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Computers and Geotechnics 36 (2009) 135–148

Seismic rock slope stability charts based on limit analysis methods

A.J. Li a,*, A.V. Lyamin b, R.S. Merifield a

a Centre for Offshore Foundations Systems, The University of Western Australia, WA 6009, Australiab Centre for Geotechnical and Materials Modelling, The University of Newcastle, NSW 2308, Australia

Received 2 October 2007; received in revised form 15 January 2008; accepted 15 January 2008Available online 5 March 2008

Abstract

Earthquake effects are commonly considered in the stability analysis of rock slopes and other earth structures. The standard approachis often based on the conventional limit equilibrium method using equivalent Mohr–Coulomb strength parameters (c and /) in a slipcircle slope stability analysis. The purpose of this paper is to apply the finite element upper and lower bound techniques to this problemwith the aim of providing seismic stability charts for rock slopes. Within the limit analysis framework, the pseudo-static method isemployed by assuming a range of the seismic coefficients. Based on the latest version of Hoek–Brown failure criterion, seismic rock slopestability charts have been produced. These chart solutions bound the true stability numbers within ±9% or better and are suited to iso-tropic and homogeneous intact rock or heavily jointed rock masses. A comparison of the stability numbers obtained by bounding meth-ods and the limit equilibrium method has been performed where the later was found to predict unconservative factors of safety forsteeper slopes. It was also observed that the stability numbers may increase depending on the material parameters in the Hoek–Brownmodel. This phenomenon has been further investigated in the paper.Crown Copyright � 2008 Published by Elsevier Ltd. All rights reserved.

Keywords: Safety factor; Earthquake; Pseudo-static; Seismic coefficient; Failure criterion

1. Introduction

In seismically active regions, earthquakes are a majortrigger for instability of natural and man-made slopes.Therefore, seismic effects are essential design consider-ations for slope stability, retaining walls, bridges and otherengineering structures. Currently, the conventionalpseudo-static (PS) approach is still widely accepted as ameans for evaluating slope stability. In the PS method,the earthquake effects are simplified as horizontal and/orvertical seismic coefficients (kh and kv). The magnitude ofthe coefficients is expressed in terms of a percentage ofgravity acceleration. Due to the simplicity of the PSapproach, it has drawn the attention of a number of inves-

0266-352X/$ - see front matter Crown Copyright � 2008 Published by Elsevie

doi:10.1016/j.compgeo.2008.01.004

* Corresponding author. Tel.: +61 8 6488 3141; fax: +61 8 6488 1044.E-mail addresses: [email protected] (A.J. Li), andrei.lyamin@-

newcastle.edu.au (A.V. Lyamin), [email protected] (R.S. Meri-field).

tigators [1–6]. In particular, Baker et al. [4] and Loukidiset al. [6] have adopted the PS method in limit equilibriumanalysis and limit analysis, respectively, to provide chartsolutions for soil slopes. It should be noted that by usingcomplicated dynamic response analysis coupled withappropriate constitutive laws, a more precise seismic eval-uation for slopes could be obtained. However, the PSmethod is still recommended as a screening procedure toidentify the requirement for more sophisticated dynamicanalyses. The pseudo-static approach has certain limita-tions [7,8], but this methodology is considered to be gener-ally conservative, and is the one most often used in currentpractice.

Since Taylor [9] proposed a set of stability charts for soilslopes, chart solutions have been presented by manyresearchers [6,10–13] and are still widely used as designand teaching tools. Unfortunately, most of the existingcharts are proposed for estimating the stability of soilslopes. This is most likely due to the fact that assessing rock

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136 A.J. Li et al. / Computers and Geotechnics 36 (2009) 135–148

mass strength is difficult and most geotechnical software iswritten in terms of the conventional Mohr–Coulomb fail-ure criterion. Hence, there are very few stability chartswhich can be used for rock slopes that are based on rockmass yield criteria. Fortunately, attractive finite elementupper and lower bound approaches have been developedby Lyamin and Sloan [14,15] and Krabbenhoft et al. [16].These techniques can be used to bracket the true stabilitysolutions for geotechnical problems from above and below,and are suited to assigning many typical failure criteria.One classical rock yield criterion, the Hoek–Brown failurecriterion [17], can be incorporated into the numerical upperand lower bound formulations. This criterion has been suc-cessfully implemented and applied to bearing capacity andslope stability problems by Merifield et al. [18] and Li et al.[19], respectively.

