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\begin{document}
%\EQE{1}{4}{00}{28}{02}
\EQE{1}{23}{\large IN PRINT\\}{28}{02}

\runningheads{B. Jeremi{\'c}}
{Time Domain SFSI in non--Uniform Soils}



\received{May 2008}
\revised{October 2008}
\accepted{December 2008}



\title{Time Domain Simulation of Soil--Foundation--Structure  Interaction in non--Uniform Soils}


 
\author{Boris Jeremi{\'c}\affil{1}\corrauth, 
Guanzhou Jie\affil{2},
Matthias Preisig\affil{3},
Nima Tafazzoli\affil{4}}

\address{
\affilnum{1}\ Department  of  Civil  and  Environmental  Engineering,
University  of California,
One Shields Ave.,
Davis, CA 95616,
Email: \texttt{Jeremic@ucdavis.edu}\\
\affilnum{2}\ 
Wachovia Corporation, 375 Park Ave, New York, NY,\\
%
%Formerly with the Department  of  Civil  and  Environmental  Engineering,
%University  of California,
%One Shields Ave.,
%Davis, CA 95616,\\
\affilnum{3}\ Ecole Polytechnique F{\' e}d{\' e}rale de Lausanne,
CH--1015 Lausanne,  Switzerland,  \\
\affilnum{4}\ Department  of  Civil  and  Environmental  Engineering,
University  of California,
One Shields Ave.,
Davis, CA 95616,
\\
}


 
\corraddr{Boris Jeremi{\'c}, Department  of  Civil  and  Environmental  Engineering,
University  of California,
One Shields Ave.,
Davis, CA 95616,
Email: \texttt{Jeremic@ucdavis.edu}}


\cgsn{NSF--CMS}{0337811}


\keywords{\emph{Time Domain, Earthquake Soil--Foundation--Structure Interaction,
Parallel Computing}}



% 
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\begin{abstract}



Presented    here    is    a  numerical    investigation    of  the influence of
non--uniform  soil  conditions  on  a prototype concrete bridge with three bents
(four  span)  where  soil beneath bridge bents is varied between stiff sands and
soft  clay.  A series of high fidelity models of the soil--foundation--structure
system  were developed and described in some details. Development of a series of
high  fidelity models was required to properly simulate seismic wave propagation
(frequency  up to $10$~Hz) through highly nonlinear, elastic plastic soil, piles
and   bridge  structure.  Eight  specific  cases  representing  combinations  of
different  soil  conditions beneath each of the bents are simulated. It is shown
that  variability  of  soil  beneath  bridge  bents has significant influence on
bridge   system   (soil-foundation-structure)  seismic  behavior.  Results  also
indicate  that  free  field  motions  differ  quite  a bit from what is observed
(simulated)  under at the base of the bridge columns indicating that use of free
field  motions  as input for structural only models might not be appropriate. In
addition  to  that,  it  is also shown that usually assumed beneficial effect of
stiff  soils underneath a structure (bridge) cannot be generalized and that such
stiff soils do not necessarily help seismic performance of structures. Moreover,
it is shown that dynamic characteristics of all three components of a triad made
up  of  of  earthquake,  soil and structure play crucial role in determining the
seismic performance of the infrastructure (bridge) system.




%

\end{abstract}





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\section{Introduction}
\label{INTRODUCTION}








Currently,   for a vast majority of  numerical  simulations  of  the response of
bridge  structures  to seismic ground motions, the input excitations are defined
either from a family of damped response spectra or as one or more time histories
of ground acceleration.
%
These input excitations are usually applied simultaneously along the entire base
of  the structure, regardless of its dimensions and dynamic characteristics, the
properties  of  the  soil  material  in foundations, or the nature of the ground
motions themselves.
%
Application of ground motions like this does not account for spatial variations of
the  traveling seismic waves that control the ground shaking.
%
In  addition  to  that,  ground  motions  applied  in  such  a  way  neglect the
soil--structure  interaction (SSI) effects,  that  can  significantly  change  ground
motions that are actually developing in such  SSI system.
%
A number of papers in recent years have investigated the influence of the SSI on
behavior  of  bridges.  
%

Even though interest in SSI effects has grown significantly in recent years, 
\citet{Tyapin2007}    notes  that after four decades of intensive studies 
there  still  exists  a  large  gap  in SSI simulation tools used  between  SSI 
specialists and practicing civil engineers.
%
Results obtained using specialized  SSI  simulation
tools  match closer  experimental and field data (validate better \citep{Oberkampf2002})
than  regular,  general  simulation  tools.
%
There  is therefor a significant need to transfer advanced simulation technology
(numerical tools, education...) to practicing engineers, so that SSI effects can
be appropriately taken into account in designing structures.
%
One  of  the  first  studies  that  has developed a three-dimensional, nonlinear
model    for    complete  soil -- skew highway bridge  system  interacting  with
their   surrounding   soils   during  strong  motion  earthquakes  was  done  by
\citet{Chen1977}.
%


Due  to a limitations of computer power, a number of studies were conducted with
a variety of modeling simplifications that usually rely on closed form solutions
for elastic material. We mention few such studies below.
%
\citet{Makris1994}  developed  a  simple    integrated  procedure to analyze 
soil-pile foundation-superstructure
interaction, based on dynamic impedance and
kinematic   seismic   response   factors  of  pile  foundations  with  a  simple
six-degree-of-freedom structural model.
% The predicted response with the proposed
%procedure was in very good agreement with recorded data, and the significance of
%considering the frequency dependent pile-foundation impedance in predicting the
%superstructure  response  was  demonstrated. 
%
\citet{Sweet1993} approximated
 geometry of pile groups to perform   finite element analysis of a bridge
system subjected  to  earthquake loads, while \citet{Dendrou1985}  resorted to 
combining      finite    element  and  boundary  integral methodology to resolve
seismic wave propagation from soil to bridge structure.



