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\title{Numerical Modeling and Simulation of Pile in Liquefiable Soil}
\author[ZC]{
Zhao Cheng
%(\raisebox{1.0mm}
%{\includegraphics[width=3.55cm]{/home/jeremic/tex/works/IMENAuOriginaly/BorisJeremic.eps}}
}
%\ead{jeremic@ucdavis.edu}
\author[BJ]{
Boris Jeremi{\'c}
%(\raisebox{1.0mm}
%{\includegraphics[width=3.55cm]{/home/jeremic/tex/works/IMENAuOriginaly/BorisJeremic.eps}}
}
\ead{jeremic@ucdavis.edu}
%\author[rvt,focal]{K.?Bazargan\fnref{fn2}}
%\ead{kaveh@rivervalley.com}
\cortext[cor1]{Corresponding Author: Boris Jeremi{\'c}}
%\cortext[cor2]{Principal corresponding author}
%\fntext[fn1]{This is the specimen author footnote.}
%\fntext[fn2]{Another author footnote, but a little more longer.}
%\fntext[fn3]{Yet another author footnote. Indeed, you can have
% any number of author footnotes.}
%\address[focal]{
%Associate Professor,
%Department of Civil and Environmental Engineering, University of
%California, Davis, CA 95616.}
\address[ZC]{Staff Engineer, Earth Mechanics Inc. Oakland, CA 94621}
\address[BJ]{
Associate Professor,
Department of Civil and Environmental Engineering, University of
California, Davis, CA 95616.}
%
% \author{
% Zhao Cheng
% (\raisebox{1.2mm}{\includegraphics[width=1.35cm]{/home/jeremic/tex/works/IMENAuOriginaly/Zhao.eps}})
% %(\raisebox{1.2mm}{\includegraphics[width=1.1cm]{Zhao.eps}})
% \footnote{{Staff Engineer, Earth Mechanics Inc. Oakland, CA 94621}}
% %%%%%%%%
% \\
% and
% \\
% %%%%%%%%
% \setcounter{footnote}{3}
% Boris Jeremi{\'c}
% (\raisebox{1.0mm}{\includegraphics[width=3.55cm]{/home/jeremic/tex/works/IMENAuOriginaly/BorisJeremic.eps}})
% %(\raisebox{1.0mm}{\includegraphics[width=3.2cm]{BorisJeremic.eps}})
% \footnote{Associate Professor,
% Department of Civil and Environmental Engineering, University of
% California, Davis, CA 95616. \texttt{jeremic@ucdavis.edu}, Coresponding Author}
% \\
% %Boris Jeremi{\'c} ({\cyr Boris Jeremi\cj{}})$^{1}$ \\
% %%%%%%%%
% }
\begin{abstract}
Presented in this paper is numerical methodology to model and simulate
behavior of piles in liquefiable
soils. Modeling relies on use of validate
elastoplastic material model for soil skeleton, verified fully coupled porous media
(soil skeleton)  pore fluid (water) dynamic finite element formulation, and
detailed load staging of FEM models. A bounding surface
elasticplastic sand model that accounts for fabric change is used to model soil
skeleton, while a fully coupled, dynamic, inelastic formulation (upU) is used
to model soil and water displacement and pore
water pressures.
Much attention is paid to accurate staged loading of the models, which start from
a zero state of stress and strain for a soil without a pile, followed by application of
self weight, then by excavation and pile installation with application of
pile selfweighting. Finally, seismic loading is applied followed by time to
dissipate excess pore pressures that have developed. A total of six cases were
modeled and simulated varying slope inclination, presence of pilecolumn and
boundary condition for pilecolumn system.
%
Presented are interesting and useful results that are used to deepen our
understanding of behavior of soilpilecolumn systems during
liquefaction (lateral deformations, pile pinning effect, ground settlement).
%
Moreover, detailed description of modeling is used to emphasize
the availability and use of high fidelity modeling tools for simulating
effects of liquefied soil on soilstructure systems.
\end{abstract}
%occurring, excess pore water buildingup and dissipation, soil skeleton lateral
%deformation, ground settlement, pile response, and pilesoil interaction were
%interpreted at some length.
\begin{keyword}
elastoplastic, fully coupled finite elements; liquefaction; soilpile dynamics;
\end{keyword}
\maketitle
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\section{Introduction}
\label{intro}
Liquefaction is one of the most complex phenomena in
earthquake engineering. Liquefaction also represents one of the biggest
contributors to damage of constructed facilities during earthquakes
\citep{Kramer96}.
%
Prediction of behaviors of liquefiable soils is difficult but achievable.
%
There are number of methods that can be
utilized to predict such behaviors.
%
Methods currently used can have varying prediction accuracy and certainty.
%
%
Of particular interest in this paper is the description of
verified and validated numerical simulation methodology based on
rational mechanics that is used to model, simulate and predict
behaviors of a single pile in liquefiable soil subjected to seismic
loading.
%
Both level and sloping ground pile systems are modeled and simulated.
%
Detailed description of background theory, formulation and implementation
were recently given by \citet{Cheng2007} and \citet{Jeremic2007e}.
It should be noted that presented development do show great promise in analyzing
a myriad of liquefaction related problems in geotechnical and structural
engineering.
%
The effectiveness and power of numerical simulation tools for analyzing
liquefaction problems becomes even more important and prominent in the light of
potential disadvantages of models used in experimental simulations.
%
These disadvantages, related to proper scaling \citep{Wood2004} and
problems in maintaining appropriate similarities \citep{Harris1999} for first
order important phenomena, can render scaled models ineffective, when used for physical
simulations (under onestep or multiplestep gravity loading).
%
In what follows, a brief literature review is provided.
%
The literature review comprises sections on observations of liquefaction
behavior in case studies, noncontinuum modeling efforts,
review of
redistribution of voids and pore fluid volume/pressures phenomena and
continuum
modeling efforts.
% %
% In addition to that, a brief comment on appropriateness of use of physical
% models for liquefaction modeling is given as well.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\subsection{Background}
%\label{background}
%
%
%
% %%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \subsection{Literature Review}
% \label{LitRev}
%
%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
% Behavior of piles in liquefiable soil has been investigated for some time now.
% However, numerical modeling and simulations of such behavior has been done only
% fairly recently.
% %
% It is noted that inadequate performance of piles in liquefied soils
% has been the main cause of a number of bridge failures during earthquakes.
% %
% It should be noted that, opposing to that, excellent performance of pile
% foundation systems in liquefied soils has saved quite a few bridges (few
% examples noted below).
%
% %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\subsection{Observation of Behavior}
\paragraph{Observation of Behavior.}
Liquefaction behavior was observed during a number of earthquakes in the past.
%Listed below are selected references that describe behavior of references that
%
%
During Alaskan Earthquake (1964), liquefaction was the main cause of
severe damage to 92 highway bridges, moderate to light damage to another 49
highway bridges, and moderate to sever damage to 75 railroad bridges \citep{Youd89}.
%
During Niigata Earthquake (1964) liquefaction induced
damage to foundation piles under Yachiya bridge \citep{Hamada92}.
%
During that same earthquake, girders of Showa Bridge toppled as the
support structure and piles moved excessively due to liquefaction
\citep{JSCE66}.
%
During Kobe Earthquake (1995), liquefaction was the primary cause of damage to
many pile supported or caisson supported bridges and structures.
%
For example, ShinShukugawa bridge was subjected to excessive pile foundation
movement due to liquefaction \citep{Yokoyama97}.
Opposed to these failures and collapses, there were a number of bridges with
pile foundations that did not suffer much or even minor damage even though there was
liquefaction around foundations.
%
For example, pile foundations of the Landing Road Bridge in New Zealand
performed quite well during Edgecumbe earthquake (1987) even with a
significant liquefaction recorded \citep{Berril97, Dobry2001}.
%
In addition to that, Second Maya Bridge piles (large steel pipes) were not
damaged during Kobe earthquake despite significant liquefaction in surrounding
soils \citep{Yokoyama97}.
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \vspace*{0.3cm}
% \paragraph{Piles in Liquefied Soils.}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\subsection{NonContinuum Modeling Efforts}
\paragraph{NonContinuum Modeling Efforts.}
Modeling and simulation of piles in liquefied grounds has been focus of a
number of recent studies.
%
The simple approach, based on scaling of py springs has been suggested early
by \citet{JRA80}, \citet{AIJ88}, \citet{Liu95}, \citet{Miura89} and
\citet{Miura91}.
%
However, large inconsistencies with material parameter selection are present when
py spring approach is used for piles in liquefied soils.
%
Since py methodology for liquefied soils is not based on rational mechanics,
appropriate choice of material parameters is primarily based on
empirical observations of behaviors of piles in liquefied soils in experimental
studies.
%
A number of experimental studies have carefully examined pile behaviors in
liquefiable soils. We mention \citet{Tokida92}, \citet{Liu95}, \citet{Abdoun97},
\citet{Horikoshi98} and \citet{Boulanger06}.
%
Studies using physical model can be used to obtain very high quality data on behavior of piles
in liquefied soils, provided that similarity of important physical phenomena is
maintained \citep{Wood2004, Harris1999}.
%
% %Comments on physical modeling of models is provided in some more
% %details below in section \ref{Comments}.
%
%
%
%
% In recent years, numerical modeling of piles in liquefied soils has
% relied on beam models on soil springs.
Some of the recent papers that discussed use of these models and gave
recommendations about parameter choices are listed for reference:
\citet{Tokimatsu98},
\citet{Martin02},
\citet{Dobry03},
\citet{Liyanapathirana05a},
\citet{Rollins05},
\citet{Cubrinovski06},
\citet{Brandenberg07a}.
%\citet{Tokimatsu03},
% %Some of the differences in
% %recommended parameter relationships and loading conditions are important to
% %practice and warrant further study to resolve.
%
% The currently proposed simulation (design) guidelines are mainly for
% two major approaches \citep{Boulanger07}: (1) piles in laterally spreading ground where the
% outofplane thickness of the spreading ground is sufficiently large
% that freefield soil displacements are relatively unaffected by the presence of the
% piles and (2) piles at approach embankments where the restraining or
% \emph{pinning} effects of the piles and bridge superstructure can reduce the
% lateral spreading demands imposed by the finite width embankment. Guidance for
% both situations is limited to equivalent static analysis using beam on nonlinear
% Winkler foundation (BNWF) analysis methods, with recognition that nonlinear
% dynamic analysis may be warranted for important bridges.
