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 The University of Western Australia
Structural Engineering School of Civil and Resource Engineering
May 2011
DECLARATION FOR THESIS CONTAINING PUBLISHED WORK AND/OR WORK PREPARED FOR PUBLICATION
This thesis contains published work and/or work prepared for publication, which has been co-authored. The bibliographical details of the work 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 2010; 36(1): 111-127. (Chapter 2) The estimated percentage contribution of the candidate is 50%. Bi K, Hao H, Chouw N. Required separation distance between decks and at abutments of a bridge crossing a canyon site to avoid seismic pounding. Earthquake Engineering and Structural Dynamics 2010; 39(3):303-323. (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. Earthquake 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 earthquake 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. Earthquake 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. Earthquake 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 02/03/2011 Print Name Signature Date
School of Civil and Resource Engineering Abstract The University of Western Australia
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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 2-4), 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 rock
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. The structural responses are formulated in
the frequency domain, and mean peak responses are estimated by the standard random
vibration method. The dynamic, quasi-static and total responses of a frame structure
(Chapter 2) and the minimum separation distances between an abutment and the adjacent
bridge deck and between two adjacent bridge decks required in the modular expansion
joint (MEJ) design to preclude pounding during strong ground motion shaking 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.
School of Civil and Resource Engineering Abstract The University of Western Australia
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A method is proposed to simulate the spatially varying earthquake ground motion time
histories at a canyon site with different soil conditions. This method takes into
consideration the local site effect on ground motion amplification and spatial variations.
The base rock 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 rock 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 rock motions consisting of out-
of-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 earthquake 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 long-
span 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 2D finite element model of bridge structure
School of Civil and Resource Engineering Abstract The University of Western Australia
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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 earthquake-induced pounding responses between
adjacent structures.
School of Civil and Resource Engineering Table of Contents The University of Western Australia
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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 1 ..........................................................................................................................................1-1
INTRODUCTION............................................................................................................................................................. 1-1 1.1 BACKGROUND .................................................................................................................................................... 1-1 1.2 RESEARCH GOALS .............................................................................................................................................. 1-6 1.3 OUTLINE ............................................................................................................................................................. 1-7 1.4 REFERENCES....................................................................................................................................................... 1-7
CHAPTER 2......................................................................................................................................... 2-1
RESPONSE OF A FRAME STRUCTURE ON A CANYON SITE TO SPATIALLY VARYING GROUND MOTIONS....... 2-1 2.1 INTRODUCTION.................................................................................................................................................. 2-2 2.2 BRIDGE AND SPATIAL GROUND MOTION MODEL ......................................................................................... 2-4
2.2.1 BRIDGE MODEL..................................................................................................................................2-4 2.2.2 BASE ROCK MOTION..........................................................................................................................2-5 2.2.3 SITE AMPLIFICATION .........................................................................................................................2-7
2.3 STRUCTURAL RESPONSE EQUATION FORMULATION..................................................................................... 2-8 2.4 MAXIMUM RESPONSE CALCULATION.............................................................................................................2-10 2.5 NUMERICAL RESULTS AND DISCUSSIONS ...................................................................................................... 2-11
2.5.1 EFFECT OF SOIL DEPTH...................................................................................................................2-13 2.5.2 EFFECT OF SOIL PROPERTIES .........................................................................................................2-17 2.5.3 EFFECT OF COHERENCY LOSS ........................................................................................................2-20
2.6 CONCLUSIONS .................................................................................................................................................. 2-22
School of Civil and Resource Engineering Table of Contents The University of Western Australia
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2.7 REFERENCES .....................................................................................................................................................2-22
CHAPTER 3......................................................................................................................................... 3-1
REQUIRED SEPARATION DISTANCE BETWEEN DECKS AND AT ABUTMENTS OF A BRIDGE CROSSING A CANYON
SITE TO AVOID SEISMIC POUNDING ............................................................................................................................3-1 3.1 INTRODUCTION ..................................................................................................................................................3-1 3.2 BRIDGE MODEL ..................................................................................................................................................3-4 3.3 SPATIAL GROUND MOTION MODEL .................................................................................................................3-5
3.3.1 BASE ROCK MOTION ......................................................................................................................... 3-5 3.3.2 SITE AMPLIFICATION......................................................................................................................... 3-7
3.4 STRUCTURAL RESPONSES ...................................................................................................................................3-8 3.5 NUMERICAL RESULTS AND DISCUSSIONS.......................................................................................................3-11
3.5.1 EFFECT OF GROUND MOTION SPATIAL VARIATIONS ................................................................. 3-12 3.5.2 EFFECT OF THE BRIDGE GIRDER FREQUENCY............................................................................ 3-16 3.5.3 EFFECT OF THE LOCAL SOIL SITE CONDITIONS .......................................................................... 3-18
3.6 CONCLUSIONS...................................................................................................................................................3-22 3.7 APPENDIX..........................................................................................................................................................3-23
3.7.1 APPENDIX A: MEAN PEAK RESPONSE CALCULATION................................................................ 3-23 3.7.2 APPENDIX B: CHARACTERISTIC MATRICES .................................................................................. 3-24
3.8 REFERENCES .....................................................................................................................................................3-25
CHAPTER 4......................................................................................................................................... 4-1
INFLUENCE OF GROUND MOTION SPATIAL VARIATION, SITE CONDITION AND SSI ON THE REQUIRED
SEPARATION DISTANCES OF BRIDGE STRUCTURES TO AVOID SEISMIC POUNDING .............................................4-1 4.1 INTRODUCTION ..................................................................................................................................................4-2 4.2 BRIDGE-SOIL SYSTEM.........................................................................................................................................4-4 4.3 METHOD OF ANALYSIS ......................................................................................................................................4-7
4.3.1 DYNAMIC SOIL STIFFNESS ................................................................................................................ 4-7 4.3.2 STRUCTURAL RESPONSE FORMULATION ........................................................................................ 4-8
4.4 NUMERICAL EXAMPLE .....................................................................................................................................4-11 4.4.1 INFLUENCE OF SITE EFFECT AND SSI .......................................................................................... 4-12 4.4.2 INFLUENCE OF GROUND MOTION SPATIAL VARIATION AND SSI............................................. 4-17
4.5 CONCLUSIONS...................................................................................................................................................4-19 4.6 APPENDIX..........................................................................................................................................................4-20
4.6.1 APPENDIX A: ELEMENT FOR [ ])( ωiZ AND )]([ ωiZg .............................................................. 4-20
4.6.2 APPENDIX B: PSDS OF THE REQUIRED SEPARATION DISTANCES ............................................ 4-22 4.7 REFERENCES .....................................................................................................................................................4-22
CHAPTER 5......................................................................................................................................... 5-1
School of Civil and Resource Engineering Table of Contents The University of Western Australia
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MODELLING AND SIMULATION OF SPATIALLY VARYING EARTHQUAKE GROUND MOTIONS AT A CANYON SITE
WITH MULTIPLE SOIL LAYERS ...................................................................................................................................... 5-1 5.1 INTRODUCTION.................................................................................................................................................. 5-2 5.2 WAVE PROPAGATION THEORY AND SITE AMPLIFICATION EFFECT ............................................................ 5-5 5.3 GROUND MOTION SIMULATION....................................................................................................................... 5-8 5.4 NUMERICAL EXAMPLES ................................................................................................................................... 5-11
5.4.1 AMPLIFICATION SPECTRA ...............................................................................................................5-12 5.4.2 EXAMPLE 1-PSD COMPATIBLE GROUND MOTION SIMULATION...............................................5-13 5.4.3 EXAMPLE 2 -RESPONSE SPECTRUM COMPATIBLE GROUND MOTION SIMULATION................5-22
5.5 CONCLUSIONS .................................................................................................................................................. 5-24 5.6 REFERENCES..................................................................................................................................................... 5-25
CHAPTER 6......................................................................................................................................... 6-1
INFLUENCE OF IRREGULAR TOPOGRAPHY AND RANDOM SOIL PROPERTIES ON COHERENCY LOSS OF SPATIAL
SEISMIC GROUND MOTIONS......................................................................................................................................... 6-1 6.1 INTRODUCTION.................................................................................................................................................. 6-2 6.2 THEORETICAL BASIS .......................................................................................................................................... 6-5
6.2.1 ESTIMATION OF COHERENCY FUNCTION.......................................................................................6-5 6.2.2 ONE-DIMENSIONAL WAVE PROPAGATION THEORY.....................................................................6-6 6.2.3 GROUND MOTION GENERATION.....................................................................................................6-7 6.2.4 RANDOM FIELD THEORY ..................................................................................................................6-9 6.2.5 MONTE-CARLO SIMULATION .........................................................................................................6-10
6.3 NUMERICAL EXAMPLE..................................................................................................................................... 6-11 6.3.1 INFLUENCE OF IRREGULAR TOPOGRAPHY...................................................................................6-15 6.3.2 INFLUENCE OF RANDOM SOIL PROPERTIES .................................................................................6-18 6.3.3 INFLUENCE OF RANDOM VARIATION OF EACH SOIL PARAMETER ............................................6-20
6.4 CONCLUSIONS .................................................................................................................................................. 6-23 6.5 REFERENCES..................................................................................................................................................... 6-23
CHAPTER 7......................................................................................................................................... 7-1
3D FEM ANALYSIS OF POUNDING RESPONSE OF BRIDGE STRUCTURES AT A CANYON SITE TO SPATIALLY
VARYING GROUND MOTIONS ...................................................................................................................................... 7-1 7.1 INTRODUCTION.................................................................................................................................................. 7-2 7.2 METHOD VALIDATION ...................................................................................................................................... 7-5 7.3 BRIDGE MODEL .................................................................................................................................................. 7-9 7.4 SPATIALLY VARYING GROUND MOTIONS...................................................................................................... 7-11 7.5 NUMERICAL EXAMPLE..................................................................................................................................... 7-15
7.5.1 LONGITUDINAL RESPONSE.............................................................................................................7-17 7.5.2 TRANSVERSE AND VERTICAL RESPONSES .....................................................................................7-19 7.5.3 TORSIONAL RESPONSE ....................................................................................................................7-22 7.5.4 RESULTANT POUNDING FORCE .....................................................................................................7-24
School of Civil and Resource Engineering Table of Contents The University of Western Australia
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7.5.5 STRESS DISTRIBUTIONS................................................................................................................... 7-26 7.6 CONCLUSIONS...................................................................................................................................................7-28 7.7 REFERENCES .....................................................................................................................................................7-28
CHAPTER 8......................................................................................................................................... 8-1
CONCLUDING REMARKS ..............................................................................................................................................8-1 8.1 MAIN FINDINGS ..................................................................................................................................................8-1 8.2 RECOMMENDATIONS FOR FUTURE WORK ......................................................................................................8-3
School of Civil and Resource Engineering Acknowledgements The University of Western Australia
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Acknowledgements
I would like 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 work to
move forward continuously. Many of the ideas in this thesis would not have taken shape
without his incisive thinking and insightful suggestions. What I learned from him will
benefit me greatly in the rest of my life.
Many thanks go to Associate Professor Nawawi Chouw from the University of Auckland
in New Zealand for his invaluable suggestions, critical and insightful reviews of some of
the papers involved in this thesis. I would also like to thank 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 like to acknowledge 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 thanks to my parents, my brothers and sisters, for their
constant love and inspiration. Without their support, I could not have done it.
School of Civil and Resource Engineering Thesis Organization and Candidate Contribution The University of Western Australia
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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 2~7 are hereby set
forth.
Paper 1
This paper is presented in Chapter 2, 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 2010; 36(1): 111-127.
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 2
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 decks and at
abutments of a bridge crossing a canyon site to avoid seismic pounding. Earthquake
Engineering and Structural Dynamics 2010; 39(3):303-323.
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
School of Civil and Resource Engineering Thesis Organization and Candidate Contribution The University of Western Australia
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and bridge decks and between two adjacent bridge decks 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. Earthquake Engineering and Structural Dynamics, published online.
This paper is an extension of Paper 2 (Chapter 3). The candidate incorporated soil-
structure interaction effect (SSI) into the program developed in Paper 2. 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 2, 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 earthquake 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
School of Civil and Resource Engineering Thesis Organization and Candidate Contribution The University of Western Australia
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• Bi K, Hao H. Influence of irregular topography and random soil properties on the
coherency loss of spatial seismic ground motions. Earthquake 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. Earthquake
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 work of others, the
contents of this thesis are original and have not been submitted to any other university.
Kaiming Bi
May 2011.
Signature:
School of Civil and Resource Engineering Publications Arising From This Thesis The University of Western Australia
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Publications Arising From This Thesis
Journal papers
1. Bi K, Hao H, Ren W. Response of a frame structure on a canyon site to spatially
varying ground motions. Structural Engineering and Mechanics 2010; 36(1): 111-127.
2. Bi K, Hao H, Chouw N. Required separation distance between decks and at
abutments of a bridge crossing a canyon site to avoid seismic pounding. Earthquake
Engineering and Structural Dynamics 2010; 39(3):303-323.
3. 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. Earthquake Engineering and Structural Dynamics, published online.
4. Bi K, Hao H. Modelling and simulation of spatially varying earthquake 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. Earthquake 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. Earthquake
Engineering and Structural Dynamics, under review.
7. Liang JZ, Hao H, Wang Y, Bi K. Design earthquake ground motion prediction for
Perth metropolitan area with microtremor measurements for site characterization.
Journal of Earthquake Engineering 2009; 13(7): 997-1028.
8. 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.
School of Civil and Resource Engineering Publications Arising From This Thesis The University of Western Australia
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Conferences papers
1. Bi K, Hao H, Chouw N. Stochastic analysis of the required separation distance to
avoid seismic pounding between adjacent bridge decks. The 14th World Conference on
Earthquake Engineering, Beijing, China, 2008; 03-03-0026.
2. Bi K, Hao H. Seismic response analysis of a bridge frame at a canyons site in
Western Australia. The 14th World Conference on Earthquake Engineering, Beijing, China,
2008; 03-03-0027.
3. Bi K, Hao H. Simulation of spatially varying ground motions with non-uniform
intensities and frequency content. Australian Earthquake Engineering Society 2008
Conference, Ballarat, Australia, 2008; Paper No 18.
4. Liang JZ, Hao H, Wang Y, Bi K. Site characterization evaluation in Perth
metropolitan area using microtremor array method. Proceedings of the 10th International
Symposium on Structural Engineering for Young Experts, Changsha, China, 2008; 1906-
1911.
5. Bi K, Hao H, Chouw N. Dynamic SSI effect on the required separation distances
of bridge structures to avoid seismic pounding. Australian Earthquake Engineering
Society 2009 Conference, Newcastle, Australia, 2009; Paper No 16.
6. Bi K, Hao H. Analysis of influence of an irregular site with uncertain soil properties
on spatial seismic ground motion coherency. Australian Earthquake Engineering Society
2009 Conference, Newcastle, Australia, 2009; Paper No 17.
7. Hao H, Bi K, Chouw N. Combined ground motion spatial variation and local site
amplification effect on bridge structure responses. 6th International Conference on Urban
Earthquake Engineering, Tokyo, Japan, 2009; 595-600.
8. Bi K, Hao H. Pounding response of adjacent bridge structures on a canyon site to
spatially varying ground motions. Australian Earthquake Engineering Society 2010
Conference, Perth, Australia, 2010; Paper No 2.
9. Bi K, Hao H, Zhang C. Analysis of coupled axial-torsional pounding response of
adjacent bridge structures. The 11th International Symposium on Structural Engineering,
Guangzhou, China, 2010; 1612-1618.
School of Civil and Resource Engineering List of Figures The University of Western Australia
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List of Figures
Figure 2-1. Schematic view of a bridge frame crossing a canyon site.................................... 2-5
Figure 2-2. Filtered ground motion power spectral density function on the base rock...... 2-6
Figure 2-3. Different coherency loss functions.......................................................................2-12
Figure 2-4. Site transfer functions for different soil depths ..................................................2-14
Figure 2-5. Power spectral densities of ground motions on site of different depths ........ 2-14
Figure 2-6. Phase difference caused by seismic wave propagation ...................................... 2-14
Figure 2-7. Normalized dynamic responses for different soil depths..................................2-16
Figure 2-8. Normalized total responses for different soil depths.........................................2-16
Figure 2-9. Dynamic, quasi-static and total responses with mhA 0= and mhB 30= ...... 2-17
Figure 2-10. Soil site transfer function for different soil properties ....................................2-17
Figure 2-11. Power spectral densities of ground motions at sites ........................................ 2-18
Figure 2-12. Phase difference owing to seismic wave propagation...................................... 2-18
Figure 2-13. Normalized dynamic responses for different soil properties.......................... 2-18
Figure 2-14. Normalized total responses for different soil properties................................. 2-19
Figure 2-15. Dynamic, quasi-static and total responses (medium soil at support B).........2-19
Figure 2-16. Normalized dynamic responses for different coherency losses ..................... 2-21
Figure 2-17. Normalized total responses for different coherency losses ............................2-21
Figure 3-1. (a) Schematic view of a bridge crossing a canyon site; (b) structural model..... 3-5
Figure 3-2. Filtered ground motion power spectral density function at base rock.............. 3-8
Figure 3-3. Effect of ground motion spatial variation on the required separation distance
(a) 2Δ , (b) 3Δ , (c) 1Δ .......................................................................................................................3-14
Figure 3-4. Left span frequency response function with respect to the frequency ratios.3-16
Figure 3-5. Effect of vibration frequency on the required separation distance.................. 3-17
Figure 3-6. Effect of soil depth on the required separation distance................................... 3-20
Figure 3-7. Effect of soil properties on the required separation distance ........................... 3-21
Figure 3-8. Ground motion power spectral density functions with.....................................3-22
School of Civil and Resource Engineering List of Figures The University of Western Australia
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Figure 4-1. (a) Schematic view of a girder bridge crossing a canyon site...............................4-6
Figure 4-2. Frequency-dependent dynamic stiffness and damping coefficients of the pile
group (a)(b) horizontal direction and (c)(d) rotational direction.......................................... 4-12
Figure 4-3. Influence of site effect and SSI on the required separation distances............. 4-13
Figure 4-4. Site effect on ground motion spatial variations: (a) transfer function,............ 4-15
Figure 4-5. Contribution of SSI to the required separation distances with different soil
conditions ( 0.11 =f Hz) (a) 3Δ , (b) 2Δ and (c) 1Δ .............................................................. 4-16
Figure 4-6. Contribution of SSI to the required separation distances with different soil
conditions ( 0.21 =f Hz ) (a) 3Δ , (b) 2Δ and (c) 1Δ ............................................................. 4-16
Figure 4-7. Influence of ground motion characteristics and SSI on the required separation
distances ( 0.11 =f Hz) (a) 3Δ , (b) 2Δ and (c) 1Δ ................................................................. 4-18
Figure 4-8. Contribution of SSI to the required separation distances with different
coherency loss functions ( 0.11 =f Hz) (a) 3Δ , (b) 2Δ and (c) 1Δ ..................................... 4-19
Figure 4-9. Contribution of SSI to the required separation distances with different
coherency loss functions ( 0.21 =f Hz) (a) 3Δ , (b) 2Δ and (c) 1Δ ..................................... 4-19
Figure 5-1. A canyon site with multiple soil layers (not to scale) ......................................... 5-12
Figure 5-2. Amplification spectra of site 3, (a) horizontal out-of-plane motion;............... 5-13
Figure 5-3. Generated base rock motions in the horizontal directions............................... 5-16
Figure 5-4. Comparison of power spectral density of the generated base rock acceleration
with model power spectral density ........................................................................................... 5-16
Figure 5-5. Comparison of coherency loss between the generated base rock accelerations
with model coherency loss function......................................................................................... 5-17
Figure 5-6. Generated horizontal out-of-plane motions on ground surface...................... 5-18
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 ... 5-19
Figure 5-8. Comparison of the coherency loss functions between base rock motions .... 5-19
Figure 5-9. Generated horizontal in-plane motions on ground surface.............................. 5-20
Figure 5-10. Comparison of power spectral density of the generated horizontal in-plane
acceleration on ground surface with the respective theoretical power spectral density ... 5-21
Figure 5-11. Generated vertical in-plane motions on ground surface................................. 5-21
Figure 5-12. Comparison of power spectral density of the generated vertical in-plane
acceleration on ground surface with the respective theoretical power spectral density ... 5-22
Figure 5-13. Generated time histories according to the specified design response spectra ..5-
23
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Figure 5-14. Comparison of the generated acceleration and the target response spectra.5-24
Figure 5-15. Comparison of coherency loss between the generated time histories with the
model coherency loss function ..................................................................................................5-24
Figure 6-1. Schematic view of a layered canyon site................................................................. 6-8
Figure 6-2. A four-layer canyon site with deterministic soil properties (not to scale)....... 6-12
Figure 6-3. Simulated acceleration time histories....................................................................6-13
Figure 6-4. Mean values and standard deviations of the lagged coherency of the horizontal
out-of-plane motion at 0.2, 2.0, 5.0 and 9.0Hz .......................................................................6-14
Figure 6-5. Comparison of the mean lagged coherency on the base rock from 600
simulations with the target model .............................................................................................6-14
Figure 6-6. Comparison of the mean lagged coherency between the surface motions (j, k)
with that of the incident motion on the base rock .................................................................6-17
Figure 6-7. Standard deviations of the lagged coherency on the ground surface .............. 6-17
Figure 6-8. Modulus of the site amplification spectral ratio of two local sites ................... 6-17
Figure 6-9. Amplitudes of the site amplification spectra of two local sites ........................ 6-17
Figure 6-10. Influence of uncertain soil properties on...........................................................6-18
Figure 6-11. Influence of uncertain soil properties on the ....................................................6-19
Figure 6-12. Influence of uncertain soil properties on the ....................................................6-19
Figure 6-13. Influence of each random soil property on the ................................................6-21
Figure 6-14. Influence of each random soil property on the ................................................6-22
Figure 6-15. Influence of each random soil property on the ................................................6-22
Figure 7-1. A typical pounding damage between bridge decks in Chi-Chi earthquake....... 7-4
Figure 7-2. Different models (not to scale): (a) lumped mass model (from [11]); ............... 7-7
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, ........ 7-10
Figure 7-5. Finite element mesh of the bridge and the nodal points for response recordings
........................................................................................................................................................ 7-11
Figure 7-6. First four vibration frequencies and mode shapes of the bridge...................... 7-11
Figure 7-7. Simulated acceleration time histories with soft soil condition and intermediately
correlated coherency loss............................................................................................................ 7-14
Figure 7-8. Simulated displacement time histories with soft soil condition and
intermediately correlated coherency loss..................................................................................7-15
Figure 7-9. Comparison of PSDs between the generated horizontal in-plane motions on
ground surface with the respective theoretical model value .................................................7-15
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Figure 7-10. Multi-component spatially varying inputs at different supports of the bridge .7-
16
Figure 7-11. Influence of pounding effect on the longitudinal displacement response ... 7-18
Figure 7-12. Influence of soil conditions on the longitudinal displacement response...... 7-18
Figure 7-13. Influence of coherency loss on the longitudinal displacement response ..... 7-18
Figure 7-14. Influence of pounding effect on the transverse displacement response ...... 7-20
Figure 7-15. Influence of soil conditions on the transverse displacement response......... 7-20
Figure 7-16. Influence of coherency loss on the transverse displacement response......... 7-20
Figure 7-17. Influence of pounding effect on the vertical displacement response ........... 7-21
Figure 7-18. Influence of soil conditions on the vertical displacement response.............. 7-21
Figure 7-19. Influence of coherency loss on the vertical displacement response.............. 7-22
Figure 7-20. Longitudinal displacements of different nodes to case 2 ground motion.... 7-24
Figure 7-21. Influence of soil conditions on the resultant pounding forces ...................... 7-25
Figure 7-22. Influence of coherency loss on the resultant pounding forces ...................... 7-26
Figure 7-23. Stress distributions in the longitudinal direction at left gap of different cases at
the time when peak resultant pounding force occur (a) Case 1 at t=6.27s, (b) Case 3 at
t=7.63s, (c) Case 4 at t=7.96s and (d) Case 5 at t=8.04s (unit: Pa)...................................... 7-27
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List of Tables
Table 2-1. Parameters for coherency loss functions............................................................... 2-12
Table 2-2. Parameters of base rock and different types of soil............................................. 2-12
Table 3-1. Parameters for coherency loss functions...............................................................3-13
Table 3-2. Parameters for local site conditions ......................................................................3-18
Table 4-1. Parameters for local site conditions. ...................................................................... 4-11
Table 5-1. First two vibration frequencies of the sites........................................................... 5-13
Table 7-1. Parameters for local site conditions. ..................................................................... 7-13
Table 7-2. Different cases studied.............................................................................................7-16
Table 7-3. Mean peak displacements in the longitudinal direction (m). .............................. 7-19
Table 7-4. Mean peak displacements in the transverse direction (m). ................................. 7-21
Table 7-5. Mean peak displacements in the vertical direction (m). ...................................... 7-22
Table 7-6. Mean peak rotational angle (degree). ..................................................................... 7-24
School of Civil and Resource Engineering Chapter 1 The University of Western Australia
1-1
Chapter 1 Introduction
1.1 Background
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 [1]. 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 earthquake. Since 1960’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 1970’s to early 1980’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
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extensively studied by engineers and seismologists, is the SMART-1 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 earthquake-induced ground motions [2]: (1) 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; (2) 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 rock 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 [2]. 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 known 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 [8-12] and semi-empirical
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[13-14] 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 [15], continuous beams [16, 17],
an arch with multiple horizontal input [18], an arch with multiple simultaneous horizontal
and vertical excitations [19], a symmetric building structure [20], an asymmetric building
structure [21], and a cable-stayed bridge [22]. 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 peaks. 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 decks owing to the large out-of-phase responses. In fact, poundings
between an abutment and bridge deck or between two adjacent bridge decks were observed
in almost all the major earthquakes [23-27]. 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 decks during strong
earthquakes 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 make 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
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system for mitigating adjacent bridge girder pounding responses and its influence on other
bridge response quantities under earthquake loading are limited. Chouw and Hao took 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 [28] and then introduced a new design philosophy for a MEJ [29]. 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 2-4), the combined influences of ground motion
spatial variation and local site effect on a frame structure (Chapter 2) 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 deck and/or the two adjacent
bridge decks 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 peak responses are calculated. In
particular, Chapter 2 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 earthquakes, 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 [11, 30-32]. Most these studies [11, 30-31] 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 [32]. 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
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mentioned above. This phenomenon, however, cannot be considered by this method [32]
since only one peak 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 earthquake ground motion time histories at sites with non-
uniform 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 flat-
lying 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 likely
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 decks during strong earthquakes is often impossible as discussed
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above. Pounding is an extreme complex phenomenon involving damage due to plastic
deformation at contact points, local cracking 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 1D,
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 earthquakes that most poundings actually occurred at corners of adjacent bridge
decks. This is because torsional responses of the adjacent decks 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
earthquakes. Owing to the difficulty in modelling ground motion spatial variation, many
studies assumed uniform excitations [38, 40-41] 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 [42-43, 45]. Study of the combined influences of ground motion spatial variation
and local site effects on earthquake-induced pounding responses of adjacent bridge
structures have not been reported. Chapter 7 investigates the pounding responses between
the abutment and the adjacent bridge deck and between two adjacent bridge decks 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.
