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{\Large \bf Seismic Behavior of NPP Structures Subjected to Realistic
3D, Inclined Seismic Motions, in Variable Layered Soil/Rock, on Surface or
Embedded Foundations}
\end{flushleft}
\vspace*{1cm}
\begin{flushleft}
{B. Jeremi{\'c}${}^{1,2}$,
N. Tafazzoli${}^{1}$,
T. Ancheta${}^{3}$,
N. Orbovi{\'c}${}^{4}$,
A. Blahoianu${}^{4}$
}\\
%\end{flushleft}
%\begin{flushleft}
\vspace*{1cm}
{\small ${}^{1}$~University of California, Davis, California, U.S.A.} \\
{\small ${}^{2}$~Lawrence Berkeley National Laboratory, Berkeley, California, U.S.A. } \\
{\small ${}^{3}$~Risk Management Solutions, Inc., Newark, California, U.S.A.}\\
{\small ${}^{4}$~Canadian Nuclear Safety Commission, Ottawa, Canada} \\
{\small Corresponding author: B. Jeremi{\'c}, phone~(530) 754-9248. fax~(530) 754-7872,
email~\texttt{jeremic@ucdavis.edu}}
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\date{}% No date.
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% % {\small \it Copyright, Boris Jeremi{\'c}}
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\section{Abstract}
Presented here is an investigation of the seismic response of
a massive NPP structures due to full 3D, inclined, un-correlated input motions
for different soil and rock profiles. Of particular interest are the effects of
soil and rock layering on the response and the changes of input motions
(frequency characteristics) due to such layering. In addition to rock/soil
layering effects, investigated are also effects of foundation embedment on
dynamic response. Significant differences were observed in dynamic response of
containment and internal structure founded on surface and on embedded
foundations. These differences were observed for both rock and soil profiles.
%
Select results are used to present most interesting findings.
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\section{Introduction}
\label{Introduction}
Seismic response of Nuclear Power Plants (NPPs) has been at the forefront of
interest from the very beginning of use of nuclear energy. Significant research
has been done over time on the subject seismic response of NPPs
\citep{Wolf1983, Finn1986, Wagenknecht1987, Chang1988, Hanada1988, Wolf1989,
Halbritter1998, Elaidi1998, VenancioFilho1997, Ueshima1997, Saxena2012, Saxena2011, Wolf1989,
Ryu2010}.
%
Seismic response of NPPs is usually quite complex, involving a number of
nonlinearities, including:
\begin{itemize}
\item Nonlinear response (elastic-plastic, damage, etc.) of soil and rock
material adjacent to the foundation,
\item Nonlinear response of contact zone between soil/rock and the
foundation (gaping, slipping, elastic-plastic, damage),
\item Full nonlinear coupling between pore fluid and soil/rock skeleton
(which strongly affects the nonlinear response of soil and rock),
\item Buoyant effects, where pore fluid pressures on foundations create
time varying, nonlinear forces on the foundation (which also strongly affect
nonlinear response of the foundation -- soil/rock contact zone),
\item Nonlinear response of the structure (containment and internal),
\item Nonlinear interaction of fluids and structures/solids within the NPP.
\end{itemize}
In addition to that,
seismic wave field features the following complexities:
\begin{itemize}
\item Seismic body waves (P and S, which are fully three dimensional (3D), and
inclined),
\item Seismic surface waves (Rayleigh, Love, etc., which produce
three translational and three rotational components of motions at point
at and near (at depth) the surface),
\item Seismic wave lack of correlations (or incoherence, as it is known in
the frequency domain)
\end{itemize}
Present study focuses on the effects soil/rock
layering has on seismic response of an NPP subjected to realistic, 3D, inclined
seismic wave field. Moreover, effects of foundation embedment are evaluated as
well for two NPP models, one on surface and the other on a $15$m
embedded foundation.
Paper is organized in two main parts, the first describing the simulation model
development, while the second part presents a selected set of results that are
used to emphasize main findings.
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\section{Simulation Model Development}
\label{SimulationModelDevelopment}
Presented here briefly is the development of the simulation model, including
the soil/rock profiles, the finite element model for soil/rock-structure
system, input motions development and the simulation platform.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Soil/Rock Profiles}
Soil and rock layering beneath the NPP was assumed to be horizontal. A total of
twelve soil (shear wave velocity $V_s \le 760$m/s) and rock (shear wave velocity
$V_s > 760$m/s) models (profiles) were developed.
%
Their spatial distribution and assumed properties (layer thicknesses and shear
wave velocities) are presented in
Table~\ref{soil_profiles_table}. One of the focuses of the study was to
investigated seismic behavior of an NPP founded on a variety of geologic
conditions, from hard rock (Case 1) all the way to soil (Case 8), with variations of uniform soil/rock
profiles (Cases 1, 2, 4, 6, 8) and layered cases (3, 5, 7, 9, 10, 11, 12).
Soil/rock profiles described in Table~\ref{soil_profiles_table} extend all the
way to a depth of $40$Km, however Table only gives the top $500$m of shear wave
velocities (and thus the soil stiffness). Beneath top $500$m the soil/rock
profiles are marked as either {\bf G} (gradual increase in soil/rock
shear wave velocity (stiffness) to values of $V_s=3200$m/s and above) or as
{\bf I} (immediate increase in soil/rock shear wave velocity
(stiffness) to a value of $V_s=4000$m/s.).