This paper aims to investigate the seismic effects on rockslope stability where the earthquake inertia force is simu-lated by using the pseudo-static method. Moreover, as per-formed by Li et al. [19], the advantage of limit theoremswill be exploited to bracket the true solutions for rockslope stability numbers and to provide a range of seismicstability charts for rock slopes. In this study, both of theupper and lower bound techniques developed by Lyaminand Sloan [14,15] and Krabbenhoft et al. [16] areemployed. The PS analyses in this study do not accountfor the effects of pore pressure, and the strength of rockmasses is assumed to be unaffected during earthquake exci-tation. As a means of comparison, the limit equilibriummethod will then be used in conjunction with equivalentMohr–Coulomb parameters for the rock and comparedwith the solutions obtained from the numerical limit anal-ysis approaches.

GROUP II

TWO DISCONTINU

SINGLE DISCONTINUITIES

INTACT ROCK

GROUP I

Fig. 1. Applicability of the Hoek–Brow

2. Applicability

2.1. Applicability of the generalised Hoek–Brown failure

criterion

Predicting the strength of large-scale rock masses is aclassical challenge for geotechnical engineers. Fortunately,the empirical failure criterion proposed by Hoek andBrown [17,20] is widely accepted as a means of estimatingthe strength of rock masses which are assumed to be isotro-pic. It is important to point out that the assumption of isot-ropy means the Hoek–Brown yield criterion is unsuitablefor slope stability problems where shear failures are gov-erned by a preferential direction imposed by a singular dis-continuity set or combination of several discontinuity sets(e.g. sliding over inclined bedding planes, toppling due tonear-vertical discontinuity, or wedge failure over intersect-ing discontinuity planes). Therefore, as pointed out byHoek [21], the Hoek–Brown yield criterion is applicableto intact rock and heavily jointed rock masses, such asthe GROUP I and GROUP III shown in Fig. 1. The slopesthat belong to GROUP II have highly anisotropic rockproperties where the Hoek–Brown failure criterion is notapplicable. The anisotropy can be induced by weatheringeffects or naturally occurring discontinuities such as joints,faults and bedding planes. In this paper, the rock masses ofall slopes have been assumed as either intact or heavilyjointed rocks as GROUP I and GROUP III so that theHoek–Brown failure criterion is applicable. The applicabil-ity and limitations of the GSI system has been discussed indetail by Marinos et al. [22].

After Hoek [20], the Hoek–Brown failure criterion canbe described by the following equations:

GROUP III

JOINTED ROCK

MASS SEVERAL DISCONTINUITIESITIES

Jointed Rockσ

ci, GSI, m

i, γ

n failure criterion to slope stability.

1.00 1.05 1.10 1.15 1.20 1.25 1.300.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

M 6.5

Seed (1979)

Recommended Pseudo-Static Safety Factor

Pseu

do-S

tatic

coe

ffic

ient

, kh

Hynes & Franklin (1984) M 8.25

Fig. 2. Design recommendations for pseudo-static analysis.

khW

WToe

Rigid Base

Jointed Rockσci , GSI, mi , γ

Hβ d

Fig. 3. Problem definition.

