Three-Dimensional Seismic Ground Motion Modeling in Inelastic Basins

Abstract

In this paper we report on the development and application of a parallel numerical methodology for simulating large-scale earthquake-induced ground motion in highly heterogeneous basins whose soil constituents can deform nonlinearly. We target sedimentary basins with large contrasts in wavelengths for which regular grid methods become ine#cient, and overcome the problem of multiple physical scales by using unstructured finite element triangulations. We illustrate the methodology with an example of an idealized basin, which contains a deep and a shallow sub-basin. The simulations show significant amplitude reduction of the ground accelerations due to inelastic soil behavior at sites above the deepest portions of the sub-basins, yet little shift in frequency. Under the assumption of linear anelastic material behavior, there is a rapid spatial distribution of the ground acceleration of the basin, which di#ers markedly from that for a one-dimensional analysis. This characteristic three-dimensional nature of the ground motion is preserved for the elastoplastic model.

Full text

Page 1

1
Three-dimensional seismic ground motion modeling in inelastic basins
Jifeng Xu1, Jacobo Bielak2,Omar Ghattas2, and Jianlin Wang3
Abstract
In this paper we report on the development and application of a parallel numerical method-
ology for simulating large-scale earthquake-induced ground motion in highly heterogeneous
basins whose soil constituents can deform nonlinearly. We target sedimentary basins with
large contrasts in wavelengths for which regular grid methods become inefficient, and over-
come the problem of multiple physical scales by using unstructured finite element triangula-
tions. We illustrate the methodology with an example of an idealized basin, which contains
a deep and a shallow sub-basin. The simulations show significant amplitude reduction of
the ground accelerations due to inelastic soil behavior at sites above the deepest portions
of the sub-basins, yet little shift in frequency. Under the assumption of linear anelastic
material behavior, there is a rapid spatial distribution of the ground acceleration of the
basin, which differs markedly from that for a one-dimensional analysis. This characteris-
tic three-dimensional nature of the ground motion is preserved for the elastoplastic model.
Concerning the ground displacement, the main qualitative difference between the elastic
and inelastic models is the occurrence of significant permanent deformations in the inelastic
case. These residual displacements can have practical implications for the design of long
structures such as bridges and structures with large plan dimensions.
Introduction
Nearly all the models used until recently in seismology for predicting ground motion induced
by earthquakes have been based on the assumption of linear elastic behavior of the soil.
On the other hand, for a number of years nonlinear soil amplification has been routinely
taken into consideration in geotechnical engineering practice (Seed and Idriss, 1969; Finn,
1991). The main reason seismologists had in the past ignored the possibility that nonlinear
phenomena could play an important role in earthquake ground motion was that compelling
evidence for nonlinear effects in the observed motion, other than in liquefied sites, was
scarce. In the last decade, however, a number of accelerograms have been recorded during
strong earthquakes that have made it possible to infer nonlinear response. The most com-
mon manifestations of inelastic soil behavior involve the reduction in shear wave velocity
and the increase in soil damping with increasing load (Hardin and Drnevich, 1972). Accord-
ingly, the corresponding nonlinear site effects include the lowering of the site amplification
factor as the amplitude of the seismic loading increases; in some cases this is accompanied
by the lowering of the resonance frequencies in the spectra of the recorded ground. Thus,
1The Boeing Company, P. O. Box 3707 MC 7L-21, Seattle, WA 98124-2207, jifeng.xu@boeing.com
2Department of Civil and Environmental Engineering, Carnegie Mellon University, Pittsburgh, PA 15213,
jbielak@cs.cmu.edu; oghattas@cs.cmu.edu
3Synopsis, Inc. 700 East Middlefield Rd., Mountain View, CA 94043, jianlin@synopsis.com