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%%%%%


It  is very important to note that assumed beneficial    role  of not performing
a full  SSI analysis has been
turned into dogma, particularly since the NEHRP-94 seismic code states that:
%
{\it  ''These    [seismic]    forces  therefore  can be evaluated conservatively
without the adjustments recommended in Sec. 2.5 [i.e. for SSI effects]''}.
%
A  number  of  studies have therefor investigated importance of performing SSI analysis. 
%
\citet{McCallen1994} developed a detailed, 3D numerical simulation of 
dynamic  response  of  a  short-span overpass bridge system and showed that even
when  structure  remains  elastic, the complete soil--structure system is highly
nonlinear due to soil interaction.
% %
% %
% Developed were  a detailed, large scale, three
% dimensional  finite element models as well as  simple reduced order models for a
% bridge  soil  system. In addition to that, attention was focused on effects soil
% embankments  will have on dynamic response 
% and on the ability of the developed models to accurately represent the bridge seismic
% response. Based on the experimental work of many researches, \citet{McCallen1994} 
% concluded  that  bridge systems  of  this type often exhibit highly nonlinear response,
% even  when  the  structural system  remains  elastic. 
% 
% 
SSI  effects  on  cable stayed bridges together with effects of foundation depth
were investigated by  \citet{Zheng1995}.
%
%  it was shown that 
%  
%  In order to investigate the effects of
%  SSI  on  the  seismic  behavior of  cable-stayed bridge placed on a moderately
%  deep  soil  stratum  overlying  rigid  bedrock,  parametric  studies  were
%  conducted  using  2-D  FEM  by \citet{Zheng1995}. The effects of
%  foundation  depth,  ratio between fundamental periods of bridge and soft deposit
%  were revealed, and the applicability of mass-spring model in the assessment
%  of  SSI has been evaluated. 
%  
\citet{Gazetas98} and \citet{Mylonakis2006} emphasized importance of proper SSI analysis on
response  of  bridges  and  provided  important  insight  on  failure  of Hanshin
Expressway bridge during Kobe earthquake.
%
% The importance of
% soil-pile-structure  interaction  in  the  seismic  behavior of bridge piers was
% investigated   by  \citet{Mylonakis1997}  with  a  multistep
% superposition  procedure. Although the response seems to be influenced by a very
% large  number of parameters, significant insight into rather complex mechanics of
% the  problem  has  been  developed  from a parameter study involving two typical
% bridge  piers  in a realistic soft layered soil deposit excited by an artificial
% and  an  actual  seismic  rock  motion.  
% % %
% % A  two-dimensional  continuum model was
% % proposed  by \citet{Koo2003} for analyzing the soil-pile-structure
% % interaction   under   seismic   shaking.  The  soil  was  modeled  as  a  linear
% % visco-elastic  layer containing a pile that was modeled as a beam and connecting
% % to  a superstructure through a rigid pile cap. Numerical results showed that the
% % ground  level  response  of  a  coupled soil-pile-structure system is in general
% % larger than that of free field shaking. 
% 
% 
% The  cause  of  partial  failure  of 
% Route~14/Interstate~5    Separation    and    Overhead    bridge  in  the  1994 
% Northridge earthquake was
% investigated  by \citet{Fenves1998} by comparing estimates of the capacities and
% demands  of  various components in the bridge. Linearized analysis were compared
% with  nonlinear  dynamic  analysis results to evaluate the capability of simpler
% models  to  predict  maximum  earthquake  displacement  demands.  The earthquake
% demands  were estimated by analyzing detailed nonlinear model of the bridge,
% including  the  inelastic  flexural  column behavior, inelastic soil properties,
% opening and closing of the intermediate hinges, and pounding and yielding of the
% the  abutments.  The demand-capacity comparison showed that shear failure of one
% of  the  piers of the bridge in a brittle-ductile mode was the most likely cause
% of  the  collapse.  
% 
% 
% A   stochastic   approach   has   been   formulated   by
% \citet{Shirkhande1999}  for  the linear analysis of suspension bridges subjected
% to earthquake excitations. The transfer functions of various responses were 
% formulated  while  including  the  effects  of  dynamic  SSI  via the use of the
% fixed-base  modes  of  the structure. It has been found that the contribution of
% the  vertical  component  of ground motion to the bridge response increases with
% increasing            soil   compliance.   
% %
% 
\citet{Small2001} developed SSI models showing how use 
of  simple spring models for the soil behavior could lead to erroneous
result    and   recommended  that  their use should be discontinued. In
addition to that, they showed  that the type of structure and its stiffness
could  have  an  effect  on the deformation of the foundation. 
%  
%  
%  that show that use of 
%  
%  Interaction of a structure
%  and  its  foundation  with  the soil was discussed by \citet{Small2001} and
%  some  of  the  numerical and analytical methods that were developed for the
%  analysis  of  raft  and piled raft foundations were presented. It was shown that
%  the  use  of  simple spring models for the soil behavior could lead to erroneous
%  result  and  it  is  recommended  that  their use should be discontinued. Simple
%  finite layer techniques were also examined, and results were compared with those
%  of  three-dimensional  finite  element  techniques  and yielded good results for
%  displacements  in the raft and for pile moments. The effect of
%  including  the stiffness of the superstructure in the analysis of the foundation
%  was  examined,  and  it  was  shown that the type of structure and its stiffness
%  could  have  an  effect  on the deformation of the foundation. 
%  % %
\citet{Tongaonkar2003}  investigated SSI effects on peak response of three-span 
continuous  deck  bridge  seismically  isolated  by the elastomeric bearings and
found  that bearing displacements at abutment locations may be underestimated if
the SSI effects are not considered.
%  
%  
%
%  
%  The   effects  of  soil-structure interaction    on  the  peak responses of
%  three-span  continuous  deck  bridge  seismically  isolated  by  the elastomeric
%  bearings  were  assessed by \citet{Tongaonkar2003}. The emphasis has been placed
%  on understanding the
%  significance  of  physical parameters that affect the response of the system and
%  identify  the  circumstances  under  which  it  is  necessary to include the SSI
%  effects  in the design of seismically isolated bridges. In order to quantify the
%  effects  of  SSI,  the  peak  responses of isolated and non-isolated bridge were
%  compared  with  the corresponding bridge ignoring these effects. It was observed
%  that  the  soil  surrounding the pier had significant effects on the response of
%  the  isolated  bridges and under certain circumstances the bearing displacements
%  at  abutment  locations  may  be  underestimated  if  the  SSI  effects  are not
%  considered  in  the  response analysis of the system. 
%  
%  
%  
%  
%
\citet{Chouw2005} studied the effect of spatial  variations  of  ground  motion 
with different wave propagation apparent
velocities    in    soft  and  medium stiff soil, and  revealed  significant SSI
effects. In addition to that, it was found that non-uniform
ground  excitation effects are significant, especially when a big difference between the
fundamental  frequency  of the bridge frames and the dominant frequencies of the
ground  motions  exists. 
%
%  
%   
%     on  the pounding response of two adjacent bridge frames. The
%   pounding behavior of girders was analyzed by using a combined finite element and
%   boundary   element  method.  The  investigation  revealed  that  the  effect  of
%   soil-structure  interaction is significant. Neglecting SSI effect will result in
%   inaccurate  prediction  of  pounding  responses  of  bridge girders. Non-uniform
%   ground  excitation effects are significant, especially when a big difference between the
%   fundamental  frequency  of the bridge frames and the dominant frequencies of the
%   ground  motions  exists.  In  almost all considered case of soft soil as well as
%   uniform and non-uniform ground excitation, the soil-structure interaction causes
%   a  larger  required  gap  to  avoid  pounding.  
%   
%   
%   
%   Seismic performance and dynamic response
%   of   bridge-embankments  during  strong  or  moderate  ground  excitations  were
%   investigated  by \citet{Kotsoglou2007} through finite
%   element    modeling  and  detailed dynamic analysis. It was shown that different
%   boundary conditions (BCs)
%   and   interactions  with  adjacent  structures  may  be  incorporated  into  the
%   mathematical  formulation  to  faithfully  represent  the actual function of the
%   embankment.  With  dynamic characteristics of bridge-embankments thus estimated,
%   the  dynamic  response  of the entire bridge system was calculated, successfully
%   reproducing  the  field  records while accounting for soil-structure interaction
%   and  embankment  flexibility. 
%   %
\citet{Soneji2008}   analyzed influence of dynamic SSI  on  
behavior of seismically isolated cable-stayed bridge and observed that the soil 
had    significant  effects  on the response of the isolated bridge. In addition
to that, inclusion  of  SSI  was found to be essential  for  effective  design of seismically
isolated  cable-stayed  bridge, especially when the towers are very rigid
and  the  soil  is soft to medium stiff. 
% 
%  \citet{Stehmeyer2008}    utilized  and  expanded  an existing  coupled BEM-FEM
%  methodology  for    investigation    of    the  effects  of  SSI    on  both  an
%  un-retrofitted and seismically isolated typical
%  bridge  structure.  A  simple numerical model of the bridge and surrounding soil
%  was    formulated and excited by an earthquake excitation, and results were used
%  to identify SSI effects. The importance of the relative rigidity between the
%  soil-foundation  system  and  the  bridge  structure  was also investigated. The
%  results  of  the  studies  indicated that the response of the complete structure
%  system considered is affected by the inclusion of SSI effects. 
%  %
%  %
%  
%  
%  
%  \citet{Saadeghvaziri2008}  presented  the  results  of an analytical study on the
%  seismic  response  of  three  Multi-Span-Simply-Supported  (MSSS) bridges in New
%  Jersey.  They  studied the capacity/demand ratio for various components in order
%  to  evaluate  the seismic vulnerability and also the effect of modeling approach
%  and appropriateness of pushover analysis using existing demand curve considering
%  the  stiffening--interaction between the bridge and the abutments. They concluded
%  that seismic response of MSSS bridges is sensitive to soil-structure interaction
%  and it should be considered in dynamic analysis of this class of bridges. 
%  %
%  
%
%
\citet{Elgamal2008}    performed    a  very  advanced  3D  analysis  of  a  full
soil--bridge system, focusing  on  interaction  of  liquefied soil in foundation and  bridge
structure. 
%One of the important contributions of this work was to show that 
%detailed  simulations  are  possible and that research and professional practice
%communities should use such simulations more often.



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In  addition  to  studies  showing  importance  of  SSI analysis, beneficial (as
suggested by the code) and possibly detrimental  effects of SSI were analyzed in
a number of studies.
For    example  \citet{Kappos2002} found  that there are advantages in
including SSI effects
in  the  seismic design of irregular R/C bridges as seismic forces are typically
lower  when  SSI  is included in the analysis. This conclusion nicely reinforces
recommendation of  NEHRP-94 seismic code, mentioned above.
On the other hand,  \citet{Jeremic2003a} found that  SSI can have either  beneficial
or  detrimental  effects  on  structural  behavior  and  is  dependent  on  the
dynamic  characteristics    of  the  earthquake  motion, the foundation soil and
the  structure.  Main conclusion was that while in some cases SSI can  
improve overall dynamic behavior of  structural system, there are
many  cases  where  SSI is detrimental to such overall seismic response of the
soil--structure    system.    However,   due   to   computational   limitations,
\citet{Jeremic2003a}  had  to  analyze  soil--pile  system  separately  from the
structure, thus reducing modeling accuracy. Present paper significantly improves
on  modeling,  treating  complete  earthquake--soil--bridge  system  as  tightly
coupled  triad,  where  interacting  components  (dynamic characteristics of the
earthquake, soil and the bridge structure) control seismic response.



Based  on  the  above (limited) literature overview, it seems that importance of
full  SSI  analysis  is  well established in the research community.
% 
Purpose  of this paper is to present a methodology for high fidelity modeling of
seismic   soil--structure   interaction. This is done in Section \ref{SM}. 
Presented   methodology   relies   on
rational    mechanics    and   aims to reduce modeling uncertainty, by employing
currently best available models and simulation procedures.
In  addition to presenting such state--of--the--art modeling, simulation results
are  used  to  illustrate influence of non--uniform soils on seismic response of
a prototype bridge system.  A    number  of  interesting  and  sometimes 
perhaps counter-intuitive
results, given in Section \ref{results}, emphasize the need for a full, detailed
SSI analysis for each particular Earthquake--Soil--Structure triad.

Analyzed    bridge  model represents prototype model that was devised as part of
a  grand  challenge,  pre--NEESR project, funded by NSF NEES program. Pre--NEESR
project, titled "Collaborative  Research:  Demonstration of NEES for Studying
Soil-Foundation-Structure  Interaction" (PI Professor Wood from UT) brought together  
researchers  from University of California at Berkeley, University of
Texas at Austin, 
University of Nevada at Reno, University of Washington, University of Kansas, 
Purdue    University   and    University of California at Davis, with the aim of
demonstrating  use of NEES facilities and use of existing and development of new
simulations tools for studying SSI problems. Presented modeling, simulations
and   developed   parallel   simulations  tools  (used  here  and  described  by
\citet{Jeremic2007d,  Jeremic2008a})  represented a small part of
this large and ambitious project.




%  
%  A  number  of  other  recent  publications testifies about 
%  the  importance  of and interest  researchers and practitioners have in  SSI
%  phenomena 
%  \citep{Spyrakos1992,
%  Saadeghvaziri2000,        
%  Ellis2001,        
%  Mylonakis2006a,
%  Song2006,     
%  Xia2006,    
%  Liao2006,    
%  Sarrazin2005,
%  Chai2003,    
%  Savin2003,    
%  Rassem1997,    
%  Zhang2002}.
%  
%\citet{Spyrakos1992},
%\citet{Saadeghvaziri2000},        
%\citet{Ellis2001},        
%\citet{Mylonakis2006a},
%\citet{Song2006},     
%\citet{Xia2006},    
%\citet{Liao2006},    
%\citet{Sarrazin2005},
%\citet{Chai2003},    
%\citet{Savin2003},    
%\citet{Rassem1997},    
%\citet{Zhang2002}.






%
%
%
%
%  A number of papers in recent years have investigated the influence of the SSI on
%  behavior                                 of                    bridges
%  \cite{Kappos2002,Makris1993,Makris1994,Sweet1993,McCallen1994,Chen1977,Dendrou1985}.
%  %
%  In     particular    Sweet    \cite{Sweet1993}    and    McCallen  and  Romstadt
%  \cite{McCallen1994}  performed  a  finite  element analysis of bridge structures
%  subjected to earthquake loads.
%  %
%  However,  Sweet    \cite{Sweet1993}  did approximate the geometry of pile groups
%  as he was unable to analyze a full model  with available computer hardware.
%  %
%  On  the  other  hand,  McCallen  and  Romstadt  \cite{McCallen1994}  performed a
%  remarkable full scale analysis of the soil--foundation--bridge system.
%  %
%  The  soil       material (cohesionless soil, sand) was  modeled using
%  equivalent  elastic  approach  (using  Ramberg--Osgood  material  model  through
%  standard  modulus  reduction  and  damping  curves  developed  by  Seed  et  al.
%  \cite{Seed1984}).
%  %
%  The  two  studies  by  Chen  and  Penzien  \cite{Chen1977} and by Dendrou et al.
%  \cite{Dendrou1985}    analyzed  the  bridge  system  including  the soil but the
%  developed models used a very coarse finite element meshes.




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\section{Model Development  and Simulation Details}
\label{SM}

The    finite    element  models  used  in  this  study have combined both solid
elements,  used  for    soils,   and    structural  elements, used  for concrete
piles,    piers,    beams  and  superstructure.  In this section described are
material and finite element models used for both soil and structural components.
In  addition  to  that,  described  is  the  methodology  used for seismic force
application   and  staged  construction  of  the  model,  followed  by  a  brief
description  of  a numerical simulation platform used for all simulations
presented here.