1.2 Research goals
This study was undertaken with the aims of:
1. Investigating the combined influences of ground motion spatial variation and local
site effect on the responses of a frame structure.
2. 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 earthquake ground
motions on a canyon site with different soil conditions.
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1-7
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 earthquakes.
1.3 Outline
This thesis comprises eight chapters. The seven chapters following this introductory
chapter are arranged as follows:
Chapters 2~4 formulate the structural responses in frequency domain based on the
stochastic method. In particular, Chapter 2 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 earthquakes, 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.
1.4 References
1. Zerva A, Zervas V. Spatial variation of seismic ground motions: an overview.
Applied Mechanics Reviews 2002; 56(3): 271-297.
2. Der Kiureghian A. A coherency model for spatially varying ground motions.
Earthquake Engineering and Structural Dynamics 1996; 25(1): 99-111.
3. Tajimi H. A statistical method of determining the maximum response of a building
structure during an earthquake. Proc. of 2nd World Conference on Earthquake Engineering,
Tokyo, Japan, 1960; 781-796.
School of Civil and Resource Engineering Chapter 1 The University of Western Australia
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4. Clough RW, Penzien J. Dynamics of Structures. New York: McGraw Hill; 1993. Joyner
WB, Boore DM. Measurement, characterization and prediction of strong ground
motion. Earthquake Engineering and Structure Dynamics II-Recent Advances in Ground
Motion Evaluation Proc (GSP 20), Park City, Utah, 1988; 43-102.
5. Joyner WB, Boore DM. Measurement, characterization and prediction of strong
ground motion. Earthquake Engineering and Structure Dynamics II-Recent Advances in
Ground Motion Evaluation Proc (GSP 20), Park City, Utah, 1988; 43-102.
6. Atkinson GM, Boore DM. Evaluation of models for earthquake source spectra in
Eastern North America. Bulletin of the Seismological Society of America 1998; 88(4): 917-
934.
7. Hao H, Gaull BA. Estimation of strong seismic ground motion for engineering use
in Perth Western Australia. Soil Dynamics and Earthquake Engineering 2009; 29(5): 909-
924.
8. Loh CH. Analysis of the spatial variation of seismic waves and ground movement
from SMART-1 data. Earthquake Engineering and Structural Dynamics 1985; 13(5): 561-
581.
9. Harichandran RS, Vanmarcke EH. Stochastic variation of earthquake ground
motion in space and time. Journal of Engineering Mechanics 1986; 112(2): 154-174.
10. Loh CH, Yeh YT. Spatial variation and stochastic modelling of seismic differential
ground movement. Earthquake Engineering and Structural Dynamics 1988; 16(4): 583-
596.
11. Hao H, Oliveira CS, Penzien J. Multiple-station ground motion processing and
simulation based on SMART-1 array data. Nuclear Engineering and Design 1989;
111(3):293-310.
12. Harichandran RS. Estimating the spatial variation of earthquake ground motion
from dense array recordings. Structural Safety 1991; 10: 219-233.
13. Luco JE, Wong HL. Response of a rigid foundation to a spatially random ground
motion. Earthquake Engineering and Structural Dynamics 1986; 14(6): 891-908.
14. Somerville PG, McLaren JP, Saikia CK, Helmberger DV. Site-specific estimation of
spatial incoherence of strong ground motion. Earthquake Engineering and Structural
Dynamics II-Recent Advances in Ground Motion Evaluation, ASCE Geotechnical Special
Publication No. 20, 1988; 188-202.
15. Harichandran RS, Wang W. Response of simple beam to spatially varying
earthquake excitation. Journal of Engineering Mechanics 1988; 114(9): 1526 - 1541.
School of Civil and Resource Engineering Chapter 1 The University of Western Australia
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16. Harichandran RS, Wang W. Response of indeterminate two–span beam to spatially
varying seismic excitation. Earthquake Engineering and Structural Dynamics 1990; 19(2):
173-187.
17. Zerva A. Response of multi-span beams to spatially incoherent seismic ground
motion. Earthquake Engineering and Structural Dynamics 1990; 19: 819-832.
18. Hao H. Arch responses to correlated multiple excitations. Earthquake Engineering and
Structural Dynamics 1993; 22(5): 389-404.
19. Hao H. Ground-motion spatial variation effects on circular arch responses. Journal
of Engineering Mechanics 1994; 120(11): 2326-2341.
20. Hao H, Duan XN. Multiple excitation effects on response of symmetric buildings.
Engineering Structures 1996; 18(9): 732-740.
21. Hao H, Duan XN. Seismic response of asymmetric structures to multiple ground
motions. Journal of Structural Engineering 1995; 121(11): 1557-1564.
22. Soyluk K, Dumanoglu AA. Spatial variability effects of ground motions on cable-
stayed bridges. Soil Dynamics and Earthquake Engineering 2004; 24(3):241-250.
23. Hall FJ, editor. Northridge earthquake, January 17, 1994. Earthquake Engineering
Research Institute, Preliminary reconnaissance report, EERI-94-01; 1994.
24. Kawashima K, Unjoh S. Impact of Hanshin/Awaji earthquake on seismic design
and seismic strengthening of highway bridges. Structural Engineering/Earthquake
Engineering JSCE 1996; 13(2): 211-240.
25. Earthquake Engineering Research Institute. Chi-Chi, Taiwan, Earthquake
Reconnaissance Report. Report No.01-02, EERI, Oakland, California. 1999.
26. Elnashai AS, Kim SJ, Yun GJ, Sidarta D. The Yogyakarta earthquake in May 27,
2006. Mid-America Earthquake Centre. Report No. 07-02, 57, 2007.
27. Lin CJ, Hung H, Liu Y, Chai J. Reconnaissance report of 0512 China Wenchuan
earthquake on bridges. The 14th World Conference on Earthquake Engineering. Beijing,
China, 2008; S31-006.
28. 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 2008; 30(1): 154-162.
29. Chouw N, Hao H. Seismic design of bridge structures with allowance for large
relative girder movements to avoid pounding. New Zealand Society for Earthquake
Engineering Conference. Wairakei, New Zealand 2008; Paper No: 10.
30. Shinozuka M. Monte Carlo solution of structural dynamics. Computers & Structures
1972; 2: 855-874.
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31. Conte JP, Pister KS, Mahin SA. Nonstationary ARMA modelling of seismic ground
motions. Soil Dynamics and Earthquake Engineering 1992; 11: 411-426.
32. Deodatis G. Non-stationary stochastic vector processes: seismic ground motion
applications. Probabilistic Engineering Mechanics 1996; 11(3): 149-167.
33. Shinozuka M, Deodatis G. Stochastic process models for earthquake ground
motion. Probabilistic Engineering Mechanics 1988; 3(3): 114-123.
34. Somerville PG, McLaren JP, Sen MK, Helmberger DV. The influence of site
conditions on the spatial incoherence of ground motions. Structural Safety 1991;
10(1):1-13.
35. 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. 170, 2007; 1-10.
36. Zerva A, Harada T. Effect of surface layer stochasticity on seismic ground motion
coherence and strain estimations. Soil Dynamics and Earthquake Engineering 1997; 16:
445-457.
37. Liao S, Li J. A stochastic approach to site-response component in seismic ground
motion coherency model. Soil Dynamics and Earthquake Engineering 2002; 22: 813-
820.
38. Malhotra PK. Dynamics of seismic pounding at expansion joints of concrete
bridges. Journal of Engineering Mechanics 1998; 124(7):794-802.
39. Jankowski R, Wilde K, Fujino Y. Pounding of superstructure segments in isolated
elevated bridge during earthquakes. Earthquake Engineering and Structural Dynamics
1998; 27:487-502.
40. Ruangrassamee A, Kawashima K. Relative displacement response spectra with
pounding effect. Earthquake Engineering and Structural Dynamics 2001; 30(10): 1511-
1538.
41. DesRoches R, Muthukumar S. Effect of pounding and restrainers on seismic
response of multi-frame bridges. Journal of Structural Engineering (ASCE) 2002; 128(7):
860-869.
42. Chouw N, Hao H. Study of SSI and non-uniform ground motion effects on
pounding between bridge girders. Soil Dynamics and Earthquake Engineering 2005;
25(10): 717-728.
43. 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 2008; 30(1):141-153.
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44. Jankowski R, Wilde K, Fujino Y. Reduction of pounding effects in elevated bridges
during earthquakes. Earthquake Engineering and Structural Dynamics 2000; 29(2): 195-
212.
45. 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 2006; 9(1):55-66.
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School of Civil and Resource Engineering Chapter 2 The University of Western Australia
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Chapter 2 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 rock
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.
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2.1 Introduction
Earthquake 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-1 array. Many
empirical spatial ground motion coherency loss functions have been derived [1-5]. 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 2-1. At a canyon site, the
spatial ground motions at base rock 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 earthquake
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 drawback is it can only approximately represent the local site effects on ground
motions. For example, it is well known 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 peaks. This phenomenon, however, cannot be considered in Der Kiureghian’s
model since only one peak corresponding to the fundamental vibration mode of the site
can be involved. In a recent study [7], derivations of earthquake ground motion spatial
variation on a site with uneven surface and different geological properties were presented.
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In the latter study, spatial base rock 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 rock 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 [10, 11], an
arch with multiple horizontal input [12], an arch with multiple simultaneous horizontal and
vertical excitations [13], a symmetric building structure [14], an asymmetric building
structure [15], and a cable-stayed bridge [16]. 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 [17], or stochastically simulated to be compatible to a selected
empirical coherency loss function [18-22]. 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 [23]
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 Soyluk
[24] 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
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consider the site effects and ground motion spatial variation was reported by Ates et al.
[25]. Similar conclusions were drawn, i.e., site effects significantly affect structural
responses. Sextos et al. [20, 21] 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.
2.2 Bridge and spatial ground motion model
2.2.1 Bridge model
Figure 2-1 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 rock
are 'A and 'B . jρ , jv , jξ and jh 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 rock are Rρ , Rv and Rξ . The deck 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 [26]. This means that each bridge frame can vibrate
independently during an earthquake 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.
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Figure 2-1. Schematic view of a bridge frame crossing a canyon site
2.2.2 Base rock motion
Assume the amplitudes of the power spectral densities at different locations on the base
rock are the same and in the form of the filtered Tajimi-Kanai power spectral density
function
Γ+−
+
+−== 222222
2224
2222
4
02
4)(4
)2()()()()(
ωωξωωωωξω
ωξωωωωωωω
ggg
ggg
fffPg SHS (2-1)
in which 2)(ωPH is a high pass filter [27], )(0 ωS is the Tajimi-Kanai power spectral
density function [8], gω and gξ 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 ωf and ξf are the central frequency and damping ratio of the high
pass filter. In this study, it is assumed that Hzf ff 25.02/ == πω , 6.0=fξ ,
Hzf gg 0.52/ == πω , 6.0=gξ and 32 /022.0 sm=Γ . These values correspond to a peak
ground acceleration (PGA) 0.5g with duration sT 20= [28]. Figure 2-2 shows the power
spectral density of the base rock motion.
A
B
'B'A
Ah
Bh
Soil AAAA v ξρ ,,
Soil BBBB v ξρ ,,
Base rockRRR v ξρ ,,
d
Ak
Bk
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Figure 2-2. Filtered ground motion power spectral density function on the base rock
Ground motion spatial variation at the base rock is modelled with a coherency loss
function [18]
appapp vdiddvdi
BABA eeeeii /)2/()(/ 2
'''' )()( ωπωωαβωωγωγ −−== (2-2)
in which
sradsradsrad
cbacba
/83.62/83.62/314.0
101.02//2
)(>
≤≤
⎩⎨⎧
++++
=ω
ωπωωπωα (2-3)
where a ,b , c and β are constants, d is the distance between the two supports, appv is the
apparent wave propagation velocity. The cross power spectral density function of the
motion at points 'A and 'B on the base rock is thus
)()()( '''' ωγωω iSiSBAgBA
= (2-4)
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 rock. This is because no information about ground motion spatial
variations at the base rock is available. It is believed that seismic wave propagation through
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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
rock motions. Further research into the influence of local site conditions on spatial ground
motion coherency loss is deemed necessary.
2.2.3 Site amplification
Using seismic wave propagation theory presented by Aki and Richards [29], Safak [30]
derived the transfer function for shear wave propagation in a horizontal layer as
( )( ) BorAjiH
iiiriiir
iUiU
jjjjj
jjjj
j
j ==−−−+
−−−+= )(
)21(2exp)(1)21(exp)1(2
)()(
'
ωξωτξξωτξ
ωω (2-5)
where )( ωiU j and )(' ωiU j is the Fourier transform of the motion )(tu j and )(' tu j
on the
ground surface and at the base rock, respectively. Qj 4/1=ξ is the damping ratio
accounting for energy dissipation owing to seismic wave propagation, and Q is the quality
factor; jjj vh /=τ is the wave propagation time from point 'j to j, and jr is the reflection
coefficient for up-going waves
BorAjvvvv
rjjRR
jjRRj =
+
−=
ρρρρ (2-6)
In engineering application, usually the outcrop motion on the rock surface is available,
instead of the base rock motion. The parameters defined above corresponding to the
Tajimi-Kanai power spectral density function in Equation (2-1) also correspond to the
outcrop motion on hard rock. Therefore, the constant 2 in Equation (2-5), which is a
measure of free surface reflection, in the transfer function is dropped. Then it has
( )( ) BorAj
iiiriiir
iHjjjj
jjjjj =
−−−+
−−−+=
)21(2exp)(1)21(exp)1(
)(ξωτξξωτξ
ω (2-7)
The auto and cross power spectral density function at point j and between points A and B
are
)()()()(
)()()(
''*
2
ωωωω
ωωω
iSiHiHiS
BorAjSiHS
BABAAB
gjj
=
== (2-8)
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in which the superscript ‘*’ represents complex conjugate.
The coherency loss between ground motions at points A and B is
( )[ ]
( )[ ]appBABA
BABA
BA
BABA
BA
ABAB
vdii
iiiHiH
iiHiH
SSiSi
/)()(exp)(
)()()(exp)()(
)()()(
)()()()(
''
''
''
22
*
ωωθωθωγ
ωγωθωθωω
ωγωω
ωωωωγ
+−=
−=== (2-9)
where ( )( ))()(Re
)()(Imtan)()( *
*1
ωωωωωθωθ
iHiHiHiH
BA
BABA
−=− is the phase difference of motions at points A
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 )('' ωγ iBA , but it
changes the phase delay between the spatial ground motion at base rock 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 (2-7) 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 [31].
2.3 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 deck 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 kinematic degrees of freedom representing the spatial excitations in the
longitudinal direction. The dynamic equilibrium equation can be written as
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⎪⎭
⎪⎬
⎫
⎪⎩
⎪⎨
⎧=
⎪⎭
⎪⎬
⎫
⎪⎩
⎪⎨
⎧
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
−−
−−++
⎪⎭
⎪⎬
⎫
⎪⎩
⎪⎨
⎧
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡+
⎪⎭
⎪⎬
⎫
⎪⎩
⎪⎨
⎧
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
000
00
00000000
00000000
B
A
t
BB
AA
BABA
B
A
t
B
A
t
uuv
kkkk
kkkk
uuvc
uuvm
&
&
&
&&
&&
&&
(2-10)
where m is the lumped mass of the bridge deck, vt is the total displacement response, and uA
and uB are the ground displacement at support A and B respectively, kA and kB are the
stiffness of the two columns. The total response consists of dynamic response and quasi-
static response
qst vvv += (2-11)
The quasi-static response can be derived as
[ ] [ ] )(1BBAA
B
ABA
B
ABA
BA
qs uuuu
uu
kkkk
v ϕϕϕϕ +=⎥⎦
⎤⎢⎣
⎡=⎥
⎦
⎤⎢⎣
⎡+
= (2-12)
in which )/( and )/( BABBBAAA kkkkkk +=+= ϕϕ . The dynamic response can be obtained
by solving the dynamic equilibrium equation
)()( BBAAqs
BA uumvmvkkvcvm &&&&&&&&& ϕϕ +−=−=+++ (2-13)
Transfer Equation (2-13) into frequency domain, the dynamic response can be obtained by
)]()()[()()()(2
1)(00
220
ωϕωϕωωωωωωξωω
ω iuiuiHiviHivi
iv BBAAsqs
sqs &&&&&&&& +−=−=
+−−
= (2-14)
in which mkk BA /)(0 +=ω is the circular natural vibration frequency of the structure, 0ξ
is the damping ratio, and )( ωiH 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
[ ]{ }[ ]{ }
[ ]( ){ })(Re2)()()(Re2)()()(
)(Re2)()(1)(
)(Re2)()()()(
222
224
222
ωϕϕωϕωϕωω
ωωω
ωϕϕωϕωϕω
ω
ωϕϕωϕωϕωω
iSSSiHSSS
iSSSS
iSSSiHS
ABBABBAAsvvv
ABBABBAAv
ABBABBAAsv
qst
qs
++−+=
++=
++=(2-15)
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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 uA. Under uniform ground motion excitation,
Equation (2-15) reduces to
[ ])()(Re2)()()(
)(1)(
)()()(
2
4
2
ωωω
ωωω
ωω
ω
ωωω
Asvvv
Av
Asv
SiHSSS
SS
SiHS
qsuu
tu
qsu
u
−+=
=
=
(2-16)
2.4 Maximum response calculation
Standard random vibration method [32] is used to calculate the mean peak displacement, it
is briefly described in the following.
For a zero mean stationary process x(t) with known power spectral density function )(ωS ,
its m th order spectral moment is defined as
ωωωλω
dSc mm ∫≈ 0
)( (2-17)
where cω 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
0
21λλ
π=v (2-18)
20
211λλλδ −= (2-19)
the mean peak response can then be calculated by
σ)ln25772.0ln2(max Tv
Tvxe
e += (2-20)
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where T is the duration of the stationary process, 0λσ = is the standard deviation of the
process, and
69.069.01.01.00
)38.063.1()2,1.2max(
45.0
≥<≤<≤
⎪⎩
⎪⎨
⎧−=
δδδ
δδ
vTvT
TTve
(2-21)
In the present study, the high cut-off frequency is taken as 25Hz since it covers the
predominant vibration modes of most engineering structures and the dominant earthquake
ground motion frequencies.
2.5 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 2-1 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 f0td [1, 9, 11, 12], in which f0 is the
structural vibration frequency, and td 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, td=d/vapp, in which vapp 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 td=d/vapp+τB-τA, in which τB and τA,
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 f0 of the structure.
The constants of coherency loss function in Equation (2-2) are obtained by processing
recorded motions during Event 45 at the SMART-1 array [18]. 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 weakly correlated ground motions, respectively. Figure 2-3 shows
different coherency loss functions corresponding to parameters given in Table 2-1. For
spatial ground motion without coherency loss, ( ) 1'' =ωγ iBA in Equation (2-2).
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Table 2-1. Parameters for coherency loss functions
Coherency loss β a b c
highly(event 45) 410109.1 −× 310583.3 −× 510811.1 −×− 410177.1 −×
intermediately 410697.3 −× 210194.1 −× 510811.1 −×− 410177.1 −×
weakly 310109.1 −× 210583.3 −× 510811.1 −×− 410177.1 −×
Figure 2-3. Different coherency loss functions
The main parameters for base rock and soil conditions can be combined together to form a
single coefficient defined as rock/soil impedance ratio [33]
SS
RRSR v
vIρρ
=/ (2-22)
This impedance coefficient reflects the differences between base rock 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 rock are given in Table 2-2.
Table 2-2. Parameters of base rock and different types of soil
Type )/( 3mkgρ )/( smv ξ SRI /
Base rock 3000 1500 0.05 /
Firm soil 2000 450 0.05 5
Medium soil 1500 300 0.05 10
Soft soil 1500 100 0.05 30
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2.5.1 Effect of soil depth
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 mhh BA 0== ,
mhh BA 30== , mhh BA 50== or locates on a canyon site with mhmh BA 300 == ,
mhmh BA 500 == and mhmh BA 5030 == . To preclude the influence of other
parameters, the soil under both site A and B are assumed to be firm soil with 5/ =SRI , and
the ground motions are assumed to be intermediately correlated.
As shown in Figure 2-4, different soil depths lead to different transfer functions. The peaks
occur at the corresponding vibration modes of the sites. Take h=50m as a example, the
resonant frequencies of the soil layer are ,...5,3,1,4/ == khkvf sk , where sv and h is the
shear wave velocity and depth of the soil layer respectively, obvious peaks can be obtained
at f=2.25, 6.75 and 11.25Hz with smvs /450= 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 rock, as shown in Figure 2-5. Motions on ground
surface have a narrower band, but higher peak, as compared to that at the base rock,
indicating the effect of site filtering and amplification on base rock motion. If the ground
surface is flat, the time lag between motions at A and B are the same as those at the base
rock (τB-τA=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 rock to
ground surface is different (τB-τA ≠0), which results in an additional phase difference
between motions at A and B, as compared to those at the base rock, as shown in Figure 2-
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 2-7 and 2-8 show the
normalized dynamic responses and total responses with respect to the dimensionless
parameter, f0td, 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 f0. When a flat site is considered, f0td=f0d/vapp, and the multiple ground
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excitations and the structural vibration mode are in-phase if L0.2,0.10 =dtf , whereas they
are out-of-phase if L5.1,5.00 =dtf for the special case [28, 34].
Figure 2-4. Site transfer functions for different soil depths
Figure 2-5. Power spectral densities of ground motions on site of different depths
Figure 2-6. Phase difference caused by seismic wave propagation
through sites of different depths
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As shown in Figure 2-7, 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 5.1,5.00 =dtf and maximum value at 0.2,0.10 =dtf
because of the out-of-phase and in-phase ground motion inputs. This observation is the
same as those reported in many previous studies [28, 34]. If a canyon site is assumed with
Point A on base rock and Point B on soil surface, the maximum responses, however, do
not occur at 0.10 =dtf . 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 mhA 0= and mhB 30= , the
first peak occurs at f0td=0.625, or f0=3.75Hz because td=d/vapp+τB=d/vapp+hB/vB=0.16667
sec. The second peak can be observed when f0=11.25Hz. As shown in Figures 2-4 and 2-5,
the resonant frequencies of the site with soil depth 30m are ,...5,3,1,75.34/ === kkhkvf sk .
