940 North- South El Centro earthquae ground motion scaled to a pea ground acceleration PGA of 0.5g. All materials are assumed as linear elastic in the simulations. Figure 7-3 shows the structural responses obtained from the different models. As shown in Figure 7-3a, the relative displacements in the longitudinal direction between the adjacent bridge decs obtained by using the lumped mass model [] are generally smaller than those based on the beam-column model and detailed 3D finite element model. These results are actually expected, since the lumped mass model only considers the fundamental vibration mode of each uncoupled system, the contribution of higher vibration modes are not involved. For the long-span bridge structure, the vibration frequencies for different vibration modes are close to each other, the contribution of higher vibration modes could be significant. Both the beam-column model and 3D model capture the influence of higher vibration modes. As a result, more high frequency oscillations can be observed in Figure 7-3a. It also can be seen that the relative displacements based on these two models are very similar. The contact forces were not presented in Reference [] due to the limitation inherent in the 7-7

176 School of Civil and Resource Engineering Chapter 7 method, but it was found that poundings occurred at.0,.7, 3.7, 4.6, 5.5, 6.3 and 7.3s. Figure 7-3b shows the pounding forces based on the beam-column model and the 3D model. It can be seen that poundings occur at the time instants observed in []. The pounding forces obtained from the 3D model are usually larger than those from the beamcolumn model. It should be noted that the pounding force obtained from the beamcolumn model depends on the pounding stiffness p of the impact element, while the selection of p is difficult since it depends on many factors and consequently the value can be varied in a wide range [8]. With a proper selection of p, closer results are expected. a b Figure 7-3. Structural responses based on different models: a relative displacement and b pounding force Based on the above analysis, it can be concluded that if earthquae ground excitation occurs only in the longitudinal direction of the bridge, all these three models can be used to calculate bridge pounding responses. However, the lumped mass model might underestimate the relative displacements between adjacent bridge decs. The beam-column model based on the contact element method can give reliable predictions of pounding responses if a proper pounding element with suitable stiffness and damping ratio is used. Therefore, if considering only uniaxial ground excitation in the longitudinal direction of the bridge, detailed 3D model is not necessary as it requires considerably more computational effort. In reality, however, earthquae ground motion is not limited to only one direction. Bridge structures inevitably subject to the excitations of multi-component and spatially varying ground motions. Spatially varying transverse ground motions induce coupled transverse and torsional responses of bridge decs even the bridge structures are symmetric. The torsional response might induce eccentric poundings between adjacent bridge decs as observed in Figure, and eccentric poundings in turn will cause more 7-8

177 School of Civil and Resource Engineering Chapter 7 torsional responses. This 3D response characteristic cannot be captured with the lumped mass model or the D beam-column model. To realistically model 3D bridge responses involving possible surface-to-surface and eccentric poundings, the use of a 3D finite element model is therefore necessary. 7.3 Bridge model Figure 7-4a shows the elevation view of a two-span simply-supported bridge crossing a canyon site considered in this study. The box-section bridge girders with the cross section shown in Figure 7-4b have the same length of 50m. The Young s modulus and density of the girders are Pa and 500 g/m 3, respectively. The L-type abutment is 8.m long in the transverse direction and its cross section is shown in Figure 7-4c. The height of the rectangular central pier is 0m, with the cross section shown in Figure 7-4d. The materials for the two abutments and the pier are the same, with Young s modulus and density of Pa and 400 g/m 3, respectively. The two bridge girders are supported by 8 high-damping rubber bearings. The cross-sectional area and height of rubber layers in a single bearing are 0.79m and 0.08m. The horizontal effective stiffness and equivalent damping ratio of a bearing are N/m and 0.4 respectively [, 9]. The stiffness of the bearing in the vertical direction is much larger than those in the horizontal directions, and is assumed to be N/m [9]. To allow for contraction and expansion of the bridge decs from creep, shrinage, temperature fluctuations and traffic without generating constraint forces in the structure, a 5cm gap is introduced between the abutments and the bridge girders and between the adjacent bridge decs. It is noted that the lateral side stoppers, which are usually installed in practice, are not considered in the model. The bridge girders can vibrate freely in the transverse direction z direction when pounding is not involved. The bridge locates on a canyon site, consisting of horizontally extended soil layers on a half-space base roc. The foundations of the bridge are assumed rigidly fixed to the ground surface and SSI is not involved. Points A, B and C are the three bridge support locations on the ground surface, the corresponding points on the base roc are A, B and C. The 3D finite element model of the bridge is constructed by using the finite element code ANSYS [6]. The bridge girders, abutments and pier are modelled by eight-node solid elements. The bearings are modelled by the spring-dashpot elements. The detailed 7-9

