Non-linear Seismic Analysis

Non-linear one-dimensional seismic ground motion propagation
in the Mississippi embayment
Youssef M.A. Hashash
a,
*
, Duhee Park
a
a
Department of Civil and Environmental Engineering, University of Illinois at Urbana-Champaign, 205 N. Mathews Avenue,
Urbana, IL, 61801 USA
Abstract
Deep unconsolidated deposits of the Mississippi Embayment overlie the New Madrid Seismic Zone, considered the most
seismically active zone in the Eastern US. The deposits range in thickness from less than 100 m in the St. Louis area to about
1 km in the Memphis area and consist of silts, clays and sands. The in¯uence of these deposits on propagation of seismic waves
remains a major source of uncertainty for site response analysis.
This paper describes the development of a new non-linear one-dimensional site response analysis model for vertical
propagation of horizontal shear waves in deep soil deposits. Soil response is modeled using a modi®ed hyperbolic model
with extended Masing criteria to represent hysteretic loading and unloading of soil. The new soil model accounts for the
in¯uence of large con®ning pressures on strain dependent modulus degradation and damping of soil. The model is calibrated
using measured shear modulus degradation and damping data from resonant column tests on sand samples under con®ning
pressures up to 3.5 MPa.
The new model is used to estimate ground motion ampli®cation and attenuation for three soil columns 100, 500 and 1000 m
thick, representative of soil thickness variability within the Embayment. The new model shows that some high frequency
components of ground motion are transmitted through these deep deposits. These components are usually ®ltered out using
conventional wave propagation methods. Longer period waves develop in deposits of 500±1000 m thickness. Spectral ampli-
®cation factors of deep deposits are greater than unity and can be as large as 5 in the longer period range of 2±10 s. Preliminary
evaluation of model results show that computed surface response spectra in the period range of 0.5±2 s are larger than the 1997
NEHRP recommended design response spectrum.
The proposed model highlights the need to account for depth dependence of modulus and damping properties of soils in
seismic wave propagation through deep soils. q 2001 Elsevier Science B.V. All rights reserved.
Keywords: Site response; Deep deposit; Frequency content; Non-linear; Ampli®cation; Con®ning pressure
1. Introduction
Earthquakes in the New Madrid Seismic Zone
(NMSZ) are characterized as low probability, high
consequence events. Estimate of ground motion char-
acteristics in the NMSZ is required to assess the seis-
mic vulnerability of structures and the susceptibility
of soils to liquefaction. The presence of very deep (up
to 1000 m) unconsolidated deposits in the Mississippi
Embayment has an important, though poorly under-
stood effect on the propagation of seismic waves.
Earthquake activity elsewhere has shown the import-
ance of local site conditions on propagated ground
motions. Strong motion records from recent earth-
quakes including the 1989 Loma Prieta, 1994
Northridge, 1995 Hyogoken±Nanbu, and 1999
Engineering Geology 62 (2001) 185±206
0013-7952/01/$ - see front matter q 2001 Elsevier Science B.V. All rights reserved.
PII: S0013-7952(01)00061-8
www.elsevier.com/locate/enggeo
* Corresponding author.
Chi±Chi events show signi®cant differences between
soil sites and nearby rock sites. Such records are not
available for the NMSZ and the Mississippi Embay-
ment.
Seismicity in the NMSZ is increasingly being charac-
terized through extensive paleo-liquefaction features
found throughout the region (e.g. Obermeier and
Pond, 1999). Interpretation of the paleo-liquefaction
record includes an estimate of ground motion levels
required to cause liquefaction. Understanding of wave
propagation characteristics of deep deposits is an
important element in developing estimates of ground
motion levels and consequently the magnitude of earth-
quakes that contributed to these liquefaction events.
In the absence of strong motion records, numerical
models can be used to develop an understanding of
wave propagation characteristics of the Mississippi
Embayment. This paper proposes an enhancement of
an existing one-dimensional, non-linear wave propa-
gation model to account for the effect of very high
con®ning pressures encountered in the Embayment.
The model extension is based in part on recent data
regarding cyclic response of soils under high con®n-
ing pressures. A series of analyses are presented to
Y.M.A. Hashash, D. Park / Engineering Geology 62 (2001) 185±206 186
Fig. 1. (a). Plan view of the Mississippi Embayment and major structures within it. Circles denote the clustered pattern of earthquake epicenters
that de®ne the New Madrid Seismic Zone within the Reelfoot Rift margins. (b) E±W section through Memphis (afterNg et al., 1989). Note:
Vertical dimension is highly exaggerated, and the Embayment trough has a shallow slope of less than 1/150.
illustrate the in¯uence of deep deposits on the ampli-
tude and frequency content of propagated weak and
strong ground motions.
2. The Mississippi Embayment and New Madrid
Seismic Zone
The Mississippi Embayment is a syncline or a
trough-like depression that plunges southward along
an axis that approximates the course of the Mississippi
River. The Embayment, beginning near the Gulf of
Mexico and extending north to the con¯uence of the
Ohio and Mississippi Rivers as shown in Fig. 1a, is
surrounded by the Illinois Basin to the north, the
Nashville dome and southern Appalachian Plateau
to the east, and the Ouachita and Ozark uplifts to the
west (Shedlock and Johnston, 1994). The Paleozoic
rock forms the bedrock ¯oor of the Mississippi
Embayment and is located about 1000 m below
Memphis and Shelby County, which is near the
central part of the Mississippi Embayment, as shown
in Fig. 1b (Ng et al., 1989). The Embayment is ®lled
with sediments of clay, silt, sand, gravel, chalk and
lignite ranging in age from Cretaceous to recent
Holocene. There is no well-consolidated rock above
the Paleozoic rock, except some local beds of ferru-
ginous and calcareous sandstone and limestone (Parks
and Lounsbury, 1976).
