Nonlinear Site Amplification Factors for Constraining the NGA Models

Abstract

Amplification factors computed from the equivalent-linear method using the program RASCALS are used to develop constraints on the nonlinear soil response for possible use by the NGA ground-motion model developers. The site response computations covered site conditions with average V-S30 values ranging from 160 to 900 m/s, soil depths from 15 to 914 m, and peak accelerations of the input rock motion (V-S30=1100 m/s) between 0.01 g and 1.5 g. Four sets of nonlinear properties of the soils are used: EPRI, Peninsular Range, Imperial Valley, and Bay Mud. The first two soil models are used for V-S30 >= 270 m/s and the later two are used for V-S30 <= 190 m/s. Simple parametric models of the nonlinear amplification factors that are functions of the PGA on rock and V-S30 are developed for the EPRI and Peninsula models.

Full text

Nonlinear Site Amplification Factors
for Constraining the NGA Models
Melanie Walling,a) M.EERI, Walter Silva,b) M. EERI, and
Norman Abrahamson,c) M.EERI
Amplification factors computed from the equivalent-linear method using
the program RASCALS are used to develop constraints on the nonlinear soil
response for possible use by the NGA ground-motion model developers. The
site response computations covered site conditions with average VS30 values
ranging from 160 to 900 m/s, soil depths from 15 to 914 m, and peak
accelerations of the input rock motion 共VS30= 1100 m/ s兲between 0.01 g and
1.5 g. Four sets of nonlinear properties of the soils are used: EPRI, Peninsular
Range, Imperial Valley, and Bay Mud. The first two soil models are used for
VS30ⱖ270 m /s and the later two are used for VS30ⱕ190 m / s. Simple
parametric models of the nonlinear amplification factors that are functions of
the PGA on rock and VS30 are developed for the EPRI and Peninsula
models. 关DOI: 10.1193/1.2934350兴
INTRODUCTION
As part of the NGA project, analytical models of the site response were exercised to
help guide the development of the ground-motion models outside the range well con-
strained by the empirical data. The purpose of this paper is to document the nonlinear
site amplification factors that were made available to the NGA developers for possible
use in their models. The decision on whether, and how, to use these results in the devel-
opers NGA model is the responsibility of each NGA developer and is described in their
model papers (Abrahamson and Silva 2008, Boore and Atkinson 2008, Campbell and
Bozorgnia 2008, Chiou and Youngs 2008, and Idriss 2008).
ANALYTICAL MODELING
The analytical model used for site response is the equivalent-linear method with ran-
dom vibration theory as implemented in the program RASCALS (Silva and Lee 1987).
Silva (2008) conducted site response simulations for a range of soil profiles parameter-
ized by the VS30 and the depth to VS=1000 m/ s. For each soil profile, amplification
factors were computed for a range of input rock PGA values 共0.001 to 1.5 g兲. For each
case, the amplification with respect to VS30= 1100 m/ s was computed. For this study,
only the cases with the soil depth randomized over a broad range were used. The subset
of site response cases used in this study are summarized in Table 1.
a) Ph.D. candidate, Civil and Environmental Engineering Department, University of California, Berkeley, CA
94720
b) Pacific Engineering and Analysis, 311 Pomona Ave, El Cerrito, CA 94546
c) Pacific Gas and Electric Company, 245 Market Street, San Francisco, CA 94105
243
Earthquake Spectra, Volume 24, No. 1, pages 243–255, February 2008; © 2008, Earthquake Engineering Research Institute

VELOCITY PROFILE
The base shear-wave velocity profiles are shown in Figure 1 for the suite of VS30
values. For each base profile, a range of soil depths are used. For each soil depth, the
base profile is truncated at the soil depth with a jump to a velocity of 1000 m / s (Figure
2). At depths greater than the soil depth, the velocity profile follows an average velocity
model for California.
The velocity profile is randomized about the base profile. For each base profile, 30
realizations of the velocity profile are generated. The profile randomization scheme,
which varies both layer velocity and thickness, is based on a correlation model devel-
oped from an analysis of variance on about 500 measured shear-wave velocity profiles
(EPRI 1993, Silva et al. 1997). To accommodate variability in the modulus reduction
and damping curves on a generic basis, the curves were independently randomized about
Tabl e 1 . Subset of site response cases from Silva (2008)
VS30 (m/s)
Depth Range to Top of Rock
共VS=1000 m / s兲Nonlinear Material Properties
270 9– 305 m EPRI, Peninsular Range
400 9– 305 m EPRI, Peninsular Range
560 9– 152 m EPRI, Peninsular Range
760 27 m EPRI, Peninsular Range
900 12 m Peninsular Range
Figure 1. Base shear-wave velocity profile.
244 WALLING, SILVA, AND ABRAHAMSON

