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%\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}}
%
% \newpage
% \tableofcontents
% \newpage
%
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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.
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\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.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\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
%--
%-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\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.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%\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.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\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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\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.
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\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.
%
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