Page 2

2
evidence of nonlinear soil response during strong earthquakes can be identified directly if
weak and strong motion records are simultaneously obtained on the ground surface and
at depth in a borehole, or through simulation, as long as records are available for strong
shaking. As these simultaneous records have been rare, nonlinear site effects is oftentimes
inferred from ground motion records obtained during strong and weak earthquakes at both
rock and soil locations by filtering out the earthquake source radiation and propagation path
effects. Identifying an appropriate reference rock site, though essential for this approach,
still represents a considerable challenge, as rock properties at the free surface differ from
those at the basement.
Nonlinear soil behavior has received increased attention from seismologists in recent years.
SMART1 (Strong Motion Accelerograph Array in Taiwan) and SMART2 (Abrahamson
et al, 1987) have provided reliable records of events available for the study of nonlinear site
effects in a particular seismic region in Taiwan in which a carefully established downhole
accelerograph array was deployed. Analyses of these record (Chang et al, 1989; Wen et al,
1994; Beresnev et al, 1995) revealed significant nonlinear soil response in both deamplifica-
tion and degradation of wave velocities in ground motions with peak ground acceleration
larger than 0.15g. In Japan, nonlinear behavior of soil sediments was identified at Kuno
in the Ashigara valley by using borehole records (Satoh et al, 1995). At Port Island, the
motion at different depths recorded during the 1995 Hyogoken–Nambu by a borehole ar-
ray demonstrated strong nonlinear features of seismic amplification in reclaimed land areas
and also in Holocene and Pleistocene soil deposits (Sato et al, 1996). The Loma Prieta
earthquake of 1989 in California presents another case in which nonlinear site effects have
been detected. Chin and Aki (1991) found a pervasive nonlinear site effect at sediment
sites in the epicentral region by eliminating the influences of the radiation pattern and to-
pography. They concluded that the site amplification factor depends on the acceleration
level and nonlinear effect may appear at levels above 0.1g to 0.3g. This is in the range
expected from geotechnical engineering studies. Darragh and Shakal (1991) estimated the
response of a soil site by studying the ratio of the smoothed Fourier amplitude spectra from
a soil site and a nearby rock site, and observed pronounced strong motion deamplification
effects. The 1994 Northridge earthquake also presents plausible evidence of nonlinear soil
response. Trifunac and Todorovska (1996) estimated an upper bound on the distance from
the earthquake source where nonlinear soil response affects peak acceleration. They ana-
lyzed the observed strong motion amplitudes in the San Fernando valley and found that
noticeable reduction in recorded horizontal peak accelerations occur at sites with shear wave
velocity less than 360 m/s and distance from the fault less than 15-20 km. By comparing
ground-motion between the Northridge earthquake and its aftershocks, Field et al (1997)
reported that sediment deamplification was up to a factor of two during the main shock
implying significant nonlinearity. More recently, O’Connell (1999) has shown that much of

Page 3

3
the same data can be explained by linear response and scattering of waves in the upper
kilometers of the earth’s crust. Strong motion deamplification effects were also observed for
the aftershocks of the 1983 Coalinga earthquake in California by Jarpe et al (1988)
Besides the well-known amplitude deamplification and changes in resonance frequency,
Archuleta (1998) has identified a characteristic spiky waveform in some strong motion
accelerograms as evidence of nonlinear response. Archuleta et al (2000) cite a number
of examples where this spiky waveform has been observed. This characteristic high fre-
quency waveform was first noticed by Porcella (1980) and later thoroughly analyzed by Iai
et al (1995).
Simulations of nonlinear earthquake response began in the late 1960s. These early studies
were conducted for horizontally layered soils and vertically incident waves, by either an
equivalent linear or a direct nonlinear method. These are the main methods that are still
used today in engineering practice. In the equivalent linear method, the soil response is
evaluated in an iterative manner. First, trial values for average strain are chosen, then soil
properties are determined in accordance with the trial values of strain, and finally the re-
sponse of the model is calculated. If the calculated strains differ significantly from the trial
values, the cycle is repeated (Seed and Idriss, 1968; Schnabel et al, 1972). Many studies have
concluded that equivalent linear approach cannot reproduce some of the important charac-
teristics of the seismic ground motion, especially for the cases of strong loadings (Streeter
et al, 1974; Finn et al, 1978). The equivalent linear method overestimates the seismic re-
sponse due to pseudo-resonance at periods corresponding to the strain-compatible stiffness
used in the final elastic iteration analysis. Also, since the method is elastic it cannot predict
the permanent deformations that occur during an earthquake.
In the direct nonlinear method, the shear modulus is modified at every time step according
to the current strain, so that the nonlinear stress-strain relationship is closely followed.
A number of representations have been used for the backbone curves and yield rules of
soil stress-strain relationships. They include, e.g., linear (Idriss and Seed, 1968), multilin-
ear (Joyner and Chen, 1975; Yu et al, 1993), and one by Archuleta et al (2000) based on a
modified Masing rule with a provision for pore pressure (Elgamal, 1991).
While one-dimensional (1D) simulations can yield reasonable estimates of nonlinear effects
under vertically incident seismic excitation, they cannot represent the effects of surface
waves and basin effects. The few 2D and 2.5D studies to date of nonlinear soil amplification
(e.g., Joyner and Chan, 1975; Joyner, 1975; Elgamal, 1991; Marsh et al, 1995; Zhang and
Papageorgiou, 1996) have confirmed the importance of nonlinearity on site response. In par-
ticular, a 2.5D study of the Marina District in San Francisco during the 1989 Loma Prieta
earthquake found that the focusing and lateral interferences often observed in studies based

