-in

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ng mtingaximpeak ground acceleration (PGA), peak ground velocity (PGV), the natural period

epresemic stive, dowing. Thationsand t

Engineering Geology 122 (2011) 5160

Contents lists available at ScienceDirect

Engineering

j ourna l homepage: www.e lsearthquake-induced sliding displacement (D) of a slope with yieldacceleration, ky (ky = seismic coefcient that when multiplied by theweight of the sliding mass and applied to the slope yields a factor ofsafety of 1.0). If the sliding mass is relatively shallow and stiff, a rigidsliding block analysis is appropriate. In this case, the natural period ofthe sliding mass (Ts) is essentially zero and the dynamic response ofthe sliding mass can be ignored. The seismic loading is simply theacceleration-time (at) history at the base of the sliding mass, withthe destabilizing force-time history (F(t))on the slope equal to the at

history can be used to directly compute D for the given ky. The Javaprogram by Jibson and Jibson (2003) makes performing thesecalculations quite easy, although the acceleration-time historiesmust be selected appropriately.

Deeper and/or softer sliding masses are exible and have naturalperiods greater than zero, such that the rigid sliding blockmodel is notappropriate. In these cases, the dynamic response of the exiblesliding mass must be taken into account (Fig. 1). Two-dimensionalnite element analysis can be used tomodel this dynamic response, orhistory (in units of gravity, g) times the weSeismic loading parameters for the slope

Corresponding author. Tel.: +1 5122323683; fax: +E-mail addresses: e.rathje@mail.utexas.edu (E.M. Ra

(G. Antonakos).

0013-7952/$ see front matter 2011 Elsevier B.V. Aldoi:10.1016/j.enggeo.2010.12.004hus has been a usefulssment.used to compute the

from empirical models (e.g., Jibson, 2007; Saygili and Rathje, 2008).Alternatively, a suite of acceleration-time histories can be selectedthat represents the expected ground shaking at the site and each timeparameter in seismic design and hazard asseFig. 1 outlines the process commonly1. Introduction

Permanent sliding displacement rparameter for evaluating the seisdisplacement represents the cumulatsliding mass due to earthquake shakdisplacement relates well with observof slopes (e.g., Jibson et al., 2000),natural period of the sliding mass. This unied framework provides a consistent approach for predicting thesliding displacement of rigid (Ts=0) and exible (TsN0) slopes.

2011 Elsevier B.V. All rights reserved.

nts a common damageability of slopes. Thisnslope movement of ae magnitude of slidingof seismic performance

represent various ground motion characteristics (GM) of an acceler-ation-time history, such as the peak ground acceleration (PGA), peakground velocity (PGV), Arias Intensity (Ia), etc. Generally, theseground motion parameters are specied based on ground motionprediction equations (e.g., Next Generation Attenuation (NGA)models) and probabilistic seismic hazard analysis. The seismic loadingparameters can be used, along with the ky of the slope, to predict Dight of the sliding mass.can be derived that

alternatively themodeled as a onRathje and Bray,the one-dimensithe seismic loadblock analysis (ecomputes the dconsideration of

1 5124716548.thje), antonakosg@gmail.com

l rights reserved.x in lieu PGA and PGV, and include a term related to the

Seismic slope stability of the sliding mass (Ts), and the mean period of the earthquake motion (Tm). The empirical predictive models

for sliding displacement utilize kmax and kvelmaSeismic hazards models are a function of theA unied model for predicting earthquakeexible slopes

Ellen M. Rathje , George AntonakosUniversity of Texas at Austin, 1 University Station C1792, Austin, TX 78712 USA

a b s t r a c ta r t i c l e i n f o

Article history:Received 26 February 2010Received in revised form 12 October 2010Accepted 17 December 2010Available online 24 December 2010

Keywords:EarthquakeLandslide

Permanent sliding displacemof slopes. Recently developehave demonstrated that inground acceleration and puncertainty. A unied framapplication to exible slidiframework includes prediccoefcient (kmax) and themduced sliding displacements of rigid and

t represents a common damage parameter for evaluating the seismic stabilitympirical models for the sliding displacement of shallow (rigid) sliding massesing multiple ground motion parameters in the predictive model (e.g., peakground velocity) improves the displacement prediction and reduces itork is developed that extends these empirical displacement models forasses, where the dynamic response of the sliding mass is important. Thisthe seismic loading for the sliding mass in terms of the maximum seismicum velocity of the seismic coefcient-time history (kvelmax). The predictive

Geology

evie r.com/ locate /enggeosliding mass at its maximum thickness can bee-dimensional soil column. Previous research (e.g.,2001; Vrymoed and Calzascia, 1978) has shown thatonal simplication provides an adequate estimate ofing for deeper sliding masses. A decoupled sliding.g., Makdisi and Seed, 1978; Bray and Rathje, 1998)ynamic response of the sliding mass without anythe sliding displacement, and then uses the results of

slidi

52 E.M. Rathje, G. Antonakos / Engineering Geology 122 (2011) 5160the dynamic response analysis to compute the sliding displacement. Acoupled analysis (e.g., Rathje and Bray, 1999, 2000) simultaneouslycomputes the dynamic and sliding responses. Within either approach,

