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\begin{document}
%
\begin{center}
{\Large \bf On Validation of Fully Coupled Behavior of Porous Media using Centrifuge Test Results}
%{\Large \bf
%Validation Procedures for
%Simulation of Fully Coupled Behavior of
%Porous Media
%}
\end{center}
\vspace*{0.5cm}
\begin{center}
{\large \bf Panagiota Tasiopoulou} \\
{PhD Student,
National Technical University of Athens,
Athens, Greece}\\ ~ \\
%%%%%%%%%%%%%%%%%%%%
{\large \bf Mahdi Taiebat} \\
Associate Professor,
Department of Civil Engineering, The University of British Columbia,
Vancouver\\ ~ \\
%%%%%%%%
%Mahdi Taiebat \scalebox{1}{\includegraphics[viewport=0 4.5 0 0]{mahdi.eps}}
{\large \bf Nima Tafazzoli} \\
Geotechnical Engineer, Tetra Tech EBA, Vancouver, BC, Canada \\ ~ \\
% %%%%%%%%
%%%%%%%%%
{\large \bf Boris Jeremi{\'c}}\footnote{Corresponding Author, Department of Civil and Environmental Engineering,
University of California, One Shields Ave, Davis, CA, 95616.
\texttt{jeremic@ucdavis.edu}}
\\
Professor,
Department of Civil and Environmental Engineering, University of
California, Davis, CA,
and Faculty Scientist, Earth Science Division, Lawrence Berkeley National
Laboratory, Berkeley, CA.
\\
\end{center}
%%%%%%%%
%and
%%%%%%%%%
keywords: verification and validation,
finite elements,
fully coupled analysis,
porous media
%\end{abstract}
%
%\linenumbers
\section*{Abstract}
Modeling and simulation of mechanical response of infrastructure
object, solids and structures, relies on the use of computational models to
foretell the state of a physical system under conditions for which such
computational model has not been validated. Verification and Validation (V\&V)
procedures are the primary means of assessing accuracy, building confidence and
credibility in modeling and computational simulations of behavior of those
infrastructure objects.
%
Validation is the process of determining a degree to which a model is
an accurate representation of the real world from the perspective of the
intended uses of the model. It is mainly a physics issue and provides evidence
that the correct model is solved \citep{Oberkampf2002}.
Our primary interest is in modeling and simulating behavior of
porous particulate media that is fully saturated with pore fluid, including
cyclic mobility and liquefaction.
Fully saturated soils undergoing dynamic shaking fall in this
category.
%
Verification modeling and simulation of fully saturated porous soils is
addressed in more detail by \citep{Tasiopoulou2014}, and
in this paper we address validation.
%
A set of centrifuge experiments is used for this purpose.
Discussion is provided assessing the effects of scaling laws
on centrifuge experiments and their influence on the validation.
%
Available validation test are reviewed in view of first and second order
phenomena and their importance to validation. For example, dynamics behavior of the
system, following the dynamic time, and dissipation of the pore fluid pressures,
following diffusion time, are not happening in the same time scale and those
discrepancies are discussed.
%
Laboratory tests, performed on soil that is used in centrifuge experiments, were
used to calibrate material models that are then used in a validation process.
%
Number of physical and numerical examples are used for validation and to
illustrate presented discussion.
%
In particular, it is shown that for the most part, numerical prediction of behavior, using
laboratory test data to calibrate soil material model, prior to centrifuge
experiments, can be validated using scaled tests. There are, of course,
discrepancies, sources of which are analyzed and discussed.
\newpage
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\section{Introduction}
Numerical predictions of behavior of civil engineering solids and structures has
gained a significant popularity in last decades, particularly with the
availability of fast digital computers and a number of commercial and research
programs (numerical modeling and simulations tools) that feature nice graphical
user interfaces (GUIs). While expansion of use of numerical prediction tools
brings great promises for improved design (improved safety and economy) there
also exists a danger of using numerical prediction tools for modeling
and simulating phenomena for which these tools have not been verified and
validated.
Verification and
Validation (V\&V) procedures are the primary means of
assessing accuracy,
building confidence and credibility in modeling and computational simulations.
%
Verification is the process of determining that a model implementation
accurately represents the developer's conceptual description and specification.
It is mainly a mathematics issue, and provides evidence that the model is solved
correctly.
%
Validation is the process of determining a degree to which a model is
an accurate representation of the real world from the perspective of the
intended uses of the model. It is mainly a physics issue and provides evidence
that the correct model is solved \citep{Oberkampf2002}.
Verification and validation has recently gained increased
attention, with the understanding that numerical prediction results can only be
trusted if proper verification and validation has been performed
\citep{Mroz88, Arulanandan93, Zienkiewicz94, Roach1998,
Oberkampf2002, Oden2004, Babuska2004, Oden2010a, Oden2010b,
Oberkampf2010, Bielak2010, Roy2011}.
In this paper, we address the issue of validating the modeling of fully coupled
behavior of particular materials (granular soil) using scaled models under
increased gravity, so called centrifuge modeling.
%
Basics of validation procedures are presented in Section~\ref{VandV_P}.
Detailed discussion on scaling laws, as they apply to our examples, centrifuge tests, and
validation results with comments are presented in Section~\ref{VCBM}.
%
Details of the u-p-U formulation are given by
\citet{Tasiopoulou2014} and will not be repeated here. In addition, validation
of the elastic-plastic material model used was presented by~\cite{Jeremic2007e}
and will not be covered here either.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Validation Procedures}
\label{VandV_P}
Validation procedures are used to provide evidence that numerical analyst have
chosen the right models for modeling phenomena in question. As such, validation
procedures are tightly coupled to the physics (mechanics) of the problem.
Validation procedures give us information about how much we can trust the
numerical simulation results.
The role of validation is graphically shown in Figure~\ref{roleVandV}.
%
\begin{figure}[!htb]
\begin{center}
%{\includegraphics[width=10.0cm]{/home/jeremic/tex/works/Conferences/2013/NRC_Short_Course_May2013/Present/Present06_figs/RoleVV_NEW_knowledge.pdf}
\includegraphics[width=7.0cm]{/home/jeremic/tex/works/Conferences/2013/NRC_Short_Course_May2013/Present/Present06_figs/RoleVV_NEW_knowledge.eps}
\caption{Role of Verification and Validation in relation to the knowledge about
reality (graphics inspired by \citet{Oberkampf2002}).}
\label{roleVandV}
\end{center}
\end{figure}
%
It is important to note that both verification and validation procedures are necessary
in order to gain confidence in numerical modeling results, and to be able to make
informed decisions about the behavior of a problem being analyzed. It is also
important to note that real physical behavior of a mechanical system is never fully known. This is
a result of
a macro scale
interpretation of the Heisenberg uncertainty principle
\citep{Heisenberg1927}, stating that one cannot obtain
position and momentum of a material particles at the same time
resulting from some deterministic or stochastic loading.
%
Hence, only an approximate knowledge (with some level of
certainty) of behavior of an object (solid and/or structure), can be gained.