%
% The static BNWF analysis methods has been proposed in two major forms: (a)
% BNWFSD with imposed soil displacements and (b) BNWFLP with imposed limit
% pressures, The first approach is preferred by some researchers, most
% notably by
% \citet{Boulanger07}. In reality, however, for both of these cases there is a
% coupling between the pile structure and the surrounding soil which in result
% determines the so called imposed soil displacements and limit pressures.
% %Moreover, there are significant uncertainties involved in predicting
% %liquefactioninduced ground displacements for lateral spreads.
%
%
% While the problem of piles in liquefiable soils has only recently started to
% receive increased attention, there is already sizable body of work, as noted
% above.
% %
% %What is missing, and is one of the main parts of this proposal, is
% %the development of methodology that will allow for simulations of problems of
% %piles in liquefied soils, using (already existing) state of the art
% %computational geomechanics formulations and implementation.
% %
% %
% %%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\subsection{Redistribution of Voids and Pore Fluid Volume/Pressures}
\paragraph{Redistribution of Voids and Pore Fluid Volume/Pressures.}
Mechanics of pile behavior in
liquefiable grounds is based on the concept of redistribution of voids and
pore fluid volume/pressures (RVPFVP).
%
It should be emphasized that geomechanics phenomena of redistribution
of voids  pore fluid
volume/pressure is used here in purely mechanistic way.
%
That is, RVPFVP is a phenomena
that occurs in saturated soils and that phenomena is responsible for (is manifested in)
liquefaction related soil behaviors with or without piles.
%
%
This is noted as in some recent publications, RVPFVP terminology
is explicitly used for problems
of liquefaction induced failures of sloping grounds without piles.
%
Our understanding of the RVPFVP phenomena is that RVFVP is responsible for
many more facets of behavior of liquefied soils,
rather than only failure of liquefied slopes.
The early investigation of the RVPFVP phenomena was related to the
behavior of infinite slopes.
%
For example, loss of shear strength in infinite slopes is one of the
early understood manifestations of RVPFVP \citep{Whitman85,NRC85,Malvick06}.
%
Laboratory investigation of sand was also used to observe the RVPFVP phenomena
\citep{Casagrande78,Gilbert84}
% In this particular case of liquefiable slopes, parts of liquefied soil will locally loosen
% while other parts will densifies \citep{Whitman85}, \citep{NRC85}. The loosening and
% subsequent strength loss (from RVPFVP) is a consequence of water
% seepage due to development of excess pore pressure and restriction of flow due to
% local permeability conditions. The slope can undergo significant deformations
% if the loss of strength associated with loosening is large enough to reduce the
% undrained strength below levels required for stability of the
% slope \citep{Malvick06}.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\subsection{Continuum Modeling Efforts}
\paragraph{Continuum Modeling Efforts.}
Continuum based formulations for modeling liquefaction problems have been
present for over two decades.
%
In a landmark paper, \citet{Zienkiewicz84} presented three possible
coupled formulations for modeling of soil skeleton  pore fluid problems.
%
The most general and complete one is the so called upU formulations while
the other two, the up and the uU have a number of restrictions on the
domain of application. Here, the unknowns are the soil skeleton displacements
$u$; the pore fluid (water) pressure $p$; and the pore fluid (water)
displacements $U$. The up formulation captures the
movements of the soil skeleton and the change of the pore pressure, and is the most
simplistic one of the three mentioned above. This formulation neglects the
differential accelerations of the pore fluid (it does account for acceleration
of pore fluid together with soil skeleton, but not the separate one if it
exists), and in one version neglects the compressibility
of the fluid (assuming complete incompressibility of the pore fluid). In the
case of incompressible pore fluid, the formulation requires
special treatment of the approximation function (shape function) for pore fluid to
prevent the volumetric locking \citep{Ziekienwicz2000a}.
The majority of the currently available implementations are based on this
formulation. For example \cite{Elgamal2002} and \cite{Elgamal2003}
developed an implementation of the up formulation with the multisurface
plasticity model by \cite{Prevost85}, while \cite{Chan88} and \cite{Zienkiewicz98}
used generalized theory of plasticity \cite{Pastor90}.
%
The uU formulations tracks the movements of both the soil skeleton and the
pore fluid. This formulation is complete in the sense of basic variables, but
might still experience numerical problems (volumetric locking) if the difference in volumetric
compressibility of the pore fluid and the solid skeleton is large.
The upU formulation resolves the issues of volumetric locking by including
the
displacements of both the solid skeleton and the pore fluid, and the pore
fluid pressure as well.
This formulation uses additional dependent unknown field of pore fluid
pressures to stabilize the solution of the coupled system. The pore fluid
pressures are connected to (dependent on) displacements of pore fluid. With known (given)
volumetric compressibility of the pore fluid, pore fluid pressure can be calculated.
%
Despite it's power, the upU formulation has rarely been implemented into
finite element code, and has never (at least to our knowledge) been used to
analyze pile  liquefied soil interaction.
%
This can be attributed in part to a sophistication of implementation that is
required, and to a sizable increase in computational cost for upU elements.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Numerical Formulation, Verification and Validation}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Finite Element Formulation }
The discretized, finite element system of equations for upU formulation,
which is based on earlier work by \citet{Zienkiewicz84},
can be written
%as
%%%
%\begin{eqnarray}
%\left[ \begin{array}{ccc}
%\bf{M_s} & 0 & 0 \\
% 0 & 0 & 0 \\
% 0 & 0 & \bf{M_f}
%\end{array} \right]
%\left[ \begin{array}{c}
%\bf{\ddot{\bar{u}}} \\
%\bf{\ddot{\bar{p}}} \\
%\bf{\ddot{\bar{U}}}
%\end{array} \right]
%+
%\left[ \begin{array}{ccc}
%\bf{C_1} & 0 & \bf{C_2} \\
% 0 & 0 & 0 \\
%\bf{C_2^T} & 0 & \bf{C_3} \\
%\end{array} \right]
%\left[ \begin{array}{c}
%\bf{\dot{\bar{u}}} \\
%\bf{\dot{\bar{p}}} \\
%\bf{\dot{\bar{U}}}
%\end{array} \right] \nonumber\\
%+
%\left[ \begin{array}{ccc}
%0 & \bf{G_1} & 0 \\
%\bf{G_1^T} & \bf{P} & \bf{G_2^T} \\
% 0 & \bf{G_2} & 0
%\end{array} \right]
%\left[ \begin{array}{c}
%\bf{\bar{u}} \\
%\bf{\bar{p}} \\
%\bf{\bar{U}}
%\end{array} \right]
%+
%\left[ \begin{array}{c}
%\bf{f_{in}^u(u)} \\
%0 \\
%0
%\end{array} \right]
%
%\left[ \begin{array}{c}
%\bf{\bar{f}^u } \\
%0 \\
%\bf{\bar{f}^U}
%\end{array} \right]
%=
%0
%\label{67}
%\end{eqnarray} %%
%or
in tensor index form
%% %%
%% \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{\bar{u}}_{Lj} \\
%% \ddot{\bar{p}}_N \\
%% \ddot{\bar{U}}_{Lj}
%% \end{array} \right]
%% +
%% \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{\bar{u}}_{Lj} \\
%% \dot{\bar{p}}_N \\
%% \dot{\bar{U}}_{Lj}
%% \end{array} \right] \nonumber \\
%% &+&
%% \left[ \begin{array}{ccc}
%% 0 & (G_1)_{KiM} & 0 \\
%% (G_1)_{LjM} & P_{MN} & (G_2)_{LjM} \\
%% 0 & (G_2)_{KiL} & 0
%% \end{array} \right]
%% \left[ \begin{array}{c}
%% \bar{u}_{Lj} \\
%% \bar{p}_M \\
%% \bar{U}_{Lj}
%% \end{array} \right] \nonumber \\
%% &+&
%% \left[ \begin{array}{c}
%% \int_{\Omega} N_{K,j}^u \sigma''_{ij} \ud \Omega \\
%% 0 \\
%% 0
%% \end{array} \right]
%% 
%% \left[ \begin{array}{c}
%% \bar{f}^u_{Ki} \\
%% 0 \\
%% \bar{f}^U_{Ki}
%% \end{array} \right]
%% = 0
%% \label{68}
%% \end{eqnarray}
%% %%
%%
%%
%%
\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{\bar{u}}_{Lj} \\
\ddot{\bar{p}}_N \\
\ddot{\bar{U}}_{Lj}
\end{array} \right]
+
\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{\bar{u}}_{Lj} \\
\dot{\bar{p}}_N \\
\dot{\bar{U}}_{Lj}
\end{array} \right] \nonumber \\
+
\left[ \begin{array}{ccc}
0 & (G_1)_{KiM} & 0 \\
(G_1)_{LjM} & P_{MN} & (G_2)_{LjM} \\
0 & (G_2)_{KiL} & 0
\end{array} \right]
\left[ \begin{array}{c}
\bar{u}_{Lj} \\