%\begin{landscape}
\begin{tiny}
\begin{table}[!h]\small
\caption{\label{soil_profiles_table} Table describing 12 soil profiles used in this
study up to a depth of $500$m. }
%\hspace*{-18.9cm}
\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|c|c|}
\hline
& \multicolumn{12}{|c|}{Soil profiles with shear wave velocity [m/s]}\\
\hline
Depth [m] & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12\\
\hline
50 &
\multirow{10}{*}{\rm 2600 } &
\multirow{10}{*}{\rm 1500 } &
\multirow{4}{*}{\rm 1000 } &
\multirow{10}{*}{\rm 1000 } &
\multirow{4}{*}{\rm 300 } &
\multirow{10}{*}{\rm 300 } &
\multirow{4}{*}{\rm 300 } &
\multirow{10}{*}{\rm 300 } &
\multirow{2}{*}{\rm 300 } &
\multirow{2}{*}{\rm 300 } &
{\rm 300} &
{\rm 300 }\\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\cline{1-1}\cline{12-12}\cline{13-13}
{100} & & & & & & & & & & & {\rm 400} & \\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\cline{1-1}\cline{10-12}
{150} & & & & & & & & & \multirow{2}{*}{\rm 500 } & & \multirow{2}{*}{\rm 500 } & \\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\cline{1-1}
{ 200} & & & & & & & & & & & & \\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\cline{1-1}\cline{4-4}\cline{6-6}\cline{8-8}\cline{10-10}\cline{12-12}
{ 250} & & & \multirow{2}{*}{\rm 1500 } & & & & \multirow{2}{*}{\rm 750 } & & \multirow{2}{*}{\rm 750 } & & \multirow{2}{*}{\rm 750 } & \\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\cline{1-1}
{ 300} & & & & & {\rm 2600} & & & & & {\rm 2600} & & {\rm 2600}\\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\cline{1-1}\cline{4-4}\cline{8-8}\cline{10-10}\cline{12-12}
{ 350} & & & \multirow{4}{*}{\rm 2000 } & & & & \multirow{4}{*}{\rm 1000 } & & \multirow{4}{*}{\rm 1000 } & & \multirow{4}{*}{\rm 1000 } & \\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\cline{1-1}
{ 400} & & & & & & & & & & & & \\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\cline{1-1}
{ 450} & & & & & & & & & & & & \\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\cline{1-1}
{ 500} & & & & & & & & & & & & \\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\cline{1-13}
%to $\infty$ & gradual & gradual & gradual & gradual & stiff & stiff & stiff & gradual & stiff & stiff & stiff & stiff\\
to $\infty$ & {\bf G} & {\bf G} & {\bf G} & {\bf G} & {\bf I} & {\bf I} & {\bf I} & {\bf G} & {\bf I} & {\bf I} & {\bf I} & {\bf I}\\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\hline
\end{tabular}
\end{table}
\end{tiny}
%\end{landscape}
It is important to note that profiles 1 and 8 represent the two extremes, while
other soil profiles fall in between.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\subsection{Seismic Input}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Seismic Motion Modeling}
\label{SMM}
Seismic motions were developed using an analytic solution, developed
using Green's functions \citep{Hisada1994, Hisada2003}. The
method takes into account fault slip mechanism, and
propagates the wave through elastic half space layered medium to the surface.
%
The fault rupture mechanics was chosen to be similar to that of
Northridge mainshock, which generated an earthquake with a moment magnitude of
$6.7$. Fault
rupture was modeled using Boore's source model, while the seismic waves
propagating from that source are modeled using Green's functions for layered
half space. As presently implemented, this model can model a minimum
period of $T=0.06$s (corresponding to the frequency of $f=16.67$Hz). Rupturing
fault parameters include: fault length of $18$km, fault width of $24$km, number of
sub-faults along the length and width is $14 \times 14$, strike of $122^o$, and
a dip of $40^o$.
Figure~\ref{Map_of_Points} shows ground motion model setup.
%
\begin{figure}[!hbt]
\begin{center}
%\includegraphics[width=14cm]{/home/jeremic/tex/works/Reports/2010/CNSC/Map_of_Points.pdf}
\includegraphics[width=12cm]{Map_of_Points.pdf}
\vspace*{-0.5cm}
\caption{\label{Map_of_Points} Analytic model (Green's functions) setup, with
locations of hypocenter, epicenter and the DRM model.}
\end{center}
\end{figure}
%
The fault was at the depth of approximately
$17$km,
while the NPP structures (the soil/rock structure model) was placed at approximately
$17$km away from the earthquake epicenter. The main motivation for a chosen setup was
to have a rupturing fault (hypocenter and epicenter) fairly close to the NPP, as
was the case of the recent earthquake at the Kashiwazaki NPP in Japan.