The axis of the Embayment is nearly coincident
with the underlying Reelfoot rift, which is the most
prominent buried structure in the northern Embay-
ment, and appears to re¯ect Cretaceous reactivation
of an ancient rift (Braile et al., 1982). The New
Madrid Seismic Zone (Fig. 1a) is a clustered pattern
of earthquake epicenters between 5 and 15 km deep
and lies mostly within the Reelfoot rift.
The presence of thick unconsolidated deposits adds
signi®cant uncertainty regarding the nature of seismic
ground motion propagation and attenuation in the
Embayment. The effect of soil deposits on propagated
ground motion is well documented in other parts of
the world (e.g. Mexico City, Stone et al., 1987).
However, limited information is available regarding
wave propagation through very thick deposits (up to
1000 m) such as those found in the Mississippi
Embayment.
In Fig. 1b the geologic layers can be considered
nearly horizontal. Analysis of wave propagation
through these deposits is approximated as one-dimen-
sional vertical propagation of horizontal shear waves.
Three pro®les, 1000, 500 and 100 m deep, shown in
Fig. 2, are selected to represent the range of soil
depths encountered in the Embayment (Ng et al.,
1989; VanArsdale et al., 1994). The 1000 m pro®le
is representative of conditions in the Memphis, Shelby
County area while the 100 m pro®le represents con-
ditions south of the St. Louis Area. The selected shear
wave velocity pro®le is based on a combination of
surface information and a few deep wells as compiled
by Romero et al. (2001). The analyses in this paper
assume that V
s
pro®les for the 100 and 500 m columns
are identical to V
s
in the upper part of the 1000 m
pro®le in Memphis despite differences in the geologic
column around the Embayment. The assumption was
necessary due to lack of detailed V
s
measurements at
considerable depths throughout the Embayment. The
density of the soil (r) in the columns is assumed to be
1.98 kg/m
3
.
3. Soil response under cyclic loading
The behavior of soil under cyclic loading is non-
linear and depends on several factors including ampli-
tude of loading, number of loading cycles, soil type
and in situ con®ning pressure. Non-linear hysteretic
soil behavior is commonly characterized by an
equivalent secant shear modulus and viscous damping
(Seed and Idriss, 1970; Hardin and Drnevich, 1972).
Secant shear modulus, normalized by maximum shear
modulus, decreases with increasing magnitude of
cyclic shear strain. Damping, which is a measure of
energy dissipation in a loading cycle, increases with
increasing magnitude of cyclic shear strain. Modulus
degradation and damping curves for a wide range of
soils have been developed by several researchers
including Seed et al. (1986) and Vucetic and Dobry
(1991). These curves have been extensively used for
estimating seismic site response in relatively shallow
deposits (,30 m). Soil parameters such as plasticity
index, number of cycles, void ratio and relative
density also in¯uence dynamic soil properties. For
cohesionless soil, the variation of dynamic curves
with change in soil properties is small and therefore
it is assumed that modulus degradation and damping
Y.M.A. Hashash, D. Park / Engineering Geology 62 (2001) 185±206 187
curves fall within a narrow range for most cohesion-
less soils (Seed and Idriss, 1970).
The effect of con®ning pressure on dynamic proper-
ties, which is signi®cant compared to other soil prop-
erties, has been recognized by Hardin et al. (1974),
Iwasaki et al. (1978) and Kokusho (1980). Ishibashi
and Zhang (1993) published relations relating modu-
lus reduction to con®ning pressure and soil plasticity
index. Hardin et al. (1994) presented high pressure (up
to 3.5 MPa) test data on sand and concluded `damping
ratios at conventional pressures are approximately
equivalent to those at pressures up to 3.5 MPa'.
They reported a damping ratio of 0.5% at strains
less than 10
25
. Hardin et al. (1994) suggested that
additional research is necessary to further understand
cyclic soil response at very high pressures.
Laird and Stokoe (1993) performed resonant
column and torsional shear tests at strain levels up
to 10
23
and con®ning pressures up to 3.5 MPa using
remolded sand specimens, as well as undisturbed
specimens of sand, silty sand, silt, lean clay, and fat
clay. Low and high amplitude cyclic torsional shear
and resonant column tests were used to determine the
effect of strain amplitude and con®nement on shear
modulus and damping curves. In this paper, only
results from remolded sand specimens (washed
mortar sand) are used. Laboratory testing was part of
the ROSRINE project (http://rccg03.usc.edu/rosrine/,
Stokoe et al., 1999), examining local site response in
the Los Angeles Basin. Fig. 3 shows shear modulus
and damping values obtained from these tests.
Measurements show that increase in con®ning pres-
sure results in lesser shear modulus degradation at a
given cyclic shear strain. Con®ning pressure increase
has a signi®cant in¯uence on damping as well. Small
strain damping decreases with an increase in con®n-
ing pressure due to an increase in number of particle
contacts, which is the main factor that dissipates
energy at low amplitude strain.