Fig. 1. Approaches for computing earthquake-inducedthe seismic loading time history for the sliding mass is related to theseismic coefcient (k)-time history, in which k represents the averageacceleration within the sliding mass as well as the shear force at thebase of the sliding mass. The destabilizing force-time history (F(t)) isthen simply equal to the k-time history times the weight of the slidingmass. For a coupled analysis, k cannot exceed ky, and the dynamicequations of equilibrium change during sliding to enforce thisequilibrium condition. For a decoupled analysis, the k-time historymay exceed ky, and the k-time history is used in a rigid sliding blockanalysis in lieu of the acceleration-time to compute displacements.

It has become common practice to use rigid sliding block analysesfor slidingmasseswith Ts below some threshold, while using a exiblesliding block analysis for Ts greater than some threshold (Jibson,2011-this issue). In actuality, the response of slopes does not abruptlychange from rigid to exible at some value of Ts, but rather there is atransition from rigid to exible behavior as Ts increases. This paperdevelops a unied framework that models the full range of dynamicresponse conditions from rigid through exible slidingmass behavior.This unied approach predicts the seismic loading parameters andpermanent displacements of sliding masses as a function of Ts (asdened by the one-dimensional period computed for the maximumthickness of the sliding mass), such that the approach tracks theresponse from rigid conditions (Ts=0) to very exible conditions(Ts0). The unied model is also built upon recently developedempirical models for rigid sliding displacement (i.e., Saygili andRathje, 2008; Rathje and Saygili, 2009), which use PGA and PGV topredict sliding displacement. The unied approach denes the seismicloading parameters in terms of these same ground motion para-meters, except that these parameters are computed from a k-timehistory rather than an acceleration-time history. Predictive models forthese seismic loading parameters are provided, and empirical modelsthat predict D for rigid and exible conditions are developed.2. Rigid sliding block displacements

A large number of empirical models are available that predict the

ng displacements for rigid and exible sliding masses.sliding displacement of rigid sliding masses. Newmark (1965) rstproposed using rigid sliding block displacement to assess the seismicperformance of slopes, and later various researchers (e.g., Sarma,1975; Franklin and Chang, 1977; Sarma, 1980; Ambraseys and Menu,1988; Yegian et al., 1991, and Ambraseys and Srbulov, 1994; Sarmaand Kourkoulis, 2004) developed charts and/or predictive equationsfor D. Recent research (e.g., Watson-Lamprey and Abrahamson, 2006;Jibson, 2007; Bray and Travasarou, 2007) includes more robustempirical models developed from larger ground motion datasets. Theunied model presented here is based on the recent empiricaldisplacement models of Saygili and Rathje (2008) and Rathje andSaygili (2009), and thus these models are discussed in detail below.

Saygili and Rathje (2008) presented a suite of empirical predictivemodels for the sliding displacement of rigid slopes, and these modelsconsidered various ground motion parameters, such as PGA, PGV, Ia,and mean period (Tm, Rathje et al., 2004), as well as combinations ofthese ground motion parameters. Rathje and Saygili (2009) slightlymodied the PGA model from Saygili and Rathje (2008) by adding aterm related to earthquake magnitude (M). The Rathje and Saygili(2009) modication is repeated in Rathje and Saygili (2011). Therecommended single (scalar) ground motion parameter model is the(PGA, M) model from Rathje and Saygili (2009), and the recom-mended two (vector) ground motion parameter model is the (PGA,PGV) model from Saygili and Rathje (2008). For simplicity, thesemodels will be called the SR08/RS09 models.

Fig. 2 plots predicted values of D from the SR08/RS09 models as afunction of ky for different earthquake scenarios of M=6, 7, and 8,each with a site-to-source distance (R) equal to 2 km. Also consideredare rock (Vs30=760 m/s) and soil (Vs30=400 m/s) site conditions.The Boore and Atkinson (2008) ground motion prediction equationwas used to predict the median values of PGA and PGV for eachscenario, and these values are listed in Table 1. Note that the PGA

53E.M. Rathje, G. Antonakos / Engineering Geology 122 (2011) 5160values become similar (i.e., saturate) at larger magnitudes, while thePGV values continue to increase (Table 1). Additionally, the soilconditions amplify PGVmore than they amplify PGA. Predicted valuesof D are shown in Fig. 2(a) for both the (PGA, M) and (PGA, PGV)models and Vs30=760 m/s. For M=6, the (PGA, M) and (PGA, PGV)models predict similar displacements, but the displacements become

Fig. 2. Rigid sliding block displacements calculated from the SR08/RS09 (PGA, M) and(PGA, PGV) models for different earthquake scenarios and site conditions.