Moreover, close proximity (in relative terms) of the seismic source will result
in the development of
significant 3D, inclined, body and surface seismic waves. The
actual NPP site, was oriented in such a way to pick up
highest horizontal motions in local X direction, as shown in
Figure~\ref{Map_of_Points}.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\paragraph{The Domain Reduction Method.}
Seismic input is modeled using the Domain Reduction Method
\citet{Bielak2001, Yoshimura2001}).
%
It is a modular, two-step dynamic procedure
aimed at reducing the large computational domain to a more manageable size.
%
The DRM replaces forces from the seismic source with dynamically consistent set
of forces applied on a single layer of finite element surrounding a
soil/rock-structure NPP system.
%
The replacement force, aka an effective force, ($P^{eff}$) and is given by :
%
%
\begin{eqnarray}
P^{eff} = \left\{\begin{array}{c} P^{eff}_i \\ P^{eff}_b \\ P^{eff}_e \end{array}\right\}
= \left\{\begin{array}{c} 0 \\ -M^{\Omega+}_{be} \ddot{u}^0_e-K^{\Omega+}_{be}u^0_e
\\ M^{\Omega+}_{eb}\ddot{u}^0_b+K^{\Omega+}_{eb}u^0_b\end{array}\right\}
\label{DRMeq09}
\end{eqnarray}
%
Accelerations ($\ddot{u}^0$) and displacements ($u^0$) are obtained from the
free field model (obtained from an analytic solution, described in section
\ref{SMM} above) and feature realistic, 3D, inclined seismic
waves including both body (P and S) and surface (Rayleigh and Love) waves.
%
Part of the model that is inside a single layer of finite elements where
effective forces are applied can be material nonlinear. Additional DRM modeling
details are available in \citet{Jeremic2012}.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsubsection{Lack of Correlation (Incoherence) for the Original Seismic Motions}
Figure~\ref{coh_comp_x_grouped.pdf} shows the analysis
of analytically developed seismic motions (using Green's functions for layered
half space) with incoherence function developed from the Lotung LSST site in
Taiwan by
\cite{Abrahamson1991,Abrahamson1992,Abrahamson2005,Abrahamson1992b}.
For each of the twelve simulations the x-components of points 1 and 2,
were used to estimate lagged coherency. To be consistent with the incoherence
function, the lagged coherency is calculated on the strong motion window using
an 11-point Hamming window.
%
%see
%Figure~\ref{Location_of_Points}.
%
%
\begin{figure}[!hbt]
\begin{center}
%\includegraphics[width=12cm]{/home/jeremic/tex/works/Reports/2010/CNSC/TimAncheta/coh_comp_x_grouped.pdf}
\includegraphics[width=12cm]{coh_comp_x_grouped.pdf}
\caption{\label{coh_comp_x_grouped.pdf} A comparison incoherence of analytically
developed seismic motions (using Green's functions for layered half-space) with
incoherence model by Abrahamson, from Lotung LSST site, for all twelve soil/rock
profiles, grouped as rock (1-4, black markers) and soil (5-12, red markers) profiles.}
\end{center}
\end{figure}
%
Lagged coherency between the pairs of the synthetic motions is biased higher
(i.e. more coherent) at frequencies up to approximately 7Hz as shown in
Figure~\ref{coh_comp_x_grouped.pdf}. The large spike in lagged coherency at 16
Hz is due to a 16 Hz frequency limit of the synthetic motions developed using
layered half space model.
It is interesting to note that even a model with quite regular layering,
simulated over half space, is able to predict incoherence, although to a smaller
degree than what the extrapolated model from Lotung LSST site shows.
These
results signify that a dimension of the model (in this case half-space) plays
an important role in development of the incoherence, since it was shown by
\cite{Walling2009}, that a limited size model (a block with sides of $1$km)
cannot account for a majority of incoherence.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\clearpage
\subsubsection{Lack of Correlation (Incoherence) for the Abrahamson Model Motions}
In addition to (partially) incoherent motions developed using layered
half space simulations, an additional (different) set of incoherent motions was
developed in order to test the effects incoherency has on NPP response.
%
In order to introduce appropriate amount of variability over the bandwidth due to
incoherence effects, each point was used as a seed motion to generate a new
motion across the grid at a separation distance of 200 meters. In other words
point 1 is modified to generate a ground motion at point 2 and vice versa. The
method of introducing incoherence effects is called Frequency Dependent
Windowing (FDW) \citep{Ancheta2011}.
%
The method is a non-stationary simulation routine that utilizes
a modified short-time Fourier transform (MSTF) routine for which spectral
modifications are made consistent with a selected coherency model. The MSTF
routine allows preservation of the non-stationary properties of the motion and
incorporation of time-varying non-linear spectral modifications.
The routine is summarized below:
%
%short time windows of various lengths where the length of the window controls the
% frequency band over which spectral modifications are made consistent with a selected
%coherency ($\gamma$) and random Fourier amplitude model ($\sigma_{\Delta A}$).
\begin{enumerate}
\item The seed time series is split into short time segments.
\item A discrete Fourier transform (DFT) is applied to the segments.
\item
Fourier amplitudes and phase angles at each frequency within a desired
frequency range (dependent on segment length) are modified consistent to a
coherency function for each segment \citep{Ancheta2010}.