178 School of Civil and Resource Engineering Chapter 7 geometric characteristics in Figure 7-4 and the material properties are implemented in the model. To reduce the required computer memory and computational time, detailed modelling with fine mesh is only applied to the areas near the contact surfaces. In particular, detailed modelling with the mesh size of 0.m is only applied to a length of m from each end of the bridge dec and to a length of 0.6m of the abutments. Beyond this region, the mesh size in the longitudinal direction is m. Figure 7-5 shows fine meshed areas of the model the numbers in the circles are the nodes examined in the present study, which will be discussed in Section 7.5. For a convergence test, a smaller mesh size of 0.m around the contact areas is also conducted. Numerical results show that the structural responses are almost the same for the two different mesh sizes. It should be noted that, only the linear elastic responses are considered in the present study, smaller mesh size might be needed if local damages are involved. Figure 7-6 shows the first four vibration frequencies and the corresponding vibration modes of the bridge. As shown, the first four vibration frequencies of the bridge equal to.08,.38,.54 and.33 Hz for the inphase longitudinal x direction, in-phase transverse z direction, out-of-phase transverse and out-of-phase longitudinal vibrations, respectively. Figure 7-4. a Elevation view of the bridge, b Cross-section of the bridge girder, c Cross-section of the abutment and d Cross-section of the pier unit: mm 7-0

179 School of Civil and Resource Engineering Chapter 7 Rayleigh damping is assumed in the model to simulate energy dissipation during structural vibrations.the first two vibration modes is chosen to determine the mass and stiffness coefficients, because the horizontal displacement in the longitudinal and transverse directions is of special interest due to its significant importance in the pounding responses. By assuming the structural damping ratio of 5%, for these two modes, the mass matrix multiplier is obtained as and the stiffness matrix multiplier is The contact algorithm of CONTACT AUTOMATIC SURFACE TO SURFACE in LS-DYNA is employed to model impact between the adjacent structures. The Coulomb friction coefficient of 0.5 is assumed in the analysis []. 3 Left abutment 4 7 Left girder 8 Left girder Right girder Right girder Right abutment Pier Figure 7-5. Finite element mesh of the bridge and the nodal points for response recordings a f =.08Hz b f =.38Hz c f 3 =.54Hz d f 4 =.33Hz Figure 7-6. First four vibration frequencies and mode shapes of the bridge 7.4 Spatially varying ground motions For the canyon site as shown in Figure 7-4, local site will significantly change the amplitudes and frequency contents of the incoming waves on the base roc owing to the amplification and filtering effect. The three sites A, B and C as shown in the figure have different influences on base roc motions, thus further intensifies the spatial variations of 7-