\bar{p}_M \\
\bar{U}_{Lj}
\end{array} \right]
+
\left[ \begin{array}{c}
\int_{\Omega} N_{K,j}^u \sigma''_{ij} \ud \Omega \\
0 \\
0
\end{array} \right]

\left[ \begin{array}{c}
\bar{f}^u_{Ki} \\
0 \\
\bar{f}^U_{Ki}
\end{array} \right]
= 0
\nonumber \\
\label{68}
\end{eqnarray}
%%
%%where
%%%%
%%\begin{equation}
%% \bf{ f_{in}^u(u)} = \int_{\Omega} N_{K,j}^u \sigma''_{ij} \ud \Omega
%%\end{equation} %%
where the components are
%%% %%%
%%% \begin{eqnarray}
%%% % \bf{M_s} &=&
%%% (M_s)_{KijL} &=& \int_{\Omega} N_K^u (1n) \rho_s \delta_{ij} N_L^u \ud \Omega \nonumber\\
%%% % \bf{M_f} &=&
%%% (M_f)_{KijL} &=& \int_{\Omega} N_K^U n \rho_f \delta_{ij} N_L^U \ud \Omega \nonumber\\
%%% % \bf{C_1} &=&
%%% (C_1)_{KijL} &=& \int_{\Omega} N_K^u n^2 k_{ij}^{1} N_L^u \ud \Omega \nonumber\\
%%% % \bf{C_2} &=&
%%% (C_2)_{KijL} &=& \int_{\Omega} N_K^u n^2 k_{ij}^{1} N_L^U \ud \Omega \nonumber\\
%%% % \bf{C_3} &=&
%%% (C_3)_{KijL} &=& \int_{\Omega} N_K^U n^2 k_{ij}^{1} N_L^U \ud \Omega \nonumber\\
%%% % \bf{G_1} &=&
%%% (G_1)_{KiM} &=& \int_{\Omega} N_{K,i}^u (\alphan) N_M^p \ud \Omega \nonumber\\
%%% % \bf{G_2} &=&
%%% (G_2)_{KiM} &=& \int_{\Omega} n N_{K,i}^U N_M^p \ud \Omega \nonumber\\
%%% % \bf{P} &=&
%%% P_{NM} &=& \int_{\Omega} N_N^p \frac{1}{Q} N_M^p \ud \Omega
%%% \label{691}
%%% \end{eqnarray}
\begin{eqnarray}
% \bf{M_s} &=&
(M_s)_{KijL} = \int_{\Omega} N_K^u (1n) \rho_s \delta_{ij} N_L^u \ud \Omega
\;\;&\mbox{;}&\;\;
% \bf{M_f} &=&
(M_f)_{KijL} = \int_{\Omega} N_K^U n \rho_f \delta_{ij} N_L^U \ud \Omega \nonumber\\
% \bf{C_1} &=&
(C_1)_{KijL} = \int_{\Omega} N_K^u n^2 k_{ij}^{1} N_L^u \ud \Omega
\;\;&\mbox{;}&\;\;
% \bf{C_2} &=&
(C_2)_{KijL} = \int_{\Omega} N_K^u n^2 k_{ij}^{1} N_L^U \ud \Omega \nonumber\\
% \bf{C_3} &=&
(C_3)_{KijL} = \int_{\Omega} N_K^U n^2 k_{ij}^{1} N_L^U \ud \Omega
\;\;&\mbox{;}&\;\;
% \bf{G_1} &=&
(G_1)_{KiM} = \int_{\Omega} N_{K,i}^u (\alphan) N_M^p \ud \Omega \nonumber\\
% \bf{G_2} &=&
(G_2)_{KiM} = \int_{\Omega} n N_{K,i}^U N_M^p \ud \Omega
% \bf{P} &=&
\;\;&\mbox{;}&\;\;
P_{NM} = \int_{\Omega} N_N^p \frac{1}{Q} N_M^p \ud \Omega \nonumber \\
% \bf{f^u} &=&
(\bar{f}_s)_{Ki} = (f_1^u)_{Ki}  (f_4^u)_{Ki} + (f_5^u)_{Ki}
% \bf{f^U} &=&
\;\;&\mbox{;}&\;\;
(\bar{f}_f)_{Ki} =  (f_1)_{Ki} + (f_2)_{Ki} \nonumber\\
%
(f_1^u)_{Ki} =
\int_{\Gamma_t} N_K^u \sigma''_{ij} r_j \ud \Gamma
\;\;&\mbox{;}&\;\;
(f_4^u)_{Ki} =
\int_{\Gamma_p} N_K^u (\alphan) p r_i \ud \Gamma \nonumber\\
%
(f_5^u)_{Ki} =
\int_{\Omega} N_K^u (1n) \rho_s b_i \ud \Omega
\;\;&\mbox{;}&\;\;
(f_1^U)_{Ki} =
\int_{\Gamma_p} N_K^U n p r_i \ud \Gamma \nonumber\\
%
(f_2^U)_{Ki} =
\int_{\Omega} N_K^U n \rho_f b_i \ud \Omega
\label{691}
\end{eqnarray}
%%
Here $N^u$, $N^p$, $N^U$ are shape functions of solid skeleton displacement, pore pressure
and fluid displacement, respectively;
$\rho$, $\rho_s$, $\rho_f$ are density of the total, solid and fluid phases, respectively;
$n$ is porosity,
the symbol $r_i$ is the direction of the normal vector on the boundary;
$\bar{u}_{Lj}$ are the nodal displacements of the solid part;
$\bar{p}_M$ are the nodal pore pressures and
$\bar{U}_{Lj}$ are the nodal displacements of the fluid part.
%
Moreover, $\Omega$ represents the domain of interest; $\Gamma_t$ is the traction boundary,
and $\Gamma_p$ is the pressure boundary.
%
%
Set of dynamic Equations (\ref{68}) represents the most general formulation
and discretization for a material nonlinear (inelastic) porous medium (soil skeleton)
that is fully saturated with linear elastic, compressible pore fluid
(water).
%
%There are number of feature of this formulation and discretization that deserve
%further comments.
%
%
Water accelerations are explicitly taken into account, both for
conforming and differential movements with respect to the soil skeleton. This
proves to be very important for models where porous soil is adjacent to
structural foundations (piles for example). In these models, the dynamics of two
model components, saturated soil and piles, are quite different and there exists
a possibility of significant relative movement (displacements, velocities and
accelerations) between soil skeleton and the concrete piles or footings. If this
relative movement exists, pore water is pumped in and out of soil skeleton, thus
creating a significant differential accelerations relative to the soils
skeleton.
Both velocity proportional and displacement proportional damping follow
directly from formulation and discretization.
%
The velocity proportional damping is taken into account through damping matrix
in Equation (\ref{68}). Of particular interest are submatrices (tensors)
$(C_1)_{KijL}$, $(C_2)_{KijL}$ and $(C_3)_{KijL}$ (Equations~(\ref{691})).
Physically, those submatrices represent coupling of pore water and solid
skeleton, which is velocity proportional. This coupling is a function of
permeability $k$ and porosity $n$. For example, from Equation~(\ref{691}), it
follows that for a soil with larger porosity $n$,
which has more pores in a soil skeleton and therefore
more pathways for pore water to travel through the soil, the damping will be
higher. Similarly, for a soil with smaller permeability, where pore water has
more difficulty in traveling through pores, where there is more friction of
flowing pore fluid with the soil skeleton, the velocity proportional
damping will be increased.
%
Displacement proportional damping is controlled by inelastic
material behavior, which in turn is controlled in the finite element
discretization by the nonlinear, elasticplastic stiffness matrix, given as
$\int_{\Omega} N_{K,j}^u \sigma''_{ij} \ud \Omega$ (see
Equation~(\ref{691})) here in order to correctly
account for general, nonlinear dynamic time integration \citep{local87}.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsubsection{Time Integration}
\label{Time Integration}
Numerical integration of Equations of motion (Equation \ref{68}) is done
using Newmark \citep{Newmark59} or HilberHughesTaylor
\citep{HHT,Hughes1978a,Hughes1978b} algorithms. Of particular importance is the
proper algorithmic treatment for nonlinear analysis which
introduces changes in a way residual forces are calculated \citep{local87,
Jeremic2007e}.
The finite element discretization in Equation~(\ref{68}) defines a damping matrix,
which takes into account physics of velocity dependent interaction of pore water
and soil skeleton. This damping matrix is more appropriately used here than damping
matrix introduced through Rayleigh damping \citep{Chopra01}.
It should be note that a small amount of numerical damping is used with both
Newmark and HilberHughesTaylor methods, in order to damp out response in higher frequencies
that is introduced by the spatial finite element discretization
\citep{local86}.
%
%
%
% Equation \ref{68} can be written in a simplified residual form
% %%%
% \begin{eqnarray}
% R = M \ddot{x} + C \dot{x} + K' x + F(x)  f = 0
% \label{71}
% \end{eqnarray}
% %%%
% where $x = \{\bar{u}_{Lj}, \; \bar{p}_M, \; \bar{u}_{Lj}\}^T$ represents the generalized unknown variables.
% Equation \ref{71} represents a general nonlinear form.
%
% Numerical damping introduced in the Newmark time integration method
% will degrade the order of accuracy.
% The HilberHughesTailor (HHT) time integration $\alpha$method
% (Hilber et al \cite{HHT}, Hughes and Liu \cite{Hughes1978a} and \cite{Hughes1978b})
% using an alternative residual form
% by introducing an addition parameter $\alpha$ to improve the performance.
% The current residual at time $n+1$ (for simplicity, the left superscripts $n+1$ are omitted)
% can be estimated from the previous variables at time $n$ by:
% %%%%
% \begin{eqnarray}
% R = M \; \ddot{x}
% + (1 + \alpha) [C \dot{x} + K' x + F(x)]
%  \alpha [C\;{}^{n}\dot{x} + K'\;{}^{n}x + F({}^{n}x)]
%  \; f
% \label{upU_HHT_R}
% \end{eqnarray}
% %%%%
% but retaining the Newmark finitedifference formulas
% and parameters $\beta$ and $\gamma$:
% %%
% \begin{eqnarray}
% x &=& {}^n x + \Delta t \; {}^n \dot{x} + \Delta^2 t \; [(\frac{1}{2}  \beta) \; {}^n \ddot{x} + \beta \; \ddot{x}]
% \label{upU_Newmark_EQ1} \\
% \dot{x} &=& {}^n \dot{x} + \Delta t \; [(1  \gamma) \; {}^n \ddot{x} + \gamma \; \ddot{x}]
% \label{upU_Newmark_EQ2}
% \end{eqnarray}
% %%
% If $\alpha = 0$,
% Equation \ref{upU_HHT_R} is reduced to
% the special case of the HHT time integration method
% which is exactly the Newmark time integration method.
% Decreasing $\alpha$ value increase numerical dissipation (Hughes \cite{local86}).
