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\title[Fully Coupled, Two Phase Behavior of Geomaterials]
% (optional, use only with long paper titles)
{Fully Coupled, Two Phase Behavior of Geomaterials}
%\subtitle
%{{\tiny full set of slides available at:}\\
%%\verb{http://sokocalo.engr.ucdavis.edu/~jeremic/}
%}
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%\author[Boris Jeremi{\'c}, CompGeoMech \includegraphics[width=8cm]{/home/jeremic/BG/amblemi/ucdavis_logo_gold_lrg}] % (optional, use only with lots of authors)
\author[Jeremi{\'c}] % (optional, use only with lots of authors)
{Boris~Jeremi{\'c} \\ Mahdi Taiebat (UBC), Zhao Cheng (EarthMechanics Inc.) }
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\institute[Computational Geomechanics Group
\pgfuseimage{universitylogo} \hspace*{0.3truecm}] % (optional, but mostly needed)
{
% \texttt{http://geomechanics.ucdavis.edu} \\
Department of Civil and Environmental Engineering\\
University of California, Davis}
%  Use the \inst command only if there are several affiliations.
%  Keep it simple, no one is interested in your street address.
\date[GheoMat] % (optional, should be abbreviation of conference name)
{GheoMat \\
{\small Masseria Salamina \\
Italy, June 2009} }
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\frametitle{Outline}
\tableofcontents
% You might wish to add the option [pausesections]
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% Structuring a talk is a difficult task and the following structure
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% \begin{frame}
% \frametitle{Motivation}
%
% \begin{itemize}
%
% \item Designers needs the best available tools for performance assessment
%
% \item Steady progress of numerical simulations over last 4 decades
%
% \item Steady progress of computer hardware
%
% \item Use of software and hardware tools for detailed (as needed) performance assessment
%
% \item In other words: simulation tools are available
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% robust, verified, validated, and ready to be used
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\section{Before We Start}
\subsection*{Before We Start}
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\begin{frame}
\frametitle{Motivation}
\begin{itemize}
\vspace*{0.5cm}
\item There is no limit to what problems one can address (can numerically
simulate)
\vspace*{0.5cm}
\item Mechanics of coupled, elasticplastic porous solid  elastic pore fluid
\vspace*{0.5cm}
\item Mechanics of infrustructure systems featuring coupled, elasticplastic
porous solid  elastic pore fluid
\vspace*{0.5cm}
\item Accurate modeling and simulation for infrustructure system design (safety
and economy)
%\vspace*{0.5cm}
% e\item
%\vspace*{2.5cm}
%\item How do we use experimental simulations to improve models
\end{itemize}
\end{frame}
% \subsection*{Fundamentals}
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\section{Modeling}
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\subsection{Formulation}
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\begin{frame}
\frametitle{Dynamic Equilibrium for Coupled Systems}
\begin{itemize}
\item Effective stress principle