If mhA 0= and mhB 50= , the first peak occurs at f0td =0.475, or f0=2.25Hz because
td=0.2111sec. Again as shown in Figures 2-4 and 2-5, 2.25 Hz is the fundamental vibration
frequency of the soil site with depth 50m. The following peaks can also be observed when
resonance occurs. If both point A and B locate on soil surface with mhA 30= and
mhB 50= , the spatial ground motion wave passage effect dominates the site effect on
dynamic structural responses, i.e., the minimum values occur around 5.1,5.00 =dtf , and the
maximum values around 0.2,0.10 =dtf . This is because, although site A and B have
different fundamental vibration modes and different peak values in their respective power
spectral density function as shown in Figure 2-5, the mean peak 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
1.0 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 [28, 34]. When the vibration frequency of the structure
coincides with the fundamental frequency of the soil layer, however, the normalized peak
dynamic responses can be larger than 1.0, 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.
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Quasi-static responses are independent of the fundamental vibration frequency of the
structure (Equations 2-15 and 2-16). The normalized quasi-static responses are therefore
constant for each case with respect to f0td. The normalized total responses are given in
Figure 2-8. As shown, when the dimensionless parameter f0td is less than 1.5, the
normalized total responses are similar to the normalized dynamic responses, indicating
dynamic response dominates the total response. When f0td 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 f0td implies the structure becomes stiffer, as f0 is increased in this
study. The dynamic response is smaller when structure is stiffer. At large f0td, quasi-static
response dominates the total response, as shown in Figure 2-9. This observation indicates
the importance of quasi-static responses for stiff structures.
Figure 2-7. Normalized dynamic responses for different soil depths
Figure 2-8. Normalized total responses for different soil depths
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Figure 2-9. Dynamic, quasi-static and total responses with mhA 0= and mhB 30=
2.5.2 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 2-2 are considered. The soil under
point A is assumed to be firm soil ( 5/ =SRI ) and unchanged in all the cases, while soil
under support B varies from firm soil ( 5/ =SRI ) to soft soil ( 30/ =SRI ). The soil depths
are assumed to be mhA 30= and mhB 50= , and the ground motions are intermediately
correlated. Figure 2-10 shows the transfer function at support B for different cases. Figure
2-11 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 2-12 shows the phase differences
between motions at Point A and B. The normalized dynamic responses and total responses
are shown in Figures 2-13 and 2-14, respectively.
Figure 2-10. Soil site transfer function for different soil properties
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Figure 2-11. Power spectral densities of ground motions at sites
with different soil properties
Figure 2-12. Phase difference owing to seismic wave propagation
through sites with different soil properties
Figure 2-13. Normalized dynamic responses for different soil properties
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Figure 2-14. Normalized total responses for different soil properties
Figure 2-15. Dynamic, quasi-static and total responses (medium soil at support B)
Figure 2-10 clearly shows again the site effects. As shown, peak 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
( 30/ =SRI ) and medium soil ( 10/ =SRI ) significantly amplifies the ground motions at its
resonant frequencies, firm soil ( 5/ =SRI ) also amplifies ground motions, but at higher
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 2-11. Soil properties also affect the
seismic wave propagation velocity and hence the phase difference between motions at
Point A and B. Figure 2-12 shows the phase differences between motions at A and B
owing to wave propagation from base rock to ground surface. It shows that the phase
School of Civil and Resource Engineering Chapter 2 The University of Western Australia
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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 2-13, 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
peak occurs at f0td =0.3, or f0=1.5Hz because td=0.2 sec. As shown in Figures 2-10 and 2-
11, the fundamental vibration frequency for the medium site is 1.5 Hz. When site B is a
soft soil site, the first peak occurs at f0td =0.267, or f0=0.5Hz because td=0.5333 sec, and
the second peak at f0td =0.8, corresponding to the second mode of the site B. Subsequent
peaks 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 2-13, soft soil amplification effect results in larger dynamic responses, normalized
dynamic responses are usually larger than 1.0 when the responses are dominated by the site
effect, and the results are always less than 1.0 when spatial ground motion phase shift
effect governs the dynamic responses.
Total responses shown in Figure 2-14 follow the similar pattern as that discussed above,
i.e., the normalized total responses are similar to the normalized dynamic responses when
f0td is less than 1.5. 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 2-15
(medium soil at support B).
2.5.3 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 2-3, i.e., highly, intermediately and weakly
correlated coherency loss functions. Moreover, two special cases, i.e., intermediate
coherency loss without considering phase shift ( 0.1)cos( =dtω ), and no coherency loss
( 1)('' =ωγ iBA
), are also considered. All the results are normalized by the corresponding
uniform excitation. For these cases, the canyon site with mhA 30= and mhB 50= is
considered, and medium soil ( 10/ =SRI ) are assumed at both sites A and B. Figure 2-16
School of Civil and Resource Engineering Chapter 2 The University of Western Australia
2-21
shows the normalized dynamic responses and Figure 2-17 shows the normalized total
responses.
Figure 2-16. Normalized dynamic responses for different coherency losses
Figure 2-17. Normalized total responses for different coherency losses
As shown in Figure 2-16, site effect governs the dynamic responses when f0td <0.5, i.e.,
peak response occurs at the resonant frequency with f0=0.2 Hz and f0td=0.3. However, if
f0td>0.5, spatial ground motion wave passage effect dominates the dynamic responses, i.e.,
the normalized dynamic responses reach their minimum value at 5.1,5.00 =dtf and their
maximum value at 0.2,0.10 =dtf because of the out-of-phase and in-phase ground motion
excitations. The more correlated are the ground motions, the more pronounced are the in-
phase 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 peak response occurs at vibration
modes of the soil site, no in-phase or out-of-phase effects are present. For total responses
School of Civil and Resource Engineering Chapter 2 The University of Western Australia
2-22
as shown in Figure 2-17, similar observations can be drawn, i.e., dynamic response
dominates total response when f0td<1.0, and quasi-static response is more significant when
the structure is stiff.
2.6 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, quasi-
static and total responses of the model structure to various cases of spatially varying ground
motions are investigated. Following conclusions can be drawn:
1. 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 rock.
2. Local site conditions significantly affect spatial surface ground motions, and hence
the structural responses. The peak 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.
2.7 References
1. Bolt BA, Loh CH, Penzien J, Tsai YB, Yeah YT. Preliminary report on the
SMART-1 strong motion array in Taiwan. Report No. UCB/EERC-82-13, University
of California at Berkeley, Berkeley, CA, 1982.
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2-23
2. Harichandran RS, Vanmarcke EH. Stochastic variation of earthquake ground
motion in space and time. Journal of Engineering Mechanics 1986; 112(2): 154-174.
3. Loh CH, Yeah YT. Spatial variation and stochastic modelling of seismic differential
ground movement. Earthquake Engineering and Structural Dynamics 1988; 16(4): 583-
596.
4. Hao H, Oliveira CS, Penzien J. Multiple station ground motion processing and
simulation based on SMART-1 data. Nuclear Engineering and Design 1989; 111(3): 293-
310.
5. Abrahamson NA, Schneider JF, Stepp JC. Empirical spatial coherency functions
for application to soil-structure interaction analyses. Earthquake Spectra 1991; 7: 1-
27.
6. Der Kiureghian A. A coherency model for spatially varying ground motions.
Earthquake Engineering and Structural Dynamics 1996; 25(1): 99-111.
7. Hao H, Chouw N. Modeling of earthquake ground motion spatial variation on
uneven sites with varying soil conditions. The 9th International Symposium on Structural
Engineering for Young Experts, Fuzhou & Xiamen, 2006.
8. Tajimi H. A statistical method of determining the maximum response of a building
structure during a earthquake. Proceedings of 2nd World Conference on Earthquake
Engineering, Tokyo, Japan 1960; 781-796.
9. Harichandran RS, Wang W. Response of simple beam to spatially varying
earthquake excitation. Journal of Engineering Mechanics 1988; 114(9): 1526 - 1541.
10. Harichandran RS, Wang W. Response of indeterminate two–span beam to spatially
varying seismic excitation. Earthquake Engineering and Structural Dynamics 1990; 19(2):
173-187.
11. Zerva A. Response of multi-span beams to spatially incoherent seismic ground
motion. Earthquake Engineering and Structural Dynamics 1990; 19: 819-832.
12. Hao H. Arch responses to correlated multiple excitations. Earthquake Engineering and
Structural Dynamics 1993; 22(5): 389-404.
13. Hao H. Ground-motion spatial variation effects on circular arch responses. Journal
of Engineering Mechanics 1994; 120(11): 2326-2341.
14. Hao H, Duan XN. Multiple excitation effects on response of symmetric buildings.
Engineering Structures 1996; 18(9): 732-740.
15. Hao H, Duan XN. Seismic response of asymmetric structures to multiple ground
motions. Journal of Structural Engineering 1995; 121(11): 1557-1564.
School of Civil and Resource Engineering Chapter 2 The University of Western Australia
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16. Dumanoglu AA, Soyluk K. Response of cable-stayed bridge to spatially varying
seismic excitation. 5th International Conference on Structure Dynamics, Munich, Germany,
2002; 1059-1064.
17. Jankowski R, Wilde K, Fujino Y. Reduction of pounding effects in elevated bridges
during earthquakes. Earthquake Engineering and Structural Dynamics 2000; 29(2): 195-
212.
18. Hao H. Effects of spatial variation of ground motions on large multiply-supported
structures. Report No. UCB/EERC-89-06, University of California at Berkeley,
Berkeley, 1989.
19. Monti G, Nuti C, Pinto E. Nonlinear response of bridges to spatially varying
ground motion. Journal of Structural Engineering 1996; 122: 1147-1159.
20. Sextos AG, Kappos AJ, Patilakis KD. Inelastic dynamic analysis of RC bridges
accounting for spatial variability of ground motion, site effects and soil-structure
interaction phenomena. Part 1: Methodology and analytical tools. Earthquake
Engineering and Structural Dynamics 2003; 32(4): 607-627.
21. Sextos AG, Kappos AJ, Patilakis KD. Inelastic dynamic analysis of RC bridges
accounting for spatial variability of ground motion, site effects and soil-structure
interaction phenomena. Part 2: Parametric study. Earthquake Engineering and
Structural Dynamics 2003; 32(4): 629-652.
22. Chouw N, Hao H. Study of SSI and non-uniform ground motion effects on
pounding between bridge girders. Soil Dynamics and Earthquake Engineering 2005;
25(10): 717-728.
23. Zembaty Z, Rutenburg A. Spatial response spectra and site amplification effect.
Engineering Structures 2002; 24(11): 1485-1496.
24. Dumanoglu AA, Soyluk K. A stochastic analysis of long span structures subjected
to spatially varying ground motions including the site-response effect. Engineering
Structures 2003; 25(10): 1301-1310.
25. Ates S, Dumanoglu AA, Bayraktar A. Stochastic response of seismically isolated
highway bridges with friction pendulum systems to spatially varying earthquake
ground motions. Engineering Structures 2005; 27(13): 1843-1858.
26. 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 2008; 30(1): 154-162.
27. Ruiz P, Penzien J. Probabilistic study of the behaviour of structures during
earthquakes. Report No. UCB/EERC-69-03, University of California at Berkeley,
Berkeley, 1969.
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28. Hao H. A parametric study of the required seating length for bridge decks during
earthquake”, Earthquake Engineering and Structural Dynamics 1998; 27(1): 91-103.
29. Aki K, Richards PG. Quantitative seismology theory and methods. WH Freeman
and Company, San Francisco, 1980.
30. Safak E. Discrete-time analysis of seismic site amplification. Journal of Engineering
Mechanics 1995; 121(7): 801-809.
31. Wolf JP. Dynamic soil-structure interaction. New Jersey: Prentice-Hall; 1985.
32. Der Kiureghian A. Structural response to stationary excitation”, Journal of Engineering
Mechanics 1980; 106: 1195-1213.
33. Roesset JM. Soil amplification of earthquakes. Numerical methods in geotechnical
engineering, McGraw-Hill, New York, 1977.
34. Hao H, Zhang S. Spatial ground motion effect on relative displacement of adjacent
building structures”, Earthquake Engineering and Structural Dynamics 1999; 28(4): 333-
349.
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Chapter 3 Required separation distance between decks and at abutments of a bridge crossing a canyon site to avoid seismic pounding
By: Kaiming Bi, Hong Hao and Nawawi Chouw
Abstract: Major earthquakes in the past indicated that pounding between bridge decks may
result in significant structural damage or even girder unseating. With conventional
expansion joints it is impossible to completely avoid seismic pounding between bridge
decks, 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 decks becomes
possible. This paper investigates the minimum total gap that a MEJ must have to avoid
pounding at the abutments and between bridge decks. 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 decks 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
3.1 Introduction
For large-dimensional structures, such as long-span bridges, earthquake 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
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collapse of adjacent bridge decks owing to the out-of-phase responses. Poundings between
an abutment and bridge deck or between two adjacent bridge decks were observed in
almost all the major earthquakes, e.g., the 1989 Loma Prieta earthquake and the 1994
Northridge earthquake [1], the 1995 Hyogo-Ken Nanbu earthquake [2], the 1999 Chi-Chi
Taiwan earthquake [3], the 2006 Yogyakarta earthquake [4], and more recently the 2008
Wenchuan earthquake [5].
More and more earthquake 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 [10] investigated the effect of
the spatially varying ground motions on the relative displacement of adjacent buildings.
Seismic design codes such as UBC [11], Australian Earthquake Loading Codes [12] and
Chinese Seismic Design Code [13] also specify the required separation distances between
buildings.
For bridge structures with conventional expansion joints, a complete avoidance of
pounding between bridge decks during strong earthquakes 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 decks. Ruangrassamee and
Kawashima [14] studied the relative displacement spectra of two SDOF systems with
pounding effect. DesRoches and Muthukumars [15] 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. Jankowski et
al. [16] and Zhu et al. [17] studied the pounding effect of an elevated bridge caused by wave
passage effect. Hao and Chouw [18] and Zanardo et al. [19] considered the pounding effect
of simply supported segmental bridges, and they confirmed that spatial ground variations
can have strong influences. Jankowski et al. [20] studied several approaches for reducing the
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3-3
damaging effects of collisions. To mitigate pounding effect some bridge design codes such
as AASHTO [21], CALTRANS [22] and JRA [23] recommended an adjustment of the
vibration properties of adjacent bridge decks so that they have the same or similar
fundamental periods.
A recent study by Chouw and Hao [24] found that even if the adjacent bridge decks 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 decks becomes possible [25]. However, only very
limited information on the required separation distances to avoid seismic pounding
between adjacent bridge structures is available. Hao [26] analysed various parameters that
influence the required seating length to prevent bridge deck unseating; Chouw and Hao
[25] 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. [27]
studied site effect on the required separation distance between two adjacent bridge decks.
In all these studies, the two adjacent bridge decks were independent and modelled as
uncoupled systems, the bearing that connects bridge pier and deck 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 decks and between the bridge
deck and adjacent abutment. The bridge model is illustrated in Figure 3-1. Bearings that
connect bridge decks to pier or abutments are included in the model. Each bridge deck is
modelled as a rigid beam with lumped mass supported on isolation bearings, and the two
adjacent bridge decks 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 deck and between two decks. This paper is a continuation
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3-4
of the previous work [26, 27]. The primary differences between the present work and
previous works include: (1) A more realistic bridge model with two adjacent decks coupled
on top of the pier instead of two independent bridge girders, and bearings that connect
bridge decks to the pier and abutments are considered in this study; (2) The required
separation distance between abutments and adjacent bridge deck is also investigated in
addition to that between bridge decks; (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.
3.2 Bridge model
Figure 3-1 (a) illustrates the schematic view of a typical bridge crossing a canyon site. Two
decks with length d1 and d2 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 rock are 'A , 'B and 'C .
jh is the depth of the soil layer under the jth support, where j represents A, B or C.
Rρ , Rv and Rξ represent density, shear wave velocity and damping ratio of the base rock,
respectively; the corresponding parameters of soil layer are jρ , jv and jξ .
A MEJ is installed between the bridge decks 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 decks to preclude pounding during strong
ground shakings. More detailed information regarding the MEJ can be found in [25]. The
gap required for a MEJ to avoid pounding between bridge decks 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: (1) The bridge decks are
assumed to be rigid with lumped mass m1 and m2; (2) The two bearings located on the
abutment and the pier for each deck have the same dynamic characteristics with stiffness
School of Civil and Resource Engineering Chapter 3 The University of Western Australia
3-5
kb1 and damping 1bc for the left span, kb2 and 2bc 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 kp and cp, 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 deck, one for pier, and three for spatial support movements as shown in Figure 3-1
(b), is used in the present study.
Figure 3-1. (a) Schematic view of a bridge crossing a canyon site; (b) structural model
3.3 Spatial ground motion model
3.3.1 Base rock motion
Assuming ground motion intensities at A’, B’ and C’ on the base rock 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
Γ+−
+
+−== 222222
222
2222
4
02
/4)/1(/41
)2()()()()(
ggg
gg
fffPg SHS
ωωξωωωωξ
ωξωωωωωωω (3-1)
)(b
1d 2d
1m 2m
3m1bc 2bc1bc
'A
A
B
'B
C
'C
pp ck
1bk 1bk 2bk 2bk
2bc
1Δ 3Δ
BSoilASoil CSoil
)(a1 2
4 36
5B
A
rockBase
2Δ
C
Ah
BhCh
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3-6
in which 2)(ωPH is a high pass filter [28], 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. )(0 ωS is the Tajimi-Kanai power spectral density function [29],
gω and gξ are the central frequency and damping ratio of the Tajimi-Kanai power spectral
density function, respectively. ωf and ξf are the central frequency and damping ratio of the
high pass filter, respectively. Without losing generality, in this study, it is assumed that
25.02/ == πω fff Hz, 6.0=fξ , 0.52/ == πωggf Hz and gξ =0.6. Γ is a scaling factor
depending on the ground motion intensity, assuming a ground acceleration of duration
T=20 s and peak value (PGA) 0.5g, 022.0=Γ m2/s3 is estimated in this study according to
the standard random vibration approach given in Appendix A. Figure 3-2 shows the power
spectral density of the base rock acceleration and displacement ( 4/ωgd SS = ).
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-
1 array data by Hao et al. [30] and is modelled in the following form
appnjnjnjappnj vdiddvdinjnj eeeeii /)2/()(/
''''''
2'''''')()( ωπωωαβωωγωγ −−== (3-2)
in which
srad
sradsradcba
cba/83.62
/83.62/314.0101.0
2//2)(
>≤≤
⎩⎨⎧
++++
=ω
ωπωωπωα (3-3)
where a ,b , c and β are constants, ''njd is the distance between points j’ and n’, 1000=appv
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-1 array. It may not be suitable
to model ground motion spatial variations on the base rock. How the base rock motion
varies spatially, however, is not known. In this study, the soil site is modelled as a
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3-7
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 rock is thus
)()()( '''' ωγωω iSiS njgnj = (3-4)
3.3.2 Site amplification
Even if the motion intensities, i.e., the power spectral density functions, at different
locations of the base rock 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 rock j’ to ground surface j is based on the one dimensional wave
propagation assumption, and is given in the following form [31]
)21(2
)21(
)(1)1(
)(jj
jj
j iijj
iijj
S eireir
iH ξωτ
ξωτ
ξ
ξω −−
−−
−+
−+= (3-5)
where jjj vh /=τ is the wave propagation time from point j’ to j, and jr is the reflection
coefficient for up-going waves
jjRR
jjRRj vv
vvr
ρρρρ
+−
= (3-6)
The power spectral density function at point j is thus
)()()(2
ωωω gSj SiHSj
= (3-7)
and the cross power spectral density function between j and n is
)()()()( ''* ωωωω iSiHiHiS njSSjn nj
= (3-8)
where the superscript ‘*’ represents complex conjugate.
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3-8
(a)
(b)
Figure 3-2. Filtered ground motion power spectral density function at base rock
(a) acceleration, (b) displacement
3.4 Structural responses
With the hypotheses mentioned above, the dynamic equilibrium equation of the system
shown in Figure 3-1 can be written as
⎭⎬⎫
⎩⎨⎧
=⎭⎬⎫
⎩⎨⎧⎥⎦
⎤⎢⎣
⎡+
⎭⎬⎫
⎩⎨⎧⎥⎦
⎤⎢⎣
⎡+
⎭⎬⎫
⎩⎨⎧⎥⎦
⎤⎢⎣
⎡00
000
000
gbbTsb
sbss
g
ss
g
ss
yy
KKKK
yyC
yyM
&
&
&&
&& (3-9)
where ][ ssM is the diagonal lumped mass matrix, ][ ssC is the viscous damping matrix and
][ ssK is the stiffness matrix corresponding to the structure degrees of freedom. ][ sbK is the
coupling stiffness matrix between the structure degrees of freedom and the support degrees
School of Civil and Resource Engineering Chapter 3 The University of Western Australia
3-9
of freedom, ][ bbK is that corresponding to the support movements, [ ] { }321 ,, yyyy T = are the
total displacements vector of the structure and [ ] { }321 ,, gggT
g yyyy = 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.
The total structural response equation can be derived from Equation (3-9) as
]][[]][[]][[]][[ gsbssssss yKyKyCyM −=++ &&& (3-10)
Equation (3-10) can be decoupled into its modal vibration equation as
][][][2 2
gkss
Tk
sbTk
kkkkkk yM
Kqqqϕϕ
ϕωωξ −=++ &&& (3-11)
where kϕ is the kth vibration mode shape of the structure, kq is the kth modal response,
kω and kξ are the corresponding circular frequency and viscous damping ratio,
respectively.
The kth modal response in the frequency domain can be obtained from Equation (3-11) as
)()()(1
ωψωω iyiHiqr
jgjjkkk ∑
=
= (3-12)
in which r is the total number of supports, and
kkk
k iiH
ωωξωωω
21)( 22 +−
= (3-13)
is the kth mode frequency response function.
kss
Tk
jsb
Tk
jk MK
ϕϕϕψ
][][
−= (3-14)
is the quasi-static participation coefficient for the kth mode corresponding to a movement
at support j, ][ jsbK is a vector in coupled stiffness matrix ][ sbK corresponding to support j.
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3-10
The structural response of the ith degree of freedom is
)()(1
tqtyl
kk
ik
i ∑=
= ϕ (3-15)
where l is the number of modes considered in the calculation, and ikϕ is the kth mode
shape value corresponding to the ith degree of freedom.
For the system shown in Figure 3-1, the relative displacement between the adjacent bridge
decks is
)()()()()(1
2
1
1212 tqtqtytyt
l
kkk
l
kkk ∑∑
==
−=−=Δ ϕϕ (3-16)
The power spectral density function of 2Δ can then be derived as:
∑∑ ∑ ∑= = = =
Δ ⎥⎦
⎤⎢⎣
⎡−−=
r
jjn
r
n
l
k
l
snssssjkkkk iSiHiHS
1 1 1 1
*21214 )()()()()(1)(
2ωψωϕϕψωϕϕ
ωω (3-17)
where )( ωiS jn is the cross power spectral density function given in Equation (3-8).
Similarly, the relative displacement between the abutment and the deck is
)()()(
)()()(
32
3
11
1
tytyt
tytyt
g
g
−=Δ
−=Δ (3-18)
The power spectral density functions of 1Δ and 3Δ can be formulated in the following form
⎥⎦
⎤⎢⎣
⎡−
⎥⎦
⎤⎢⎣
⎡=
∑∑
∑∑ ∑ ∑
= =
= = = =Δ
r
j
l
kjjkkk
r
jjn
r
n
l
k
l
snsssjkkk
iSiH
iSiHiHS
1 11
14
1 1 1 1
*114
)()(Re2
)()()(1)(1
ωψωϕω
ωψωϕψωϕω
ω (3-19)
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3-11
⎥⎦
⎤⎢⎣
⎡−
⎥⎦
⎤⎢⎣
⎡=
∑∑
∑∑ ∑ ∑
= =
= = = =Δ
r
j
l
kjjkkk
r
jjn
r
n
l
k
l
snsssjkkk
iSiH
iSiHiHS
1 13
24
1 1 1 1
*224
)()(Re2
)()()(1)(3
ωψωϕω
ωψωϕψωϕω
ω (3-20)
where ‘Re’ denotes the real part of a complex quantity. The mean peak responses can be
obtained based on Equations (3-17), (3-19) and (3-20) 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 deck, and between the two adjacent bridge decks of the bridge model shown
in Figure 3-1 subjected to spatially varying ground motions at a canyon site. It should be
noted that in the case of strong earthquake non-linear behaviour of piers, bearings and
foundations might occur, which will strongly affect the structural responses. However, the
current work 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 [10, 24-27]. In the present study,
though the two adjacent bridge decks are coupled with each other on the top of the pier,
the uncoupled frequency ratio 12 / ff of the two spans is still used to approximately quantify
the frequency difference of the two decks. 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 12 / ff , where π2//2 111 mkf b= and π2//2 222 mkf b= , are the
uncoupled frequency (in Hz) of the left and right spans, respectively.