%
% The iteration procedure is
% \begin{eqnarray}
% \left[ M + (1+\alpha) \gamma \Delta t C + (1+\alpha) \beta \Delta t^2 K \right] \Delta \ddot{x} =  R
% , \;\;\;
% \ddot{x} = \; {}^n \ddot{x} + \Delta \ddot{x}
% \label{upU_HHT_Newton}
% \end{eqnarray}
% %%%
% where
% \begin{eqnarray}
% K = K' + \frac{\partial F}{\partial x}
% \end{eqnarray}
%
% If the parameters $\alpha$, $\beta$ and $\gamma$ satisfy
% %%%%
% \begin{equation}
% \frac{1}{3} \le \alpha \le 0, \;\;\; \gamma = \frac{1}{2}(12\alpha), \;\;\; \beta = \frac{1}{4}(1\alpha)^2
% \label{upU_HHT_ab}
% \end{equation}
% %%%
% it is unconditionally stable and secondorder accurate (Hughes \cite{local86}).
%
% % There are different denotation meaning of the parameter $\alpha$,
% % e.g. the parameter $\alpha$ in the HHT integrator codes of OpenSees
% % equals to the conventional $\alpha$ plus one.
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsubsection{Material Model}
\label{Material_Model}
Material model plays one of the key roles in numerical simulation of the dynamic
behaviors of liquefiable soil. Correct modeling of volumetric response by a good
model on the constitutive level allows for accurate modeling of the boundary
value problems where liquefaction is involved.
%
In this work, a critical state soil mechanics based model developed by
\citet{Manzari97} and \citet{Dafalias04} is used.
%
%This model considered the effects of the state parameter on the behaviors
%of the dense or loose sands.
Among many excellent features of this model we note the capability to
utilize a single set of material parameters for a wide range of void
ratios and stress states for the same soil. This feature allows
the same material parameters to be used from the very
beginning of loading (from zero stress/strain state), through self
weight, pile construction and finally dynamic shaking. In addition to that,
model validation for Toyoura sand, used in this study, shows excellent
agreement with test data \citep{Jeremic2007e}.
%
It is emphasized again that DafaliasManzari set of models used here, together
with powerful constitutive integrations,
upU formulation and proper dynamic equation of motion time integration
\citep{Jeremic2007e}, is capable to follow soil material from zero
stress/strain state (before applying self weight) all the way to
preliquefaction, liquefaction and post liquefaction behaviors. This transition
%%
\begin{itemize}
%
\item from preliquefaction, where mean effective stress
$p^{\prime}(=\sigma^{\prime}_{kk}/3)$ is being continuously reduced (as pore water
pressure is increased) to become numerically almost zero,
%
\item through liquefaction, where mean effective stress $p^{\prime}$ is numerically
almost zero,
%
\item to post liquefaction, where mean effective stress $p^{\prime}$ is
increasing (as pore water pressure is decreasing due to diffusion),
%
\end{itemize}
%%
is successfully handled by the material model with the help of proper
constitutive and dynamic level finite element integrations. Each of the above
phases results in proper change (evolution) of the void
ratio and the soil skeleton fabric, representing material internal variables.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Verification and Validation }
\label{VandV}%
Confidence in numerical prediction is firmly based on a valid
process of verification and validation \citep{Oberkampf2002}.
%
Verification provides
evidences that the model is solved correctly. Verification is also meant to
identify and remove errors in computer coding and verify numerical algorithms
and is desirable in quantifying numerical errors in computed solution.
%
Validation provides evidences
that the correct model is solved. It is custom to identify verification
with mathematics of the problem and validation with the physics of the problem.
Computational simulation tools used in this study underwent detailed
verification and validation program.
%
Validation consisted of applying a comprehensive set of software verification
tools, available in public domain and/or developed at the Computational
Geomechanics Lab at UC Davis to developed libraries and programs.
%%for removing errors in implementation.
%
In addition to that, a number of closed form solutions were used to verify that
our models were solved correctly. A set of problems with solutions
developed by \citet{Coussy95} were numerically modeled and
simulated during validation process \citep{Jeremic2007e}.
%
Mentioned are verification problems of 1D consolidation,
line injection of fluid in a reservoir, and cavity expansion in saturated medium
(2D and 3D).
In addition to those problems which dealt with relatively slow phenomena
without significant influence of inertial forces, a truly dynamic problem of
shock wave propagation in porous medium was also used for verification
\citep{Gajo1995,Gajo1995b}.
%
Verification examples using shock wave propagation are most
sever tests of the upU formulation and the numerical integration
algorithms for the dynamic equations of motions.
%
It should be noted that these verification test provided excellent, close
matching of numerical and closed for solutions, within the known limitations
of numerical accuracy of finite element discretization.
%
This limitation in accuracy is a simple and expected consequence of
approximate nature of the finite element method \citep{FEMIV1, FEMIV2}, and
the finite precision arithmetic used in computer calculations
\citep{Schnabel83}.
Validation was done by simulating constitutive behaviors of sand material used in
predictions (Toyoura sand). Comparison of validation predictions with physical
test data \citep{Jeremic2007e} shows very good predictive
capabilities of the material model as well as of the underlying numerical
integration algorithms on the constitutive level.
\section{Staged Simulation Model Development}
%\section{Simulation Model Development}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Model development for a pile in the liquefiable soil follows physics (mechanics) of
the problem as close as possible.
%
Numerical simulation of such problems in geomechanics is usually based on stages of
loading and increments within those stages.
%
%Hence, modeling follows the load staging in detail.
%
% Load stages are centered around the assumed past history of loading on a soil
% pile system.
%
%
%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \subsection{Staged Model Construction}
All load stages are applied to a series of finite element models, all of
which share features of an initial soil model.
%
This initial soil model consists of a soil block with dimension of
$12 \times 12 \times 15$~m (length $\times$ width $\times$ depth).
%
Due to the symmetry of the model, only half of the block is modeled.
%
Symmetry assumptions is based on assumption that all the loads, dynamic
shaking and other influences are symmetric with respect to the plane of symmetry.
%
This specialization
to symmetric model reduces model generality (for example this use of symmetry will
preclude analysis of dynamic shaking perpendicular to sloping ground dip).
%
However, as our goal is to present a methodology of analyzing behavior of piles in
liquefying ground, this potential drawback is not deemed significant in this
study.
%
Finite element mesh for the model is presented in Figure~(\ref{FEmesh3D}).
The initial mesh consists of 160 eight node upU elements.
%
%
\begin{figure}[!htbp]
\begin{center}
\includegraphics[width=0.43\textwidth]{NewFiga/FEmesh3D.eps}
\hfill
\includegraphics[width=0.52\textwidth]{NewFiga/FEmeshPileBeam.eps}
\caption{
\label{FEmesh3D}
{Left: Three dimensional finite element mesh featuring initial soil setup, where all
the soil elements are present. The gray region of elements is excavated
(numerically) and replaced by a pile during later stages of loading; Right:
Side view of the pilesoil model with some element and node annotation, used
to visualize results.}
}
\end{center}
\end{figure}
%
Each node of the mesh has 7 degrees of freedom, three for soil
skeleton displacements ($u_i$), one for pore water pressure ($p$), and three
for pore water displacement ($U_i$).
%
While it can be argued that the mesh is somewhat coarse, it is well refined
around the pile, yet to be installed, in place of gray region in the middle.
%
%%
\begin{table}[tbh]
\begin{center}
\caption{ Material parameters used for DafaliasManzari elasticplastic model. }
% {\normalsize
\begin{tabular}
[c]{lcclcc}\hline
\multicolumn{2}{l}{Material Parameter} & Value &
\multicolumn{2}{l}{Material Parameter} & Value\\\hline
Elasticity & $G_{0}$ & 125~kPa & Plastic modulus & $h_{0}$ & 7.05\\
& $v$ & 0.05 & & $c_{h}$ & 0.968\\
Critical sate & $M$ & 1.25 & & $n_{b}$ & 1.1\\
& $c$ & 0.8 & Dilatancy & $A_{0}$ & 0.704\\
& $\lambda_{c}$ & 0.019 & & $n_{d}$ & 3.5\\
& $\xi$ & 0.7 & Fabricdilatancy & $z_{max}$ & 4.0\\
& $e_{r}$ & 0.934 & & $c_{z}$ & 600.0\\
Yield surface & $m$ & 0.02 & & & \\\hline
\end{tabular}
% }
\end{center}
\label{DMTest_parameters}%
\end{table}
%%
%%
%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{table}[tbh]
\begin{center}
\caption{ \label{OneC_Material_Parameter}
Additional parameters used in FEM simulations.}
\begin{tabular}[c]{ccc} \hline
\multicolumn{2}{c}{Parameter} & Value \\ \hline
Solid density & $\rho_s$ & $2800~kg/m^3$ \\
Fluid density & $\rho_f$ & $1000~kg/m^3$ \\
Solid particle bulk modulus & $K_s$ & $1.0 \times 10^{12}~kN/m^2$ \\
Pore fluid bulk modulus & $K_f$ & $2.2 \times 10^{6}~kN/m^2$ \\
%Biot coefficient & $\alpha$ & $1.0$ \\
permeability & $k$ & $1.0\times 10^{4}$~$m/s$ \\
Gravity & $g$ & 10~m/s$^2$ \\ \hline
%
\end{tabular}
\end{center}
\end{table}
%
A single set of parameters is used with the DafaliasManzari material model.
Soil is modeled as Toyoura sand and material parameters (summarized in Table
\ref{DMTest_parameters}) are calibrated using tests by
\citet{Verdugo1996}, while initial void ration was set to $e_0 = 0.80$.
%
It is very important to emphasize that the state of stress
and internal variables from initial state (zero for stress and given value for
void ratio and fabric) will evolve through all stages of loading
by proper modeling and algorithms,
by using single set of material parameters.
%
%
%
Table \ref{OneC_Material_Parameter} presents additional parameters, other than material
parameters presented in Table \ref{DMTest_parameters}, used for numerical
simulations.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{First Loading Stage: Self Weight}
The initial stage of loading is represented by the application of self weight on
soil, including both the soil skeleton and the pore water.
%
Initial state in soil before application of self
weight is of a zero stress and
strain while void ratio and fabric are given initial
values.
%
The state of stress/strain, void ratio and fabric will evolve upon application of self weight.
%
At the end of self weight loading stage, soil is under appropriate state of
stress ($K_0$ stress), the void ratio corresponds to the void ratio after self
weight (redistributed such that soil is denser at lower layers),
while soil fabric has evolved with respect to stress induced anisotropy.