Table 1Ground motion parameters for each earthquake scenario and site conditions.

Vs30=760 m/s Vs30=400 m/s

M R (km) PGA (g) PGV (cm/s) Tm (s) PGA (g) PGV (cm/s) Tm (s)

6.0 2 0.30 19 0.37 0.34 27 N/A7.0 2 0.43 42 0.44 0.47 59 N/A8.0 2 0.48 74 0.46 0.52 102 N/Adifferent as earthquake magnitude, and the associated PGV, increases.The (PGA, PGV) model generally predicts smaller displacements forthese scenarios, on the order of 30 to 40% smaller. These differencesare caused by the fact that the empirical models were developed usingrock and soil motions from the Next Generation Attenuation (NGA)ground motion dataset. Because soil motions tend to have larger PGVvalues than rock motions and the (PGA, M) model does not includethe effects of PGV, the (PGA, M) model predicts larger displacementsthan the (PGA, PGV) model for rock sites. This effect is demonstratedin Fig. 2(b),whichuses soil groundmotionparameters (Vs30=400 m/s)to predict D.When utilizing groundmotion parameters for soil sites, thedifferences between the (PGA, M) and (PGA, PGV) models are muchsmaller.

In addition to the differences in median displacements from the(PGA, M) and (PGA, PGV) models, there are signicant differences inthe standard deviations of the predictions. The standard deviation(lnD) for each model increases with increasing ky/PGA, with valuesranging between 0.75 and 1.0 (in natural log units) for the (PGA, M)model (Rathje and Saygili, 2009), and values ranging between 0.4 and0.9 for the (PGA, PGV) model (Saygili and Rathje, 2008). To illustratethese differences, themedian and1lnD displacements for the (PGA,M) and (PGA, PGV)models are shown in Fig. 3 for theM=7, R=2 kmscenario event with Vs30=760 m/s. At larger ky, the 1lnD range indisplacement is close to a factor of 10 for the (PGA, M)model, while atsmaller ky the 1lnD range represents a factor of about 5. For the(PGA, PGV)model, the1lnD displacement range is much smaller bycomparison, with the range representing a factor of 4.0 at larger kyand a factor of 2.5 at smaller ky. Thus, there is signicantly lessuncertainty in the displacement prediction when PGV is used in thedisplacement calculation.

One limitation of the SR08/RS09 empirical models is that they onlyrepresent rigid sliding block conditions, yet exible sliding blockconditions are very common. It is proposed to use a framework similarto SR08/RS09 for exible sliding conditions, but application of thisframework to exible sliding conditions rst requires appropriatequantication of the seismic loading.

3. Seismic loading parameters for rigid and exible sliding masses

First consider the seismic loading parameters for rigid sliding blocks.The SR08/RS09models use (PGA,M) and (PGA, PGV) to characterize theseismic loading for these systems. The recorded acceleration-timehistory from the GIL067 station during the 1989 Loma Prieta (M=6.9)earthquake is shown in Fig. 4(a) (PGA=0.36 g), and the velocity-timehistory derived from numerical integration of the acceleration-timehistory is shown in Fig. 4(b) (PGV=29 cm/s). For a rigid sliding masssubjected to the GIL067 motion, the acceleration-time history repre-sents the seismic loading and the characteristics of the both theacceleration- and velocity-time histories will inuence the level ofinduced displacement.

The seismic loading for a exible sliding mass subjected to theGIL067motion is not the acceleration-time history due to the dynamicresponse of the sliding mass. Rather, the seismic loading is the k-timehistory (e.g., Seed and Martin, 1966; Bray and Rathje, 1998), whichrepresents the average acceleration within the sliding mass as well asthe shear force at the base of the sliding mass. Consider the dynamicresponse of a 30-m thick sliding mass (H=30 m) with a shear wavevelocity of 250 m/s (Vs=250 m/s) and associated site period of 0.5 s(Ts=4 H/Vs=0.5 s). The k-time history for this site, computed usingone-dimensional, equivalent-linear site response analysis, is shown inFig. 4(c). The k-time history displaysmuch less high frequencymotionthan the acceleration-time history due to the averaging of accelera-tions within the sliding mass. Additionally, its peak value (kmax) issmaller than the input PGA (kmax=0.12 g versus PGA=0.36 g). Thek-time history and its associated kmax represent the appropriate

seismic loading for this exible sliding mass.