\item
The new set of Fourier phase angles is combined with the new set of Fourier
amplitudes and transformed into the time domain with an inverse Fourier
transform (IFT).
\item The modified short time segments are recombined to form a modified time
series.
\item Steps 1-5 are performed multiple times for multiple segment lengths, with
each segment length having a specified frequency range over which phase angles
are modified. Hence, multiple modified time series are created. Segment lengths
and corresponding frequency limits used are shown in Table \ref{AnchetaTable1}.
\item The multiple modified time series are match-filtered (i.e., band-pass
filtered with the limits of the pass-band matching the band of the
modification) to combine the modified frequency bands in the frequency domain.
\item The non-overlapping frequency bands are inverted with an IFT to create
the final broadband modified time series.
\end{enumerate}
\begin{table}[!htb]
\begin{center}
\caption{\label{AnchetaTable1} Segment duration (L) and frequency bands (b) used in
the FDW routine. $T$ is the duration of the seed series.}
%\hspace*{-18.9cm}
\begin{tabular}{|c|c|}
\hline
Segment duration, $L_i$ [sec] & Frequency limits , $b_i$ [Hz] \\
\hline
1.28 & 2-Nyquist \\
2.56 & 1-2 \\
5.12 & 0.5-1 \\
10.24 & 0.25-0.5 \\
20.48 & 0.12-0.25 \\
$T$ & 0-0.12 \\ \hline
\end{tabular}
\end{center}
\end{table}
%A description of the incoherence models used is presented next.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\paragraph{Description of Selected Incoherence Models.}
Synthetic models created by analytic solution for layered half space, with
different profiles but similar source were used as seed motions for incoherence
motion development. Soil/Rock profiles as described in
Table~\ref{soil_profiles_table} were used
and grouped as rock cases (1-4) with $V_s \ge 760$m/s and soil cases (5-12)
with $V_s < 760$m/s.
Separate coherency and random Fourier amplitude models are selected for
rock (1-4) and soil (5-12) profiles separately. For soil cases 1-4, a plane wave
coherency ($\gamma_{pw}$) and $\sigma_{\Delta A}$
developed by \cite{Abrahamson2007} from the Pinyon Flat array recordings located
in California was selected. These models are shown in
equations~\ref{Ancheta01} and \ref{Ancheta02} and Table~\ref{Ancheta03}
%
\begin{eqnarray}
\gamma_{pw} (f, \xi) =
\left[
1+
\left(
\frac{f \tanh( a_3 \xi)}{a_1 f_c(\xi)}
\right)^{n1(\xi)}
\right]^{-1/2}
%
\left[
1+
\left(
\frac{f \tanh( a_3 \xi)}{a_2}
\right)^{n2}
\right]^{-1/2}
\label{Ancheta01}
\end{eqnarray}
%
%
\begin{eqnarray}
\sigma_{\Delta A} (f, \xi)
=
0.79 (1-e^{-0.45 - 0.0017 \xi)f})
\label{Ancheta02}
\end{eqnarray}
\begin{table}[!htb]
\begin{center}
\caption{\label{Ancheta03} Plane-wave coherency model coefficients for the
horizontal component.}
%\hspace*{-18.9cm}
\begin{tabular}{|l|l|}
\hline
Coefficient name & Horizontal coefficient value\\
\hline
$a_1$ & $1.0$ \\ \hline
$a_2$ & $40$ \\ \hline
$a_3$ & $0.4$ \\ \hline
$n_1(\xi)$ & $3.80-0.040 \ln{(\xi+1)}+0.0105 \left[\ln{(\xi+1)}-3.6\right]^2$ \\ \hline
$n_2$ & $16.4$ \\ \hline
$f_c(\xi)$ & $27.9-4.82 \ln{(\xi+1)}+1.24 \left[\ln{(\xi+1)}-3.6\right]^2$\\ \hline
\end{tabular}
\end{center}
\end{table}
For soil cases (5-12), a lagged coherency ($|\gamma|$) and $\sigma_{\Delta A}$
developed by \cite{Ancheta2010} from the
Lotung LSST (Large Scale Seismic Test) and BVDA (Borrego Valley Differential Array)
recordings located in Taiwan and California, respectively, was selected. The models are
shown in equations~\ref{Ancheta04} and \ref{Ancheta05}.
%
\begin{eqnarray}
\tanh^{-1}{|\gamma(f,\xi)|}
=
\left(3.79-0.499(\ln{\xi})\right)
\exp{\left((-0.115 - 0.0084 \xi)f\right)}
+
\frac{1}{3}(\xi) f^{-0.878(\xi)}
+
0.35
\nonumber \\
\label{Ancheta04}
\end{eqnarray}
%
\begin{eqnarray}
\sigma_{\Delta A} (f, \xi)
=
(1-e^{-0.1005 - 0.0025 \xi)f})
\nonumber
\\
\label{Ancheta05}
\end{eqnarray}
Plane wave coherency is typically lower than lagged coherency as it includes random
variations in the wave passage effect. The plane wave coherency model was selected
instead of a lagged coherency model as there are no available lagged coherency models
available for rock sites comparable to eastern North America rock sites.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\paragraph{Range of Incoherence Model Applicability}
Based on a report by \cite{Abrahamson2007} the incoherence models
developed from the Pinyon Flat array appears to be applicable to separation
distances of $5$ to $150$~meters and a frequency range of from $5$ to $40$~Hz.