$\sigma^{\prime}_{ij} = \sigma_{ij} + \alpha \delta_{ij} p$ ; ($p=1/3 \sigma_{kk}$)
\vspace*{0.3cm}
\item Equilibrium of the mixture
$ \sigma_{ij,j}\rho \ddot{u}_i\rho_f[\ddot{w}_i+
\underline{\dot{w}_j\dot{w}_{i,j}}]+\rho b_i=0$ ; ($
\rho=n\rho_f+(1n)\rho_s$)
\vspace*{0.3cm}
\item Equilibrium of the fluid
$p_{,i} R_i  \rho_f \ddot{u}_i\rho_f[\ddot{w}_i+
\underline{\dot{w}_j \dot{w}_{i,j}}]/n+\rho_f b_i=0$;
(Darcy:
$n\dot{w}_j = Ki$; $i=h_{,j}$;
$R_i=k_{ij}^{1} \dot{w}_j$;
$k_{ij}=K_{ij}/\rho_f g$ $[m]^3[s]/[kg]$)
\vspace*{0.3cm}
\item Flow conservation
$\dot{w}_{i,i}+\alpha \dot{\varepsilon}_{ii}+{\dot{p}}/{Q}+\underline{n
{\dot{\rho_f}}/{\rho_f}+\dot{s}_0}=0$;
${1}/{Q}\equiv {n}/{K_f}+({1n})/{K_s}$
\end{itemize}
%\vspace*{2.0cm}
\end{frame}
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\begin{frame}
\frametitle{Dynamic Equilibrium for Coupled Systems (cont.)}
After neglecting convective accelerations, density variations and assuming
isothermal process (no volume expansion):
\begin{itemize}
\item Equilibrium of the mixture \\
$ \sigma_{ij,j}\rho \ddot{u}_i\rho_f \ddot{w}_i+\rho b_i=0$
\vspace*{0.3cm}
\item Equilibrium of the fluid \\
$p_{,i} R_i  \rho_f \ddot{u}_i\rho_f\ddot{w}_i/n+\rho_f b_i=0$
\vspace*{0.3cm}
\item Flow conservation \\
$\dot{w}_{i,i}+\alpha \dot{\varepsilon}_{ii}+{\dot{p}}/{Q}=0$
\end{itemize}
%\vspace*{2.0cm}
\end{frame}
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\begin{frame}
\frametitle{Dynamic Equilibrium for Coupled Systems (cont.)}
Replace relative pseudodisplacement $w_i$ with real displacement
$U_i=u_i+U_i^R=u_i+{w_i}/{n}$
\begin{figure}[!hbpt]
\begin{center}
\includegraphics[width=0.7\textwidth]{/home/jeremic/tex/works/LectureNotes/Figures/Darcy_vs_Real_01.pdf}
\end{center}
\end{figure}
%\vspace*{2.0cm}
\end{frame}
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\begin{frame}
\frametitle{Dynamic Equilibrium for Coupled Systems (cont.)}
After some manipulations we obtain
\begin{eqnarray*}
\sigma_{ij,j}^{''}(\alphan) p_{,i}+(1n) \rho_s b_i(1n) \rho_s \ddot{u}_i +
n R_i=0
\label{34}
\end{eqnarray*}
\begin{eqnarray*}
n p_{,i}+n \rho_f b_in \rho_f \ddot{U}_i
 n R_i=0
\label{35}
\end{eqnarray*}
\begin{eqnarray*}
n \dot{U}_{i,i}=(\alphan) \dot{\varepsilon}_{ii}+ \dot{p}/Q
\label{36}
\end{eqnarray*}
%\vspace*{2.0cm}
\end{frame}
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%\subsection{Fully Coupled Formulation}
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\begin{frame}
\frametitle{Fully Coupled $upU$ Formulation}
\begin{itemize}
\item Formulation: fully coupled by Zienkiewicz and Shiomi 1984),
nonlinear dynamics by Argyris and Mlejnek (1991)
\vspace*{0.3cm}
\item Physical, velocity proportional damping from solidfluid interaction
(not using Rayleigh damping)
\vspace*{0.3cm}
\item Accelerations of pore fluid not neglected
\begin{itemize}
\item important for SFSI
\item inertial forces of fluid allow liquefaction modeling
\end{itemize}
\vspace*{0.3cm}
\item Stable formulation for near incompressible pore fluid
% \item Formulation and implementation verified on a number of available closed
% form solutions
\end{itemize}
%\vspace*{2.0cm}
\end{frame}
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\begin{frame}
