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1- Prof. Jonathon Bray – Simplified Seismic Slope Displacement Procedures

Empirical Predictive Models for Earthquake-Induced Sliding
Displacements of Slopes
Gokhan Saygili1 and Ellen M. Rathje, M.ASCE2
Abstract: Earthquake-induced sliding displacement is the parameter most often used to assess the seismic stability of slopes. The
expected displacement can be predicted as a function of the characteristics of the slope 共yield acceleration兲 and the ground motion 共e.g.,
peak ground acceleration兲, yet there is significant aleatory variability associated with the displacement prediction. Using multiple ground
motion parameters to characterize the earthquake shaking can significantly reduce the variability in the prediction. Empirical predictive
models for rigid block sliding displacements are developed using displacements calculated from over 2,000 acceleration–time histories
and four values of yield acceleration. These empirical models consider various single ground motion parameters and vectors of ground
motion parameters to predict the sliding displacement, with the goal of minimizing the standard deviation of the displacement prediction.
The combination of peak ground acceleration and peak ground velocity is the two parameter vector that results in the smallest standard
deviation in the displacement prediction, whereas the three parameter combination of peak ground acceleration, peak ground velocity, and
Arias intensity further reduces the standard deviation. The developed displacement predictive models can be used in probabilistic seismic
hazard analysis for sliding displacement or used as predictive tools for deterministic earthquake scenarios.
DOI: 10.1061/共ASCE兲1090-0241共2008兲134:6共790兲
CE Database subject headings: Earthquakes; Landslide; Slope stability; Probability; Seismic effects.
Introduction
The seismic performance of slopes and earth structures is often
assessed by calculating the permanent, downslope sliding displacement expected during earthquake shaking. Newmark 共1965兲
first proposed a rigid sliding block procedure, and this procedure
is still the basis of many analytical techniques used to evaluate the
stability of slopes during earthquakes. Newmark 共1965兲 realized
that accelerations generated by earthquake shaking could impart a
destabilizing force sufficient to temporarily reduce the factor of
safety of a slope below 1, leading to sliding episodes and the
accumulation of permanent, downslope sliding displacement. The
original Newmark procedure models the sliding mass as a rigid
block and utilizes two parameters: the yield acceleration 共ky, the
acceleration in units of g that initiates sliding for the slope兲 and
the acceleration–time history of the rigid foundation beneath the
sliding mass. A sliding episode begins when the acceleration exceeds ky and continues until the velocity of the sliding block and
foundation again coincide. The relative velocity between the rigid
block and its foundation is integrated to calculate the relative
1
Graduate Research Assistant, Dept. of Civil, Architectural, and
Environmental Engineering, Univ. of Texas, Austin, TX 78712. E-mail:
gokhansaygili@mail.utexas.edu
2
J. Neils Thompson Associate Professor, Dept. of Civil, Architectural,
and Environmental Engineering, Univ. of Texas, Austin, TX 78712.
E-mail: e.rathje@mail.utexas.edu
Note. Discussion open until November 1, 2008. Separate discussions
must be submitted for individual papers. To extend the closing date by
one month, a written request must be filed with the ASCE Managing
Editor. The manuscript for this paper was submitted for review and possible publication on May 15, 2007; approved on October 4, 2007. This
paper is part of the Journal of Geotechnical and Geoenvironmental
Engineering, Vol. 134, No. 6, June 1, 2008. ©ASCE, ISSN 1090-0241/
2008/6-790–803/\$25.00.
sliding displacement for each sliding episode and the sum of the
displacements for each sliding episode represents the cumulative
sliding displacement 共Fig. 1兲.
The original rigid sliding block procedure is applicable to thin,
veneer slope failures. This failure mode is common in natural
slopes 共Keefer 1984兲, whereas deeper sliding surfaces are more
common in engineered earth structures 共although shallow failure
surfaces can also occur in engineered slopes兲. The sliding block
displacement methodology has been extended to account for the
deformable response of deeper sliding masses in earth structures