180 School of Civil and Resource Engineering Chapter 7 the ground motions. However, traditional method e.g., Hao et al. [9] to simulate the spatially varying ground motions is based on the flat-lying site assumption and the influence of local site effect is not considered. With such an assumption, ground motions at the three sites on ground surface have the same intensity and frequency contents. More recently, Bi and Hao [5] developed an approach to stochastically simulate the spatially varying motions on the ground surface of a canyon site. In the method, the base roc motions are assumed to consist of out-of-plane SH wave and in-plane combined P and SV waves propagating into the site with an assumed incident angle. The power spectral density function on the base roc is assumed to be the same, and is modelled by a filtered Tajimi- Kanai power spectral density function [30]. The spatial variation of ground motions at base roc is modelled by an empirical coherency function. Local site effect is modelled using the one-dimensional wave propagation theory [3]. The power spectral density functions of the horizontal in-plane, horizontal out-of-plane and vertical in-plane motions on the ground surface can thus be formulated by considering local site effect in the corresponding directions. The multi-component spatially varying ground motions can then be simulated by using the approach similar to the traditional method. This approach directly relates site amplification effect with local soil conditions, and can capture the multiple vibration modes of local site, is believed more realistically simulating the multi-component spatially varying motions on surface of a canyon site. The ground motion intensities at points A, B and C on the base roc are assumed to be the dame and have the following form: 4 = + 4ξ gg S g Γ ξ + 4ξ f f f g g g where g and ξ g are the central frequency and damping ratio of the Tajimi-Kanai power spectral density function, f and ξ f are the corresponding central frequency and damping ratio of the high pass filter function. Γ is a scaling factor depending on the ground motion intensity. In the analysis, the out-of-plane horizontal motion is assumed to consist of SH wave only, while the in-plane horizontal and vertical motions are assumed to be combined P and SV waves. The parameters for the horizontal motion are assumed as = 0π rad/s, ξ g = 0.6, f = 0. 5π, ξ = 0. 6 and Γ = m /s 3. These parameters correspond to a f ground motion time history with duration T=6s and PGA of 0.5g based on the standard random vibration method [3]. The vertical motion on the base roc is also modelled with 7- g

181 School of Civil and Resource Engineering Chapter 7 the same filtered Tajimi-Kanai power spectral density function, but the amplitude is assumed to be /3 of the horizontal component. The Sobczy model [33] is selected to describe the coherency loss between the ground motions at points j and where j, represents A, B or C on the base roc: γ ' ' ' ' ' ' ' ' ' ' j j j app j app j i = γ i exp id cosα / v = exp βd / v exp id cosα / v 7-4 app where β is a coefficient reflecting the level of coherency loss. β =0.0, 0.00 and 0.00 are considered in the present paper, which represent perfectly correlated spatial ground motions, or spatial ground motion with wave passage effect only, intermediately and wealy correlated motions, respectively. d j is the distance between the points j and. For the analysed bridge structure, d A B =d B C =50m, and d A C =00m. v app is the apparent wave velocity on the base roc, which is related to the base roc property and incident angle α. With the given properties of local site shown in Table 7- and assumed incident angle o α = 60, v app equals 697m/s in the present study. Not to further complicate the problem, only one single layer resting on the base roc is considered, and the soil properties at sites A, B and C are assumed to be the same, the only difference is the soil depth. In the present study, the depths for the three local sites are 48.6, 30 and 48.6m respectively. To study the influence of local soil conditions, two types of soil, i.e. firm and soft soils, are considered. Table 7- gives the corresponding parameters for the soils and base roc. It should be noted that to limit the considered influence factors, SSI is not considered even when the bridge model locates on a soft soil site. Table 7-. Parameters for local site conditions Type Density g/m 3 Shear modulusmpa Damping ratio Poisson s ratio Base roc Firm soil Soft soil With the proposed approach in [5] and the given parameters of local site, the horizontal in-plane, horizontal out-of-plane and vertical in-plane motions on the ground surface can be simulated. It should be noted that a series of random phase angles uniformly distributed over the range of [0, π] are included in the simulation. For each realization of the phase 7-3