%
All of these changes are modeled using DafaliasManzari
material model and using constitutive and finite element level integration
algorithms developed within UC Davis Computational Geomechanics group in recent
years.
%
Boundary conditions (BC) for self weight stage of loading are set in the following
way:
%
\begin{itemize}
\item Soil skeleton displacements ($u_i$), are fixed in all three directions
at the bottom of the model.
At the side planes, nodes move only vertically to mimic selfweight effect.
All other nodes are free to move in any direction.
%
%free to move only vertically at four
%sides of the model, and are free to move in all three directions at all other
%nodes.
\item Pore water pressures ($p$), are free to develop at the bottom plane and at
all levels of the models except at the top level at soil
surface where they are fixed (set to zero, replicating drained condition),
\item Pore water displacements ($U_i$), are fixed in all three directions at
the bottom, are free to move only vertically at four sides of the
model and are free to move in any direction at all other nodes.
\end{itemize}
%
%
These boundary condition are consistent with initial selfweighting deformation
condition for soil and pore water at the site.
For the case of sloping ground, an additional load substage is applied after self
weight loading, in order to mimic self weight of inclined (sloping) ground.
%
This is effectively achieved by applying a resultant of total self weight of
the soil skeleton times the sine of the inclination angle at uphill side of
the model. This load is applied only to the solid skeleton DOFs, and not on the
water DOFs. Physically it would be correct to consider the sloping ground
effects on the pore water as well. This will create a constant flow field of
the water downstream, which, while physically accurate, is small enough that it
does not have any real effect on modeling and simulations performed here.
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Second Loading Stage: PileColumn Installation}
\label{pile_installation}
After the first loading stage, comprising self weight applications (for level or sloping
ground, as discussed above), second loading stage includes installation
(construction) of the pilecolumn.
Modeling changes performed during loading stage included:
\begin{itemize}
\item Excavation of soil occupying space where the pile will be installed. This
was done by removing elements, nodes and loads on elements shown
in gray in Figure~(\ref{FEmesh3D}).
\item These elements were replaced by very soft set of
elements with small
stiffness, low permeability. This was done in order to prevent water from
rushing into the newly opened hole in the ground after original soil elements
(used in the first loading stage) are removed.
\item Installation of a pile in the ground and a superstructure (column) above the
ground. Nonlinear beancolumn elements were used for both pile and column
together with addition of appropriate nodal masses at each beamcolumn node,
and with the addition of a larger mass at the top representing lumped
mass of a bridge superstructure. Pile beamcolumn elements were connected with
soil skeleton part of soil elements using a specially devised technique.
\end{itemize}
As mentioned earlier, the volume that would be physically occupied by the pile
in the pile hole,
is ``excavated" during this loading stage.
%
Beamcolumn elements, representing piles, are then placed in the middle of this
opening. Pile (beamcolumn) elements are then connected to the
surrounding soil elements by means of stiff elastic beamcolumn
elements. These ``connection" beamcolumn elements extend from each pile
node to surrounding nodes of soil elements. The
connectivity of nodes to soil
skeleton nodes is done only for three beamcolumn translational DOFs,
while the three rotational DOFs from the
beamcolumn element are left unconnected. These three DOFs from the
beamcolumn side are connected to first three DOFs of the upU soil
elements, representing displacements of the soil skeleton ($u_i$).
%
Water displacements ($U_i$) and pore water pressures ($p$) are not connected in
any way. Rather, these two sets of DOFs representing pore water behave in a
physical manner (cannot enter newly created hole around pile beamcolumn
elements) because of the addition of a soft, but very impermeable set of upU
elements, replacing excavated soil elements. By using this method, both solid
phase (pile with soil skeleton) and the water phase (pore water within
the soil) are appropriately modeled.
%
%
Figure~(\ref{coupling}) shows in some detail schematics of coupling between
the pile and soil skeleton part finite elements.
%
%
\begin{figure}[!htbp]
\begin{center}
%{\includegraphics[width=8cm]{/home/jeremic/tex/works/Papers/2008/Pile_in_liquefied_soil_upU/SFSIModelSetup_upU_01.eps}}
{\includegraphics[width=9cm]{NewFiga/SFSIModelSetup_upU_01.eps}}
\caption{\label{coupling} Schematic description of coupling of displacement DOFs
($u_i$) of beamcolumn element (pile) with displacement DOFs ($u_i$) of upU elements (soil).}
\end{center}
\end{figure}
%
Nonlinear force based beamcolumn elements \citep{Spacone1996a, Spacone1996b} were used for modeling the
pilecolumn.
Pile was assumed to be made of aluminum. This was done
in order to be able to validate simulations with centrifuge experiments (when
they become available). Presented models were all done in prototype scale, while
for (possible future) validation, select results will be carefully scaled and
compared with appropriate centrifuge modeling. Pile and the column were assumed to
have a diameter of $d=1.0$~m, with Young's modulus of $E=68.5$~GPa, yield
strength $f_y = 255$~kPa, and the density $\rho =2.7~{\rm kg/m^3}$. Wall
thickness of prototype pilecolumn is $t = 0.05$~m. Lumped mass of pile and
column was distributed along the beamcolumn nodes, while an additional mass
was added on top $(m = 1200$~kg) that represents (small) part of the
superstructure mass. This particular mass $(m = 1200$~kg) comes from a
standard (scaled up in our case) centrifuge model for pilecolumnmass used at
UCD.
%
% %The critical state characteristics of the material model
% %and thin thickness soil elements around the pile hole can numerically
% %simulate the effects of gapping and slipary between the pile and soil \citep{Desai1984}.
% %The single pile is assumed made of aluminum with
% %the Young's modulus of 68.5 GPa,
% %yield strength 255 kPa,
% %and the density 2.7 kg/$m^3$.
% %The pile cross section is an aluminum tub wrapped with a shrinkage layer.
% %The pile hole diameter is 1.2 m.
% %The aluminum tubbing has an outer diameter of 1.0 m and the wall thickness is 0.05 m.
% %In calculation, the aluminum tubing dimensions are used to define the pile fiber section.
% %The outer shrinkage layer is 0.1 m.
% %The mass of pile are distributed over nodes as lumped masses.
%
%
% %
% % In addition, at the top of the pile has a total structure mass of 1200 Mg.
% % The fiber section is defined by the OpenSees Steel01 material.
% % The pile section is subdivided into 20 segments in the circumferential direction and 10 segments in the radial direction.
% % The selfweight analysis of the pile is followed by soil selfweight analysis.
% % This analysis altogether with the previous stage
% % provide stable stress and pore pressure fields
% % with the pile inside the soils before the earthquake shaking.
%
Figure~(\ref{FEmesh3D}) (right side) shows side view of the columnpilesoil model
after second stage of loading.
%
% \begin{figure}[!htbp]
% \begin{center}
% \includegraphics[width=0.6\textwidth]{NewFiga/FEmeshPileBeam.eps}
% \caption{
% \label{FEmeshPileBeam}
% { Side view of the pilesoil model with some element and node
% annotation, used to visualize results.}
% }
% \end{center}
% \end{figure}
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Third Loading Stage: Seismic Shaking}
After the application of self weight on the uniform soil profile, excavation
and construction of the single pile with column and super structure mass on top
and application of their self weight, the model is at the appropriate initial
state for further application of loading. In this case, this additional loading
comprises seismic shaking.
%
%
%
For this stage, fixed horizontal DOFs used on the side planes
during the first stage are removed (set free).
The input acceleration time history, shown in Figure~(\ref{SinglePileGM})
was taken from the recorded horizontal
acceleration of Model No.1 of VELACS project \cite{Arulanandan93} by Rensselaer
Polytechnic Institute (\url{http::/geoinfo.usc.edu/gees/velacs/}).
%
%%%%%
\begin{figure}[!htbp]
\begin{center}
\includegraphics[width=0.65\textwidth]{NewFiga/GM.eps}
\caption{
\label{SinglePileGM}
{Input earthquake ground motions.}
}
\end{center}
\end{figure}
%%%%%
%
The magnitude of the motion is close to $0.2$~g, while main shaking lasts
for about 12 seconds (from $1$~s to $13$~s).
%
Although the input earthquake motions lasts until approx. $13$~seconds,
simulations are continued until $120$~seconds so that both liquefaction
(dynamic) and pore water dissipation (slow transient) can be appropriately
simulated during and after earthquake shaking \citep{Jeremic2007e}.
%
% In addition to the horizontal level soil ground,
% cases with a small slope soil ground are investigated as well.
% This small slope plays an importance role during and after the earthquake excitation.
% The effects of the small slope are approximately simulated
% by a horizontal distributing load loaded at the left side surface of the soil block,
% which is assumed 3\% of the soil vertical effective pressure at the same levels.
% This side load is added after the analysis of the pile driving and takes
% effects during and after the earthquake shaking.
% The lateral load pushes the soils and causes the soil lateral spread.
% The soil lateral spread thus further pushes the pile and causes the pile deflects.
% The movement of the pile further results in the displacements of the upper structure.
% However, the real mechanism is not so simple during the earthquake,
% because there are interactions between the soils, pile and upper structure.
% it is also interesting to see the soil behaviors subjected to the earthquake loading
% with different pile head conditions, namely, free head and fix head.
% The free pile head condition assumes the pile head have free horizontal and rotational DOFs
% in the lateral direction, which is to simulate condtionas
% similar to the bridge pier movement in the transverse direction.
% The fixed pile head condition assumes the pile head have free horizontal but no rotational DOFs
% in the lateral direction, which is to mimic conditionas
% such as the bridge pier shaking in the longitude direction.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Free Field, Lateral and Longitudinal Models}
\label{six_models}
Six models were developed during the course of this study. First three models
(model numbers I,
II and III) were for level ground, while last three models (model numbers IV, V,
and VI) were for sloping ground. First in each series of models (model I for
level ground and model IV for sloping ground) were left without the second
loading stage, without a pilecolumn system. Other four models (numbers II,
III, V and VI)
were analyzed for all three loading stages. Second in each series of models
(models number II and V) had all displacements and rotations of pilecolumn top
(where additional mass representing superstructure was placed) left free,
without restraints. Thus, these two models represent lateral behavior of a bridge.
