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Al-Defae, A. H. et al. (2013). Geotechnique 63, No. 14, 12301244 [http://dx.doi.org/10.1680/geot.12.P.149]

Aftershocks and the whole-life seismic performance of granular slopes

A . H . A L - D E F A E , K . C A UC I S a n d J . A . K N A P P E T T

Shallow embankment slopes are commonly used to support elements of transport infrastructure inseismic regions. In this paper, the seismic performance of such slopes in non-liquefiable granular soilsis considered, focusing on permanent movement and dynamic motion at the crest, which would formkey inputs into the aseismic design of supported infrastructure. In contrast to previous studies, theevolution of this behaviour under multiple sequential strong ground motions is studied throughdynamic centrifuge, numerical (finite-element, FE) and analytical (sliding-block) modelling, thecentrifuge tests being used to validate the two non-physical approaches. The FE models focus on thespecification of model parameters for existing non-linear constitutive models using routine siteinvestigation data, allowing them to be used routinely in design and analysis. Soil-specific constitutive

parameters are derived from shearbox and oedometer test data, and are found to significantlyoutperform existing empirical correlations based on relative density, highlighting the importance ofspecifying a suitably detailed site investigation. An improved sliding-block (Newmark) approach isalso developed for estimating permanent deformations during preliminary design, in which theformulation of the yield acceleration is fully strain-dependent, incorporating both material hardening/

softening and geometric hardening (re-grading). The site-specific (improved) FE models and the newsliding-block approach are shown to outperform considerably existing FE parameters and sliding-blockmodels in capturing the permanent deformations of the slope under virgin conditions, and further,only the improved FE and sliding-block models are found to capture correctly the behaviour of thedamaged slope under subsequent earthquakes (e.g. strong aftershocks). The FE models canadditionally accurately replicate the settlement profile at the crest and quantify the dynamic motionsthat would be input to supported structures, although these were generally overpredicted. The FE

procedures and sliding-block models are therefore complementary, the latter being useful forpreliminary design and the former for later detailed design and analysis.

KEYWORDS: centrifuge modelling; earthquakes; embankments; numerical modelling; sands; slopes

INTRODUCTIONTransport infrastructure is a vital lifeline that must be safe-guarded in seismic regions. Shallow embankment slopes andcuttings will commonly be used along the length of a roador railway line to allow changes in gradient, and damage tothis type of infrastructure could inhibit the movement ofemergency services and rebuilding in the aftermath of anearthquake. Although the infrastructure generally supportedon such constructions is relatively light (low bearing pres-sure) and flexible, significant damage can be caused by largepermanent seismic slip within the slope.

A popular method of analysis for predicting seismic slip isthe Newmark sliding-block technique (Newmark, 1965). In

this method a yield acceleration is calculated under pseudo-static conditions using limit analysis techniques for simplecases (e.g. Kim & Sitar, 2004), or numerical methods suchas finite-element limit analysis (FELA) for more complexcases (e.g. Loukidis et al., 2003). A strong ground motion(either a historically recorded motion or a syntheticallyproduced accelerogram) is then used to determine the netdownslope acceleration, which can then be integrated nu-merically to obtain slip velocity and slip displacement (Jib-son, 1993). Since it was originally proposed in the 1960s,this method has been substantially improved to incorporate

material strength characteristics, which may be either strain-hardening or strain-softening (Matasovic et al., 1997; Wart-man et al., 2005), and to incorporate the dynamic behaviour(e.g. ground motion amplification) within the sliding massitself in the case of deep rotational slips, as may be commonin landfill slopes (Rathje & Bray, 1999; Wartman et al.,2003). Experimental work to validate such methods has beenconducted using 1g shaking tables on cohesive soils (Wart-man et al., 2005); however, there remains a need to demon-strate the approachs validity at realistic stress levels (e.g. byconducting centrifuge modelling). Although such analyticaltools are useful for parametric/comparative study, particularlyin preliminary design, they are not able to predict the

dynamic ground motions or settlement distribution (angulardistortion) behind the slope crest, both of which may have acontrolling influence on the design of any infrastructuresupported at the top of the slope. They are therefore not wellsuited to detailed analysis or design.

Recent periods of seismic activity in Japan and NewZealand (20112012) have demonstrated that civil engineer-ing infrastructure may be subjected to several successivestrong ground motions within a short period (short heremeaning that there has been insufficient time to completeremediation or reinstatement of damage caused by earlierground shaking). This means that there is a further need tounderstand the behaviour of seismically damaged infrastruc-ture under successive periods of ground shaking. This fea-

ture is not currently incorporated in existing sliding-blockmethods.

This paper will consider the behaviour of shallow cohe-sionless slopes under a sequence of strong earthquake

Manuscript received 5 October 2012; revised manuscript accepted 7June 2013. Published online ahead of print 15 July 2013.

Discussion on this paper closes on 1 April 2014, for further details seep. ii.University of Dundee, UK, and Wasit University, Iraq.yUniversity of Dundee, Dundee, UK.


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ground motions, and develop improved analysis tools for theprediction of such behaviour. These tools will be applicableto the analysis of seismically damaged slopes under strongaftershocks, and to determination of the whole-life seismicperformance of slopes (i.e. under a range of successivemotions of different strengths that the slope may see duringits design life).

Fully dynamic numerical modelling in the time domain,

using the finite-element method (FEM), will first be appliedto the problem, with the aim of producing a single analysisin the time domain that captures both the dynamic vibrationeffects and the permanent slope deformations. Existing non-linear soil constitutive models will be used that encapsulatethe strain history (seismic memory) of the soil. Emphasiswill be placed on the efficient parameterisation of such amodel: (a) using previously published correlations based ondatabases of element test data (which require only relativedensity as an input); and (b) through the use of routinelaboratory test data to produce improved soil-specific cali-brations. The ability to model soil response realistically,without requiring an excessive number of empirical param-eters derived from non-standard tests, will allow the vali-dated FEM procedures developed to be used with confidencein routine geotechnical practice.

An improved sliding-block method is also developed thatcan fully capture the strain history of the soil, particularlyby incorporating the effects of slope deformation (re-grad-ing) during slip within the formulation of the yield accelera-tion. This allows the slip of a seismically damaged(deformed) slope to be predicted analytically under subse-quent strong ground motion.

Both the FEM and sliding-block models are validatedagainst centrifuge test data. The computationally simplersliding-block model, which requires no specialist software,can then be used to undertake rapid parametric analyses

(such as may be necessary during the early stages of design,or in the immediate aftermath of an earthquake) or usedwithin a probabilistic, performance-based earthquake engin-eering framework if desired (Kramer, 2008). The FEM ap-proach can subsequently be used to undertake a moredetailed analysis of specific cases, which can incorporate thedynamic behaviour of any infrastructure located at the slopecrest.