This argument emphasizes the need for stochastic treatment of both physical and
numerical modeling and simulations \citep{Oberkampf2002}.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\paragraph{Independence of Computational Confidence Assessment}
%\begin{enumerate}
% \item V\&V conducted by the computational tool developer, {\it No Independence}
% \item V\&V conducted by a user from same organization
% \item V\&V conducted by a computational tool evaluator contracted by developer's organization
% \item V\&V conducted by a computational tool evaluator contracted by the customer
% \item V\&V conducted by a computational tool evaluator contracted by the a legal authority {\it High Independence}
%\end{enumerate}
A more detailed analysis of validation reveals the importance
of a hierarchy of experimental data. Figure~\ref{VerifValidFund01} shows
relationship of real world behavior with verification and (emphasis on)
validation.
%--
\begin{figure}[!htb]
\begin{center}
{\includegraphics[width=11cm]{/home/jeremic/tex/works/Papers/2011/Validation_Coupled_Systems/VerifValidFund01.eps}}
\caption{Relationship of verification and validation to the real world, with
emphasis on validation and experimental data (inspired by \citet{Oberkampf2002}).}
\label{VerifValidFund01}
\end{center}
\end{figure}
%
As noted by \citet{Oberkampf2002}, validation can be understood as a process to
determine how accurately the model (focusing on its intended use) represents the real world,
%
The validation experiments\footnote{As opposed to traditional experiments which
are used to improve (a) understanding of physics and (b) mathematical models
of/for a phenomena in question.} used for this purpose are designed and
performed to estimate computational model's ability to model
defined physical behavior/phenomena. In a sense, the numerical modeling tool
(computational simulation tool) becomes the main customer of designed validation
experiment.
%
Ideally, a validation experiments should be jointly designed and performed by
physical modeler (experimentalist) and numerical modeler (numerical analyst).
Validation experiment should be able to capture relevant/important physics,
where physical effects of primary importance are properly modeled while
secondary effects might be modeled using some level of approximation.
It is important to note that the validation domain is almost always exclusive of
the application domain. That is, real physical phenomena that we are interested in,
cannot be fully physically modeled due to complexity, cost, size, etc.
%
For example,
civil engineering systems like bridges, buildings, port facilities, dams,
nuclear power plants, etc. are to complex, expensive and large to be tested for
all loading scenarios of interest.
%
Even if the engineering system is small (less expensive, complex), environmental
influences (generalized loads, conditions, wear and tear) are hard to model
physically.
%
Validation domain thus represents a simplification of application domain.
Figure~\ref{VPI03} shows a relationship of validation and application domain for
civil engineering applications (structural and soil mechanics) that are almost
always exclusively non-overlapping in their systems parameters and system
complexity.
%
\begin{figure}[!htb]
\begin{center}
{\includegraphics[width=4.0cm]{/home/jeremic/tex/works/Presentation/2004/VandV/VPI03.eps}}
\caption{Relationship of the validation domain to the application domain, which, in general
for civil engineering application (structural and soil mechanics) are
exclusively non-overlapping \citep{Oberkampf2002}).}
\label{VPI03}
\end{center}
\end{figure}
%
The inference from validation to application domain is done numerically.
Such inference is based on physics while uncertainties in material behavior, loads,
geometry, etc. have to be addressed as well.
%
The importance of uncertainty quantification in experiments and numerical
predictions cannot be overstated. All relevant sources of uncertainty in
physical models need to be
identified and uncertainties estimated. Those uncertainties then need to be
propagated through modeling and simulation process.
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Validation of Coupled Behavior Modeling}
\label{VCBM}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Scaling Laws}
\label{Discussion of Scaling laws}
%\vspace{10 mm}
Scaling laws are of great importance, not only for the centrifuge modeling
itself, but also for the accurate numerical reproduction of the centrifuge
tests.
The important scaling laws for higher gravity modeling of liquefaction are
concerning the dynamic time and the permeability and consequently the diffusion time.
%
The Darcy permeability of the centrifuge model (under increased gravity field of
$N\times g$) is $N$ times larger than permeability that was measured
in the laboratory (under gravity field of $1\times g$).
%
This leads to a difference between the scaling factors for the dynamic time
and the diffusion time if the same materials (water and soil) are used in the
model and prototype. This conflict in time scales is essential to scaling the
centrifuge measurements up to the prototype scale since both generation of
excess pore fluid pressures (dynamic time) and dissipation of excess pore fluid
pressure (diffusion time) happen at the same time throughout shaking and are
equally important to the modeling (physical and/or numerical) of liquefaction.
%
In order to analyze the appropriate scaling of the permeability and the
diffusion time, we consider three different cases: A, B and C, as illustrated in
Figure~\ref{scale}. It is assumed that water fills the voids of the soil in all
cases.
\begin{itemize}
\item[] Case A is the object in prototype (original) scale representing
the original soil conditions,
\item[] Case B is the centrifuge model in
model scale, and
\item[] Case C is the scaled up prototype model derived from scaling the centrifuge model up to
prototype scale.
\end{itemize}
\begin{figure}[!hbt]
%\begin{Large}
%\begin{sffamily}
\begin{center}
\includegraphics[width=8cm]{/home/jeremic/tex/works/Thesis/PanagiotaTasiopoulou/Thesis/validation/Figures/scale.eps}
\end{center}
\caption{\label{scale}{Schematic illustration of the centrifuge
modeling concepts and scaling laws described in detail in Section \ref{Discussion of
Scaling laws}. Case A represents the object for simulation, Case B represents
the centrifuge model and Case C represents what has been actually simulated in
prototype scale. The symbols, $k_D$, $K$ and $T$, correspond to Darcy's
permeability, specific permeability and consolidation time respectively.}}
%\end{sffamily}
%\end{Large}
\end{figure}
In literature, the term "prototype model" is sometimes confusingly used
to characterize either Case A or Case C,
without further clarifications. Herein, Case A is characterized as "Original
Model" and Case C as "Prototype Model". Scaling factors for all the important
quantities related to our study are presented in Table \ref{Scaling} \citep{Wood2004}.
%
\begin{table}[!htb]
\caption{ \label{Scaling}
Scaling factors for a case of water filling the pore space (voids) in the soil.}
\begin{center}
%\begin{spacing}{1.4}
\begin{tabular}{c|c|c|c}
& \bf{Case A} & \bf{Case B} & \bf{Case C} \\
\hline
& Original Model & Centrifuge Model & Prototype Model \\
quantity & 1g & Ng & 1g \\
& & Numerical Model 1 & Numerical Model 2 \\
\hline
length & N & 1 & N \\
mass density & 1 & 1 & 1 \\
stress & 1 & 1 & 1 \\
strain & 1 & 1 & 1 \\
displacement & N & 1 & N \\
acceleration & $1/N$ & 1 & $1/N$ \\
Darcy's permeability & $1/N$ & 1 & 1 \\
specific permeability & 1 & 1 & N \\
time (diffusion) & $N^2$ & 1 & N \\
time (dynamic) & N & 1 & N \\
frequency & $1/N$ & 1 & $1/N$ \\
\end{tabular}
%\end{spacing}
\end{center}
\end{table}
The base, to which all the scaling factors refer to, in Table
\ref{Scaling}, is the model scale. The column corresponding to the "Original
Model" presents the scaling factors needed to be applied to the quantities in
model scale in order to reproduce what has been intended to be simulated from
the beginning - the "Original Model". The conflict in time scales is related to
the "Original Model". The column corresponding to the "Prototype Model" presents
the scaling factors needed to be applied to the quantities in model scale in
order to reproduce what has been actually simulated in prototype scale -
the "Prototype Model" . Thus, comparison between what was intended to be
studied (Case A) and what was studied in reality (Case C) can reveal scaling problems.