182 School of Civil and Resource Engineering Chapter 7 angles, one set of ground motion time histories can be simulated. Since most design codes require to 4 independent analyses with independently simulated ground motions as input and tae the averaged structural responses, in this study, three sets of multi-component spatially varying ground motions are independently simulated and used as input in the analysis. In the simulation, the sampling frequency and the upper cut-off frequency are set to be 00 and 5 Hz respectively, and the time duration is assumed to be T=6s. Figures 7-7 and 7-8 show the simulated three-dimensional spatially varying acceleration and displacement time histories on ground surface corresponding to the soft soil conditions with intermediate coherency loss. Figure 7-9 shows the comparisons of the simulated power spectral densities with the theoretical values of the horizontal in-plane motions, good agreements are observed. For conciseness, the comparisons of the horizontal out-ofplane and vertical in-plane motions are not plotted. Good agreements for these two ground motion components are also observed. For the coherency loss function between the motions on the ground surface, Reference [5] indicates that it is different from that on the base roc. The spatial ground motions on ground surface are least correlated when the spectral ratios of two local sites differ from each other significantly. Discussion of the influence of local soil condition on spatial ground motion coherency loss is out of the scope of the present study. More detailed information can be found in Reference [5]. It should be noted that the simulated spatial ground motions corresponding to the firm soil condition also match the model values very well. Figure 7-7. Simulated acceleration time histories with soft soil condition and intermediately correlated coherency loss 7-4

183 Chapter 7 School of Civil and Resource Engineering Figure 7-8. Simulated displacement time histories with soft soil condition and intermediately correlated coherency loss Figure 7-9. Comparison of PSDs between the generated horizontal in-plane motions on ground surface with the respective theoretical model value 7.5 Numerical example The earthquae-induced pounding responses of the two-span simply-supported bridge as shown in Figure 7-4 are discussed in detail in this section. The simulated horizontal inplane, horizontal out-of-plane and vertical in-plane motions are applied simultaneously along the longitudinal, transverse and vertical directions of the bridge respectively as shown in Figure 7-0, where d xa, d ya and d za represents input displacement time histories in the x, y and z directions at site A. So as for sites B and C. All the calculations are carried out by using the transient dynamic finite element code LS-DYNA. The time step is automatically selected by the code so that converged results can be obtained. To investigate the 7-5

184 School of Civil and Resource Engineering Chapter 7 influences of pounding effect, local soil conditions and ground motion spatial variations on the structural responses, five different cases as shown in Table 7- are studied. In which, the case without pounding Case is simulated by adjusting the model to mae the separation gaps between the abutment and the girder and between two adjacent girders large enough so that pounding phenomenon can be completely precluded and the structure vibrates freely. Abutment Girder Girder dya dza dxa Pier dyc dzc dxc dyb dzb dxb Figure 7-0. Multi-components spatially varying inputs at different supports of the bridge Table 7-. Different cases studied Case Soil conditions Coherency loss With/without pounding Firm intermediately without Firm intermediately with 3 Soft intermediately with 4 Firm wave passage effect with 5 Firm wealy with Poundings may occur between the abutments and the adjacent bridge girders and between two adjacent bridge girders as mentioned above. Although the bridge considered is a symmetrical structure, the responses of different parts will be different owing to the ground motion spatial variations and pounding effects. To obtain a general idea of the earthquaeinduced structural responses, the nodes as indicated in Figure 7-5 are selected to record the results. Three simulations using the three sets of independently simulated spatially varying ground motions as inputs for each case are carried out in the present study, the mean pea responses, which are mostly concerned in engineering practice, are calculated and discussed. For a better understanding of the results, the time histories of the structural response corresponding to a particular set of ground motions are also plotted when necessary. 7-6