To simplify the analysis, the cross sections of the two decks of the bridge model shown in
Figure 3-1 are assumed to be the same, with mass per unit length 4102.1 × kg/m, the lengths
of the left and right spans are assumed to be d1=d2=100 m, so the masses of the two decks
are 621 102.1 ×== mm kg. The lumped mass at the top of the pier is 5
3 102×=m kg. The actual
bearing stiffness of a bridge depends on many factors such as the deck dimension and
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3-12
weight, bearing types and dimensions, etc. In practice, most commonly used bearings have
stiffness in the range of 6102× N/m to 7106× N/m. For parametrical study, the bearing
stiffness of the left span is assumed to be 61 106×=bk N/m, which corresponds to the
uncoupled frequency of the left span 5.01 =f Hz. The bearing stiffness of the right span
varies from 5102× N/m to 8106× N/m in the present paper to obtain different frequency
ratios 12 / ff . The stiffness of the pier used in the study is 810=pk 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.
3.5.1 Effect of ground motion spatial variations
To investigate the influence of spatially varying ground motions on the required separation
distance, highly, intermediately and weakly correlated ground motions are considered. The
parameters are given in Table 3-1. For comparison, spatial ground motions with
intermediate coherency loss without considering phase shift ( 0.1)/cos( =appvdω ), spatial
ground motions without considering coherency loss ( 0.1)('' =ωγ iBA
, wave passage effect
only) and uniform ground motion ( 0.1)('' =ωγ iBA) are also considered. To preclude the
effect of site amplification, the analysis in this section assumes that the bridge is located on
the base rock, i.e. 0=== CBA hhh m. Previous papers [24-27] considered the required
separation distance between adjacent bridge decks ( 2Δ in the present paper), no paper
regarding the required separation distance between the abutment and the bridge deck ( 3Δ
and 1Δ ) has been reported. For discussion purpose, the sequences of the required
separation distances discussed in the paper are 2Δ , 3Δ and then 1Δ . 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 peak responses. Therefore,
only the mean peak responses will be presented and discussed hereafter.
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Table 3-1. Parameters for coherency loss functions
Coherency loss β a b c
Highly 410109.1 −× 310583.3 −× 510811.1 −×− 410177.1 −×
Intermediately 410697.3 −× 210194.1 −×
510811.1 −×− 410177.1 −×
Weakly 310109.1 −× 210583.3 −× 510811.1 −×−
410177.1 −×
As shown in Figure 3-3(a), with an assumption of uniform excitation, the relative
displacement between the two adjacent bridge decks ( 2Δ ) is relatively small when the
fundamental frequencies of the adjacent structures are similar, and is zero when 12 ff = .
This is because the vibration modes of the two adjacent spans are exactly the same when
the frequencies coincide with each other [26]. 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 12 / ff is close to unity, weakly 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 decks. When 5.1/ 12 >ff , 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 3-3(a), there is one obvious peak
occurring at 3.0/ 12 =ff . This is because at this frequency ratio, the first modal vibration
frequency of the coupled system is 0.15 Hz, which coincides with the predominant
frequency of ground displacement as shown in Figure 3-2(b). The above observations also
indicate that adjusting the frequencies of adjacent bridge decks alone is not sufficient to
preclude pounding because ground motion spatial variation also induces relative
displacement of adjacent decks.
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(a)
(b)
(c)
Figure 3-3. Effect of ground motion spatial variation on the required separation distance
(a) 2Δ , (b) 3Δ , (c) 1Δ
Very few researchers studied the required separation distance between bridge deck and
abutment to avoid pounding although pounding damages between bridge deck and
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abutment have been observed in many earthquakes in the past. In this study, the relative
displacement between decks 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 ( 3Δ ) as shown in Figure 3-3(b), one
obvious peak can be observed when the modal frequency of the coupled system coincides
with the predominant frequency of the base rock 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
1000=appv 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 deck, although the
stiffness of the left span remains unchanged, 1Δ in Figure 3-3(c) is not a constant and
varies with the change of f2 because of the coupling through the centre pier. As can be seen
in Figure 3-3(c), when the frequency ratio is slightly smaller than 1, 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 f2/f1 is close to unity. It is
interesting to observe that the spatially varying ground motions have positive effects on 1Δ ,
i.e., weakly 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 1Δ will be a constant.
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Figure 3-4. Left span frequency response function with respect to the frequency ratios
3.5.2 Effect of the bridge girder frequency
The bearing stiffness of the left span studied above is 61 106×=bk N/m, which makes the
left deck rather flexible ( 5.01 =f Hz). To cover a larger range of possible cases, two other
bearing stiffness for the left span, i.e., 71 104.2 ×=bk N/m, and 7106.9 × N/m, are also
considered. The corresponding frequencies of the left span are =1f 1.0 Hz and 2.0 Hz,
respectively, which represent intermediate and stiff isolated bridge deck for longitudinal
movements. Figure 3-5 shows the results corresponding to the intermediately correlated
spatial ground motions.
As shown in Figures 3-5(a) and (b), the largest separation distance is required when the
modal vibration frequency of the coupled system coincides with the base rock 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), 5.01 =f Hz and 1.0 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
distance almost reaches a constant when 5.0/ 12 >ff if 0.21 =f Hz as shown in Figure 3-
5(a). 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 1Δ , 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 3-5(c). As
expected, the higher is the uncoupled frequency of the left span, the smaller separation
School of Civil and Resource Engineering Chapter 3 The University of Western Australia
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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) 2Δ , (b) 3Δ , (c) 1Δ
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3.5.3 Effect of the local soil site conditions
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 3-2 gives the corresponding parameters of site conditions. Figure 3-6 shows the
required separation distances 2Δ , 3Δ , and 1Δ corresponding to the intermediately correlated
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 mhh CA 50== ,
30=Bh 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 3-2. Parameters for local site conditions
Type Density (kg/m3) Shear wave velocity (m/s) Damping ratio
Base rock 3000 1500 0.05
Firm soil 2000 450 0.05
Medium soil 1500 300 0.05
Soft soil 1500 200 0.05
Based on the discussion above, one obvious peak occurs when the structural frequency
coincides with the central frequency of the ground displacement if the bridge locates on the
base rock. When one or all the bridge supports are located on the soil site, additional peaks
can be observed as shown in Figure 3-6(a), (b) and Figure 3-7(a), (b). One example is
shown in Figure 3-6(a) and (b), when 50== CA hh m ( Bh varies from 0 to 50 m), another
obvious peak occurs at 3/ 12 =ff . This is because when 3/ 12 =ff , the corresponding bearing
stiffness of the right span is 72 104.5 ×=bk N/m. It is found that with this stiffness the
second modal vibration frequency of the coupled system is 1.4 Hz, which coincides with
the predominant frequency of the ground motion on the 50m soil site as can be seen in
Figure 3-8(a), 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 2.4 Hz.
Another example is shown in Figure 3-7(a) and (b), when soft soil (SFS) is considered, the
second peak appears when 2/ 12 =ff which corresponds to the second modal frequency of
the coupled system at 0.95 Hz. As can be seen from Figure 3-8(b), 0.95Hz is the
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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 3-6(a) with Figure 3-
6(b), Figure 3-7(a) with Figure 3-7(b), the results show that the local soil site conditions
have a more significant effect on 3Δ than 2Δ , especially when 2.1/ 12 >ff . This is because
3Δ depends on the absolute response of the structure, while 2Δ depends on the relative
response of the bridge girders. The softer site results in larger absolute structural response
( 3Δ ), which slightly increases the relative displacement ( 2Δ ). As for 1Δ , fluctuations can be
seen when 1f and 2f 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 rock directly (e.g., [26]), only one
peak can be observed, for a bridge locates on a canyon site, more peaks 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 [26]
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.
School of Civil and Resource Engineering Chapter 3 The University of Western Australia
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(a)
(b)
(c)
Figure 3-6. Effect of soil depth on the required separation distance
(a) 2Δ , (b) 3Δ , (c) 1Δ
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(a)
(b)
(c)
Figure 3-7. Effect of soil properties on the required separation distance
(a) 2Δ , (b) 3Δ , (c) 1Δ
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(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 decks during strong earthquake 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 decks 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
decks, between the abutment and the bridge deck are considered. Following conclusions
can be obtained based on the numerical results:
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1. 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 decks have similar vibration frequencies. Less
correlated ground motions require a larger separation distance to avoid pounding
between bridge decks. The coupling effect is significant on the required separation
distance between the deck and abutment when the uncoupled frequency ratio of
the adjacent spans is close to unity.
2. 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 make 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 work) 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 earthquakes.
3.7 Appendix
3.7.1 Appendix A: Mean peak response calculation
Standard random vibration method is used to calculate the mean peak displacement, it is
briefly described in the following [32].
For a zero mean stationary process x(t) with known power spectral density function )(ωS ,
its m th order spectral moment is defined as
ωωωλω
dSc mm ∫≈ 0
)( (A3-1)
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where cω 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
0
21λλ
π=v (A3-2)
20
211λλλδ −= (A3-3)
the mean peak response can then be calculated by
σ)ln25772.0ln2(max Tv
Tvxe
e += (A3-4)
where T is the duration of the stationary process, 0λσ = is the standard deviation of the
process, and
69.069.01.01.00
)38.063.1()2,1.2max(
45.0
≥<≤<≤
⎪⎩
⎪⎨
⎧−=
δδδ
δδ
vTvT
TTve
(A3-5)
In the present study, the high cut-off frequency is taken as Hzc 25=ω since it covers the
predominant vibration modes of most engineering structures and the dominant earthquake
ground motion frequencies.
3.7.2 Appendix B: Characteristic matrices
For the bridge model shown in Figure 3-1, the mass matrix is
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡=
3
2
1
000000
][m
mm
M ss (B3-1)
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where 1m , 2m and 3m is the lumped mass of the two bridge decks and the pier,
respectively.
The stiffness matrices can be formulated as
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
++−−−−
=
pbbbb
bb
bb
ss
kkkkkkkkk
K
2121
22
11
2002
][ (B3-2)
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
−−
−=
0000
00][ 2
1
p
b
b
sb
kk
kK (B3-3)
where 1bk and 2bk are the bearing stiffness of the left and right spans, respectively, and pk
the corresponding stiffness of the pier.
3.8 References
1. Yashinsky M, Karshenas MJ. Fundamentals of seismic protection for bridges. Earthquake
Engineering Research Institute, 2003.
2. Kawashima K, Unjoh S. Impact of Hanshin/Awaji earthquake on seismic design
and seismic strengthening of highway bridges. Structural Engineering/Earthquake
Engineering JSCE 1996; 13(2):211-240.
3. Earthquake Engineering Research Institute. Chi-Chi, Taiwan, Earthquake
Reconnaissance Report. Report No.01-02, EERI, Oakland, California. 1999.
4. Elnashai AS, Kim SJ, Yun GJ, Sidarta D. The Yogyakarta earthquake in May 27,
2006. Mid-America Earthquake Centre. Report No. 07-02, 57; 2007.
5. Lin CJ, Hung H, Liu Y, Chai J. Reconnaissance report of 0512 China Wenchuan
earthquake on bridges. The 14th world conference on earthquake engineering, Beijing,
China, 2008; S31-006.
6. Jeng V, Kasai K, Maison BF. A spectral difference method to estimate building
separations to avoid pounding. Earthquake Spectra 1992; 8(2):201-223.
7. Kasai K, Jagiasi AR, Jeng V. Inelastic vibration phase theory for seismic pounding
mitigation. Journal of Structural Engineering 1996; 122(10):1136-1146.
School of Civil and Resource Engineering Chapter 3 The University of Western Australia
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8. Penzien J. Evaluation of building separation distance required to prevent pounding
during strong earthquakes. Earthquake Engineering and Structural Dynamics 1997;
26(8):849-858.
9. Lin JH. Separation distance to avoid seismic pounding of adjacent buildings.
Earthquake Engineering and Structure Dynamics 1997; 26(3):395-403.
10. Hao H, Zhang SR. Spatial ground motion effect on relative displacement of
adjacent building structures. Earthquake Engineering and Structural Dynamics 1999;
28(4):333-349.
11. Uniform Building Code (UBC). International Building Officials. Whittier,
California; 1999.
12. AS 1170.4 SAA Earthquake Loading codes. Stands Association of Australia. 1993.
13. Seismic design code for building and structures-GB11-89. Chinese Academy of Building
Research, Beijing, 1989.
14. Ruangrassamee A, Kawashima K. Relative displacement response spectra with
pounding effect. Earthquake Engineering and Structural Dynamics 2001; 30(10):1151-
1138.
15. DesRoches R, Muthukumar S. Effect of pounding and restrainers on seismic
response of multi-frame bridges. Journal of Structural Engineering, ASCE 2002;
128(7):860-869.
16. Jankowski R, Wilde K, Fujino Y. Pounding of superstructure segments in isolated
elevated bridge during earthquakes. Earthquake Engineering and Structural Dynamics
1998; 27(5):487-502.
17. Zhu P, Abe M, Fujino Y. Modelling of three-dimensional non-linear seismic
performance of elevated bridges with emphasis on pounding of girders. Earthquake
Engineering and Structural Dynamics 2002; 31(11): 1891-1913.
18. Hao H, Chouw N. Response of a RC bridge in WA to simulated spatially varying
seismic ground motions. Australian Journal of Structural Engineering 2008; 8(1):85-97.
19. Zanardo G, Hao H, Modena C. Seismic response of multi-span simply supported
bridges to spatially varying earthquake ground motion. Earthquake Engineering and
Structural Dynamics 2002; 31(6): 1325-1345.
20. Jankowski R, Wilde K, Fujino Y. Reduction of pounding in elevated bridges during
earthquakes. Earthquake Engineering and Structural Dynamics 2000; 29(2): 195-212.
21. LRFD bridge design specifications and commentary. American Association of State
Highway and Transportation Officials (AASHTO), 1994.
School of Civil and Resource Engineering Chapter 3 The University of Western Australia
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22. Caltrans Seismic Design Criteria Version l.2. Department of Transportation.
Sacramento, California, 2001.
23. Design specifications for highway bridges—Part V: Seismic Design. Japan Road
Association (JRA), Tokyo, Japan, 2004.
24. 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 2008; 30(1):141-153.
25. 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 2008; 30(1):154-162.
26. Hao H. A parametric study of the required seating length for bridge decks during
earthquake. Earthquake Engineering and Structural Dynamics 1998; 27(1):91-103.
27. Bi K, Hao H, Chouw N. Stochastic analysis of the required separation distance to
avoid seismic pounding of adjacent bridge decks. The 14th world conference on
earthquake engineering, Beijing, China, 2008; 03-03-0026.
28. Ruiz P, Penzien J. Probabilistic study of the behaviour of structures during
earthquakes. Report No. UCB/EERC-69-03, University of California at Berkeley.
1969.
29. Tajimi H. A statistical method of determining the maximum response of a building
structure during an earthquake. Proceedings of 2nd World Conference on Earthquake
Engineering, Tokyo, 1960; 781-796.
30. Hao H, Oliveira CS, Penzien J. Multiple-station ground motion processing and
simulation based on SMART-1 array data. Nuclear Engineering and Design 1989;
111(3):293-310.
31. Hao H, Chouw N. Modeling of earthquake ground motion spatial variation on
uneven sites with varying soil conditions. The 9th International Symposium on Structural
Engineering for Young Experts, Fuzhou, China, 2006; 79-85.
32. Der Kiureghian A. Structural response to stationary excitation. Journal of Engineering
Mechanics 1980; 106(6): 1195-1213.
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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 earthquake ground excitations at multiple-
supports 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 earthquake shaking 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 [1], 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 peak structural
responses are estimated by using the standard random vibration method. The minimum
total gap between two adjacent bridge decks or between bridge deck 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
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4.1 Introduction
Observations from past strong earthquakes 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 known as soil-structure
interaction (SSI) effect. Seismic ground motion variations may result in pounding or even
collapse of adjacent bridge decks owing to the out-of-phase responses. In fact, poundings
between an abutment and bridge deck or between two adjacent bridge decks were observed
in almost all the major earthquakes, e.g. the 1994 Northridge earthquake [2], the 1995
Hyogo-Ken Nanbu earthquake [3], the 1999 Chi-Chi Taiwan earthquake [4], the 2006
Yogyakarta earthquake [5], and more recently the 2008 Wenchuan earthquake [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
Muthukumar [8] investigated pounding effect on the global response of a multiple-frame
bridge. After installation of the SMART-1 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. Jankowski et al. [9] and Zhu et al. [10] studied the pounding effect of an
elevated bridge caused by spatial ground motions with only wave passage effect. Taking
combined wave passage effect and coherency loss effect into consideration, Hao [11]
investigated the required seating length to prevent bridge deck unseating; Zanardo et al.
[12] 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 [13] proposed a transfer function that implicitly
School of Civil and Resource Engineering Chapter 4 The University of Western Australia
4-3
models the site effect on spatial seismic ground motions. Using this model, Dumanogluid
and Soyluk [14] analysed the stochastic responses of a cable-stayed bridge to spatially
varying ground motions with site effect; Ates et al. [15] 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 [16] presented a uniform approach in the frequency domain to analyse the free-
field response of local site with multiple soil layers and SSI effect on structural responses;
Spyrakos and Vlassis [17] assessed the effect of SSI on the response of seismically isolated
bridge piers; Makris et al. [18] 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 [19, 20] 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. Shrikhande and Gupta
[21] proposed a stochastic approach for the linear analysis of suspension bridges subjected
to earthquake excitations with consideration of ground motion spatial variation and SSI.
The studies with the broadest scope known to the authors are that by Sextos et al. [22, 23],
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 decks during strong
earthquakes 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 decks becomes possible [24]. 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 took 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 [24] and then introduced a new design
philosophy for a MEJ [25]; In a recent study [1], 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 decks. It should
School of Civil and Resource Engineering Chapter 4 The University of Western Australia
4-4
be noted that these studies either neglected site effect [24, 25] or SSI [1]. To the best
knowledge 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 work [1] in which SSI effect is not considered. The present work
therefore focuses on the SSI effect on bridge structural responses. Random vibration
method is adopted in the study. Spatial ground motions on the base rock 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 rock 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
decks and between bridge deck and abutment are derived and their mean peak responses
are estimated. The minimum total gaps between abutment and bridge deck and two
adjacent bridge decks 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.
4.2 Bridge-soil system
Figure 4-1 (a) illustrates the schematic view of a typical girder bridge crossing a canyon site.
The superstructure of the bridge is adopted from [1]. The length and total mass for each
deck are d1=d2=100m and m1=m2=1.2×106kg, respectively. The two bearings for each
deck have the same dynamic properties with an effective stiffness kb1 and an equivalent
viscous damping 1bc for the left span, and kb2 and 2bc for the right span. The concrete pier
with a height of L=20m is modelled as an elastic column with a lumped mass m3=2×105kg
at pier top, the lateral stiffness of the pier is kp=108N/m. To simplify the analysis, a
constant damping ratio of 5% is used for bearings and the pier. Different from Reference
[1], where all the foundations are assumed rigidly fixed to the ground surface, the pier in
School of Civil and Resource Engineering Chapter 4 The University of Western Australia
4-5
the present study is founded on a rigid cap which is supported by a 2×2 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=12m. 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 1, 2 and 3 as
shown in Figure 4-1(a), the corresponding points at base rock are 1', 2' 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 [26]. 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 hk , hc in the
horizontal direction and rk , rc 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 4.3.1. 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 4-1(b): the dynamic displacements 1u and 2u of the bridge deck
movement relative to the free field motion 1gu and 2gu ; the horizontal displacement 0u of
the pile foundation at site 3 relative to the free field motion 3gu ; the rotational response φ
of the pier at the foundation level and the dynamic response 3u at the pier top.
School of Civil and Resource Engineering Chapter 4 The University of Western Australia
4-6
2m2gu2u1gu 1u
1m
)(b
1d 2d
1m 2m
3m1bc 2bc1bc
'1
1
3
'3
2
'2
pp ck
1bk 1bk 2bk 2bk
2bc
1Δ
3Site1Site 2Site
)(a
rockBase
3Δ
hc
hk
s d
2Δ
L
3gu 0u φL 3u3m
φ
rcrk
Figure 4-1. (a) Schematic view of a girder bridge crossing a canyon site
and (b) structural model
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4-7
4.3 Method of analysis
In the present paper, the structural responses are calculated in the frequency domain. Free-
field 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 weak. The support
motions therefore can be approximately considered equal to the free-filed motions [18, 27].
Similar to Reference [1], the free-field spatial ground motions in the present study are
derived from the base rock 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
[1]. Substructure method is used to analyse SSI effect. The dynamic impedances of the
foundations are defined in Section 4.3.1. The mean peak responses of the system are
estimated based on random vibration method after the power spectral densities of the
relative displacements are derived in Section 4.3.2.
4.3.1 Dynamic soil stiffness
As mentioned above, the soil-abutment interaction is ignored at sites 1 and 2, the only SSI
considered in the paper is the 2×2 pile group embedded in a uniform stratum at site 3.
The dynamic stiffness of a pile group ( GK ) can be calculated using the dynamic stiffness of
a single pile ( SK ) 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
[18].
The dynamic stiffness and damping of a single pile can be described in terms of complex
stiffness
SSS cikK ω+= (4-1)
where superscript S represents the values for a single pile, ω is circular frequency, Sk and Sc are the stiffness and equivalent viscous damping of the pile. The corresponding
expressions for Sk and Sc can be readily obtained for use from the previous studies [28,
29], and the coefficients suggested by Gazetas [29] are used in the present paper.
School of Civil and Resource Engineering Chapter 4 The University of Western Australia
4-8
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
qp
ww
== )('' ωαα (4-2)
where qpw is the dynamic displacement of pile q caused by pile p and qqw 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 [31] 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 [31] is
adopted herein.
The frequency-dependent dynamic stiffness and damping coefficient of the pile group then
can be estimated as
( )Ghh Kk Re= , ( ) ω/Im G
hh Kc = (4-3a)
in the horizontal direction, and
( )Grr Kk Re= , ( ) ω/Im G
rr Kc = (4-3b)
in the rotational direction, where GhK and G
rK 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.
4.3.2 Structural response formulation
With the hypotheses mentioned above, the total displacements of bridge decks and the pier
are
111 uuu gt += , 222 uuu g
t += , 3033 uLuuu gt +++= φ (4-4)
School of Civil and Resource Engineering Chapter 4 The University of Western Australia
4-9
The dynamic equilibrium equations of the idealized model in Figure 4-1(b) can be
expressed in the matrix form as follows:
⎪⎭
⎪⎬
⎫
⎪⎩
⎪⎨
⎧
⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢
⎣
⎡
−−−−−−
−−
+⎪⎭
⎪⎬
⎫
⎪⎩
⎪⎨
⎧
⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢
⎣
⎡
−−−−−−
−−
+⎪⎭
⎪⎬
⎫
⎪⎩
⎪⎨
⎧
⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢
⎣
⎡
−−−
−−
=
⎪⎪⎪
⎭
⎪⎪⎪
⎬
⎫
⎪⎪⎪
⎩
⎪⎪⎪
⎨
⎧
⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢
⎣
⎡
++++−−++++−−++++−−
−−−−−−
+
⎪⎪⎪
⎭
⎪⎪⎪
⎬
⎫
⎪⎪⎪
⎩
⎪⎪⎪
⎨
⎧
⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢
⎣
⎡
++++−−++++−−++++−−
−−−−−−
+
⎪⎪⎪
⎭
⎪⎪⎪
⎬
⎫
⎪⎪⎪
⎩
⎪⎪⎪
⎨
⎧
⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢
⎣
⎡
3
2
1
2121
2121
2121
22
11
3
2
1
2121
2121
2121
22
11
3
2
1
3
3
3
2
1
3
2
1
221212121
21212121
21212121
2222
1111
3
2
1
221212121
21212121
21212121
2222
1111
3
2
1
333
333
333
2
1
00
00
000000
0000
/
2002
/
2002
000000
00000000
g
g
g
bbbb
bbbb
bbbb
bb
bb
g
g
g
bbbb
bbbb
bbbb
bb
bb
g
g
g
o
rbbbbbbbb
bbhbbbbbb
bbbbpbbbb
bbbb
bbbb
o
rbbbbbbbb
bbhbbbbbb
bbbbpbbbb
bbbb
bbbb
o
uuu
kkkkkkkkkkkk
kkkk
uuu
cccccccccccc
cccc
uuu
mmm
mm
Luuuu
Lkkkkkkkkkkkkkkkkkkkkkkkkkkk
kkkkkkkk
Luuuu
Lccccccccccccccccccccccccccc
cccccccc
Luuuu
mmmmmmmmm
mm
&
&
&
&&
&&
&&
&
&
&
&
&
&&
&&
&&
&&
&&
φ
φφ
(4-5)
We define the circular frequencies of the structure and the subsoil system in the following
form
1
121
2mkb=ω ,
2
222
2mkb=ω ,
321
23 mmm
k p
++=ω (4-6a)
321
2
mmmkh
h ++=ω , 2
321
2
)( Lmmmkr
r ++=ω (4-6b)
the corresponding viscous damping ratio of the bridge deck, pier and the subsoil system are
expressed as
1
111 2 b
b
kcωξ = ,
2
222 2 b
b
kcωξ = ,
p
p
kc
23
3
ωξ = (4-7a)
h
hhh k
c2ωξ = ,
r
rrr k
c2ωξ = (4-7b)
With the aid of Equations (4-6) and (4-7), and by assuming the mass ratios
School of Civil and Resource Engineering Chapter 4 The University of Western Australia
4-10
3
1
mm
=α , 3
2
mm
=β (4-8)
Equation (4-5) can be expressed in the frequency domain as
{ } { })()]([)()]([ ωωωω iuiZiuiZ gg= (4-9)
where
{ } { }TiLiuiuiuiuiu )()()()()()( 0321 ωφωωωωω = (4-10a)
and
{ } { }Tgggg iuiuiuiu )()()()( 321 ωωωω = (4-10b)
are the dynamic response vector and the input ground motion vector, respectively. [ ])( ωiZ
and )]([ ωiZg are the impedance matrices of the system, which are in the following form
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡=
)()(
)()()]([
5551
1511
ωω
ωωω
iziz
iziziZ
L
MM
L
(4-11a)
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
=)()(
)()()]([
5351
1311
ωω
ωωω
iziz
iziziZ
gg
gg
g
L
MM
L
(4-11b)
The elements of these two matrixes are given in Appendix A. The dynamic response of the
bridge structure can then be calculated by
{ } { } { })()]([)()]([)]([)( 1 ωωωωωω iuiHiuiZiZiu ggg == − (4-12)
For the bridge model shown in Figure 4-1, 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
School of Civil and Resource Engineering Chapter 4 The University of Western Australia
4-11
)()( 11 ωω iui =Δ , )()( 22 ωω iui =Δ , )()()( 213 ωωω iuiui tt −=Δ (4-13)
The power spectral density functions of 1Δ , 2Δ and 3Δ thus can be derived, and the
corresponding expressions are given in Appendix B. It should be noted that Δ1 and Δ2, i.e.,
the relative displacement between bridge deck and abutments, only depend on the
respective dynamic response of the bridge deck, whereas the relative displacement response
between two adjacent decks, Δ3, depend on both bridge deck dynamic response and spatial
ground displacements.