A detailed analysis of scaling for each of three cases (A, B and C) follows.
%\vspace{10 mm}
\paragraph{Case A} is the object for simulation and represents the prototype soil
conditions and properties that are intended to be simulated through centrifuge
modeling. Darcy's permeability, $k_{DA}$, of this type of soil has been measured
in the lab. Case A could be simply described as a model N times larger than the
centrifuge model with the same soil under 1g. The specific permeability, $K_A$
and the time needed for completion of the 1D consolidation process, $T_A$, can
be estimated using $k_{DA}$ through Equation \ref{case_A_k} and Equation
\ref{case_A_T} respectively.
\begin{eqnarray}
K_A=\frac{k_{DA}}{\rho_f\times g}
\label{case_A_k}
\end{eqnarray}
\begin{eqnarray}
T_A=\frac{H^2}{C_v}=\frac{H^2\times \rho_f \times g}{k_{DA}\times E_{eod}}=\frac{H^2}{K_A\times E_{oed}}
\label{case_A_T}
\end{eqnarray}
%
Here $\rho_f$ is the mass density of the fluid
(water), $g$ is the acceleration of gravity ($9.81$~$\rm{ m/s^2}$) in this case,
$H$ is the thickness of the soil layer in real/original scale and $E_{oed}$ is
the one dimensional soil stiffness:
\paragraph{Case B} represents the centrifuge model which consists of the same type
of soil as in case A with the same relative density. However, Darcy's
permeability is proportional to the gravity, as it is indicated by Equation
\ref{case_A_k}.
%
The centrifuge gravity field (model) is $N\times g$,
and the specific permeability, $K$, is a soil constant
(independent of the permeant). It follows that by neglecting the changes
of void ratio and the gravity level, the actual Darcy's
permeability of the
centrifuge model is N times larger that that of
Case A, as shown in Equation \ref{case_B_k} \citep{Wood2004}.
%
\begin{eqnarray}
k_{DB}=K_B\times \rho_f\times N\times g=K_A\times \rho_f \times N\times g=N\times k_{DA}
\label{case_B_k}
\end{eqnarray}
%
Furthermore, the actual
consolidation time of Case B becomes $N^2$ smaller than that of Case A, as shown
in Equation \ref{case_B_T}.
%
\begin{eqnarray}
T_B=\frac{(H/N)^2}{K_B\times E_{oed}}=\frac{H^2}{N^2\times K_A\times E_{oed}}=\frac{T_A}{N^2}
\label{case_B_T}
\end{eqnarray}
%
In other words, the appropriate time scaling factor
for the centrifuge measurements for diffusion is $N^2$, in order to get a
similar response with Case A. On the other hand, the appropriate scaling factor
for the dynamic time is $N$. This fact leads to a problem with realistic simulation
of the Case A using centrifuge model. This stems from the fact that it is
difficult to separate the dynamic and the diffusion times since they are both
contributing to first rate physical effects and cannot be separated (both are of
the same importance for liquefaction and cyclic mobility phenomena).
%\vspace{10 mm}
\paragraph{Case C} represents what has been actually simulated and tested
in (scaled up to) prototype scale, through the centrifuge modeling process. The
centrifuge model under a
gravity field of $N\times g$ corresponds to a soil layer N times larger in size
in prototype scale, so that the stress field is common in both models. Case C
can be simply described as a model N times larger than the centrifuge model
consisting of soil with the same Darcy permeability (N times less than that in
Case A). Darcy's permeability of the prototype model (Case C) is equal to that
of the centrifuge model (Case B), which means N times larger than that in Case
A. Theoretically, the fact that $k_{DB}=k_{DC}$ leads to different values of
specific permeability between the centrifuge model and the prototype one. In
detail, the specific permeability of Case C, $K_C$, is N times larger than that
of Case B and Case C, as described by Equation \ref{case C_k}.
%
\begin{eqnarray}
K_C=\frac{k_{DC}}{\rho_f\times g}=\frac{k_{DB}}{\rho_f\times g}=N\times K_B=N\times K_A
\label{case C_k}
\end{eqnarray}
%
This is the
recommended value of the specific permeability that should be used in numerical
simulation of the centrifuge tests in prototype scale. Moreover, the diffusion
time of the prototype model is estimated (Equation \ref{case C_T}) to be N times
less than that of the centrifuge model, which coincides with the scaling factor
for the dynamic time.
%
\begin{eqnarray}
T_C=\frac{H^2}{K_C\times E_{oed}}=\frac{H^2}{N\times K_B\times E_{oed}}=\frac{T_B}{N}=N\times T_A
\label{case C_T}
\end{eqnarray}
%
However, the diffusion time of the prototype model is N
times larger when compared to Case A.
Since the permeability is proportional to gravity, this presents a problem for
modeling real scale case (Case A) using changed (larger) gravity field (Case B),
if water is used as pore fluid.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\paragraph{Use of Higher Viscosity Replacements Fluids.}
%
In order to overcome this scaling discrepancy and achieve the same value of
permeability in all cases, fluids with larger viscosity are chosen to fill the
voids within the soil, according to Equation \ref{1000} below:
%
\begin{eqnarray}
k_D=\frac{K\times \rho_f g}{\mu}
\label{1000}
\end{eqnarray}
%
where $\mu$ is pore fluid viscosity.
%
The Equations \ref{v_A} and \ref{v_B}
%
\begin{eqnarray}
k_{DA}=K\times \rho_w g
\label{v_A}
\end{eqnarray}
%
\begin{eqnarray}
k_{DB}=\frac{K\times \rho_f\times N\times g}{\mu}
\label{v_B}
\end{eqnarray}
%
indicate that the pore fluid
viscosity should be equal to $\rho_f\times N/\rho_w$, where $\rho_w$ is the mass
density of the water, so that $k_{DA}=k_{DB}$. In that case, the scaling factor
for the diffusion time is equal to $\rho_f\times N/\rho_w$ instead of $N^2$ when
water is used. In particular, if the pore fluid has the same mass density as
water, then the time scaling factor for diffusion is the same as the one for
dynamic events.
%\vspace{10 mm}
\cite{Wood2004} suggests the use of silicone fluid as a replacement fluid.