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\subsection{Soil Model}
\label{soilmodel}

Two types of soil were used in modeling. First type was based on soil found at the
Capitol  Aggregates site, a local quarry located south--east of
Austin,      Texas.    This  soil was chosen as part of modeling requirement for
above mentioned pre--NEES project.
Site characterization has been preformed to collect information
on   soil by \citet{texas:site}.  
%Triaxial test data was used (single monotonic
%loading test) to calibrate  an elastic plastic   soil  constitutive  model.
%The undrained  triaxial  compression  test  has been carried out on
% a  test specimen trimmed from
%an  undisturbed  soil  sample  obtained  from borehole at an
%approximate  depth  of  $10.6ft$.  
%Detailed   description   of   testing   procedures  and  results  are  given  by
%\citep{texas:site}.
%
%  The initial
%  size and index
%  properties    of    the    soil    specimen    are   given   in
%  Table~\ref{tab:texas}~\citep{texas:site}.
%  %
%  \begin{table}[!htbp]
%  \begin{center}
%  \caption{\label{tab:texas}  Index Properties of the Undisturbed
%  Triaxial Test Specimen}
%  \begin{tabular}{|c|c|c|c|}\hline
%  Soil Index Property & Initial & After Consolidation & Failure \\\hline
%  Diameter $D$ ($inch$) & 1.50 & 1.48 & 1.56 \\\hline
%  Height $H$ ($inch$) & 3.00 & 2.87 & 2.56 \\\hline
%  Total Unit Weight $\gamma_t$ ($pcf$) & 107.3 & 111.1 & 112.8 \\\hline
%  Water Content $w$ ($\%$)& 18 & 18 & 18 \\\hline
%  Dry Unit Weight $\gamma_d$ ($pcf$) & 90.9 & 94.3 & 95.7 \\\hline
%  Void Ratio $e$\footnotemark & 0.84 & 0.77 & 0.75 \\\hline
%  Degree of Saturation $S_r$\addtocounter{footnote}{-1}\footnotemark ($\%$) & 57 & 62 & 64 \\\hline
%  \end{tabular}
%  \begin{minipage}[b]{\textwidth}
%  \addtocounter{footnote}{-1}
%  \footnotemark Specific gravity $G_s$ is assumed to be 2.68.
%  \end{minipage}
%  \end{center}
%  \end{table}

%   In the
%   triaxial   cell,   the   specimen  was  allowed  to  come  into
%   equilibrium  (compress/consolidate  with  drainage  lines open)
%   under  an  isotropic pressure equal to the assumed in-situ mean
%   total  stress, which is about $5.6psi$. Upon equilibrating, the
%   specimen was
%   sheared under undrained conditions with a strain rate of $\%1$ per
%   hour.  No  pore pressure readings were taken since the specimen
%   was unsaturated. The resulting stress-strain curve is presented
%   in  Figure~\ref{fig:comparison_austin}(a). An estimate of the undrained shear strength in
%   terms  of  total stresses was measured as $13.41psi$ ($1931psf$)
%   at about $9\%$ strain. The specimen failed in a bulging mode. The
%   index  properties  of  the specimen at failure are presented in
%   Table~\ref{tab:texas}.
%   %\begin{figure}[!htbp]
%   %\begin{center}
%   %\includegraphics[width=1.0\textwidth]{/home/jeremic/tex/works/Thesis/GuanzhouJie/thesis/Verzija_Februar/Images/triaxial_test_austin.eps}
%   %\caption{\label{fig:texas} Total Stress Strain Curve Determined
%   %from  Undrained  Triaxial  Compression Test (Undisturbed Sample
%   %from Depth $10.6ft$)}
%   %\end{center}
%   %\end{figure}
Based      on      stress-strain  curve  obtained  from  a  triaxial  test
\citep{texas:site},        as       shown       in      
Figure~\ref{fig:comparison_austin}(a), a nonlinear, kinematic hardening, elastic-plastic soil
model   has   been   developed   using   Template   Elastic   plastic  framework
\citep{Jeremic2000f}.
%
It  should  be noted that an isotropic hardening model would have been enough to
fit  monotonic  lab  test data. However, for cyclic loading, only kinematic
hardening  (in  this  case, rotational kinematic) is able to appropriately model
Bauschinger effect.
%
Developed   model   consists  of  a  Drucker-Prager  yield  surface,
Drucker--Prager plastic flow directions (potential surface) and a nonlinear Armstrong-Frederick
(rotational) kinematic    hardening    rule  \citep{Armstrong66}.
Model  calibration  was performed using limited data set resulting in a very good match
(see Figure~\ref{fig:comparison_austin}(b)).
%
\begin{figure}[!htbp]
\begin{center}
%\includegraphics[width=.9\textwidth]{triaxial_test_austin}
%\mbox{a)}\includegraphics[width=0.50\textwidth]{/home/jeremic/tex/works/Thesis/GuanzhouJie/thesis/Verzija_Februar/Images/triaxial_test_austin.eps}
\mbox{a)}\includegraphics[width=0.50\textwidth]{figs/triaxial_test_austin.eps}
\hfill
%\mbox{b)}\includegraphics[width=0.40\textwidth]{/home/jeremic/tex/works/Thesis/GuanzhouJie/thesis/Verzija_Februar/Images/Triaxial_Austin.eps}
\mbox{b)}\includegraphics[width=0.40\textwidth]{figs/Triaxial_Austin.eps}
\caption{\label{fig:comparison_austin} (a) Stress--strain curve obtained
from  triaxial  compression test (b) Stress--strain
curve by obtained by calibrated model (from Depth $10.6ft$)}
\end{center}
\end{figure}
%
Initial  opening  of  a    Drucker--Prager cone was set at approximately $5^{o}$
only   (in   normal--shear   stress   space).   This  makes  for  a  very  sharp
Drucker--Prager  cone,  with  a  very  small elastic region (similar to Dafalias
Manzari models \citet{Dafalias2004}). The actual deviatoric hardening is produced
using   Armstrong--Frederick   nonlinear   kinematic  hardening  with  hardening
constants $a=116.0$ and $b=80.0$.




%\begin{figure}[!htbp]
%\begin{center}
%%\includegraphics[width=.9\textwidth]{triaxial_test_austin}
%\includegraphics[width=1.\textwidth]{cyclic_16psi}
%\includegraphics[width=1.\textwidth]{cyclic_56psi}
%\includegraphics[width=1.\textwidth]{cyclic_160psi}
%\caption{\label{fig:other_test}  Total   Stress  Strain
%Curve   by   OpenSees   Simulation   of  Cyclic  Triaxial  Test
%(Confinement $1.6psi$, $5.6psi$ and $16.0psi$)}
%\end{center}
%\end{figure}

Second  type  of soil used in modeling was soft clay, Bay Mud). This type of soil was
modeled  using  a  total  stress  approach with an elastic perfectly plastic von
Mises  yield surface and plastic potential function. The shear strength for such
(very      soft)      Bay    Mud  material  was  chosen  to  be  $C_u  =5.0$~kPa
\citep{Dames1989}. Since this soil  is
treated   as fully saturated and there is not enough time during shaking for any
dissipation to occur, elastic--perfectly plastic model provides enough
modeling accuracy.




%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\paragraph{Soil Element Size Determination}


The    accuracy   of   a   numerical   simulation   of  seismic wave propagation
in a  dynamic
Soil-Structure--Foundation Interaction) (SFSI) problems is
controlled by two main parameters~\cite{matthias:thesis}:
%
\begin{enumerate}
\item The spacing of nodes in finite element model $\Delta h$
\item The length of time step $\Delta t$.
\end{enumerate}
%
Assuming  that numerical method converges toward
exact solution as $\Delta t$ and $\Delta h$ tend toward zero,
desired accuracy of solution can be obtained as long as
sufficient computational resources are available.

In order to represent a
traveling  wave  of a given frequency accurately about 10 nodes
per wavelength are required. Fewer than 10 nodes can lead to
numerical damping as discretization misses certain peaks of
seismic wave.  In  order to determine appropriate maximum grid
spacing  the highest relevant frequency $f_{max}$ that is present in
model needs to be found by performing a Fourier analysis of
input motion. Typically, for seismic analysis one can assume $f_{max}= 10Hz$. 
By choosing wavelength $\lambda_{min} = v/f_{max}$,
where  $v$  is (shear) wave velocity, to be represented by 10 nodes,
smallest wavelength that can still be captured with any confidence is
$\lambda = 2\Delta h$, corresponding to a frequency of $5f_{max}$.
%
The maximum grid spacing should therefor not be larger than 
%
\begin{equation}
\label{eq:gridsize}
\Delta h \le \frac{\lambda_{min}}{10} = \frac{v}{10f_{max}}
\end{equation}
%
where $v$ is smallest wave velocity that is of interest in
simulation (usually wave velocity of softest soil layer).
%
%In    this study, for  the  prototype site (Capitol Aggregates site, Austin Texas),
%and in accordance with a study by~\cite{texas:site}, maximal finite element sizes were  obtained
%and tabulated in Table~\ref{tab:element_size1}.
%%
%\begin{table}
%\begin{center}
%\caption{\label{tab:element_size1}    Maximum    Element   Size
%Determination ($f_{max}=10HZ$)}
%\begin{tabular}{|r|r|r|r|}\hline
%Depth ($ft$) & Layer thick. ($ft$) & $v_{shear}$ ($fps$) & $\Delta h_{max}$ ($ft$) \\\hline\hline
%0.0          & 1.0           & 320                 & 3.2 \\\hline
%1.0          & 1.5           & 420                 & 4.2 \\\hline
%2.5          & 4.5           & 540                 & 5.4 \\\hline
%7.0          & 7.0           & 660                 & 6.6 \\\hline
%14.0         & 7.5           & 700                 & 7.0 \\\hline
%21.5         & 17.0          & 750                 & 7.5 \\\hline
%38.5         & half-space    &2200                 & 22.0\\\hline
%\end{tabular}
%\end{center}
%\end{table}

