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\title[EarthquakeSoilStructure Systems]
% (optional, use only with long paper titles)
{EarthquakeSoilStructure Systems}
%\subtitle
%{{\tiny full set of slides available at:}\\
%%\verb{http://sokocalo.engr.ucdavis.edu/~jeremic/}
%}
\pgfdeclareimage[height=0.2cm]{universitylogo}{/home/jeremic/BG/amblemi/ucdavis_logo_blue_sm}
%\author[Boris Jeremi{\'c}, CompGeoMech \includegraphics[width=8cm]{/home/jeremic/BG/amblemi/ucdavis_logo_gold_lrg}] % (optional, use only with lots of authors)
\author[Boris Jeremi{\'c}] % (optional, use only with lots of authors)
{Boris~Jeremi{\'c} and
Sashi~Kunnath \\ with\\
Kallol~Sett and
Nima~Tafazzoli}
%  Give the names in the same order as the appear in the paper.
%  Use the \inst{?} command only if the authors have different
% affiliation.
%\institute[Computational Geomechanics Group
\institute[Structural/Geotechnical Engineering
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{ \vspace*{1cm} \\
Department of Civil and Environmental Engineering\\
University of California, Davis}
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\date[] % (optional, should be abbreviation of conference name)
{ }
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\titlepage
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\begin{frame}
\frametitle{Outline}
\tableofcontents
% You might wish to add the option [pausesections]
\end{frame}
% Structuring a talk is a difficult task and the following structure
% may not be suitable. Here are some rules that apply for this
% solution:
%  Exactly two or three sections (other than the summary).
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% you think necessary.
%  If you omit details that are vital to the proof/implementation,
% just say so once. Everybody will be happy with that.
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\section{Goals}
\subsection{}
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\begin{frame}
\frametitle{Focus}
\begin{itemize}
\item Steady progress in software and hardware allows
high fidelity, detailed performance assessment (simulations) of
critical (and other) infrastructure systems (bridges, dams, buildings,
ports...)
\item Interplay of Earthquake Soil Structure systems seems to play
a major role in catastrophic failures (and successes)
\item Quantify uncertainty and variablity in soil and structural behavior
\item Provide methodology (formulation, implementation) for probabilistic
performance based engineering (PEER type)
\item Overcome traditional performance assessment approaches used in
engineering practice (design using prescriptive code!)
\end{itemize}
\end{frame}
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\begin{frame}
\frametitle{Historical Note (Full Circle?)}
\begin{itemize}
\item SoilStructure Interaction phenomena first realized and described by
Professor Kyoji Suyehiro
\begin{itemize}
\item Ship engineer (Professor of Naval Arch. at U. of Tokyo),
\item Earthquake engineer (First Director of the Earthquake Research
Institute at U. of Tokyo),
\item Was in Tokyo during Great Kant{\= o} earthquake
(11:58am (12:08pm! slow?), 1st. Sept. 1923)
\item Saw earthquake surface waves travel and buildings sway
(ships in the ocean)
\item Presented his new SSI work in the USA (Caltech,
UCB, Stanford, MIT) in 1931...
\end{itemize}
\item Slow and Fast earthquakes
\item Uncertainty and variability (source, material...)
\end{itemize}
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%%% the intended uses of the model. Physics issue. {\it Validation provides
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%%% \item Prediction: use of computational model to foretell the state of a
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%%%%\item Models available (some now, some later)
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\section{ESS Systems}
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\subsection{High Fidelity, 3D Models}
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\begin{frame}
\frametitle{Detailed 3D, FEM model}
\begin{itemize}
\item Construction process
\item Two types of soil: stiff soil (UT, UCD), soft soil (Bay Mud)
\item Deconvolution of given surface ground motions
\item Use of the DRM (Prof. Bielak et al.) for seismic input
\item Piles $\rightarrow$ beamcolumn elements in soil holes
\item No artificial damping (only mat. dissipation, radiation)
\item Structural model: collaboration UCD, UCB and UW
\item Element size issues (filtering of frequencies)
\vspace*{0.3cm}
\begin{table}[!htbp]
%\caption{ }
\begin{center}
\begin{tabular}{ccccr}
\hline
model size (el) & el. size & $f_{cutoff}$ & min. $G/Gmax$ & $\gamma$ \\
\hline
%
%
% 100cm element model, f_cuttof=10HZ, G/Gmax=1.0, 12K elements, Gmax model
12K & 1.0~m & 10~Hz & 1.0 & <0.5~\% \\
%
%
%~90cm element model, f_cutoff ~= 3Hz, G/Gmax~=0.08, for epsilon ~= 1%, 15K
%elements, now running...
15K & 0.9~m & >3~Hz & 0.08 & 1.0~\% \\
%
%
%27.6cm element model, f_cutoff= 10HZ, G/Gmax=0.08, for epsilon = 1%, 150K
%elements, finest model
150K & 0.3~m & 10~Hz & 0.08 & 1.0~\% \\
500K & 0.15~m & 10~Hz & 0.02 & 5.0~\% \\
% 27.6cm element model, f_cutoff= 10HZ, G/Gmax=0.08, for epsilon = 1%, 150K
%elements, finest model
\hline
\end{tabular}
\end{center}
\end{table}
\end{itemize}
\end{frame}
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\begin{frame}
\frametitle{FEM Mesh (one of)}
%
\begin{figure}[!htbp]
\begin{center}
\includegraphics[width=7cm]{/home/jeremic/tex/works/Reports/2006/NEESDemoProject/PrototypeMesh.jpg}
%\caption{\label{BridgeSFSI01} FEM model for seismic response of a three bend
%bridge.}
\end{center}
\end{figure}
%
%\vspace*{0.5cm}
{%
\begin{beamercolorbox}{section in head/foot}
\usebeamerfont{framesubtitle}\tiny{B. Jeremi\'{c} and G. Jie.} "Parallel
SoilFoundationStructure Computations",
Chapter in Book: \textit{ Progress in Computational Dynamics and
Earthquake Engineering}; {Taylor and Francis Publishers}, 2008.
%\vskip2pt\insertnavigation{\paperwidth}\vskip2pt
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\begin{frame}
\frametitle{Parallel Computer GeoWulf}
\begin{itemize}
%\vspace*{0.2cm}
\item Distributed memory \\
parallel computer
%\vspace*{0.2cm}
\item Multiple generation \\
compute nodes \\
and networks
%\vspace*{0.2cm}
\item Very cost effective!
%\vspace*{0.2cm}
\item Same architecture as \\
large parallel supercomputers \\
(SDSC, TACC, EarthSimulator...)
%\vspace*{0.2cm}
\item Local design, construction, \\
available at all times!