CENTRIFUGE MODELLINGDynamic centrifuge testing was conducted using the 3.5 m

diameter beam centrifuge and servo-hydraulic earthquakesimulator at the University of Dundee. Two models wereflown, representing identical slopes of 0 288 (1:2) at1:50 scale in dry sand at 50g. This slope angle was selectedto ensure that the soil was statically stable (0 , 9cs, thecritical state friction angle of the soil), but with a suffi-ciently low factor of safety (and therefore low yield accel-eration, khy) to ensure that large slip displacements would begenerated during strong ground motion, such that the FEMand sliding-block models could be validated to large dis-placements. All subsequent dimensions and properties aregiven at prototype scale at 50g, unless otherwise stated.

The arrangement and instrumentation of the slope modelsis shown in Fig. 1(a). The slopes were prepared at a relativedensity of ID 5560% (the range accounts for the accu-racy in being able to measure and replicate ID), were 8 mtall from toe to crest, and were underlain by a further 6 mof sand at the same relative density. HST95 (Congleton)silica sand was used, which has the basic properties given inTable 1 (after Lauder, 2011). This uniformly rounded sand isvery fine, and has been used by other researchers at Dundeeto study other seismic phenomena (e.g Bertalot & Brennan,2012). The values of emax and emin reported in Table 1 were

14(280)

LVDTs

6(120)

8(160)

(a)

Accelerometers

1

2

3

4

5

9

7

8

6

10

15

14

11

12

13

Size of model in centrifuge Fig. 1(a)

Positive acceleration(downslope)

Absorbent boundary(Lysmer & Kuhlmeyer, 1969)

(b)

11 (220) 15 (300) 75 (150)

Fig. 1. Slope configuration: (a) centrifuge model layout and location of virtualinstruments in numerical models; (b) finite-element mesh, showing boundary conditions

AFTERSHOCKS AND THE WHOLE-LIFE SEISMIC PERFORMANCE OF GRANULAR SLOPES 1231


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determined in accordance with BS1377 Part 4 (BSI, 1990).The sand was pluviated in air using a slot pluviator into anequivalent shear beam (ESB) container having flexible walls,the construction of which is described in Bertalot (2012).This container was used to reduce potential boundary effectsdue to shear-wave reflection at the container walls. Themodel was instrumented within the soil by 15 type ADXL78MEMS accelerometers (70g range) manufactured by Ana-log Devices, and four external linear variable differentialtransformers (LVDTs), with a pair of instruments measuringsettlement at and behind the crest of the slope along thecentreline of the model, and a further pair placed adjacentto one of the side walls to measure any boundary effects(this latter pair will not be discussed further in this paper).All settlements shown subsequently in this paper are takenfrom the rightmost instrument in Fig. 1(a).

Ground motions were applied to the models using theActidyn QS67-2 servo-hydraulic earthquake simulator re-cently installed at the University of Dundee. The perform-ance of this actuator is described in Bertalot et al. (2012).Earthquake ground motions were downloaded from thePEER (Pacific Earthquake Engineering Research) NGA data-base. One of the models (test AA01) was subjected to a

horizontal strong ground motion recorded during theMw 7.6 Chi-Chi earthquake in 1999 (Station TCU072,PGA 0.41g), and a second (test AA02) was subjected to ahorizontal motion recorded at the Nishi-Akashi recordingstation in the Mw 6.9 Kobe earthquake in 1995(PGA 0.43g). These motions had approximately the samepeak acceleration, but different characteristics in the timeand frequency domains, as shown in Figs 2(a) and 2(b) andFigs 3(a) and 3(b). Both records were recorded in groundwith Vs . 450 m/s, representing shaking from stiffer layersbeneath the soil profile tested, such that any site amplifica-tion occurs solely from the soil layer tested in the model.The former motion was used, as the Chi-Chi earthquakecaused a particularly high number of slope failures (Khazai

& Sitar, 2004), and is strongly directional (in these tests, the

stronger shaking was directed in the downslope direction,which is represented by positive values of acceleration). Thelatter motion is well known to be particularly destructive tocivil engineering infrastructure having a broad frequencyband below 3 Hz. The demand motions were bandpass-filtered between 0.8 Hz and 8 Hz (40400 Hz at modelscale) using a zero-phase-shift digital filter to remove com-ponents of the signal that were outside the range that can beaccurately controlled by the earthquake simulator. In bothtests, four nominally identical earthquake motions wereapplied to the model in succession, to investigate the behav-iour of the slope under strong aftershocks following initialstrong shaking causing substantial slip.

Details of the model tests are summarised in Table 2. Allground motions were initially calibrated on a dummy modelidentical to that shown in Fig. 1(a), but without instrumenta-tion, to train the programmable logic controller within theearthquake simulator to achieve a faithful and repeatablereplication of the demand motions. As a result, the ground

motions applied in each successive earthquake are felt to beas close to identical as could realistically be achieved inpractice, this being an idealisation of the successive motionshaving the same source (depth, faulting mechanism andposition).

FINITE-ELEMENT MODELLINGThe centrifuge tests were modelled using PLAXIS 2D in

plane strain with the mesh and boundary conditions shownin Fig. 1(b). Compared with the centrifuge model shown inFig. 1(a), the dimensions of the model domain were ex-tended laterally and combined with non-reflecting boundaryelements controlling the dynamic stresses along the vertical

boundaries (after Lysmer & Kuhlmeyer, 1969) to representsemi-infinite soil conditions, that is, boundary deformationsat the location of the centrifuge container wall that arecontrolled by the dynamic deformation of the adjacent soil.This boundary condition can also be modelled by horizontalnode-to-node ties between the two vertical boundaries of amodel the width of the soil tested in the centrifuge. Com-pared with this alternative, the method used has a higherelement requirement for the same mesh density, but allows

Table 1. State-independent physical properties of HST95 silicasand (after Lauder, 2011)

Property Value

Specific gravity,Gs 2.63D10: mm 0.09D30: mm 0.12D60: mm 0.17

Cu 1.9Cz 1.06Maximum void ratio,emax 0.769Minimum void ratio,emin 0.467

05

0

05

0 5 10 15 20 25 30 35 40 45 50Acceleration:g

Time: s(a)

05

0

05

0 5 10 15 20 25 30 35 40 45 50Acceleration:g

Time: s(b)

Fig. 2. Input bedrock motions in time domain: (a) Chi-Chi (testAA01); (b) Kobe (test AA02)

0

50

100

150

0 05 10 15 20 25 30 35 40 45 50Magnitude:

/Hz

g

Frequency: Hz(a)

0

1020

30

40

0 05 10 15 20 25 30 35 40 45 50Magnitud

e:

/Hz

g

Frequency: Hz(b)

Fig. 3. Input bedrock motions in frequency domain: (a) Chi-Chi(test AA01); (b) Kobe (test AA02)

Table 2. Summary of centrifuge models tested

Test ID : degrees ID: % Input motion (no.) Peak inputacceleration:g

AA01 28 56 Chi-Chi, 1999 (4) 0.41AA02 28 59 Kobe, 1995 (4) 0.43

1232 AL-DEFAE, CAUCIS AND KNAPPETT


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information to subsequently be collected for points far fromthe crest and toe of the slope (although these are notreported in this paper). A dynamic ground displacement wasapplied along the bottom edge of the finite-element (FE)model, shown by the repeating arrows in Fig. 1(b). Theinput motion applied to the model was that measured atinstrument 8 in the centrifuge model: that is, the motion thatthe slope in the centrifuge actually saw, accounting for any

losses between the shaking table and the container, andbetween the container bottom and the soil. The motions wereinput as ground displacements, determined by high-passfiltering and integration of the accelerometer records: filter-ing before integration to obtain velocity, and again beforeintegrating velocity to obtain displacement, ensures thatthere is no permanent wander due to any offset in theaccelerometer recordings or integration of random noisewithin the signal. Displacement data were extracted from theFE models at the locations of the instruments in thecentrifuge tests shown in Fig. 1(a). At points 115, accelera-tions were subsequently determined from double differentia-tion of the displacements.