\frametitle{Finite Element Discretization}
\begin{small}
\begin{eqnarray*}
& &\left[ \begin{array}{ccc}
(M_s)_{KijL} & 0 & 0 \\
0 & 0 & 0 \\
0 & 0 & (M_f)_{KijL}
\end{array} \right]
\left[ \begin{array}{c}
\ddot{\overline{u}}_{Lj} \\
\ddot{\overline{p}}_N \\
\ddot{\overline{U}}_{Lj}
\end{array} \right]
+
\nonumber\\
+
& &\left[ \begin{array}{ccc}
(C_1)_{KijL} & 0 & (C_2)_{KijL} \\
0 & 0 & 0 \\
(C_2)_{LjiK} & 0 & (C_3)_{KijL} \\
\end{array} \right]
\left[ \begin{array}{c}
\dot{\overline{u}}_{Lj} \\
\dot{\overline{p}}_N \\
\dot{\overline{U}}_{Lj}
\end{array} \right]
+
\nonumber\\
+
& &\left[ \begin{array}{ccc}
(K^{EP})_{KijL} & (G_1)_{KiM} & 0 \\
(G_1)_{LjM} & P_{MN} & (G_2)_{LjM} \\
0 & (G_2)_{KiL} & 0
\end{array} \right]
\left[ \begin{array}{c}
\overline{u}_{Lj} \\
\overline{p}_M \\
\overline{U}_{Lj}
\end{array} \right]
=
\left[ \begin{array}{c}
\overline{f}_{Ki}^{solid} \\
0 \\
\overline{f}_{Ki}^{fluid}
\end{array} \right] \nonumber\\
\label{68}
\end{eqnarray*}
\end{small}
\end{frame}
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\begin{frame}
\frametitle{Finite Element Discretization}
%
\begin{small}
\begin{eqnarray*}
(M_s)_{KijL} =\int_{\Omega} N_K^u (1n) \rho_s \delta_{ij} N_L^u d\Omega
\;\; &\mbox{;}& \;\;
(M_f)_{KijL} =\int_{\Omega} N_K^U n \rho_f \delta_{ij} N_L^U d\Omega
%%%%%%%%
\\
%%%%%%%%
(C_1)_{KijL} =\int_{\Omega} N_K^u n^2 k_{ij}^{1} N_L^u d\Omega
\;\; &\mbox{;}& \;\;
(C_2)_{KijL} =\int_{\Omega} N_K^u n^2 k_{ij}^{1} N_L^U d\Omega
%%%%%%%%
\\
%%%%%%%%
(C_3)_{KijL} =\int_{\Omega} N_K^U n^2 k_{ij}^{1} N_L^U d\Omega
\;\; &\mbox{;}& \;\;
(K^{EP})_{KijL}=\int_{\Omega} N_{K,m}^u D_{imjn} N_{L,n}^u d\Omega
%%%%%%%%
\\
%%%%%%%%
(G_1)_{KiM} =\int_{\Omega} N_{K,i}^u (\alphan) N_M^p d\Omega
\;\; &\mbox{;}& \;\;
(G_2)_{KiM} =\int_{\Omega} n N_{K,i}^U N_M^p d\Omega
%%%%%%%%
\\
%%%%%%%%
P_{NM} =\int_{\Omega} N_N^p \frac{1}{Q} N_M^p d\Omega
& &
\end{eqnarray*}
\end{small}
%
%
%
%
%\newpage
\end{frame}
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\begin{frame}
\frametitle{Finite Element Discretization}
%
%
%\newpage
\begin{eqnarray*}
\overline{f}_{Ki}^{solid}
&=&
\int_{\Gamma_t} N_K^u n_j \sigma_{ij}^{''} d\Gamma

%
\\ & &
%
\int_{\Gamma_p} N_K^u (\alphan) n_i p d\Gamma
%
\\ & &
%
+
\int_{\Omega} N_K^u (1n) \rho_s b_i d\Omega
%
\nonumber\\
~
\nonumber\\
%
\overline{f}_{Ki}^{fluid}
&=&

\int_{\Gamma_p} n N_K^U n_i p d\Gamma
%
\\& &
%
+
\int_{\Omega} n N_K^U \rho_f b_i d\Omega
\end{eqnarray*}
%
\end{frame}
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\subsection{ElasticPlastic
Material Model}
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\begin{frame}
\frametitle{Dafalias Manzari Material Model}
\begin{itemize}
%\item Constitutive model is the necessary component of any rational numerical analysis.
\item Dafalias \& Manzari (2004): critical state compatible elastoplastic
constitutive model for sands.
\vspace*{0.3cm}
\item Systematic and relatively simple calibration process.
\vspace*{0.3cm}
\item Capable of simulating different feature of sand response such as
\begin{itemize}
\item hardening
\item softening
\item consolidation
\item dilation
\end{itemize}
\vspace*{0.3cm}
\item Single set of parameters for all stages of loading (self weight, cycling...)