Another possibility is to use a solution of Hydroxypropyl methylcellulose in
water, which, when mixed in right proportion, increases the viscosity of water
from $1 mm^2/s$ to approximately $25 mm^2/s$ \citep{Kulasingam2004}. Increasing
viscosity more than that amplifies the problem of proper (full) saturation of
the sample, which than significantly affects other aspects of the experiment. By
using scaled up viscosity of (up to) $\mu=25 mm^2/s$, the geometric scaling of
model is also limited to $25$. That means that any centrifuge experiment that
uses $\mu=25 mm^2/s$ and is modeled in a centrifuge spinning at 50g level, using
original soil, will have a scaling factor for the diffusion time equal to
$25({1}/{50})^2/{1} = 0.01 =1/100 $ and for the dynamic time equal to
$({1}/{50})\sqrt{{1}/{1}} = 0.02 = 1/50$. That is, the geometric scaling will properly
match the dynamic time scale while diffusion time scale will be half of what it
is supposed to be. This means that the diffusion is occurring twice as fast in
comparison with the original model. Similarity of the original and the
centrifuge model does not exist in this case. The larger the geometric scaling
is, when compared to the fixed value of viscous scaling (usually not more than
25), the larger the
discrepancy in time scale is. This inconsistency in scaling creates the so
called distorted models as they inappropriately scale one of the first order
phenomena.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Numerical Simulation of RPI Centrifuge Test (Model No1, Test 2) by Taboada and Dobry (1993)}
\label{Numerical Simulation of RPI centrifuge test (Model No1, Test 2) by Taboada and Dobry (1993)}
Numerical modeling and simulation of the liquefaction is
performed using the $u-p-U$ \citep{Zienkiewicz84, Jeremic2007e} formulation
in combination with the
constitutive model by \cite{Dafalias2004}. For validation, results
from the centrifuge tests of Model No. 1 (test 2) presented by \cite{Dobry1993}
from VELACS project are used.
A schematic configuration of the centrifuge model No.1 is illustrated in
Figure~\ref{conf_cent}. The soil consists of a uniform layer of Nevada sand with
relative density $D_r\approx40\%$ and is fully-saturated with the pore fluid.
The thickness of the soil layer is 20 cm in model scale and the field of gravity
applied to the model is 50g. The input motion applied to the base of the model
is also shown in Figure~\ref{conf_cent} in real time (model) scale. According to
the scaling laws provided by \cite{Wood2004}, the prototype model is 10 m thick
and its Darcy permeability is 50 times greater than the value obtained from
lab tests, as reported by \cite{Arulmoli1992}. Accelerations, displacements and
pore pressures
were measured during testing at select locations. The locations and the type of
measurements recorded are shown in Table \ref{Location_of_Instrumentation}.
\begin{figure}[!hbt]
\begin{Large}
\begin{sffamily}
\begin{center}
\includegraphics[width=10cm]{/home/jeremic/tex/works/Thesis/PanagiotaTasiopoulou/Thesis/validation/Figures/conf_cent.ps}
\end{center}
\caption{\label{conf_cent}{Schematic configuration of Model No. 1, Test 2, RPI
\citep{Dobry1993}.}}
\end{sffamily}
\end{Large}
\end{figure}
\begin{table}[!htb]
\caption{ \label{Location_of_Instrumentation}
Location and Type of Measurement in Model No. 1, Test 2, RPI }
\begin{center}
\begin{spacing}{1.4}
\begin{tabular}{c|c|c|c}
\bf & & {Depth in} & {Depth in} \\
Measurement & {Instrument ID} & {model scale} & {prototype scale} \\
\hline
horizontal acceleration & $AH 1$ & $20 cm$ & $10 m$ \\
horizontal acceleration & $AH 3$ & $0 cm$ & $0 m$ \\
horizontal acceleration & $AH 4$ & $5.2 cm$ & $2.6 m$ \\
horizontal acceleration & $AH 5$ & $10 cm$ & $5 m$ \\
horizontal displacement & $LVDT 3$ & $0.9 cm$ & $0.45 m$ \\
horizontal displacement & $LVDT 4$ & $5 cm$ & $2.5 m$ \\
horizontal displacement & $LVDT 5$ & $10 cm$ & $5 m$ \\
horizontal displacement & $LVDT 6$ & $15 cm$ & $7.5 m$ \\
vertical displacement & $LVDT 1$ & $0 cm$ & $0 m$ \\
pore fluid pressures & $P 1$ & $2.9 cm$ & $1.45 m$ \\
pore fluid pressures & $P 2$ & $5.2 cm$ & $2.6 m$ \\
pore fluid pressures & $P 3$ & $10 cm$ & $5 m$ \\
pore fluid pressures & $P 4$ & $15 cm$ & $7.5 m$ \\
\end{tabular}
\end{spacing}
\end{center}
\end{table}
%\vspace{10 mm}
Prior to the numerical simulation, calibration of SANISAND \citep{Dafalias2004}
constitutive model was performed for Nevada sand, using soil data reported by
\cite{Arulmoli1992} in the framework of VELACS project. \cite{Taiebat2009a}
present the values of the material parameters of SANISAND for Nevada sand, shown
in Table \ref{DM04}. Fig. \ref{nevada test} compares the
stress paths and stress-strain loops for Nevada sand with $D_r\approx40\%$
obtained from lab tests with those obtained from calibration process.