Since the model is developed from
Pinyon Flat recordings alone. it is unknown how well this model fits similar rock site
array recordings or possible future array recordings located in the eastern North America
region.
The incoherence models developed by \cite{Ancheta2010} are a slight modification of
incoherence models developed by \cite{Abrahamson1992} and are applicable to separation
distances of $6$ to $160$~meters and frequencies greater than $1$~Hz.
Both models have been extrapolated to $200$~meters for the process of including
incoherence using the FDW routine. In a comparison of coherency from the SMART
1 and LSST array in Taiwan, \cite{Abrahamson1992} concluded that the decay in
coherency was similar up to $200$m.
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Finite Element Model}
Full 3D finite element models were developed for both surface foundation and
embedded foundation NPPs. Structural models (containment and internal
structures) consisted of modal equivalent stick model.
\paragraph{Containment Structure Modal Equivalent Stick Model} was modeled using
twelve (12) 3D elastic beam elements, where each stick has its properties
calculated from the cylindrical and/or half sphere part of the containment
structure. Three moments of inertia were used (two bending and one torsional),
while lumped masses included both translational and mass moments of inertia.
\paragraph{Internal Structure Modal Equivalent Stick Model} consists of sixteen
nodes, connected with elastic beam elements as well as rigid connections (where
center of mass is not on the beam axes). Stiffness properties were characterized
with bending moments of inertia (two in 3D), coordinates of stiffness center,
moment of inertia in torsion and shear area. Since for the internal structure,
centers of mass do not coincide with centers of the stiffness, rigid
connection elements were used to place lumped mass elements away from stiffness
centers.
\paragraph{Foundation Slab} was placed at the bottom of both containment and
internal structures (connection at the same node). Since foundation slab was
modeled using brick (solid) elements, a rectangular/cross beam structure was placed on top
of the foundation slab to provide for stability of above beam models
(containment and internal stick models). In addition to providing for the
stability of containment and internal stick models, this rectangular/cross beam
structure also provided additional stiffness for the foundation slab that was
missing since containment structure (in reality a thick shell with
certain horizontal spatial extent) is replaced with a stick model. Addition of
rectangular/cross beam structure was done for both surface and embedded
foundation models.
%
The foundation slab has dimensions of $100$m $\times$ $150$m
and a thickness of $3.5$m. For the surface foundation, the lower level of
concrete is at the depth of $5$m while for the embedded it is at the depth of $15$m.
%
The 3D finite element models for a soil/rock block have dimensions of
$200$m (lenght) $\times$ $140$m (width) $\times$ $60$m (depth).
They consist of eight node brick elements with linear interpolation of
displacements and dimensions of $5{\rm m} \times 5{\rm m} \times 5{\rm m}$.
With chosen dimension of the finite element, different types of soil/rock
material models (stiffness) will be able to accurately model different
frequencies of motions:
\begin{itemize}
\item Rock with shear wave velocity of $v_s=2600$m/s, and with the element
size of $h=5$m is able to model (with small error, below 10\%) frequencies
of up to $f_{max}=65$Hz, while any frequencies above that will be modeled
with increased error,
\item Similar to the above case for rock with shear wave velocity of
$v_s=1500$m/s, and element dimension of $h=5$m, accurate modeling of
frequency of $f_{max}=37$Hz is expected, while anything above that will
introduce larger error,
\item Rock with $v_s=1000$m/s, and element size $h=5$m, will accurate model $f_{max}=25$Hz;
\item While soil with shear wave velocity of $v_s=300$m/s, and element size
of $h=5$m, will accurately model $f_{max}=7$Hz, while for higher
frequencies, the error will be larger.
\end{itemize}
For all the finite element models, Newmark numerical time stepping algorithm was
used (\cite{Newmark1959}), with time step increment of $\delta t = 0.015$s. A
small amount of numerical damping was introduced through Newmark algorithms
constants $\gamma = 0.6000$ $\beta= 0.3025$ in order to damp out high
frequencies that are present due to the finite element discretization process
\cite{local-86}, \cite{Argyris91}.
Figures~\ref{DRMSurfaceModel05} and \ref{DRMModel04} show a 3D finite element model for a case of
surface and embedded foundation, respectively.
%
\begin{figure}[!hbt]
\begin{center}
%\includegraphics[width=10cm]{/home/jeremic/tex/works/Reports/2010/CNSC/ModelPlots/VisIt_Mesh_Surface/NPP_surface01.jpeg}
\includegraphics[width=10cm]{NPP_surface01.jpeg}
\caption{\label{DRMSurfaceModel05} Surface foundation model with underlying
rock/soil and the equivalent elastic stiffness and mass superstructure model
representing NPP.}
\end{center}
\end{figure}
%
\begin{figure}[!hbt]
\begin{center}
%\includegraphics[width=10cm]{/home/jeremic/tex/works/Reports/2010/CNSC/ModelPlots/VisIt_Mesh_Excavated/NPP_embedded02.jpeg}
\includegraphics[width=10cm]{NPP_embedded02.jpeg}
\caption{\label{DRMModel04} Embedded foundation model with underlying rock/soil
and the equivalent elastic stiffness and mass superstructure model representing
NPP.}
\end{center}
\end{figure}
%
In addition to the modal equivalent stick models for a containment and
internal structures, model includes a concrete slab, the DRM layer, the soil/rock inside the DRM
layer and the material outside the DRM layer.