CONSTITUTIVE MODELLINGAn elasto-plastic soil model with isotropic hardening

(Schanz et al., 1999) is used in the modelling presentedherein, in which the elastic behaviour incorporates strain-dependent stiffness variation following the model proposedby Hardin & Drnevich (1972), as modified by Santos &Correia (2001)

G

G0

1

10:385s=s,0:7j j (1)

where s is shear strain ands,0:7 is the shear strain at whichG/G0 is 70%. Plastic failure is modelled using a cap-typeyield surface combined with the MohrCoulomb failure

criterion. This model is included in the PLAXIS FE suiteused for the work described herein as the hardening soilmodel with small-strain stiffness (Benz, 2006). Althoughthis model captures only strain-hardening (and not strain-softening) behaviour, it will subsequently be shown that theeffects of strain-softening on slope movement are minorwhen the earthquake is strong enough to induce significantslip, as in the motions used in the centrifuge testing. Theselection of appropriate soil strength (peak or critical statefriction angles) will be discussed in greater detail later inthis section.

The model requires 13 input parameters: unit weightsunder saturated and dry conditions; three measurable effec-tive stress strength parameters, 9pk, c9 and 9 (angle of

dilation), for use in the MohrCoulomb failure criterion; sixmeasurable stiffness parameters (which are stress dependent)describing the response to deviatoric loading (E50), compres-sive loading (Eoed), unloadreload cycles (Eur, ur), small-strain stiffness (G0) and a shear strain for describing theshape of the Gs relationship (s,0:7); and finally, twoempirical parameters Rf and m, the former controlling thedeviatoric stress at failure, and the latter controlling thevariation of the stiffness parameters with effective confiningstress.

E 93 Eref c9 cos9 93sin 9

c9 cos9 prefsin 9

m(2)

Eoed 93 E

ref

oed

c9 cos9 93=K0 sin9

c9 cos9 prefsin 9

m

(3)

where E may be E50, Eur or G0; Eref is the value of the

parameter at a reference stress of pref; and K0 1 sin 9.

A reference pressure of pref100 kPa is used throughoutthe remainder of this paper. Although the model has severalinput parameters, all except Rf can be measured in someform through routine laboratory testing.

In the first set of FEM simulations that will be describedin the subsequent sections, the empirical correlations be-tween the input parameters and relative density for coarse-grained soils presented by Brinkgreve et al. (2010) were

used. Use of these parameters required no additional soiltesting, and all the constitutive parameters can be definedusing relative density, ID, alone (which was uniform withinthe centrifuge model slopes). Use of this model representedthe case in practice where no detailed laboratory test dataare available, but where relative density can be determinedfrom standard penetration tests (SPTs) or a cone penetrationtest (CPT) profile (e.g. as outlined in Knappett & Craig,2012).

A second set of simulations was then conducted in whichthe key strength and stiffness parameters were calibrated forthe soil used in the centrifuge tests using laboratory testdata, the stiffness parameters being important in modellingthe dynamic effects and shear-wave propagation, and thestrength properties controlling the permanent deformations.This model is henceforth termed HST95. A large amountof published shearbox test data was available from previousstudies that used the HST95 sand (Lauder, 2011; Bransby etal., 2011); this was used herein to derive soil-specificstrength parameters, supplemented with oedometer tests todetermine soil-specific stiffness parameters. In a practicalcase it would be necessary only to undertake tests appro-priate for the in situ soil states (e.g. density in this case);however, in this paper a range of test data were collated/obtained for reconstituted samples covering a wide range ofrelative densities such that a more complete model could bedeveloped for use in future simulations of physical testresults that use this material.

Strength parametersFigure 4 shows a summary of shearbox data from a total

of 38 tests conducted over a range of confining normaleffective stresses between 5 kPa and 200 kPa, as summarisedin Table 3. Fig. 4(a) shows shear stress measurements atcritical state (when volumetric change had stopped), indi-cating that the critical state friction angle is 9cs 328. Fig.4(b) shows the secant peak friction angles measured over thestress ranges considered (stress-independent values weredetermined by straight-line fits to the peak strength data, as9pk is stress-independent within the constitutive model). Astraight-line fit to the data as a function of relative densityappeared to be appropriate; focusing on the data points

below ID 80% gave

9pk 20ID29 (degrees) (4)

Dilation angles are also shown in Fig. 4(b); the straight-line fit for this data was found to be

9 25ID 4 (degrees) (5)

These simple linear fits to 9pk and9 satisfy the dilatancyrelationship given by Bolton (1986), and predict the value of9cs 328 with an error of , 1% using either this relation-ship, or that by Rowe (1962), which is incorporated in theconstitutive model formulation.

Whereas the strength properties given by equations (4)and (5) match the element test data well, their use within a

strain-hardening model would imply that the peak strength isappropriate to the analysis. It will subsequently be demon-strated that, for large-strain slope problems, the permanentdeformations are governed by the critical state strength.



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Hence, in modelling the centrifuge tests, 9pk 9cs 328and9 08 were used.

Stiffness parametersOedometer tests were conducted on dry samples of sand

prepared by air pluviation within a Clockhouse EngineeringLtd J550 oedometer (sample height 19 mm; sample diam-eter76 mm) at a range of relative densities, ID, between5% and 83%. The tests were conducted up to an effectivestress of 600 kPa, and included three unloadreload cycles(200 100200 kPa; 400200400 kPa; 600400 600 kPa)to ensure that Eur was well calibrated (this parameter islikely to be important during cyclic seismic loading). Theconstitutive model parameters were then determined by con-

ducting virtual oedometer tests using FEM, starting with theparameters suggested by Brinkgreve et al. (2010), andmodifying them as necessary to achieve a reasonable fit tothe data. Fig. 5 shows the axisymmetric model geometryand boundary conditions employed. To develop a completedensity-dependent model in this paper, calibrated parameters

were determined for loose and dense samples, from whichlinear interpolations were made as a function of relativedensity, as presented by Brinkgreve et al. (2010). Theinterpolated parameters were subsequently checked againstfurther tests at intermediate densities. Fig. 6(a) shows acomparison of some results for the Brinkgreve et al. (2010)model, which overpredicts stiffness in dense sand and dra-matically underpredicts stiffness in loose sand. Fig. 6(b)shows the markedly improved results using fitted values ofEoed, E50, Eur and m. To reduce the number of independentparameters, E50 1.25Eoed and Eur 3Eoed were assumed,

0

20

40

60

80

100

120

140

0 50 100 150 200 250

Shearstressatcriticalstate,

:kPa

Normal effective stress, : kPa(a)

n

Lauder (2011)

Bransby (2012)et al.