\end{itemize}
%\vspace*{2.0cm}
\end{frame}
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%
%
% \begin{frame}
% \frametitle{Dafalias Manzari Material Model}
%
% Model Formulation in Triaxial Space%
% \[
% \left\{
% \begin{array}
% [c]{l}%
% \dot{\varepsilon}_{q}^{{}}=\dot{\varepsilon}_{q}^{e}+\dot{\varepsilon}_{q}%
% ^{p}\\
% \dot{\varepsilon}_{v}^{{}}=\dot{\varepsilon}_{v}^{e}+\dot{\varepsilon}_{v}^{p}%
% \end{array}
% \right. \ \ \ ;\ \ \ \eta=\dfrac{q}{p}\ \ \ \ ;\ \ \ \
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% \begin{align*}
% \dot{\varepsilon}_{q}^{p} & =\dfrac{\dot{\eta}}{h%
% %TCIMACRO{\TeXButton{blue}{\color{blue}}}%
% %BeginExpansion
% %\color{blue}%
% %EndExpansion
% b}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
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% %EndExpansion
% d%
% %TCIMACRO{\TeXButton{black}{\color{black}}}%
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% \left\vert \dot{\varepsilon}_{q}^{p}\right\vert \ \ \ \ \ \ \ \
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% %BeginExpansion
% %\color{red}%
% %EndExpansion
% d=(M_{{}}^{d}\eta)
% \end{align*}
%
%
% %\vspace*{2.0cm}
%
% \end{frame}
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\begin{frame}
\frametitle{Multiaxial Representation}
\vspace*{2.5cm}
%TCIMACRO{\TeXButton{B}{\begin{figure}[htbp] \centering}}%
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\end{frame}
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%
%
% \begin{frame}
% \frametitle{Dafalias Manzari Material Model}
%
% Fabric Dilatancy Tensor%
% \begin{align*}
% \dot{\varepsilon}_{v}^{p} & =A_{d}d\left\vert \dot{\varepsilon}_{q}%
% ^{p}\right\vert \\
% A_{d} & =A_{0}\left( 1+\langle\mathbf{z}\mathbf{:}\mathbf{n}\rangle\right)
% \text{ \ \ \ ; \ \ \ }\mathbf{\dot{z}}=c_{z}\langle\dot{\varepsilon}_{v}%
% ^{p}\rangle\left( z_{max}\mathbf{n}+\mathbf{z}\right)
% \end{align*}%
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\section{Examples}
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\subsection{Seismic Isolation by Liquefaction}
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\begin{frame}
\frametitle{Model}
\vspace*{0.2cm}
\begin{figure}[!htbp]
\begin{center}
\includegraphics[width=6cm,angle=90]{/home/jeremic/tex/works/Papers/2009/SeismicIsolationLiquefaction/MeshIsolation.pdf}
\end{center}
\end{figure}
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\begin{frame}
\frametitle{Stress Variation}
\vspace*{0.1cm}
\begin{figure}[!htbp]
%\begin{center}
\includegraphics[width=6cm]{/home/jeremic/tex/works/Papers/2009/SeismicIsolationLiquefaction/variationsigmaxzsigmaz.jpg} \\
%\end{center}
\end{figure}
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%\hspace*{2.3cm} EPPR \hspace{1.5cm} $\gamma$ \hspace{1.7cm} $u_{hor}$
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\begin{frame}
\frametitle{Stress Strain Response}
\vspace*{0.1cm}
\begin{figure}[!htbp]
%\begin{center}
\includegraphics[width=6cm]{/home/jeremic/tex/works/Papers/2009/SeismicIsolationLiquefaction/variationsigmaxzgamma.jpg} \\
%\end{center}
\end{figure}
%\vspace*{0.3cm}
%\hspace*{2.3cm} EPPR \hspace{1.5cm} $\gamma$ \hspace{1.7cm} $u_{hor}$
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\begin{frame}
\frametitle{Void Ratio Variation}
\vspace*{0.1cm}
\begin{figure}[!htbp]
%\begin{center}
\includegraphics[width=6cm]{/home/jeremic/tex/works/Papers/2009/SeismicIsolationLiquefaction/variationet.jpg} \\
%\end{center}
\end{figure}
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%\hspace*{2.3cm} EPPR \hspace{1.5cm} $\gamma$ \hspace{1.7cm} $u_{hor}$
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\begin{frame}
\frametitle{Acceleration Time History}
\vspace*{0.1cm}
\begin{figure}[!htbp]