In addition to that, mechanical   properties of soil changes with (cyclic)
loadings  as plastification develops. In order to quantify those changes in soil
stiffness, a number of laboratory and in situ
tests were performed by \citet{texas:site}. 
Moduli reduction  curve  ($G/G_{max}$)  and damping ratio relationship
were then used to capture determine soil element size while taking 
into account soil stiffness degradation (plastification).
% 
%   and are shown Figure~\ref{fig:g_gmax_austin}.
%  %
%  \begin{figure}[!htbp]
%  \begin{center}
%  %\includegraphics[width=0.45\textwidth]{/home/jeremic/tex/works/Thesis/GuanzhouJie/thesis/Verzija_Februar/Images/g_gmax_austin_1.eps}
%  %\includegraphics[width=0.45\textwidth]{/home/jeremic/tex/works/Thesis/GuanzhouJie/thesis/Verzija_Februar/Images/g_gmax_austin_2.eps}
%  %\includegraphics[width=0.45\textwidth]{/home/jeremic/tex/works/Thesis/GuanzhouJie/thesis/Verzija_Februar/Images/damping_ratio_austin_1.eps}
%  %\includegraphics[width=0.45\textwidth]{/home/jeremic/tex/works/Thesis/GuanzhouJie/thesis/Verzija_Februar/Images/damping_ratio_austin_2.eps}
%  \includegraphics[width=0.485\textwidth]{figs/g_gmax_austin_1.eps}
%  \includegraphics[width=0.485\textwidth]{figs/g_gmax_austin_2.eps}
%  \includegraphics[width=0.485\textwidth]{figs/damping_ratio_austin_1.eps}
%  \includegraphics[width=0.485\textwidth]{figs/damping_ratio_austin_2.eps}
%  \caption{\label{fig:g_gmax_austin} Comparison of the Variation
%  in  Normalized  Shear  Modulus  and Damping Ratio with Shearing
%  Strain  from  the  Resonant Column Tests with Modulus Reduction
%  Curves \citep{texas:site}}
%  \end{center}
%  \end{figure}
%  \clearpage
%  %
%  The  degradation  of  dynamic  soil  properties  as observed in
%  experiments  has to be considered in finite element analysis in
%  order  to  capture  more  accurate behaviors. 
%  
Using shear wave
velocity relation  with shear modulus 
%
\begin{equation}
\label{eq:g_v}
v_{shear} = \sqrt{\frac{G}{\rho}}
\end{equation}
%
one      can   readily  obtain  dynamic  degradation  of  wave velocities. 
This  leads  to  smaller  element  size required for detailed simulation of wave
propagation in soils which have stiffness degradation (plastification). The addition of 
stiffness  degradation effects (plastification) of soil on soil finite size are listed
in Table~\ref{tab:new_size}.
%
\begin{table}
\begin{center}
\caption{\label{tab:new_size} Soil finite element   size
determination  with  shear  wave velocity and stiffness degradation effects  for
assumed  seismic  wave  with    $f_{max}=$10~HZ,  (minimal  value of $G/G_{max}$
corresponding to $0.2\%$ strain level.)}
\begin{tabular}{|r|r|r|r|r|r|}\hline
Depth  ($ft$)  &  Layer thick. ($ft$) & $v_s$ ($fps$) & 
$G/G_{max}$ &  $v_s^{min}$(fps) & $\Delta h_{max}$ ($ft$) \\\hline\hline
0    & 1      & 320 & 0.36 & 192 & 1.92 \\\hline
1    & 1.5    & 420 & 0.36 & 252 & 2.52 \\\hline
2.5  & 4.5    & 540 & 0.36 & 324 & 3.24 \\\hline
7    & 7      & 660 & 0.36 & 396 & 3.96 \\\hline
14   & 7.5    & 700 & 0.36 & 420 & 4.20 \\\hline
21.5 & 17     & 750 & 0.36 & 450 & 4.50 \\\hline
38.5 & half-space &2200 & 0.36 &1320 & 13.20\\\hline
\end{tabular}
\begin{minipage}[b]{\textwidth}
\addtocounter{footnote}{-1}
\end{minipage}
\end{center}
\end{table}
%
Based  on  above  soil finite element size determination, a three bent prototype
finite element model has been developed and is shown in Figure~\ref{fig:prototype}.
%
\begin{figure}[!htbp]
\begin{center}
%\includegraphics[width=1.0\textwidth]{/home/jeremic/tex/works/Thesis/GuanzhouJie/thesis/Verzija_Februar/Images/PrototypeMesh.eps}
\includegraphics[width=0.65\textwidth]{figs/PrototypeMesh.eps}
\caption{\label{fig:prototype}  
Detailed  Three  Bent Prototype SFSI Finite Element Model, 484,104 DOFs, 151,264
Elements.}
\end{center}
\end{figure}
%
The  model  features    484,104  DOFs,  151,264 soil and beam--column elements, 
it  is  intended to model appropriately seismic waves of up to 10Hz, for minimal
stiffness  degradation  of  $G/G_{max} =  0.08$, maximum shear strain of $\gamma =
1$\% and with maximal element size $\Delta h = 0.3$~m.
% 
It is noted that even larger set of models was created, that was able to capture
10~Hz motions, for $G/G_{max} =  0.02$, and maximum shear strain of $\gamma =
5$\%.  This  (our largest to date) set of models features over 1.6 million DOFs
and  over  half  a  million finite elements. However, results from this very
detailed  model  were  almost  same as results for model with half a
million  DOFs  (484,104  to  be precise) and it was decided to continue analysis
with this smaller model. 
%
However, development of this more detailed model (featuring 1.6 million DOFs), 
that  did  not  add much (anything) to our results brings another very important issue.
%
It  proves  very  important  to  develop  a  hierarchy of models that will, with
refinement,  improve  our  simulations.  When  model  refinements (say mesh
refinement)      does    not  improve  simulation  results any more (there is no
observable difference), model can be
considered  mature \citep{Oberkampf2002} and no further refinement is necessary. 
This maturation of model allows us use of
immediate   lower  level  (lower  level  of  refinement)  model  for  production
simulations. It is therefore always advisable to develop a hierarchy of
models,  and  to  potentially  settle  for model that is one level below the
most detailed model. This most detailed models is chosen as model 
which  did not improve accuracy of simulation significantly enough to warrant its
use.  For our particular example, the most detailed model used, did
not    improve  results  (displacements,  moments...) significantly (actually it
almost  did  not  change  them  at  all)  implying  that  accurate  modeling  of
frequencies  up  to  10~Hz  for  this Earthquake--Soil--Structure system did not
affect seismic response.




%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\paragraph{Time Step Length Requirement}


The time step $\Delta t$
used for numerically solving nonlinear
vibration  or wave propagation problems has to be limited for
two  reasons  \citep{local-87}.  The  stability  requirement depends on time
integration scheme  in use and it restricts the size of $\Delta t=T_n/10$.  
Here, $T_n$  denotes  smallest  fundamental  period  of the
system.  Similar to spatial discretization, $T_n$ needs to be
represented  with  about  10  time  steps.  While   accuracy
requirement  provides  a  measure  on  which  higher  modes  of
vibration   are  represented  with  sufficient  accuracy,  
stability  criterion needs to be satisfied for all modes. If 
stability criterion is not satisfied for all modes of vibration,
then the solution may diverge. In many cases it is necessary to
provide an upper bound to frequencies that are present in a
system by including frequency dependent damping to time integration scheme.


The  second  stability criterion results from the nature of
finite  element  method.  As a wave front progresses in space, it
reaches  one  point (node) after  the  other. If time step in 
finite  element  analysis  is too large, than  wave front can reach
two consecutive points (nodes) at the same time. This would violate
a  fundamental  property  of  wave  propagation and can lead to
instability. The time step therefore needs to be limited to
%
\begin{equation}
\Delta t < \frac{\Delta h}{v}
\end{equation}
%
where $v$ is the highest wave velocity.
%
Based on values determined in Table~\ref{tab:new_size}, time step requirement
can be written as 
%
\begin{equation}
\Delta t < \frac{\Delta h}{v} = 0.00256 s
\end{equation}
%
thus limiting effective time step size used in numerical simulations of this
particular soil--structure model.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \subsection{Piles Model}
% \label{Piles_Model}
% 
% 
% 
% 









%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Structural Model}


The  nonlinear  structure model (piers and superstructure) used in this study were 
initially developed by \citet{Dryden2005}.
%
Model calibration was performed using experimental data from UNR shaking table tests
\citep{Dryden2008}.
%
The  original  model  was  developed  with fully fixed conditions at the base of
piers.  This  choice  of  boundary  conditions  influences  location of possible
plastic hinges in piers. 
%
This  is  important  as   model  predetermines location of plastic hinges by
placing zero--length elements at bottom and top of piers.
%
In reality, piers extend into piles and possible
plastic    hinges might form below ground surface in piles as well as at top
of piers, and not necessarily at bottom of piers.
%
The  structural  model  was  subsequently updated to reflect this more realistic
condition.
%
%The zero--length elements were removed from the bottom of the pier. 
%
In  addition  to  that, beam elements used for piles were modeled using nonlinear fiber beam
element which allows for  development of (distributed) plastic hinges.
%




%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\subsubsection{Material Modeling}


\paragraph{Concrete Modeling.}
Concrete material was modeled using Concrete01 uniaxial
material      as  available  in  OpenSees framework and  is  fully  described  by
\citet{Dryden2005, Dryden2006}. Basic description is provided here for completeness.
%
Concrete model is based on work by
\citet{kent1971}  and includes linear unloading/reloading stiffness that degrades
with  increasing  strain. No tensile strength is included in the model. The peak
strength  for  unconfined  concrete  model  is  based  on  test of concrete
cylinders   performed  at  UNR. 
% The  confined  concrete  compressive  strength,
%$f_{cc}^{\prime}$, has been calculated
%based on work by \citet{Mander1988}. 
%The ultimate strain for
%the confined concrete was determined using the energy balance approach developed
%by  \citet{Mander1988}. 
Material model parameters used for  unconfined  concrete
in simulation models are $f_{co}^{\prime}=5.9$~ksi,  
$\epsilon_{co}=0.002$,    $f_{cu}^{\prime}=0.0$~ksi,  and $\epsilon_{cu}=0.006$,
while material   parameters   for  confined  concrete  used are
$f_{co}^{\prime}=7.5$~ksi,  $\epsilon_{co}=0.0048$,  $f_{cu}^{\prime}=4.8$~ksi, and
$\epsilon_{cu}=0.022$. 

%  The  response  of these models under cyclic loading is
%  shown in Figure~\ref{fig:F4.1}.
%  
%  %\begin{table}[!htbp]
%  %\begin{center}
%  %\caption{\label{T4.1}  Concrete material properties \cite{Dryden2006}}
%  %\begin{tabular}{|c|c|c|c|c|}\hline
%  %Concrete  Type & $f_{co}^{\prime}$ ($ksi$) & $\epsilon_{co}$ & $f_{cu}^{\prime}$($ksi$) & $\epsilon_{cu}$ \\\hline
%  %Unconfined & $5.9$ & 0.002 & 0.0 & 0.006 \\\hline
%  %Confined & 7.5 & 0.0048 & 4.8 & 0.022 \\\hline
%  %\end{tabular}
%  %\end{center}
%  %\end{table}
%  \begin{figure}[!htbp]
%  \begin{center}
%  \includegraphics[width=0.6\textwidth]{figs/f4-1.eps}
%  \caption{\label{fig:F4.1} Cyclic response of concrete material model \citep{Dryden2006}}
%  \end{center}
%  \end{figure}
%  
%  

\paragraph{Steel Modeling.}
Hysteretic        uniaxial   material  model available within OpenSees framework
used to model response of steel reinforcement to match the monotonic
response observed during the steel coupon tests. Parameters included in this
model  are  $F_1=67$~ksi, $\epsilon_1=0.0023$, $F_2=92$~ksi, $\epsilon_2=0.028$,
$F_3=97$~ksi, and  $\epsilon_3=0.12$.  No  allowance  for  pinching or damage under
cyclic   loading  has  been  made  ($pinchX=pinchY=1.0$,  $damage1=damage2=0.0$,
$beta=0$).