Fig. 3 includes data from Laird and Stokoe (1993),
the measured range of sand properties from Seed
and Idriss (1970) and curves obtained from the
Y.M.A. Hashash, D. Park / Engineering Geology 62 (2001) 185±206 188
Fig. 2. Soil and shear wave velocity pro®les used in the analysis. (a) V
s
and soil pro®le at Memphis, TN, (b) V
s
pro®le, 100 and 500 m columns.
model proposed by Ishibashi and Zhang (1993) for
cohesionless soils. The speci®c effect of con®ning
pressure is not available in the curves proposed by
Seed and Idriss (1970). The data from Laird and
Stokoe (1993) show signi®cantly less degradation of
shear modulus compared with the modulus degrada-
tion range presented by Seed and Idriss (1970). Even
at a moderate con®ning pressure (221 kPa), the test
Y.M.A. Hashash, D. Park / Engineering Geology 62 (2001) 185±206 189
Fig. 3. In¯uence of con®ning pressure on modulus degradation and damping ratio of soil. Data points and curves are developed from laboratory
measurements and proposed models published in the literature.
data exceed the boundaries of that range. Damping
falls below the proposed range at a con®ning pressure
of 55 kPa or larger. Modulus degradation curves by
Ishibashi and Zhang (1993) give a much stiffer
response of sand compared to the measured values
from Laird and Stokoe (1993). Similarly, the Ishibashi
and Zhang (1993) damping curves do not capture
measured behavior. The data from Laird and Stokoe
(1993) is considered more reliable since it is derived
from experiments designed to account for the in¯u-
ence of con®ning pressure. This data is used to cali-
brate the new non-linear soil model proposed in this
paper.
4. One-dimensional wave propagation in soil
deposits
One-dimensional site response analysis is used to
solve the problem of vertical propagation of horizon-
tal shear waves (SH waves) through a horizontally
layered soil deposit. Horizontal soil layer behavior
is approximated as a Kelvin±Voigt solid whereby
constant elastic shear moduli and viscous damping
characterize soil properties. Solution of wave propa-
gation equations is performed in the frequency
domain.
Seed, Idriss and co-workers introduced the equiva-
lent linear approximation method to capture non-
linear cyclic response of soil. For a given ground
motion time series (T.S., also referred to as time
history) and an initial estimate of modulus and damp-
ing values, an effective shear strain (equal to about
65% of peak strain) is computed for a given soil layer.
Modulus degradation and damping curves are then
used to obtain revised values of shear modulus and
damping. An iterative scheme is required to arrive at a
converged solution (e.g. shake, Schabel et al., 1972).
This approach has provided results that compare well
with ®eld measurements and is widely used in engine-
ering practice. More recently, Sugito (1995) and
Assimaki et al. (2000) extended the equivalent linear
approach to include frequency and pressure depend-
ence of soil properties. Analyses show that some high
frequency components of ground motion that would
be otherwise ®ltered in a conventional equivalent
linear analysis are preserved using their proposed
extended equivalent linear procedure.
The equivalent linear approach is computationally
easy to use and implement. However, it does not
capture the full range of cyclic behavior of soil,
including modulus degradation due to number of
loading cycles, permanent (residual) straining of soil
and excess pore pressure generation. Non-linear
analysis is used to capture these important aspects
of soil behavior. In this approach, equations of motion
and equilibrium are solved in discrete time increments
in the time domain. The following dynamic equation
of motion is solved:
‰MŠ{ u} 1‰CŠ{_ u} 1‰KŠ{u} ˆ 2‰MŠ{I} u
g
…1†
where [M] ˆ mass matrix; [C] ˆ damping matrix;
[K] ˆ stiffness matrix; { u} ˆ vector of nodal relative
acceleration; { u} ˆ vector of nodal relative veloci-
ties; and {u} ˆ vector of nodal relative displace-
ments.  u
g
is the acceleration at the base of the soil
column and {I} is the unit vector. The [M.], [C]
and [K] matrices are assembled using the incremental
properties of the soil layers. The properties are
obtained from a constitutive model that describes
the cyclic behavior of soil.
The earliest constitutive relations use a simple
model relating shear stress to shear strain, whereby
the backbone curve is represented by a hyperbolic
function. Modulus degradation curves such as those
shown in Fig. 3 are used to de®ne the backbone curve.
The Masing criteria (Masing, 1926) and extended
Masing criteria (Pyke, 1979; Vucetic, 1990) de®ne
unloading±reloading criteria and behavior under
general cyclic loading conditions as shown in Fig. 4.
Lee and Finn (1978) developed a one-dimensional
seismic response analysis program using the hyper-
bolic model. Matasovic (1993) and Matasovic and
Vucetic (1995) further extended the model with a
modi®cation of the hyperbolic equation. Plasticity
models have also been used to represent cyclic soil
behavior. For example, Borja et al. (1999) used a
bounding surface plasticity model to represent cyclic
soil response at the Lotung Site in Taiwan. These
models do not explicitly account for the in¯uence of
con®ning pressure on strain dependent stiffness and
damping characteristics of soil for very deep soils and
have been used for soil columns less than 100 m thick.
In a non-linear soil model, soil damping is captured
through hysteretic loading±unloading cycles in the
soil model. The use of the damping matrix [C] may
Y.M.A. Hashash, D. Park / Engineering Geology 62 (2001) 185±206 190
become unnecessary. The damping matrix may be
used as a mathematical convenience or to include
damping at very small strains where response of
many constitutive models is nearly linear elastic.
5. Non-linear pressure dependent cyclic soil model
The backbone curve for the model proposed by
Matasovic (1993) and implemented in a widely used
code D_MOD, is described by the following equation:
t ˆ
G
mo
g
1 1b
G
mo
t
mo
g

s
ˆ
G
mo
g
1 1b
g
g
r

s
…2†
where t is the shear stress; g the shear strain; G
mo
the
initial shear modulus; t
mo
the shear stress at approxi-
mately 1% shear strain; and g
r
ˆ t
mo
=G
mo
is the refer-
ence shear strain (Hardin and Drnevich, 1972) and
is considered a material constant. The model is a
modi®cation of the hyperbolic model by Konder and
Zelasko, 1963, through the addition of two parameters
b and s that adjust the shape of the backbone curve to
represent a wider range of measured soil behavior.