%\vspace*{0.2cm}
% \item
%%\vspace*{0.2cm}
\end{itemize}
%
\vspace*{5.6cm}
\begin{figure}[!htbp]
\hspace*{6cm}
%\begin{center}
\includegraphics[width=4truecm]{/home/jeremic/public_html/GeoWulf/Dec2006/IMG_0907.jpg}
\\
\hspace*{6cm}
\includegraphics[width=4truecm]{/home/jeremic/public_html/NSFNuggets/Students_develop_parallel_computer/StudentsConstructingGeoWulf.jpg}
%\end{center}
\end{figure}
\end{frame}
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\subsection{Behavior for Short and Long Period Motions}
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\begin{frame}
\frametitle{Northridge and Kocaeli Input Motions}
\begin{figure}[!htbp]
\begin{center}
\hspace*{1cm}
\includegraphics[width=6.0truecm]{/home/jeremic/tex/works/Thesis/GuanzhouJie/thesis/Verzija_Februar/Images/InputMotion_Northridge.pdf}
\includegraphics[width=6.0truecm]{/home/jeremic/tex/works/Thesis/GuanzhouJie/thesis/Verzija_Februar/Images/LongMotion/InputMotion_Kocaeli.pdf}
\hspace*{1cm}
\\
\hspace*{1cm}
\includegraphics[width=6.0truecm]{/home/jeremic/tex/works/Thesis/GuanzhouJie/thesis/Verzija_Februar/Images/InputMotion_Northridge_Spectrum.pdf}
\includegraphics[width=6.0truecm]{/home/jeremic/tex/works/Thesis/GuanzhouJie/thesis/Verzija_Februar/Images/LongMotion/InputMotion_Kocaeli_Spectrum.pdf}
\hspace*{1cm}
%\caption{\label{BridgeSFSI01} FEM model for seismic response of a three bend
%bridge.}
\end{center}
\end{figure}
%
\end{frame}
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\begin{frame}
\frametitle{Parametric Simulation Results}
\vspace*{0.5cm}
%\begin{landscape}
\begin{figure}[!htbp]
\begin{center}
\hspace*{1cm}
\includegraphics[width=1.8cm]{/home/jeremic/tex/works/Thesis/GuanzhouJie/thesis/Verzija_Februar/Images/DispSoilBlock1.pdf}
\includegraphics[width=1.8cm]{/home/jeremic/tex/works/Thesis/GuanzhouJie/thesis/Verzija_Februar/Images/DispBent1.pdf}
\includegraphics[width=1.8cm]{/home/jeremic/tex/works/Thesis/GuanzhouJie/thesis/Verzija_Februar/Images/DispSoilBlock2.pdf}
\includegraphics[width=1.8cm]{/home/jeremic/tex/works/Thesis/GuanzhouJie/thesis/Verzija_Februar/Images/DispBent2.pdf}
\includegraphics[width=1.8cm]{/home/jeremic/tex/works/Thesis/GuanzhouJie/thesis/Verzija_Februar/Images/DispSoilBlock3.pdf}
\includegraphics[width=1.8cm]{/home/jeremic/tex/works/Thesis/GuanzhouJie/thesis/Verzija_Februar/Images/DispBent3.pdf}
\hspace*{1cm}
\\
\hspace*{1cm}
\includegraphics[width=1.8cm]{/home/jeremic/tex/works/Thesis/GuanzhouJie/thesis/Verzija_Februar/Images/AccelSoilBlock1.pdf}
\includegraphics[width=1.8cm]{/home/jeremic/tex/works/Thesis/GuanzhouJie/thesis/Verzija_Februar/Images/AccelBent1.pdf}
\includegraphics[width=1.8cm]{/home/jeremic/tex/works/Thesis/GuanzhouJie/thesis/Verzija_Februar/Images/AccelSoilBlock2.pdf}
\includegraphics[width=1.8cm]{/home/jeremic/tex/works/Thesis/GuanzhouJie/thesis/Verzija_Februar/Images/AccelBent2.pdf}
\includegraphics[width=1.8cm]{/home/jeremic/tex/works/Thesis/GuanzhouJie/thesis/Verzija_Februar/Images/AccelSoilBlock3.pdf}
\includegraphics[width=1.8cm]{/home/jeremic/tex/works/Thesis/GuanzhouJie/thesis/Verzija_Februar/Images/AccelBent3.pdf}
\hspace*{1cm}
\\
\hspace*{1cm}
\includegraphics[width=1.8cm]{/home/jeremic/tex/works/Thesis/GuanzhouJie/thesis/Verzija_Februar/Images/DispSoilBlock1_Spectrum.pdf}
\includegraphics[width=1.8cm]{/home/jeremic/tex/works/Thesis/GuanzhouJie/thesis/Verzija_Februar/Images/DispBent1_Spectrum.pdf}
\includegraphics[width=1.8cm]{/home/jeremic/tex/works/Thesis/GuanzhouJie/thesis/Verzija_Februar/Images/DispSoilBlock2_Spectrum.pdf}
\includegraphics[width=1.8cm]{/home/jeremic/tex/works/Thesis/GuanzhouJie/thesis/Verzija_Februar/Images/DispBent2_Spectrum.pdf}
\includegraphics[width=1.8cm]{/home/jeremic/tex/works/Thesis/GuanzhouJie/thesis/Verzija_Februar/Images/DispSoilBlock3_Spectrum.pdf}
\includegraphics[width=1.8cm]{/home/jeremic/tex/works/Thesis/GuanzhouJie/thesis/Verzija_Februar/Images/DispBent3_Spectrum.pdf}
\hspace*{1cm}
\\
\hspace*{1cm}
\includegraphics[width=1.8cm]{/home/jeremic/tex/works/Thesis/GuanzhouJie/thesis/Verzija_Februar/Images/AccelSoilBlock1_Spectrum.pdf}
\includegraphics[width=1.8cm]{/home/jeremic/tex/works/Thesis/GuanzhouJie/thesis/Verzija_Februar/Images/AccelBent1_Spectrum.pdf}
\includegraphics[width=1.8cm]{/home/jeremic/tex/works/Thesis/GuanzhouJie/thesis/Verzija_Februar/Images/AccelSoilBlock2_Spectrum.pdf}
\includegraphics[width=1.8cm]{/home/jeremic/tex/works/Thesis/GuanzhouJie/thesis/Verzija_Februar/Images/AccelBent2_Spectrum.pdf}
\includegraphics[width=1.8cm]{/home/jeremic/tex/works/Thesis/GuanzhouJie/thesis/Verzija_Februar/Images/AccelSoilBlock3_Spectrum.pdf}
\includegraphics[width=1.8cm]{/home/jeremic/tex/works/Thesis/GuanzhouJie/thesis/Verzija_Februar/Images/AccelBent3_Spectrum.pdf}
\hspace*{1cm}
\\
\hspace*{1cm}
\includegraphics[width=1.8cm]{/home/jeremic/tex/works/Thesis/GuanzhouJie/thesis/Verzija_Februar/Images/MomentBent1Pile1.pdf}
\includegraphics[width=1.8cm]{/home/jeremic/tex/works/Thesis/GuanzhouJie/thesis/Verzija_Februar/Images/MomentBent1Pile2.pdf}
\includegraphics[width=1.8cm]{/home/jeremic/tex/works/Thesis/GuanzhouJie/thesis/Verzija_Februar/Images/MomentBent2Pile1.pdf}