Page 4

4
on linear soil behavior are still present for strong excitation though not as prominently as
for weak excitation (Zhang and Papageorgiou, 1996). The use of an equivalent linearization
technique meant that no permanent deformations could be detected.
In this study we describe a finite element methodology for modeling three-dimensional
ground motion in basins, including inelastic behavior of the soil, due to arbitrary seismic
excitation, and illustrate it for an idealized basin. The soil within the basin is idealized
as a Drucker-Prager elastoplastic material (Drucker and Prager, 1952). Results show that
whereas the ground motion decreases due to soil nonlinearity, the spatial variation of this
motion follows closely that of the linear model, showing clear basin effects. Also, despite
the significant reduction in peak response, there is only a small change in the dominant
frequencies.
Method of Analysis
The mathematical problem under consideration is one of earthquake-induced waves trav-
eling from a flat-layered halfspace of rock into a basin with heterogeneous and possibly
nonlinear soils and irregular geometry. We assume small deformations throughout. We will
use standard finite element techniques to solve this problem. Topics of special importance
in the development of the model, such as the governing equations, the constitutive laws
governing the materials, the type of loading, and the type of elements are discussed briefly
next; details can be found in Xu (1998).
The governing equation for the balance of momentum is given by:
ρ ¨ u − ∇ · σ = f
(0.1)
where ρ is the density of the material, u is the displacement vector field, σ is the stress
tensor field, and f is the body force, which as will be explained subsequently, is introduced
to represent the seismic excitation.
Standard Galerkin discretization in space by finite elements produces a system of ordinary
differential equations of the form:
M¨U(t) +
?
e
?
VeBTσ dVe= F(t)(0.2)
where M is the global mass matrix; U is the nodal displacement time-dependent vector;
F is a time-dependent vector of applied nodal forces; Veis the element volume; B is the
element matrix of shape function derivatives; and σ is the current stress tensor within
each element computed by an appropriate procedure from the soil constitutive law. This
system of equations is nonlinear because the stress σ depends nonlinearly on the current
displacement U and the loading history.

Page 5

5
The main factor influencing the reliability of numerical calculations of nonlinear dynamic
soil behavior is the implementation of the stress-strain law. For this initial analysis of the
ground motion of soil deposits in 3D basins we idealize the soil, both clays and sands, as a
Drucker-Prager material. We selected this material because: (1) the implementation of its
constitutive law is similar to that required for more complex constitutive laws; (2) it can
represent soil dilatancy and its parameters can be related to the physical soil properties
(cohesion and friction angle) in a rather straightforward way; and (3) despite its relative
simplicity, it can lead to reasonable agreement between the results of simulations and ob-
servations for problems that do not involve soil liquefaction. This satisfactory performance
of the Drucker-Prager model was observed, in particular, in a study aimed at predicting
the monotonic and cyclic response of pile foundations to axial and lateral loads (Trochanis
et al, 1991).
In our implementation of the finite element method we use lumped mass diagonal matrices
for each element, and the integral in the second term in (2) is evaluated by Gauss numerical
integration. As is usually done in displacement-based finite element formulations, the algo-
rithmic framework for the elastoplastic analysis is strain driven. Given a prescribed strain
increment at a step n + 1, the problem is to update the stress tensor at each Gauss point
at the new time tn+1given its value at the previous time tn. We use a return mapping
algorithm (Ortiz et al, 1983; Simo and Taylor, 1986), which is capable of accommodating
arbitrary yield criteria, flow rules, and hardening laws.
In order to treat efficiently large-scale 3D problems that involve elastoplastic laws, we use
an element-by-element procedure to implement equation (2) into our software. That is, the
product of the mass matrix and the corresponding nodal accelerations, and the second term
in (2) are evaluated separately for each finite element and then assembled. Notice that no
global matrices need to be stored; only vectors need to be assembled.
We have built a parallel elastoplastic wave propagation simulation code on top of Archimedes,
an environment for solving unstructured mesh finite element problems on parallel computers
(”Archimedes”, 1988), as a generalization of our earlier parallel elastic wave propagation
simulation code (Bao et al, 1998). Archimedes includes 2D and 3D mesh generators, a
mesh partitioner, a parceler and a parallel code generator. We favor finite elements for
their ability to efficiently resolve multiscale phenomena by tailoring the mesh size to local
wavelengths, and the ease with which they handle traction interface and boundary condi-
tions, and complex geometries, including arbitrary topography. Since the mesh generator
in Archimedes builds meshes that are made up of tetrahedra with straight edges, we use
10-node subparametric quadratic tetrahedral for our calculations. The reason we introduce
a quadratic approximation of the displacement field within each element, rather than the
linear approximation we used for our elastic models, is to be able to represent exactly linear