A.J. Li et al. / Computers and Geotechnics 36 (2009) 135–148 137

r01 ¼ r03 þ rci mbr03rci

þ s� �a

ð1Þ

where

mb ¼ mi expGSI� 100

28� 14D

� �

s ¼ expGSI� 100

9� 3D

� �

a ¼ 1

2þ 1

6e�GSI=15 � e�20=3� �

From the above equations, it can be seen that the magni-tudes of mb, s and a rely on the geological strength index(GSI) which describes the rock mass quality. Guidelineson how to determine the value of GSI and other parame-ters within the Hoek–Brown failure criterion can be foundin [23], and will not be discussed in this paper. In addition,D is defined as a disturbance coefficient that ranges from 0for undisturbed in situ rock masses and 1 for disturbedrock mass properties. For the analyses presented in thispaper, a value of D = 0 has been adopted to simulatenatural slopes and can be compared to the results withoutseismic effects from Li et al. [19].

2.2. Applicability of the pseudo-static method

In general, the seismic coefficients are determined fromexperience by using the maximum horizontal accelerationor peak ground acceleration of a design earthquake. Itshould be noted that, current design using PS analysis isoften based on a horizontal seismic coefficient (kh). There-fore, this study is primarily focused on investigating theearthquake effects on rock slope stability by using a rangeof horizontal seismic coefficients.

With reference to the magnitude of kh, Seed [2]suggested that the PS method is applicable in assessingthe performance of embankments constructed of materialswhich do not suffer significant strength loss during earth-quakes. It is recommended to utilise k = 0.1 for earth-quakes of Richter’s magnitude 6.5, and k = 0.15 forearthquakes of Richter’s magnitude 8.5. For both cases, asafety factor F P 1.15 is required for design.

The suggestion proposed by Hynes-Griffith and Frank-lin [24] is one of the widely used and accepted methods fordetermining an appropriate value of kh. They recom-mended that a PS analysis can be used for preliminary eval-uation of slope stability, where a seismic coefficient equal toone-half the measured bedrock acceleration is adopted.Provided the obtained factor of safety is greater than 1.0,the slope design can be accepted. For factors of safety ofless than 1, Hynes-Griffith and Franklin [24] proposed thata more thorough numerical analysis need to be performed.However, due to the determination of kh magnitude isrelated to the measured bedrock acceleration as discussedabove, the PS method may not account for the site ampli-fication induced by the underlain stratum [25] or topogra-phy [26], etc.

In order to select an appropriate PS coefficient for agiven site, a diagram (Fig. 2) summarized by the CaliforniaDivision of Mines and Geology [27] provides the recom-mendations in regards to the seismic coefficient (kh), versusa required factor of safety. From this diagram, it can beseen that the recommended kh values do not exceed0.375. Therefore, the range of the seismic coefficientsadopted in the present paper will be between kh = 0.0and kh = 0.375.

3. Problem definition

A plane strain illustration of the upper and lower boundslope stability analyses is shown in Fig. 3 where the jointedrock mass has an intact uniaxial compressive strength rci,geological strength index GSI, intact rock yield parametermi, and unit weight c. All the quantities are assumed con-stant throughout the slope. In the limit analyses, the seis-mic force is assumed as a horizontal internal body forceand its magnitude is represented by the horizontal seismiccoefficient (kh). The direction of khW, is considered to bepositive when acting outward with respect to the slope(see Fig. 3).

For given slope geometry (H, b) and rock mass(rci GSI, mi), the optimised solutions of the upper boundand lower bound programs can be carried out with


respect to the unit weight (c). In this study, slope inclina-tions of b = 30�,45�,60� and 75� are analysed. The non-dimensional stability number proposed by Li et al. [19]has been adopted, and is given as

N ¼ rci=cHF ð2Þwhere F is the safety factor of the slope.

4. Previous studies

To overcome the problem of estimating rock slopestrength and stability governed by the complicated failuremechanisms, Jaeger [28] and Goodman and Kieffer [29]had outlined several simple methods and emphasized theirlimitations. Buhan et al. [30] found that the rock massscale-effect may influence the results of stability analysis.Hoek and Bray [11] and Zanbak [31] proposed chart solu-tions for stability and toppling problems of rock slopes,respectively. However all of the above studies were basedon a static analysis. This means that the seismic effects werenot taking into account.

