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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.) }
% - Give the names in the same order as the appear in the paper.
% - Use the \inst{?} command only if the authors have different
%   affiliation.

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\pgfuseimage{university-logo} \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.
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\date[GheoMat] % (optional, should be abbreviation of conference name)
{GheoMat \\
 {\small Masseria Salamina  \\
 Italy, June 2009} }
% - Either use conference name or its abbreviation.
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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
%
%   \item Need to convince practice (you) that these tools are useful,
%         robust, verified, validated, and ready to be used
%
%
% \end{itemize}
% \end{frame}






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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, elastic--plastic porous solid -- elastic pore fluid

\vspace*{0.5cm}
\item  Mechanics  of  infrustructure systems featuring coupled, elastic--plastic
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+(1-n)\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 
  $\dot{w}_{i,i}+\alpha       \dot{\varepsilon}_{ii}+{\dot{p}}/{Q}+\underline{n
{\dot{\rho_f}}/{\rho_f}+\dot{s}_0}=0$; 
${1}/{Q}\equiv {n}/{K_f}+({1-n})/{K_s}$

\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      pseudo--displacement   $w_i$   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}^{''}-(\alpha-n) p_{,i}+(1-n) \rho_s b_i-(1-n) \rho_s \ddot{u}_i +
n R_i=0
\label{34}
\end{eqnarray*}

\begin{eqnarray*}
-n p_{,i}+n \rho_f b_i-n \rho_f \ddot{U}_i
- n R_i=0
\label{35}
\end{eqnarray*}

\begin{eqnarray*}
-n \dot{U}_{i,i}=(\alpha-n) \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 $u-p-U$ 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 solid--fluid 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 (1-n) \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 (\alpha-n) 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 (\alpha-n) n_i p d\Gamma
%
 \\ & & 
%
+
\int_{\Omega} N_K^u (1-n) \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{Elastic--Plastic  
Material Model}

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


\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 elasto-plastic
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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%-- f=\left\vert \eta-\alpha\right\vert -m=0
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%-- 
%-- 
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%-- 
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%-- 
%-- \vspace*{-1cm}%
%-- \begin{align*}
%-- \dot{\varepsilon}_{q}^{p}  &  =\dfrac{\dot{\eta}}{h%
%-- %TCIMACRO{\TeXButton{blue}{\color{blue}}}%
%-- %BeginExpansion
%-- %\color{blue}%
%-- %EndExpansion
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%-- %\color{blue}%
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%-- %\color{red}%
%-- %EndExpansion
%-- d%
%-- %TCIMACRO{\TeXButton{black}{\color{black}}}%
%-- %BeginExpansion
%-- %\color{black}%
%-- %EndExpansion
%-- \left\vert \dot{\varepsilon}_{q}^{p}\right\vert \ \ \ \ \ \ \ \
%-- %TCIMACRO{\TeXButton{red}{\color{red}}}%
%-- %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}

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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*}%
% %TCIMACRO{\FRAME{ftbpF}{2.8383in}{2.188in}{0pt}{}{}{fabric_effect.pdf}%
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% %filename 'fabric_effect.pdf';file-properties "XNPEU";}}}%
% %BeginExpansion
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% [ptb]
% \begin{center}
% \includegraphics[
% trim=0.000000in 0.000000in 0.388321in 0.194938in,
% %height=2.188in,
% width=2.8383in
% ]%
% {/home/jeremic/tex/works/Conferences/2007/GeoDenver/Coupled/Present/Mahdi/fabric_effect.pdf}%
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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/Mesh-Isolation.pdf}
\end{center}
\end{figure}





\end{frame}

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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/variation-sigmaxz-sigmaz.jpg} \\
%\end{center}
\end{figure}
%\vspace*{-0.3cm}
%\hspace*{2.3cm} EPPR \hspace{1.5cm}  $\gamma$ \hspace{1.7cm}  $u_{hor}$




\end{frame}

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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/variation-sigmaxz-gamma.jpg} \\
%\end{center}
\end{figure}
%\vspace*{-0.3cm}
%\hspace*{2.3cm} EPPR \hspace{1.5cm}  $\gamma$ \hspace{1.7cm}  $u_{hor}$



\end{frame}

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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/variation-e-t.jpg} \\
%\end{center}
\end{figure}
%\vspace*{-0.3cm}
%\hspace*{2.3cm} EPPR \hspace{1.5cm}  $\gamma$ \hspace{1.7cm}  $u_{hor}$



\end{frame}

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\begin{frame}
 \frametitle{Acceleration Time History}


\vspace*{-0.1cm}
\begin{figure}[!htbp]
%\begin{center}
\includegraphics[width=6cm]{/home/jeremic/tex/works/Papers/2009/SeismicIsolationLiquefaction/time-history-acc.jpg} \\
%\end{center}
\end{figure}
%\vspace*{-0.3cm}
%\hspace*{2.3cm} EPPR \hspace{1.5cm}  $\gamma$ \hspace{1.7cm}  $u_{hor}$