the base case values. A truncated log normal distribution was assumed with a
␴
ln of 0.35
at a cyclic shear strain of 3⫻10−2%. The distribution was truncated at ±2
␴
to prevent
modulus reduction or damping models that are not physically possible. The random
curves are generated by sampling the modulus reduction or damping at 3⫻10−2%shear
strain, and applying the ratio of the sample value to the average value at all strains. The
random perturbation factor is reduced or tapered near the ends of the strain range to
preserve the general shape of the median curves (Silva 1992).
NONLINEAR DYNAMIC MATERIAL PROPERTIES
From the full set of results given by Silva (2008), a subset based on two sets of
G/Gmax and hysteretic damping curves are used: EPRI (1993) and Peninsular Range
(Silva et al. 1997). The G/Gmax and hysteric damping curves for EPRI and Peninsular
Range (PEN) models are shown in Figures 3a and 3b, respectively.
INPUT MOTIONS
The stochastic point-source model is used to compute the input motions at the sur-
face of the reference rock (Vs30 of 1100 m / s). Validations of the stochastic point-source
model (Hanks and McGuire 1981, Boore 1983, McGuire et al. 1984, Boore and Atkin-
son 1987, Silva and Lee 1987, Toro and McGuire 1987, EPRI 1993, Schneider et al.
1993, Silva and Darragh 1995, Silva et al. 1997) have demonstrated that it provides ac-
curate response spectral estimates. The earthquake magnitude and stress drop are fixed
at M6.5 and 60 bars, respectively, since they have little affect on the amplification fac-
tors. The model used for the anelastic Q(f) model is 176f0.6. For rock sites, the kappa is
0.04 sec. For soil sites, the rock kappa is adjusted to maintain a total small strain total
kappa value of 0.04 sec for deep soil sites as described in Silva (2008).
The point-source distance is selected such that the PGA on the rock 共VS30
Figure 2. Example of truncated base profile at a depth of 76 m.
NONLINEAR SITE AMPLIFICATION FACTORS FOR CONSTRAINING THE NGA MODELS 245

=1100 m / s兲matches the desired level. In all, 11 distances corresponding to the follow-
ing 11 PGA values were used: 0.001 g,0.05 g,0.1 g,0.2 g,0.3 g,0.4 g,0.5 g,0.75 g,
1.0 g,1.25 g,1.5 g. For the soil sites, this distance is held fixed and soil velocity profile
is used in place of the rock velocity profile. The ratio of the spectral acceleration on soil
to the spectral acceleration on rock is computed, resulting in a suite of amplification fac-
tors as a function of reference rock outcrop peak acceleration values.
(
b)
(
a)
Figure 3. Generic G/Gmaz and hysteretic damping curves for the North Coast cohesionless soil
sites (EPRI 1993). Generic G/Gmaz and hysteretic damping curves for the Peninsular Range
cohesionless soil sites (EPRI 1993).
246 WALLING, SILVA, AND ABRAHAMSON

EXAMPLE OF ANALYTICAL RESULTS
As an example, the amplification factors of the analytic model for T=0.2 sec. and
VS30 of 270, 398, 560, 750 and 850 m/ s are shown in Figure 4.
PARAMETERIZATION OF THE ANALYTICAL RESULTS
In the linear range, Boore et al. (1997) found that shear-wave velocity dependence of
the site amplification can be modeled by the following:
ln共Amp兲=aln
冉
Vs30
VREF
冊
,共1兲
where VREF is a reference shear-wave velocity. To account for nonlinear site response,
Abrahamson and Silva (1997) developed an empirical model for site amplification of
generic soil with respect to generic rock. They used the following model for the nonlin-
ear response:
ln共Amp兲=a+bln共PG
ˆArock +c兲,共2兲
where PG
ˆArock is the expected value of the peak acceleration on generic rock. Since this
model was only for a single site condition (generic soil), it did not include a VS30 de-
pendence. Choi and Stewart (2005) developed empirical amplification factors for
NEHRP categories based on the VS30 and the PGA on rock using the following form:
Figure 4. Example of the amplification factors for different VS30 values at T = 0.2 sec based on
the Peninsular Range soil model.
NONLINEAR SITE AMPLIFICATION FACTORS FOR CONSTRAINING THE NGA MODELS 247