In order to estimate rock slope stability under earth-quake effects, Seed’s [2] recommendation of seismic coeffi-cient based on the PS analysis can be used. Newmark [1]applied and extended the PS method to evaluate theground movement induced by an earthquake. Thisapproach has been accepted and extensively used to studyearthquake triggered landslides and rockslides [26,32,33].Pradel et al. [34] in particular, obtained a good agreementof slope crest displacement between the calculated andobserved results. In their study, the strength parametersused in analyses are determined by repeated direct sheartesting and back analysis. Bhasin and Kaynia [35] utilisedthe distinct element method to investigate a rock slope pro-gressive failure under initial static loading, climatic effects,and dynamic loading. Based on finite element and limitequilibrium analyses, Luo et al. [36] found that groundwater may significantly reduce slope stability during earth-quake excitation where the obtained maximum seismiccoefficient change by up to 60%. Sepulveda et al. [37]pointed out that the topographic amplification effects suchas slope orientation and seismic wavelength may influencethe rock slope stability assessment. In the case study ofChen et al. [38], the vertical ground acceleration was foundto be an important factor leading to rockslide under nearfield conditions. Reliability analysis was employed by Shouand Wang [39], Wang et al. [40] and Hack et al. [41] to pre-dict the failure risk of rock slopes and investigate the influ-ence of earthquakes.

Currently, practising engineers typically use stabilitycharts originally derived for soils when attempting to pre-dict the stability of rock slopes (i.e Hoek and Bray [11]).Zanbak [31] proposed a set of stability charts for rockslopes susceptible to toppling, whilst Siad [42] producedcharts based on the upper bound approach that can beused for rock slopes with earthquake effects. However,the design charts presented by these authors require con-

ventional Mohr–Coulomb soil parameters, cohesion (c)and friction angle (/), as input.

Recently, Collins et al. [43] and Drescher and Chrosto-poulos [44] and Yang et al. [45] adopted tangential strengthparameters (ct and /t) from nonlinear failure criteria toestimate the slope stability. However, only the study ofYang et al. [45] were based on the latest version of theHoek–Brown yield criterion and obtained optimised stabil-ity factors using the upper bound approach and a failuremechanism originally formulated by Chen [46].

A review of the literature reveals that few numericalslope stability analyses have been performed which arebased on the native form of the Hoek–Brown failure crite-rion (where strength parameters rci, GSI, mi and D areinput variables). The only exception is the stability chartsproposed by Li et al. [19]. However, in their study, the seis-mic effects were not taken into consideration. Therefore,the purpose of this paper is to provide a set of rock slopestability charts which incorporate the earthquake effectsand can be used by engineers in the assessment of seismicrock slope stability.

5. Results and discussion

5.1. Numerical limit analysis solutions

Figs. 4–6 present three sets of stability charts obtainedfrom the numerical upper and lower bound formulationswith horizontal seismic coefficients of kh = 0.1, 0.2 and0.3, respectively. The magnitude of kh in current designcodes is generally within this range. For all analyses per-formed in this study, the maximum difference betweenthe bound solutions was found to be less than ±9%. As aconsequence, average values of the upper and lower boundstability numbers will be used in the following discussions.

Referring to Figs. 4–6, it can be observed that the stabilitynumber N decreases when GSI or mi increases, as expected.Remembering from Eq. (2) that the stability number N isproportional to the inverse of the factor of safety F. A lowerfactor of safety equates to a higher stability number and vise-versa. The direction of increasing stability is shown by thearrow in Fig. 4a. A decrease in stability number with increas-ing GSI or mi is not unexpected because, based on the defini-tion of the Hoek–Brown yield criterion, the largermagnitudes of GSI or mi signify that the rock masses havegreater overall strength for any given normal stress. How-ever, one exception to this trend is shown in Fig. 6d whereN increases slightly with increasing mi. This phenomenonwill be discussed in more detail below.