185 Chapter Longitudinal response School of Civil and Resource Engineering Figures 7-, 7- and 7-3 show the longitudinal displacement response time histories at nodes and of different ground motion cases. For conciseness, the response time histories of other nodes are not plotted. The mean pea displacements at different nodes are listed in Table 7-3. As shown in Figure 7-a, the longitudinal displacement response of node is almost unaffected by the poundings owing to the fact that the abutment is quite rigid as compared to the adjacent girder. Similar observations were obtained by Maragais et al. [34], who investigated the influences of abutment and dec stiffness, gap, and dec to abutment mass ratio on the pounding responses between abutments and bridge decs, and concluded that pounding effect on rigid abutment is not evident. The influence of collisions on the girder response is, however, significant. As shown in Figure 7-b, the pea displacements of node in the longitudinal direction with and without pounding effect are 0.0 and 0.74m respectively, poundings result in a reduction of displacement response by 3.4%. This is because the rigid abutment acts as a constraint to the flexible girder. Comparing the mean pea responses of different nodes of Cases and in Table 7-3, same conclusions can be obtained. The influence of local soil conditions on the structural response is shown in Figure 7-. As shown, softer soil results in lager longitudinal displacement. Taing node for example, the pea displacements are 0.0 and 0.76m for firm and soft soil respectively. This is because softer soil usually leads to larger ground displacements at the foundations of the structure, which results in larger total structural displacement responses. Comparing the mean pea responses of cases and 3 in Table 7-3, same conclusions can be drawn. The influence of coherency loss on the longitudinal displacement is shown in Figure 7-3. As shown in Figure 7-3a, the influence of coherency loss on node displacement is insignificant. This is because the ground motions propagate from left to right in the present study, the simulated ground motion time histories at site A are the same for the three sets of ground motions of each considered cases. The influence is expected for nodes at the girders and right abutment. As shown in Figure 7-3b, different coherency loss results in different longitudinal displacements of node. By examining the mean pea responses of cases, 4, and 5 in Table 7-3, it is generally true that the higher is the correlation between spatial ground motions, the larger is the longitudinal mean pea responses. 7-7

186 Chapter 7 School of Civil and Resource Engineering Figure 7-. Influence of pounding effect on the longitudinal displacement response Figure 7-. Influence of soil conditions on the longitudinal displacement response Figure 7-3. Influence of coherency loss on the longitudinal displacement response 7-8

187 School of Civil and Resource Engineering Chapter 7 Table 7-3. Mean pea displacements in the longitudinal direction m Node Case Transverse and vertical responses As will be demonstrated, the influences of different site and ground motion parameters on the transverse and vertical displacement responses of the bridge follow the same pattern, so they are discussed together in this section. Figures 7-4, 7-5 and 7-6 show the response time histories in the transverse direction of nodes and, and the corresponding time histories in the vertical direction are plotted in Figures 7-7, 7-8 and 7-9. The mean pea responses in the transverse and vertical directions are listed in Table 7-4 and Table 7-5, respectively. Similar to the responses in the longitudinal direction, the influence of poundings on displacement response of the abutments can be neglected. However, the influence on responses of the bridge girder is evident. Poundings usually result in smaller pea transverse and vertical displacements. This is because of the friction forces between the adjacent surfaces during poundings, which reduce the displacement responses of the bridge structures in the transverse and vertical directions. As shown in Figures 7-5 and 7-6, Tables 7-4 and 7-5 or the responses in the transverse and vertical directions, softer soil condition always results in larger displacement responses as discussed above. Ground motion spatial variations affect bridge responses, especially the responses of bridge decs. As shown in Tables 7-4 and 7-5, wealy correlated ground motions, among the three spatial ground motion cases, usually lead to the largest mean pea responses in the two directions. It also can be seen from Figures 7-7, 7-8 and 7-9 that more high frequency contents are involved in responses in the vertical direction as compared to those in the longitudinal and transverse directions. This is because the stiffness of the bridge in the vertical direction is much higher than that in the longitudinal and transverse directions. For 7-9

188 School of Civil and Resource Engineering Chapter 7 the considered bridge model, the first vertical vibration mode is the 7th mode and the vibration frequency is.37 Hz. It should be noted that the lateral side stoppers are not considered in the present study. If the stoppers are considered, the transverse responses might be altered. Figure 7-4. Influence of pounding effect on the transverse displacement response Figure 7-5. Influence of soil conditions on the transverse displacement response Figure 7-6. Influence of coherency loss on the transverse displacement response 7-0

189 Chapter 7 School of Civil and Resource Engineering Table 7-4. Mean pea displacements in the transverse direction m Node Case Figure 7-7. Influence of pounding effect on the vertical displacement response Figure 7-8. Influence of soil conditions on the vertical displacement response 7-