The model D_MOD has been used to estimate seismic
response of deep sites in Los Angeles (Chang et al.,
1997) and displacements of solid waste-®lls (Bray and
Rathje, 1998). D_MOD was also used in the back-
analysis of several well-documented case histories
including the Wildlife site during several shaking
events and Treasure Island during the Loma Prieta
earthquake (Matasovic 1993). The back-analysis
case histories represent a limited validation of
D_MOD for soil pro®les up to 88 m in depth. In
this model, there is no coupling between con®ning
pressure and shear stiffness. The Matasovic (1993)
model is extended in this paper to capture the in¯u-
ence of con®ning pressure on modulus degradation
and damping as presented in Fig. 3. The following
paragraphs present the proposed equations and
Table 1 lists values of model parameters as used in
the analyses presented in this paper.
5.1. Effect of con®ning pressure on shear modulus
Hardin and Drnevich (1972) used laboratory test
data on clean dry sand to show that the reference
strain, g
r
, is dependent on con®ning pressure and
that it can be used as a normalizing strain to capture
modulus degradation and damping variation with
con®ning pressure. Shibata and Soelarno (1975)
proposed the use of g
r
proportional to s
0:5
0
to capture
the con®ning pressure effect in the classical hyper-
bolic model (b ˆ s ˆ 1). However, in the model
Y.M.A. Hashash, D. Park / Engineering Geology 62 (2001) 185±206 191
Fig. 4. Hyperbolic, non-linear soil model with extended Masing rule to de®ne loading and unloading behavior.
used in D_MOD, g
r
is considered a constant material
property. This paper introduces a new formulation
for the reference strain, g
r
, to capture the in¯uence
of con®ning pressure on modulus degradation and
damping ratio as shown in Eq. (3):
g
r
ˆ a
s
0
s
ref
!
b
…3†
where a and b are curve ®tting parameters and s
ref
is
a reference con®ning pressure of 0.18 MPa. Fig. 5
shows that using the proposed equation; the model
can capture the variation in shear modulus measured
in laboratory experiments. Fig. 5 plots the extra-
polated modulus degradation curve to an effective
stress of 10000 kPa (equivalent to a depth of
1000 m, with the water table at ground surface).
5.2. Effect of con®ning pressure on damping ratio
Hysteretic damping of the soil model de®ned by
Matasovic (1993) can capture damping at strains
larger than 10
24
±10
22
%, depending on the value of
reference strain. Use of Eq. (3) captures the hysteretic
damping dependency on con®ning pressure.
However, the hyperbolic model is nearly linear at
small strains (less than 10
24
±10
22
%) with practically
no damping, which can cause unrealistic resonance
during wave propagation. The model described by
Matasovic (1993) incorporates additional damping
to the dynamic equations in the form of [C] matrix,
as shown in Eq. (1). The [C] matrix is a combination
of the mass matrix and the stiffness matrix (Rayleigh,
1945). For practical purposes, small strain viscous
damping effects are assumed proportional only to
the stiffness of the soil layers. The damping matrix
is expressed as:
‰CŠ ˆ
2j
v
‰KŠ …4†
where v is circular frequency and j the equivalent
damping ratio. [C] is assumed to be independent of
strain level and therefore, the effect of hysteretic
damping induced by non-linear soil behavior can
be separated from (but added to) viscous damping.
The value of the equivalent damping ratio j is
obtained from the damping ratio curves at small
strains. A constant small strain viscous damping is
used in D_MOD with a recommended upper bound
value of 1.5±4% for most soils, independent of
con®ning pressure (Matasovic, 1993; Lanzo and
Vucetic, 1999).
Laird and Stokoe (1993) data show a dependency of
very small strain soil damping on con®ning pressure.
Eq. (5) is proposed in this paper to describe the
dependency of zero strain equivalent damping ratio
on con®ning pressure, as follows:
j ˆ
c
…s
0
†
d
…5†
Y.M.A. Hashash, D. Park / Engineering Geology 62 (2001) 185±206 192
Table 1
Material properties used in the proposed extended hyperbolic model
Symbol Description Source Value
G
mo
Initial shear modulus As proposed in D_MOD rV
2
s
Fig. 2
b Dimensionless coef®cient used to modify the classical
hyperbolic model
As proposed in D_MOD 1.4
S Dimensionless coef®cient used to modify the classical
hyperbolic model
As proposed in D_MOD 0.8
.s
ref
Reference stress New: proposed in this paper 0.18 MPa
a Dimensionless constant used in the computation of
reference strain variation with con®ning pressure
New: proposed in this paper 0.163
b Dimensionless constant used in the computation of
reference strain variation with con®ning pressure
New: proposed in this paper 0.63
c Dimensionless constant used in the computation of
small strain variation with con®ning pressure.
New: proposed in this paper 1.5
d Dimensionless constant used in the computation of
small strain variation with con®ning pressure
New: proposed in this paper 0.3
V
SE
Shear wave velocity of elastic bedrock 3 km/s
where c and d are material parameters and s
0
is the
vertical effective stress.
Eq. (4) is modi®ed to account for the change of
j with depth. The new formulation can be written
as:
‰CŠ ˆ
2
v
‰j
i
K
i
Š
ˆ
2
v
j
1
K
1
2j
1
K
1
2j
1
K
1
j
1
K
1
1j
2
K
2
2j
2
K
2
2j
2
K
2
¼
2
6
6
4
3
7
7
5
…6†
Fig. 6 shows a comparison between the small strain
damping from Eq. (5) Laird and Stokoe (1993) data.