54 E.M. Rathje, G. Antonakos / Engineering Geology 122 (2011) 5160In the same way that an acceleration-time history can benumerically integrated to generate a velocity-time history, the k-time history can be numerically integrated to generate a velocity-timehistory of the k-time history. This velocity is called kvel, and while itdoes not represent the average velocity of motion within the slidingmass, it does provide information regarding the frequency content ofthe k-time history. The maximum value of the kvel-time history iscalled kvelmax. As expected, the kvel-time history contains less highfrequency motion than the velocity-time history. Surprisingly,however, the value of kvelmax (31 cm/s) is similar to the value ofPGV (29 cm/s). Because the integrated kvel-time history is inu-enced by both the amplitude and frequency content of the k-timehistory, the increase in long period motion in the k-time history isbalanced by the reduction in its peak such that kvelmax is similar inamplitude to PGV.

Fig. 3. Median and 1lnD rigid sliding block displacements predicted by the SR08/RS09 (PGA, M) and (PGA, PGV) models for M=7, and R=2 km.To use the SR08/RS09 predictivemodels for exible slidingmasses,the appropriate seismic loading parameters must be specied. Basedon the above descriptions, kmax should be used to replace PGA in theSR08/RS09 models and kvelmax should be used to replace PGV.Earthquake magnitude does not need to be modied. Predictivemodels for kmax and kvelmax are required such that engineers do notneed to perform dynamic response analysis to estimate these seismicloading parameters. These predictive models are along the same linesas the design charts for kmax developed by Bray and Rathje (1998) andBray et al. (1998).

Predictive models for kmax and kvelmax are developed based onone-dimensional site response calculations of ve sites subjected to80 input motions using the equivalent-linear site response code Strata(Kottke and Rathje, 2008). The sites consist of one 15-m prole(Vs=400 m/s), two 30-m proles (Vs=400 m/s and 250 m/s) andtwo 100-m proles (Vs=400 m/s and 265 m/s). The resulting valuesof site period (Ts) are 0.15 s, 0.30 s, 0.48 s, 1.0 s, and 1.5 s. Thenonlinear soil properties are modeled with the curves of Darendeliand Stokoe (2001) using PI=0 and appropriate values of conningpressure based on the thicknesses of the proles. The 80 inputmotions represent motions from M=6 to 7.9 earthquakes recorded adistances between 0.1 and 60 km with Vs30=200 to 1000 m/s.However, most of the Vs30 values are between 400 and 800 m/s. Theinput PGA values range from 0.02 to 1.0 g, and the input PGV valuesrange from 1.2 cm/s to 70 cm/s. k-time histories were computed atthe base of each one-dimensional site prole, from which kmax andkvelmax values were derived. Further details about the analysesperformed can be found in Antonakos (2009).

The computed kmax values are plotted versus input PGA in Fig. 5(a)for the 400 analyses performed. There is trend of increasing kmax withincreasing PGA, although at a decreasing rate and withmore scatter atlarger values of PGA. Bray and Rathje (1998) investigated the ratio ofkmax to PGA and showed that the period ratio (Ts/Tm) has a stronginuence on this value. kmax/PGA is plotted versus Ts/Tm in Fig. 5(b),and several important observations can be made. First, kmax/PGAapproaches 1.0 as Ts/Tm approaches 0.1. This trend is consistent withkmax=PGA for rigid sliding masses, and indicates that Ts/Tm=0.1essentially represents rigid sliding conditions. Next, kmax is greaterthan PGA at moderate period ratios (Ts/Tm=0.1 to 0.7), while kmax isless than PGA at larger period ratios. The data in Fig. 5(b) are shownfor different ranges of input PGA. These data indicate that the ratio ofkmax/PGA decreases with increasing input PGA.

A predictive equation for kmax/PGA is developed to model thesetrends. This model assumes a log-normal distribution for kmax/PGA,and predicts ln(kmax/PGA) as a function of ln[Ts/Tm] and PGA.

ln kmax = PGA = 0:4590:702PGA ln Ts = Tm = 0:1 f g+ 0:228 + 0:076PGA ln Ts =Tm =0:1 f g2

for Ts = Tm0:1

ln kmax = PGA = 0 for Ts = Tmb 0:1 1

The standard deviation for this model in natural log units is 0.25.Given the predicted value of kmax/PGA and the inputmotion PGA, kmaxcan be estimated.

Fig. 6 presents the model predictions of kmax/PGA as a function ofinput PGA and Ts/Tm. Generally, kmax/PGA is greater than 1.0 atsmaller values of Ts/Tm, and then falls below 1.0 at larger period ratios.The range of Ts/Tm values that predict kmax greater than PGA (i.e.,kmax/PGAN1.0) decreases with increasing PGA, and at large inputintensities kmax is less than PGA at all period ratios. All curves predictkmax/PGA=1.0 for Ts/Tm0.1, i.e., rigid sliding conditions.