ln共Amp兲=aln
冉
VS30
VREF
冊
+bln
冉
PGArock
0.1
冊
共3兲
In the Choi and Stewart model, the b coefficients vary for each NEHRP category (e.g.,
b is a function of the Vs30).
A more general form of the amplification factor can be written as:
ln共Amp兲=aln
冉
VS30
VLIN
冊
+fb共VS30,VLIN兲ln共PGArock +fc共VS30,VLIN兲兲 +d,共4兲
where VLIN is a reference velocity above which the site response is linear. The d term is
needed because the VLIN may not be the reference velocity, VREF. We assume that in the
linear range, the site response should reduce to the form used by Boore et al. (1997).
That is, as PGArock goes to zero or as VS30 goes to VLIN, the ln(Amp) becomes propor-
tional to ln共VS30兲.
There are two simple forms of Equation 4 that satisfy these constraints. In the first
form, the fbterm includes a Vs30 dependence and the fcis constant. This results in the
following model:
ln共Amp兲=
冦
aln
冉
VS30
VLIN
冊
+bln
冉
VS30
VLIN
冊
ln共PGArock +c兲+dfor VS30 ⬍VLIN
aln
冉
VS30
VLIN
冊
+dfor VS30 艌VLIN
冧
共5兲
While this form meets the constraints, it has the undesired consequence that there exists
aPGArock value at which the amplification factor becomes independent of the VS30 for
VS30⬍VLIN. In particular, if the PGArock= exp共−a/b兲−c, then the amplification is con-
stant for all VS30 values (Walling and Abrahamson 2006). This feature of the model is
considered to be unrealistically restrictive.
As an alternative, we considered the case where fbis independent of VS30 and fcis
dependent on VS30. The following model meets the constraints described above:
ln共Amp兲=
冦
aln
冉
VS30
VLIN
冊
−bln共PGArock +c1兲
+bln
冉
PGArock +c
冉
VS30
VLIN
冊
n
冊
+dfor VS30 ⬍VLIN
共a+bn兲ln
冉
VS30
VLIN
冊
+dfor VS30 艌VLIN
冧
共6兲
This form avoids the feature of the amplification being independent of Vs30 atagiven
PGA level.
248 WALLING, SILVA, AND ABRAHAMSON

EXAMPLE FITTING
The amplification factors are modeled using Equation 6 and the parameters n, c,
VLIN, and b were estimated using ordinary least squares. The n and c coefficients were
constrained to be independent of frequency and the parameters VLIN and b were esti-
mated at each frequency. An example of the parametric fit of the analytic model for
VS30= 270 m/ s and 560 m/ s at T = 0.2 sec is seen in Figure 5.
SMOOTHING OF PARAMETERS
A smoothed model of the period dependence of the b and VLIN terms was developed.
The b and VLIN term are correlated parameters. This correlation is addressed by first
smoothing the VLIN values and using the smoothed VLIN values to recompute the b
terms. These resulting b terms were then smoothed. The functional form of the model
used for smoothing the b and VLIN term is given by
x=
冦
␤
2T艌T2
␣
0+兺
i=1
6
␣
i关ln共T/T0兲兴iT1⬍T⬍T2
␤
1T艋T1
冧
,共7兲
where xis either ln共VLIN兲or band Tis the spectral period. The resulting coefficients for
the smoothed b and VLIN models listed in Table 2. The period dependence of the b and
VLIN terms is shown in Figures 6 and 7 for both the EPRI and Peninsular Range models.
Figure 5. Example of the parametric fit of the analytic model for VS30 = 270 and 560 m /s using
Eq. 6 at T= 0.2 s based on the PEN model.
NONLINEAR SITE AMPLIFICATION FACTORS FOR CONSTRAINING THE NGA MODELS 249