Fig. 6 includes plots of the total damping ratio equal to
hysteretic plus small strain damping. The proposed
equation captures measured damping at very small
strains, as shown in the inset. Total damping curves
fall within range of measured data but do not provide
an exact ®t.
5.3. Numerical implementation of new model in one-
dimensional wave analysis: DEEPSOIL
The problem of wave propagation is approximated
as one-dimensional vertical wave propagation of hori-
zontally propagating shear (SH) waves. The dynamic
equilibrium equation, Eq. (1), is solved numerically at
each time step using the Newmark (1959) b method.
The geologic column is discretized into individual
layers using a multi-degree-of freedom lumped para-
meter model shown in Fig. 7 (Matasovic, 1993).
Each individual layer i is represented by a corre-
sponding mass, non-linear spring, and dashpot for
viscous damping. Lumping half the mass of each of
two consecutive layers at their common boundary
forms the mass matrix. The stiffness matrix is updated
at each time increment to incorporate non-linearity of
the soil. At each time increment, the stiffness k
i
for
layer i, is de®ned as:
k
i
ˆ
G
i
h
i
ˆ
Dt
i
…g
i
†
h
i
Dg
i
…7†
Y.M.A. Hashash, D. Park / Engineering Geology 62 (2001) 185±206 193
Fig. 5. In¯uence of con®ning pressure on normalized shear modulus degradation curves in the proposed non-linear model. Data of Laird and
Stokoe (1993) shown for comparison.
where G
i
is the current tangent shear modulus and h
i
is
the thickness of layer i. The viscous damping matrix is
formed using Eq. (6).
The geologic material (soil or rock) is represented
either as a linear elastic material with constant value
of damping or using the proposed extended hyperbolic
model (pressure-dependent, non-linear soil model) as
described in previous paragraphs.
The base soil column can be modeled as either an
in®nitely stiff or a visco-elastic half space. For a
visco-elastic half space, the following energy trans-
mission model is incorporated (Joyner and Chen,
Y.M.A. Hashash, D. Park / Engineering Geology 62 (2001) 185±206 194
Fig. 6. In¯uence of con®ning pressure on damping ratio curves in proposed non-linear model. Data of Laird and Stokoe shown for comparison.
Legend shown in Fig. 5.
Fig. 7. Multi-degree-of freedom lumped parameter model representation of horizontally layered soil deposit shaken at the base. (a) Computed
surface response spectra, (b) Number of global computation steps.
1975):
C
E
ˆ r
E
V
SE
…8†
where C
E
is the viscous damping coef®cient, r
E
is
the density and V
SE
is the shear wave velocity of the
elastic bedrock. The viscous damping coef®cient of
elastic bedrock is assumed to be proportional to the
product of the mass density and shear wave velocity
of underlying rock.
A deconvolution procedure is needed to correctly
impose a recorded surface rock input motion at the
bottom of the soil column. When performing decon-
volution of a linear elastic system, some practical
dif®culties have to be resolved (Kramer, 1996).
In this program, the deconvolution procedure imple-
mented in SHAKE (Schabel et al., 1972) is used. A
shear wave velocity of 1.2 km/s is assumed in the rock
column above the rock base, which has a shear wave
velocity of 3 km/s.
Solution of the Newmark-b method assuming a
linear material response can be implemented without
the need for an iterative scheme. For non-linear
material response, an iterative solution scheme is
required to arrive at a set of compatible stresses and
strains in the soil column. For a given acceleration
increment, Eq. (1) is solved using the Newmark-b
method and an initial estimate of the stiffness matrix.
A set of strains and stresses are computed and the
stiffness matrix is then updated. Eq. (1) is solved
again using the updated stiffness matrix. The process
is repeated until the maximum difference in computed
strain for two consecutive iterations is less than a speci-
®ed residual (10
28
is used in this implementation).
The input acceleration time series is given as accel-
eration (increment) values at constant time intervals.
Corresponding strain and stress values are computed
throughout the soil column. It is important to sub-
divide acceleration increments to accurately compute
the non-linear stress-strain response of soil. Flexible
and ®xed sub-incrementation schemes are imple-
mented in the code. In a ®xed sub-incrementation
scheme each time-step is divided into equal N sub-
increments throughout the time series. In a ¯exible
sub-incrementation scheme, a time increment is
subdivided only if computed strains in the soil exceed
a speci®ed maximum strain increment.
Fig. 8 presents a comparison of the computed 5%
damped surface response spectra using the ®xed and
the ¯exible sub-incrementation schemes. Four analyses
were performed to compare the accuracy and effect-
iveness of the computation. Except for the N ˆ 1 ®xed
scheme, all of the other schemes give similar results.
The ¯exible sub-incrementation scheme with Dg
max
ˆ
0.05% has been used throughout this study since
Y.M.A. Hashash, D. Park / Engineering Geology 62 (2001) 185±206 195
Fig. 8. Numerical accuracy of computed surface response using ®xed and ¯exible time incrementation solution schemes, input time history
No. 9, time interval 0.02 s, 3400 points.
it gives accurate results at a minimal computation
cost.
The proposed numerical scheme is implemented in
a C11 computer program called Deepsoil. Deep-
soil allows the user any desired number of layers to
model very deep deposits. For the deepest soil pro®le
modeled in this paper (see Fig. 2), the analyses use
84 layers, where layers are 5 m thick near the ground
surface and 20 m at the bottom of the 1000 m
column.