Bray and Rathje (1998) developed a predictive model for kmax that

uses a power law relationship to predict a normalized kmax (kmax/

[NRFPGA]) as a function of Ts/Tm. The power law relationship resultsin a log-linear relationship between kmax and Ts/Tm for a constantvalue of PGA. The PGA normalization effectively scales kmax linearlywith PGA, although the nonlinear response factor (NRF) recom-mended by Bray and Rathje (1998) takes into account some nonlinearscaling. The NRF is a parameter that decreases with increasing input

Fig. 4. (a) Acceleration and (b) velocity-time histories for a rigid sliding block. (c) k-time history and (d) kvel-time history for a exible sliding mass with Ts=0.5 s.

55E.M. Rathje, G. Antonakos / Engineering Geology 122 (2011) 5160Fig. 5. (a) Variation of kmax with PGA, and (b) kmax/PGA verus Ts/Tm.PGA, such that for a given Ts/Tm kmax/PGA decreases with increasinginput PGA. Bray and Rathje (1998) state that their model isappropriate for Ts/Tm greater than 0.5. The predictive model fromEq. (1) is compared to the predictions from Bray and Rathje(1998) inFig. 7 for input PGA values of 0.2 and 0.8 g. For PGA=0.2 g, the Brayand Rathje (1998) predictions agree favorably with Eq. (1) in theperiod range of 0.5 to 2.0 where the log-linear shape is most valid. Atlarger period ratios the Bray and Rathje (1998) model predicts largervalues of kmax because the log-linear shape cannot represent thenonlinear relationship. The second-order polynomial used in Eq. (1)more accurately models the variation of kmax/PGA over a wide range ofperiod ratios. For PGA=0.8 g, the Bray and Rathje (1998) model isconsistently larger than Eq. (1), although the difference is mostpronounced at large period ratios. This difference indicates that the NRFfactor incorporated inBray andRathje (1998)doesnotmodel asmuch soilnonlinearity as the model developed in this study.

The additional information required to use the SR08/RS09predictive models is k-velmax. Fig. 8(a) shows the computed valuesof kvelmax versus PGV. Based on the example shown in Fig. 4, weshould not expect signicant differences in kvelmax and PGV. Asignicant amount of the data in Fig. 8(a) centers about a 1:1 line, butthere are some considerably smaller values. To further explore thisvariability, the ratio of kvelmax to PGV was computed for each datapoint and plotted versus Ts/Tm (Fig. 8(b)) for different ranges on input

PGA. The kvelmax/PGV data display similar trends to the kmax/PGA

Fig. 6. kmax/PGA model predictions from Eq. (1).

data. The data indicate kvelmax equal to PGV at very small periodratios (Ts/Tm0.2 in this case), kvelmax greater than PGV at smallerperiod ratios (Ts/Tm=0.2 to 1.5) and kvelmax less than PGV at largerperiod ratios. The range of period ratios where kvelmax/PGVN1.0 islarger than the range of period ratios where kmax/PGAN1.0. Again,there is an input amplitude effect, with smaller values of kvelmax/PGV observed at larger values of input PGA. However, this inputamplitude effect is not pronounced at smaller period ratios.

A predictive model for kvelmax/PGV was developed with a similarfunctional form to Eq. (1). Because the input PGA effect for kvelmax/

PGV is not signicant at small period ratios, only the coefcient for thesecond-order term is a function of PGA. The predictive model for kvelmax/PGV is given by:

ln kvelmax = PGV = 0:240 ln Ts = Tm = 0:2 f g+ 0:0910:171PGA ln Ts =Tm =0:2 f g2

for Ts=Tm0:2

ln kvelmax = PGV =0 for Ts = Tmb0:2 2

The standard deviation for this model in natural log units is 0.25.Fig. 9 presents themodel predictions of kvelmax/PGV as a function

of input PGA and Ts/Tm. At period ratios less than 0.3 the predictedvalues of kvelmax/PGV are similar for all input intensities. At largerperiod ratios, kvelmax/PGV is smaller for larger input intensities. Themodel predicts kvelmax/PGV=1.0 (i.e., rigid sliding conditions) for

Fig. 7. Comparisons of kmax predictions from Eq. (1) and from Bray and Rathje (1998).

56 E.M. Rathje, G. Antonakos / Engineering Geology 122 (2011) 5160Fig. 8. (a) Variation of kvelmax with PGV, and (b) kvelmax/PGV versus Ts/Tm.Ts/Tm0.2.

4. Displacement predictions for rigid and exible sliding masses

The objective of this study is to modify the SR08/RS09 rigid blockempirical models such that they can be used to predict the decoupleddisplacements of rigid and exible sliding systems. The initialhypothesis is that the original SR08/RS09 empirical models can beused, but with PGA replaced by kmax and PGV replaced by kvelmax. Totest this hypothesis, decoupled sliding displacements were calculatedusing the computed k-time histories for the ve sites and 80 inputmotions (400 time histories). Displacements were calculated forky=0.04, 0.08, 0.12, and 0.16. The resulting dataset included 569 non-zero values of displacement (i.e., instances where kybkmax). Thesevalues of displacement were compared with the median valuespredicted by the SR08/RS09 empirical models given the computedvalues of kmax and kvelmax for each calculated k-time history.Additionally, rigid sliding block displacements were computed for the80 input time histories and the four values of ky for comparison withthe median values predicted by the SR08/RS09 empirical models.