Third in each series of models (model numbers III and VI) had rotations in $y$
directions fixed at the pilecolumn top, thus representing
longitudinal behavior of a bridge. Modeling longitudinal behavior of a bridge by
restraining rotations perpendicular to the bridge superstructure is appropriate
if the stiffness of a bridge superstructure is large enough, which in this case
it was, as it was assumed to be a posttensioned concrete box girder, so that realistically,
the top of a column does not rotate (much) during application of loads.
%
Table \ref{CasesOfSiglePile} summarizes models described above.
%
%
%%%%%
\begin{table}[!htbp]
\caption{ \label{CasesOfSiglePile}
Cases descriptions.
}
\begin{center}
\begin{tabular}{ccc}
\hline
Case & Model sketch & Descriptions \\
\hline \hline
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
I
&
\includegraphics[width=0.025\textwidth,angle=0]{Model_I.eps}
&
horizontal ground, no pile \\ \hline
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
II
&
\includegraphics[width=0.025\textwidth,angle=0]{Model_II.eps}
&
horizontal ground, single pile, free column head \\ \hline
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
III
&
\includegraphics[width=0.025\textwidth,angle=0]{Model_III.eps}
&
horizontal ground, single pile, no rotation at column head \\
\hline
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
IV
&
\includegraphics[width=0.025\textwidth,angle=0]{Model_IV.eps}
&
sloping ground, no pile \\ \hline
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
V
&
\includegraphics[width=0.025\textwidth,angle=0]{Model_V.eps}
&
sloping ground, single pile, free column head \\ \hline
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
VI
&
\includegraphics[width=0.025\textwidth,angle=0]{Model_VI.eps}
&
sloping ground, single pile, no rotation at column head \\
\hline
\end{tabular}
\end{center}
\end{table}
%%%%%
%
% Totally six cases are listed in Table \ref{CasesOfSiglePile}.
% For ground conditions,
% cases of horizontal ground without any slope and small slope ground with a slope angle 3\% are considered.
% For pile head conditions,
% cases of no pile, single pile with free pile top, and single pile without rotation at the top are taken into account.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Simulation Results}
\label{Single_Pile_Liquefiable_Soil}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Pore Fluid Migration}
\label{Numerical_Results}
Figures (\ref{Ru_14}) through (\ref{Ru_36}) show the Ru time history for up to
$30$~seconds, for elements (at one of Gauss point) e1, e3, e5 and e7 (refer to
right side of Figure~(\ref{FEmesh3D})).
%
It is important to note that $R_u$ is defined as the ratio of the
difference of initial mean and current mean effective stresses over the initial mean
effective stress:
%
\begin{equation}
R_u = \frac{p_{initial}^{\prime}  p_{current}^{\prime}}{p_{initial}^{\prime}}
\nonumber
\end{equation}
%
where mean effective
stress is defined as $p^{\prime} = \sigma^{\prime}_{kk}/3$.
%
This is different from traditional definition for $R_u$, that uses ratio of
excess pore pressure
over the initial mean
effective stress ($p_{initial}^{\prime}$). However, these two definitions are
essentially equivalent, as soil is in the state of liquefaction for
$R_u=1$ (so that $p_{current}^{\prime}=0$),
while there is no excess pore pressure for $R_u=0$ (so that
$p_{initial}^{\prime} = p_{current}^{\prime}$).
%
However, the former
definition is advocated here as it avoids the interpolation of
pore pressure or extrapolation of the stresses (as the latter definition
requires), since for the upU element, stresses are available at Gauss
points while pore pressures are available element nodes.
%
\begin{figure}[!htbp]
\begin{center}
\includegraphics[width=0.8\textwidth]{NewFiga/Ru_case_1_4.eps}
\vspace*{0.6cm}
\caption{\label{Ru_14} $R_u$ times histories for elements e1 (top element),
e3, e5, and e7 (bottom element)
Gauss point) for Cases I (level ground, no pile) and IV (sloping ground, no
pile).}
\end{center}
\end{figure}
%
In particular, Figure \ref{Ru_14} shows $R_u$ time histories for four points for
models I (level ground without pile) and model IV (sloping ground without pile).
%
It is noted that differences are fairly small.
%
It is interesting to observe that lower layers do not liquefy as supply
of pore fluid for initial void ratio of $e_0=0.8$ is too small, and the pore
fluid dissipation upward seems to be to rapid.
%
On the other hand, the upper soil layers do reach close to or
liquefaction state ($R_u=1$).
%
This is primarily due to the propagation of pore fluid pressure/volume from
lower layers upward (pumping effect) and, in addition to that, to a local excess
pore fluid production.
%
These results can also be
contrasted with those of \citet{Jeremic2007e}, where similar pumping scenario
has been observed.
%
The main difference between soil
used by \citet{Jeremic2007e} and here is in the coefficient of permeability.
%
Namely, here $k = 1.0\times 10^{4}$~$m/s$ was used
\citep{Cubrinovski2008,Uzuoka2008} while
\citet{Jeremic2007e} used $k = 5.0\times 10^{4}$~$m/s$.
%
It is important to note that other values of permeability for Toyoura
sand have also been reported \citep{Sakemi1995}, but current value was chosen
based on \citet{CubrinovskiPrivateCommunications}.
%
In addition to that, similar to \citet{Jeremic2007e}, sloping ground case shows
larger $R_u$ spikes, since there is static shear force (stress) that is always
present from gravity load on a slope. This static gravity on a slope will result
in an asymmetric
horizontal shear stresses in the downslope direction during cycles of
shaking.
%
This asymmetric shear stress induces a more dilative response for downslope
shaking which will help soil regain its
stiffness in the dilative parts of the loading cycles. This observation can be
used to explain smaller $R_u$ spikes for the sloping ground case.
%
Of course, this asymmetry in loading will result in larger accumulation
of downslope deformation.
%
While $R_u$ ratios for level and sloping ground cases are fairly similar
along the depth of the model, the response changes when the pile is present.
Figure~(\ref{Ru_25}) shows $R_u$ responses at four different points (along the depth)
approximately midway between the pile and the model boundary, in the plane of
shaking (see location of those elements in Figure~(\ref{FEmesh3D}) on page
\pageref{FEmesh3D}).
%
In comparison to behavior without the pile (Figure~(\ref{Ru_14})),
it is immediately obvious that addition of a pile
with a mass on top reduces
$R_u$ during shaking for the top element (e1). This is to be expected as presence
of a pilecolumnmass (PCM) system changes the dynamics of the top layers of soil
significantly enough to reduce total amount of shear. This is particularly true
for the top layers of soil as effects of columnmass tend to create
compressive and extensive movements (compression when the PCM
system moves toward soil and extension, and possibly even tension, when PCM
system moves away from soil). However, this extension, or possible
tension, is not
directly observable in presented plots since array of elements where we follow
$R_u$ (e1, e3, e5, e7) is some distance away from the pilesoil interface.
% and possibly forming
%tension gap between pile and adjacent soil elements.
%
Middle layers (e3 and e5), on the other hand, display similar response to that of
Cases I and IV, as shown in Figure~(\ref{Ru_14}). It is
noted that in a case with of sloping ground with pile, the $R_u$ measurements are
always larger that those for level ground (this is also observed for Cases III
and VI, as shown in Figure~(\ref{Ru_36})). This is
expected as presence of a pile in loose sand, and particularly the dynamic movement
of a pile during seismic shaking, create an additional shearing deformation
field (in the soil adjacent to the pile) that provides for additional
(faster) compression of soil skeleton and thus creates additional volume of pore
fluid, that is then distributed to adjacent soil (adjacent to the pile).
%
%
\begin{figure}[!htbp]
\begin{center}
\includegraphics[width=0.8\textwidth]{NewFiga/Ru_case_2_5.eps}
\vspace*{0.6cm}
\caption{\label{Ru_25} $R_u$ times histories for elements e1, e3, e5, and e7 (upper
Gauss point) for Cases II (level ground, with pilecolumn, free column head) and V
(sloping ground, with pilecolumn, free column head).}
\end{center}
\end{figure}
%
Particularly interesting are $R_u$ results for soil element e7, which is
located below pile tip level (see Figure~(\ref{FEmesh3D})).
Observed $R_u$ for Case V in element e7 is significantly larger than for
the same element for Case II. Similarly, simulated $R_u$ is larger than what was observed
in cases without a pile (see bottom of Figure~(\ref{Ru_14})). This
increase in $R_u$ for Case V (sloping ground with pile) is
explained by noting that the pile ``reinforces" upper soil layers
and thus prevents
excess shear deformation in the upper $12.0$~m of soil (above pile tip).
% , when compared with sloping ground Case without the pile (IV).
%This reduced deformation (and subsequently shearing).
The reduction of deformation
in upper layers of soil (top $12.0$~meters) results in transfer of excessive soil deformation
to soil layers below pile tip (where element e7 is located). This, in
turn, results in a much larger and faster shearing of those lower loose soil layers. This
significantly larger shearing results in a much higher $R_u$. Deformed shape,
shown in Figure~(\ref{Snapshots123456}) for Case V, reinforces this
explanation, showing much
large shearing deformation in lower soil layers, below pile tip. Same
observation can be made for Case VI, shown in
Figure~(\ref{Snapshots123456}).
%
Observation similar to the above, for Cases II and V can be made for Cases III
and VI, results for which are shown in Figure~(\ref{Ru_36}).
%
\begin{figure}[!htbp]
\begin{center}
\includegraphics[width=0.8\textwidth]{NewFiga/Ru_case_3_6.eps}
\vspace*{0.6cm}
\caption{\label{Ru_36}
$R_u$ times histories for elements e1, e3, e5, and e7 (upper
Gauss point) for Cases III (level ground, with pile, no rotation of pile
head) and VI (sloping ground, with pilecolumn, no rotation of column head).}
\end{center}
\end{figure}
%
One noticeable difference in $R_u$ results between cases with free column
head (Cases II and V) and cases with fixed rotation
column head (Cases III and VI) is in significantly
larger (and faster) development of $R_u$ close to soil surface for a stiffer, no
rotation column cases (Cases III and VI).
% Similar observation is made for sloping ground
%cases (Cases V, sloping ground with pile, restricted rotation on
%top of column) and VI (sloping ground with pile, restricted rotation on top
%of column).