%\vspace{10 mm}
\begin{table}[tbh]
\caption{ \label{DM04} Material parameters of Dafalias-Manzari model. }
\begin{center}
\begin{spacing}{1.4}
\begin{tabular}
[c]{l|c|c||l|c|c}
\multicolumn{2}{ l|}{\bf{Material Parameter}} & \bf Value &
\multicolumn{2}{|l|}{\bf{Material Parameter}} & \bf Value\\
\hline
Elasticity & $G_{0}$ & 150 kPa & Plastic modulus & $h_{0}$ & 9.7\\
& $v$ & 0.05 & & $c_{h}$ & 1.03\\
Critical sate & $M$ & 1.14 & & $n_{b}$ & 2.56\\
& $c$ & 0.78 & Dilatancy & $A_{0}$ & 0.81\\
& $\lambda_{c}$ & 0.027 & & $n_{d}$ & 1.05\\
& $\xi$ & 0.45 & Fabric-dilatancy & $z_{max}$ & 5.0\\
& $e_{r}$ & 0.83 & & $c_{z}$ & 800.0\\
Yield surface & $m$ & 0.05 & & & \\
\end{tabular}
\end{spacing}
\end{center}
\end{table}
%%\vspace{10 mm}
%======================
\begin{figure}[!htb]
\begin{center}
{\normalsize
\mbox{
\subfigure{Experiment} \quad
\hspace{4cm}
\subfigure{Simulation}
} \\
\setcounter{subfigure}{0}%
\mbox{
\subfigure[]{\includegraphics[width=0.48\textwidth]{CIUC-Cyclic-Nevada-test-1.eps}} \quad
\hspace{-1cm}
\subfigure[]{\includegraphics[width=0.48\textwidth]{CIUC-Cyclic-Nevada-model-1.eps}}
} \\
\mbox{
\subfigure[]{\includegraphics[width=0.48\textwidth]{CIUC-Cyclic-Nevada-test-2.eps}} \quad
\hspace{-1cm}
\subfigure[]{\includegraphics[width=0.48\textwidth]{CIUC-Cyclic-Nevada-model-2.eps}}
}\\
}
\end{center}
\caption{
\label{nevada test}
{Simulations versus experiments in undrained triaxial tests on Nevada sand with
$D_r\approx40\%$} published by \cite{Taiebat2009a}. The experimental data can be
found in \cite{Arulmoli1992}.}
\end{figure}
%======================
%
%\begin{figure}[!hbt]
%\begin{Large}
%\begin{sffamily}
%\begin{center}
%\includegraphics[width=16cm]{/home/jeremic/tex/works/Thesis/PanagiotaTasiopoulou/Thesis/validation/Figures/nevada_test.ps}
%\end{center}
%\caption{
%\label{nevada test}
%{Simulations versus experiments in undrained triaxial tests on Nevada sand with
%$D_r\approx40\%$} published by \cite{Taiebat2009a}. The experimental data can be
%found in \cite{Arulmoli1992}.}
%\end{sffamily}
%\end{Large}
%\end{figure}
%\vspace{10 mm}
After the calibration of the model, numerical simulation of direct shear test
was conducted so as to find the relationship between the cyclic resistance of
Nevada sand with relative density $D_r=40\%$, versus the number of cycles to
liquefaction, and compare it with the experimental data by \cite{Arulmoli1992}.
The numerical model consisted of a single $u-p-U$ brick element subjected to
cyclic horizontal shear loading under undrained conditions. Two different
initial vertical effective stresses were considered, $20$kPa and $70$kPa, in
order to investigate the effect of the initial confining stress, as
captured by the model. The cyclic stress
ratio, CSR, is defined as the ratio of horizontal shear stress to the initial vertical
effective stress. The number of cycles to liquefaction both in the experiment
and the analysis is determined according to the following criteria: (i) either
the axial strain exceeds 1.5\% or (ii) the excess pore water pressure ratio
becomes equal to one. Figure~\ref{CSR} illustrates the comparison between the
numerical and experimental data, showing, a fairly satisfactory agreement.
%, although the effect of
%the vertical effective stress on number of cycles cannot be properly captured by the model
% pis negligible after normalizing the shear stress to
% CSR.
%\vspace{10 mm}
\begin{figure}[!hbt]
\begin{Large}
\begin{sffamily}
\begin{center}
\includegraphics[width=12cm]{/home/jeremic/tex/works/Thesis/PanagiotaTasiopoulou/Thesis/validation/Figures/CSR.ps}
\end{center}
\caption{\label{CSR}{Cyclic shear stress ratio for Nevada sand with relative
density $D_r=40-45\%$ versus the number of cycles to liquefaction. Comparison
between the numerical results using material model from \cite{Dafalias2004} and the experimental
data by \cite{Arulmoli1992}.}}
\end{sffamily}
\end{Large}
\end{figure}
%\vspace{10 mm}
Most frequently, the numerical simulation of centrifuge tests is conducted in
prototype scale. However, due to conflicts related to the appropriate scaling of
dynamic time and dissipation-time, when water is used in tests
(\cite{Wood2004}), a different approach was adopted in the present study.
%
%Details on the scaling factors are presented in Section \ref{Discussion of
%Scaling laws}.
Two different numerical models were used:
%\vspace{10 mm}
\begin{enumerate}
\item Numerical Model \#1 in model scale.
The numerical model \#1 consists of a soil column of twenty 8-node $u-p-U$
brick elements with dimensions $1cm \times 1cm \times 1cm$. The total height
of the soil column is 20 cm. The gravity applied to the model is equal to 50g.
The input motion is shown in Fig. \ref{conf_cent}. The specific permeability,
$K$, is $3.2\times10^{-6} cm^3s/g$. The results of the analysis are
illustrated in Fig. \ref{disp_model} and \ref{acc_model} in model scale in
terms of time histories of displacements and accelerations respectively.
\begin{figure}[!hbt]
\begin{Large}
\begin{sffamily}
\begin{center}
\includegraphics[width=9cm]{/home/jeremic/tex/works/Thesis/PanagiotaTasiopoulou/Thesis/validation/Figures/model/disp_model.ps}
\end{center}
\caption{\label{disp_model}{Time histories of horizontal displacements at the depths of 0, 5 ,10 and 20 cm from the surface in model scale, as obtained from Numerical Model 1.}}
\end{sffamily}
\end{Large}
\end{figure}
\begin{figure}[!hbt]
\begin{Large}
\begin{sffamily}
\begin{center}
\includegraphics[width=9cm]{/home/jeremic/tex/works/Thesis/PanagiotaTasiopoulou/Thesis/validation/Figures/model/acc_model.ps}
\end{center}
\caption{\label{acc_model}{Time histories of horizontal acceleration at the depths of 0, 5 ,10 and 20 cm from the surface in model scale, as obtained from Numerical Model 1.}}
\end{sffamily}
\end{Large}
\end{figure}
\item Numerical Model \#2 in prototype scale.
The numerical model \#2 consists of a soil column of twenty 8-node u-p-U brick
elements with dimensions $0.5m\times0.5m\times0.5m$. The total thickness of
the soil column is 10 m. The gravity applied to the model is equal to $1$g.
The input motion has been modified to prototype scale (the dynamic time has
been multiplied by 50 and the acceleration has been divided by 50). The
specific permeability, $K$, is $1.6\times10^{-4} cm^3s/g$. The results of the
analysis are illustrated in Figs. \ref{disp_pm_comp} to \ref{pe_pm_comp}.
Results are presented in prototype scale and are comparing Numerical Models
\#1 and \#2 after scaling laws between Case B and C of Table~\ref{Scaling}
were applied - apart from strain and stresses which are in the same scale by
default for both models.