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%\clearpage
\subsection{Simulation Platform}
Finite element program used in this study was developed using a
number of numerical libraries, algorithms, material models and finite
elements available through one of open source licenses.
%
%
The finite element domain was managed using Modified OpenSees Services (MOSS)
library
\citep{Jeremic2004d, Jeremic2007d, Jeremic2008a, McKenna97}.
%
On a lower functional level, a set of NewTemplate3Dep numerical libraries
\citep{Jeremic2000f}
was used for constitutive level modeling, while nDarray numerical libraries
\citep{Jeremic97d} are used to handle vector, matrix
and tensor manipulations, and element libraries from UCD
CompGeoMech FEMtools \citep{Jeremic2004d} are used to supply other
necessary libraries and components.
%
%
%
The solution of system of equations was provided by UMFPACK solver
\citep{Davis97b, Davis97, Davis1999, Davis2004, Davis2004b}.
%
Numerical simulation system used in this study is a precursor to the NRC ESSI
Simulator modeling and simulation system that is currently undergoing a very
active development (expansion, documentation, verification and validation, etc.) \citep{Jeremic2011a}.
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\section{Select Results}
\label{ComparissonResults}
A large number of test cases were analyzed and a wealth of results developed.
Presented here is a select number of results that are used to emphasize main
findings related to the influence of variable layered soil/rock under the
surface and embedded foundation slab.
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%\clearpage
\subsection{Comparison of Responses Between NPP Structures on a Variable
Stiffness Single Layer Base Soil/Rock}
Presented here are results for behavior of the NPP on surface and embedded
foundation with variable thickness of a uniform soil/rock layer.
%
%
In particular, four rock/soil profiles (all with uniform rock/soil up to the
depth of $500$m) are used to emphasize differences.
%
The following profiles are used:
%
\begin{itemize}
\item Case 1 ($v_s=2600$m/s to a depth of $500$m),
\item Case 2 ($v_s=1500$m/s to a depth of $500$m),
\item Case 4 ($v_s=1000$m/s to a depth of $500$m),
\item Case 8 ($v_s=300$m/s to a depth of $500$m)
\end{itemize}
%
where a more detailed description of profiles is given in
Table~\ref{soil_profiles_table}.
Results of four different uniform profiles (from hard rock, to
soil, Cases 1, 2, 4 and 8) are used to show that softer foundation soil/rock
does filter out higher frequencies as well as de-amplify motions for higher
frequencies at the top of both containment and internal structures. Behavior
is quite the opposite at lower frequencies, where softer
soil/rock amplify motions and contributes to (in some cases) significant
amplification of motions (particularly for frequencies below $1$Hz).
Figures
\ref{NPP_surface_containmentTop_Node_fft_frequency_log_Case_1_2_4_8_X.pdf}
and
\ref{NPP_embedded_containmentTop_Node_fft_frequency_log_Case_1_2_4_8_X.pdf}
show response for surface and embedded foundation models, respectively.
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\clearpage
%\subsubsection{Containment and Internal Structure on Embedded Foundation,
%Variable Stiffness Single Layer Base Soil/Rock}
%\clearpage
\begin{figure}[!htb]
\vspace*{-4.0cm}
\begin{center}
\includegraphics[width=13.5cm]{NPP_surface_containment_Comparison_Top_Node_fft_frequency_log_Case_1_2_4_8_X.pdf}
\\
\vspace*{-5.0cm}
\includegraphics[width=13.5cm]{NPP_surface_internal_Comparison_Top_Node_fft_frequency_log_Case_1_2_4_8_X.pdf}
\\
\vspace*{-5.0cm}
\includegraphics[width=13.5cm]{NPP_surface_containment_Comparison_Base_Node_fft_frequency_log_Case_1_2_4_8_X.pdf}
\vspace*{-1.5cm}
\caption{\label{NPP_surface_containmentTop_Node_fft_frequency_log_Case_1_2_4_8_X.pdf}
Comparison of acceleration FFTs for cases 1, 2, 4, 8 for horizontal direction at the
(upper) {\bf top of containment structure},
(middle) {\bf top of internal structure}, and
(bottom) {\bf the base of containment and internal structures} on a {\bf surface foundation}.}
\end{center}
\end{figure}
%
%\clearpage
\begin{figure}[!htb]
\vspace*{-4.0cm}
\begin{center}
\includegraphics[width=13.5cm]{NPP_embedded_containment_Comparison_Top_Node_fft_frequency_log_Case_1_2_4_8_X.pdf}
\\
\vspace*{-5.0cm}
\includegraphics[width=13.5cm]{NPP_embedded_internal_Comparison_Top_Node_fft_frequency_log_Case_1_2_4_8_X.pdf}
\\
\vspace*{-5.0cm}
\includegraphics[width=13.5cm]{NPP_embedded_internal_Comparison_Base_Node_fft_frequency_log_Case_1_2_4_8_X.pdf}
\vspace*{-1.5cm}
\caption{\label{NPP_embedded_containmentTop_Node_fft_frequency_log_Case_1_2_4_8_X.pdf}
Comparison of acceleration FFTs for cases 1, 2, 4, 8 in horizontal direction at the
(upper) {\bf top of containment structure},
(middle) {\bf top of internal structure},
(bottom) {\bf base of internal and containment structures} on an {\bf embedded
foundation}.}
\end{center}
\end{figure}
%
There are a number of observations that can be made:
\begin{itemize}
\item There is a slight increase in base motions for all four cases for surface
foundation (when compared to embedded foundations) for lower frequencies, below
$4$Hz, while there is a larger decrease (for surface foundations when
compared to embedded foundations) for frequencies above $4$Hz for all
cases, with particularly large decrease for soil (Case 8) (compare bottom
in Figures
\ref{NPP_surface_containmentTop_Node_fft_frequency_log_Case_1_2_4_8_X.pdf}
and
\ref{NPP_embedded_containmentTop_Node_fft_frequency_log_Case_1_2_4_8_X.pdf}).