%  A  comparison  of  the monotonic response of the steel material model
% with  the  results  of  the  coupon  tests  is  shown  in Figure~\ref{fig:F4.2}.
% Figure~\ref{fig:F4.3}  shows  the  response  of  the  steel material model under
% cyclic loading.
% 
% %\begin{table}[!htbp]
% %\begin{center}
% %\caption{\label{T4.2}  Reinforcing steel material properties \cite{Dryden2006}}
% %\begin{tabular}{|c|c|c|c|c|c|}\hline
% %$F_1$ ($ksi$) & $\epsilon_1$ & $F_2$ ($ksi$) & $\epsilon_2$ & $F_3$ ($ksi$) & $\epsilon_3$ \\\hline
% %67 & 0.0023 & 92 & 0.028 & 97 & 0.12 \\\hline
% %pinchX & pinchY & damage1 & damage2 & beta \\\hline
% %1.0 & 1.0 & 0.0 & 0.0 & 0.0 \\\hline
% %\end{tabular}
% %\end{center}
% %\end{table}
% \begin{figure}[!htbp]
% \begin{center}
% \includegraphics[width=0.6\textwidth]{figs/f4-2.eps}
% \caption{\label{fig:F4.2} Comparison of monotonic response of steel 
% material model calibrated to the results of 
% steel coupon tests \citep{Dryden2006}}
% \end{center}
% \end{figure}
% \begin{figure}[!htbp]
% \begin{center}
% \includegraphics[width=0.6\textwidth]{figs/f4-3.eps}
% \caption{\label{fig:F4.3} Cyclic response of steel material 
% model calibrated to the results of steel coupon tests \citep{Dryden2006}}
% \end{center}
% \end{figure}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\paragraph{Pier and Pile Modeling}

The   finite element  model for piers and piles features   a   nonlinear   fiber
beam--column element
\citep{Spacone1996}.  In  addition to that, a zero-length  elements is introduced
at top of  piers in order to capture effect of rigid body
rotation at joints due to elongation of anchored reinforcement.
%
%
%  
%  \begin{figure}[!htbp]
%  \begin{center}
%  \includegraphics[width=0.6\textwidth]{figs/f4-4.eps}
%  \caption{\label{fig:F4.4} Structural model for bent~\citep{Dryden2005}}
%  \end{center}
%  \end{figure}
%  
%  The  series  model  include  a  nonlinear beam-column element based on the force
%  formulation   presented  by  Spacone  et  al.  \cite{Spacone1996}  as  shown  in
%  Figure~\ref{fig:F4.4}. Based on the discussion of frame elements by Filippou and
%  Fenves  \cite{Bozorgnia2004}, the frame element is represented by a line element
%  with   basic   forces,   q,   for   the   two-dimensional   case   as  shown  in
%  Figure~\ref{fig:F4.5}.
%  
%  
%  \begin{figure}[!htbp]
%  \begin{center}
%  \includegraphics[width=0.6\textwidth]{figs/f4-5.eps}
%  \caption{\label{fig:F4.5} Basic forces for nonlinear beam-column 
%  element \cite{Bozorgnia2004}}
%  \end{center}
%  \end{figure}
%  
%  The element deformation , v, can be obtained by the following equation:

Cross section of both piers and piles was discretized 
using  $4 \times 16$ subdivisions of core
and  $2 \times 16$  subdivisions  of  cover  for  radial  and  tangential direction
respectively. 
%    which    gives    the   moment-curvature   response   shown   in
%Figure~\ref{fig:F4.6}.
%
%%\begin{table}[!htbp]
%%\begin{center}
%%\caption{\label{T4.3}  Cross-section discretization \cite{Dryden2005}}
%%\begin{tabular}{|c|c|c|}\hline
%%  & Radial & Tangential \\\hline
%%Core & 4 & 16 \\\hline
%%Cover & 2 & 16 \\\hline
%%\end{tabular}
%%\end{center}
%%\end{table}
%\begin{figure}[!htbp]
%\begin{center}
%\includegraphics[width=0.6\textwidth]{figs/f4-6.eps}
%\caption{\label{fig:F4.6} Moment-curvature analysis of reinforced 
%concrete circular column section~\cite{Dryden2005}}
%\end{center}
%\end{figure}
%
%
%
%
%
%The series model represents the columns by using a nonlinear beam-column element
%with  zero-length  elements  at  the  ends  to model the bar slip. In the series
%model,  the  element flexibility matrix is evaluated using 4-point Gauss-Lobatto
%quadrature   with   the   integration   points   and   weights   as   shown   in
%Figure~\ref{fig:F4.7}.
%
%\begin{figure}[!htbp]
%\begin{center}
%\includegraphics[width=0.6\textwidth]{figs/f4-7.eps}
%\caption{\label{fig:F4.7}  Integration  points  and  weights using Gauss-Lobatto
%quadrature  for  nonlinear  beam-column  element  (based  on  \citet{Scott2006b}}
%\end{center}
%\end{figure}
%
Additional    deformation  that  can  develop at the upper pier end results from
elongation of steel
reinforcement at beam--column joint with superstructure.
To    model    this    phenomenon,    a  simplified  hinge  model  is  developed
\citep{Mazzoni2004}. 
In that model, bar slip occurs
in  two modes: elongation due to variation in strain along length of 
anchored  bar  resulting  from  bond to surrounding concrete, and rigid body
slip of  bar that is resisted by friction from  surrounding concrete.
%
A  bi-uniform bond stress distribution was
assumed  along  length  of  anchored bar based on simplified model
developed  by  \citet{Lehman1998}. Two sets of parameters were considered
for  this  bond  stress distribution, namely $u_e=12\sqrt{f_c^{\prime}}$ and
$u_e=6\sqrt{f_c^{\prime}}$ for assuming bond stress distribution, and $u_e=8\sqrt{f_c^{'}}$
and $u_e=6\sqrt{f_c^{\prime}}$ determined based on strain gauge  data  from  tests. 
First  set is based on recommendations given by \citet{Lehman1998} while  
second  set  is based on a calibration done by \cite{Ranf2006} to match 
bond  stress  distribution to strain gauge data recorded along length of 
anchored reinforcement during shaking table tests at UNR.

% 
%  %\begin{table}[!htbp]
%  %\begin{center}
%  %\caption{\label{T4.4}  Bond stress assumptions \cite{Lehman1998} and \cite{Ranf2006}}
%  %\begin{tabular}{|c|c|c|}\hline
%  %Bond Stress Model & $u_e$ & c \\\hline
%  %Assumed bond stress distribution & $12\sqrt{f_c^{\prime}}$ & $6\sqrt{f_c^{\prime}}$ \\\hline
%  %Determined   based   on   strain  gauge  data  from  the  experimental  tests  &
%  %$8\sqrt{f_c^{'}}$ & $4\sqrt{f^{'}_c}$ \\\hline
%  %\end{tabular}
%  %\end{center}
%  %\end{table}
%  \begin{figure}[!htbp]
%  \begin{center}
%  \includegraphics[width=1.0\textwidth]{figs/f4-8.eps}
%  \caption{\label{fig:F4.8} Rigid-body rotation due to bar slip 
%  \citep{Mazzoni2004}.}
%  \end{center}
%  \end{figure}
%  
%  
%  The hinge model is implemented in OpenSees as a zero-length element with degrees
%  of  freedom  describing  the  moment-rotation  response  of  the hinge about the
%  longitudinal  and  transverse  axes  of the bridge. The material response of the
%  hinge  is  modeled  using  the hysteretic uniaxial material, which consists of a
%  backbone  curve  that  is described by three points in the positive and negative
%  directions,   respectively.   The  hinge  model  developed  by  Mazzoni  et  al.
%  \cite{Mazzoni2004}  accommodates  the  limitations of the hysteretic material by
%  basing  the  moment-rotation  response on three stages from the moment-curvature
%  analysis  of  the  cross-section.  These stages correspond to first yield of the
%  reinforcement,  nominal  moment where the outermost fiber in compression reaches
%  epsc  =  0:003, and ultimate moment when the confined concrete crushes. For each
%  stage, the stress in the outer reinforcing bar in tension is determined, and the
%  corresponding section rotation is calculated.

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\paragraph{Bent Cap}


The  bent  cap  beams  are  modeled  as linear elastic beam-column elements with
geometric  properties  developed using effective width of 
cap  beam  and  reduction of its stiffness due to cracking. The cap beams at all
bents are assumed to have a depth of $15$~in. and an effective width of $15$~in. The
effective  width  is  selected  based  on   observation  that  the amount of
longitudinal  reinforcement  outside  this effective width is small. A reduction
factor is applied to gross stiffness to account for cracking in a 
member.  Based  on  recommendations of Seismic Design Criteria
(SDC)  developed  by \citet{Caltrans2004}, value of this reduction factor is
selected   within   the   range   of   0.5-0.75,  where  0.5  corresponds  to  a
lightly-reinforced   section,  and  0.75  corresponds  to  a  heavily-reinforced
section.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\paragraph{Superstructure}

The superstructure consists of prismatic prestressed concrete members, which are
prestressed  in both longitudinal and transverse directions. Each segment of
superstructure  is  modeled  with  a linear elastic beam-column element. No
stiffness  reduction  has  been done  for these elements in accordance with 
recommendations  of  SDC. In addition to that, no reduction of  torsional
moment of inertia is
done  since  this  bridge  meets  Ordinary  Bridge requirements of SDC
\citep{Caltrans2004}. Superstructure ends were left free, as it was assumed that
structure was disconnected from approach abutments. 