Rathje 1998兲, and to account for the coupled interaction between
sliding and dynamic responses 共Rathje and Bray 1999, 2000兲.
Nonetheless, for natural slopes, rigid sliding block analysis is the
most common analytical procedure used to predict the potential
for earthquake-induced landslides and will be the focus of this
work.
In practice, the expected permanent displacement for a slope is
often assessed by either 共1兲 selecting a suite of earthquake ground
motions appropriate for the design event, computing the sliding
displacement for each motion using the yield acceleration of the
slope, and computing the median and standard deviation of the
computed displacements, or 共2兲 using design charts and equations
that predict sliding displacement based on various ground motion
parameters and the yield acceleration. Although these approaches
attempt to deal with the variabilities inherent in earthquake engineering, they do not rigorously account for the variability in
earthquake ground motions or the variability in the predicted sliding displacement.
Alternatively, following the context of performance-based
earthquake engineering, seismically induced permanent sliding
displacement can be considered an engineering demand parameter
that describes the performance of slopes. A probabilistic assessment of the sliding displacement 共D兲 can be computed, with the
790 / JOURNAL OF GEOTECHNICAL AND GEOENVIRONMENTAL ENGINEERING © ASCE / JUNE 2008
J. Geotech. Geoenviron. Eng., 2008, 134(6): 790-803
parameters that includes multiple characteristics of the ground
motion 共Bazzurro and Cornell 2002; Baker and Cornell 2005兲
␭D共x兲 =

i
j
⫻P关兩D ⬎ x兩GM1 = zi,GM2 = y j兴P关GM1 = zi,GM2 = y j兴

Here, the sliding displacement is conditioned on two ground
motion parameters, GM1 and GM2, and the joint probability of
occurrence of the two ground motion levels GM1 = zi and
GM2 = y j, P关GM1 = zi , GM2 = y j兴. Similar to the scalar approach, a
predictive model for displacement as a function of GM1 and
GM2 is required, along with the standard deviation in the prediction. However, the joint occurrence of GM1 and GM2 cannot be
derived from traditional PSHA hazard curves, and thus a more
complex approach is required 共Bazzurro and Cornell 2002兲.
Toward the development of sliding displacement hazard
curves, this paper describes rigid block displacement models appropriate for scalar and vector PSHA. The implementation of
these models in scalar and vector PSHA is discussed in Rathje
and Saygili 共2008兲.
Previous Work
Fig. 1. Acceleration–time history, sliding velocity–time history, and
sliding displacement–time history for a rigid sliding block and
ky = 0.1
result being a displacement hazard curve that describes the annual
rate of exceedance 共␭兲 as a function of displacement levels. The
use of a displacement hazard curve allows for a more rational
assessment of the displacement hazard at a site because it considers the probability that different performance levels are exceeded.
The mean annual rate of exceedance of the sliding displacement conditioned on a single ground motion parameter 共GM兲 is
defined as
␭D共x兲 =

i
where ␭D⫽mean annual rate of exceedance of a sliding displacement level of x, P关D ⬎ x 兩 GM= zi兴 represents the probability the
displacement level x is exceeded when the ground motion level is
equal to zi; and P关GM= zi兴⫽probability of occurrence of ground
motion level zi. The term P关D ⬎ x 兩 GM= zi兴 is computed from a
predictive model for displacement as a function of the GM and
slope parameters, along with the aleatory variability 共standard deviation兲 in the prediction. The probability of occurrence for the
ground motion is derived from the derivative of the ground motion hazard curve 共␭GM兲. This derivative is also known as the
mean rate density of the ground motion, MRDGM共z兲 共Bazzurro
and Cornell 2002兲. Essentially, Eq. 共1兲 calculates the annual frequency of exceeding a given value of displacement by combining
the evaluation of the seismic hazard in terms of the ground motion parameter with the evaluation of sliding displacement conditioned on the value of the ground motion parameter.
Eq. 共1兲 is only applicable for a single ground motion parameter
and thus is considered scalar probabilistic seismic hazard assessment 共PSHA兲, but it can be extended to a vector of ground motion
Many researchers have proposed models that predict rigid block
sliding displacement as a function of ground motion parameters