This paper

cs 32

10

0

10

20

30

40

50

60

70

0 20 40 60 80 100Peakfrictionangle,

;dila

tionangle,

:degrees

pk

Relative density, : %(b)

ID

Lauder (2011)

Bransby (2012)et al.

This paper

Brinkgreve (2010)et al.

Fig. 4. DSA (shearbox) test data used in soil-specific calibration ofconstitutive model: (a) 9cs; (b) 9pk and 9

Table 3. DSA test data for HST95 silica sand

Source ID: % No. of tests Effective normal stresses: kPa

Lauder (2011) 17 8 5, 8, 11, 16, 30, 35, 70, 12540 6 16, 30, 55, 100, 135, 15075 4 11, 16, 30, 70

Bransbyet al.(2011) 9 5 10, 25, 50, 100, 20041 5 15, 35, 55, 100, 15093 5 10, 25, 50, 100, 180

This paper (see Fig. 17) 55 5 5, 8, 13, 16, 25

CL Applied pressure

h 19 mm

Radius 38 mm ( 2 ) h

Fig. 5. FE mesh used in simulating oedometer tests, showingboundary conditions

0

100200

300

400

500

600

700

0 1 2 3 4 5Appliedveriticaleffectivestress:kPa

Vertical strain: %(a)

Oedometer, 20%ID

FEM, 20%ID

Oedometer, 73%ID

FEM, 73%ID

0

100

200

300

400

500

600

700

0 1 2 3 4 5Appliedveriticaleffectivestress:kPa

Vertical strain: %(b)

Oedometer, 20%ID

FEM, 20%ID

Oedometer, 73%ID

FEM, 73%ID

Fig. 6. Comparison of one-dimensional compression curves forloose and dense samples: (a) using Brinkgreve et al. (2010)parameters; (b) using HST95 (soil-specific) parameters



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so that Eoed could be used to simulate the strain magnitudecorrectly, and m used to control the shape of the stressstrain curve. E50 Eoed was proposed by Brinkgreve et al.(2010), but this is not exactly true from comparison ofequations (2) and (3). The value of 1.25 ensured that nounrealistic values of K0 would be implied in denser soils(including that used in the centrifuge tests); the analysesappeared to be relatively insensitive to this assumption at

lower densities. Following iteration, the best-fit stiffnessparameters were determined to be

Erefoed 25ID 20:22 (MPa) (6)

m 0:60:1ID (7)

The power coefficient m is the inclination of stiffnessstress curves normalised by pref in double logarithmic scale.The proposed calibration for m (equation (7)) gives a power-law exponent for stress dependence of stiffness that is be-tween 0.5 and 0.6 at all densities. This is consistent withprevious studies (e.g. Lo Presti et al., 1998), while maintain-ing the slight negative correlation between the two param-eters noted by Brinkgreve et al. (2010).

As neither of the aforementioned tests measures small-strain parameters, G0 was estimated using the relationshipbased on void ratio (e) proposed by Hardin & Drnevich(1972),

Gref0 33(2:97e)2

1e (MPa) (8)

for pref100 kPa. Equation (8) can be linearised, ignoringterms of order e2 and above; expressing e in terms ofrelative density with the values ofe max andemin gives

Gref0 50ID 88:80 (MPa) (9)

The shear strain parameter s,0:7 was assumed to increaselinearly from 0.01% at ID 20% to 0.02% at ID 80%: thatis

s,0:7 1:7ID 0:67 (3104) (10)

Other parameters and commentsOf the remaining parameters, the default value of Rf0.9

was used, and the unit weight parameters were determinedfrom standard relationships as a function of relative densityand linearised, giving

dry 3ID 14:5 (kN=m3

) (11)

sat

1:8ID 18:8 (kN=m3

) (12)

The particular choice of direct shear apparatus (DSA) andoedometer tests for determining the parameters, as describedabove, was guided by available test data. Triaxial test datacould equally be used to determine both strength andstiffness data, by simulating triaxial compression on a virtualtest sample using FEM in a similar way to that described forthe oedometer tests. Damping will be discussed later in thepaper.

VALIDATION OF FEMDynamic shear modulus and damping

Shear modulus and damping as functions of cyclic shear

strain within both the centrifuge models and the numericalsimulations are shown in Fig. 7. The data points representingthe centrifuge models were determined from second-orderestimates using the accelerometer data, following the method

proposed by Brennan et al. (2005). The data points for thenumerical simulations were determined in the same way,using data from virtual accelerometers at homologouspoints within the FE mesh. Fig. 7 demonstrates that theoperative shear modulus at a given cyclic shear strain iscomparable to that measured in the centrifuge tests, but that

the material hysteretic damping within the model is gener-ally smaller than that inferred from the centrifuge tests. Thiswill be revisited in the following section. Nonetheless, itwould appear that the use of simple monotonic test data iseffective in defining a constitutive model that can representthe principal dynamic behaviour of the soil.

AccelerationsFigure 8 shows a comparison of the measured and simu-

lated acceleration at the toe of the slope (instrument 14) inthe first earthquake of test AA01 (Chi-Chi motion) in boththe time and frequency domains (Figs 8(a) and 8(b) respec-tively). In this figure three cases are considered: (i) the use

of Brinkgreve et al. (2010) constitutive parameters; (ii) theuse of HST95 parameters; and (iii) the use of HST95parameters with additional Rayleigh damping. The constitu-tive model implicitly includes material hysteretic damping

0

02

04

06

08

10

0001 001 01 1

G

G/

0

Shear strain: %(a)

Chi-Chi (centrifuge data)

Kobe (centrifuge data)

Chi-Chi (FEM data)

Kobe (FEM data)

Hardin & Drnevich (1972)

Ishibashi & Zhang (1993)

Santos & Correia (2001)

Chi-Chi (centrifuge data)Kobe (centrifuge data)

Chi-Chi (FEM data)

Kobe (FEM data)

Hardin & Drnevich (1972)

Ishibashi & Zhang (1993)

Santos & Correia (2001)

0

01

02

03

04

05

0001 001 01 1

Damping

Shear strain: %(b)

Fig. 7. Comparison of shear modulus degradation and dampingin centrifuge tests and FE simulations, along with other publishedcorrelations



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within in its formulation; however, these models generallyoverpredicted accelerations (as a result of underpredictingdamping; see Fig. 7). The Rayleigh damping formulationallows additional mass and/or stiffness-proportional (modal,frequency-dependent) damping (add) to be included

add cm1

4fn

ck(fn) (13)

where add is the additional equivalent viscous dampingratio, and fn is the natural frequency of modes within thesoil. Values ofcm 0.0005 andck 0.005 were selected foruse in case (iii), and the justification for these will be givenlater.