\begin{figure}[!hbt]
\begin{Large}
\begin{sffamily}
\begin{center}
\includegraphics[width=9cm]{/home/jeremic/tex/works/Thesis/PanagiotaTasiopoulou/Thesis/validation/Figures/proto_model_comp/disp_proto_model_comp.ps}
\end{center}
\caption{\label{disp_pm_comp}{Time histories of horizontal displacement at the
depths 0, 2.5, 5, 7.5 m from the surface. Comparison between the results
obtained from Numerical Model 1 and Numerical Model 2, in prototype scale.}}
\end{sffamily}
\end{Large}
\end{figure}
\begin{figure}[!hbt]
\begin{Large}
\begin{sffamily}
\begin{center}
\includegraphics[width=9cm]{/home/jeremic/tex/works/Thesis/PanagiotaTasiopoulou/Thesis/validation/Figures/proto_model_comp/acc_proto_model_comp.ps}
\end{center}
\caption{\label{acc_pm_comp}{Time histories of horizontal acceleration at the
depths 0, 2.5, 5, 10 m from the surface. Comparison between the results obtained
from Numerical Model 1 and Numerical Model 2, in prototype scale.}}
\end{sffamily}
\end{Large}
\end{figure}
\begin{figure}[!hbt]
\begin{Large}
\begin{sffamily}
\begin{center}
\includegraphics[width=9cm]{/home/jeremic/tex/works/Thesis/PanagiotaTasiopoulou/Thesis/validation/Figures/proto_model_comp/pp_proto_model_comp.ps}
\end{center}
\caption{\label{pp_pm_comp}{Time histories of excess pore water pressure at the
depths 1.5, 5, 7.5, 9 m from the surface. Comparison between the results
obtained from Numerical Model 1 and Numerical Model 2, in prototype scale.}}
\end{sffamily}
\end{Large}
\end{figure}
\begin{figure}[!hbt]
\begin{Large}
\begin{sffamily}
\begin{center}
\includegraphics[width=9cm]{/home/jeremic/tex/works/Thesis/PanagiotaTasiopoulou/Thesis/validation/Figures/proto_model_comp/settle_proto_model_comp.ps}
\end{center}
\caption{\label{settle_pm_comp}{Time histories of vertical displacement at the
depths 0, 3, 6, 9 m from the surface. Comparison between the vertical outward
movement of the water and the settlement of the soil, in prototype scale.}}
\end{sffamily}
\end{Large}
\end{figure}
\begin{figure}[!hbt]
\begin{Large}
\begin{sffamily}
\begin{center}
\includegraphics[width=9cm]{/home/jeremic/tex/works/Thesis/PanagiotaTasiopoulou/Thesis/validation/Figures/proto_model_comp/pq_proto_model_comp.ps}
\end{center}
\caption{\label{pq_pm_comp}{Horizontal shear stress versus effective vertical
stress at the depths of 1.25, 5, 7.5, 9 m. Comparison between the stress paths
obtained from Numerical model 1 and Numerical Model 2.}}
\end{sffamily}
\end{Large}
\end{figure}
\begin{figure}[!hbt]
\begin{Large}
\begin{sffamily}
\begin{center}
\includegraphics[width=9cm]{/home/jeremic/tex/works/Thesis/PanagiotaTasiopoulou/Thesis/validation/Figures/proto_model_comp/pe_proto_model_comp.ps}
\end{center}
\caption{\label{pe_pm_comp}{Shear stress-strain loops at the depths of 1.25, 5,
7.5, 9 m. Comparison between the stress paths obtained from Numerical model 1
and Numerical Model 2.}}
\end{sffamily}
\end{Large}
\end{figure}
\end{enumerate}
The soil properties used for the both numerical models are shown in Table
\ref{Soil_Properties_for_centrifuge_test}. The Newmark time integration method is used
since it was shown in verification study \citep{Tasiopoulou2014} to better dissipate high
frequencies introduced in the coupled system by the discretization
process. The Newmark parameters
used for all the numerical analysis for both models are: $\gamma=0.7$,
$\beta=0.42$. The column is horizontally excited during a second stage of
loading, after first stage self-weight loading. It
should be noted that the self weight loading is performed on an initially zero
stress (unloaded) soil column and that the material model and constitutive
integration algorithm is versatile enough to
follow through this early loading with proper parameter evolution. The boundary
conditions are such that the soil and water displacement degrees at the bottom
surface are fixed, the pore fluid pressure degrees are free; the soil and water
displacement degrees at the upper surface are free, however the pore pressure
degrees are fixed (zero pore fluid pressure) to simulate the upward drainage. In order to simulate one
dimensional shaking (shear box), all the degrees of freedom at the same level are constrained
in a master-slave fashion.
%\vspace{10 mm}
\begin{table}[!htb]
\caption{ \label{Soil_Properties_for_centrifuge_test}
Soil Properties for the centrifuge test, Model No. 1, test 2, RPI}
\begin{center}
\begin{spacing}{1.4}
\begin{tabular}{c|c|c}
Parameter & Symbol & Value \\
\hline
gravity acceleration & $g$ & 9.81~$m/s^2$ \\
solid particle density & $\rho_s$ & $2.65\times 10^3$~$kg/m^3$ \\
water density & $\rho_f$ & $1.0\times 10^3$~$kg/m^3$ \\
solid particle bulk modulus & $K_s$ & $36.0\times 10^6$~$kN/m^2$ \\
fluid bulk modulus & $K_f$ & $2.2\times 10^6$~$kN/m^2$ \\
porosity & $n$ & 0.4253 \\
initial void ratio & $e_0$ & 0.74 \\
Darcy permeability & $k_D$ & $1.6\times\ 10^{-3}$m/s \\
Biot coefficient & $\alpha$ & $1.0$ \\
\end{tabular}
\end{spacing}
\end{center}
\end{table}
The Darcy permeability of Nevada sand with $D_r\approx40\%$, varies in the
range of $k_D = 2.1\times10^{-5} m/s$ to $k_D = 3.3\times10^{-5} m/s$, as reported by
\cite{Arulmoli1992}. The prototype model is considered to have 50 times larger
Darcy permeability that the one measured in the lab. The Darcy permeability used
in the present study for both Numerical Models \#1 and \#2 is $1.6\times10^{-3} m/s$.
It is noted that the input parameter in our numerical code is the specific permeability,
$K$, given by Equation \ref{kspec}:
\begin{eqnarray}
k=\frac{k_D}{\rho_f\times{G}}
\label{kspec}
\end{eqnarray}
%
where $G$ is equal to $50$g for Numerical Model \#1 and $1$g for Numerical Model \#2.
%\vspace{10 mm}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Discussion of Numerical Results}
\label{Conclusions}
%\vspace{10 mm}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\paragraph{Comparison between Numerical Model \#1 and Numerical Model \#2.}
\label{numerical models}
The results obtained from the Numerical Model \#1 are shown in Figures
\ref{disp_model} to \ref{acc_model} in model scale in order to illustrate that
this type of analysis, using very small size of elements and time step can be
easily performed by the numerical code and also to illustrate/present results that
are obtained in model scale in a centrifuge experiment.
The numerical
results obtained from Numerical Model \#1 were scaled to prototype scale
according to the scaling laws (Table, \ref{Scaling} between Column 2
and 3), so that they can compared with those of Numerical Model \#2.
This comparison between Numerical Models \#1 and \#2, illustrated in
Figures~\ref{disp_pm_comp} to \ref{pe_pm_comp}, is used to verify the scaling
factors derived in section
\ref{Discussion of Scaling laws}, while also giving a better insight in
the intrinsic differences that occur due to the difference in scale of the
two numerical models.
Numerical results between the two models compare well
for displacements, accelerations and excess pore water
pressure. In particular, Figure~\ref{pp_pm_comp}
shows a complete agreement of the numerical results in the dissipation of the excess
pore pressure during and after the end of shaking. Here the excess pore
water pressure, $r_u$, is defined as the ratio of excess pore water pressure over
the initial
vertical effective stress.