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \subsection{Mass Distribution}
% 
% Point masses are placed along the longitudinal axis of the bridge model. Sources
% of mass that are modeled include the dead weight of the bridge deck and columns,
% and additional concrete and lead blocks.
% 
% 
% 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Coupling of Structural and Soil Models}
\label{couplingSection}

In    order  to  create  a  model  of  a complete soil--structure system, it was
necessary to couple structural and soil (solid) finite elements.
%
Figure  \ref{coupling}  shows  schematically  how  was  this coupling performed,
between bridge foundation (piles) and surrounding soil.
%
%                          
%
\begin{figure}[!htbp]
\begin{center}
{\includegraphics[width=8cm]{/home/jeremic/tex/works/Conferences/2005/OpenSeesWorkshopAugust/UsersExample/SFSIModelSetup02.eps}}
\caption{\label{coupling} Schematic description of coupling of structural elements (piles)
with solid elements (soil).} 
\end{center}
\end{figure}
%
The volume that would be physically occupied by pile is left open within 
solid  mesh  that  models  foundation soil. This opening (hole) is excavated
during  a staged construction process (described later in section \ref{staged}).
Beam--column elements (representing piles) are then placed in middle of this
opening.   Beam--column   elements   representing  pile  are  connected  to  
surrounding  solid  (soil) elements be means of stiff short elastic beam--column
elements.  These  short "connection" beam--column elements extend from each pile
beam--column   node   to  surrounding  nodes  of  solids  (soil)  elements.  The
connectivity   of  short, connection beam--column element nodes to nodes of soil
(solids)  is  done  only for translational degrees of freedom (three of them for
each  node),  while  rotational degrees of freedom (three of them) from 
beam--column element are left unconnected. 
Connecting piles to soil using above described method has a number of advantages
and  disadvantages.  On a positive side, geometry of soil--pile system
is  modeled  very  accurately.  A thin layer of elements next to pile is used to
mimic  frictional  behavior  soil--pile  interface.  In  addition  to  that, 
deformation modes of a pile (axial, bending, shearing) are accurately transferred
to surrounding soil by means of connection beam--column elements. In addition to
that,  both  pile  and  soil  are  modeled using best available finite
elements (nonlinear beam--column for pile and elastic--plastic solids for soil).
On  a  negative  side,   discrepancy  of displacement approximation fields
between  pile  (  a  nonlinear  beam--column)  and  soil  (a  linear solid brick
elements)  will  lead  to  incompatibility  of  displacements  between  nodes of
pile--soil system. 
However,  this  incompatibility  was deemed acceptable in view of advantages
described above.




%--

%--
%--
%--
%-- The nonlinear structure model developed in this paper is
%-- a  joint  effort  of  UCB  and  UCD. Experimental data has been
%-- collected  from  UNR  shaking  table  tests  to  calibrate  the
%-- structural  models developed using OpenSees. The effort in this
%-- paper  has  been  focused  on how to integrate advanced
%-- structure  model  with  geotechnical model to enable full-scale
%-- prototype  simulations.  The  assumption that the plastic hinge
%-- forms  either  on the top of column or at the fixed bottom does
%-- not  hold  for SFSI problems. This restriction has been removed
%-- as   the  geotechnical  and  structural  models  are  connected
%-- together.
%-- %\begin{landscape}
%-- \begin{figure}[!htbp]
%-- \begin{center}
%-- \includegraphics[width=1.0\textwidth]{/home/jeremic/tex/works/Thesis/GuanzhouJie/thesis/Verzija_Februar/Images/HingeModel.eps}
%-- \caption{ Simplified Hinge Model Developed
%-- for SFSI Prototype Simulations~\cite{dryden:nees}}
%-- \end{center}
%-- \end{figure}
%-- %\end{landscape}
%-- \begin{figure}[!htbp]
%-- \begin{center}
%-- \includegraphics[width=1.0\textwidth]{/home/jeremic/tex/works/Thesis/GuanzhouJie/thesis/Verzija_Februar/Images/Hinge_MomentRotation.eps}
%-- \caption{\label{fig:hinge2} Moment-Rotation Relationship of
%-- Structural Hinge Model~\cite{dryden:nees}}
%-- \end{center}
%-- \end{figure}
%-- \begin{figure}[!htbp]
%-- \begin{center}
%-- \includegraphics[width=1.0\textwidth]{/home/jeremic/tex/works/Thesis/GuanzhouJie/thesis/Verzija_Februar/Images/StructureModel.eps}
%-- \caption{ Structural Model Developed Using
%-- OpenSees~\cite{dryden:nees}}
%-- \end{center}
%-- \end{figure}
%-- \clearpage
%--
%-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\subsection{Domain Reduction Method}

Seismic  ground  motions  were  applied  to  finite element model of SSI
system using Domain Reduction Method~\citep{Bielak2001,Yoshimura2001}.
%
The  DRM represents the only method that can consistently (analytically)
apply free field ground motions to finite element model.
%
The method
features a two-stage strategy for complex, realistic three dimensional
earthquake  engineering  simulations. The first is an auxiliary
problem  that  simulates  earthquake source and propagation
path  effects  with  a  model that encompasses source and a
background  structure from which soil--structure system has been
removed.  The  second  problem  models  local, soil-structure effects. Its
input    is a set of equivalent forces (so called effective forces) derived from
the 
first  step.  These  forces  act  only within a single layer of
elements  adjacent to interface between exterior region
and  geological  feature  of  interest. 
%
While  DRM allows for application of arbitrary, 3D wave fields to the finite
element  model,  in  this  study  a  vertically propagating wave field was used.
Given  surface, free field ground motions were de-convoluted to a depth of 100~m.
Then,  a vertically propagating wave field was (re--) created and used to create
effective forces for DRM~\citep{Bielak2001,Yoshimura2001}. Deconvolution and
(back)    propagation  of  vertically  propagating  wave field was performed using
closed form solution as implemented in Shake
program \citep{SHAKE91}.




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\subsection{Time Integration}
\label{Time Integration}


Numerical    integration   of  equations  of motion was done
using   Hilber-Hughes-Taylor \citep{HHT,Hughes1978a,Hughes1978b}  algorithm. 
Proper  algorithmic  treatment  for  nonlinear analysis follows methodology 
described by \citet{local-87}.
%
No    Rayleigh    damping    was  used  here,  and modeling completely relies on
displacement proportional damping 
\citep{local-87},  provided  by    elastic--plastic  behavior of material (soil,
piles and structure) while small amount of numerical damping was used
 to damp out response in higher frequencies 
that   are introduced   by    spatial   finite   element   discretization
\citep{local-86}.





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\subsection{Staged Simulations}
\label{staged}

A    very  important modeling issue is that of staged
construction.  Initial state of stress in soil significantly affects its response.
Similar observation can be made about concrete structures as well.


In  general,    nonlinear   finite  element  analysis  can  be split up in three
nested loops (levels). This is true for both geometric and/or material nonlinear
finite element analysis \citep{Felippa93}.
%
Top    loop    comprises  load stages, which represent realistic loading sequence on
solids  and  structures.  Within loading stage loop is an  incremental loading loop,
which splits load in each stage into increments. Split into increments
is not only important from numerical stability standpoint, but is also essential
from  proper  modeling  of  elastic--plastic  materials.  Within each increment,
equilibrium  iterations  are  advisable  but  not  necessary  for advancement of
solution.
%  
%  
% 
% 
% 
% 
% The  top,  outermost  loop  is  the  stage  loop.  This analysis level will
% feature  stages of loading, for example,  self  weight  loading,  service loads,
% hazard loads. It is very
% important   to  properly  sequence  the  loading  stages  as the state  of  stress,
% deformation and  internal  variables  at  the  end of one of the stages will become
% initial state (of stress, deformation, internal variables) for the next stage.
% 
% Within each stage loop, there exists an incremental loading loop. For nonlinear,
% inelastic  materials, it is very important to apply loads in a (appropriate, large) number of
% increments  in  order  to  properly simulate the material behavior within solids
% and  structures.  Theoretically  the  more  increments are used, the smaller the
% error  will be. Time constraints, of course, limit the number of increments that
% are  usually used. For examples, it has been shown by \citet{Jeremic96b} the for
% elastic--plastic  simulations,  the  error of implicit constitutive integrations
% grows  as the step size increases. The error in implicit integrations is smaller
% than that for explicit integrations, however it still exists and has to be taken
% into considerations when decisions about number if incremental steps are made.
% 
% The  inner  most  loop  is  the  iterative  loop,  which  is  used to return the
% current  state  of  the  generalized displacements and generalized forces to the
% equilibrium path on the finite element level. In addition to that, the iterative
% loop  is  also  present  on  the constitutive level, where the stress / internal
% variables  state  is updated so that yield condition is satisfied. The iterative
% loop  can be skipped altogether, in which case one deals with the pure explicit,
% forward  stepping simulations. However, for these pure explicit simulations, the
% error accumulation can quickly render produced results unusable.
%
Simulations    presented  in  this  studies were performed in three main stages,
number of increments and equilibrium iterations.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\paragraph{Soil Self--Weight Stage.}
%
During   this  stage,   finite element model for soil (only, no structure) is
loaded with soil
self--weight.  The  finite  element model for this stage excludes any structural
elements,    and opening  (hole)    where  the  pile  will  be  placed,  is full of soil.
Displacement  boundary conditions on sides of three soil blocks are  allowing  
vertical displacement,  and  allow horizontal in boundary plane
displacement,  while they prohibit out of boundary plane displacement of soil. All
displacements  are  suppressed  at  bottom of all three soil blocks (for
a model shown in Figure~\ref{fig:prototype}). The
soil  self weight is in our case applied in 10 incremental steps.  While such small number of
steps  is  not  advisable  in  general,  initial finite element model was simple
enough  (three  soil blocks without any interactions between them) that only ten
increments of load were sufficient to obtain initial state of stress, strain and
internal variables for soil.



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\paragraph{Piles, Columns and Superstructure Self--Weight Stage.}
%
In  this,  second  stage, number of modeling changes happen to occur.
Firstly, soil elements where piles will be placed are removed (excavated), then
concrete    piles  (beam--column  elements)  are  placed    in  holes (while
appropriately  connecting  structural and solids degrees of freedom, see section
\ref{couplingSection}),  columns  are  placed  on  top  of  piles  and  finally  
superstructure  is placed on top of columns. All of this construction is done at
once.   With   all    components   in   place,    self  weight analysis of  
piles--columns--superstructure  system is performed. Ideally, it would have been
better  to  perform "construction" process in few stages, but even by adding all
structural  elements  at  once and performing their self weight analysis in
(this)  one stage (using 100 increment of load) an accurate initial state of
section       forces  (stress)  and  deformation (strains) has been obtained for
prototype bridge model.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\paragraph{Seismic Shaking Stage.}

The  last  stage  in our analysis consists of applying seismic shaking, by means
of effective forces through use of DRM. It is important to note that seismic
shaking  is  applied  to  already  deformed  model,  with all stresses,
internal  variables  and  deformation  that  resulted  from  first two stages of
loading.