\includegraphics[width=1.8cm]{/home/jeremic/tex/works/Thesis/GuanzhouJie/thesis/Verzija_Februar/Images/MomentBent2Pile2.pdf}
\includegraphics[width=1.8cm]{/home/jeremic/tex/works/Thesis/GuanzhouJie/thesis/Verzija_Februar/Images/MomentBent3Pile1.pdf}
\includegraphics[width=1.8cm]{/home/jeremic/tex/works/Thesis/GuanzhouJie/thesis/Verzija_Februar/Images/MomentBent3Pile2.pdf}
\hspace*{1cm}
\end{center}
\end{figure}
\vspace*{0.8cm}
\begin{figure}[!htbp]
\begin{center}
\includegraphics[width=3.2cm]{/home/jeremic/tex/works/Reports/2006/NEESDemoProject/PrototypeMesh.jpg}
%\caption{\label{BridgeSFSI01} FEM model for seismic response of a three bend
%bridge.}
\end{center}
\end{figure}
\vspace*{2cm}
\clearpage
\end{frame}
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\begin{frame}
\frametitle{Northridge Input Motions}
\vspace*{0.7cm}
\begin{figure}[!htbp]
\begin{center}
\hspace*{1cm}
\includegraphics[width=8.0truecm]{/home/jeremic/tex/works/Thesis/GuanzhouJie/thesis/Verzija_Februar/Images/InputMotion_Northridge.pdf}
\hspace*{1cm}
\\
\hspace*{1cm}
\includegraphics[width=8.0truecm]{/home/jeremic/tex/works/Thesis/GuanzhouJie/thesis/Verzija_Februar/Images/InputMotion_Northridge_Spectrum.pdf}
\hspace*{1cm}
%\caption{\label{BridgeSFSI01} FEM model for seismic response of a three bend
%bridge.}
\end{center}
\end{figure}
\vspace*{1.0cm}
%
\end{frame}
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\begin{frame}
\frametitle{Short Period E.: Left Bent, Structure and Soil, Disp.}
\vspace*{1cm}
\begin{figure}[!htbp]
\begin{center}
\includegraphics[width=8cm]{/home/jeremic/tex/works/Thesis/GuanzhouJie/thesis/Verzija_Februar/Images/DispBent1.pdf}
%\includegraphics[width=8cm]{/home/jeremic/tex/works/Thesis/GuanzhouJie/thesis/Verzija_Februar/Images/DispBent1.pdf}
\\
\includegraphics[width=8cm]{/home/jeremic/tex/works/Thesis/GuanzhouJie/thesis/Verzija_Februar/Images/DispSoilBlock1.pdf}
%\includegraphics[width=8cm]{/home/jeremic/tex/works/Thesis/GuanzhouJie/thesis/Verzija_Februar/Images/DispSoilBlock1.pdf}
%\caption{\label{BridgeSFSI01} FEM model for seismic response of a three bend
%bridge.}
\end{center}
\end{figure}
\vspace*{2.9cm}
\hspace*{0.2cm}
\begin{figure}[!htbp]
\begin{flushright}
\includegraphics[width=1.5cm]{/home/jeremic/tex/works/Reports/2006/NEESDemoProject/PrototypeMesh.jpg}
%\caption{\label{BridgeSFSI01} FEM model for seismic response of a three bend
%bridge.}
\end{flushright}
\end{figure}
\hspace*{0.5cm}
\vspace*{2.0cm}
\end{frame}
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%
%
%
%
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% \begin{frame}
% \frametitle{Short Period E.: Left Bent, Structure and Soil, Acc.}
%
%
%
%
% \vspace*{1cm}
% \begin{figure}[!htbp]
% \begin{center}
% \includegraphics[width=8cm]{/home/jeremic/tex/works/Thesis/GuanzhouJie/thesis/Verzija_Februar/Images/AccelBent1.pdf}
% \\
% \includegraphics[width=8cm]{/home/jeremic/tex/works/Thesis/GuanzhouJie/thesis/Verzija_Februar/Images/AccelSoilBlock1.pdf}
% %\caption{\label{BridgeSFSI01} FEM model for seismic response of a three bend
% %bridge.}
% \end{center}
% \end{figure}
%
%
% \vspace*{2.9cm}
% \hspace*{0.2cm}
% \begin{figure}[!htbp]
% \begin{flushright}
% \includegraphics[width=1.5cm]{/home/jeremic/tex/works/Reports/2006/NEESDemoProject/PrototypeMesh.jpg}
% %\caption{\label{BridgeSFSI01} FEM model for seismic response of a three bend
% %bridge.}
% \end{flushright}
% \end{figure}
% \hspace*{0.5cm}
% \vspace*{2.0cm}
%
% \end{frame}
% %
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%
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\begin{frame}
\frametitle{Short Period E.: Left Bent, Structure and Soil, Acc.Sp.}
\vspace*{1cm}
\begin{figure}[!htbp]
\begin{center}
\includegraphics[width=8cm]{/home/jeremic/tex/works/Thesis/GuanzhouJie/thesis/Verzija_Februar/Images/AccelBent1_Spectrum.pdf}
\\
\includegraphics[width=8cm]{/home/jeremic/tex/works/Thesis/GuanzhouJie/thesis/Verzija_Februar/Images/AccelSoilBlock1_Spectrum.pdf}
%\caption{\label{BridgeSFSI01} FEM model for seismic response of a three bend
%bridge.}
\end{center}
\end{figure}
\vspace*{2.9cm}
\hspace*{0.2cm}
\begin{figure}[!htbp]
\begin{flushright}
\includegraphics[width=1.5cm]{/home/jeremic/tex/works/Reports/2006/NEESDemoProject/PrototypeMesh.jpg}
%\caption{\label{BridgeSFSI01} FEM model for seismic response of a three bend
%bridge.}
\end{flushright}
\end{figure}
\hspace*{0.5cm}
\vspace*{2.0cm}
\end{frame}
%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{frame}
% \frametitle{Short Period E.: Left Bent, Structure and Soil, M.}
%
%
%
%
% \vspace*{1cm}
% \begin{figure}[!htbp]
% \begin{center}
% \includegraphics[width=8cm]{/home/jeremic/tex/works/Thesis/GuanzhouJie/thesis/Verzija_Februar/Images/MomentBent1Pile1.pdf}
% \\
% \includegraphics[width=8cm]{/home/jeremic/tex/works/Thesis/GuanzhouJie/thesis/Verzija_Februar/Images/MomentBent1Pile2.pdf}
% %\caption{\label{BridgeSFSI01} FEM model for seismic response of a three bend
% %bridge.}
% \end{center}
% \end{figure}
%
%
%
%
% \vspace*{2.9cm}
% \hspace*{0.2cm}
% \begin{figure}[!htbp]
% \begin{flushright}