At point 14 the match between the centrifuge test andFEM is good for all soil parameter sets considered, althoughthe models without the additional Rayleigh damping slightlyoverpredict the peak acceleration values in some of thecycles. This match is not surprising, given that the shearwave has propagated only a short distance in level ground at

this point, and has not yet interacted with the slope.Figure 9 shows similar plots at the crest of the slope

(instrument 5), representing a more stringent test of thecapabilities of the numerical model. This would be of sig-

nificant interest for being able to determine the input motionto any (infra)structure located at the crest of the slope. Forboth cases (i) and (ii) without any additional damping,significant overprediction of acceleration is observed. From

Fig. 9(b) it is clear that this appears to arise from increasedamplification of the higher-frequency components (above3 Hz). The Rayleigh damping parameters reported earliergive damping that is predominantly stiffness-proportional,such that the motions at higher frequencies would be moresignificantly damped, without over-damping the lower fre-quencies where the match is better. Case (iii) in Fig. 9(b)shows a markedly improved prediction of the accelerationtime history at this point, owing to reduced contribution ofthe higher-frequency modes. Similar results were found forsimulations of test AA02 (Kobe motion). Increases in the ckparameter, to increase and further improve the damping athigher frequencies, resulted in poorer predictions of perma-nent movements. The values for cm and ck given above

appear to represent the best compromise, enabling bothground accelerations and permanent slip (see later) to bereasonably estimated within the same analysis.

The ratios of crest peak acceleration (at instrument 5) to

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Brinkgreve (2010) FEMet al.


HST95 silica sand FEM


HST95 sand Rayleigh FEM


Case (i)

Case (ii)

Case (iii)

Time: s(a)

0

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2030

40

1 2 3 4 5Magnitude:

/Hz

gCase (i)

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1 2 3 4 5Magnitude:

/Hz

g Case (ii)

0

20

40

1 2 3 4 5Mag

nitude:

/Hz

g

Frequency: Hz(b)

Case (iii)

Fig. 8. Comparison of measured and predicted accelerations atslope toe during test AA01: (a) time domain; (b) frequencydomain

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/Hz

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/Hz

g

Case (ii)

0

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40

1 2 3 4 5Magnitude

:

/Hz

g

Frequency: Hz(b)

Case (iii)

Fig. 9. Comparison of measured and predicted accelerations atslope crest during test AA01: (a) time domain; (b) frequencydomain



8/15

the peak value of the input motion were found to beinsensitive to the number of earthquakes, and of the order of1.3 (highest value recorded was 1.4). This amplificationfactor clearly contains two distinct effects: the site effect (inessence a material property effect, due to the dynamicproperties of the soil); and a topographic effect (a geometriceffect due to the ground surface profile). In Eurocode 8, Part5 (BSI, 2005b), the ground motion at the crest (instrument

5), for use in constructing response spectra, is calculatedusing

PGA SST ag (14)

where ag is the peak acceleration in the underlying bedrock(input motion in the centrifuge tests); S is the soil factordescribing the site effect (1.4 for the ground type E soilin this study, as classified using Eurocode 8, Part 1 (BSI,2005a)); and ST is the topographic amplification factor(> 1.2 for shallow slopes). The overall amplification factoris then PGA/ag: Based on Eurocode 8, the overall amplifica-tion within the centrifuge test would be predicted to bebetween 1.4 a n d 1.7, which would be conservative whencompared with the centrifuge observations (amplification1.3) for the two motions considered. Ashford et al. (1997)and Bouckovalas & Papadimitriou (2005) have conducteddetailed numerical studies based on harmonic ground shak-ing, and have demonstrated that the topographic amplifica-tion is dependent on the ratio of slope height to wavelengthand position from the crest. In light of this, the centrifugetests of Brennan & Madabhushi (2009) are a useful compari-son with the test data in this paper, as they considered aslope of similar (although not identical) height, slope angleand sand. Their results support the value of 1.3 measured intests AA01 and AA02.

However, the hazard posed to infrastructure at the crest ismore usefully represented by a response spectrum, ratherthan a ratio of peak accelerations, which represents thespectral response at a period T 0 s only. Fig. 10 shows a

comparison of predicted crest spectra for 5% nominal struc-tural damping from FEM and measured data from thecentrifuge tests, also including design spectra based onEurocode 8 for context. For the cases including Rayleighdamping (HST95, case (iii)), a very good match to thecentrifuge data is obtained, although there is a tendency forthe response to be slightly underpredicted for periods above0.5 s. Use of the other constitutive models results in a much

more significant underprediction of response above 0.5 s,and overprediction below this. Fig. 10 also suggests that

there is a range of natural periods (between approximately0.4 s and 1.0 s) over which the FEM may substantiallyunderpredict the response, compared with the centrifuge data(this is particularly noticeable for test AA02). This rangecould be used as a simple screening tool to identify keypieces of infrastructure atop slopes that may be more vulner-able to seismic damage than FEM would suggest, and towhich extra consideration should be paid in design. It isclear that care should be taken in interpreting the hazard tosupported infrastructure from analyses using FEM, and thatthis will be significantly dependent on the natural period ofthe supported structure. Further comparison between the firstand last earthquake motions demonstrated that the spectralmagnitudes appear to be relatively insensitive to repeatedstrong earthquake shaking.

Permanent deformationsFigure 11 shows the permanent crest settlements across

the four earthquakes as predicted by FEM (all three casesare shown), and as measured in the centrifuge. It is clearthat, for both suites of earthquake motions, use of theBrinkgreve et al. (2010) parameters hugely overpredictssettlement at the crest. This would lead to significant over-prediction of the risk posed to the slope, and hence, poten-tially, uneconomic design. In contrast, use of the HST95parameters (either case) gives a much better prediction,

although inclusion of the additional damping, which wasbeneficial for modelling the dynamic behaviour accurately,results in an underprediction of permanent deformation(which is unsafe). The Brinkgreve et al. (2010) parameterswere initially envisaged as being useful for cases wherethere is extremely limited site investigation data, but theimplication of Fig. 11 is that it is always important to obtainsoil parameters from high-quality laboratory (or in situ) testdata to achieve an accurate prediction of movement. It islikely that the extra cost of this additional investigationwould be significantly offset if a lower amount of remedia-tion/repair were to be required (as a result of a moreaccurate prediction of seismic response). When consideredalongside the dynamic performance discussed in the previous

section, it is clear that neither case (ii) or case (iii) can giveconsistently better performance of both dynamic and perma-nent movements simultaneously; this would appear to sug-gest that the additional Rayleigh damping may not be amaterial characteristic, but may be masking an effect of thesloping ground geometry in which wave reflection at thesloping ground surface is not modelled correctly within theFE model.

Figure 12 shows a comparison (drawn to scale) of theground surface profile measured at the end of the centrifugetest (a negligible amount of movement was recorded duringspin-down), and as predicted at the end of the last earth-quake (EQ4) from the FE model (case (ii)) for test AA01. Itcan be seen that the numerical model captures the deformed

shape of the slope well, particularly the angular distortion atthe crest, which is likely to be of greatest significance forsupported infrastructure. A similar result was obtained fortest AA02 (not shown).