%
%set NumSteps 3000
%set Dt 0.02
%


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\subsection{Simulation Platform}
\label{SP}


Numerical  simulations  described  in  this  paper  were  done  using a parallel
computer   program   based   on    Plastic   Domain   Decomposition (PDD) method
\citep{Jeremic2007d,Jeremic2008a}.  Program was developed using a number of
publicly available numerical libraries. They are briefly described below.
%
Graph  partitioning  is  achieved  using ParMETIS libraries \citep{Karypis98c}).
%
Parts    of   OpenSees framework \citep{McKenna97}  were  used to
connect
the finite  element  domain.  In  particular, Finite Element Model Classes from
OpenSees (namely, class abstractions Node, Element, Constraint, Load, Domain and
set of Analysis classes)
where  used to describe finite element model and to store results of 
analysis
performed     on    a  model.    
%
An  excellent adoption  of  Actor  model
\citep{Hewitt73,Agha84}    and  addition  of  a  Shadow,  Chanel, MovableObject,
ObjectBroker,        MachineBroker        classes  within OpenSees framework
\citep{McKenna97} also provided an excellent basis for our development.
%
%
On      a   lower   level,   a   set   of   Template3Dep   numerical   libraries
\citep{Jeremic2000f}
were
used  for  constitutive  level  integrations, nDarray numerical libraries
\citep{Jeremic97d} were used to handle vector, matrix
and  tensor  manipulations, while FEMtools  element  libraries from UCD
CompGeoMech      toolset    \citep{Jeremic2004d}      were   used to supply other
necessary libraries and components.
%
%
%
Parallel  solution  of system of equations has been provided by PETSc set of
numerical libraries \citep{petsc-web-page, petsc-user-ref, petsc-efficient}).
%
Large part of  simulation was carried out on our local parallel computer GeoWulf.
Only the largest models (too big to fit on GeoWulf system) were simulated
on  TeraGrid  machine  at  SDSC and TACC. 
%
%It should be noted that the software
%part  of  the  simulation  platform  (executable  program)  is available through
%Author's web site.




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\section{Seismic Simulation Results}
\label{results}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%
%\section{Earthquake Simulations - 1994 Northridge}
%


Effects  of  varying soil conditions under prototype soil--bridge system are
main    focus    of  this study. Since one of concerns with 
prototype  bridge  structure  was  the  effect higher frequencies have on bridge
subcomponents (like short structural elements...), Northridge (1994) earthquake motions at
Century             City          were    chosen    for  this  study  (shown  in
Figure~\ref{fig:input}).  This  particular  ground  motion  contains frequencies
that were deemed potentially detrimental to parts of the structure.
Ground  motions were propagated in a vertical direction, and were polarized in a
plane  transversal  to  main  bridge  axes.  That  is,  incident motions are
perpendicular to main bridge axis, thus exciting mainly transversal motions of
bridge structure. Such transversal motions put highest demand on 
foundation--bridge    system,   particularly in non--uniform soils. It should be
noted    that    DRM    can    be    used  to  apply  any  3D seismic motions to
soil--foundation--bridge (SFB)  system,   however   for  this  study  1D,  vertically
propagating  motions were chosen for analysis. It is also important to note that
while 1D vertically propagating transversal motions were used as input for DRM,
SFB  system  has  been  subjected  to  a  full 3D motions, as difference in soil
conditions, difference  in  pier  length and difference in soil profile depth, create 
conditions for incoherence,
which  results  in  full 3D, incoherent bridge input motions 
(including longitudinal  and vertical components).

%
%\begin{landscape}
\begin{figure}[!htbp]
\begin{center}
%\includegraphics[width=0.95\textwidth]{/home/jeremic/tex/works/Thesis/GuanzhouJie/thesis/Verzija_Februar/Images/InputMotion_Northridge.eps}
\includegraphics[width=0.75\textwidth]{figs/InputMotion_Northridge.eps}
\includegraphics[width=0.73\textwidth]{figs/InputMotion_Northridge_Spectrum.ps}
\caption{\label{fig:input} Input Motion - Century City, Northridge Earthquake 1994}
\end{center}
\end{figure}
%\end{landscape}



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Simulation Scenarios}


In  order  to investigate effects variable soil  condition beneath bridge
bents have on seismic behavior a parametric study was performed. 
%
Base  soil  finite  element  model  (to which structural components are added in
loading   stage~2,   as   described  in  section  \ref{staged})  is  shown  in
Figure~\ref{fig:3bent}.    Boundary  conditions for first and second loading
stages (soil self weight and structure construction/self weight) are full support
at the bottom of a model, while vertical sides are allowed to slide down but
not  to  displace  perpendicular  to  the  plane.    For a dynamic loading stage, 
applied using DRM, and due to analytic nature of DRM, support
condition outside single  layer  of  elements  that is used to apply
effective      DRM     forces  will  not have much  effect on  behavior  of  the  model
\citep{Bielak2001,Yoshimura2001}.  Nevertheless,  minimal  amount  of  support is
needed    to    remove    rigid   body  modes for a model. In our case, for this
particular  stage of loading (seismic), same support conditions were used as for
the first two stages.
%This is indeed what was
%used,  that  is  lower  plane was partially supported while vertical planes were
%left
%
%
%\begin{landscape}
\begin{figure}[!htbp]
\begin{center}
%\includegraphics[width=0.8\textwidth]{/home/jeremic/tex/works/Thesis/GuanzhouJie/thesis/Verzija_Februar/Images/3BentModel.eps}
\includegraphics[width=0.7\textwidth]{figs/3BentModel.eps}
\caption{\label{fig:3bent} Finite Element Model for 3 Bent Prototype
Bridge System}
\end{center}
\end{figure}
%\end{landscape}
%
%

Soil    beneath    each of the bents was varied between stiff sand and soft clay
(models for which were described in section \ref{soilmodel})
%
There   were   total   of   8   soil   condition   scenarios,  as  described  in
Table~\ref{tab:scenarios}.
%
%  
\begin{table}[!htbp]
\begin{center}
\caption{\label{tab:scenarios}  Simulation  scenarios for prototype soil--bridge
system study}
\begin{tabular}{|c|c|c|c|}\hline
Simulation Cases &  Soil Block 1 & Soil Block 2 & Soil Block 3 \\\hline
Case 1 (SSS) & Stiff Sand & Stiff Sand &  Stiff Sand \\\hline
Case 2 (SSC) & Stiff Sand & Stiff Sand &  Soft Clay \\\hline
Case 3 (SCS) & Stiff Sand & Soft Clay &  Stiff Sand\\\hline
Case 4 (SCC) & Stiff Sand & Soft Clay  & Soft Clay  \\\hline
Case 5 (CSS) & Soft Clay  & Stiff Sand & Stiff Sand \\\hline
Case 6 (CSC) & Soft Clay  & Stiff Sand &  Soft Clay \\\hline
Case 7 (CCS) & Soft Clay  & Soft Clay  & Stiff Sand \\\hline
Case 8 (CCC) & Soft Clay  & Soft Clay &  Soft Clay \\\hline
\end{tabular}
\end{center}
\end{table}
%
It  is important to note that in each of eight cases, there was a stiff soil
layer  at  the  base  of  piles. This stiff layer was needed in order to provide
enough  carrying  capacity for cases where soft clay was used beneath bridge
bents. 


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\subsection{Input Motion}

%



%\clearpage

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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Displacement Response}


The  study  presented  here  produced  a  very  rich  dataset. While space restriction
prevents  us  from  showing  all  results,  a  subset of results is used to
emphasize    main    findings  related to effects of soil variability on seismic
response of the soil--bridge system.
%
In  order to illustrate main findings, results for top of the  first (left most) bent
and top of first soil block (next to pile/pier) are used. 
%
Figure~(\ref{fig:disp_B1})    shows    displacement  time  histories for that first
bent and soil block,  for all eight cases.
%
\begin{figure}[!htbp]
\begin{center}
%\includegraphics[width=0.85\textwidth]{/home/jeremic/tex/works/Thesis/GuanzhouJie/thesis/Verzija_Februar/Images/DispBent1.eps}
\includegraphics[width=0.99\textwidth]{figs/DispBent1.eps}
\\
%\includegraphics[width=0.85\textwidth]{/home/jeremic/tex/works/Thesis/GuanzhouJie/thesis/Verzija_Februar/Images/DispSoilBlock1.eps}
\includegraphics[width=0.99\textwidth]{figs/DispSoilBlock1.eps}
\caption{\label{fig:disp_B1}
Simulated Displacement Time Series, Northridge 1994, Century City input motions, 
Comparison   of  Eight  Cases  (upper:  structure,  top of bent \#~1; lower: top
of soil block, next to pier/pile \#~1).}
\end{center}
\end{figure}
%
In  addition  to that, Figure~(\ref{fig:disp_B1_spec}) shows displacement response
spectra for the same, first bent and soil block.
%
\begin{figure}[!htbp]
\begin{center}
%\includegraphics[width=0.85\textwidth]{/home/jeremic/tex/works/Thesis/GuanzhouJie/thesis/Verzija_Februar/Images/DispBent1_Spectrum.eps}
\includegraphics[width=0.99\textwidth]{figs/DispBent1_Spectrum.eps}
\\
%\includegraphics[width=0.85\textwidth]{/home/jeremic/tex/works/Thesis/GuanzhouJie/thesis/Verzija_Februar/Images/DispSoilBlock1_Spectrum.eps}
\includegraphics[width=0.99\textwidth]{figs/DispSoilBlock1_Spectrum.eps}
\caption{\label{fig:disp_B1_spec}
Simulated Displacement Response Spectra, Northridge 1994, Century City input motions,
Comparison of Eight Cases (upper:  structure,  top of bent \#~1; lower: top
of soil block, next to pier/pile \#~1).}
\end{center}
\end{figure}
%
A number of observations can be made. 
%

The  first observation is that displacement time histories for bent and
soil  response  are quite a bit different from each other. 
These  differences are present irrespective of what are local soil condition
beneath  bent~1. For example, Figure~\ref{fig:disp_B1}  shows that even for 
four  cases  when  soil  beneath   observed bent number~1 is stiff (all full
lines)  or  soft  (all dot--dashed lines),  response is very variable.
This  is  true  for  both displacement time histories of  top of bent and
for top of soil (near  pier/pile
connection).  Both   amplitude  (Figure~(\ref{fig:disp_B1}))       
and        frequency      (Figure~(\ref{fig:disp_B1_spec}))  
show great variability.  This  is  particularly  true  for 
structural    response,    while   soil response (lower set of time histories in
Figure~(\ref{fig:disp_B1})) is
somewhat   less   variable.   Smaller  variability  of   soil  is
understandable  as  dynamic  characteristic of soil at soil block~1 is very
much  dependent  on  its  own  stiffness,  while structure is significantly 
affected by soil  conditions at other soil block as well.