% \includegraphics[width=1.5cm]{/home/jeremic/tex/works/Reports/2006/NEESDemoProject/PrototypeMesh.jpg}
% %\caption{\label{BridgeSFSI01} FEM model for seismic response of a three bend
% %bridge.}
% \end{flushright}
% \end{figure}
% \hspace*{0.5cm}
% \vspace*{2.0cm}
%
%
% \end{frame}
% %
%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Short Period E.: Left Bent, Structure and Soil, M.}
\vspace*{1cm}
\begin{figure}[!htbp]
\begin{center}
\includegraphics[width=11cm]{/home/jeremic/tex/works/Thesis/GuanzhouJie/thesis/Verzija_Februar/Images/MomentBent1Pile1_25s_SC.pdf}
%\caption{\label{BridgeSFSI01} FEM model for seismic response of a three bend
%bridge.}
\end{center}
\end{figure}
\vspace*{2.9cm}
\hspace*{0.2cm}
\begin{figure}[!htbp]
\begin{flushright}
\includegraphics[width=1.5cm]{/home/jeremic/tex/works/Reports/2006/NEESDemoProject/PrototypeMesh.jpg}
%\caption{\label{BridgeSFSI01} FEM model for seismic response of a three bend
%bridge.}
\end{flushright}
\end{figure}
\hspace*{0.5cm}
\vspace*{2.0cm}
\vspace*{2.0cm}
{%
\begin{beamercolorbox}{section in head/foot}
\usebeamerfont{framesubtitle}\tiny{B. Jeremi\'{c}, G. Jie,
M. Preisig and N. Tafazzoli.} "SoilFoundationStructure Interaction
in nonUniform Soils",
in review in \textit{Earthquake Engineering and Structural Dynamics}, 2008.
%\vskip2pt\insertnavigation{\paperwidth}\vskip2pt
\end{beamercolorbox}%
}
\end{frame}
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{frame}
% \frametitle{Short Period E.: Left Bent, Free Field vs Real Disp.}
%
%
%
%
% \vspace*{1cm}
% \begin{figure}[!htbp]
% \begin{center}
% \includegraphics[width=11cm]{/home/jeremic/tex/works/Thesis/GuanzhouJie/thesis/Verzija_Februar/Images/DispSoilBlock1_15cm_25s_SC.pdf}
% %\caption{\label{BridgeSFSI01} FEM model for seismic response of a three bend
% %bridge.}
% \end{center}
% \end{figure}
%
%
% \vspace*{2.9cm}
% \hspace*{0.2cm}
% \begin{figure}[!htbp]
% \begin{flushright}
% \includegraphics[width=1.5cm]{/home/jeremic/tex/works/Reports/2006/NEESDemoProject/PrototypeMesh.jpg}
% %\caption{\label{BridgeSFSI01} FEM model for seismic response of a three bend
% %bridge.}
% \end{flushright}
% \end{figure}
% \hspace*{0.5cm}
% \vspace*{2.0cm}
%
%
% \end{frame}
% %
%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\begin{frame}
\frametitle{Kocaeli Input Motions}
\vspace*{0.7cm}
\begin{figure}[!htbp]
\begin{center}
\hspace*{1cm}
\includegraphics[width=8.0truecm]{/home/jeremic/tex/works/Thesis/GuanzhouJie/thesis/Verzija_Februar/Images/LongMotion/InputMotion_Kocaeli.pdf}
\hspace*{1cm}
\\
\hspace*{1cm}
\includegraphics[width=8.0truecm]{/home/jeremic/tex/works/Thesis/GuanzhouJie/thesis/Verzija_Februar/Images/LongMotion/InputMotion_Kocaeli_Spectrum.pdf}
\hspace*{1cm}
%\caption{\label{BridgeSFSI01} FEM model for seismic response of a three bend
%bridge.}
\end{center}
\end{figure}
%
\vspace*{1cm}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{frame}
% \frametitle{Long Period E.: Left Bent, Structure and Soil, Disp.}
%
%
%
%
% \vspace*{1cm}
% \begin{figure}[!htbp]
% \begin{center}
% \includegraphics[width=8cm]{/home/jeremic/tex/works/Thesis/GuanzhouJie/thesis/Verzija_Februar/Images/LongMotion/DispBent1.pdf}
% \\
% \includegraphics[width=8cm]{/home/jeremic/tex/works/Thesis/GuanzhouJie/thesis/Verzija_Februar/Images/LongMotion/DispSoilBlock1.pdf}
% %\caption{\label{BridgeSFSI01} FEM model for seismic response of a three bend
% %bridge.}
% \end{center}
% \end{figure}
%
%
% \vspace*{2.9cm}
% \hspace*{0.2cm}
% \begin{figure}[!htbp]
% \begin{flushright}
% \includegraphics[width=1.5cm]{/home/jeremic/tex/works/Reports/2006/NEESDemoProject/PrototypeMesh.jpg}
% %\caption{\label{BridgeSFSI01} FEM model for seismic response of a three bend
% %bridge.}
% \end{flushright}
% \end{figure}
% \hspace*{0.5cm}
% \vspace*{2.0cm}
%
%
%
% \end{frame}
% %
%
%
%
%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{frame}
% \frametitle{Long Period E.: Left Bent, Structure and Soil, Acc.}
%
%
%
%
% \vspace*{1cm}
% \begin{figure}[!htbp]
% \begin{center}
% \includegraphics[width=8cm]{/home/jeremic/tex/works/Thesis/GuanzhouJie/thesis/Verzija_Februar/Images/LongMotion/AccelBent1.pdf}
% \\
% \includegraphics[width=8cm]{/home/jeremic/tex/works/Thesis/GuanzhouJie/thesis/Verzija_Februar/Images/LongMotion/AccelSoilBlock1.pdf}
% %\caption{\label{BridgeSFSI01} FEM model for seismic response of a three bend
% %bridge.}
% \end{center}
% \end{figure}
%
%
%
% \vspace*{2.9cm}
% \hspace*{0.2cm}
% \begin{figure}[!htbp]
% \begin{flushright}
% \includegraphics[width=1.5cm]{/home/jeremic/tex/works/Reports/2006/NEESDemoProject/PrototypeMesh.jpg}
% %\caption{\label{BridgeSFSI01} FEM model for seismic response of a three bend
% %bridge.}
% \end{flushright}
% \end{figure}
% \hspace*{0.5cm}
% \vspace*{2.0cm}
%
%
% \end{frame}
% %
%
%
%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{frame}
% \frametitle{Long Period E.: Left Bent, Structure and Soil, Acc.Sp.}
%
%
%
%
% \vspace*{1cm}
% \begin{figure}[!htbp]
% \begin{center}
% \includegraphics[width=8cm]{/home/jeremic/tex/works/Thesis/GuanzhouJie/thesis/Verzija_Februar/Images/LongMotion/AccelBent1_Spectrum.pdf}
% \\