%
%These differences can be explained by the fact that surface waves (in this case
%the Rayleigh waves) do influence motions at depth but decay fast, so that a
%single wave length (in depth) is where these influences usually disappear.
%Hence, for the embedded foundation, less high frequency surface motions affect
%the NPP.
\item For structural response on embedded foundation
(compare middle and upper in figures
\ref{NPP_surface_containmentTop_Node_fft_frequency_log_Case_1_2_4_8_X.pdf})
and
Figures~\ref{NPP_embedded_containmentTop_Node_fft_frequency_log_Case_1_2_4_8_X.pdf})
the internal and containment structure react quite differently at frequencies above
$2$Hz, for different soil/rock stiffness beneath the foundation.
%
It is particularly important to note that the embedded foundation seems to
reduce motions of both containment and internal structures for frequency range
of $1$Hz to $5$Hz (significantly) while motions remain almost the same for a
frequency range above $5$Hz. This can be explained by the fact that for the embedded
foundation, the energy of surface waves (Rayleigh waves in this case) is much
smaller for embedded foundation and with that, the rotational components of
motions are not exciting structure (containment and internal) as much as for the
surface foundation case. This conclusion, however cannot be drawn in general as
there might be cases where the lack of surface waves (at depth) might
not be beneficial to the dynamic response.
\item The soil subsurface condition (Case 8) does consistently show the lower
response for higher frequencies, and higher response for lower frequencies,
This is, of course to be expected, and will only be amplified if real soil/rock
non-linearities are taken into account (only linear elastic material properties
were used in this study).
\item Response of the structures on surface foundations
is fairly similar for containment structure
with significant divergence of response only after about $4$Hz (again the soil case does
diverge the most).
%
On the other hand, the internal structure has a very different response on
different soil/rock layers.
%
Soil profile shows again the most divergence for
both containment and internal structure
reduction in response only for frequencies higher than $6$Hz.
\end{itemize}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\clearpage
\subsection{Comparison of Responses Between NPP Structures on Variable
Thickness Soft Soil Layer}
In addition to the above uniform profiles, analyzed were profiles with a varying
thickness of a soil layer ($v_s=300$m/s) (cases 6, 5, 10 and 12)
where the thickness was varying:
\begin{itemize}
\item Case 6, top $500$m of soil with $v_s=300$m/s, below that is a hard rock ($v_s=2600$m/s),
\item Case 5, top $200$m of soil with $v_s=300$m/s, below that is a hard rock ($v_s=2600$m/s),
\item Case 10, top $100$m of soil with $v_s=300$m/s, below that is a hard rock ($v_s=2600$m/s),
\item Case 12, top $50$m of soil with $v_s=300$m/s, below that is a hard rock ($v_s=2600$m/s).
\end{itemize}
Figure~\ref{NPP_surface_containment_Comparison_Top_Node_fft_frequency_log_Case_5_6_10_12_X.pdf},
shows seismic response at the top and base of containment and internal
structures
for the surface foundation, while
Figure~\ref{NPP_embedded_containment_Comparison_Top_Node_fft_frequency_log_Case_5_6_10_12_X.pdf}
show response of the top and base of containment and internal structures on embedded foundation.