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\includegraphics[width=6cm]{/home/jeremic/tex/works/Papers/2009/SeismicIsolationLiquefaction/timehistoryacc.jpg} \\
%\end{center}
\end{figure}
%\vspace*{0.3cm}
%\hspace*{2.3cm} EPPR \hspace{1.5cm} $\gamma$ \hspace{1.7cm} $u_{hor}$
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% % \frametitle{Excess Pore Pressure}
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% %\includegraphics[width=10cm]{/home/jeremic/tex/works/Papers/2009/SeismicIsolationLiquefaction/contoursexcporepress.jpg} \\
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\begin{frame}
\frametitle{Excess Pore Pressure Ratio}
\vspace*{0.2cm}
\begin{figure}[!htbp]
%\begin{center}
\includegraphics[width=10cm]{/home/jeremic/tex/works/Papers/2009/SeismicIsolationLiquefaction/contoursexcporepressratio.jpg} \\
%\end{center}
\end{figure}
%\vspace*{0.3cm}
%\hspace*{2.3cm} EPPR \hspace{1.5cm} $\gamma$ \hspace{1.7cm} $u_{hor}$
\vspace*{0.3cm}
\begin{center}
%\begin{center}
\includegraphics[width=3cm]{/home/jeremic/tex/works/Conferences/2009/GheoMat/Coupledbehavior_04/NiigataEarthquaketiltedbuildings.jpg} \\
%\end{center}
\end{center}
%\vspace*{0.3cm}
%\hspace*{2.3cm} EPPR \hspace{1.5cm} $\gamma$ \hspace{1.7cm} $u_{hor}$
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\begin{frame}
\frametitle{ElasticPlastic Energy Dissipation: Uniform Soil}
%\vspace*{0.2cm}
\begin{figure}[!htbp]
%\begin{center}
\hspace*{1.2cm}
\includegraphics[width=8cm]{/home/jeremic/tex/works/Conferences/2009/CompDyn/Present/Uniform_Percentage.pdf}
\hspace*{1.2cm}
%\end{center}
\end{figure}
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\begin{frame}
\frametitle{ElasticPlastic Energy Dissipation: Layered Soil}
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\begin{figure}[!htbp]
%\begin{center}
\hspace*{1.2cm}
\includegraphics[width=8cm]{/home/jeremic/tex/works/Conferences/2009/CompDyn/Present/Layered_Percentage.pdf}
\hspace*{1.2cm}
%\end{center}
\end{figure}
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\begin{frame}
\frametitle{Kinetic Energy at the Top}
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\begin{figure}[h]
%\begin{center}
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\includegraphics[width=13cm]{/home/jeremic/tex/works/Conferences/2009/CompDyn/Present/StackElementsCompare.pdf}
\hspace*{0.5cm}
%\end{center}
\end{figure}
%\vspace*{0.3cm}
%\hspace*{2.3cm} EPPR \hspace{1.5cm} $\gamma$ \hspace{1.7cm} $u_{hor}$
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\subsection{Piles in Liquefying Soils}
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\begin{frame}
\frametitle{Bridge PierPile Model}
\vspace*{0.2cm}
\begin{figure}[!htbp]
\begin{center}
\includegraphics[width=6cm]{/home/jeremic/tex/works/Conferences/2007/PEERAnnualMeeting/Liquefaction/PileBridgeModel01.jpg} \\
\includegraphics[width=7cm]{/home/jeremic/tex/works/Conferences/2007/PEERAnnualMeeting/Liquefaction/PileBridgeModel02.jpg}
\includegraphics[width=4cm]{/home/jeremic/tex/works/Papers/2008/Pile_in_liquefied_soil_upU/NewFiga/GMklot02.pdf}\end{center}
\end{figure}
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\begin{frame}
\frametitle{Bridge PierPile Staged Construction}
\begin{itemize}
\item Soil self weight (no pile)
\item Excavations for pile
\item Pile installation
\begin{itemize}
\item impermeable filler material,
\item connecting solids and structure,
\end{itemize}
\item Pile self weight,
\item Construction of pier structure and self weight
\item Seismic shaking
\item Excess pore pressure dissipation
\end{itemize}
\end{frame}
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\begin{frame}
\frametitle{Bridge Pier in Level Ground}
\vspace*{0.3cm}
\begin{figure}[!htbp]
%\begin{center}
\hspace*{1.0cm}