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EC8 ( 14, 12)S S T

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Fig. 10. Measured (centrifuge), predicted (FEM) and design(Eurocode 8) response spectra at top of slope (instrument 5) for5% structural damping: (a) Chi-Chi (AA01); (b) Kobe (AA02)



9/15

DEVELOPMENT OF AN IMPROVED SLIDING-BLOCKMETHOD

The horizontal yield acceleration of a shallow translational

(infinite) slip can be determined using standard limit equi-librium techniques, incorporating pseudo-static accelerationcomponents due to the seismic ground motion (see Fig. 13).For a slip plane at depth z beneath the slope surface under

uniaxial horizontal shaking (plane strain), the applied down-slope shear stress is

applied zsin coskhzcos2 (15)

where the first term relates to the static shear stress due tothe ground slope, and the second term relates to the addi-

tional peak dynamic shear stress induced by the earthquake.The shear strength of the soil along the slip plane, assumingthat the soil failure can be described by the MohrCoulombfailure criterion, is given by

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Settle

ment:m

Time: s(a)

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Brinkgreve (2010) FEM (505 m)et al.

HST95 sand FEM (079 m)

HST95 sand FEM Rayleigh (041 m)

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Settlement:m

Time: s

(b)

Centrifuge settlement (102 m)

Brinkgreve (2010) FEM (33 m)et al.

HST95 sand FEM (076 m)

HST95 sand FEM Rayleigh (032 m)

Fig. 11. Comparison of permanent crest settlements from FEM and centrifuge modelling:(a) test AA01; (b) test AA02

14(280)

11 (220) 15 (300) 75 (150)

6(120)

8

(160)

Original profile

079m

(HST95FEM)

091m

(centrifuge)

After centrifuge

After FEM

(Centrifuge 195 m)

Fig. 12. Comparison of slope profile after EQ4 as predicted byFEM and as measured in centrifuge, test AA01

g

khgL

k zLh cos

zLcos

ultL

( ) u L

z

Fig. 13. Forces acting within infinite slope under horizontalshaking



10/15

ult c9 9 tan9

c9 (zcos2khzsin cosu) tan9(16)

The soil yields when applied ult: The value of kh atwhich this occurs (the yield acceleration, khy) can be deter-mined from equations (15) and (16) as

khy

c9 (zcos2u)tan9 zsin cos

zcos2zsin cos tan9 (17)

In a standard Newmark analysis, when the horizontalcomponent of the ground acceleration (a) exceeds khy in thedownslope direction, the slope will start to accelerate underthe slip acceleration, aslip a khy (positive value impliesdownslope movement). This acceleration is numerically in-tegrated with respect to time to obtain the slip velocity,which is then itself integrated to obtain the slip displace-ment. Once a , khy, the sliding block will begin to decele-rate (as aslip , 0) until the slip velocity reaches zero, atwhich point the block comes to rest until aslip is againpositive. This procedure is shown schematically in Fig.14(a).

In a dry cohesionless soil, c9 0, u 0, and z cancelsin equation (17): that is, khy is independent of the depth of

the slip plane (so long as it continues to be parallel to theslope surface). Equation (17) then simplifies to

khy cos tan9 sin

cossin tan9 (18)

In equation (18) the only parameters affecting khy are theslope angle (geometric) and the soil friction angle 9

(constitutive). In a standard analysis, both and 9

areconstant. In reality, however, the soil may be strain-softening,in which case 9 will depend on the magnitude of the shearstrain (s) on the slip plane and the density of the soil.Matasovicet al. (1997) presented a simplified model for this,which is shown schematically in Fig. 14(b). If9pk 9cs, themodel reduces to the standard case of a strain-softeningmaterial. To incorporate this into an analysis, the value ofkhy at a particular time step can be estimated, based on thecurrent permanent downslope displacement, computed in theprevious time step, divided by the thickness of the shearband/slip plane to obtain an estimate of the shear strain.

The slope angle will also change during an analysis, asslip will cause settlement at the crest and accumulation ofmaterial at the toe: that is, the slope will become shallower(re-grading, RG). A simplified model for this geometriceffect is developed in this paper, and is shown in Fig. 15.Fig. 15(a) shows the kinematically admissible failure me-chanism assumed for an increment of slip, di, in whichinfinite sliding is the predominant component. This leads todownward vertical movement of the material at the slopecrest and a horizontal translation of the position of the toe,resulting in a new, shallower slope with an approximateslope angle of i1 in Fig. 15(b). Ambrayses & Srbulov(1995) present an alternative mechanism for shallow slidingthat is similar to Fig. 15(a), but with more complex changesin geometry (and for post-seismic sliding only, i.e. whenthere is no seismic inertia). The new (simpler) mechanismproposed here has the advantage that the calculations are

much more straightforward, relying on only a single param-eter (the initial slope angle ) to define the slope geometry,and more closely match the deformations observed visuallyin the centrifuge tests (e.g. see Fig. 12).

Provided that is relatively small (such that the slope islong compared with its height), the equilibrium of thismechanism will be well approximated by infinite slopetheory (i.e. equations (17) or (18) will adequately describe

Acceleration

Including strain-softening (SS):( )k fhy

Including strain-softening (SS):( )k fhy

Classical Newmark:const.khy

Classic Newmark:const.khy

Time

Time

Time

Ground motion

Slipvelocity Classical

Including SS

(a)

Slipdisplacement

k h

y

and

Shearbox test curvetan ( / ) vs

1s

s,pk s,cs Shear strain,s

pk

cs

(b)

Fig. 14. Newmark sliding-block procedure, and effect of strain-softening (after Matasovic et al., 1997)

Settlement incrementsind

i i

Crest Toe

Slip increment,d

i

di icos

(a)

(b)

New slopesurface ( 1)i

Hi H d

i i isin

i

i1

Hi icot d

i icos

Fig. 15. (a) New incremental slope re-grading mechanism;(b) incremental changes in geometry



11/15

khy). It will be demonstrated later that even a slope as steepas ,1:2 satisfies this condition to a reasonable degree ofaccuracy. It is assumed that any volumetric change in thematerial in the sliding block is negligible. From Fig. 15(b),the instantaneous slope angle i1 can be determined from

i1 tan1 Hi disini

Hicoti dicosi

(19)

where Hi is the height of the slope at the previous time step.For the initial time step, d0 0, Hi H andi 0 (initialslope angle). It should be noted that the deformations in Fig.15(b) are shown at exaggerated scale for clarity. The slopeangle can therefore be recalculated at each time step toaccount for the re-grading of the slope based on theincrement of slip occurring in the previous time step, as 9was previously to account for strain-softening. The yieldacceleration (equation (18)) will thus be fully strain-depen-dent. This method assumes that once the slope has beendeformed to a new, smaller value of , the failure mechan-ism will continue to be of the infinite type, with a new slipsurface forming parallel to the new slope surface. It alsoassumes that the strain-dependent effects on and 9 areindependent, to simplify the calculations. In reality, thislatter assumption may not be wholly true, as the changingangle of a softened slip plane (i.e. with 9 9cs) may pushat least part of the slip surface into previously undisturbedsoil. If this effect and the effects of strain-softening aresignificant, it is expected that the model will overestimatemovements; however, this would be conservative for use inanalysis and design. It should also be noted that the modelas formulated can be used even for the case of large totalslope movements (such as may accrue during a series ofstrong aftershocks), as the displacement increment in eachindividual time step will remain small, and therefore theinstantaneous failure mechanism will be represented by Fig.15(a) for small displacement increments.