190 Chapter 7 School of Civil and Resource Engineering Figure 7-9. Influence of coherency loss on the vertical displacement response Table 7-5. Mean pea displacements in the vertical direction m Node Case Torsional response With the lumped mass model or beam-column element model, the torsional response of the structure cannot be considered because they are D models. With the detailed 3D finite element model, the torsional responses can be readily estimated. In this study, the torsional responses are estimated by the rotational angle of the corresponding nodes on both sides of the same section, i.e., between nodes and 3, nodes and 4, etc. These can be achieved by dividing the relative longitudinal displacement of these corresponding nodes by the dec width, which is 8.m in the present study. Table 7-6 shows the mean pea rotational angles for different cases. Different from the longitudinal, transverse and vertical displacement responses, poundings increase the torsional responses. This is because pounding imposes a 7-

191 School of Civil and Resource Engineering Chapter 7 restraint to the bridge spans, thus reduces lateral responses. However, eccentric poundings induced by spatially varying ground motions generate large eccentric impact forces that enhance the torsional responses. Comparing Case 3 with Case, it is obvious again that softer soil results in larger torsional responses. Comparing the responses obtained from spatial ground motions with different coherency losses, it is difficult to draw a general conclusion. Although highly correlated ground motions usually lead to the largest longitudinal displacements as discussed in Section 7.5., they do not necessarily yield the largest torsional response. This is probably because the torsional response is related to the relative displacement between nodes on the same cross section of the bridge structure instead of the absolute displacement. To examine the occurrence of poundings, the longitudinal displacements of nodes and and nodes 3 and 4 are plotted in the same figure with the displacements of nodes and 3 shifted by the initial gap of 5cm. Thus, in the figure, the instants when the displacements of the two adjacent points coinciding with each other indicate the occurrence of poundings. As shown in Figure 7-0a, node and node come into contacts 5 times, at the time instants 3.6, 5.9, 6.9, 6.68, 7.30, 7.7, 8.0, 8.63, 9.3, 9.66,.3,.89,.44, 3.70 and 4.6s. Whereas between nodes 3 and 4 as shown in Figure 7-0b, the poundings at 6.9,.89 and.44s do not occur, but two more collisions can be observed at 3.76 and 3.0s. Since these points locate at the opposite corners of the bridge dec cross section, pounding at these points occurring simultaneously implies the entire cross sections are in contact, i.e. surface to surface pounding occurs. Otherwise, they are torsional response induced eccentric poundings. In this example, pounding occurring at 6.9,.89 and.44s are eccentric poundings between nodes and, and those at 3.76 and 3.0s are eccentric poundings between nodes 3 and 4. Torsional response induced eccentric poundings between other corner points shown in Figure 7-5 are also observed. Owing to page limit, they are not shown here. These observations indicate that if 3D model with tri-axial ground motion inputs are considered, more number of poundings will be observed than the lumped mass and D beam-column element model because the two letter models cannot capture the possible eccentric poundings induced by torsional responses. 7-3

192 Chapter 7 Table 7-6. Mean pea rotational angle degree School of Civil and Resource Engineering Node Case and and and and and and Initial gap=5cm Initial gap=5cm Figure 7-0. Longitudinal displacements of different nodes to case ground motion Resultant pounding force Resultant pounding force in the longitudinal direction can be obtained by integrating the normal stresses over the entire cross section of the contact surface. Though torsional response induced eccentric poundings may result in the noncollinear impacts on the contact surface, the components of pounding forces in the transverse and vertical directions, which are induced owing to frictional forces during contact, are relatively small as compared to the component in the longitudinal direction. In this paper, only the influences of site conditions and coherency losses on the resultant pounding forces in the longitudinal direction are discussed. Figures 7- and 7- show the pounding forces at different time instants corresponding to different ground motion cases. It can be seen from Figure 7- that soft soil condition results in larger pea pounding forces than firm soil condition. This is because soft soil leads to larger displacement response in the longitudinal direction as shown in Figure 7-, which also results in larger relative displacement between the adjacent components of the bridge and maes the poundings more severe than that on the firm site. Comparing Figure 7-a and 7-c with 7-b, it is obvious 7-4