6. Parametric study
A series of analyses are presented using Deepsoil
and are summarized in Table 2. Input ground motion
time series include a range of earthquake events at
rock outcrop and synthetic time series using the
program smsim (Boore, 2000) and parameters for the
NMSZ (Frankel et al., 1996). The peak accelerations,
a
max
, range from 0.0073 to 1.16 g. Three soil columns,
100, 500 and 1000 m thick, shown in Fig. 2 are used in
the analyses. The non-linear, con®ning pressure-
dependent (NLPD) soil model proposed in this
paper is used in most analyses. Some analyses use
the non-linear pressure independent (NLPI) soil
model as described by Matasovic (1993) for compari-
son purposes. The results of analyses are presented in
the form of (a) surface acceleration time series, (b)
surface response spectra with 5% damping, (c) surface
Fourier amplitude spectra, and (d) pro®les of maxi-
mum shear strains.
Y.M.A. Hashash, D. Park / Engineering Geology 62 (2001) 185±206 196
Table 2
Summary of site response parametric study
Input motion time series a
max
(g) Non-linear pressure
independent model (NLPI)
Non-linear pressure dependent
model (NLPD)
T.S. no. Strong motion location
earthquake event
Thickness of soil column (m)
100 500 1000 100 500 1000
Ground motion recording from rock sites (deconvolution procedure used for rock base input)
1 Big Tujunga station (61)
Hector Mine, 1999, M ˆ 7.0
0.0073 X X X
2 OCTT
Michoacan,1985, M ˆ 8.1
0.02 X X X
3 Yerba Buena
Loma Prieta 1989, M ˆ 7.1
0.067 X X X X X X
4 Gilroy #1, NS
Coyote Lake, 1979, M ˆ 5.9
0.12 X X X
5 Ofunato:
N41 Miyagi-Oki,1978, M ˆ 7.4
0.23 X X X
6 University Hospital EW
Northridge, 1994, M ˆ 6.7
0.23 X X X
7 LA City Terrace Northridge,
1994, M ˆ 6.7
0.23 X X X
8 Gilroy #1
Loma Prieta, 1989, M ˆ 7.1
0.43 X X X
9 JMA NS
Hyogoken-Nanbu (Kobe), 1995,
M ˆ 7.2
0.83 X X X X X
Synthetic ground motion generated using SMSIM (input at rock base)
10 M ˆ 4.5, R ˆ 20 km 0.03 X X X
11 M ˆ 6,
R
ˆ 20 km 0.29 X X X X
12 M ˆ 7, R ˆ 20 km 0.59 X X X
13 M ˆ 8, R ˆ 20 km 1.16 X X X
7. In¯uence of con®ning pressure on one-
dimensional site response analysis
The in¯uence of con®ning pressure on one-dimen-
sional site response is demonstrated through com-
parisons of analyses using non-linear, con®ning
pressure-dependent (NLPD) and independent (NLPI)
soil models. Figs. 9 and 10 present such comparisons
using ground motion No. 3 and all three soil columns.
Fig. 9 compares computed time series at the ground
surface using the two soil models. For the 1000 m
column, the NLPD model gives a response with a larger
a
max
and greater number of high frequency peaks than
the NLPI model. The 500 m column shows a similar
response. For the 100 m column, the NLPI and NLPD
time series are quite similar, though the NLPD model
shows greater amplitude of high frequency components
early in the time series (t ˆ 0±8 s).
The difference between the two models can be
observed better by examining the surface response
spectra shown in Fig. 10. For a soil thickness of
1000 m, the in¯uence of the con®ning pressure is
very pronounced. Short period spectral accelerations
are much larger for the NLPD model compared to the
NLPI model. For both models, motion ampli®cation is
computed at a period of about T ˆ 5.0 s which corre-
sponds to the theoretical characteristic site period for
the 1000 m soil column. Similar observations can be
made for the 500 m column. For the 100 m column,
response spectra are also similar for T $0.9 s.
However, for shorter periods the NLPD model spec-
tral acceleration is larger than that of the NLPI model.
The in¯uence of pressure-dependent behavior is still
signi®cant for the 100 m thick column.
The higher response computed using the NLPD
model compared to the NLPI model is a result of
stiffer soil modulus degradation curves and lower
damping in the soil column in the NLPD model. A
measure of cumulative damping in the column is the
quantity kappa:
K ˆ
Z
2j
V
s
dz …9†
Fig. 11 plots values of k versus strain level for
NLPD and NLPI model and the three soil columns
used in the analyses. The values of the damping
ratio j and V
s
are obtained from modulus degradation
and damping curves such as shown in Figs. 5 and 6.
The NLPD model has consistently lower k than that
for the NLPI model. Laboratory data from Laird and
Stokoe (1993) support the lower values of damping
used in the NLPD model. However, more investiga-
tion of damping characteristic of soils at depth is
needed. Improved estimates of damping might be
obtained by further constraining the NLPD model
using ground motion recordings in the Embayment
for small events. In this paper, the model is
constrained using laboratory data only.
Fig. 12 shows a comparison of the soil model effect
on surface response spectra using time series No. 9 for
the 1000 and 500 m soil columns. Observations simi-
lar to those made for analyses using input time series
No. 3 can be made. The NLPD model gives a signi®-
cantly larger spectral response for periods less than 1 s
compared to NLPI. For periods larger than 1 s, the
NLPD model also gives larger or equal spectral
response compared to the NLPI model.
Surface response spectra are commonly used by
engineers to examine ground motion content and its
effect on a hypothetical one-dimensional oscillator,
while Fourier amplitude spectra are used to examine
the frequency content of the ground motion. Fig. 13
shows a comparison of the soil model effect on
surface response spectra and Fourier amplitude spec-
tra using time series No. 11 for the 1000 m soil
column. Both spectra show similar in¯uences with
respect to the soil model as observed in Figs. 10 and
12. The NLPD model results in greater ground motion
content compared to the NLPI model, in particular for
periods less than 1 s (frequencies larger than 1 Hz).