% \includegraphics[width=8cm]{/home/jeremic/tex/works/Thesis/GuanzhouJie/thesis/Verzija_Februar/Images/LongMotion/AccelSoilBlock1_Spectrum.pdf}
% %\caption{\label{BridgeSFSI01} FEM model for seismic response of a three bend
% %bridge.}
% \end{center}
% \end{figure}
%
%
% \vspace*{2.9cm}
% \hspace*{0.2cm}
% \begin{figure}[!htbp]
% \begin{flushright}
% \includegraphics[width=1.5cm]{/home/jeremic/tex/works/Reports/2006/NEESDemoProject/PrototypeMesh.jpg}
% %\caption{\label{BridgeSFSI01} FEM model for seismic response of a three bend
% %bridge.}
% \end{flushright}
% \end{figure}
% \hspace*{0.5cm}
% \vspace*{2.0cm}
%
%
% \end{frame}
% %
%
%
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Long Period E.: Left Bent, Structure and Soil, M.}
\vspace*{1cm}
\begin{figure}[!htbp]
\begin{center}
\includegraphics[width=11cm]{/home/jeremic/tex/works/Thesis/GuanzhouJie/thesis/Verzija_Februar/Images/LongMotion/MomentBent1Pile1.pdf}
%\\
%\includegraphics[width=8cm]{/home/jeremic/tex/works/Thesis/GuanzhouJie/thesis/Verzija_Februar/Images/LongMotion/MomentBent1Pile2.pdf}
%\caption{\label{BridgeSFSI01} FEM model for seismic response of a three bend
%bridge.}
\end{center}
\end{figure}
\vspace*{2.9cm}
\hspace*{0.2cm}
\begin{figure}[!htbp]
\begin{flushright}
\includegraphics[width=1.5cm]{/home/jeremic/tex/works/Reports/2006/NEESDemoProject/PrototypeMesh.jpg}
%\caption{\label{BridgeSFSI01} FEM model for seismic response of a three bend
%bridge.}
\end{flushright}
\end{figure}
\hspace*{0.5cm}
\vspace*{2.0cm}
\end{frame}
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{frame}
% \frametitle{Soft Soils: Left Bent, Structure and Soil, Moments}
%
%
%
%
% \vspace*{1cm}
% \begin{figure}[!htbp]
% \begin{center}
% \includegraphics[width=8cm]{/home/jeremic/tex/works/Thesis/GuanzhouJie/thesis/Verzija_Februar/Images/LongMotion/MomentBent1Pile1_25s_SC.pdf}
% %\caption{\label{BridgeSFSI01} FEM model for seismic response of a three bend
% %bridge.}
% \end{center}
% \end{figure}
% \vspace*{2cm}
%
%
% \end{frame}
% %
%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{frame}
% \frametitle{Soft Soils: Left Bent, Free Field vs Real Displacements}
%
%
%
%
% \vspace*{1cm}
% \begin{figure}[!htbp]
% \begin{center}
% \includegraphics[width=8cm]{/home/jeremic/tex/works/Thesis/GuanzhouJie/thesis/Verzija_Februar/Images/LongMotion/DispSoilBlock1_15cm_25s_SC.pdf}
% %\caption{\label{BridgeSFSI01} FEM model for seismic response of a three bend
% %bridge.}
% \end{center}
% \end{figure}
% \vspace*{2cm}
%
%
% \end{frame}
% %
%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Uncertain Seismic Motions}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Constitutive and Spatial Uncertainties}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Problem Setup}
\begin{itemize}
\vspace*{0.3cm}
\item Incr. 3D elpl:
%
%
% \begin{equation}
$
\nonumber
d\sigma_{ij} = \left \{
D^{el}_{ijkl}

\frac{\displaystyle D^{el}_{ijmn} m_{mn} n_{pq} D^{el}_{pqkl}}
{\displaystyle n_{rs} D^{el}_{rstu} m_{tu}  \xi_* r_*}
\right \}
{d\epsilon_{kl}}
$
% \end{equation}
\vspace*{0.3cm}
\item phase density $\rho$ of $\sigma(x,t)$ varies in time according to a continuity
Liouville equation (Kubo 1963)
\vspace*{0.3cm}
\item
Continuity equation written in ensemble average form (eg. cumulant
expansion method (Kavvas and Karakas 1996))
\vspace*{0.3cm}
\item
van Kampen's Lemma (van Kampen 1976) $\rightarrow$ $ <\rho(\sigma,t)>=P(\sigma,t) $,
ensemble average of phase density is the probability density
%
%
% \item Focus on 1D $\rightarrow$ a nonlinear ODE with random coefficient
% (material) and random forcing ($\epsilon$)
% %
% %
% %
% \begin{eqnarray}
% \nonumber
% \frac{d\sigma(x,t)}{dt} &=& \beta(\sigma(x,t),D^{el}(x),q(x),r(x);x,t) \frac{d\epsilon(x,t)}{dt} \\
% \nonumber
% &=& \eta(\sigma,D^{el},q,r,\epsilon; x,t)
% \end{eqnarray}
% %
% with initial condition $\sigma(0)=\sigma_0$
\end{itemize}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{frame}
%
%
%
% \frametitle{Evolution of the Density $\rho(\sigma,t)$ }
%
%
%
% \begin{itemize}
%
% \vspace*{1cm}
% \item From each initial point in \\
% $\sigma$space a trajectory \\
% starts out describing \\
% the corresponding solution \\
% of the stochastic process
%
% \vspace*{0.3cm}
% \item Movement of a cloud of initial\\
% points described by density \\
% $\rho(\sigma,0)$ in $\sigma$space, \\
% is governed by the \\
% constitutive equation,
%
% \end{itemize}
%
%
% %\begin{figure}[!hbpt]
% %\begin{center}
% \vspace*{5cm}
% \hspace*{6cm}
% \includegraphics[height=4.5cm,angle=90]{/home/jeremic/tex/works/Conferences/2007/USC_seminar/Present/Cloud_of_Points.pdf}
% %\end{center}
% %\end{figure}
%
%
%
% \end{frame}
%
%
%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{frame}
%
%
% \frametitle{Stochastic Continuity (Liouville) Equation}
%
%
%
% \begin{itemize}
%
%
% \item phase density $\rho$ of $\sigma(x,t)$ varies in time according to a continuity
% Liouville equation (Kubo 1963):
% %
% \begin{eqnarray}
% \frac{\partial \rho (\sigma(x,t),t)}{\partial t}
% =
% \nonumber
% \\
% \frac{\partial \eta (\sigma(x,t), D^{el}(x), q(x), r(x), \epsilon(x,t)) }{\partial \sigma}