displacements for four earthquake motions and showed that displacement was a function of ky, peak ground acceleration 共PGA兲,
and peak ground velocity 共PGV兲. Franklin and Chang 共1977兲,
Ambraseys and Menu 共1988兲, Yegian et al. 共1991兲, and Ambraseys and Srbulov 共1994兲 developed charts and/or predictive equations for rigid sliding block displacements using different ground
motion data sets. However, these models were developed based
on somewhat limited data sets and the resulting predictive equations displayed very large variability 共␴ln ⬎ 1.0兲.
Recent research has used larger ground motion data sets to
develop displacement predictive models and developed better estimates of the variability 共␴ln兲 in the predictions. WatsonLamprey and Abrahamson 共2006兲 developed a model for rigid
block displacement using a large data set consisting of 6,158 recordings scaled with seven different scale factors and computed
for three values of yield acceleration. Their displacement model is
a function of various parameters including PGA, spectral acceleration at a period of 1 s 共SaT=1 s兲, root mean square acceleration

standard deviation for the predictive model was not presented,
although this information was available for preliminary versions
of the model and ranged from 0.3 to 0.7 in natural log space 共N.
Abrahamson, personal communication, 2005兲.
Jibson 共2007兲 developed predictive models for rigid block displacements using 2,270 strong motion recordings from 30 earthquakes. A total of 875 values of calculated displacement, evenly
distributed between four values of ky, were used to develop the
predictive models. Models were developed as a function of
ky / PGA 共called the critical acceleration ratio兲, ky / PGA and earthquake magnitude 共M兲, ky and Arias intensity 共Ia兲, and ky / PGA
and Ia. The standard deviations 共␴ln兲 for each of their proposed
models is close to 1.0 关reported as ␴log = 0.5 in Jibson 共2007兲兴.
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J. Geotech. Geoenviron. Eng., 2008, 134(6): 790-803
Bray and Travasarou 共2007兲 presented a predictive relationship
for earthquake-induced displacements of rigid and deformable
slopes. Displacements were calculated using the equivalent-linear,
fully coupled, stick-slip sliding model of Rathje and Bray 共1999,
2000兲. 688 earthquake records 共2 orthogonal components per
record兲 obtained from 41 earthquakes were used to compute displacements for ten values of ky and eight site geometries 共i.e.,
fundamental site periods, Ts兲. Displacements for the two components of orthogonal motion were averaged and values less than
1 cm were set equal to zero because they were assumed to be of
no engineering significance. The model input parameters include
ky, the initial fundamental period of the sliding mass 共Ts兲, the
magnitude of the earthquake 共M w兲, and the spectral acceleration
at a degraded period equal to 1.5Ts, called Sa共1.5Ts兲. When considering shallow, rigid sliding surfaces, Sa共1.5Ts兲 is taken as PGA
and Ts = 0. The standard deviation 共␴ln兲 for the predictive model is
0.66.
Fig. 2 presents a comparison of several current models

2007, and the Jibson 2007 ky / PGA model兲 for a shallow, rigid
sliding mass, a deterministic earthquake scenario of M w = 7 and
R = 5 km, and rock site conditions 共Vs30 = 760 m / s兲. The predicted
ground motion parameters for this scenario are listed in Fig. 2.
The values of PGA and SaT=1 s are from Boore and Atkinson

et al. 共2004兲. Note that Jibson 共2007兲 and Bray and Travasarou

whereas Watson-Lamprey and Abrahamson 共2006兲 use four
parameters 共PGA, ARMS, SaT=1 s, and Durky兲. The Bray and
Travasarou 共2007兲 model predicts the largest displacement, the
Watson-Lamprey and Abrahamson 共2006兲 model predicts the
smallest, and the Jibson 共2007兲 ky / PGA model falls in between.
The difference between the Bray and Travasarou 共2007兲 and Jibson 共2007兲 models can be attributed to the fact that the Jibson