The residuals (i.e., ln(data)ln(predicted)) of the computed valuesof D (i.e., data) with respect to the empirically predicted values of Dwere calculated for both the (PGA, M) model and the (PGA, PGV)model. For both models, the average residuals over the completedataset are greater than 0.0, with an average of 0.24 for the (PGA, M)model and an average of 0.42 for the (PGA, PGV)model. These positivevalues indicate that the computed values of D for these exible slidingmasses are larger, on average, than the values predicted by the SR08/RS09 empirical models. The difference is caused by the fact that thefrequency content of a k-time history is signicantly different than forFig. 9. kvelmax/PGV model predictions from Eq. (2).

an acceleration-time history (Fig. 4), which results in larger displace-ments. While kvelmax attempts to take into account this difference infrequency content, the time histories in Fig. 4 demonstrate that PGVand kvelmax do not vary signicantly from one another although thek-time histories display signicantly different frequency contents.Thus, the original SR08/RS09 empirical models require an additionalmodication to capture this effect.

The residuals for the rigid sliding block displacement wereinvestigated to evaluate how the selected ground motion dataset maybe inuencing the results. The average residuals for rigid sliding blockconditions should be equal to 0.0, because the SR08/RS09 modelrepresents rigid sliding conditions. For the (PGA, M) model the averageresidual for rigid sliding (Ts=0.0 s) was0.8, which signies that, onaverage, the computed values of D from the 80 motion dataset aresmaller than those predicted by SR08/RS09. The computed values of Dare smaller than predicted by SR08/RS09 because the average Vs30 forthemotionsused in this study (Vs30~550 m/s) is larger than the averagefor those used in the SR08/RS09 studies (Vs30~400 m/s). Motions fromsites with larger Vs30 display less long period energy, which results insmaller displacements. For the (PGA, PGV) model, the average residualfor rigid sliding conditions was essentially zero. The Vs30 effect is notapparent for this model because the inclusion of PGV takes into accountthe different frequency contents for rock and soil motions.

The residuals for the exible sliding masses were investigatedtogether with the residuals for rigid slidingmasses to identify the site/ground motion parameters that inuence the difference between thecomputed and predicted displacements. Fig. 10 plots the residualsversus site period for the two displacement models. These data

57E.M. Rathje, G. Antonakos / Engineering Geology 122 (2011) 5160Fig. 10. (a) Displacement residuals versus Ts for the SR08/RS09 (PGA, M) model and

(b) displacement residuals versus Ts for the SR08/RS09 (PGA, PGV) model.indicate that the residuals increase with increasing site period (Ts),but at a decreasing rate. The residuals increase with Ts because slidingmasses with larger values of Ts generate k-time histories with morelong period energy that lead to larger displacements. The scatter atany one period in Fig. 10 is larger for the (PGA, M) model than the(PGA, PGV) model, and this observation is consistent with the relativevalues of lnD reported for the two models.

For the (PGA, M) model (Fig. 10(a)), the average residual is equal to0.8at Ts=0.0 s (rigid conditions) and increases to 1.95 at Ts=1.5 s. Apositive residual of 1.95 corresponds to computed displacements that are7 times larger than those predicted by the empirical model. A secondorder polynomial was t to the average residuals, and this expression canbe used to modify the SR08/RS09 (PGA, M) rigid sliding block model forthe effects of decoupled, exible sliding. However, the residuals in Fig. 10(a) are inuenced by the fact that the ground motion dataset is not fullyconsistent with the dataset used in the SR08/RS09 studies (i.e., differentVs30) which causes the average residual to be non-zero at Ts=0.0 s.Therefore, the recommended modication involves translating the curveshown in Fig. 10(a) such that the average residual is equal to zero atTs=0.0 s. The resultingmodication to the SR08/RS09 (PGA,M)model toaccount for exible sliding is:

ln Dflexible = ln DPGA;M

+ 3:69Ts1:22 Ts 2 for Ts1:5

ln Dflexible = ln DPGA;M

+ 2:78 for Ts N 1:5 3

where DPGA,M represents the median displacement predicted by the(PGA, M) SR08/RS09 rigid sliding block model and Ts is the naturalperiod of the slidingmass. For the calculation of DPGA,M, kmax is used inlieu of PGA.