%
This much larger $R_u$ observed in a ``stiffer" PCM system setup, is
due to larger shearing deformation that develops in soils adjacent to the
pile during shaking. The stiffer PCM system can displace less (because of
additional no rotation condition on column top) while the
soil beneath is undergoing shaking (same demand in all cases), thus resulting
in larger relative shearing of soil, which then results in larger and faster
pore pressure development close to the soil surface, where the column no
rotation effect is most pronounced.
% %
% In order to compare the slope effects on $R_u$,
% $R_u$ for
% Case I and IV (Figure \ref{Ru_14}),
% Case II and V (Figure \ref{Ru_25}),
% Case III and VI (Figure \ref{Ru_36})are plotted together up to 30 seconds..
% Form these figures, $R_u$ at all elements are found to increase shortly after earthquake shaking.
% At upper layers, $R_u$ is up to as big as close to 1.0,
% which means that liquefaction have occurred at upper layers.
% The $R_u$ time histories are almost overlapped for Case I and IV,
% which indicates that the slope effect on $R_u$ is not distinct
% for free field without pile (Figure \ref{Ru_14}).
% However, $R_u$ curves with level ground are generally lower than those with slope ground,
% which suggests strong slope effect on $R_u$ for the cases with the pile in soil,
% whichever the pile head is fixed or not (Figure \ref{Ru_25} and Figure \ref{Ru_36}).
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Soil Skeleton Deformation}
%%
A number of deformation modes is observed for both level and sloping ground
cases, with or without PCM system. Figure~(\ref{Snapshots123456}) shows
deformation patterns and excess pore pressures in symmetry plane for all six
cases over a period of eighty seconds.
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[!htbp]
\begin{center}
\begin{tabular}{rlllllll}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
I
&
\includegraphics[width=0.04\textwidth,angle=0]{Model_I.eps}
&
\includegraphics[height=0.12\textwidth]{NewFiga/Snap_1_T002.eps}
&
\includegraphics[height=0.12\textwidth]{NewFiga/Snap_1_T005.eps}
&
\includegraphics[height=0.12\textwidth]{NewFiga/Snap_1_T010.eps}
&
\includegraphics[height=0.12\textwidth]{NewFiga/Snap_1_T015.eps}
&
\includegraphics[height=0.12\textwidth]{NewFiga/Snap_1_T020.eps}
&
\includegraphics[height=0.12\textwidth]{NewFiga/Snap_1_T080.eps}
\\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
II
&
\includegraphics[width=0.04\textwidth,angle=0]{Model_II.eps}
&
\includegraphics[height=0.12\textwidth]{NewFiga/Snap_2_T002.eps}
&
\includegraphics[height=0.12\textwidth]{NewFiga/Snap_2_T005.eps}
&
\includegraphics[height=0.12\textwidth]{NewFiga/Snap_2_T010.eps}
&
\includegraphics[height=0.12\textwidth]{NewFiga/Snap_2_T015.eps}
&
\includegraphics[height=0.12\textwidth]{NewFiga/Snap_2_T020.eps}
&
\includegraphics[height=0.12\textwidth]{NewFiga/Snap_2_T080.eps}
\\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
III
&
\includegraphics[width=0.04\textwidth,angle=0]{Model_III.eps}
&
\includegraphics[height=0.12\textwidth]{NewFiga/Snap_3_T002.eps}
&
\includegraphics[height=0.12\textwidth]{NewFiga/Snap_3_T005.eps}
&
\includegraphics[height=0.12\textwidth]{NewFiga/Snap_3_T010.eps}
&
\includegraphics[height=0.12\textwidth]{NewFiga/Snap_3_T015.eps}
&
\includegraphics[height=0.12\textwidth]{NewFiga/Snap_3_T020.eps}
&
\includegraphics[height=0.12\textwidth]{NewFiga/Snap_3_T080.eps}
\\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
IV
&
\includegraphics[width=0.04\textwidth,angle=0]{Model_IV.eps}
&
\includegraphics[height=0.12\textwidth]{NewFiga/Snap_4_T002.eps}
&
\includegraphics[height=0.12\textwidth]{NewFiga/Snap_4_T005.eps}
&
\includegraphics[height=0.12\textwidth]{NewFiga/Snap_4_T010.eps}
&
\includegraphics[height=0.12\textwidth]{NewFiga/Snap_4_T015.eps}
&
\includegraphics[height=0.12\textwidth]{NewFiga/Snap_4_T020.eps}
&
\includegraphics[height=0.12\textwidth]{NewFiga/Snap_4_T080.eps}
\\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
V
&
\includegraphics[width=0.04\textwidth,angle=0]{Model_V.eps}
&
\includegraphics[height=0.12\textwidth]{NewFiga/Snap_5_T002.eps}
&
\includegraphics[height=0.12\textwidth]{NewFiga/Snap_5_T005.eps}
&
\includegraphics[height=0.12\textwidth]{NewFiga/Snap_5_T010.eps}
&
\includegraphics[height=0.12\textwidth]{NewFiga/Snap_5_T015.eps}
&
\includegraphics[height=0.12\textwidth]{NewFiga/Snap_5_T020.eps}
&
\includegraphics[height=0.12\textwidth]{NewFiga/Snap_5_T080.eps}
\\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
VI
&
\includegraphics[width=0.04\textwidth,angle=0]{Model_VI.eps}
&
\includegraphics[height=0.12\textwidth]{NewFiga/Snap_6_T002.eps}
&
\includegraphics[height=0.12\textwidth]{NewFiga/Snap_6_T005.eps}
&
\includegraphics[height=0.12\textwidth]{NewFiga/Snap_6_T010.eps}
&
\includegraphics[height=0.12\textwidth]{NewFiga/Snap_6_T015.eps}
&
\includegraphics[height=0.12\textwidth]{NewFiga/Snap_6_T020.eps}
&
\includegraphics[height=0.12\textwidth]{NewFiga/Snap_6_T080.eps}
\\
&
t=
&
2~sec
&
5~sec
&
10~sec
&
15~sec
&
20~sec
&
80~sec
\end{tabular}
\includegraphics[width=10cm,height=1cm]{NewFiga/GMklot02.ps}\hspace*{1cm}
\\
\includegraphics[angle=90,width=0.5\textwidth]{NewFiga/Snap_scale.ps}
\caption{
\label{Snapshots123456}
{Time sequence of deformed shapes and excess pore pressure in symmetry
plane of a soil system. Deformation is exaggerated 15 times; Color scale for
excess pore pressures (above) is in $kN/m^2$. Graph of ground motions
used (also shown in Figure~(\ref{SinglePileGM})) is placed below
appropriate time snapshots and is matching for $t=2, 5, 10, 15, 20$~seconds while
at $t=80$~seconds there is no seismic shaking.}
}
\end{center}
\end{figure}
%
A number of observation can be made on both deformation patterns, excess
pore fluid patterns and their close coupling.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\paragraph{Level Ground without Pile (Case I).}
%
Excess pore pressures and deformations in symmetry plane for level ground without a
pile are shown in Figure~(\ref{Snapshots123456}) (I).
%
At the very beginning (at $t=2$~s) there is initial
development of excess pore fluid pressure in the middle soil layers. This
expected, as the self weight loading stage has densified lower soil
layers enough so that their response is not initially contractive enough to produce
excess pore pressure. Top soil layers, on the other hand, have a drainage boundary
(top surface) too close to develop any significant excess pore pressures. As
seismic shaking progresses (for $t = 5, 10$~s), the excess pore pressure
increases, and starts developing in lower soil layers as well. It should be
noted that a small nonuniformity in results is present. For example, zones of
variable, nonuniform excess pore pressures on the lower mid and right side for Case I at
$t=10$~s develop.
%This nonuniformity was not initially expected, as the
%boundary conditions and the model should be similar to a 1D soil column (Case I).
%However, nonuniform mesh (many small, long elements in the middle,
%large elements outside this middle zone) introduce small errors in results
%(which can be observed by slightly nonuniform results at $t=12$~s and $t=10$~s).
%%
Nonuniform mesh (many small, long elements in the middle,
large elements outside this middle zone) may introduce small numerical errors in results
which can be observed by slightly nonuniform results at $t=10$~s and $t=15$~s.
%%
It should be noted that results for excess pore pressure shown for
first $13$~seconds (during shaking) in Figure~(\ref{Snapshots123456}) (I) are
transient in nature, that is,
seismic waves are traveling throughout the domain (model) during shaking (first
$13$~seconds) and slight oscillations in vertical stresses are possible. This
oscillations will contribute to the (small) nonuniformity of excess pore pressure
results. After the shaking (after $15$~seconds) resulting excess pore pressure
field is quite uniform.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\paragraph{Level Ground with Pile (Cases II and III).}
%
Excess pore pressures and deformations in symmetry plane for models with PCM
system and with two different boundary conditions at
top of column (see model description in section~\ref{six_models})
in level ground are shown in Figures~(\ref{Snapshots123456}) (II and III).
%
One of the interesting observations is significant shearing
and excess pore pressure generation adjacent to the pile tip. The reason for
this is that pile is too short, that is, pile tip has significant
horizontal displacements during shaking. Those pile tip displacements
shear the soil, resulting in excess pore pressure generation.
As soon as there is time for dissipation, this localized excess pore pressure
dissipates to adjacent soil, and then, after shaking has ceased (at $t=13$~s and
later), it slowly dissipates upward.
%
Addition of pile into the model (construction), with a highly impermeable
elements (that mimic permeability of concrete) is apparent as there is a low
excess pore pressure region in the middle of model, where pile is located.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\paragraph{Sloping Ground without Pile (Case IV).}
%
Excess pore pressures and deformation in symmetry plane for sloping ground
without pile is shown in Figures~(\ref{Snapshots123456}) (IV).
%
It is noted that initially the excess pore pressure starts developing in
middle soil layers,similar to the Case I above. Bottom layers start
developing excess pore
pressure only after significant shear deformation occurs (at
$t=10$~s) at approximately $2/3$ of the model depth. Lower layers have
densified enough during self weight stage of loading that initial shaking is not
strong enough to create excess pore water pressure, rather, those layers are fed
by the excess pore pressure from above. Lower soil layers also do not develop much
deformation, while middle and upper layers together develop excessive
horizontal deformation.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\paragraph{Sloping Ground with Pile (Cases V and VI).}
%
Excess pore pressures and deformation in symmetry plane for sloping ground
with PCM system are shown in Figures~(\ref{Snapshots123456}) (V and VI).