193 School of Civil and Resource Engineering Chapter 7 that the pounding forces between two bridge girders are generally smaller than those between the left or right abutment and the adjacent girder. This is because the bridge analysed in the present study is a symmetric structure, the left and right girders have the same dynamic characteristics and tend to vibrate in phase. If the spatially varying ground motions and the restraints from the abutments are not considered, the two spans will vibrate fully in phase and no pounding will be observed [7]. At the left and right gaps between abutment and girder, the abutments are much rigid than the adjacent bridge girders, the relative displacement is induced not only by spatially varying ground motions, but also by out of phase vibrations owing to different vibration frequencies of abutment and bridge span. In this case, the out of phase vibration induced relative displacement response dominates the responses. Therefore, larger pounding forces between abutments and girders are observed. Figure 7- illustrates the consequence of coherency loss between spatial ground motions for the pounding force development. As shown, spatially varying ground motions with wave passage effect only lead to larger pounding forces. This also can be explained by its influence on the longitudinal displacements as shown in Figure 7-3 and Table 7-3, where wave passage effect results in larger relative displacement responses. Same conclusion was also drawn in [6], in which the two adjacent bridge girders were simplified as two lumped masses. Figure 7-. Influence of soil conditions on the resultant pounding forces 7-5

194 Chapter 7 School of Civil and Resource Engineering Figure 7-. Influence of coherency loss on the resultant pounding forces Stress distributions By using the traditional lumped-mass model or beam-column element model, the stress on the entire contact surface will be the same. However, the use of 3D finite element model allows a more detailed prediction of the largest stresses and their locations, and thus where earthquae-induced damage may occur. Figure 7-3 shows the stress distributions in the longitudinal direction at left expansion joint of the bridge corresponding to the different cases considered in this study at the time instant when pea resultant pounding force occurs. As shown in Figure 7-3a, when bridge is on the firm soil site, the maximum compressive stress appears at the bottom outside corner of the girder. However, when it is on the soft soil site, the maximum compressive stress appears at the top inside corner of the girder. Although surface to surface pounding occurs, the largest stresses always occurs at the corners of the bridge girders corresponding to eccentric poundings because the pounding forces are distributed in a smaller area. This is why most observed pounding damages occurred at corners of bridge girders. It also can be seen that larger resultant pounding force not necessarily results in larger compressive stress. Taing the results from different soil conditions as example, the pea resultant pounding force for firm and soft soil are 55 and 80 MN, respectively as shown in Figure 7-. The resultant pounding force corresponding to the soft site condition is much larger than that corresponding to the firm site condition. However, the maximum compressive stresses are 88.8 and 59.3 MPa, 7-6

195 School of Civil and Resource Engineering Chapter 7 respectively for these two particular pounding events. This is again because the stress development is not only related to the pounding force but also related to the actual contact area at each pounding instant. The lumped mass and beam-column element models, which estimate the stress by dividing the pounding forces by the cross sectional area of the bridge girder, may not lead to correct predictions of stresses. As also shown in Figure 7-3, the maximum stresses can reach as high as 05.4 MPa Figure 7-3d. It is much larger than the compressive strength of normal concrete used in bridge construction, which is usually MPa under impact loading [35], thus concrete damages are expected although the concrete compressive strength increases owing to strain rate effect. These results are consistent with the observations in the past major earthquaes, in which the damages around the corners of the structure were usually the most serious as shown in Figure 7-. However, it should be noted that only linear elastic responses are considered in this study. Further study by modelling concrete damage is necessary as concrete damage will affect the subsequent bridge responses. a b c d Figure 7-3. Stress distributions in the longitudinal direction at left gap of different cases at the time when pea resultant pounding force occur a Case at t=6.7s, b Case 3 at t=7.63s, c Case 4 at t=7.96s and d Case 5 at t=8.04s unit: Pa 7-7