In addition, the chart solutions can be presented as analternative form which is a function of the slope angle (b)as shown in Fig. 7 where N can be seen to increase whenthe inclination of a slope increases. For a given slope incli-nation, the stability number can be obtained by estimatingGSI and mi. For the same rock mass properties of a slope,the difference in stability numbers between various slopeangles can provide the corresponding variation in factor

5 10 15 20 25 30 35

0.01

0.1

1

10

AverageSLIDE-Hoek-Brown ModelTangential Method

β = 30, kh

= 0.1

N=

σ ci/γ

HF

N=

σ ci/γ

HF

N=

σ ci/γ

HF

N=

σ ci/γ

HF

GSI=10

GSI=50

GSI=100

mi

Hβ

Increasing Stability

5 10 15 20 25 30 350.01

0.1

1

10

100

mi

AverageSLIDE-Hoek-Brown ModelTangential method

β = 45, kh = 0.1

GSI=10

GSI=50

GSI=100Hβ

5 10 15 20 25 30 350.01

0.1

1

10

100AverageSLIDE-Hoek-Brown ModelTangential Method

mi

β = 60, kh = 0.1

GSI=10

GSI=50

GSI=100

Hβ

5 10 15 20 25 30 350.01

0.1

1

10

100


mi

β = 75, kh = 0.1

GSI=50

GSI=100

GSI=10

Hβ

Fig. 4. Average finite element limit analysis solutions of stability numbers (kh = 0.1).


of safety. For instance, it can be observed in Fig. 7 thatdecreasing slope angle from b = 75� to 60� can increasethe safety factors by 50–100% for mi = 5. This trend is sim-

ilar to the results of Li et al. [19] in which reducing theslope angle was found to increase the factor of safety bymore than 50%.

5 10 15 20 25 30 350.01

0.1

1

10


mi mi

β = 30, kh

= 0.2

GSI=10

GSI=50

GSI=100

N=

σ ci/γ

HF

N=

σ ci/γ

HF

N=

σ ci/γ

HF

N=

σ ci/γ

HF

Hβ

5 10 15 20 25 30 350.01

0.1

1

10

100

AverageSLIDE-Hoek-Brown ModelTangential method

β = 45, kh

= 0.2

GSI=10

GSI=50

GSI=100

Hβ

5 10 15 20 25 30 350.01

0.1

1

10

100


mi

β = 60, kh

= 0.2

GSI=10

GSI=50

GSI=100

Hβ

5 10 15 20 25 30 350.01

0.1

1

10

100

1000AverageSLIDE-Hoek-Brown ModelTangential Method

mi

β = 75, kh

= 0.2

GSI=50

GSI=10

GSI=100

Hβ



An analysis of static rock slope stability using the Hoek–Brown criterion has been recently performed by Li et al.[19]. Figs. 8 and 9 display the stability numbers from the

PS analyses compared to the results of these static analyses.The average lines shown in these figures with kh = �0.1represent the stability numbers obtained when the PS force

5 10 15 20 25 30 350.01

0.1

1

10

100

mi

AverageSLIDE-Hoek-Brown Model

β = 30, kh

= 0.3

GSI=100

GSI=50

GSI=10

N=

σ ci/γ

HF

N=

σ ci/γ

HF

N=

σ ci/γ

HF

N=

σ ci/γ

HF

Hβ

5 10 15 20 25 30 35

0.1

1

10

100

mi


β = 45, kh = 0.3

GSI=10

GSI=50

GSI=100Hβ

5 10 15 20 25 30 350.1

1

10

100

mi


β = 60, kh = 0.3

GSI=10

GSI=50

GSI=100Hβ

5 10 15 20 25 30 350.1

1

10

100

1000

mi


β = 75, kh = 0.3

GSI=10

GSI=50

GSI=100

Hβ



acts toward the rock slope face. Understandably, when thePS force acts toward the slope, this was found to increasethe stability of the slope as seen in Fig. 8. In general, therock slopes and the potential epicentres such as faults or

volcanos can scatter around in seismically active region,so adopting the more critical khW direction (away formthe slope) in the analyses is more appropriate from a designperspective.