\begin{figure}[!htb]
\vspace*{-4.0cm}
\begin{center}
\includegraphics[width=13.5cm]{NPP_surface_containment_Comparison_Top_Node_fft_frequency_log_Case_5_6_10_12_X.pdf}
\\
\vspace*{-5.0cm}
\includegraphics[width=13.5cm]{NPP_surface_internal_Comparison_Top_Node_fft_frequency_log_Case_5_6_10_12_X.pdf}
\\
\vspace*{-5.0cm}
\includegraphics[width=13.5cm]{NPP_surface_containment_Comparison_Base_Node_fft_frequency_log_Case_5_6_10_12_X.pdf}
\vspace*{-1.5cm}
\caption{\label{NPP_surface_containment_Comparison_Top_Node_fft_frequency_log_Case_5_6_10_12_X.pdf}
Comparison of acceleration FFTs for cases 5, 6, 10, 12 for horizontal direction
at the
(upper) {\bf top of a containment structure},
(middle) {\bf top of an internal structure}, and
(bottom) {\bf base of containment and internal structures} on a {\bf
surface foundation}.}
\end{center}
\end{figure}
%
\begin{figure}[!htb]
\vspace*{-5.0cm}
\begin{center}
\includegraphics[width=13.5cm]{NPP_embedded_containment_Comparison_Top_Node_fft_frequency_log_Case_5_6_10_12_X.pdf}
\\
\vspace*{-5.0cm}
\includegraphics[width=13.5cm]{NPP_embedded_internal_Comparison_Top_Node_fft_frequency_log_Case_5_6_10_12_X.pdf}
\\
\vspace*{-5.0cm}
\includegraphics[width=13.5cm]{NPP_embedded_internal_Comparison_Base_Node_fft_frequency_log_Case_5_6_10_12_X.pdf}
\vspace*{-1.5cm}
\caption{\label{NPP_embedded_containment_Comparison_Top_Node_fft_frequency_log_Case_5_6_10_12_X.pdf}
Comparison of acceleration FFTs for cases 5, 6, 10, 12 for horizontal direction at
the
(upper) {\bf top of a containment structure},
(middle) {\bf top of a internal structure}, and
(bottom) {\bf base of containment and internal structures} on an {\bf embedded
foundation}.}
\end{center}
\end{figure}
%
%
A number of observation can be made:
%
\begin{itemize}
\item A significant differences are observed in responses between surface and embedded
foundation structures (both for containment and internal structures).
\item For example,
internal structure (middle figures in Figures
\ref{NPP_surface_containment_Comparison_Top_Node_fft_frequency_log_Case_5_6_10_12_X.pdf}
and
\ref{NPP_embedded_containment_Comparison_Top_Node_fft_frequency_log_Case_5_6_10_12_X.pdf})
show significant differences, and in general the embedded
foundation internal structure has amplified motions when compared with the surface
foundation case.
\item On the other hand, containment structure (upper figures in Figures
\ref{NPP_surface_containment_Comparison_Top_Node_fft_frequency_log_Case_5_6_10_12_X.pdf}
and
\ref{NPP_embedded_containment_Comparison_Top_Node_fft_frequency_log_Case_5_6_10_12_X.pdf})
show amplification for embedded foundation case only below about $1.5$Hz and
above $4$Hz, while in the range $1.5-4$Hz, the surface foundation has larger
amplifications.
\item The thickness of the soft soil layer (it varies from $50$m
to $500$m) does not affect much the response of either containment or internal
structures at higher frequencies. Presented Fourier Amplitudes suggest that
a small variation of response is to be expected for variable thickness of soft
soil layer, particularly for frequencies above approximately $2$Hz. Below such
frequency, response does differ, in some cases more significant that in other.
This behavior is somewhat expected, as the soft soil layer "isolates" the
structures from higher frequencies, while the lower frequencies do get certain
amplification, depending on the layer thickness, stiffness and mass. This is
most obvious for the frequencies of about $0.8$Hz where the amplification
factors seems to differ by an order of magnitude between various cases.
\item Independent of the soft soil layer thickness, significant reduction of
higher frequencies is noticeable, which is apparent in Figure
\ref{NPP_embedded_containment_Comparison_Top_Node_fft_frequency_log_Case_5_6_10_12_X.pdf}.
\end{itemize}
%OVDE
% Time Ancheta Comments
% When comparing the relative amplification of the base motion to the:
%
% 1. I see the same pattern of amplification/de-amplification in the uniform Vs
% profiles, and the variable thickness cases. It seems however, that the input
% motion at the base for the uniform Vs profile has a relatively flat spectrum
% between 0.1-10 Hz (except case 8 in Figure 6/7) while the input motion at the
% base for the variable thickness profiles has more spectral peaks and troughs.
%
% 2. Are the relative spectral peaks and troughs dependent on a property of the
% profiles?
%
%\clearpage
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Summary}
Presented in this paper was an investigation of the seismic response of a
massive NPP soil/rock-structure system (both containment and internal structures, with the foundation slab
(surface and/or embedded) and with a significant soil/rock volume) due to full
3D, inclined, un-corellated body and surface seismic motions for a different
soil/rock profiles. Presented was an investigation of the importance of soil/rock
layering on response of containment and internal structures to realistic seismic
motions. In addition to that, an investigation of effects of foundation
embedment was also performed for all soil/rock layering cases.
A number of specific conclusions can be made for each of the soil/rock layer
cases, and such conclusions were made in the above results section. It is,
however, very important to note that presented results do suggest that each
soil/rock layering system features behavior specific for that particular
layering, for particular structural system (containment and/or internal) and for
a particular (realistic) earthquake motion, landing to the conclusion that the
interaction of all three components, the earthquake, the soil/rock and the
structure play crucial role in dynamic response of an NPP.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Acknowledgment}
%
Work presented here was funded by a grant from the Canadian Nuclear Safety
Commission (SNSC) and such support is greatly appreciated.
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\newpage
\newpage
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