The pressure dependent model, NLPD, shows that
there is signi®cantly less damping of ground motion
even for very deep soil columns. The analyses show
that ignoring pressure dependent cyclic soil behavior
may lead to signi®cant underestimates of propagated
ground motion. The analyses suggest that a larger
portion of the seismic input motion (or energy)
reaches the surface of very deep deposits than would
be obtained from conventional analyses. Estimates of
ground motion parameters of events that caused
paleo-liquefaction features should account for higher
levels of energy transmitted through deep Embayment
deposits. The following paragraphs present prelimin-
ary interpretation of site response analysis results
using the NLPD model and input ground motions
listed in Table 2.
Y.M.A. Hashash, D. Park / Engineering Geology 62 (2001) 185±206 197
Y
.
M
.
A
.
H
a
s
h
a
s
h
,
D
.
P
a
r
k
/
E
n
g
i
n
e
e
r
i
n
g
G
e
o
l
o
g
y
6
2
(
2
0
0
1
)
1
8
5
±
2
0
6
1
9
8
Fig. 9. Comparison of computed surface acceleration time series from site response analyses using pressure independent and pressure dependent soil models. Input time series No.
3, Yerba Buena, Loma Prieta. The pressure dependent model (lower row) shows higher amplitude and greater high frequency content compared to the pressure independent model
(upper row).
Y.M.A. Hashash, D. Park / Engineering Geology 62 (2001) 185±206 199
Fig. 10. Surface response spectra, with 5% damping, from same analyses presented in Fig. 9 Note the overall higher spectral acceleration in the
pressure dependent model analyses (dark solid lines) compared to pressure independent analyses (light solid lines).
Fig. 11. Cumulative damping factor kappa in soil columns for pressure dependent and presure independent soil models. Kappa increases with
increasing strain level and is consistently smaller for the pressure dependent model compared to the pressure independent model. The lower
kappa explains the higher response computed using the pressure dependent model.
Fig. 12. Comparison of computed surface response spectra with 5% damping from site response analyses using pressure independent and
pressure dependent soil models. Input time series No. 9, Kobe. Note the overall higher spectral acceleration using the pressure dependent model
compared to the pressure independent model.
Y.M.A. Hashash, D. Park / Engineering Geology 62 (2001) 185±206 200
Fig. 13. Effect of con®ning pressure dependency on computed surface response spectra (5% damping) and surface Fourier spectra for the
1000 m column, time series No. 11 (synthetic ground motion). Note the overall higher spectral acceleration and Fourier amplitude using the
pressure dependent model compared to the pressure independent model. (a) 5% Damped response spectra. (b) Fourier amplitude spectra.
Fig. 14. Computed responses of the 100, 500 and 1000 m soil columns for input motion No. 12 (synthetic ground motion) and pressure
dependent model only. (a) Computed surface response spectra, (b) Shear strain versus depth.
8. Layer thickness and extent of non-linear
response
Fig. 14 presents surface response spectra and maxi-
mum strain pro®les through soil deposits for time
series No. 12 and three soil columns. Maximum
shear strain pro®les for analyses with the 500 and
1000 m columns are nearly identical below a depth
of 100 m. The maximum shear strain is nearly
constant between a depth of 600 and 1000 m and
has a value of about 0.04 %. The surface response
spectra, however, show that there are differences
between surface response for each of the soil columns
considered up to and including a depth of 1000 m.
Greater damping and some shift in the dominant
periods of the computed spectra occur with increasing
soil column thickness. This is especially true when the
period is close to the site period for the soil column.
Similar trends are observed for analyses with other
time histories listed in Table 2. The analyses suggest
the need to consider the entire depth of the soil
column when performing site response analysis in
the Embayment. The use of an arbitrary cutoff
depth in the Embayment short of the bottom of the
soil column in a site response analysis may lead to
incorrect results.
9. Range of computed spectral ampli®cation
factors
A measure of the effect of deep soil deposits on
propagated ground motion is the spectral ampli®cation
factor. The spectral ampli®cation factor is de®ned as
the ratio of surface spectral acceleration to input
motion spectral acceleration for a given period or
frequency.
Fig. 15 plots Fourier amplitude spectral ampli®-
cation factors for the 1000 m soil column analyses.
The plots include spectra for all thirteen time series
listed in Table 2 using the NLPD model. It is not
possible to distinguish individual spectra in these
plots. However, it is possible to establish ranges of
ampli®cation factors that can be interpreted from
these analyses. Most time series results fall within a
well-de®ned band. The ampli®cation factor has a
peak at approximately 0.2 Hz, which corresponds to
the characteristic site period. Other peaks are
observed that correspond to higher order natural
frequencies of the soil column. A gray trend line is
sketched through the data, which shows the general
change of ampli®cation factor with frequency. In the
frequency range 0.1±4 Hz the ampli®cation factor is
greater than unity and can be up to a value of 5. At
higher frequencies, the ampli®cation factor is less
than one. This interpretation represents the general
trend, but as Fig. 15 shows, there are numerous
exceptions.
Fig. 16 plots the 5% damped response spectral
ampli®cation factors. The plot does show less data
scatter due to damping assumed in a one-dimensional
oscillator. The plot shows trends similar to those
observed in Fig. 15. The ampli®cation at the charac-
teristic site period can be observed, however the
Y.M.A. Hashash, D. Park / Engineering Geology 62 (2001) 185±206 201
Fig. 15. Ratio of computed surface to input Fourier amplitude spectra for all ground motions using pressure dependent model and 1000 m soil
column. The gray line shows the general trend for the data sets. The peaks correspond to the site natural frequencies.
ampli®cation at higher order frequencies is damped
out.