For the (PGA, PGV) model (Fig. 10(b)), the average residual is zeroat Ts=0.0 s and increases to a value as large as 0.70. However, theaverage residuals become relatively constant at periods greater than0.5 s. A linear relationship was t through the average residuals atTs0.5 s, with no further increase modeled at larger periods. Theresulting modication to the SR08/RS09 (PGA, PGV) model to accountfor exible sliding is:

ln Dflexible = ln DPGA;PGV

+ 1:42Ts for Ts 0:5

ln Dflexible = ln DPGA;PGV

+ 0:71 for Ts N 0:5 4

where DPGA,PGV represents the median displacement predicted by the(PGA, PGV) SR08/RS09 rigid sliding block model and Ts is the naturalperiod of the sliding mass. For the calculation of DPGA,PGV, kmax is usedin lieu of PGA and kvelmax is used in lieu of PGV.

After correcting the biases observed in the residuals shown inFig. 10, the standard deviation of lnD (lnD) was computed from thecorrected residuals. Considering that the SR08/RS09 models display avariation of lnD with ky/PGA, the models from this study shoulddisplay a variation of lnD with ky/kmax. The computed values of lnDare plotted versus ky/kmax in Fig. 11 for the (PGA, M) and (PGA, PGV)models. The lnD values for the (PGA, M) model follow a linear trend(Fig. 11(a)), and are about 10% smaller than the lnD values from theSR08/RS09 (PGA, M) model. The reduction in standard deviation forexible sliding masses is expected because the dynamic responsecalculation lters out any high frequency peaks that contribute to thevariability in rigid block displacements. The recommended lnDrelationship for the (PGA, M) model for exible sliding masses isgiven by:

lnD = 0:694 + 0:322ky = kmax for PGA;M model 5

58 E.M. Rathje, G. Antonakos / Engineering Geology 122 (2011) 5160The lnD values for the (PGA, PGV) model (Fig. 11(b)) are alsosmaller than those from SR08/RS09 (0 to 25% smaller), particularly atlarge values of ky/kmax. A revised linear relationship is used to predictlnD for exible sliding masses for the (PGA, PGV) model:

lnD = 0:40 + 0:284ky = kmax for PGA;PGV model 6

5. Example applications

To illustrate the application of the unied model for predicting thedynamic response and sliding displacement of slopes, consider thefollowing example. The critical sliding mass for a slope is 20-m thickwith an average Vs=400 m/s and resulting Ts=0.2 s. The ky is equalto 0.1. The design event is M=8 and R=2 km, with the input rockmotions described by Table 1 (PGA=0.48 g, PGV=74 cm/s) andwithTm=0.46 s (Rathje et al., 2004). Based on the site and ground motioncharacterizations, Ts/Tm=0.43.

Eqs. (1) and (2) are used to predict kmax and kvelmax based onthe PGA (0.48 g), PGV (74 cm/s), and Ts/Tm (0.43). Using these values,Eq. (1) predicts kmax/PGA=0.79, while Eq. (2) predicts kvelmax/PGV=1.08. Thus, the seismic loading for this sliding mass is predictedas: kmax= 0.38 g (=0.790.48 g) and kvelmax= 80 cm/s(=1.0874 cm/s).

Using the seismic loading parameters of kmax=0.38 g and M=8along with ky=0.1, the SR08/RS09 (PGA, M) model predicts 63.1 cmwhen kmax is used in place of PGA. This valuemust be adjusted using themodication for exible sliding given in Eq. (3). For Ts=0.2 s, this

Fig. 11. (a) Standard deviation of lnD for exible sliding masses using revised (PGA, M)model, (b) standard deviation of lnD for exible sliding masses using revised (PGA,PGV) model.expression predicts a median displacement value of 126 cm for exiblesliding conditions. For the SR08/RS09 (PGA, PGV)model, the appropriateseismic loading parameters are kmax=0.38 g and kvelmax=80 cm/s.Using kmax in lieu of PGA and kvelmax in lieu of PGV in the SR08/RS09(PGA, PGV) model generates a displacement of 36.9 cm. Adjusting thisvalue for exible sliding and Ts=0.2 s (Eq. (4)), the median predictedvalue of displacement is 49 cm for exible sliding conditions. Note thatthis value is less thanhalf thevaluepredictedby the SR08/RS09(PGA,M)model.

It is interesting to note the signicant difference between thedisplacements predicted by the (PGA, M) and (PGA, PGV) models.Depending on the seismic conditions and slope parameters, the (PGA,M)model may predict displacements as much as 2.5 times larger thanthe (PGA, PGV) model. The larger displacements for the (PGA, M)model are a result of two issues: the neglected Vs30 effect (Fig. 2) andthe proposed modication in Eq. (3). As noted previously, Vs30inuences the frequency content of shaking and leads to largerdisplacements. Because the dataset used to develop the (PGA, M)model included both rock and soil motions and because the (PGA, M)model does not take into account Vs30, this model tends to predictlarger displacements. The modication in Eq. (3) was generated bytranslating up the residuals in Fig. 10(a) so that the mean residualwould be zero for rigid sliding conditions. This signicant translationleads to even larger displacements. There is uncertainty in thedecision to translate the entire curve in Fig. 10(a) based on theresiduals at Ts=0.0 s, and thus there is less condence in the (PGA,M) model for exible sliding. Therefore, the (PGA, PGV) model isrecommended over the (PGA, M) model for use in practice.