%This confirms the
%correctness of the scaling factors applied to the numerical analysis.
The slight difference in the results, shown in time histories of displacement
(Figure~\ref{disp_pm_comp}) is resulting from differences in the stress
paths and stress-strain loops shown in Figures~\ref{pq_pm_comp} and
\ref{pe_pm_comp} respectively.
Numerical Model \#1 demonstrates a more
dilative behavior giving larger negative strains than Numerical Model \#2.
%
These differences are related to the velocity proportional
damping associated with the $u-p-U$ formulation.
Velocity proportional damping is related to permeability \citep{Jeremic2007e},
and in this particular case, it is related to intrinsic
permeability which does not scale.
As discussed by \citet{Tasiopoulou2014}, intrinsic
(isotropic) permeability ${k}$ has dimensions of [$L^{3}TM^{-1}$] and is
different from Darcy permeability (hydraulic conductivity), ($k_{D}$), which has
the dimension of velocity, i.e. [$LT^{-1}$]. They are related by $k = k_{D} / g
\rho_f$, where $g$ is the gravitational acceleration and $\rho_f$ is the density
of the pore fluid which is slightly changed here.
%
However,
effects are small and are mostly apparent at lower levels of the centrifuge
model. It should be noted that there are two types of energy dissipation in
fully-coupled systems: (i) velocity proportional damping due to viscous coupling
and (ii) displacement proportional damping due to frictional damping and
elasto-plasticity \citep{local-87}. In this case displacement proportional
damping (energy dissipation) related to elasto-plasticity is much larger than
velocity proportional damping, and controls the response.
%OVDE
Figure~\ref{pq_pm_comp}, shows the reduction of the vertical effective stress
as a function of cyclic shear stress. It is also noted that vertical effective
stress rebounds after shaking stops.
%
The $r_u$ time
histories in Figure~\ref{pp_pm_comp} give clearer effective stress reduction and
rebounding.
% Here the excess pore water pressure, $r_u$, is defined as the ratio
%of excess pore water pressure over the initial vertical effective stress.
We define that liquefaction occurred for values of $r_u \ge 0.90$.
%
%The calculated maximum horizontal shear strain can reach up to
%$3.2\%$.
%
It is also apparent (Figure~\ref{pe_pm_comp}) that the shear modulus decreases as the shaking
progresses and the pore fluid pressure increases. It is important to note
that not all soil layers
experience liquefaction, as $r_u$ decreases with the depth increase.
The excess pore fluid pressure dissipation at the lower layers is quicker than
that of the upper layers. The upper layers have to dissipate their own excess
pore
pressure, however they also receive additional volume of water
from lower levels (dissipation is upward). This leads to increase in pore
fluid pressure at top layers even after the shaking
has stopped (see upper three plots in Figure~\ref{pp_pm_comp}). The soil
settlement and water drainage continue after the initial shaking is over mainly
due to the continuous pore fluid movement upward.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\paragraph{Comparison between numerical and centrifuge results.}
\label{n_vs_c_comp}
The comparison of the numerical and the centrifuge results are presented in
Figures~\ref{disp_comp} to \ref{pp_comp}. The time and displacements are
multiplied by 50 and the accelerations are divided by 50, according to the
scaling laws provided in section \ref{Discussion of Scaling laws}.
Scaled results are validated using centrifuge results, as presented by
\cite{Dobry1993}.
\begin{figure}[!hbt]
\begin{Large}
\begin{sffamily}
\begin{center}
\includegraphics[width=9cm]{/home/jeremic/tex/works/Thesis/PanagiotaTasiopoulou/Thesis/validation/Figures/disp_comp.ps}
\end{center}
\caption{\label{disp_comp}{Time histories of horizontal displacements at the
locations LVTD3, LVDT4, LVDT5 and LVDT6. Comparison between the numerical and
the experimental results from the centrifuge test, in prototype scale.}}
\end{sffamily}
\end{Large}
\end{figure}
\begin{figure}[!hbt]
\begin{Large}
\begin{sffamily}
\begin{center}
\includegraphics[width=9cm]{/home/jeremic/tex/works/Thesis/PanagiotaTasiopoulou/Thesis/validation/Figures/acc_comp.ps}
\end{center}
\caption{\label{acc_comp}{Time histories of horizontal accelerations at the
locations AH1, AH3, AH4 and AH5. Comparison between the numerical and the
experimental results from the centrifuge test, in prototype scale.}}
\end{sffamily}
\end{Large}
\end{figure}
\begin{figure}[!hbt]
\begin{Large}
\begin{sffamily}
\begin{center}
\includegraphics[width=9cm]{/home/jeremic/tex/works/Thesis/PanagiotaTasiopoulou/Thesis/validation/Figures/pp_comp.ps}
\end{center}
\caption{\label{pp_comp}{Time histories of excess pore water pressure at the
locations P1, P2, P3 and P4. Comparison between the numerical and the
experimental results from the centrifuge test, in prototype scale.}}
\end{sffamily}
\end{Large}
\end{figure}
In general, it is shown that the physical mechanism of liquefaction is captured by the numerical
analysis, providing validation of numerical modeling.
%
In particular, the displacement time histories, and
especially the residual values of displacement (see Fig.~\ref{disp_comp})
indicate good agreement with the experimental results. The magnitude of
cyclic component, however, is underestimated by the numerical results, a phenomena which
was also observed in other studies \citep{Elgamal2002, Taiebat2007}.
%
The excess pore pressure time histories (Fig.~\ref{pp_comp}) indicate
good agreement between numerical and centrifuge results, except for the deepest layer.
%
Earlier rise
in excess pore pressures is observed in the numerical analysis when compared to the
centrifuge test.
Differences in liquefaction initiation and timing
affect the acceleration time histories,
as can be observed in Fig. \ref{acc_comp}.
%
The cut-off of acceleration, indicative of liquefaction\footnote{and
providing for base isolation by liquefaction, as described by
\citet{Taiebat2009a}.},
start to occur deep in
the soil column (location AH5), and this contrasts the experimental
results.
%
These results are compatible with the development of excess pore pressure shown in
Figure~\ref{pp_comp}.
However, there is a good agreement between the numerical and the
centrifuge results in terms of the acceleration time history at the surface of
the soil layer.
The fact that liquefaction seems to occur deep in the soil column for the
numerical case, may be attributed to the calibration of the constitutive model.
By observing the stress paths in Figs.~\ref{pq_pm_comp} and
\ref{pe_pm_comp}, it can be noted that there is a quick decrease of the
effective vertical stress at all depths. This may be attributed to the fact that the current
constitutive model is based only on changes in stress ratio (see Figure~\ref{CSR}) , whereas in reality
the rate of excess pore pressure generation depends on the initial mean
effective stress as well. This could be modified by adjusting the values of the
parameters of the model with depth in order to capture this specific case. However, the
intention of this study is to calibrate the constitutive model using the soil
data obtained from laboratory (triaxial) tests, conducted in the
framework of a VELACS project, and thus perform a real numerical prediction of
behavior observed in physical tests, without using centrifuge test results
to (back/re-) calibrate (change) material parameters.