After the derivation of the power spectral density functions of the required separation
distances, the mean peak responses can be estimated based on the standard random
vibration method [32]. In the present study, the high cut-off frequency is assumed to be 25
Hz since it covers the predominant vibration modes of most engineering structures and the
dominant earthquake 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 decks or
at abutments of the bridge shown in Figure 4-1 must provide. Three types of soil, i.e., firm,
medium and soft soil, are considered. Table 4-1 gives the corresponding parameters of soil
and base rock. Figure 4-2 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 [1], 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 weakly correlated ground
motions are studied. The corresponding parameters can be found in [1].
Table 4-1. Parameters for local site conditions
Type Density (kg/m3) Shear wave velocity (m/s) Damping ratio Poisson’s ratio
Base rock 3000 1500 0.05 0.4
Firm soil 2000 400 0.05 0.4
Medium soil 2000 200 0.05 0.4
Soft soil 1500 150 0.05 0.4
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4-12
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 [1, 11,
24, 25]. In the present study, though the two adjacent bridge decks are coupled with each
other on the top of the pier, the uncoupled frequency ratio 12 / ff of the two spans is still
used to approximately quantify the frequency difference of the two decks. 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 12 / ff . The stiffness of the left span is assumed to be
a constant with 71 104.2 ×=bk N/m, which gives an uncoupled frequency
0.12//2 111 == πmkf b Hz. The bearing stiffness of the right span varies from 5102×
N/m to 8105.1 × N/m to obtain different frequency ratios 12 / ff . 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 keep the damping ratio
unchanged in the calculations.
Figure 4-2. Frequency-dependent dynamic stiffness and damping coefficients of the pile
group (a)(b) horizontal direction and (c)(d) rotational direction
4.4.1 Influence of site effect and SSI
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4-13
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 1000=appv m/s. Results corresponding to soft, medium and firm soils with SSI
(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) 3Δ , (b) 2Δ and (c) 1Δ
As shown in Figure 4-3, for 3Δ and 2Δ the largest required separation distance occurs at
18.0/ 12 =ff , or when the uncoupled vibration frequency of the right span is 0.18 Hz.
This is because 0.18 Hz coincides with the central frequency of ground displacement as
shown in Figure 4-4(c). This significantly increases the displacement response of the right
span and hence the relative displacement response 3Δ and 2Δ .
When soft soil (bold solid line) is considered, additional amplification in 3Δ and especially
in 2Δ occurs at 75.0/ 12 =ff 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 150 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 peak occurs at 0.1/ 12 ≈ff in 2Δ owing to the same reason. However, this peak is
not observed in 3Δ , this is because 3Δ measures the relative displacement between the two
adjacent bridge decks, 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 peak can be observed, because the fundamental
School of Civil and Resource Engineering Chapter 4 The University of Western Australia
4-14
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 [11].
As for the relative displacement 1Δ between the left abutment and the deck, although the
stiffness of the left span remains unchanged, 1Δ in Figure 4-3(c) is not a constant and
varies with the change of 2f because of the coupling through the pier. 1Δ experiences a
sudden significant variation when 12 / ff is close to unity. This is because changing 2f has
only insignificant influence on the vibration frequency of the left span when 2f differs
pronouncedly from 1f . However, when 2f is close to 1f , or 12 / ff 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 peak responses also occur at respective frequency ratios when the
right span resonates with the site. However, the peak response at 18.0/ 12 =ff is not
observed in 1Δ . This is because 1f is a constant in the simulations, and this peak 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 [1].
Larger 2Δ in Figure 4-3(b) is usually obtained when the site is soft. However, for 3Δ and
1Δ , medium soil (bold dash line) might give larger responses than the soft soil condition.
This is because the frequency of the left span is 1.0 Hz, which coincides with fundamental
frequency of the medium soil site with a depth of 50 m and shear wave velocity 200 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.
School of Civil and Resource Engineering Chapter 4 The University of Western Australia
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Compared with the results without SSI effect in Figure 4-3, SSI only slightly changes the
frequency content of the bridge, i.e., the peaks 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
( )( ωiS ) and the frequency response function of the structure ( )( ωiH ) 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. Take soft site for example, at the site fundamental frequency of 0.75 Hz the
soft soil amplifies ground motions on the base rock 24 times as shown in Figure 4-4(a),
which alters the ground motions on ground surface significantly from the base rock motion
as can be seen in Figure 4-4(b). 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 4-5(b), when soft or medium soil is considered, the influence of SSI on
2Δ increases and becomes most pronounced when the structure resonates with local site.
The contribution of SSI on relative displacement response is nearly 0.2 m for soft soil
when 12 / ff is around 0.75. As previously discussed, the right span resonates with local site
at this frequency ratio. When 75.0/ 12 >ff , the influence of SSI on the total responses
School of Civil and Resource Engineering Chapter 4 The University of Western Australia
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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 3Δ , similar observations of different SSI effect
can be obtained. It should be noted that no obvious peak can be observed when resonance
occurs for medium soil, owing to the two spans tend to vibrate in phase as discussed
above. For 1Δ , 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 ( 0.11 =f Hz) (a) 3Δ , (b) 2Δ and (c) 1Δ
Figure 4-6. Contribution of SSI to the required separation distances with different soil
conditions ( 0.21 =f Hz ) (a) 3Δ , (b) 2Δ and (c) 1Δ
Reference [1] 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
0.21 =f Hz is additionally investigated. Figure 4-6 shows the contribution of SSI to the
total responses. For 3Δ and 2Δ , similar conclusions can be obtained as in Figure 4-5, i.e.,
School of Civil and Resource Engineering Chapter 4 The University of Western Australia
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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 1Δ , however, the influence of site
conditions is less prominent as compared to the case with 0.11 =f 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 weakly coupled through the bearings at the
pier cap, and the coupling effect is less pronounced when the left span bearings are stiff.
4.4.2 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 weakly correlated ground motions are
considered. The soils under the three foundations are assumed to be medium soil. The
apparent wave velocity is 1000=appv m/s. As shown in Figure 4-7, whether or not SSI is
considered, the influence pattern of ground motion spatial variation on the required
separation distances does not change too much. Take 3Δ in Figure 4-7(a) for example,
when SSI is not considered, ground motion spatial variation effect is most significant when
12 / ff is close to unity, where weakly 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 decks. When SSI is
included, similar results can be observed. Similar observations can be made on 2Δ and 1Δ .
These observations are also similar to those reported in [1] where SSI is not considered. As
shown, the influence of changing spatial ground motions from highly correlated to weakly
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., [11]).
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Figure 4-7. Influence of ground motion characteristics and SSI on the required separation
distances ( 0.11 =f Hz) (a) 3Δ , (b) 2Δ and (c) 1Δ
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
( 0.11 =f Hz) and stiff ( 0.21 =f Hz) left span bearings, respectively. It is obvious in Figure
4-9 that the effect of SSI is most significant at 5.0/ 12 ≈ff , or 0.12 =f . As discussed
above, natural vibration frequency of the 50 m medium site is about 1.0 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 2Δ in Figure 4-8(b), similar observation can be obtained when right
span resonates with the site at 0.1/ 12 ≈ff or 0.12 =f . For 3Δ and 1Δ in Figure 4-8(a) and
(c), however, no such peaks can be observed. This is because both the left and right spans
resonate with the site at 0.1/ 12 ≈ff . Since the two adjacent spans tend to vibrate in-phase
at this vibration frequency, the relative displacement response 3Δ is the smallest, and the
effect of SSI is also insignificant. As for 1Δ , it shows a fluctuation when 12 / ff 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 Shrikhande and Gupta [21], 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
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with those in [24], in which it was concluded that SSI effect is more prominent only when
the spatial ground motions are highly correlated in the range of 12 / ff between 0.55 and
1.0. The later study [24] was carried out in the time and Laplace domain by using 20 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 ( 0.11 =f Hz) (a) 3Δ , (b) 2Δ and (c) 1Δ
Figure 4-9. Contribution of SSI to the required separation distances with different
coherency loss functions ( 0.21 =f Hz) (a) 3Δ , (b) 2Δ and (c) 1Δ
4.5 Conclusions
Based on the fixed base assumption, Reference [1] 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 [1] 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
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and SSI, combined effect of ground motion spatial variation and SSI are investigated, and
the SSI effect is highlighted. Following conclusions are obtained:
1. 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.
2. 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
4.6.1 Appendix A: Element for [ ])( ωiZ and )]([ ωiZg
1121
211 2)( ξωωωωω iiz ++−=
0)(12 =ωiz 2/)2()( 11
2113 ξωωωω iiz +−= (A4-1)
2/)2()( 112114 ξωωωω iiz +−=
2/)2()( 112115 ξωωωω iiz +−=
0)(21 =ωiz
2222
222 2)( ξωωωωω iiz ++−=
2/)2()( 222223 ξωωωω iiz +−= (A4-2)
2/)2()( 222224 ξωωωω iiz +−=
2/)2()( 222225 ξωωωω iiz +−=
2/)2()( 11
2131 ξωωωαω iiz +−=
2/)2()( 222232 ξωωωβω iiz +−=
)2)(1(2/)2(2/)2()( 332322
2211
21
233 ξωωωβαξωωωβξωωωαωω iiiiz ++++++++−=
2/)2(2/)2()( 222211
21
234 ξωωωβξωωωαωω iiiz ++++−= (A4-3)
2/)2(2/)2()( 222211
21
235 ξωωωβξωωωαωω iiiz ++++−=
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2/)2()( 112141 ξωωωαω iiz +−=
2/)2()( 222242 ξωωωβω iiz +−=
2/)2(2/)2()( 222211
21
243 ξωωωβξωωωαωω iiiz ++++−= (A4-4)
)2)(1(2/)2(2/)2()( 222
2211
21
244 hhh iiiiz ξωωωβαξωωωβξωωωαωω ++++++++−=
2/)2(2/)2()( 222211
21
245 ξωωωβξωωωαωω iiiz ++++−=
2/)2()( 11
2151 ξωωωαω iiz +−=
2/)2()( 222252 ξωωωβω iiz +−=
2/)2(2/)2()( 222211
21
253 ξωωωβξωωωαωω iiiz ++++−= (A4-5)
2/)2(2/)2()( 222211
21
254 ξωωωβξωωωαωω iiiz ++++−=
)2)(1(2/)2(2/)2()( 222
2211
21
255 rrr iiiiz ξωωωβαξωωωβξωωωαωω ++++++++−=
2/)2()( 11
21
211 ξωωωωω iizg +−=
0)(12 =ωizg (A4-6)
2/)2()( 112113 ξωωωω iizg +=
0)(21 =ωizg
2/)2()( 2222
222 ξωωωωω iizg +−= (A4-7)
2/)2()( 222223 ξωωωω iizg +=
2/)2()( 11
2131 ξωωωαω iizg +=
2/)2()( 222232 ξωωωβω iizg += (A4-8)
222
2211
2133 2/)2(2/)2()( ωξωωωβξωωωαω ++−+−= iiizg
2/)2()( 11
2141 ξωωωαω iizg +=
2/)2()( 222242 ξωωωβω iizg += (A4-9)
222
2211
2143 2/)2(2/)2()( ωξωωωβξωωωαω ++−+−= iiizg
2/)2()( 11
2151 ξωωωαω iizg +=
2/)2()( 222252 ξωωωβω iizg += (A4-10)
222
2211
2153 2/)2(2/)2()( ωξωωωβξωωωαω ++−+−= iiizg
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4-22
4.6.2 Appendix B: PSDs of the required separation distances
[ ]
[ ] [ ])()()(Re2)()()(Re2
)()()(Re2)()(1
)()(1)()(1)(
23131241313114
1212114332
134
222
124112
1141
ωωωω
ωωωω
ωωωω
ωωω
ωωω
ωωω
ω
iSiHiHiSiHiH
iSiHiHiSiH
iSiHiSiHS
∗∗
∗
Δ
++
++
+=
(B4-1)
[ ]
[ ] [ ])()()(Re2)()()(Re2
)()()(Re2)()(1
)()(1)()(1)(
23232241323214
1222214332
234
222
224112
2142
ωωωω
ωωωω
ωωωω
ωωω
ωωω
ωωω
ω
iSiHiHiSiHiH
iSiHiHiSiH
iSiHSiHS
∗∗
∗
Δ
++
++
+=
(B4-2)
[ ][ ]{ }[ ][ ]{ }[ ][ ]{ })()()(1)()(Re2
)()()(1)()(Re2
)(1)()(1)()(Re2
)()()(1
)(1)()(1
)(1)()(1)(
23231322124
13231321114
12221221114
332
23134
222
22124
112
211143
ωωωωωω
ωωωωωω
ωωωωωω
ωωωω
ωωωω
ωωωω
ω
iSiHiHiHiH
iSiHiHiHiH
iSiHiHiHiH
iSiHiH
iSiHiH
iSiHiHS
∗
∗
∗
Δ
−−−+
−+−+
−−+−+
−+
−−+
+−=
(B4-3)
where )( ωiS jk 3,2,1, =kj is the cross power spectral density function between points j
and k on the ground surface, which can be obtained from Reference [1]. )( ωiH jk is the
frequency response function, and can be determined by Equation (4-12).
4.7 References
1. Bi K, Hao H, Chouw N. Required separation distance between decks and at
abutments of a bridge crossing a canyon site to avoid seismic pounding. Earthquake
Engineering and Structural Dynamics 2010; 39(3): 303-323.
2. Hall FJ, editor. Northridge earthquake, January 17, 1994. Earthquake Engineering
Research Institute, Preliminary reconnaissance report, EERI-94-01; 1994.
School of Civil and Resource Engineering Chapter 4 The University of Western Australia
4-23
3. Kawashima K, Unjoh S. Impact of Hanshin/Awaji earthquake on seismic design
and seismic strengthening of highway bridges. Structural Engineering/Earthquake
Engineering JSCE 1996; 13(2): 211-240.
4. Earthquake Engineering Research Institute. Chi-Chi, Taiwan, Earthquake
Reconnaissance Report. Report No.01-02, EERI, Oakland, California. 1999.
5. Elnashai AS, Kim SJ, Yun GJ, Sidarta D. The Yogyakarta earthquake in May 27,
2006. Mid-America Earthquake Centre. Report No. 07-02, 57, 2007.
6. Lin CJ, Hung H, Liu Y, Chai J. Reconnaissance report of 0512 China Wenchuan
earthquake on bridges. The 14th World Conference on Earthquake Engineering. Beijing,
China, 2008; S31-006.
7. Ruangrassamee A, Kawashima K. Relative displacement response spectra with
pounding effect. Earthquake Engineering and Structural Dynamics 2001; 30(10): 1511-
1538.
8. DesRoches R, Muthukumar S. Effect of pounding and restrainers on seismic
response of multi-frame bridges. Journal of Structural Engineering ASCE 2002; 128(7):
860-869.
9. Jankowski R, Wilde K, Fujino Y. Pounding of superstructure segments in isolated
elevated bridge during earthquakes. Earthquake Engineering and Structural Dynamics
1998; 27(5): 487-502.
10. Zhu P, Abe M, Fujino Y. Modelling of three-dimensional non-linear seismic
performance of elevated bridges with emphasis on pounding of girders. Earthquake
Engineering and Structural Dynamics 2002; 31(11): 1891-1913.
11. Hao H. A parametric study of the required seating length for bridge decks during
earthquake. Earthquake Engineering and Structural Dynamics 1998; 27(1): 91-103.
12. Zanardo G, Hao H, Modena C. Seismic response of multi-span simply supported
bridges to spatially varying earthquake ground motion. Earthquake Engineering and
Structural Dynamics 2002; 31(6): 1325-1345.
13. Der Kiureghian A. A coherency model for spatially varying ground motions.
Earthquake Engineering and Structural Dynamics 1996; 25(1): 99-111.
14. Dumanogluid, AA, Soyluk K. A stochastic analysis of long span structures
subjected to spatially varying ground motions including the site-response effect.
Engineering Structures 2003; 25(10): 1301-1310.
15. Ates S, Bayraktar A, Dumanogluid, AA. The effect of spatially varying earthquake
ground motions on the stochastic response of bridges isolated with friction
pendulum systems. Soil Dynamics and Earthquake Engineering 2006; 26: 31-44.
16. Wolf JP. Dynamic soil-structure interaction. Englewood Cliffs, NJ: Prentice Hall; 1985.
School of Civil and Resource Engineering Chapter 4 The University of Western Australia
4-24
17. Spyrakos CC, Vlassis AG. Effect of soil-structure interaction on seismically isolated
beiges. Journal of Earthquake Engineering 2002; 6(3): 391-429.
18. Makris N, Badoni D, Delis E, Gazetas G. Prediction of observed bridge response
with soil-pile-structure interaction. Journal of Structural Engineering 1994; 120(10):
2992-3011.
19. Chouw N, Hao H. Study of SSI and non-uniform ground motion effect on
pounding between bridge girders. Soil Dynamics and Earthquake Engineering 2005; 25:
717-728.
20. 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 2008; 30(1): 141-153.
21. Shrikhande M, Gupta VK. Dynamic soil-structure interaction effects on the seismic
response of suspension bridges. Earthquake Engineering and Structural Dynamics 1999;
28(11): 1383-1403.
22. Sextos AG, Pitilakis KD, Kappos AJ. Inelastic dynamic analysis of RC bridges
accounting for spatial variability of ground motion, site effects and soil-structure
interaction phenomena. Part 1: Methodology and analytical tools. Earthquake
Engineering and Structural Dynamics 2003; 32(4): 607-627.
23. Sextos AG, Pitilakis KD, Kappos AJ. Inelastic dynamic analysis of RC bridges
accounting for spatial variability of ground motion, site effects and soil-structure
interaction phenomena. Part 2: Parametric study. Earthquake Engineering and
Structural Dynamics 2003; 32(4): 629-652.
24. 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 2008; 30(1): 154-162.
25. Chouw N, Hao H. Seismic design of bridge structures with allowance for large
relative girder movements to avoid pounding. New Zealand Society for Earthquake
Engineering Conference. Wairakei, New Zealand 2008; Paper No: 10.
26. Tongaonkar NP, Jangid RS. Seismic response of isolated bridges with soil-structure
interaction. Soil Dynamics and Earthquake Engineering 2003; 23: 287-302.
27. Fan K, Gazetas G, Kaynis A, Kausel E, Ahmad S. Kinematic seismic response of
single piles and pile groups. Journal of Geotechnique Engineering 1991; 117(12): 1860-
1879.
28. Novak M, Sharnouby BL. Stiffness constants of single piles. Journal of Geotechnical
Engineering 1983; 109(7): 961-974.
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29. Gazetas G. In: Fang HY, editor. Foundation vibrations, foundation engineering handbook,
2nd edition 1991; 553-593.
30. Gazetas G, Fan K, Kaynia A, Kausel E. Dynamic interaction factors for floating
pile groups. Geotechnical Engineering 1991; 117(10): 1531-1548.
31. Dobry R, Gazetas G. Simple method for dynamic stiffness and damping of floating
pile groups. Geotechinique 1988; 38(4): 557-574.
32. Der Kiureghian A. Structural response to stationary excitation. Journal of Engineering
Mechanics 1980; 106(6): 1195-1213.
School of Civil and Resource Engineering Chapter 5 The University of Western Australia
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Chapter 5 Modelling and simulation of spatially varying earthquake 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
earthquake 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 rock 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 rock 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
School of Civil and Resource Engineering Chapter 5 The University of Western Australia
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5.1 Introduction
In the design of structures to resist strong earthquake 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 earthquake 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 [1] and the Clough-Penzien model [2]. Both of them were proposed by
assuming the base rock 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 Safak [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 earthquake are
inevitably different, which is known 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-1 array
in Lotung, Taiwan. Zerva and Zervas [9] overviewed these models. It should be noted that
School of Civil and Resource Engineering Chapter 5 The University of Western Australia
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most of these models were proposed based on the seismic data recorded from the relatively
flat-lying sites. Taking different soil conditions into consideration, Der Kiureghian [10]
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 drawback is that it can only approximately
represent the local site effects on ground motions. For example, it is well known 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 peaks. This phenomenon, however, cannot
be considered in Der Kiureghian’s model since only one peak 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. [11] 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 [12] 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 [13, 14] to generate sample functions of a
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5-4
non-stationary, multivariate stochastic process with evolutionary power spectrum. Similar
to the Der Kiureghian [10] 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
[15].
On the other hand, many studies of site amplifications of seismic waves have also been
reported. Taking the site amplification effect into consideration, Hao [16] 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 [17] 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 [18], EERA [19] and NERA [20], 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 [13, 14] 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 rock 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 rock motions are assumed to have the same intensity and frequency contents and are
modelled by the filtered Tajimi-Kanai power spectral density function [2]. The spatial
variation effect is modelled by an empirical coherency loss function [21]. 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
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5-5
used to generate spatially varying ground motion time histories compatible to the derived
auto and cross power spectral density functions. Compared to the work by Deodatis [12],
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 rock 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 rock 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 known 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.
5.2 Wave propagation theory and site amplification effect
The one-dimensional (1D) 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 rock and soil layer
interface, and then propagate vertically in the soil layers in the 1D wave propagation
theory, the scattering and diffraction of waves by canyons, which is a 2D wave propagation
problem, are not considered. Further studies are needed to incorporate the scattering and
diffraction effect into the simulation technique. For completeness, the 1D wave
propagation theory is briefly introduced here. More detailed information can be found in
Reference [6].
For a harmonic excitation with frequencyω , the dynamic equilibrium equations can be
written as
ec
ep2
22 ω
−=∇ or { } { }Ω−=Ω∇ 2
22
scω (5-1)
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where e2∇ and { }Ω∇2 are the Laplace operator of the volumetric strain amplitude e and
rotational-strain-vector { }Ω . pc and sc 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-
plane motion, ][ LSHS and ][ L
SVPS − , can be formulated independently by analysing the
relations of shear stresses and displacements at the boundary of each soil layer. Assembling
the matrices of each soil layer and the base rock, the dynamic stiffness of the total system is
obtained and denoted by ][ SHS and ][ SVPS − , respectively. The dynamic equilibrium
equation of the site in the frequency domain is thus
{ } { }SHSHSH PuS =][ or { } { }SVPSVPSVP PuS −−− =][ (5-2)
where { }SHu and { }SHP are the out-of-plane displacements and load vector corresponding
to the incident SH wave, { }SVPu − and { }SVPP − are the in-plane displacements and load
vector of the combined P and SV waves. The stiffness matrices ][ SHS and ][ SVPS − depend
on soil properties, incident wave type, incident angle and circular frequency ω . The
dynamic load { }SHP and { }SVPP − depend on the base rock properties, incident wave type,
incident wave frequency and amplitude. By solving Equation (5-2) in the frequency domain
at every discrete frequency, the relationship of the amplitudes between the base rock and
each soil layer can be formed, and the site transfer function )]([ ωH at each soil layer can be
estimated. In the present study, only the motion on the ground surface is of interest.