To illustrate the dynamic and displacement responses of slopesunder rigid through exible conditions using the developed frame-work, consider sliding masses with site periods (Ts) ranging from 0.0to 1.0 s. Fig. 12 shows both the dynamic responses (i.e., kmax and kvelmax) and sliding displacements for these sliding masses subjectedto motions fromM=6, 7, and 8 earthquakes at a distance of 2 km. Forthese earthquake scenarios, themedian values of PGA, PGV, and Tm forVs30=760 m/s are given in Table 1. Fig. 12(a) demonstrates that kmaxfor a exible system is generally smaller than kmax for a rigid system(Ts=0.0), except for sliding masses with very small site periods (Tsless than about 0.1 s). The reduction in kmax with increasing Ts issignicant, with kmax decreasing by 80% at large periods and largeinput intensities. Alternatively, the reduction in kvelmax with siteperiod is not as dramatic. Flexible systems with periods of up to 0.4 sdisplay larger values of kvelmax than rigid systems (Fig. 12(b)), andat larger periods kvelmax is never more than 35% smaller than therigid value. At larger periods kvelmax gets smaller, but the rate ofreduction is much smaller than for kmax.

Fig. 12(c) and (d) shows the variation of displacement as afunction of Ts for ky=0.05 and 0.1 for the revised (PGA, PGV) model.At shorter periods (Tsb0.3 s for ky=0.05, Tsb0.15 s for ky=0.1), theexible systems displace more than the rigid systems due to theenhanced amplitudes and increase in long period content of theseismic loading induced by the dynamic response. At longer periods,the displacements for exible systems are smaller than for rigidsystems due to the nonlinear response of the soil and the reductions inthe amplitude of the seismic loading (i.e., kmax). The period range overwhich exible systems displace more than rigid systems depends onky, and generally this range is larger for smaller values of ky.

6. Conclusions

Evaluating the seismic performance of slopes involves predictingsliding block displacements for critical sliding masses. Currentpractice typically uses a rigid sliding block approach for shallowsliding masses and a decoupled, exible sliding block approach fordeeper/softer sliding masses. Empirical predictive models are avail-

able to predict the sliding displacements of rigid sliding masses and

59E.M. Rathje, G. Antonakos / Engineering Geology 122 (2011) 5160exible sliding masses, but these models do not adequately model thetransition from rigid to exible behavior.

This paper presents a unied empirical model to predict the slidingdisplacements of rigid and exible sliding masses. The unied modelis an extension of the empirical models for rigid sliding massesdeveloped by Saygili and Rathje (2008) and Rathje and Saygili (2009).The main advancements contributed by the SR08/RS09 modelsinclude: (1) the use of a large ground motion dataset, (2) the additionof a frequency content parameter (PGV) to better predict displace-ments, and (3) a better description of the standard deviationassociated with each model.

The unied approach involves rst predicting the seismic loadingparameters for a potential sliding mass, and then using these seismic

Fig. 12. (a) Predicted values of kmax as a function of Ts, (b) predicted values of kvelmax asky=0.05 for the revised (PGA, PGV)model developed in this study, and (d) predicted valuesdeveloped in this study.loading parameters to predict sliding displacement. The seismicloading parameters are given by kmax and kvelmax, dened as themaximum value of the k-time history and the maximum velocity ofthe k-time history, respectively. A predictive model for kmax wasdeveloped as a function of PGA and Ts/Tm, and a predictive model forkvelmax was developed as a function of PGV, PGA, and Ts/Tm. Topredict sliding displacement, kmax is used in lieu of PGA and kvelmaxis used in lieu of PGV in the SR08/RS09 models.

In addition to the change in seismic loading parameters, the SR08/RS09 models must be further modied to account for the differencesin frequency characteristics between acceleration-time histories andk-time histories. This modication is a function of Ts and increases thepredicted displacement. Modication for both the (PGA, M) and (PGA,

a function of Ts, (c) predicted values of sliding displacement as a function of Ts withof sliding displacement as a function of Ts with ky=0.1 for the revised (PGA, PGV)model

PGV) models are developed, but the (PGA, PGV) model is recom-mended because of the signicant frequency content informationprovided by PGV (for rigid sliding) and by k-velmax (for exiblesliding).

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A unified model for predicting earthquake-induced sliding displacements of rigid and flexible slopesIntroductionRigid sliding block displacementsSeismic loading parameters for rigid and flexible sliding massesDisplacement predictions for rigid and flexible sliding massesExample applicationsConclusionsReferences