From the form of equation (19) it is clear that can onlyever reduce during an earthquake, resulting in an increase ofkhy from equation (17) or (18): that is, the slope willgeometrically harden during an earthquake, and the slip in asubsequent (identical) earthquake will be less than thatoccurring in the original. The behaviour of a seismicallydamaged slope during a subsequent earthquake can thereforebe determined by starting from the initial conditions (amountof slip, accumulated strain, re-graded slope angle and currentfriction angle) obtained at the end of the analysis for theprevious ground motion.

VALIDATION OF SLIDING-BLOCK MODEL

Predictions of permanent deformationVisual observations from the centrifuge tests suggested

that the 1:2 slopes tested failed in a shallow translationalmechanism consistent with that shown in Fig. 15. The depthof the shear plane (0.5 m) was estimated using the dis-continuity layout optimisation (DLO) technique (Smith &Gilbert, 2007) to obtain a minimum upper-bound mechanismfor the actual limited geometry of the centrifuge model.DLO calculations were carried out using LimitState:GEO,v2.0. The position of the slip plane was not affected by thevalue of 9 used, and is shown in Fig. 16(a). The idealisedgeometry assumed in the analysis (cf. Fig. 15(a)) is shownin Fig. 16(b), overlaid onto the accumulated shear strainwithin the FE simulation of test AA01 (case (ii)), for

comparison. Shearbox testing was then conducted on sam-ples of dry sand prepared to the same relative density as inthe centrifuge tests, using a standard 60 mm 3 60 mm DSA,to obtain 9pk and9cs: These tests were conducted at a range

of effective confining stresses representing those within thetop 1 m of the soil. The test data are shown in Fig. 17, fromwhich values of 9pk 408, 9cs 328, s;pk 3.5% ands;cs 7.5% were estimated for 0.5 m depth. Using thesefriction angles and limiting shear strains, the initial yieldaccelerations of the slope were computed using equation(18) and DLO; the results are shown in Table 4, alongsidevalues for the static factor of safety of the slope (F),calculated using equations (15) and (16) with kh 0, andDLO.

The shear band thickness (required for converting slipdisplacement in the sliding-block model into an approximate

shear strain) was estimated at t 16D50 2.4 mm, based ona range of previous studies (e.g. Mulhaus & Vardoulakis,1987; Oda & Kazama, 1998; Muir Wood, 2002). Thecalculations were conducted at prototype scale, and so theshear strain was estimated using 50t 120 mm to model thecorrect ratio between the slope geometry and the grain sizewithin the model. For application to a true field case wherethe grains are smaller compared with the overall size of theslope, the true shear band thickness should be used instead.For the tests presented herein, changing the shear bandthickness from 120 mm to 2.4 mm resulted in less than 1%change in crest settlement (the actual value varied slightlywith the input motion considered), confirming that the grain-

05 m

(a)

(b)

0 1 2 3 4 5 6 7 8

Shear strain: %

400360

320

280

240

200

160

120

80

40

0

Fig. 16. (a) Failure mechanism computed from DLO for seismiccase; (b) accumulated shear strain within FE model (test AA01)and assumed approximate infinite slip mechanisms used inNewmark analyses

55 kPa (034 m)

82 kPa (051 m)

136 kPa (085 m)

Analytical model

0

02

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Stressratio(

tan

)

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Fig. 17. Soil test data from direct shear apparatus (DSA) at loweffective confining stress



12/15

size scaling effect is negligible, and that centrifuge model-ling is therefore an appropriate technique for modellingslope failure problems in coarse-grained soil. A similar con-clusion was reached by Anastasopoulos et al. (2007) forfault rupture (shear band) propagation through the samesand.

Figure 18 shows the effect of the geometric re-grading(change in ), using the first earthquake (EQ1) of test AA01as an example. Only the positive (downslope) accelerationshave been shown, for clarity. No slip occurs until the groundmotion exceeds the yield acceleration based on the peakstrength. Once the ground motion exceeds this value, how-ever, and the slope begins to slip, the shear strain rapidlyaccrues, resulting in softening to critical state conditionsafter the first large pulse. Motion of the slope also causesre-grading (geometric hardening), and the yield accelerationcan subsequently be seen to increase non-linearly throughoutthe remainder of the earthquake, leading to reduced slipvelocities (and hence reduced permanent slip) comparedwith the case with no geometric hardening.

Sliding-block analyses were subsequently conducted foreach of the centrifuge tests (for all four earthquakes in thecases of tests AA01 and AA02). Simulations were conductedusing the bedrock input motion (this was taken from thebottom-most accelerometer in the model, instrument 8), asused in the FEM. Additional simulations using instrument 6

did not show significant differences in the overall cumulativeslip predicted, possibly as there was some shear decouplingacross/around the shear band, which counteracted (at leastpartially) any increase in acceleration due to soil amplifica-tion. In slopes with deep rotational failure surfaces (e.g. inmunicipal solid waste or steep cohesive slopes), the dynamic

behaviour of the material within the sliding block (due to itsthickness) may be significant (e.g. Rathje & Bray, 1999);this was not the case for the extremely shallow translationalslips that occurred within the cohesionless slopes testedherein.

Figures 19 and 20 show the results of simulations ofcumulative crest displacement both with (SS + RG) andwithout re-grading (SS) with the centrifuge test data, fortests AA01 and AA02 respectively. It can be seen that, ineach case, the improved model presented in this paper(SS + RG) tracks the settlement at the crest of the slopemuch more closely than the model that incorporates only theconstitutive effect (SS). These latter models increasinglydiverge from the measured values with further strong shak-ing, as they always start with the initial (steeper) slopegeometry, and therefore overpredict the slip. If the inputmotions were identical, the slip in each subsequent earth-quake would be identical for the case of no re-grading(although the movements in the first earthquake might beslightly smaller, owing to the strain-softening effect). Theimproved models are not perfect, and in each case over-predict the measured movement; as the slope re-grades, thenew position of the slip plane may cause it to pass at least

Strain-softening

Geometric hardening(re-grading)

Input ground motion

khy SS RG

khy SS

0

010

020

030

040

0 10 20 30 40 50

A

cceleration:g

Time: s(a)

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02

04

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10

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Time: s(b)

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Strain-softening only (SS)

Fig. 18. Application of new sliding-block model, showing keyfeatures (Chi-Chi EQ1 used, test AA01)

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Crestsettlement:m

Centrifuge (AA01)

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Sliding block (SS)

EQ1 EQ2 EQ3 EQ4

Inputacceleration:g

05

0

05

021016

020 023025

0 50 100 150 200

Ground acceleration (AA01)

khy (SS RG)

khy (SS)

Fig. 19. Comparison of predicted cumulative crest settlements(with and without re-grading) with centrifuge test measurements:test AA01 (Chi-Chi)

Table 4. Static and dynamic slope stability data

Soil strength Static stability parameters Dynamic stability parameters

F(equations (15)and (16))

F(DLO) khy(equation (18))

khy(DLO)

9 328 1.17 1.20 0.07g 0.07g9 408 1.56 1.61 0.21g 0.22g



13/15

partially through undisturbed soil, thereby leading to anenhanced average frictional resistance along the slip plane.