To illustrate Equation (5-2), a site consisting of a single homogeneous layer resting on a
half-space is used as an example. The input on the base rock is assumed to be SH wave. It
can be directly extended to more soil layers or combined P and SV waves. Assembling the
dynamic stiffness matrix of the layer ][ LSHS and of the half-space R
SHS , the stiffness matrices
][ SHS , out-of-plane displacements { }SHu and load vector { }SHP can be expressed as [6]
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5-7
{ }
{ } TRRSH
TbtSH
LL
L
L
LL
SH
vGiktP
vvu
dktipdktdkt
dktGktS
],0[
],[
sincos11cos
sin][
0*
*
=
=
⎥⎦
⎤⎢⎣
⎡
+−−
=
(5-3)
where k 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 rock and soil
layer respectively, vt and vb 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 (5-2), the ratio of surface motion tv to
outcropping motion 0v is
dktpidktv
vHLL
t
sincos
1)(0 +==ω (5-4)
in which LLGR GtGtp ** /= 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
njiidSiHiHiS
niSiHS
jijigjiij
giii
,...,2,1,),()()()()(
,...,2,1)()()(
''''*
2
==
==
ωγωωωω
ωωω (5-5)
where )( ωiHi , )( ωiH j are the site transfer function at support i and j, respectively;
superscript ‘*’ denotes complex conjugate; )(ωgS is the ground motion power spectral
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5-8
density at the base rock; ),( '''' ωγ idjiji
is the coherency loss function of spatial ground
motions at the base rock, 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 5-1.
5.3 Ground motion simulation
Spatial earthquake ground motions on the base rock 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:
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅
⋅⋅⋅⋅⋅⋅
=
)()()(
)()()()()()(
)(
21
22221
11211
ωωω
ωωωωωω
ω
nnnn
n
n
SiSiS
iSSiSiSiSS
iS (5-6)
where )(ωiiS and njiiSij ,...,2,1,),( =ω are the auto and cross power spectral density
function respectively, defined in Equation (5-5).
The matrix )( ωiS is Hermitian and positive definite, it can be decomposed into the
multiplication of a complex lower triangular matrix )( ωiL and its Hermitian )( ωiLH :
)()()( ωωω iLiLiS H= (5-7)
The decomposition can be performed by using the Cholesky’s method. The lower
triangular matrix )( ωiL is in the following form:
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅
⋅⋅⋅⋅⋅⋅
=
)()()(
0)()(00)(
)(
21
2221
11
ωωω
ωωω
ω
nnnn LiLiL
LiLL
iL (5-8)
and
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ijS
iSiSiSiL
niiSiSSL
jj
i
kjkikij
ij
i
kikikiiii
,...,2,1)(
)()()()(
,...,2,1)()()()(
1
1
*
2/11
1
*
=−
=
=⎥⎦
⎤⎢⎣
⎡−=
∑
∑
−
=
−
=
ω
ωωωω
ωωωω
(5-9)
After obtaining )( ωiL , the stationary time series nitui ,...,2,1),( = , can be simulated in the
time domain as [11]
)]()(cos[)()(1 1
nmnnimnn
i
m
N
nimi tAtu ωϕωβωω ++=∑∑
= =
(5-10)
where
Nim
imim
Nimim
iLiL
iLA
ωωωωωβ
ωωωωω
≤≤=
≤≤Δ=
− 0),)](Re[)](Im[(tan)(
0,)(4)(
1
(5-11)
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);
)( nmn ωϕ is the random phase angles uniformly distributed over the range of ]2,0[ π , mnϕ
and rsϕ should be statistically independent unless rm = and sn = ; Nω 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; ωΔ is the resolution in the
frequency domain, and ωω Δ= nn is the nth discrete frequency.
Directly use Equation (5-10) 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 )(tui is in the following
form [11]
NniBiU nimnimn
i
mimni ,...,2,1)],(sin)()[cos()(
1
=+= ∑=
ωαωαωω (5-12)
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where )( nimB ω is the amplitude at frequency nω , and )( nim ωα is the corresponding phase
angle, defined by:
)()()(
2/)()(
nmnnimnim
nimnim AB
ωϕωβωα
ωω
+=
= (5-13)
The corresponding time series )(tui can be obtained by inverse transforming )( ni iU ω into
the time domain.
The time series generated by Equation (5-10) or (5-12) are stationary processes. In order to
obtain the non-stationary time histories, an envelope function )(tζ is applied to )(tui , the
non-stationary time histories at different locations are obtained by
nituttf ii ,...,2,1),()()( == ζ (5-14)
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
njiidSiSjijigij ,...,2,1,),,()()( '''' == ωγωω (5-15)
This is because 1)()( == ωω iHiH ji when the site amplification effect is not considered. In
this case the spatial ground motions will have the same power spectral density function
Sg(ω), 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. [11]. 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 works of generating spatially varying ground motion
time histories [11, 12], 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 [11, 12]. In this paper, a similar
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5-11
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 )(ωRSA , the corresponding power spectral
density )(ωS can be estimated by [22]
)lnln(/)()( 2 pT
RSASωπω
πωξω −−= (5-16)
where ξ is the damping ratio, T the time duration and p the probability coefficient,
usually 85.0≥p [22].
Using the above approach, the generated time histories usually match well with the multiple
target response spectra. If the response spectra of the generated time histories )()( ωifRSA
do not match satisfactorily the target spectra, iterations need be carried out by adjusting the
power spectral density function, which is done by multiplying )(ωS by the ratio
2)( )](/)([ ωω ifRSARSA , 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 [12], in which good match can be
obtained usually only after more than 10 iterations.
5.4 Numerical examples
An alluvium canyon site with multiple soil layers shown in Figure 5-1 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 rock and on
ground surface at three different locations indicated in the figure will be simulated in the
study.
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5-12
Figure 5-1. A canyon site with multiple soil layers (not to scale)
5.4.1 Amplification spectra
The site amplification effect is studied first. For conciseness, only the amplification spectra
at site 3 are plotted. Figure 5-2(a) shows the amplification spectra for the horizontal out-of-
plane motions when SH wave propagates into the site with different incident angles. Figure
5-2(b) and Figure 5-2(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 2/3 of that of the horizontal
component.
As shown in Figure 5-2, 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 rock 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 peaks 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 peaks. Table 5-1 gives the first two horizontal and vertical vibration
frequencies of the site.
3
No.1 Sandy fill, h=5m, G=30MPa, 3/1900 mkg=ρ , %5=ξ , 45.0=υ
1
2
1’ 2’ 3'
No.2 Soft Clay, h=15m, G=20MPa, 3/1600 mkg=ρ , %5=ξ , 40.0=υ
No.3 Silt sand, h=6m, G=220MPa, 3/2000 mkg=ρ , %5=ξ , 33.0=υ
No.4 Firm clay, h=7m, G=30MPa, 3/1600 mkg=ρ , %5=ξ , 40.0=υ
Base rock, G=1800MPa, 3/2300 mkg=ρ , %5=ξ , 33.0=υ
No.3 Silt sand, h=6m, G=220MPa, 3/2000 mkg=ρ , %5=ξ , 33.0=υ
α α α
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5-13
Figure 5-2. Amplification spectra of site 3, (a) horizontal out-of-plane motion;
(b) horizontal in-plane motion; and (c) vertical in-plane motion
Table 5-1. First two vibration frequencies of the sites in the
horizontal and vertical directions
Site Horizontal (Hz) Vertical (Hz)
2.55 5.25 Site 1
10.15 21.10
4.85 10.20 Site 2
14.60 30.70
1.05 2.20 Site 3
2.65 5.65
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 5-1. 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.
5.4.2 Example 1-PSD compatible ground motion simulation
In this example, the ground motion time histories at different locations of the ground
surface shown in Figure 5-1 are generated. The motion on the base rock is assumed to have
the same intensity and frequency contents and is modelled by the filtered Tajimi-Kanai
power spectral density function as
Γ+−
+
+−== 222222
222
2222
4
0 4)(41
)2()()()()(
ωωξωωωωξ
ωξωωωωωωω
ggg
gg
fffPg SHS (5-17)
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where )(ωPH is a high pass filter function [23], 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. )(0 ωS is the Tajimi-Kanai power spectral
density function [1], gω and gξ are the central frequency and damping ratio of the Tajimi-
Kanai power spectral density function, ωf and ξf 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
sradg /10πω = , 6.0=gξ , πω 5.0=f , 6.0=fξ and 32 /0034.0 sm=Γ . These parameters
correspond to a ground motion time history with duration T=20s and peak ground
acceleration (PGA) g2.0 and peak ground displacement (PGD) 0.082m based on the
standard random vibration method [24]. The vertical motion on the base rock is also
modelled with the same filtered Tajimi-Kanai power spectral density function, but the
amplitude is assumed to be 2/3 of the horizontal component. It should be noted that if a
specific site and an earthquake 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 rock motion.
The Sobczyk model [21] is selected to describe the coherency loss between the ground
motions at points 'i and 'j ( ji ≠ ) at the base rock:
)/cosexp()/exp()/cosexp()()( ''''''''''2
appjiappjiappjijiji vdivdvdiii αωβωαωωγωγ −⋅−=−= (5-18)
in which, )('' ωγ iji
is the lagged coherency loss, β is a coefficient which reflects the level of
coherency loss, 0005.0=β is used in the present paper, which represents highly correlated
motions; '' jid is the distance between the points 'i and 'j , and mdd 100'''' 3221
== is
assumed; α is the incident angle of the incoming wave to the site, and is assumed to be
60°; appv is the apparent wave velocity at the base rock, which is 1768m/s according to the
base rock property and the specified incident angle.
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To model the temporal variation of the simulated ground motions, the simulated stationary
time histories are multiplied by the Jennings envelope function [25], which has the
following form
⎪⎩
⎪⎨
⎧
≤<−−≤<≤≤
=Tttttttttttt
t
nn
n
)](155.0exp[1
0)/()( 0
02
0
ζ (5-19)
with st 20 = and stn 10= in this study.
In the simulation, the sampling frequency and the upper cut-off frequency are set to be 100
Hz and HzN 25=ω , and the time duration is assumed to be T=20s. To improve the
computational efficiency, the ground motions are generated in the frequency domain by
using the FFT technique as discussed above, and N=2048 is used in the paper.
The three generated horizontal base rock motions are shown in Figure 5-3(a) and 5-3(b) for
acceleration and displacement respectively. The PGAs and PGDs of the simulated motions
are 2.27, 2.21, 2.33m/s2 and 0.0861, 0.0857, 0.0839m respectively, which are close to the
theoretical PGA of g2.0 and PGD of 0.082 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 Sobczyk model, good match can also be
observed except for '3'1γ 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., [26]]) 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.
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Figure 5-3. Generated base rock motions in the horizontal directions
(a) acceleration; and (b) displacement
Figure 5-4. Comparison of power spectral density of the generated base rock acceleration
with model power spectral density
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Figure 5-5. Comparison of coherency loss between the generated base rock accelerations
with model coherency loss function
Assuming the incoming motions at the base rock consist of SH wave with an incident
angle of o60=α , 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 4.32, 6.53, 3.31m/s2 and 0.0565, 0.0540,
0.0712m at the three different locations as shown in Figure 5-6. As compared with the
motions at the base rock, site 2 significantly amplifies the horizontal out-of-plane ground
acceleration. This is because the fundamental vibration frequency of site 2 is 4.85Hz as
given in Table 5-1, which is very close to the central frequency of the filtered Tajimi-Kanai
power spectral density function of ground motions at the base rock, so resonance occurs.
Site 1 and 3 also amplify the base rock 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 15%
respectively. This might attribute to the fact that local soil layers also filter the frequency
content of the incoming waves. Although PGA of site 2 is the largest, the PGD is the
smallest because site 2 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.
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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
[27, 28]. Further studies revealed that local site effect not only causes phase difference of
the coherency function [10], but also affects its modulus [29, 30]. Figure 5-8 shows the
comparison of the lagged coherency loss functions of the base rock 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 rock
motion. These results are consistent with those obtained from recorded surface ground
motions [27, 28]. Same results of coherency loss between in-plane surface ground motions,
which are not shown here, are also obtained. They indicate wave propagation through non-
uniform paths cause further coherency loss between spatial ground motions. More detailed
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5-19
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 rock motions
(solid line) and those between surface motions (dashed line)
Assuming the incoming motions on the base rock are combined P and SV waves with the
P wave incident angle o60 and SV wave incident angle o4.75 , the horizontal and vertical in-
plane 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 5-10. 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 5-11 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 5-12.
Good agreements are observed again. As expected, the simulated vertical motions are
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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 5-11,
the PGAs and PGDs of the vertical motions are 3.11, 2.78, 2.67m/s2 and 0.0293, 0.0289,
0.0349m respectively for the three sites. Site 1 amplifies the base rock motion most because
the fundamental vertical vibration frequency of site 1 is 5.25Hz as given in Table 5-1,
which is close to the central frequency of the filtered Tajimi-Kanai power spectral density
function of ground motion at the base rock. Resonance results in the significant site
amplification.
Figure 5-9. Generated horizontal in-plane motions on ground surface
(a) acceleration; and (b) displacement
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Figure 5-10. Comparison of power spectral density of the generated horizontal in-plane
acceleration on ground surface with the respective theoretical power spectral density
Figure 5-11. Generated vertical in-plane motions on ground surface
(a) acceleration; and (b) displacement
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Figure 5-12. 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.
5.4.3 Example 2 -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 Earthquake Loading Code [31]. The sub-soil classes at the three sites are assumed
to be shallow soil (Class C), rock (Class B) and deep/soft soil (Class D), respectively. The
peak ground acceleration (PGA) for the three sites is assumed to be 2m/s2. The
corresponding response spectra normalized to PGA of 2m/s2 in the New Zealand
Earthquake Loading Code for the three sites are plotted in Figure 5-14 (dashed line).
The Sobczyk model [21] 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 1000m/s. It should be noted that the Sobczyk model for spatial ground motion
coherency is suitable for a flat site. As observed above and in a few previous studies [29,
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., [10]) also adopted the coherency model for ground motions on flat-laying site to
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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 5-13. Generated time histories according to the specified design response spectra
(a) acceleration; and (b) displacement
The shape function in the form of Equation (5-19) is applied to modulate the simulated
stationary time histories. Figure 5-13 shows the generated acceleration and displacement
time histories at the three locations on the ground surface after 4 iterations with the
damping ratio 05.0=ξ and probability coefficient 85.0=p . The sampling frequency and
the upper cut-off frequency are set to be 100 and 25Hz, respectively. The time duration is
T=20s. As shown in Figure 5-13, though the PGAs for the three sub-soils are almost the
same, about 2m/s2, 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
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conditions. The PGD of surface motion increases with the decrease in soil stiffness. In this
example, the PGD of the three sites reaches 0.0805, 0.0624 and 0.124m respectively. Figure
5-14 and Figure 5-15 show the response spectra and coherency loss function of the
generated time histories and the prescribed models, good matches are observed.
Figure 5-14. Comparison of the generated acceleration and the target response spectra
Figure 5-15. 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 earthquake ground
motion time histories at sites with non-uniform conditions. It takes into consideration of
the local site effects on ground motion amplification and spatial variation. The base rock
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 rock motions consisting of out-of-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 spatially varying earthquake 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
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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
1. Tajimi H. A statistical method of determining the maximum response of a building
structure during an earthquake. Proc. of 2nd World Conference on Earthquake Engineering,
Tokyo, Japan, 1960; 781-796.
2. Clough RW, Penzien J. Dynamics of Structures. New York: McGraw Hill; 1993.
3. Joyner WB, Boore DM. Measurement, characterization and prediction of strong
ground motion. Earthquake Engineering and Structure Dynamics II-Recent Advances in
Ground Motion Evaluation Proc (GSP 20), Park City, Utah, 1988; 43-102.
4. Atkinson GM, Boore DM. Evaluation of models for earthquake source spectra in
Eastern North America. Bulletin of the Seismological Society of America 1998; 88(4): 917-
934.
5. Hao H, Gaull BA. Estimation of strong seismic ground motion for engineering use
in Perth Western Australia. Soil Dynamics and Earthquake Engineering 2009; 29(5): 909-
924.
6. Wolf JP. Dynamic Soil-structure Interaction, Englewood Cliffs, NJ: Prentice Hall; 1985.
7. Wolf JP. Soil-structure interaction analysis in time domain, Englewood Cliffs, NJ: Prentice
Hall; 1988.
8. Safak E. Discrete-time analysis of seismic site amplification. Journal of Engineering
Mechanics 1995; 121(7): 801-809.
9. Zerva A, Zervas V. Spatial variation of seismic ground motions: An overview.
Applied Mechanics Reviews 2002; 56(3): 271-297.
10. Der Kiureghian A. A coherency model for spatially varying ground motions.
Earthquake Engineering and Structural Dynamics 1996; 25(1): 99-111.
11. Hao H, Oliveira CS, Penzien J. Multiple-station ground motion processing and
simulation based on SMART-1 Array data. Nuclear Engineering and Design 1989; 111:
293-310.
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12. Deodatis G. Non-stationary stochastic vector processes: seismic ground motion
applications. Probabilistic Engineering Mechanics 1996; 11(3): 149-167.
13. Shinozuka M. Monte Carlo solution of structural dynamics. Computers and Structures
1972; 2: 855-874.
14. Shinozuka M, Jan CM. Digital simulation of random processes and its applications.
Journal of Sound and Vibration 1972; 25(1): 111-128.
15. Shinozuka M, Deodatis G. Stochastic process models for earthquake ground
motion. Probabilistic Engineering Mechanics 1988; 3(3): 114-123.
16. Hao H. Input seismic motions for use in the seismic structural response analysis.
The Sixth International Conference on Soil Dynamics and Earthquake Engineering, 1993; 87-
100.
17. Wang S, Hao H. Effects of random vibrations of soil properties on site
amplification of seismic ground motions. Soil Dynamics and Earthquake Engineering
2002; 22(7): 551-564.
18. Idriss IM, Sun JI. User’s manual for SHAKE91, in User’s Manual for SHAKE91,
Department of Civil and Environmental Engineering, University of California at
Davis, 1992.
19. Baedet JP, Ichii K, Lin CH. EERA, a computer program for equivalent linear
earthquake site response analysis of layered soil deposits, in EERA, A Computer
Program for Equivalent Linear Earthquake Site Response Analysis of Layered Soil Deposits,
University of Southern California, 2000.
20. Baedet JP, Tobita T. NEAR, a computer program for nonlinear earthquake site
response analysis of layered soil deposits, in NEAR, A Computer Program for
Equivalent Linear Earthquake Site Response Analysis of Layered Soil Deposit, University of
Southern California, 2001.
21. Sobczky K. Stochastic Wave Propagation, Netherlands: Kluwer Academic Publishers;
1991.
22. Kaul MK. Stochastic characterization of earthquake through their response
spectrum,” Earthquake Engineering and Structural Dynamics 1978; 6:187-196.
23. Ruiz P, Penzien J. Probabilistic study of the behaviour of structures during
earthquakes. Report No. UCB/EERC-69-03, University of California, Berkeley,
1969.
24. Der Kiureghian A. Structural response to stationary excitation. Journal of Engineering
Mechanics 1980; 106: 1195-1213.
School of Civil and Resource Engineering Chapter 5 The University of Western Australia
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25. Jennings PC, Housner GW, Tsai NC. Simulated earthquake motions. Report of
Earthquake Engineering Research Laboratory, EERL-02, California Institute of
Technology, 1968.
26. Hao H. Effects of spatial variation of ground motions on large multiply-supported
structures. Report No. UCB/EERC-89-06, University of California, Berkeley, 1989.
27. Somerville PG, McLaren JP, Sen MK, Helmberger DV. The influence of site
conditions on the spatial incoherence of ground motions. Structural Safety 1991;
10(1): 1-13.
28. 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 2007; pp. 1-10.
29. Lou L, Zerva A. Effects of spatially variable ground motions on the seismic
response of s skewed, multi-span, RC highway bridge. Soil Dynamics and Earthquake
Engineering 2005; 25: 729-740.
30. Bi K, Hao H. Influences of irregular topography and random soil properties on
coherency loss of spatial seismic ground motions. Earthquake Engineering and
Structural Dynamics 2010, published online.
31. Standards New Zealand. Structural design actions, Part 5: Earthquake actions in
New Zealand (NZS1170.5-2004), 2004.
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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 [1]. 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 one-
dimensional 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
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6.1 Introduction
For large dimensional structures, such as long-span bridges, pipelines, communication
transmission systems, their supports inevitably undergo different seismic motions during an
earthquake 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 [2]. Ground motion spatial variation effect has
been extensively studied by many researchers especially after the installation of strong
motion arrays (e.g. the SMART-1 array in Lotung, Taiwan). Many empirical [3-7] and semi-
empirical [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 [10] 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. [1] investigated the
coherency function of ground motions on a site located on folded sedimentary rocks (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.
[11], based on the seismic data recorded at the Parkway 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 rock station. They concluded that the lagged coherency between the
sediment and rock 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 [10] might not be able to reliably describe the influence of
local site effect on the coherency function.
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Although it was observed that the heterogeneity of site conditions strongly affect the
ground motion spatial variations [1, 11], 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 likely to have influence on the coherency function. Zerva
and Harada [12] modelled horizontal soil layers at a site as a 1-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 rock 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 [12]. However, it should be noted
that a 1-DOF system cannot realistically represent the multiple predominate frequencies
that may exist at a site with multiple layers and multiple modes. Liao and Li [13] 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 [14].
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,
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to the best knowledge of the authors, no more theoretical/analytical analysis in this field
can be found except for the studies mentioned above [10, 12, 13].
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 rock). The
base rock 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 rock is accounted for by an empirical coherency function for
spatial ground motions on a flat-lying site. Using the one-dimensional wave propagation
theory [15], 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 rock 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 take 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 [16]. 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: 1) random generation of soil properties; 2)
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.
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6.2 Theoretical basis
6.2.1 Estimation of coherency function
Let )(tu j and )(tuk be the recorded (simulated) acceleration time histories at locations j
and k of a site, and the corresponding Fourier transform of the time histories are )(ωjU
and )(ωkU , respectively. The smoothed auto spectral density function of ground motion at
location j or k is then
( ) ( ) ( ) kjimUmWSM
Mmninii or 2 =Δ+Δ= ∑
−=
ωωωω (6-1)
and the cross power spectral density function between motions at stations j and k is
( ) ( ) ( ) ( )∑−=
∗ Δ+Δ+Δ=M
Mmnknjnjk mUmUmWS ωωωωωω (6-2)
where ( )ωW is the spectral smoothing window, ωΔ is the frequency step, ωω Δ= nn is
the n-th discrete frequency, and ∗ denotes the complex conjugate.
The coherency function of the spatial ground motions can be obtained as [4]
)()()(
)(ωω
ωωγ
kkjj
jkjk SS
S= (6-3)
The coherency function in Equation (6-3) is generally a complex function and can be
written as
[ ])(exp)()( ωθωγωγ jkjkjk i= (6-4)
in which )(ωγ jk is the lagged coherency, ( )( )( )( )⎥⎥⎦
⎤
⎢⎢⎣
⎡= −
ωω
ωθjk
jkjk S
SReIm
tan)( 1 is the phase angle,
‘Im’ and ‘Re’ denote the imaginary and real part of a complex number.
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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 [17] and the
spectral representation method. These two parts are briefly introduced in Sections 6.2.2
and 6.2.3, more detailed information can be found in Reference [15].
6.2.2 One-dimensional wave propagation theory
For a site with horizontally extended multiple soil layers on a half space (base rock), the
base rock 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 frequencyω , the dynamic equilibrium equations can be written as
[17]
ec
ep2
22 ω
−=∇ or { } { }Ω−=Ω∇ 2
22
scω (6-5)
where e2∇ and { }Ω∇2 are the Laplace operator of the volumetric strain amplitude e and
rotational-strain-vector { }Ω . pc and sc 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-
plane motion, ][ LSHS and ][ L
SVPS − , can be formulated independently by analysing the
relations of shear stresses and displacements at the boundary of each soil layer. Assembling
the matrices of each soil layer and the base rock, the dynamic stiffness of the total system is
obtained and denoted by ][ SHS and ][ SVPS − , respectively. The dynamic equilibrium
equation of the site in the frequency domain is thus [17]
{ } { }SHSHSH PuS =][ or { } { }SVPSVPSVP PuS −−− =][ (6-6)
where { }SHu and { }SHP are the out-of-plane displacements and load vector corresponding
to the incident SH wave, { }SVPu − and { }SVPP − are the in-plane displacements and load
vector of the combined P and SV waves. The stiffness matrices ][ SHS and ][ SVPS − depend
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6-7
on soil properties, incident wave type, incident angle and circular frequency ω . The
dynamic load { }SHP and { }SVPP − depend on the base rock properties, incident wave type,
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 rock and
each soil layer can be formed, and the site transfer function )]([ ωH in the out-of-plane and
in-plane directions can be estimated.