Fig. 17 plots Fourier amplitude spectral ampli®-
cation factors for the 500 m soil column analyses.
The plot shows that the ampli®cation factor has peaks
at the deposit natural frequencies. The ampli®cation
factor exceeds unity over a frequency range of 0.1±
5 Hz. Higher frequency ampli®cation factors are
generally less than unity.
Fig. 18 plots Fourier amplitude spectral ampli®-
cation factors for the 100 m soil column analyses.
The plot shows that there is signi®cantly less ampli-
®cation of ground motion at low frequency (long
periods) compared to 1000 m and 500 m soil columns.
The plot shows that greater ampli®cation of high
frequency components is computed compared to
deeper soil pro®les.
Ampli®cation factor plots show that for deeper soil
columns there is greater ampli®cation of low
frequency (long period) components. The analyses
for the three soil columns show that ampli®cation
factors at short periods/high frequency generally
increase with decreasing soil deposit thickness. The
fundamental frequency of the site and higher order
natural frequencies in¯uence the ampli®cation factor.
The deposit natural frequencies are related to the
Y.M.A. Hashash, D. Park / Engineering Geology 62 (2001) 185±206 202
Fig. 16. Ratio of computed surface to input 5% damped response spectra for all ground motions using pressure dependent model and 1000 m
soil column.
Fig. 17. Ratio of computed surface to input Fourier amplitude spectra for all ground motions using pressure dependent model and 500 m soil
column. The gray line shows the general trend for the data sets. The peaks correspond to the site natural frequencies.
shear wave velocity of the site. It is necessary to
obtain more extensive measurements of shear wave
velocity pro®les at depth at various locations within
the Embayment.
10. Comparison with selected NEHRP response
spectrum for NMSZ
Response spectra from the present analyses (NLPD
Y.M.A. Hashash, D. Park / Engineering Geology 62 (2001) 185±206 203
Fig. 18. Ratio of computed surface to input Fourier amplitude spectra for all ground motions using pressure dependent model and 100 m soil
column. The gray line shows the general trend for the data sets. The peaks correspond to the site natural frequencies.
Fig. 19. Comparison of computed surface response spectra with the maximum NEHRP recommended spectrum for the NMSZ. At periods
greater than 0.7 s, computed spectra exceed NEHRP spectra.
only) are compared with response spectra proposed in
the NEHRP Recommended FEMA, 1997. The NEHRP
response spectrum is developed for Site Class D (Stiff
Soil). The soil column de®ned in Fig. 2 and used in the
analyses falls within the NEHRP Site Class D.
NEHRP spectral accelerations are obtained from
Maps 13 and 14, Maximum Considered Earthquake
Ground Motion for the New Madrid Area. Maximum
values of spectral parameters in NMSZ S
a
(0.2 s) ˆ
3.69 g and S
a
(1 s) ˆ 1.23 g at the B±C boundary are
used.
Fig. 19 shows a plot of surface response spectrum
for NEHRP Site Class D and for a 1000 m soil column
using the proposed model and Time Series 9 and 13.
Time Series 9 and 13 have spectral parameters less
than those used in developing NEHRP spectrum. The
computed response in the proposed model gives spec-
tral accelerations larger than those for NEHRP Site
Class D at periods longer than 0.7 s. A detailed study
is planned to re-examine NEHRP spectral values
systematically within the Mississippi Embayment
using the new proposed model to represent the under-
lying soil column.
11. Conclusions and future work
Analyses presented in this paper show the import-
ance of the in¯uence of con®ning pressure on seismic
site response analysis. The analyses show that:
1. Signi®cant portions of high frequency components
of ground motion are propagated through deep soil
deposits.
2. The entire depth of the soil column should be consid-
ered when performing site response analysis in the
Embayment. The use of an arbitrary cutoff depth in
the Embayment short of the bottom of the soil
column may give incorrect dominant period.
3. Propagation of seismic waves through very deep
deposits result in the development of long period
ground motion.
4. Spectral amplitudes of propagated ground motions
are higher than what would be obtained using
conventional wave propagation analyses. Therefore
estimates of ground motion that are derived from
paleo-liquefaction features should consider the
higher propagated ground motions.
The paper presents preliminary results of ongoing
model development. Further work is underway to
improve model calibration. The analyses presented in
this paper have several limitations:
1. There is a need to calibrate the proposed model using
ground motion data fromthe Embayment. The place-
ment of a deep strong motion/broad band instrument
array in the Embayment will provide important
information regarding wave propagation through
the thick deposits. More work is underway to use
surface ground motion recordings for small earth-
quake events in the Embayment to further calibrate
the model.
2. There is a need to better characterize the soil column
in the Embayment. A deep boring can be used to
retrieve soil/rock samples and can later be used for
a vertical instrument array. The analyses are based in
part on laboratory high-pressure test data on soil
taken from locations other than the Embayment.
There is a need to test samples taken from the
Embayment at various depths and at pressures repre-
sentative of those found in the Embayment.
Acknowledgements
This work was supported primarily by the Earth-
quake Engineering Research Centers Program of the
National Science Foundation under Award Number
EEC-9701785; the Mid-America Earthquake Center.
The authors gratefully acknowledge this support. The
authors thank Dr David Boore, Dr Joan Gomberg,
Dr Scott Olson and anonymous reviewers for provid-
ing insightful comments that greatly enhanced this
paper. The authors also thank Dr Neven Matasovic
of GeoSyntec Consultants for providing an executable
version of the program D-Mod_2. All opinions
expressed in this paper are solely those of the authors.
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