%
This approach contrasts an approach (often used) in which material model parameters are
re-calibrated (changed) so that numerical results fit the centrifuge test data,
while neglecting prior calibration from laboratory (triaxial) tests.
%
Possible improvement to the material model would be to render some parameters of the constitutive model
dependent on the confining stress, for example the parameters that control the
dilatancy, such as $A_d$.
%
Comparison of vertical displacements, shown in Figure~\ref{SETTLE_comp}, shows
again differences in magnitude.
%
\begin{figure}[!hbt]
\begin{Large}
\begin{sffamily}
\begin{center}
\includegraphics[width=10cm]{/home/jeremic/tex/works/Papers/2011/Validation_Coupled_Systems/SETTLE_comp.ps}
\end{center}
\caption{\label{SETTLE_comp}{Time histories of vertical displacements (settlement).
Comparison between the numerical and the
experimental results (at two surface locations, middle -- LVDT1 and quarter
width from the side -- LVDT2 ) from the centrifuge test, in prototype scale.}}
\end{sffamily}
\end{Large}
\end{figure}
%
%Some of these differences were observed by other researchers as well.
%
%
These differences are significant and have been occasionally reported in
technical meetings, but to our knowledge, have never been presented or published
in an archival journal.
There are a number possible effects that can be used to explain the
observed discrepancies between the numerical and centrifuge results.
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\subsection{Comments on Validation Testing Using Centrifuge Modeling Results}
%\label{Discussion of Scaling laws}
Provided here are comments related to potential issues with our
numerical modeling and with centrifuge modeling results use for validation of
numerical modeling and simulation.
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\paragraph{Use of Constant Permeability.} Permeability of soil changes during
liquefaction \citep{Shahiri2008}. However, the effect of changing
permeability is not modeled in this study. This modeling
simplification can potentially have large effects on results, as
with an increase in permeability during liquefaction, water moves more freely
and thus more settlement occurs.
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\paragraph{Boundary Conditions.}
%
Idealized boundary
conditions used in numerical simulation of centrifuge experiments
simulate a 1D shear beam.
%
These boundary conditions mimic a laminar box around the contained soil.
%
For numerical model, these boundary conditions allow the (separate) vertical movement of
the soil and water (while allowing horizontal shaking of soil and water) and
prevent any lateral expansion. However, the real conditions in the centrifuge model
involve 3D effects that, for example, may allow for a lateral
flow and consequently quicker dissipation which leads to slower built up of
excess pore pressures.
%
While we know that full 3D behavior is present in a centrifuge laminar box,
limitations in instrumentation and pre-assumption of 1D behavior leave us with
no data (measurements) of such 3D behavior.
%
The only options thus is to numerically model a 1D shear box.
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\paragraph{Variability of Material in Laboratory Test and Centrifuge Model.}
Laboratory tests performed on same (similar) soil were used for calibration of
material model that was then used in modeling and simulation validation, using
centrifuge tests. It is not clear how similar soil used in those laboratory
tests is to the soil used in a centrifuge experiment. Only relative density is
used to describe both soils (same Nevada sand), however, other factors do affect
soil behavior. For example, sand fabric and anisotropy, both initial and
induced, have certainly been different for laboratory and centrifuge tests, and
yet information about those differences is not available (not measured).
Differences in material behavior can significantly affect the final numerical
simulation and test results, and that might be one of the reasons for some of
the discrepancies in this validation study.
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\paragraph{Scaling of Dynamic and Dissipation Time.}
It is common to perform numerical modeling of centrifuge
tests in prototype scale. This is due to the fact that centrifuge results are
published in prototype scale, while, unfortunately, the scaling factors that
have been used are not always reported. Scaling factors used to
present the experimental measurements in prototype scale correspond to
Case C (see Table \ref{Scaling}), resulting in similarity of diffusion and
dynamic time. These are the scaling factors applied to the current study.
However, the scaling relationships were not enforced in the centrifuge. This can
lead to a possible concussion that since diffusion happens faster than pore
water pressure generation, more diffuses that is the reason for slower
pore fluid pressure rises and larger settlements in centrifuge.
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\paragraph{Scaling of Failure Zones -- Shear Bands.}
Scaling problems for failure of soils, where localization of deformation occurs,
can bring additional challenges. Sand (coarse grained particulate material) develop shear
zone that is usually 5-20 sand
particles wide \citep{Alshibli2000}. That means that if model and original soil
are the same, i.e. Nevada sand with mean grain size of $0.15$~mm, the original
shear zone width will be between $5 \times 0.15{\rm mm} = 0.75$mm and $20 \times
0.15{\rm mm} = 3.00$mm wide, while the same width of shear zone will be carried to
the model
scale. Ideally, in model scale the shear zone would have to be scaled down $N$
times, which is not the case here. This can have effect on
any phenomena modeled in centrifuge where localized deformation occurs, as
for example is the case of vertically propagating shear waves, which plastify
(and fail) soil as they propagate. This lack of scaling of the plastified/shear
band zone can influence results, particularly when soil in such
shear zone dilates and compresses, while pore fluid is being pumped out and in.
%
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\paragraph{Uncertainty in Measurement Locations.}
The accuracy of location of measurements in the centrifuge can be
ambiguous. For example, if the location of the instrumentation is off by 1 cm in
the centrifuge model, this error translates to half a meter location error for
a centrifuge experiment performed at $50$g level.
%
In addition, dense
instrumentation and connecting wires within centrifuge model affect behavior
of soils, but are not explicitly taken into account for validation.
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\section{Conclusions}
Presented here was a validation study that aimed to validate numerical modeling
and simulation of fully coupled behavior of fully saturated sand, using
u-p-U formulation. Numerical
models of centrifuge tests were used for validation. An extensive discussion of
scaling laws as they apply to validation of numerical modeling using centrifuge
test data was provided.
Numerical modeling was, for the most part, successfully validated.
Observed differences have been explained using scaling laws, mechanics of
coupled systems, and inconsistencies in boundary conditions (assumed versus actual).
%
It is noted that centrifuge modeling can be very useful in modeling of
coupled system, provided that scaling laws are carefully taken into account and
that phenomena of first and second order importance are prioritized and
separated.
%
On of the main conclusions is that numerical modeling and simulation can be
successfully used to predict behavior of fully coupled solids (saturated soil)
and thus used to improve design (safety and economy), provided that there is a
clear trail of verification and validation for the numerical tool used.
%
Without such documented verification and validation, software errors and
modeling uncertainty that are (potentially) present in numerical modeling tools,
can render results unreliable and thus unusable for design.
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\section{Acknowledgement}
Author's would like to thanks Professor Bruce Kutter for many useful comments
and guidance provided during this research.
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\end{document}
\bye