6.2.3 Ground motion generation
Consider a canyon site with horizontally extended multiple soil layers resting on an elastic
half-space as shown in Figure 6-1, in which hm, Gm, mρ , mξ and mυ is the depth, shear
modulus, mass density, damping ratio and Poisson’s ratio of layer m. The spatially varying
base rock 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 rock are assumed
to have the same power spectral density, and are modelled by a filtered Tajimi-Kanai [18]
power spectral density function. The spatial variation of ground motions at base rock 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:
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅
⋅⋅⋅⋅⋅⋅
=
)()()(
)()()()()()(
)(
21
22221
11211
ωωω
ωωωωωω
ω
nnnn
n
n
SiSiS
iSSiSiSiSS
iS (6-7)
where
nkjidSiHiHiS
njSiHS
kjkjgkjjk
gjjj
,...,2,1,),()()()()(
,...,2,1)()()(
''''*
2
==
==
ωγωωωω
ωωω (6-8)
are the auto and cross power spectral density function respectively. In which )(ωgS is the
ground motion power spectral density on the base rock; ),( '''' ωγ idkjkj
is the coherency
function between location 'j and 'k on the base rock; )( ωiH j , )( ωiH k are the site transfer
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6-8
function at locations j and k on the ground surface, which can be formulated based on one-
dimensional wave propagation theory discussed in Section 6.2.2.
Figure 6-1. Schematic view of a layered canyon site
Decomposing the Hermitian, positive definite matrix )( ωiS into the multiplication of a
complex lower triangular matrix )( ωiL and its Hermitian )( ωiLH
)()()( ωωω iLiLiS H= (6-9)
the stationary time series njtu j ,...,2,1),( = , can be simulated in the time domain directly
[6]
)]()(cos[)()(1 1
nmnnjmnn
j
m
N
njmj tAtu ωϕωβωω ++= ∑∑
= =
(6-10)
where
Njm
jmjm
Njmjm
iLiL
iLA
ωωωω
ωβ
ωωωωω
≤≤=
≤≤Δ=
− 0),)](Re[)](Im[
(tan)(
0,)(4)(
1
(6-11)
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);
)( nmn ωϕ is the random phase angles uniformly distributed over the range of ]2,0[ π , mnϕ
and rsϕ should be statistically independent unless rm = and sn = ; Nω 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.
k
j
j’ k’
Layer l: ,lh lG , lρ , lξ , lυ
M
M
Layer m: ,mh mG , mρ , mξ , mυ
Layer m-1: ,1−mh 1−mG , 1−mρ , 1−mξ , 1−mυ
Layer 1: ,1h 1G , 1ρ , 1ξ , 1υ
Base rock: BG , Bρ , Bξ , Bυ
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The generated time series by Equation (6-10) are stationary processes. In order to obtain
the non-stationary time histories, an envelope function )(tζ is applied to )(tu j . The non-
stationary time histories at different locations are then obtained by
njtuttf jj ,...,2,1),()()( ==ζ (6-12)
6.2.4 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 [16] is widely used to describe the
variability of soil properties. In this theory the random soil property )(zu is characterized
by the mean value u , standard deviation uσ and the correlation distance uδ . uσ measures
the intensity of fluctuation or degree to which actual value of )(zu may deviate from. uδ
measures the correlation level or persistence of the property from one point to another in a
site, small values of uδ suggest rapid fluctuation about the average, while large values of uδ
imply a slowly varying component is superimposed on the average value of u .
Consider a one-dimensional random field )(zu with mean value )(zu and standard
deviation uσ , its local average process )(zuZ of )(zu over the interval Z centered at z is
defined as:
')'(1)(2/
2/dzzu
Zzu
Zz
ZzZ ∫+
−= (6-13)
It can be seen that the local average )(zuZ depends on the specific location of the interval
z within the statistically homogeneous soil layer. The mean and variance of )(zuZ are [16]
[ ] [ ]
[ ] )()(
)()()(
2 ZzuVar
zuzuEzuE
uZ
Z
λσ=
== (6-14)
where )(Zλ is a variance reduction function of )(zu , which measures the reduction of
point variance 2uσ under local average. The variance function )(Zλ can be derived from
auto-correlation function )( zu Δρ in the following form
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)()()1(2)(0
zdzZz
ZZ u
ZΔΔ
Δ−= ∫ ρλ (6-15)
By using the exponential auto-correlation function [19]
)/2exp()( uu zz δρ Δ−=Δ (6-16)
the variance reduction function can be derived as [19]
[ ]1)/(2)/(2
1)( )/(22 −+= − uZ
uu
eZZ
Z δδδ
λ (6-17)
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. Take shear
modulus as an example
( )φλφλσ )(1)( ZCOVGZGG G ×+=+= (6-18)
where G and Gσ are the mean value and standard deviation of shear modulus, )(Zλ is
the variance reduction function and φ is a normal distributed random process with zero
mean and unity variance. GCOV G /σ= is the coefficient of variation.
6.2.5 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 [20]. 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.
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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 rock is selected as an example as shown in Figure 6-2. The mean
values of the corresponding soil properties of each soil layer and base rock are also given in
the Figure.
The motions on the base rock 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:
Γ+−
+
+−== 222222
222
2222
4
0 4)(41
)2()()()()(
ωωξωωωωξ
ωξωωωωωωω
ggg
gg
fffPg SHS (6-19)
where )(ωPH is a high pass filter function [21], 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. )(0 ωS is the Tajimi-Kanai power spectral density
function [18], gω and gξ are the central frequency and damping ratio of the Tajimi-Kanai
power spectral density function, ωf and ξf 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 πω 10=g rad/s,
6.0=gξ , πω 5.0=f , 6.0=fξ and 0034.0=Γ m2/s3. These parameters correspond to a
ground motion time history with duration 20=T s and peak ground acceleration (PGA)
0.2g based on the standard random vibration method [22]. The vertical motion on the base
rock is also modelled with the same filtered Tajimi-Kanai power spectral density function,
but the amplitude is assumed to be 2/3 of the horizontal component of PGA 0.2g.
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Figure 6-2. A four-layer canyon site with deterministic soil properties (not to scale)
The Sobczyk model [23] is selected to describe the coherency loss between the ground
motions at points j’ and k’ on the base rock:
)/cosexp()/exp()/cosexp()()( ''''''''''2
appkjappkjappkjkjkj vdivdvdiii αωβωαωωγωγ −⋅−=−= (6-20)
where β is a coefficient reflecting the level of coherency loss, 001.0=β is used in the
present paper, which represents intermediately correlated motions; ''kjd is the distance
between the points j’ and k’, and =''kjd 100 m is assumed; α is the incident angle of the
incoming wave to the site, and is assumed to be 60°; appv is the apparent wave velocity on
the base rock, which is 1768 m/s according to the base rock property and the specified
incident angle. Seismic waves are assumed propagating vertically from the base rock to the
ground surface.
Take 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 6-2, the
acceleration time histories on the base rock and the ground surface are simulated based on
the procedures presented in Sections 6.2.2 and 6.2.3. The sampling frequency and the
upper cut-off frequency are set to be 100 Hz and 20=Nω Hz, respectively. 2048 sampling
points are used in each set of ground motion time histories. As mnϕ in Equation (6-10) is a
random variable uniformly distributed over the range of ]2,0[ π , any realization of a
random angle mnϕ 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.
No.3 Soft Clay, h=15m, G=20MPa, 3/1600 mkg=ρ , %5=ξ , 40.0=υ
No.2 Silt sand, h=16m, G=220MPa, 3/2000 mkg=ρ , %5=ξ , 33.0=υ
No.1 Firm clay, h=12m, G=30MPa, 3/1600 mkg=ρ , %5=ξ , 40.0=υ
Base rock: G=1800MPa, 3/2300 mkg=ρ , %5=ξ , 33.0=υ
j’ k’
j
No.4 Sandy fill, h=5m, G=30MPa, 3/1900 mkg=ρ , %5=ξ , 40.0=υ
k
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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. [24] 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 11-point Hamming window, if
the coherency estimates is to be used in structural analysis, for time windows less than
approximately 2000 samples and for structural damping coefficient 5% of critical [24]. 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 rock and ground surface,
Monte-Carlo simulation method is used as discussed in Section 6.2.5. Convergence test
needs to be conducted to check 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
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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 6.3.2, 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 0.2Hz, 2.0Hz, 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 in-
plane 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 rock smoothed by the 11-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 0.2, 2.0, 5.0 and 9.0Hz
Figure 6-5. Comparison of the mean lagged coherency on the base rock from 600
simulations with the target model
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6.3.1 Influence of irregular topography
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 k on the
ground surface of the canyon site. For comparison purpose, the lagged coherency between
incident motion on the base rock at j’ and k’ 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 0.13. 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 rock motions significantly. At all frequencies, the
coherency loss functions on the ground surface are smaller than those on the base rock,
i.e., the coherency function on the base rock 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 [25], and Liao et al. [11]. 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 rock. As shown, there are many obvious peaks
and troughs in the coherency function of surface motions. These peaks and troughs
directly relate to the modulus of the spectral ratio of two local sites, namely
( ) ( )ωω iHiH jk / , as shown in Figure 6-8. ( )ωiH j and ( )ωiH k are the transfer functions of
site j and k respectively. They are the spectral ratio of the surface motion at j or k to the
corresponding bedrock motion at j’ or k’, which can be calculated based on the one-
dimensional wave propagation theory as discussed in Section 6.2.2. Figure 6-9 shows the
modulus of the transfer functions at sites j and k. It is obvious that site amplifies the
motions on the base rock significantly, which makes 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 1-DOF model [12]. With a 1-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.
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Taking the horizontal out-of-plane motion as an example, four obvious minima can be
observed around the frequencies 0.78, 1.90, 4.20 and 7.10 Hz, which correspond to the
four evident peaks in the spectral ratio as shown in Figure 6-8(a). 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 [1] and the Wainuiomata Valley
in New Zealand [11]. These observations also indicate that site effect will not only cause
phase difference of the coherency function [10], but will also affect its modulus.
Liao and Li [13] 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 1.25 Hz as shown in Figure 6-9(a). 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 peak value in the lagged frequency at this
frequency as shown in Figure 6-6(a). This contradicts with Liao and Li’s conclusion. This is
because in the present example, both wave paths from j’ to j and k’ to k or both sites
amplify the bedrock 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 bedrock 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.
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Figure 6-6. Comparison of the mean lagged coherency between the surface motions (j, k)
with that of the incident motion on the base rock
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
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6.3.2 Influence of random soil properties
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-
2. According to a more specific review and summary [26], 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 10% to 100%. 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 20%,
40% and 60% for all soil layers, while the COV of soil density is assumed to be 5% in all
the cases. Vanmarcke [16] studied the scale of soil fluctuation, and concluded that the
correlation distance of various soils vary from 0.16 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 6-10. Influence of uncertain soil properties on
the mean values of lagged coherency functions
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Figure 6-11. Influence of uncertain soil properties on the
standard deviations of lagged coherency functions
Figure 6-12. Influence of uncertain soil properties on the
mean spectral ratios of two local sites
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Figure 6-10 and Figure 6-11 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 rock 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 rock as observed above. When the COV of soil properties is
0.2, 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 6-12. In this case, larger COV leads to smaller spectral ratios, which results
in the relatively larger lagged coherency values. As shown in Figure 6-11, 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 rock 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 rock
(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.
6.3.3 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 6-13 and Figure 6-14 shows the mean values and the
corresponding standard deviations of the lagged coherency respectively. The corresponding
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values with deterministic soil properties, and that between the incident motions on the base
rock 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 6-15,
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 6-13 and
Figure 6-14.
It should be noted that all the results obtained above are based on the assumption of a
correlation distance uδ of 4 m for a typical clay site. In fact, the correlation distance varies
in a relatively wide range [16], 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 6-13. Influence of each random soil property on the
mean values of lagged coherency functions
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Figure 6-14. Influence of each random soil property on the
standard deviations of lagged coherency functions
Figure 6-15. Influence of each random soil property on the
mean spectral ratios of two local sites
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6.4 Conclusions
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:
1. The coherency function between surface ground motions on a canyon site is
different from that between base rock motions. The lagged coherency function on
the base rock is the upper bound of that on the ground surface.
2. 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 check 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
1. Somerville PG, McLaren JP, Sen MK, Helmberger DV. The influence of site
conditions on the spatial incoherence of ground motions. Structural Safety 1991;
10(1):1-13.
2. Saxena V, Deodatis G, Shinozuka M. Effect of spatial variation of earthquake
ground motion on the nonlinear dynamic response of highway bridges. Proceeding of
12th World Conference on Earthquake Engineering, Auckland, New Zealand, 2000.
School of Civil and Resource Engineering Chapter 6 The University of Western Australia
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3. Loh CH. Analysis of the spatial variation of seismic waves and ground movement
from SMART-1 data. Earthquake Engineering and Structural Dynamics 1985; 13(5): 561-
581.
4. Harichandran RS, Vanmarcke EH. Stochastic variation of earthquake ground
motion in space and time. Journal of Engineering Mechanics 1986; 112(2): 154-174.
5. Loh CH, Yeh YT. Spatial variation and stochastic modelling of seismic differential
ground movement. Earthquake Engineering and Structural Dynamics 1988; 16(4): 583-
596.
6. Hao H, Oliveira CS, Penzien J. Multiple-station ground motion processing and
simulation based on SMART-1 array data. Nuclear Engineering and Design 1989;
111(3):293-310.
7. Harichandran RS. Estimating the spatial variation of earthquake ground motion
from dense array recordings. Structural Safety 1991; 10: 219-233.
8. Luco JE, Wong HL. Response of a rigid foundation to a spatially random ground
motion. Earthquake Engineering and Structural Dynamics 1986; 14(6): 891-908.
9. Somerville PG, McLaren JP, Saikia CK, Helmberger DV. Site-specific estimation of
spatial incoherence of strong ground motion. Earthquake Engineering and Structural
Dynamics II-Recent Advances in Ground Motion Evaluation, ASCE Geotechnical Special
Publication No. 20, 1988; 188-202.
10. Der Kiureghian A. A coherency model for spatially varying ground motions.
Earthquake Engineering and Structural Dynamics 1996; 25(1): 99-111.
11. 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. 170, 2007; 1-10.
12. Zerva A, Harada T. Effect of surface layer stochasticity on seismic ground motion
coherence and strain estimations. Soil Dynamics and Earthquake Engineering 1997; 16:
445-457.
13. Liao S, Li J. A stochastic approach to site-response component in seismic ground
motion coherency model. Soil Dynamics and Earthquake Engineering 2002; 22: 813-
820.
14. Chen Y, Li J. Effect of random media on coherency function of seismic ground
motion. World Earthquake Engineering 2007; 23(3):1-6 (in Chinese).
15. Bi K, Hao H. Simulation of spatially varying ground motions with non-uniform
intensities and frequency content. Australia Earthquake Engineering Society 2008
Conference, Ballart, Australia, 2008; Paper No. 18.
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16. Vanmarcke EH. Probabilistic modelling of soil profiles. Journal of the Geotechnical
Engineering Division 1977; 103(11): 1227-1246.
17. Wolf JP. Dynamic soil-structure interaction. Prentice Hall: Englewood Cliffs, NJ, 1985.
18. Tajimi H. A statistical method of determining the maximum response of a building
structure during an earthquake. Proceedings of 2nd World Conference on Earthquake
Engineering, Tokyo, 1960; 781-796.
19. Vanmarcke EH. Random fields: analysis and synthesis. Cambridge: MIT Press, 1983.
20. Yeh CH, Rahman MS. Stochastic finite element methods for the seismic response
of soils. Internal Journal for Numerical and Analytical Methods in Geomechanics, 1998;
22(10): 819-850.
21. Ruiz P, Penzien J. Probabilistic study of the behaviour of structures during
earthquakes. Report No. UCB/EERC-69-03, University of California at Berkeley;
1969.
22. Der Kiureghian A. Structural response to stationary excitation. Journal of the
Engineering Mechanics Division 1980; 106(6): 1195-1213.
23. Sobczky K. Stochastic wave propagation. Netherlands: Kluwer Academic Publishers,
1991.
24. Abrahamson NA, Schneider JF, Stepp JC. Empirical spatial coherency functions
for applications to soil-structure interaction analysis. Earthquake Spectra 1991; 7(1):1-
28.
25. Lou L, Zerva A. Effects of spatially variable ground motions on the seismic
response of a skewed, multi-span, RC-highway bridge. Soil Dynamics and Earthquake
Engineering 2005; 25: 729-740.
26. Baecher GB, Chan M, Ingra TS, Lee T, Nucci LR. Geotechnical reliability of
offshore gravity platforms. Report MITSG 80-20, Sea Grant College Program,
Cambridge: MIT Press; 1980.
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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 beam-
column element model. Consequently, only point to point pounding in 1D, 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 decks and these response will cause
eccentric poundings. That is why many pounding damage occurred at corners of the
adjacent decks as observed in many previous earthquakes. A simplified 1D 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 decks, which may induce local damage around
the corners of bridge decks. It demonstrates the necessity of detailed 3D modelling for
realistic simulation of pounding responses of adjacent bridge decks to earthquake
excitations.
Keywords: pounding response; eccentric pounding; torsional responses; 3D FEM; local
site effect; spatially varying ground motions
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7.1 Introduction
For bridge structures with conventional expansion joints, a complete avoidance of
pounding between bridge decks during strong earthquakes 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 earthquakes. In the 1971 San Fernando earthquake, it was
found that impacts between bridge decks and abutments were the source of extensive
damages to highway bridges with seat type abutments [1]. In the 1989 Loma Prieta
earthquake, poundings between the lower roadway and columns supporting the upper-lever
deck of the Southern viaduct section at the China Basin in California led to significant
damage to the decks and column sides [2]. Reconnaissance reports from the 1995 Kobe
earthquake identified pounding as a major cause of fracture of bearing supports, which
subsequently led to the unseating of bridge decks [3]. Surveys conducted after the 1999
Chi-Chi Taiwan earthquake 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 2006 Yogyakarta earthquake [5] and 2008 Wenchuan
earthquake [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 decks 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 makes avoiding pounding
possible. Hao [7] analysed the effect of various bridge and ground motion parameters on
the relative displacement between adjacent bridge decks, and defined the required seating
length for bridge decks 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 [10] on the required separation distances
between bridge structures crossing a canyon site to avoid seismic pounding.
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Pounding is an extremely complex phenomenon involving damage due to plastic
deformation, local cracking or crushing, fracturing due to impact, and friction when two
adjacent bridge decks are in contact with each other. To simplify the analysis, many
researchers modelled a bridge girder as a lumped mass. For example, Malhotra [11]
investigated a concrete bridge that experienced significant pounding during California
earthquakes with a lumped mass model; Jankowski et al. [12] presented an analysis of
pounding between superstructure segments of an isolated elevated bridge induced by the
seismic wave passage effect; Ruangrassamee and Kawashima [13] calculated the relative
displacement spectra of two single-degree-of-freedom (SDOF) systems with pounding
effect; DesRoches and Muthukumar [14] examined the factors affecting the global
response of a multiple-frame bridge due to pounding of adjacent frames; Chouw and Hao
[15, 16] 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, Jankowski et al. [17] 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. [18] modelled the girders and piers as 2D 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 1D
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 take place along the entire
surfaces of the adjacent structures. Moreover, it was observed from previous earthquakes
that most poundings actually occurred at corners of adjacent bridge decks as shown in
Figure 7-1. This is because torsional responses of the adjacent decks 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. [19]
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. [20] evaluated the
effectiveness of cable restrainers to mitigate earthquake 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 [19, 20], neither the surface to surface nor eccentric
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pounding was considered, instead the pounding was simulated by the contact elements
which linked the external nodes of adjacent segments together. Zhu et al. [21] 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 decks. The drawback of the method is that it could not model material
non-linearities during contacts. The task to search contact pairs is also very time consuming
and the searching algorithm is complicated. More recently, Jankowski [22] analyzed the
earthquake-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 makes the study of earthquake-induced pounding more reliable than
using the discrete lumped mass or beam-column element models. To the best knowledge
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 7-1. A typical pounding damage between bridge decks in Chi-Chi earthquake
Pounding between adjacent bridge decks 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 [11, 13, 14, 20-22] or assumed variation was caused by wave passage effect only
[12, 17]. Only a few studies considered the combined wave passage effect and coherency
loss effect in analyzing relative displacement responses of adjacent bridge structures [15,
16, 18, 19]. 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.
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Studies revealed that local site effect not only causes further phase difference [23, 24], but
also affects the coherency loss between spatial ground motions [25]. These differences will
significantly affect the structural responses [9, 10, 24]. 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 deck and
between two adjacent bridge decks 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 [26], and then LS-DYNA [27] is employed to calculate the
structural responses. To model the local site effect on spatial ground motions, the base
rock 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.
7.2 Method validation
A multi-span concrete bridge studied by Malhotra [11] 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 [11], 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 7-2(a). The length, mass,
column stiffness and damping ratio for the short span are Ls=100m, ms=1.2×106kg,
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ks=107MN/m and ξs=0.05, respectively. The corresponding parameters for the long span
are Ll=200m, ml=2.4×106kg, kl=94MN/m and ξl =0.05. These parameters correspond to
the vibration frequencies for the short and long span of fs=1.5 and fl=0.996Hz, respectively
[11]. 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 known
properties of the bridge [11]. They are: Young’s modulus of the bridge decks and piers
E=35GPa; densityρ=2400kg/m3; rectangular cross section of the decks m5.22× with
m5.2 in the transverse direction of the bridge; heights of the bridge piers mh 9= , with
cross section m5.2963.0 × and m5.2922.0 × for the short and long span, respectively.
Figure 7-2(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 (kp) and damping (cp) of the impact element are
two important parameters that need to be determined. Previous investigation suggested a
kp varying from 10 to 40 times of the lateral stiffness of the stiffer adjacent structures [28].
kp is assumed to be mMN /5000 in the present study as suggested in [16]. 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 [12]:
ls
lsppp mm
mmkc+
= ζ2 (7-1)
with
( )22 ln
ln
e
ep
+
−=
πζ (7-2)
In the present study, e=0.46 is used [11], which corresponds to a damping ratio of
24.0=pζ .
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 0.1m are used for both decks and
piers in the model. The treatment of sliding and impact along contact surfaces is an
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important issue in the modelling. To realistically consider the poundings between entire
surfaces of adjacent bridge decks, 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 7-2. Different models (not to scale): (a) lumped mass model (from [11]);
(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 1940 North-
South El Centro earthquake ground motion scaled to a peak 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 7-3(a), the
relative displacements in the longitudinal direction between the adjacent bridge decks
obtained by using the lumped mass model [11] 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 7-3(a). 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 [11] due to the limitation inherent in the
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method, but it was found that poundings occurred at 2.0, 2.7, 3.7, 4.6, 5.5, 6.3 and 7.3s.
Figure 7-3(b) 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 [11]. The
pounding forces obtained from the 3D model are usually larger than those from the beam-
column model. It should be noted that the pounding force obtained from the beam-
column model depends on the pounding stiffness kp of the impact element, while the
selection of kp is difficult since it depends on many factors and consequently the value can
be varied in a wide range [28]. With a proper selection of kp, 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 earthquake 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 decks. 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, earthquake 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 decks even the bridge structures are
symmetric. The torsional response might induce eccentric poundings between adjacent
bridge decks as observed in Figure 1, and eccentric poundings in turn will cause more
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torsional responses. This 3D response characteristic cannot be captured with the lumped
mass model or the 2D 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 7-4(a) 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 7-4(b) have the same length of 50m. The Young’s modulus and density of
the girders are 3.45×1010 Pa and 2500 kg/m3, respectively. The L-type abutment is 8.1m
long in the transverse direction and its cross section is shown in Figure 7-4(c). The height
of the rectangular central pier is 20m, with the cross section shown in Figure 7-4(d). The
materials for the two abutments and the pier are the same, with Young’s modulus and
density of 3.0×1010 Pa and 2400 kg/m3, 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 0.7921m2 and 0.082m. The horizontal effective stiffness and equivalent
damping ratio of a bearing are 2.33×107 N/m and 0.14 respectively [12, 19]. The stiffness
of the bearing in the vertical direction is much larger than those in the horizontal
directions, and is assumed to be 1.87×1010 N/m [19]. To allow for contraction and
expansion of the bridge decks from creep, shrinkage, 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 decks. 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 rock). 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 rock are A’, B’ and
C’.
The 3D finite element model of the bridge is constructed by using the finite element code
ANSYS [26]. The bridge girders, abutments and pier are modelled by eight-node solid
elements. The bearings are modelled by the spring-dashpot elements. The detailed
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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 0.2m is only applied to a length of 1m
from each end of the bridge deck and to a length of 0.6m of the abutments. Beyond this
region, the mesh size in the longitudinal direction is 1m. 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 0.1m
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 1.081, 1.138, 1.254 and 1.313 Hz for the in-
phase 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)
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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 0.3483 and the stiffness matrix multiplier is 0.0072. 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 [22].