%
Similar to the above cases (II and III), pile is too short and there
is again excessive shearing of soil at the pile tip, suggesting large movement
of that pile tip.
%
In addition to that, pile introduces significant stiffness to upper $12$~meters
of soil (along the length of pile) and helps reduce deformation of those upper
soil layers. Downslope gravity load is thus transferred to lower soil layers
(below pile tip) which exhibit most of the deformation. It should be noted that
soil in middle and upper layers (adjacent to pile) does deform, just not as much as the
soil below pile tip. The predominant mode of deformation of middle soil layers is
shearing in horizontal plane, around the pile. Deformation in horizontal
plane is not significant as the pile is short in this examples (as mentioned
above) and does not have enough horizontal support at the bottom.
The deformation pattern of a soil  pile system is such that pile experiences
significant rotation, and deforms with the soil that moves downslope.
%
If the pile was longer, and if it had significant horizontal support at the
bottom, the middle and upper soil layers would have showed more significant flow
around the pile in horizontal planes.
%
Upper layers undergo significant settlement, as seen in
Figure~(\ref{Settlements}).
This settlement is mainly caused by the above mentioned rotation of pilesoil
system, where soil in general settles (compacts) but also undergoes differential
settlement, between left (upslope from pile) and right (downslope from
pile) side of the model.
%
As significant shearing with excess pore pressure generation develops in lower soil
layers, below pile tip, those lower layers contribute to most of downslope
horizontal deformation.
%
In a sense, all the demand from downslope gravity forces
and seismic shaking is now responded to by lower soil layers,
which contribute to most of the excess pore pressure generation and
consequently, to most of the soil deformation.
%
Soil surface horizontal deformation is thus strongly influenced by significant
shearing of the bottom layers and by rotation of the middle and upper soil
layers with the pile.
%
%
%
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[!htbp]
\begin{center}
\begin{tabular}{rrr}
%\hline
I
&
II
&
III
\\
\includegraphics[height=0.12\textwidth]{NewFiga/settlement_case1.ps}
&
\includegraphics[height=0.12\textwidth]{NewFiga/settlement_case2.ps}
&
\includegraphics[height=0.12\textwidth]{NewFiga/settlement_case3.ps}
\\
\\
% \hline
IV
&
V
&
VI
\\
\includegraphics[height=0.12\textwidth]{NewFiga/settlement_case4.ps}
&
\includegraphics[height=0.12\textwidth]{NewFiga/settlement_case5.ps}
&
\includegraphics[height=0.12\textwidth]{NewFiga/settlement_case6.ps}
%\\
%
%\hline
\end{tabular}
\includegraphics[angle=90,width=0.50\textwidth]{NewFiga/settlement_scale.ps}
\caption{
\label{Settlements}
{Soil surface settlements at $120$~s for all six cases. Color scale given in
meters}
}
\end{center}
\end{figure}
%
%
It is interesting to note that the largest settlement is observed just
downslope from pile for Cases V and VI.
%
% It is also very interesting to note that, even though soil achieves cyclic
% mobility (for example for Cases II and III, see $R_u$ values in
% Figures~(\ref{Ru_25}),
% and (\ref{Ru_36})) and in some cases full
% liquefaction (for example for Cases V and VI, see $R_u$ values in
% Figures~(\ref{Ru_25})
% and (\ref{Ru_36})), pile is still able to
% hold the vertical loads (from mass on top and from the its own and
% column's self weights) by utilizing skin friction.
%
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Pile Response}
%
%
%
% for plots the bending moment distribution at each calculation
% time step for Case II, III, V and VI.
% During the structurepilefoundation interaction,
% the moment envelope upon the time duration is important to pile performance.
% The pile moment distributions between the free and fixed pile head are quite different.
% For free pile head, the maximum pile bending moments are at depth between 0 and 2 feet
% with a magnitude about 800 kN,
% while for fixed pile head, the maximum pile bending moment are at the pile head
% with a magnitude close to 2000 kN.
% The slope effect on the bending moment envelop is not distinct.
%
Figure~(\ref{Moments}) shows bending moment envelops for pilecolumnmass
(PCM) system for all four cases (II, III, V and VI).
%
It should be noted that bending moment diagrams are plotted on compression side
of the beamcolumn.
%
A number of observations can be made about bending moment envelopes.
%
For cases with free pile head (shaking transverse to the bridge main axes,
Cases II and V)
the maximum moments are attained in soil, at depths of approximately $0.6D 
1.2D$, where $D$~($=1.0$~m in this case) is the pile diameter.
%
Opposed to that are cases for PCM systems with restricted rotations at
the pile top which (Cases III and VI), which, of course feature largest moment
at the column top.
Maximum bending moments for section of PCM system in soil (pile) in these two
cases are now attained at the depth of approximately $1.8D  2.0D$.
It is noted that bending moment envelopes are mostly symmetric. Slight
nonsymmetry is introduced for cases on sloping ground (Case V and VI).
It is also noted that moments do exist (are not zero) all the say to the bottom of the pile.
Theoretically, moments should be zero at the pile tip, but since physical volume
of the pile is considered (see note on that in section \ref{pile_installation}
and Figure~(\ref{coupling})), differential pressure on pile bottom from soil
will produce small (nonzero) moments even at the pile tip. More importantly,
nonzero moments at the bottom and along the lower part of the pile show that
pile is indeed too short, and thus changing curvatures are present along the
whole length of the pile.
%
\begin{figure}[!htbp]
%\vspace*{1cm}
\begin{center}
%\begin{tabular}{rrrr}
\begin{tabular}{rr}
%II
%&
\includegraphics[width=0.3\textwidth]{NewFiga/M/M2.eps}
&
%III
%&
\includegraphics[width=0.3\textwidth]{NewFiga/M/M5.eps}
\\
%V
%&
\includegraphics[width=0.3\textwidth]{NewFiga/M/M3.eps}
&
%VI
%&
\includegraphics[width=0.3\textwidth]{NewFiga/M/M6.eps}
\end{tabular}
\caption{
\label{Moments}
{Envelope of bending moments for pilecolumn system for Cases II, III, V
and VI.}
}
\end{center}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Pile Pinning Effects}
Piles in sloping liquefying ground can also be used to resists movement of soil
(all liquefied or liquefied with hard crust on top) downslope. For models
developed in this paper, pile pinning effect can be investigated for
Cases IV, V and VI. In particular, deformation of sloping ground without the
pile (Case IV) can be compared with either of the cases of piles in sloping
ground, Cases V and VI.
%
It is very important to note, again, that models developed here had relatively
short pile, and that major soil shearing developed below the pile
tip. This apparent shortcoming of a short pile results in reduced pile pinning
capacity, thus reducing the downslope movement by only approximately
half, from
$0.35$~m (Case IV) to $0.22$~m (Case V) and to $0.18$~m (Case VI) as seen
in Figure~(\ref{Dx}). It would have
been expected that, had the pile been longer and had it penetrated in deeper,
nonliquefiable layers, it would have reduced downslope movement of the soil to
a much larger extent. However, had the pile been longer and had it penetrated
nonliquefiable layers, it would have had a much firmer horizontal support at the
bottom and would have thus attracted much larger forces too, potentially leading
to pile damage and yielding.
% %
% OVDE
%
%
% Figure \ref{Dx} plots the lateral displacements of the ground center
% (the middle point at the ground surface in the middle plain,
% or the point of pile at the ground surface).
% For all cases, the ultimate lateral displacements of the ground center,
% there are some residual lateral displacement after the earthquake exciting.
% It is obviously that the pile has much bigger lateral displacement for the slope ground (Case I, II and III)
% than those for the level ground (Case IV, V, and VI), which is as expected.
% No matter for level ground, or for slope ground,
% the the lateral displacements of the ground center without pile (Case I and IV) are much higher
% than those with pile (Case II, III and V and VI),
% which suggest strong pile pinning effects on the ground lateral displacement.
%
%
\begin{figure}[!htbp]
\begin{center}
%\includegraphics[width=0.6\textwidth]{NewFiga/Dx123.eps}
%\\
\vspace*{1cm}
\includegraphics[width=0.7\textwidth]{NewFiga/Dx456.eps}
\\
\caption{
\label{Dx}
{Downslope movement at the ground surface (model center) for Cases IV (no
pile), V and VI (with pilemass system).}
}
\end{center}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Summary}
\label{Summary}
Presented in this paper was methodology for numerical modeling and simulation of
piles in liquefiable soil. Of particular interest was the detailed description
of modeling which aimed at replicating the prototype model as close as possible.
High fidelity modeling included use of verified and validated models, detailed
model development, including use of realistic loading stages.
%
Detailed application of loading staged, starting from a zero state of stress
and strain for a soil without a pile, followed by application of soil self
weight, excavation and pilecolumn installation with application of
pilecolumn selfweighting is finally followed by seismic loading with extended
time after that for dissipation of excess pore pressures that have developed.
%
An implementation for a bounding surface elasticplastic sand model that
accounts for fabric change, and for a fully coupled porous media
(soil skeleton)  pore fluid (water) dynamic finite element formulation were
developed and used in simulation of soil and water displacement and pore
water pressure.
%
Six models were developed and simulated, feature level and sloping ground
without and with pilecolumn systems.
%
Results of simulations are presented with the aim of increasing our
understanding of behavior of soilpilecolumn systems during
liquefaction events, including lateral soil deformation, effects of pile
pinning, and ground settlement.
%
In addition to detailed presentation of useful and interesting results, one of
the main aims of this paper was to emphasize the need for, importance and
availability of high fidelity modeling tools for simulating effects of liquefied
soil on soilstructure systems.
%occurring, excess pore water buildingup and dissipation, soil skeleton lateral
%deformation, ground settlement, pile response, and pilesoil interaction were
%interpreted at some length.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\noindent
\section*{Acknowledgment}
%{\Large \bf Acknowledgment}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{samepage}
%
Work presented in this paper was supported in part by the Earthquake
Engineering Research Centers Program of the National Science Foundation under
Award Number EEC9701568.
%
\end{samepage}
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