Another  important  observation  is  that  free field motions (that are often
used  as  input motions for structural only models) that are shown in lower part
of  Figures~(\ref{fig:disp_B1})  and~(\ref{fig:disp_B1_spec})  are  also quite a bit
different  than  what is actually recorded (simulated) at same location with
structure present. In particular, displacement time history of free field motions
(seen   as  gray  line  in  Figure~(\ref{fig:disp_B1}))  has  significantly  lower
amplitudes. In  some  cases  (for example Clay--Sand--Clay (CSC) case) 
free    field    amplitude  is  only  half    of  what  is  observed beneath the
structure. This
difference between free field motions and the ones observed (simulated) with SSI
effects    is    also    obvious    in    displacement   response   spectra   in
Figure~(\ref{fig:disp_B1_spec}).
%
The  difference  between free field motions and the ones observed (simulated) is
even  more  apparent for two cases with all stiff soil (Case 1, SSS) and all 
soft soil (Case 8, CCC). Such comparison is shown for the first 25
seconds of shaking in Figure~(\ref{fig:disp_scb1}).
%
\begin{figure}[!htbp]
\begin{center}
%\includegraphics[width=0.85\textwidth]{/home/jeremic/tex/works/Thesis/GuanzhouJie/thesis/Verzija_Februar/Images/DispSoilBlock1_15cm_25s_SC.eps}
\includegraphics[width=0.99\textwidth]{figs/DispSoilBlock1_15cm_25s_SC.eps}
\caption{\label{fig:disp_scb1}
Simulated Displacement Time Series, Northridge 1994, Century City input motions,
Comparison of Two Cases (SSS and CCC) - First 25s (Soil Block 1)}
\end{center}
\end{figure}
%
It  is  interesting  to note a very large discrepancy of free field motions with
those   that  were  simulated  in  soil  under  the  structure,  as  shown  in  
Figure~(\ref{fig:disp_scb1}).
%
This  discrepancy emphasizes the importance of soil--structure interaction (SSI)
on response of bridge and other infrastructure objects.
%
%
It     is     also     apparent     from     displacement    response    spectra
(Figure~\ref{fig:disp_B1_spec})  that  amplification  of  ground  motions (those
measured  with  SSI  effects) can be significant. In some cases (for example for
Sand--Clay--Clay  (SCC)  case),  amplification  at  period of 2.5 seconds is
almost  five. 
%This large amplification corresponds to time domain response at
%about        15   seconds  where, for that particular case (SCC) the structure is
%almost  in  resonance  with  base  motions, and only energy dissipation, through
%elastic--plastic response, prevents further amplification.


It  is  also  very  interesting to observe that response of structure in stiffer
soil  is much larger than response of structure founded on soft soil. This might
seem  to  contradict  usual assumption that structures founded on soft soils are
much  more  prone  to  large  deformation,  and  subsequently  more damage. This
particular  amplification of response for structure founded on stiff soil is 
due to  (positive,   amplifying)   interaction   of  dynamic
characteristics  of  earthquake, stiff soil
and    stiff  structure  (both  will have natural periods in somewhat higher
frequency range).
Positive  interaction  of  dynamic  characteristics of  earthquake, soil  and
structure  are  producing  larger amplifications and are thus detrimental to 
behavior of the structural system.
%
%
%
%
Similar observations can be made about responses of other bents and soil blocks.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Bent Bending Moments}

Time   domain  bending  moment history, recorded at   top  of  bent  number~1,
shown in Figure~(\ref{fig:accel_B12}) presents another set of interesting results.
%
It    is    important    to    note    that   bending   moments   presented   in
Figure~\ref{fig:accel_B12} are resulting for seismic shaking
stage of loading, which comes after previous structural self weight stage.
The effects of previous loading stage
is observed as initial
bending       moment of approximately $\pm 750$~kNm. 
%
\begin{figure}[!htbp]
\begin{center}
%\includegraphics[width=0.85\textwidth]{/home/jeremic/tex/works/Thesis/GuanzhouJie/thesis/Verzija_Februar/Images/MomentBent1Pile1.eps}
\includegraphics[width=0.99\textwidth]{figs/MomentBent1Pile1.eps}
\\
%\includegraphics[width=0.85\textwidth]{/home/jeremic/tex/works/Thesis/GuanzhouJie/thesis/Verzija_Februar/Images/MomentBent1Pile2.eps}
\includegraphics[width=0.99\textwidth]{figs/MomentBent1Pile2.eps}
\caption{\label{fig:accel_B12}
Simulated Maximum Moment Time Series, Northridge 1994, Century City input motions,
Comparison  of  Eight Cases
(Structure Bent 1, upper: column \#~1; lower: column \#~2).}
\end{center}
\end{figure}
%

A number of interesting observation can be made.
%
First  observation is that, similar to displacement results, bending moments are
quite  variable for different local soil conditions. 
%
This  variability  is  expected  as  support condition beneath bents change with
changes     in   local soil, however, what is striking is that the  magnitude of
variability is quite large.
For  example, moment differences at 10 seconds (time of highest moments for most
cases, see Figure~(\ref{fig:accel_B12})) between cases SCC (${\rm M}=3,750~{\rm kN/m^2}$) and CSS 
(${\rm M}=750~{\rm kN/m^2}$)
are on the order of 5 times.
%The variability of response
%is larger for four cases when soil beneath bent number~1 is either stiff sand
%(depicted by full lines), or soft clay (depicted by dot--dashed lines).

It  is also noted that in some, but not all, cases  top of  piers reach
yield   plateau. This is observed for a  number of response curves between $8$~s
and $17$~s marks. It is important to note that these yielding points are reached
first  by  structure that is founded on stiff
soil.   This   observation   accentuates   previous   observation  about  larger
displacements in those cases with stiff soil  underneath bridge.
There  are  two  main  reasons this attraction of large moments happens. First,
when  soil is stiffer beneath bridge bent number~1, that bent attracts more
forces,  since  bridge bent -- pile -- soil in foundation system is stiffer than
other  bents (all or some of them). This force attraction naturally results as 
stiff  components  of   bridge  system  will resist more seismic demand from
earthquake motions, and will thus produce stronger shaking and larger moments at
bent  number~1.  Secondly,  for  an  earthquake with fairly short period motions, 
combined with a
fairly  stiff  structure  and  a  stiff  soil  (sand),  bent  number~1  might be
experiencing    condition    close  to  resonance.  Occurrence  of  (or close to)
resonance amplifies response (motions and bending moments) significantly and 
contribute to observed
stronger    response.  The only reason resonance is not amplifying response even
more is that
displacement proportional damping (through elasto--plastic behavior of soil, pile
and  structure, cf. \citep{local-87}) is dissipating enough seismic (input)
energy thus damping out motions.


%
Figure~(\ref{fig:m_scB1})  is used to emphasize the effect of uniform sand (stiff)
and clay (soft) soil on bending moments in one of piers of bent number~1.
%
\begin{figure}[!htbp]
\begin{center}
%\includegraphics[width=0.85\textwidth]{/home/jeremic/tex/works/Thesis/GuanzhouJie/thesis/Verzija_Februar/Images/MomentBent1Pile1_25s_SC.eps}
\includegraphics[width=0.99\textwidth]{figs/MomentBent1Pile1_25s_SC.eps}
\caption{\label{fig:m_scB1}
Simulated moment time series, Northridge 1994, Century City input motions,
comparison  of  stiff  sand (SSS) and soft clay (CCC) cases -- First 25s (bridge
bent number~1).}
\end{center}
\end{figure}
%
%  
The  absolutely  largest  damage  (longest occurrence of  plastic bending moment) 
is observed for a case
where bridge is founded on stiff soil 
(SSS case).  For this case, interplay of dynamic characteristics of earthquake,
soil  and  structure  creates  close  to  resonance  condition,  thus amplifying
response.
%both of the above mentioned reasons for
%attracting  more  influences  to  this bent are present. That is, with other two
%bents founded in soft clay, bent number~1 is stiffer than both other bents, thus
%attracting forces. Moreover, dynamic interaction of stiff soil, stiff structure 
%and  short  period earthquake create condition for resonance, further amplifying
%response for bent number~1.
On  the other hand, when bridge is founded on soft clay, response is smaller and
in  fact,  only  partial yielding occurs at time $t=14$s. These results emphasize
the  importance of ESS interaction and can be used  as counterargument against
usually  made  claim  that  bridges  founded in soft soils are experiencing more
damage than those founded in stiff soils during seismic loading.





In   addition   to   dynamic   effects  discussed  above,  bending  moments  are
redistributed as a result of  partial  or  full  yielding  (plastic  hinge formation) 
during main shaking  event.  This  is  observed  toward  the end of bending moment time
history,  as bending moment  results tend to oscillate around zero moment instead around
their   respective  initial  values  ($\pm 750$~kNm).  
This  redistribution  of bending moments  becomes
important  as  it indicates change of structural system for bent and thus a bridge
structure. That is, at the beginning  of loading stage~3 (seismic shaking) 
bent behaves as monolithic, full moment bearing frame. After seismic shaking and
consequent yielding (formation of full  or partial plastic hinges) 
monolithic structural system has changed to
a couple of piers (consoles) with a simple  beam on top, representing cross beam. 
This change of   structural   system   might   significantly  affect   response  of  
bridge  system  during  next seismic event  or aftershock.  
In addition to that, dynamic characteristics of foundation soil during 
future seismic events (aftershocks or new earthquakes) might change as soil
might have become denser and thus stiffer.





%\clearpage



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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Summary and Conclusions}
\label{SUMMARY}


Presented  in this paper  was   simulation methodology and numerical investigation of seismic
soil--structure interaction (SSI) for bridge structure on variable soil.
Number of high fidelity models were developed and used to assess the influence
of SSI on response of a prototype soil--bridge system.
%
Detailed    description    was provided of high fidelity  modeling approach that
emphasizes low modeling uncertainty.
%
%
Number  of  interesting  conclusions  were made. First, SSI effects cannot be
neglected  and  should  be  modeled  and simulated as much as possible. 
%
This  observation becomes    even    more    important    as  it  seems  that  a
triad  of dynamic characteristic of earthquake, soil and structure (ESS) plays crucial
role in determining the seismic behavior of infrastructure objects.
%
%That is, the
%dynamic characteristics of all three
%components  (earthquake,  soil,  structure, ESS)  will  play  important role in final
%seismic performance of the system. 
% 
In    addition to that, results show that stiffer soil does not necessarily help
seismic behavior of the structure. This emphasizes above mentioned interplay
of ESS as the main mechanism that ultimately controls seismic performance of any
object.
%
It  is  also important to observe that nonlinear response of the soil--structure
system changes the dynamics characteristics of its components where soil might become
denser,  stiffer,  while the structure might become softer. This change, that is
happening  during  shaking,  might  significantly  affect  the  response  of 
soil--structure system for possible aftershocks and future seismic events.
%
On  a  final note, since tools (both software and hardware) are 
available  and  affordable,  it is our hope that professional practice will use
the  opportunity  and start using advanced, detailed models in assessing seismic
performance  of  infrastructure  objects  in  order  to make them safer and more
economical.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{samepage}
%
\section*{Acknowledgment}


The  work presented in this paper was supported by a grant from
%
the Civil    and  Mechanical  System  program,
Directorate  of  Engineering  of  the  National  Science Foundation, under Award
NSF--CMS--0337811 (cognizant program director Dr. Steve McCabe).
%
The Authors are grateful for this support.
%
The    Authors  would  also  like  to  thank Professor Fenves from University of
California at Berkeley for providing initial detailed
finite element model for  concrete bents and  bridge deck.
%
In  addition  to  that, Authors would like to thank anonymous reviewers for very
useful comments that have helped us improve our paper.



%
\end{samepage}


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