COMPARISON OF SLIDING-BLOCK AND FEMFigure 21 presents a summary of the predictive ability of

the FEM and the sliding-block models for determiningpermanent settlements. These have been grouped into ex-isting models, representing the current state-of-the-art,namely sliding block with strain-softening and FEM usingthe previously published parameters of Brinkgreve et al.(2010), and improved models, consisting of the fully

strain-dependent sliding-block model (strain-softening andgeometric hardening) and the laboratory-test-calibrated FEMdeveloped in this paper (HST95, case (ii)). Although theimproved models are not perfect, they give a much im-proved prediction of the response under the initial earth-quake on virgin soil, and are subsequently able to give abetter prediction of behaviour, even after several previousstrong earthquakes during which significant deformation has

accrued. Both existing models overpredict settlements in thefirst earthquake and then get progressively worse withfurther shaking, as they are unable to correctly capture theeffect of the previous seismic (strain) history on the slope.In the case of the sliding-block model this is associatedwith incorrect description of the deformed slope; in thecase of the FEM, the case (i) constitutive parameters donot appear to correctly capture the strain history or mech-anical response of the soil. The development of the im-proved sliding-block and FEM tools in this paper providesan improved means of quantifying the response of shallowcohesionless slopes under strong earthquake shaking, andthe ability to consider behaviour under multiple successiveearthquakes. With further development, these tools willallow civil engineers to obtain a better estimate of thehazard associated with aftershocks, and lead to new ap-proaches to quantifying seismic performance and managingcritical transport infrastructure, in which whole-life per-formance can be considered.

CONCLUSIONSIn this paper, improved procedures for modelling the

seismic response of dry granular shallow slopes using theFEM and a fully strain-dependent Newmark sliding-blockprocedure have been developed and validated against dy-namic centrifuge test data. In the FE modelling, a non-linear elasto-plastic constitutive model was used, the para-meters of which could be estimated based only on relative

density using existing correlations, or using routine labora-tory tests to develop a soil-specific model. These methods,as they do not require specialist testing, offer significantbenefits for use in routine design, although the predictionsobtained are of an approximate nature. It was demonstratedthat the use of soil-specific parameters gave far improvedpredictions, particularly of permanent settlement, comparedwith the existing correlations, which overpredicted settle-ments, particularly in subsequent earthquakes. This high-lights the value of performing adequate site investigation.In the improved sliding-block model, reduction in slopeangle with slip/strain (re-grading or geometric hardening)has been incorporated alongside an existing strain-softening/hardening formulation.

The sliding-block and FE models gave comparable predic-tions of permanent slip, capturing the significant decay inground displacement (geometric hardening) with subsequentshaking observed in the centrifuge tests. They thereforepermit both the response of slopes under strong aftershocks,and the whole-life performance of a slope, to be quantified.Whereas the sliding-block models are useful in preliminarydesign, owing to the limited soil property data required andthe reduced computational effort, the FEM is additionallyable to quantify the dynamic performance of the soil and theground deformation profile (angular distortion) at the crest.This would provide the information necessary to make adetailed study of the seismic hazard posed to infrastructurelocated at the slope crest, without requiring an excessive

amount of specialist laboratory testing, and is thereforecomplementary to the sliding-block models, being useful inthe later stages of detailed design. The FE models generallyoverpredicted the magnitude of dynamic ground motions in

4

2

00 50 100 150 200

Time: s

Cr

estsettlement:m

Centrifuge (AA02)


Sliding block (SS)

EQ1 EQ2 EQ3 EQ4

Inputacceleration:g

05

0

05

021

012016 018

020

0 50 100 150 200

Ground acceleration (AA02)

khy (SS RG)

khy (SS)

Fig. 20. Comparison of predicted cumulative crest settlements(with and without re-grading) with centrifuge test measurements:test AA02 (Kobe)

0

2

4

6

8

10

12

Predicted/measured(crestsettlement)

Earthquake no.

1 2 3 4

Improved methods:


FEM (ii)

Existing methods:

Sliding block (SS)

FEM (i)

Parity

Fig. 21. Accuracy of existing models/procedures, compared withthose proposed in this and the companion paper (improvedmodels) for predicting permanent crest settlement



14/15

the slopes (particularly for components above 3 Hz). Addingin additional damping using the Rayleigh formulation im-proved this, but had an adverse effect on the prediction ofpermanent slope movements.

ACKNOWLEDGEMENTSThe authors would like to express their sincere gratitude

to Mark Truswell and Colin Stark at the University ofDundee for their assistance in performing the centrifugetests. The first author would also like to acknowledge thefinancial support of his PhD studies from the Ministry ofHigher Education and Scientific Research (MOHESR) of theRepublic of Iraq.

NOTATIONa ground acceleration

ag underlying bedrock peak accelerationaslip slip acceleration

Cu coefficient of uniformityC2 coefficient of curvaturec9 cohesion intercept

ck stiffness-proportional Rayleigh damping coefficientcm mass-proportional Rayleigh damping coefficient

D10 particle diameter at which 10% is smallerD30 particle diameter at which 30% is smallerD60 particle diameter at which 60% is smaller

d slope parallel slipEoed oedometric tangent stiffness (in compression)

Eur unloading reloading stiffnessE50 triaxial secant stiffness (at 50% of deviatoric failure stress

in drained triaxial compression)e natural void ratio

emax maximum void ratioemin minimum void ratio

F static factor of safetyfn natural frequency

G shear modulusG0 small-strain modulusGs specific gravityg acceleration due to gravity (9.81 m/s2)

H slope height above toeID relative densityK0 lateral earth pressure coefficient (at rest)

kh pseudo-static seismic horizontal accelerationkhy yield acceleration

Mw moment magnitudem power-law index for stress-level dependence of stiffness

PGA peak ground acceleration (at soil surface)pref reference stress (100 kPa)

Rf ratio of deviatoric failure stress to asymptotic limitingdeviator stress

S soil factor (Eurocode 8)

ST topographic amplification factor (Eurocode 8)t shear band thickness

u pore water pressureVs shear wave velocity

z depth of slip plane slope angle0 initial slope angle (pre-earthquake) soil unit weightd dry unit weightsat saturated unit weights shear strain

s,cs shear strain at critical states,pk shear strain at peak stateadd viscous damping ratiour Poissons ratio (unload reload)9 normal effective stress

applied applied shear stressult shear strength9 effective angle of friction9cs critical state angle of friction

9pk (secant) peak angle of friction9 effective angle of dilation

REFERENCESAmbrayses, N. & Srbulov, M. (1995). Earthquake induced displace-

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