% \;\;
% \rho[\sigma(x,t),t]
% \nonumber
% \end{eqnarray}
%
% \vspace{0.5cm}
% \item with initial conditions $\rho(\sigma,0) = \delta(\sigma\sigma_0)$
%
%
% \end{itemize}
%
% \end{frame}
%
%
%
%
%
%
%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{frame}
%
%
% \frametitle{{Ensemble Average form of Liouville Equation}}
%
%
%
%
%
%
%
% \noindent
% Continuity equation written in ensemble average form (eg. cumulant
% expansion method (Kavvas and Karakas 1996)):
%
%
% %\vspace*{0.5cm}
%
% \begin{footnotesize}
%
% \begin{eqnarray}
% \nonumber
% &&\displaystyle \frac{\partial \left < \rho(\sigma(x_t,t), t) \right >}{\partial t}=
%  \displaystyle \frac{\partial}{\partial \sigma} \left[ \left\{\left< \vphantom{\int_{0}^{t} d\tau} \eta(\sigma(x_t,t), D^{el}(x_t),
% q(x_t), r(x_t), \epsilon(x_t,t)) \right> \right. \right. \\
% \nonumber
% &+& \left. \left. \int_{0}^{t} d\tau Cov_0 \left[ \displaystyle \frac{\partial \eta(\sigma(x_t,t), D^{el}(x_t), q(x_t), r(x_t),
% \epsilon(x_t,t))}{\partial \sigma}; \right. \right. \right. \\
% \nonumber
% & & \left. \left. \left. \eta(\sigma(x_{t\tau},t\tau), D^{el}(x_{t\tau}), q(x_{t\tau}), r(x_{t\tau}),
% \epsilon(x_{t\tau},t\tau) \vphantom{\int_{0}^{t} d\tau} \right] \right \} \left < \rho (\sigma(x_t,t),t) \right > \right] \\
% \nonumber
% &+& \displaystyle \frac{\partial^2}{\partial \sigma^2} \left[ \left\{ \int_{0}^{t} d\tau Cov_0 \left[ \vphantom{\int_{0}^{t}}
% \eta(\sigma(x_t,t), D^{el}(x_t), q(x_t), r(x_t), \epsilon(x_t,t)); \right. \right. \right. \\
% \nonumber
% & & \left. \left. \left. \eta (\sigma(x_{t\tau},t\tau), D^{el}(x_{t\tau}), q(x_{t\tau}), r(x_{t\tau}),
% \epsilon(x_{t\tau},t\tau)) \vphantom{\int_{0}^{t}} \right] \vphantom{\int_{0}^{t}} \right\} \left < \rho (\sigma (x_t,t),t) \right > \right] \\
% \nonumber
% \end{eqnarray}
%
% \end{footnotesize}
%
%
%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%
% \end{frame}
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{EulerianLagrangian FPK Equation}
%
%\begin{itemize}
\begin{footnotesize}
% %\noindent
% van Kampen's Lemma (van Kampen 1976) $\rightarrow$ $ <\rho(\sigma,t)>=P(\sigma,t) $,
% ensemble average of phase density
% %(in stress space here)
% is the probability density;
%
\begin{eqnarray}
\nonumber
&&\displaystyle \frac{\partial P(\sigma(x_t,t), t)}{\partial t}=
 \displaystyle \frac{\partial}{\partial \sigma} \left[ \left\{\left< \vphantom{\int_{0}^{t} d\tau} \eta(\sigma(x_t,t), D^{el}(x_t),
q(x_t), r(x_t), \epsilon(x_t,t)) \right> \right. \right. \\
\nonumber
&+& \left. \left. \int_{0}^{t} d\tau Cov_0 \left[ \displaystyle \frac{\partial \eta(\sigma(x_t,t), D^{el}(x_t), q(x_t), r(x_t),
\epsilon(x_t,t))}{\partial \sigma}; \right. \right. \right. \\
\nonumber
& & \left. \left. \left. \eta(\sigma(x_{t\tau},t\tau), D^{el}(x_{t\tau}), q(x_{t\tau}), r(x_{t\tau}),
\epsilon(x_{t\tau},t\tau) \vphantom{\int_{0}^{t} d\tau} \right] \right \} P(\sigma(x_t,t),t) \right] \\
\nonumber
&+& \displaystyle \frac{\partial^2}{\partial \sigma^2} \left[ \left\{ \int_{0}^{t} d\tau Cov_0 \left[ \vphantom{\int_{0}^{t}}
\eta(\sigma(x_t,t), D^{el}(x_t), q(x_t), r(x_t), \epsilon(x_t,t)); \right. \right. \right. \\
\nonumber
& & \left. \left. \left. \eta (\sigma(x_{t\tau},t\tau), D^{el}(x_{t\tau}), q(x_{t\tau}), r(x_{t\tau}),
\epsilon(x_{t\tau},t\tau)) \vphantom{\int_{0}^{t}} \right] \vphantom{\int_{0}^{t}} \right\} P(\sigma (x_t,t),t) \right] \\
\nonumber
\end{eqnarray}
\end{footnotesize}
\vspace*{0.5cm}
{%
\begin{beamercolorbox}{section in head/foot}
\usebeamerfont{framesubtitle}\tiny{B. Jeremi\'{c}, K. Sett, and M. L. Kavvas, "Probabilistic
ElastoPlasticity: Formulation in 1D", \textit{Acta Geotechnica}, Vol.~2,
No.~3, pp~197210, 2007.}
%\vskip2pt\insertnavigation{\paperwidth}\vskip2pt
\end{beamercolorbox}%
}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{EulerLagrange FPK Equation}
\begin{itemize}
\item Advectiondiffusion equation
%
\begin{equation}
\nonumber
\frac{\partial P(\sigma,t)}{\partial t} = \frac{\partial}{\partial \sigma}\left[N_{(1)}P(\sigma,t)\frac{\partial}{\partial \sigma}
\left\{N_{(2)} P(\sigma,t)\right\} \right]
\end{equation}
%
\item Complete probabilistic description of response
\item Solution PDF is secondorder exact to covariance of time (exact mean and variance)
\item It is deterministic equation in probability density space
\item It is linear PDE in probability density space
$\rightarrow$ Simplifies the numerical solution process
\item Template FPK diffusionadvection equation is applicable to any material model $\rightarrow$
only the coefficients $N_{(1)}$ and $N_{(2)}$ are different for different material models
%\vspace*{0.2truecm}
\end{itemize}
{%
\begin{beamercolorbox}{section in head/foot}
\usebeamerfont{framesubtitle}\tiny{K. Sett, B. Jeremi{\'c} and M.L. Kavvas, "The Role of Nonlinear
Hardening/Softening in Probabilistic ElastoPlasticity", \textit{International Journal for Numerical
and Analytical Methods in Geomechanics}, Vol.~31, No.~7, pp~953975, 2007}
%\vskip2pt\insertnavigation{\paperwidth}\vskip2pt
\end{beamercolorbox}%
}
\end{frame}