Figure 7-5. Finite element mesh of the bridge and the nodal points for response recordings
(a) f1=1.081Hz (b) f2=1.138Hz
(c) f3=1.254Hz (d) f4=1.313Hz
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 rock owing to the
amplification and filtering effect. The three sites (A, B and C) as shown in the figure have
different influences on base rock motions, thus further intensifies the spatial variations of
1 2
3 4 Left abutment
Left girder 12
10
11
9
7
5 6
8 Left girder Right girder Right girder
Pier
Right abutment
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the ground motions. However, traditional method (e.g., Hao et al. [29]) 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 [25] developed an approach to stochastically simulate the spatially
varying motions on the ground surface of a canyon site. In the method, the base rock
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 rock 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
rock is modelled by an empirical coherency function. Local site effect is modelled using the
one-dimensional wave propagation theory [31]. 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 rock are assumed to be
the dame and have the following form:
Γ+−
++−
= 222222
222
2222
4
4)(41
)2()()(
ωωξωωωωξ
ωξωωωωω
ggg
gg
fffgS (7-3)
where ωg and ξg are the central frequency and damping ratio of the Tajimi-Kanai power
spectral density function, ωf and ξf 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 πω 10=g rad/s,
6.0=gξ , πω 5.0=f , 6.0=fξ and 0232.0=Γ m2/s3. These parameters correspond to a
ground motion time history with duration T=16s and PGA of 0.5g based on the standard
random vibration method [32]. The vertical motion on the base rock is also modelled with
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the same filtered Tajimi-Kanai power spectral density function, but the amplitude is
assumed to be 2/3 of the horizontal component.
The Sobczyk model [33] is selected to describe the coherency loss between the ground
motions at points j’ and k’ (where j’, k’ represents A’, B’ or C’) on the base rock:
)/cosexp()/exp()/cosexp()()( ''''''''''2
appkjappkjappkjkjkj vdivdvdiii αωβωαωωγωγ −⋅−=−= (7-4)
where β is a coefficient reflecting the level of coherency loss. β =0.0, 0.001 and 0.002 are
considered in the present paper, which represent perfectly correlated spatial ground
motions, or spatial ground motion with wave passage effect only, intermediately and weakly
correlated motions, respectively. dj’k’ is the distance between the points j’ and k’. For the
analysed bridge structure, dA’B’=dB’C’=50m, and dA’C’=100m. vapp is the apparent wave
velocity on the base rock, which is related to the base rock property and incident angle α .
With the given properties of local site (shown in Table 7-1) and assumed incident angle o60=α , vapp equals 1697m/s in the present study.
Not to further complicate the problem, only one single layer resting on the base rock 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 7-1 gives the corresponding parameters
for the soils and base rock. 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 7-1. Parameters for local site conditions
Type Density (kg/m3) Shear modulus(MPa) Damping ratio Poisson’s ratio
Base rock 2500 1800 0.05 0.33
Firm soil 2000 320 0.05 0.4
Soft soil 1600 40 0.05 0.4
With the proposed approach in [25] 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, 2π] are included in the simulation. For each realization of the phase
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angles, one set of ground motion time histories can be simulated. Since most design codes
require 2 to 4 independent analyses with independently simulated ground motions as input
and take 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 100 and 25 Hz respectively, and the time duration is assumed to be T=16s. 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-of-
plane 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 [25] indicates that it is different from that on the
base rock. 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 [25]. 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
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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 earthquake-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 in-
plane, 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 7-10, where xAd , yAd and zAd 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
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influences of pounding effect, local soil conditions and ground motion spatial variations on
the structural responses, five different cases as shown in Table 7-2 are studied. In which,
the case without pounding (Case 1) is simulated by adjusting the model to make 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.
dxA
Abutm
ent
dzA
dxC
dzC
dxB
dzB
Girder Girder
Pier
dyB
dyA dyC
Figure 7-10. Multi-components spatially varying inputs at different supports of the bridge
Table 7-2. Different cases studied
Case Soil conditions Coherency loss With/without pounding
1 Firm intermediately without
2 Firm intermediately with
3 Soft intermediately with
4 Firm wave passage effect with
5 Firm weakly 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 earthquake-
induced structural responses, the 12 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 peak 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.
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7.5.1 Longitudinal response
Figures 7-11, 7-12 and 7-13 show the longitudinal displacement response time histories at
nodes 1 and 2 of different ground motion cases. For conciseness, the response time
histories of other nodes are not plotted. The mean peak displacements at different nodes
are listed in Table 7-3. As shown in Figure 7-11(a), the longitudinal displacement response
of node 1 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
Maragakis et al. [34], who investigated the influences of abutment and deck stiffness, gap,
and deck to abutment mass ratio on the pounding responses between abutments and
bridge decks, 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
7-11(b), the peak displacements of node 2 in the longitudinal direction with and without
pounding effect are 0.210 and 0.274m respectively, poundings result in a reduction of
displacement response by 23.4%. This is because the rigid abutment acts as a constraint to
the flexible girder. Comparing the mean peak responses of different nodes of Cases 1 and 2
in Table 7-3, same conclusions can be obtained.
The influence of local soil conditions on the structural response is shown in Figure 7-12.
As shown, softer soil results in lager longitudinal displacement. Taking node 2 for example,
the peak displacements are 0.210 and 0.276m 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 peak responses of cases 2 and 3 in Table 7-3, same conclusions can be drawn.
The influence of coherency loss on the longitudinal displacement is shown in Figure 7-13.
As shown in Figure 7-13(a), the influence of coherency loss on node 1 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 7-13(b), different coherency loss results in
different longitudinal displacements of node 2. By examining the mean peak responses of
cases 2, 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 peak responses.
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Figure 7-11. Influence of pounding effect on the longitudinal displacement response
Figure 7-12. Influence of soil conditions on the longitudinal displacement response
Figure 7-13. Influence of coherency loss on the longitudinal displacement response
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Table 7-3. Mean peak displacements in the longitudinal direction (m)
Case Node
1 2 3 4 5
1 0.147 0.148 0.204 0.149 0.151
2 0.216 0.168 0.220 0.177 0.163
3 0.147 0.148 0.203 0.148 0.147
4 0.227 0.176 0.233 0.190 0.173
5 0.200 0.167 0.235 0.182 0.161
6 0.203 0.169 0.234 0.183 0.157
7 0.215 0.173 0.233 0.191 0.158
8 0.226 0.176 0.244 0.177 0.165
9 0.206 0.168 0.231 0.177 0.151
10 0.140 0.140 0.191 0.146 0.138
11 0.205 0.170 0.223 0.169 0.168
12 0.140 0.139 0.191 0.146 0.138
7.5.2 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 7-14, 7-15 and 7-16 show the response
time histories in the transverse direction of nodes 1 and 2, and the corresponding time
histories in the vertical direction are plotted in Figures 7-17, 7-18 and 7-19. The mean peak
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
peak 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 7-15 and 7-
16, 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 decks.
As shown in Tables 7-4 and 7-5, weakly correlated ground motions, among the three
spatial ground motion cases, usually lead to the largest mean peak responses in the two
directions. It also can be seen from Figures 7-17, 7-18 and 7-19 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
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the considered bridge model, the first vertical vibration mode is the 7th mode and the
vibration frequency is 2.237 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 7-14. Influence of pounding effect on the transverse displacement response
Figure 7-15. Influence of soil conditions on the transverse displacement response
Figure 7-16. Influence of coherency loss on the transverse displacement response
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Table 7-4. Mean peak displacements in the transverse direction (m)
Case Node
1 2 3 4 5
1 0.119 0.119 0.202 0.119 0.119
2 0.186 0.183 0.265 0.191 0.188
3 0.119 0.119 0.202 0.119 0.119
4 0.185 0.182 0.265 0.189 0.187
5 0.270 0.252 0.349 0.259 0.274
6 0.272 0.230 0.342 0.264 0.271
7 0.270 0.252 0.348 0.259 0.272
8 0.272 0.230 0.341 0.263 0.271
9 0.192 0.189 0.289 0.192 0.201
10 0.123 0.123 0.196 0.119 0.125
11 0.191 0.189 0.289 0.190 0.199
12 0.123 0.123 0.196 0.119 0.125
Figure 7-17. Influence of pounding effect on the vertical displacement response
Figure 7-18. Influence of soil conditions on the vertical displacement response
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Figure 7-19. Influence of coherency loss on the vertical displacement response
Table 7-5. Mean peak displacements in the vertical direction (m)
Case Node
1 2 3 4 5
1 0.072 0.072 0.079 0.072 0.072
2 0.162 0.111 0.131 0.116 0.114
3 0.072 0.072 0.080 0.072 0.072
4 0.145 0.117 0.136 0.118 0.119
5 0.152 0.106 0.127 0.118 0.117
6 0.127 0.115 0.129 0.115 0.121
7 0.131 0.112 0.142 0.117 0.121
8 0.131 0.121 0.130 0.116 0.132
9 0.132 0.109 0.118 0.108 0.122
10 0.073 0.073 0.081 0.073 0.073
11 0.125 0.110 0.121 0.105 0.119
12 0.073 0.073 0.081 0.073 0.072
7.5.3 Torsional response
With the lumped mass model or beam-column element model, the torsional response of
the structure cannot be considered because they are 2D 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 1 and 3, nodes 2 and 4, etc. These can be achieved
by dividing the relative longitudinal displacement of these corresponding nodes by the deck
width, which is 8.1m in the present study. Table 7-6 shows the mean peak 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
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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 2, 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 7.5.1, 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 1 and 2
and nodes 3 and 4 are plotted in the same figure with the displacements of nodes 1 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 7-20(a), node 1 and node 2 come into contacts 15 times, at the time
instants 3.26, 5.29, 6.29, 6.68, 7.30, 7.72, 8.20, 8.63, 9.13, 9.66, 11.13, 11.89, 12.44, 13.70
and 14.26s. Whereas between nodes 3 and 4 as shown in Figure 7-20(b), the poundings at
6.29, 11.89 and 12.44s do not occur, but two more collisions can be observed at 3.76 and
13.20s. Since these points locate at the opposite corners of the bridge deck 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 6.29, 11.89 and 12.44s
are eccentric poundings between nodes 1 and 2, and those at 3.76 and 13.20s 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 2D beam-column element model because the two letter models cannot
capture the possible eccentric poundings induced by torsional responses.
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Table 7-6. Mean peak rotational angle (degree)
Case Node
1 2 3 4 5
1 and 3 0.0014 0.0177 0.0219 0.0262 0.0149
2 and 4 0.2638 0.2957 0.3459 0.2745 0.3027
5 and 7 0.2150 0.2504 0.3374 0.2879 0.2844
6 and 8 0.2271 0.2624 0.3317 0.2214 0.2766
9 and 11 0.2624 0.3211 0.3572 0.2822 0.2872
10 and 12 0.0007 0.0170 0.0198 0.0113 0.0127
Figure 7-20. Longitudinal displacements of different nodes to case 2 ground motion
7.5.4 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 7-21 and 7-22 show the pounding forces at
different time instants corresponding to different ground motion cases. It can be seen from
Figure 7-21 that soft soil condition results in larger peak pounding forces than firm soil
condition. This is because soft soil leads to larger displacement response in the longitudinal
direction as shown in Figure 7-12, which also results in larger relative displacement
between the adjacent components of the bridge and makes the poundings more severe
than that on the firm site. Comparing Figure 7-21(a) and 7-21(c) with 7-21(b), it is obvious
Initial gap=5cm
Initial gap=5cm
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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 7-22 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 7-13 and
Table 7-3, where wave passage effect results in larger relative displacement responses. Same
conclusion was also drawn in [16], in which the two adjacent bridge girders were simplified
as two lumped masses.
Figure 7-21. Influence of soil conditions on the resultant pounding forces
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Figure 7-22. Influence of coherency loss on the resultant pounding forces
7.5.5 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
earthquake-induced damage may occur. Figure 7-23 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 peak resultant pounding force
occurs. As shown in Figure 7-23(a), 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. Taking the results from
different soil conditions as example, the peak resultant pounding force for firm and soft
soil are 55 and 80 MN, respectively as shown in Figure 7-21. 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,
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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 7-23, the
maximum stresses can reach as high as 105.4 MPa (Figure 7-23(d)). It is much larger than
the compressive strength of normal concrete used in bridge construction, which is usually
30-65 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 earthquakes, in which the damages
around the corners of the structure were usually the most serious as shown in Figure 7-1.
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 7-23. Stress distributions in the longitudinal direction at left gap of different cases at
the time when peak resultant pounding force occur (a) Case 1 at t=6.27s, (b) Case 3 at
t=7.63s, (c) Case 4 at t=7.96s and (d) Case 5 at t=8.04s (unit: Pa)
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7.6 Conclusions
Based on a detailed 3D finite element model, the earthquake-induced pounding responses
between adjacent components of a two-span simply-supported bridge structure located at a
canyon site are studied in the present paper. The influences of local soil conditions and
ground motion spatial variations on the pounding responses are investigated in detail.
Following conclusions are obtained based on the numerical results:
1. The lumped mass model and beam-column element model can be used to calculate
bridge pounding responses if only longitudinal ground excitation is considered. The
detailed 3D finite element model is necessary to model the torsional response
induced by spatially varying transverse ground motions and the corresponding
eccentric poundings.
2. The influence of pounding effect on the displacement response of the stiff
abutments can be neglected. Its influence on the bridge girder displacement is
evident. Poundings usually result in smaller mean peak displacements in the
longitudinal, transverse and vertical directions, but larger mean peak torsional
responses.
3. Local soil conditions significantly influence the structural responses. The softer is
the local site, the larger are the structural responses.
4. Spatially varying ground motions with wave passage effect only usually lead to
larger longitudinal displacement. Weakly correlated ground motions result in larger
transverse and vertical responses.
5. Maximum stress usually appears at the corners of the contact surfaces owing to
eccentric poundings.
6. 3D FE model is needed for more realistic predictions of pounding responses and
pounding induced bridge girder damages.
7.7 References
1. Jennings PC. Engineering features of the San Fernando Earthquake of February 9,
1971. Report No. EERL-71-02, California Institute of Technology, 1971.
2. Priestley MJN, Seible F, Calvi GM. Seismic design and retrofit of bridges. Wiley: New
York, 1996.
3. Kawashima K, Unjoh S. Impact of Hanshin/Awaji earthquake on seismic design
and seismic strengthening of highway bridges. Structural Engineering/Earthquake
Engineering JSCE 1996; 13(2):211-240.
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4. Earthquake Engineering Research Institute. Chi-Chi, Taiwan, Earthquake
Reconnaissance Report. Report No. 01-02, EERI, Oakland, CA, 1999.
5. Elnashai AS, Kim SJ, Yun GJ, Sidarta D. The Yogyakarta earthquake in May 27,
2006. Mid-America Earthquake Centre. Report No. 07-02, 2007.
6. Lin CJ, Hung H, Liu Y, Chai J. Reconnaissance report of 0512 China Wenchuan
earthquake on bridges. The 14th world conference on earthquake engineering, Beijing, China,
2008; S31-006.
7. Hao H. A parametric study of the required seating length for bridge decks during
earthquake. Earthquake Engineering and Structural Dynamics 1998; 27(1):91-103.
8. 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 2008; 30(1):154-162.
9. Bi K, Hao H, Chouw N. Required separation distance between decks and at
abutments of a bridge crossing a canyon site to avoid seismic pounding. Earthquake
Engineering and Structural Dynamics 2010; 39(3):303-323.
10. 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. Earthquake Engineering and Structural Dynamics 2010 (published
online).
11. Malhotra PK. Dynamics of seismic pounding at expansion joints of concrete
bridges. Journal of Engineering Mechanics 1998; 124(7):794-802.
12. Jankowski R, Wilde K, Fujino Y. Pounding of superstructure segments in isolated
elevated bridge during earthquakes. Earthquake Engineering and Structural Dynamics
1998; 27:487-502.
13. Ruangrassamee A, Kawashima K. Relative displacement response spectra with
pounding effect. Earthquake Engineering and Structural Dynamics 2001; 30(10): 1511-
1538.
14. DesRoches R, Muthukumar S. Effect of pounding and restrainers on seismic
response of multi-frame bridges. Journal of Structural Engineering (ASCE) 2002; 128(7):
860-869.
15. Chouw N, Hao H. Study of SSI and non-uniform ground motion effects on
pounding between bridge girders. Soil Dynamics and Earthquake Engineering 2005;
23:717-728.
16. 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 2008; 30(1):141-153.
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17. Jankowski R, Wilde K, Fujino Y. Reduction of pounding effects in elevated bridges
during earthquakes. Earthquake Engineering and Structural Dynamics 2000; 29: 195-212.
18. 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 2006; 9(1):55-66.
19. Zanardo G, Hao H, Modena C. Seismic response of multi-span simply supported
bridges to spatially varying earthquake ground motion. Earthquake Engineering and
Structural Dynamics 2002; 31(6): 1325-1345.
20. Julian FDR, Hayashikawa T, Obata T. Seismic performance of isolated curved steel
viaducts equipped with deck unseating prevention cable restrainers. Journal of
Constructional Steel Research 2006; 63:237-253.
21. Zhu P, Abe M, Fujino Y. Modelling three-dimensional non-linear seismic
performance of elevated bridges with emphasis on pounding of girders. Earthquake
Engineering and Structural Dynamics 2002; 31:1891-1913.
22. Jankowski R. Non-linear FEM analysis of earthquake-induced pounding between
the main building and the stairway tower of the Olive View Hospital. Engineering
Structures 2009; 31:1851-1864.
23. Der Kiureghian A. A coherency model for spatially varying ground motions.
Earthquake Engineering and Structural Dynamics 1996; 25(1): 99-111.
24. Bi K, Hao H, Ren W. Response of a frame structure on a canyon site to spatially
varying ground motions. Structural Engineering and Mechanics 2010; 36(1): 111-127.
25. Bi K, Hao H. Influence of irregular topography and random soil properties on
coherency loss of spatial seismic ground motions. Earthquake Engineering and
Structural Dynamics 2010 (published online).
26. ANSYS. ANSYS user’s manual revision 12.1. ANSYS Inc, USA, 2009.
27. LS-DYNA. LS-DYNA user manual. Livermore Software Technology Corporation:
California, USA, 2007.
28. Hao H, Ma G. An investigation of the coupled torsional-pounding responses of
adjacnet asymmetric structures. Proceeding of the 7th East Asian-Pacafic Conference on the
Structural Engineering and Constructuion, Kochi, Japan, 1999; 788-793.
29. Hao H, Oliveira CS, Penzien J. Multiple-station ground motion processing and
simulation based on SMART-1 array data. Nuclear Engineering and Design 1989;
111(3): 293-310.
30. Tajimi H. A statistical method of determining the maximum response of a building
structure during an earthquake. Proceedings of 2nd World Conference on Earthquake
Engineering, Tokyo, 1960; 781-796.
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31. Wolf JP. Dynamic soil-structure interaction. Prentice Hall: Englewood Cliffs, NJ, 1985.
32. Der Kiureghian A. Structural response to stationary excitation. Journal of the
Engineering Mechanics Division 1980; 106(6): 1195-1213.
33. Sobczky K. Stochastic wave propagation. Netherlands: Kluwer Academic Publishers,
1991.
34. Maragakis E, Douglas B, Vrontinos S. Classical formulations of the impact between
bridge deck and abutments during strong earthquake. Proceedings of the 6th Canadian
Conference on Earthquake Engineering, Toronto, Canada, 1991; 205-212.
35. Bischoff PH, Perry SH. Impact behaviour of plain concrete loaded in uniaxial
compression. Journal of Engineering Mechanics 1995; 121(6): 685-693.
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Chapter 8 Concluding Remarks
8.1 Main findings
This thesis has focused 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 SSI effect,
and dynamic response modelling of bridge structures with pounding effect. The major
contributions and findings made in this research are summarised below.
1. A stochastic method is adopted and further developed in Chapter 2 to investigate the
combined ground motion spatial variation effect and local site effect on the responses
of a bridge frame located on a canyon site. In the proposed approach, the spatial ground
motions are modelled in two steps. Firstly, the base rock 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. The structural responses are formulated in the frequency domain,
and the mean peak responses are estimated based on the standard random vibration
method. Numerical results show that wave propagation through multiple sites with
different site conditions cause further variations of spatial ground motions, and thus
significantly influence the structural responses.
2. Chapters 3 and 4 investigate the minimum total gaps between abutment and bridge deck
and between two adjacent bridge decks connected by MEJs to avoid seismic pounding
during strong earthquakes. In particular, Chapter 3 focuses on the combined ground
motion spatial variation and local site effect and Chapter 4 highlights the SSI effect. The
stochastic structural responses are also formulated in the frequency domain. Numerical
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results show that the required MEJ total gap depends on the dynamic properties of the
participating adjacent structure and the dynamic behaviour of the supporting subsoil and
the spatially varying ground excitations. Sufficient total gap of a MEJ needed in the
bridge design to preclude possible pounding during strong earthquakes is presented. The
numerical results obtained in these studies can be used as references in designing the
total gap of MEJs.
3. Chapter 5 presents a method to model and simulate spatially varying earthquake ground
motion time histories at sites with non-uniform conditions. It takes into consideration
the local site effect on ground motion amplification and spatial variations. The base
rock motions are modelled by a filtered Tajimi-Kanai power spectral density function or
a stochastic ground motion attenuation model. The specific site ground motion power
spectral density function is derived by considering seismic wave propagations through
the local site by assuming the base rock motions consisting of out-of-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 spatially varying earthquake 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 model coherency loss function between any two of them.
This method 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 simulated time histories can be used as inputs to multiple supports of long-span
structures on non-uniform sites in engineering practice.
4. Chapter 6 evaluates the influence of local site irregular topography and random soil
properties on the coherency function between spatial surface motions based on the
method proposed in Chapter 5. In the analysis, the 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 mean coherency loss functions
are derived. Numerical results show that the coherency function between surface
ground motions on a canyon site is different from that between base rock motions. The
lagged coherency function on the base rock 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. The coherency function models
for motions on a flat-lying site cannot be used to model that of motions on a canyon
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site. The influence of random soil properties on the lagged coherency function depends
on the level of variations of soil properties.
5. Chapter 7 investigates the pounding responses between the abutment and the adjacent
bridge deck and between two adjacent bridge decks of a two-span simply-supported
bridge located on a canyon site based on a detailed 3D finite element model. The multi-
component spatially varying ground motions are stochastically simulated as inputs based
on the method proposed in Chapter 5, and the numerical analysis is carried out by using
the transient dynamic finite element code LS-DYNA. The influences of local soil
conditions and ground motion spatial variations on the pounding responses are
investigated in detail. Numerical results indicate that the torsional response of bridge
structures resulted from the spatially varying transverse motions induces eccentric
poundings between adjacent bridge structures. Traditionally used SDOF model and 2D
finite element model of bridge structures could not capture the torsional response
induced eccentric poundings, therefore might lead to inaccurate pounding response
predictions. Detailed 3D finite element model clearly captures the eccentric poundings
of bridge decks and the potential bridge deck damages, thus is needed for a more
reliable prediction of earthquake-induced pounding responses between adjacent
structures.
8.2 Recommendations for future work
The modelling of seismic ground motion spatial variations and its effect on bridge
structural responses have been carried out in this research. Further investigations can be
made in the future study as outlined below:
1. Three different computer programs have been developed to investigate the structural
responses in the frequency domain in Chapters 2-4. These programs are based on the
simplified structural models and stochastic structural response analysis. They are thus
only suitable for linear elastic response analysis of the particular problems discussed in
the corresponding chapters. Furthermore, only one-dimensional seismic excitation,
which is along the longitudinal direction of the structure is considered because they are
the primary sources for relative displacement responses of bridge structures in the
longitudinal direction.
2. In Chapter 5, the surface motions of a canyon site with multiple soil layers are derived
based on the one-dimensional wave propagation theory, the scattering and diffraction
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effect of waves by canyons are not involved. How to incorporate this 2D wave
propagation phenomenon into the simulation technique needs be studied.
3. For a canyon site, the coherency function of spatial surface ground motions is found to
be related to the spectral ratios of two local sites in Chapter 6, but analytical relation that
can be straightforwardly used to model the influences of local site conditions on spatial
ground motion coherency loss is not derived yet.. Further study is needed to develop
the analytical relation for easy use in engineering application to predict local site effects
on ground motion spatial variations.
4. Random soil properties are modelled by the independent one-dimensional random fields
in the vertical direction in Chapter 6, the soil nonlinearities are not considered. Soil
nonlinearities also affect the surface motion spatial variations. Further study to
investigate the influence of soil nonlinearities on the surface motion spatial variations is
needed.
5. Surface to surface pounding and torsional response induced eccentric pounding
between different components of a two-spam simply-supported bridge are investigated
in Chapter 7. The material non-linearities and pounding induced local damage are not
considered, which needs to be included in the subsequent studies.
6. Shaking table tests on scaled bridge models to spatially varying earthquake ground
motions to verify the numerical results are necessary.
7. Further study to develop design guides to mitigate pounding and unseating damage of
bridge decks is needed.