representing the absolute value of PGA兲 and thus cannot distinguish the effect of large PGA intensities on displacements. The
Bray and Travasarou 共2007兲 model does not predict any displacement for ky ⬎ 0.17 because at these values of ky the median displacement is less than 1 cm and their model assumes that D
⬍ 1 cm can be considered equal to zero. The smaller values from
Watson-Lamprey and Abrahamson 共2006兲 are due to the supplementary information provided by the additional ground motion
parameters, particularly the parameter Durky. The ⫾1 standard
deviation displacements for the Jibson 共2007兲 and Bray and
Travasarou 共2007兲 models are shown in Fig. 2共b兲. Generally, the
Bray and Travasarou 共2007兲 model displays less variability.
Of the current models, the Jibson 共2007兲 and Bray and
Travasarou 共2007兲 models are the most appropriate for PSHA
because they were developed using large data sets and rigorous
regression techniques, and they provide estimates of the ␴ln of the
displacement prediction. Yet, the Jibson 共2007兲 model has significant aleatory variability 共␴ln兲 such that a large range of displacements is predicted for the given ground motion and site/slope
parameters. The ␴ln for the Bray and Travasarou 共2007兲 model is
smaller than for Jibson 共2007兲; however, it does not take advantage of multiple ground motion parameters to predict the displacement and further reduce ␴ln. The goal of this study is to develop
rigid block displacement predictive equations that utilize multiple
ground motion parameters in an effort to reduce the aleatory variability 共␴ln兲. These displacement predictive equations can be used
to develop hazard curves for sliding displacement using either the
scalar or vector approach.
Fig. 2. Comparison of predictive models for sliding displacement for
a deterministic scenario of M w = 7 and R = 5 km: 共a兲 median
predictions; 共b兲 ⫾␴ predictions
Framework for Model Development
Rigid sliding block displacements were computed using the rigid
sliding block programs developed by Jibson and Jibson 共2003,
Personal communications 2006兲. The computed displacement values were then used to develop predictive relationships for displacement as a function of ky and different combinations of
ground motion parameters, with the goal of identifying the com-
792 / JOURNAL OF GEOTECHNICAL AND GEOENVIRONMENTAL ENGINEERING © ASCE / JUNE 2008
J. Geotech. Geoenviron. Eng., 2008, 134(6): 790-803
bination共s兲 of ground motion parameters that produce the smallest
variability in the prediction of sliding displacement. The developed displacement models can be used to compute hazard curves
for earthquake-induced permanent displacement, can be used as
predictive tools for deterministic earthquake scenarios, or can be
used to rapidly predict the likelihood of earthquake-induced landslides after an earthquake using recorded ground motions.
Ground Motion Database
Variability in the expected ground motion is the biggest contributor to the variability in the displacement prediction; thus, using a
large, high-quality data set of strong motion records is essential to
developing a robust displacement model. Currently the number of
available records is considerable and one significant data set is
readily available from the Next Generation Attenuation 共NGA兲
strong motion database of the Pacific Earthquake Engineering Research Center 具http://peer.berkeley.edu/nga典.
The initial data set included motions from earthquakes M w
ranging from 5 to 7.9 and distances from 0.1 to 100 km. Motions
recorded at soft soil sites, on the crest or abutments of dams,
underground, not at the ground floor of a building, or in buildings
larger than four stories were removed from the database. Additionally, motions with high-pass filter corner frequencies larger
than 0.25 Hz or low-pass filter corner frequencies less than 10 Hz
were removed. The resulting data set included 2,383 motions.
Rigid sliding block displacements were computed for ky values of
0.05, 0.1, 0.2, and 0.3 g, which encompasses typical values for
earth slopes. For each motion, displacements were calculated for
positive and negative polarities, with the largest displacement
used for the model development. Displacements computed from
orthogonal components recorded at the same station during the
same earthquake were treated as separate data points for the
model.
Approximately 25% of the initial ground motion data set had
PGA values of less than 0.05 g, and thus these motions do not
predict any displacement for the ky values used. To further populate the database at larger values of PGA, displacements were also
calculated for each motion scaled by factors of 2.0 and 3.0. To
ensure that unreasonable PGA values were not used when scaling
the motions, the motions were capped at PGA= 1.0 g. However,
regressions were also performed using motions up to PGA= 2.0 g,
and similar results were obtained. The final displacement data set
when capping PGA= 1.0 g included approximately 14,000 nonzero displacements. Only these nonzero displacements were used
in the regression analysis. The distribution of records in the final
ground motion data set in terms of earthquake magnitude, closest
distance, and various ground motion parameters is given in Fig. 3.
Ground Motion Parameter Selection
The sliding displacement of earth slopes is a function of various
features of the expected ground motion such as intensity, f …