% 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Spectral Stochastic ElasticPlastic FEM}
\vspace*{0.6truecm}
\begin{flushright}
\begin{equation}
\nonumber
\sum_{n = 1}^N K_{mn} d_{ni} + \sum_{n = 1}^N \sum_{j = 0}^P d_{nj} \sum_{k = 1}^M C_{ijk} K'_{mnk} = \left< F_m \psi_i[\{\xi_r\}] \right >
\end{equation}
\end{flushright}
\vspace*{0.5cm}
\begin{equation}
\nonumber
K_{mn} = \int_D B_n \textcolor{mycolor}{D} B_m dV \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ K'_{mnk} = \int_D B_n {\sqrt \lambda_k h_k} B_m dV
\end{equation}
\vspace*{0.5cm}
\begin{equation}
\nonumber
C_{ijk} = \left < \xi_k(\theta) \psi_i[\{\xi_r\}] \psi_j[\{\xi_r\}] \right > \ \ \ \ \ \ \ \ \ \ \ \ F_m = \int_D \phi N_m dV \ \ \ \ \ \ \ \ \ \ \ \
\end{equation}
\begin{itemize}
\item SFEM: Ghanem and Spanos 2003
\item Material variables random field represented through a finite number of
random variables using KLexpansion
\item Unknown solution random variables represented using polynomial chaos of
(known) input random variables
\item FokkerPlanckKolmogorov approach based probabilistic constitutive
integration at Gauss integration points
\end{itemize}
%% \end{itemize}
%
\end{frame}
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% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%\section{An Application}
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\subsection{Seismic Wave Propagation Through Uncertain Soils}
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\begin{frame}
\frametitle{``Uniform'' CPT Site Data}
\vspace*{0.7cm}
%\begin{figure}
\begin{center}
\includegraphics[height=6.7cm]{/home/jeremic/tex/works/Thesis/KallolSett/Dissertation/figures/CPT_DataAnalysis_Plots/EastWestProfileEdited.pdf}
\end{center}
%\end{figure}
\end{frame}
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\begin{frame}
\frametitle{Random Field Parameters from Site Data}
\begin{itemize}
%\item maximizing the loglikelihood of observing the spatial data under assumed joined distribution (for finite
%scale model) or maximizing the loglikelihood of observing the periodogram estimates (for fractal model)
\item Maximum likelihood estimates of correlation length
\vspace*{0.3truecm}
%\begin{figure}
\begin{flushleft}
\hspace*{1.7cm}
\includegraphics[height=4.0cm]{/home/jeremic/tex/works/Thesis/KallolSett/Dissertation/figures/CPT_DataAnalysis_Plots/SamplingPlanEdited.jpg}
\hspace*{0.0cm}
\includegraphics[height=4.0cm]{/home/jeremic/tex/works/Thesis/KallolSett/Dissertation/figures/CPT_DataAnalysis_Plots/TypicalDataPlotBH1Edited.jpg} \\
\small{~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~Typical CPT $q_T$}
\end{flushleft}
%\end{figure}
\vspace*{4.9truecm}
%\begin{figure}
\begin{flushright}
\includegraphics[width=4.0cm]{/home/jeremic/tex/works/Thesis/KallolSett/Dissertation/figures/CPT_DataAnalysis_Plots/TypicalAutoCovariancePlotBH1_FiniteScaleEdited.jpg} \\
\vspace*{0.01truecm}
\small{Finite Scale}
\end{flushright}
%\end{figure}
\vspace*{0.02truecm}
%\begin{figure}
\begin{flushright}
\includegraphics[width=4.0cm]{/home/jeremic/tex/works/Thesis/KallolSett/Dissertation/figures/CPT_DataAnalysis_Plots/TypicalAutoCovariancePlotBH1_FractalEdited.jpg} \\
\small{Fractal}
\end{flushright}
%\end{figure}
\end{itemize}
\end{frame}
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\begin{frame}
\frametitle{Seismic Wave Propagation through Stochastic Soil}
%\begin{flushleft}
%\includegraphics[height=5.0cm]{PEER2007_3.jpg}
%\end{flushleft}
%\vspace*{0.5truecm}
\begin{itemize}
\item Soil as 12.5 m deep 1D soil column (von Mises Material)
\begin{itemize}
\item Properties (including testing uncertainty) obtained through random field modeling of CPT $q_T$
%
$\left = 4.99 ~MPa;~~Var[q_T] = 25.67 ~MPa^2; $\\
Cor. ~Length $[q_T] = 0.61 ~m; $ Testing~Error $= 2.78 ~MPa^2$
\end{itemize}
\vspace*{0.2cm}
\item $q_T$ was transformed to obtain $G$: ~~$G/(1\nu)~=~2.9q_T$
\begin{itemize}
\item Assumed transformation uncertainty = 5\%
%
$\left = 11.57MPa; Var[G] = 142.32 MPa^2$ \\
Cor.~Length $[G] = 0.61 m$
\end{itemize}
%\begin{center}
%\hspace*{1.7cm}
%\includegraphics[height=3.5cm]{Chapter9_Schematic.jpg}
%\hspace*{0.0cm}
%\includegraphics[height=3.5cm]{Chapter9_BaseDisplacement.jpg} \\
%\small{~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~Base Displacement}
%\end{center}
\vspace*{0.2cm}
\item Input motions: modified 1938 Imperial Valley
% \vspace*{0.2cm}
% \begin{center}
% \includegraphics[height=2.0cm]{Chapter9_BaseDisplacement.jpg}
% \end{center}
\end{itemize}
\end{frame}
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% \subsection{Seismic Wave Propagation Through Uncertain Soils}
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% BJtempout\setbeamercovered{invisible}
\begin{frame}
\frametitle{Surface Displacement Time History}
\vspace*{12.0truecm}
%\hspace*{0.7cm} \uncover<1>{\includegraphics[width=12.0cm]{Chapter9_ElasticPlasticResponse_MeanNew.pdf}}
%%
%\hspace*{6.0cm} \uncover<2>{\includegraphics[width=12.0cm]{Chapter9_ElasticPlasticResponse_SDNew.pdf}} \hspace*{2.0cm} \\
%%
%\uncover<1>{
\hspace*{0.7cm}
\includegraphics[width=12.0cm]{/home/jeremic/tex/works/Conferences/2008/EM/Present/Chapter9_ElasticPlasticResponse_MeanNew.pdf}