COMPARISON OF RESULTS
The smoothed nonlinear amplification model from this study is compared to the lin-
ear amplification model from Boore et al. (1997) and the nonlinear amplification model
from Choi and Stewart (2005). The Choi and Stewart (2005) model was derived using
the empirical models of Abrahamson and Silva (1997), Sadigh et al. (1997), and Camp-
bell and Bozorgnia (2003) for generic rock. The Choi and Stewart (2005) model refer-
ence VS30 derived with the Abrahamson and Silva (1997) model is 532±93 m/s for a
Tabl e 2 . Parameterization of the nonlinear site response from the Silva simulations
EPRI Model Peninsular Model
VLIN bVLIN b
T00.0133 0.02 0.025 1.00
T10.020 0.025 0.025 0.0125
T21.1 2.5 1.25 2.5
␣
07.244 −1.1050 6.763 −1.9546
␣
1−1.6411 −0.42439 0.20784 1.9097
␣
22.7107 1.482073 0.400139 1.16744
␣
3−1.42332 −1.329229 −0.5196731 −0.3716778
␣
40.294717 0.45954657 0.1566076 −0.3893755
␣
5−0.0216321 −0.0705797 −0.0144830 −0.0931755
␣
60.0 0.00418515 0.0 −0.007279
␤
16.9431 −1.139 6.7628 −1.190
␤
26.0380 −0.650 5.9964 0.1504
n 1.30 1.18
c 1.38 1.88
Figure 6. Comparison of the estimated VLIN terms with the smoothed models.
250 WALLING, SILVA, AND ABRAHAMSON

frequency= 0.3 Hz and 519 ±69 m / s for a frequency=1.0 Hz (Choi and Stewart 2005).
This is consistent with other estimates of the reference velocity for Abrahamson and
Silva (1997) rock as approximately 550 m / s.
To allow for comparisons of the analytical model results with the Choi and Stewart
(2005) model, the amplification factors from the analytical model were normalized by
the amplification for Vs30= 550 m/ s and the PGA1100 was converted to PGA550.Asan
example, the amplification for VS30= 274 m/ s, corresponding to generic soil is shown as
a function of the PGA550 in Figures 8–10 for spectral periods of 0.01, 0.2, and 1.0 sec,
Figure 7. Comparison of the estimated b terms with the smoothed models.
Figure 8. Comparison of site amplification models for VS30 = 274 m / s for PGA.
NONLINEAR SITE AMPLIFICATION FACTORS FOR CONSTRAINING THE NGA MODELS 251

respectively. The linear amplification from the Boore et al. (1997) model is also shown
for comparison.
As seen in each figure, the three models are in general agreement at the low levels of
shaking, becoming approximately equal at PGA550= 0.1 g.ForPGA550 values greater
than 0.2 g, the Choi and Stewart (2005) model shows less nonlinearity than the models
derived in this paper. These trends are more noticeable at shorter periods where soil re-
sponse is more nonlinear, as seen in Figure 9 for T= 0.2 s, and but less at the longer
periods T⬎1.0 s, where soil response is primarily linear as seen in Figure 10 for T
=1.0 s.
Figure 9. Comparison of site amplification models for VS30 = 274 m / s for T = 0.2.
Figure 10. Comparison of site amplification models for Vs30 = 274 m / s for T = 1.0 s.
252 WALLING, SILVA, AND ABRAHAMSON

USE OF RESULTS BY NGA DEVELOPERS
Of the five NGA developers, two used the analytical site response model results de-
scribed in this paper to constrain the nonlinear site response. Abrahamson and Silva
(2008) and Campbell and Bozorgnia (2008) used the form of the amplification given in
Equation 6 and constrained the b, VLIN,c1, and n terms to be given by the values from
the PEN model (Table 2). The linear site response (a term) was derived from the em-
pirical data. They also constrained the impact of the nonlinear response on the standard
deviation of the ground motion using the PEN model. The Boore and Atkinson (2008)
model constrained the nonlinear amplification in their model using the Choi and Stewart
(2005) model results. Chiou and Youngs derived the nonlinear response based on the em-
pirical data and checked these results against the analytical results shown in this paper.
The fifth model, Idriss (2008), is for rock only, so it does not include a nonlinear site
response term.
CONCLUSIONS
The analytical site response results based on the equivalent-linear method with RVT
were provided to the NGA developers as one possible approach for constraining the non-
linear site response in their empirical ground-motion models. Two of the four developers
that included site response used the analytical results based on the PEN soil to constrain
their ground-motion models. With these constraints, the site amplification from these
two ground motion-models will, in general, be consistent with site-specific site response
approaches commonly used for engineering projects. In particular, hazard studies con-
ducted using the empirical soil ground-motion models are expected to be generally con-
sistent with site-specific studies using empirical rock ground motions models as input to
site-specific site response analyses.
ACKNOWLEDGMENTS
This study was sponsored by the Pacific Earthquake Engineering Research Center’s
Program of Applied Earthquake Engineering Research of Lifelines Systems supported
by the California Department of Transportation, the California Energy Commission, and
the Pacific Gas and Electric Company.
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