\end{frame}

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% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %\begin{frame}
% % \frametitle{Excess Pore Pressure}
% %
% %
% %\vspace*{-0.3cm}
% %\begin{figure}[!htbp]
% %%\begin{center}
% %\includegraphics[width=10cm]{/home/jeremic/tex/works/Papers/2009/SeismicIsolationLiquefaction/contours-exc-pore-press.jpg} \\
% %%\end{center}
% %\end{figure}
% %%\vspace*{-0.3cm}
% %%\hspace*{2.3cm} EPPR \hspace{1.5cm}  $\gamma$ \hspace{1.7cm}  $u_{hor}$
% %
% %
% %
% %\end{frame}
% %
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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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/contours-exc-pore-press-ratio.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/Coupled-behavior_04/Niigata-Earthquake-tilted-buildings.jpg} \\
%\end{center}
\end{center}
%\vspace*{-0.3cm}
%\hspace*{2.3cm} EPPR \hspace{1.5cm}  $\gamma$ \hspace{1.7cm}  $u_{hor}$
               
\end{frame}

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% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\begin{frame}
 \frametitle{Elastic--Plastic 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}
%\vspace*{-0.3cm}
%\hspace*{2.3cm} EPPR \hspace{1.5cm}  $\gamma$ \hspace{1.7cm}  $u_{hor}$



               
\end{frame}

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% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\begin{frame}
 \frametitle{Elastic--Plastic Energy Dissipation: Layered Soil}

%\vspace*{-0.2cm}
\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}
%\vspace*{-0.3cm}
%\hspace*{2.3cm} EPPR \hspace{1.5cm}  $\gamma$ \hspace{1.7cm}  $u_{hor}$



               
\end{frame}

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% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\begin{frame}
 \frametitle{Kinetic Energy at the Top}


%\vspace*{-0.3cm}
\begin{figure}[h]
%\begin{center}
\hspace*{-0.5cm}
\includegraphics[width=13cm]{/home/jeremic/tex/works/Conferences/2009/CompDyn/Present/StackElements-Compare.pdf} 
\hspace*{-0.5cm}
%\end{center}
\end{figure}
%\vspace*{-0.3cm}
%\hspace*{2.3cm} EPPR \hspace{1.5cm}  $\gamma$ \hspace{1.7cm}  $u_{hor}$
               
\end{frame}

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\subsection{Piles in Liquefying Soils}

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\begin{frame}
 \frametitle{Bridge Pier--Pile 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}





\end{frame}

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\begin{frame}
 \frametitle{Bridge Pier--Pile 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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% \includegraphics[height=3.78cm]{/home/jeremic/tex/works/Papers/2008/Pile_in_liquefied_soil_upU/NewFiga/Snap_1_T005.pdf}
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% \includegraphics[height=3.78cm]{/home/jeremic/tex/works/Papers/2008/Pile_in_liquefied_soil_upU/NewFiga/Snap_1_T010.pdf}
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% \includegraphics[height=3.78cm]{/home/jeremic/tex/works/Papers/2008/Pile_in_liquefied_soil_upU/NewFiga/Snap_1_T015.pdf}
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% \\
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% \\
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% %\includegraphics[width=10cm,height=1cm]{/home/jeremic/tex/works/Papers/2008/Pile_in_liquefied_soil_upU/NewFiga/GMklot02.pdf}\hspace*{1cm}
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% \begin{center}
% \begin{tabular}{rlllllll} 
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%  \includegraphics[angle=-90,width=0.5\textwidth]{/home/jeremic/tex/works/Papers/2008/Pile_in_liquefied_soil_upU/NewFiga/Snap_scale.pdf}
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 \frametitle{Bridge Pier in Level Ground}





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%\begin{center}
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%\begin{center}
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\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}
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\vspace*{-0.9cm}
\begin{figure}[!htbp]
%\begin{center}
\includegraphics[width=5cm]{/home/jeremic/tex/works/Papers/2009/SeismicIsolationLiquefaction/SlopeLayersModel.jpg} \\
%\end{center}
\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/SlopeUniform-amax-02g.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/SlopeLayers-amax-02g.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}
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\vspace*{-0.1cm}
\begin{figure}[!htbp]
%\begin{center}
\includegraphics[width=7.3cm]{/home/jeremic/tex/works/Papers/2009/SeismicIsolationLiquefaction/SlopeLayers-amax-04g.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/SlopeLayers-disp-time-history.jpg} \\
%\end{center}
\end{figure}



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\section*{Summary}
\subsection*{Summary}

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 \frametitle{Summary}

\begin{itemize}


%\vspace*{0.5cm}
%     \item Importance of consistent formulation, material modeling and implementation

%\vspace*{0.5cm}
%     \item Verified, validate models and simulations tools used for prediction of behavior

%\vspace*{0.2cm}
    \item  High  fidelity  numerical  models  (verified  and  validated)      of
    Earthquake--Soil--(Structure) systems
    

%\vspace*{0.2cm}
%    \item High performance 
%    computational tools (software and 
%    hardware) developed and available

\vspace*{0.5cm}
    \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



%
% \vspace*{0.5cm}
%      \item Program and examples available in public domain (Author's web site)

%
%\item Models available (some now, some later) 

%\vspace*{0.5cm}
\end{itemize}



\end{frame}






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\end{document}