\includegraphics[width=13cm]{/home/jeremic/tex/works/Conferences/2007/PEERAnnualMeeting/Liquefaction/PileBridgeLevelGroundEPP.jpg} \\
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\end{figure}
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\begin{frame}
\frametitle{Bridge Pier in Sloping Ground}
\vspace*{0.3cm}
\begin{figure}[!htbp]
%\begin{center}
\hspace*{1.0cm}
\includegraphics[width=13cm]{/home/jeremic/tex/works/Conferences/2007/PEERAnnualMeeting/Liquefaction/PileBridgeSlopingGroundEPP.jpg} \\
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\subsection{Seismic Shearing of a Mild Slope with Liquefaction}
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\begin{frame}
\frametitle{Slope Models}
\vspace*{0.9cm}
\begin{figure}[!htbp]
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\includegraphics[width=5cm]{/home/jeremic/tex/works/Papers/2009/SeismicIsolationLiquefaction/SlopeLayersModel.jpg} \\
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\end{figure}
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\begin{frame}
\frametitle{Uniform Slope with $a_{max}=0.2$g}
\vspace*{0.1cm}
\begin{figure}[!htbp]
%\begin{center}
\includegraphics[width=7.5cm]{/home/jeremic/tex/works/Papers/2009/SeismicIsolationLiquefaction/SlopeUniformamax02g.jpg} \\
%\end{center}
\end{figure}
\vspace*{0.5cm}
\hspace*{2.3cm} EPPR \hspace{1.5cm} $\gamma$ \hspace{1.7cm} $u_{hor}$
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\begin{frame}
\frametitle{Layered Slope with $a_{max}=0.2$g}
\vspace*{0.1cm}
\begin{figure}[!htbp]
%\begin{center}
\includegraphics[width=7.1cm]{/home/jeremic/tex/works/Papers/2009/SeismicIsolationLiquefaction/SlopeLayersamax02g.jpg} \\
%\end{center}
\end{figure}
\vspace*{0.5cm}
\hspace*{2.3cm} EPPR \hspace{1.5cm} $\gamma$ \hspace{1.7cm} $u_{hor}$
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\begin{frame}
\frametitle{Layered Slope with $a_{max}=0.4$g}
\vspace*{0.1cm}
\begin{figure}[!htbp]
%\begin{center}
\includegraphics[width=7.3cm]{/home/jeremic/tex/works/Papers/2009/SeismicIsolationLiquefaction/SlopeLayersamax04g.jpg} \\
%\end{center}
\end{figure}
\vspace*{0.5cm}
\hspace*{2.3cm} EPPR \hspace{1.5cm} $\gamma$ \hspace{1.7cm} $u_{hor}$
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\begin{frame}
\frametitle{Surface Displacements}
\vspace*{0.3cm}
\begin{figure}[!htbp]
%\begin{center}
\includegraphics[width=10.5cm]{/home/jeremic/tex/works/Papers/2009/SeismicIsolationLiquefaction/SlopeLayersdisptimehistory.jpg} \\
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\end{figure}
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\section*{Summary}
\subsection*{Summary}
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\frametitle{Summary}
\begin{itemize}
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% \item Importance of consistent formulation, material modeling and implementation
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% \item Verified, validate models and simulations tools used for prediction of behavior
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\item High fidelity numerical models (verified and validated) of
EarthquakeSoil(Structure) systems
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% hardware) developed and available
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\item Space and time distribution of the matching triad:
\underline{Earthquake},
\underline{Soil} and
\underline{Structure}
({\bf ESS}) and its interaction determines
possible benefits or detriments
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% \item Program and examples available in public domain (Author's web site)
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%\item Models available (some now, some later)
%\vspace*{0.5cm}
\end{itemize}
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\end{document}