%}
% BJtempout\uncover<3>{\hspace*{6.0cm} \includegraphics[width=12.0cm]{Chapter9_ElasticPlasticResponse_SDNew.pdf} \hspace*{6.0truecm}}
{\hspace*{6.0cm} \includegraphics[width=12.0cm]{/home/jeremic/tex/works/Conferences/2008/EM/Present/Chapter9_ElasticPlasticResponse_SDNew.pdf} \hspace*{6.0truecm}}
\vspace*{0.25truecm}
% BJtempout\hspace*{0.70cm} Deterministic\uncover<2>{/Mean} \uncover<3>{\hspace*{2.35cm} Standard Deviation \hspace*{1.0cm}}
\hspace*{0.70cm} Deterministic/Mean {\hspace*{2.35cm} Standard Deviation \hspace*{1.0cm}}
\end{frame}
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\begin{frame}
\frametitle{Mean $\pm$ Standard Deviation}
\vspace*{19.0truecm}
\hspace*{0.15truecm} \includegraphics[width=19.0cm]{/home/jeremic/tex/works/Conferences/2008/EM/Present/Chapter9_ElasticPlasticResponseNew.pdf}
\end{frame}
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%
%\begin{frame}
%\frametitle{Evolution of Statistics of Top Node Displacement}
%
%\vspace*{9.0truecm}
%
%\hspace*{0.5cm} \includegraphics[width=9.0cm]{Chapter9_ElasticPlasticResponse_MeanNew.pdf}
%\hspace*{4.0cm} \includegraphics[width=9.0cm]{Chapter9_ElasticPlasticResponse_SDNew.pdf}
%
%%\vspace*{0.5truecm}
%%\hspace*{1.0cm}
%\tiny{~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~Mean~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~Standard Deviation}
%
%\vspace*{8.35truecm}
%
%
%\hspace*{0.05cm} \includegraphics[width=9.0cm]{Chapter9_ElasticPlasticResponseNew.pdf}
%\hspace*{3.55cm} \includegraphics[width=9.0cm]{Chapter9_ElasticPlasticResponse_COVNew.pdf}
%
%%\vspace*{0.1truecm}
%%\hspace*{0.5cm}
%\tiny{~~~~~~~~~~~~~~~~~~~~~~~~~~Mean$\pm$ Standard Deviation~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~COV}
%
%\end{frame}
%
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\begin{frame}
\frametitle{PDF of Surface Displacement Time History}
\begin{itemize}
\item PDF at the finite \\
element nodes can be \\
obtained using, e.g., \\
Edgeworth expansion \\
(Ghanem and Spanos \\
2003)
\vspace*{0.1truecm}
\item Numerous applications, \\
especially where extreme \\
statistics are critical
\end{itemize}
\vspace*{11.0cm}
\hspace*{5.5truecm} \includegraphics[height=12.0cm]{/home/jeremic/tex/works/Conferences/2008/EM/Present/Chapter9_ElasticPlasticResponse_PDFNewEdited.pdf} \hspace*{2.0cm}
\end{frame}
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\begin{frame}
\frametitle{Most Probable Solution}
\vspace*{10.25cm}
\includegraphics[height=16.0cm]{/home/jeremic/tex/works/Conferences/2008/EM/Present/SamplePDF_DynamicProblem_Mode.pdf} \hspace*{5.0truecm} At $t~=~1.1$ sec \hspace*{2.0truecm}
\end{frame}
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\begin{frame}
\frametitle{Tails of PDF}
\vspace*{10.25cm}
\includegraphics[height=16.0cm]{/home/jeremic/tex/works/Conferences/2008/EM/Present/SamplePDF_DynamicProblem_TailsOfPDF.pdf} \hspace*{5.0truecm} At $t~=~1.1$ sec \hspace*{2.0truecm}
\end{frame}
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\begin{frame}
\frametitle{Probability of Exceedance}
\vspace*{10.25cm}
\includegraphics[height=16.0cm]{/home/jeremic/tex/works/Conferences/2008/EM/Present/SampleCDF_DynamicProblem.pdf} \hspace*{5.0truecm} At $t~=~1.1$ sec \hspace*{2.0truecm}
\end{frame}
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\begin{frame}
\frametitle{Derivative Applications}
%\vspace*{0.25truecm}
\begin{itemize}
\item Performance based engineering
\begin{itemize}
\item Reliability index ($\beta$)
\item Probability of damage and/or failure ($p_f$)
\end{itemize}
\vspace*{10.5truecm}
\hspace*{1.0truecm} \includegraphics[height=13.0cm]{/home/jeremic/tex/works/Conferences/2008/EM/Present/SchematicReliability.pdf} \hspace*{1.5truecm}
%\vspace*{0.5truecm}
\item Sensitivity analysis
\item Financial risk analysis
\item In general, useful for applications where mean, mode and extreme
statistics are important
%\vspace*{8.0truecm}
%\hspace*{4.0truecm} \includegraphics[angle=90,height=10.0cm]{SchematicFinancialRisk.pdf} \hspace*{1.5truecm}
\end{itemize}
\end{frame}
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\section{Summary}
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\begin{frame}
\frametitle{Summary}
\begin{itemize}
\item {\bf Steady progress} in software and hardware allows for high fidelity Model
Based Simulations for performance assessment of infrastructure systems
\vspace*{0.2cm}
\item {\bf Interplay} of {\bf Earthquake(s)} with {\bf Soil} and {\bf Structure}
\underline{\bf Systems} plays a major role in catastrophic failures (and successes)
\vspace*{0.2cm}
\item {\bf Probabilistic} performance based engineering (uncertainty in soil and
structural behavior, earthquake motions...)
\vspace*{0.2cm}
\item {\bf Overcome} traditional performance assessment approaches used in
engineering practice (design using prescriptive code!)
%
% \item {\bf Matching Triad}:
% \underline{Earthquake},
% \underline{Soil} and
% \underline{Structure}
% ({\bf ESS}) interaction determines
% possible benefits or detriments of SFSI
\end{itemize}
\end{frame}
%
\end{document}