5 10 15 20 25 30 350.01

0.1

1

10

β = 75β = 60β = 45β = 30

GSI=80, kh

= 0.1

N=

σ ci/γ

HF

N=

σ ci/γ

HF

N=

σ ci/γ

HF

mi

5 10 15 20 25 30 35

mi

5 10 15 20 25 30 35

mi

Hβ

0.1

1

10

GSI=50, kh

= 0.2

β = 75β = 60β = 45β = 30

Hβ

1

10

100

GSI=20, kh

= 0.3

β = 75β = 60β = 45β = 30

Hβ

Fig. 7. Average finite element limit analysis solutions of stability numbers.

5 10 15 20 25 30 35

0.01

0.1

kh = -0.1

kh = 0.3

kh = 0.2

kh = 0.1

kh = 0.0

β =30, GSI =100

Hβ

N=

σ ci/γ

HF

mi

5 10 15 20 25 30 35m

i

Decreasing Stability

0.01

0.1

1

kh = -0.1

β =45, GSI =70

N=

σ ci/γ

HF

kh = 0.0

kh = 0.1

kh = 0.2

kh = 0.3

Hβ


Fig. 8. Comparisons of stability numbers between the static and pseudo-static analyses (b = 30� and 45�).


Referring to Figs. 8 and 9, it can be found that Nincreases with increasing kh. This means that the factorsof safety will be smaller as the earthquake loading

increases. In addition, from these figures, the factor ofsafety is found to decrease by 30% (mi = 5) or more whenkh increases by 0.1 for all slope angles.

Fig. 10 shows the several of the observed upper boundplastic zones for different GSI. In general, the modes offailure consisted of shallow toe type mechanisms for allanalyses. It can be noticed that the depth of slip surfaceincreases only slightly with increasing GSI. But this phe-nomenon is not observed when the slope angle b 6 45�.A similar trend is found where the depth of slip surfaceincreases slightly with the reduction of mi. Moreover,Fig. 11 indicates that the depth of the plastic zones isalmost unchanged for various seismic coefficients. Thismeans that the shape of the potential failure surface isalmost independent of the magnitude of kh for a givengeometry and rock strength parameters.

5.2. Analytical upper bound solutions

Using the generalised tangential technique proposed byYang et al. [45] in conjunction with the assumed failuremechanism proposed by Chen [46] shown in Fig. 12, theoptimised height of a slope with Hoek–Brown rock

0.1

1

10

kh = -0.1

β =60, GSI =50

N=

σ ci/γ

HF

kh = 0.0

kh = 0.1

kh = 0.2

kh = 0.3

Hβ


5 10 15 20 25 30 351

10

100

kh = -0.1

N=

σ ci/γ

HF

β =75, GSI =20

kh = 0.3

kh = 0.2

kh = 0.1

kh = 0.0

mi

5 10 15 20 25 30 35m

i

Hβ


Fig. 9. Comparisons of stability numbers between the static and pseudo-static analyses (b = 60� and 75�).

10=GSI

50=GSI

100=GSI

Fig. 10. Comparisons between the upper bound plastic zones and failuresurfaces of the tangential method for various GSI (kh = 0.1 and mi = 10).


strength parameters can be obtained using tangentialparameters (ct and /t). Yang et al. [45] proposed that, fora given friction angle (/t), ct can be expressed as

ct

rci

¼ cos /t

2

mbað1� sin /tÞ2 sin /t

� �ða=1�aÞ

� tan /t

mb1þ sin /t

a

� �

� mbað1� sin /tÞ2 sin /t

� �ð1=1�aÞ

þ smb

tan /t ð3Þ

where s, mb and a are from the latest version of Hoek–Brown yield criterion shown in Eq. (1). Therefore, thenew tangential Mohr–Coulomb yield criterion can be ex-pressed as

srci

� �¼ rn

rci

tan /t þct

rci

� �ð4Þ

In addition, to transfer the yield surface from the (s/rci � rn/rci) plane to the major and minor principal stressplane (r1/rci�r3/rci), we can use Eq. (5)

r1

rci

¼ 2ðc=rciÞ cos /1� sin /

þ 1þ sin /1� sin /

� r3

rci

ð5Þ

where c and / are the cohesion and friction angle in thes � rn plane. It should be noted that /t in Eq. (4) is

unknown and determined by the optimisation that thesmallest slope height is obtained which is given as [46]:

H ¼ sin b0

2c sinðb0 � aÞ tan /t

� ct exp 2ðhh � h0Þ tan /t½ � � 1f gðf1 � f2 � f3 � f4Þ þ khðf5 � f6 � f7 � f8Þ� sinðhh þ aÞ exp ðhh � h0Þ tan /t½ � � sinðh0 þ aÞf g

ð6ÞEq. (6) has been formulated in FORTRAN and optimisedin this study using the Hooke–Jeeves algorithm which is afunction of four variables, namely hh, h0 and b

0for specify-

0.1=hk

2.0=hk

3.0=hk

Fig. 11. Comparisons between the upper bound plastic zones and failuresurfaces of the tangential method for various kh (GSI = 50 and mi = 15).

ct

Shea

r st

ress

(τ)

Normal stress (σn)

φt

(a) Tangential method

(b) Logarithmic spiral failure mechanism

H

hθoθ

ββ ′

Wkh

Wkv

α

Fig. 12. Illustration of adopted tangential method and failure mechanism.


ing the assumed failure mechanism and /t for specifyingthe location of the tangency point. More details on howthe parameters (hh, h0 and b

0) are determined and the def-

initions of the parameters f1–f8 can be found in Chen [46]and Yang et al. [45].

To examine the results of PS analysis employed in thepresent study, the newly obtained average lower and upperbound solutions are compared with the optimised upperbound solutions using Eq. (6) in Figs. 4 and 5. The tangen-tial upper bound solutions are shown as cross symbols

which are stability numbers obtained from Eq. (2) forGSI = 10, 50 and 100. It can be observed that most ofthe cross symbols plot below the newly obtained averagelower and upper bound solutions. This means that usingthe tangential approach may overestimate the factor ofsafety for the rock slope.

The difference in stability numbers between the averagebounding solutions and the results of the tangentialmethod is found to increase sharply when GSI and slopeinclination increase. For the case of b = 75� andGSI = 100 (Figs. 4d and 5d), the difference can be by uparound 80% which is quite significant and cannot beignored. Referring to these two figures, the value of N

obtained using the tangential method with GSI = 50 isfound to be smaller than that of newly obtained lowerand upper bound average solutions with GSI = 60. In par-ticular, for larger mi (mi P 25), they are close to the aver-age bound solutions with GSI = 70. It was found that thestability number estimate of tangential technique tends tobe unconservative, particularly for larger GSI.

In order to determine the source of overestimation in thefactor of safety when using the tangential method, the fail-ure surface information from the numerical lower boundresults is observed more closely. The stress componentsin the vicinity of the lower bound failure zone/mechanismhave been extracted and the location of each point on theHoek–Brown yield envelope can be observed. This isshown in Fig. 13 for b = 75� and kh = 0.1. For comparison

-0.0005 0.0000 0.0005 0.00100.00

0.01

σ 1 / σci

σ 1 / σci

σ 1 / σci

σ3 / σ

ci

σ3 / σ

ci

σ3 / σ

ci

Hoek-Brown (GSI=10, mi=10)

Tangential method Lower bound (k

h=0.1)

Lower bound fitted yield surface σ

1 / σ

ci=15.784 (σ

3 / σ

ci) + 0.0025

-0.01 0.00 0.01 0.02 0.03 0.04 0.05

0.0

0.1

0.2

0.3

0.4

0.5



h=0.1)

Fitted yield surfaceσ

1 / σ

ci=13.276 (σ

3 / σ

ci) + 0.0511

-0.15 -0.10 -0.05 0.00 0.05 0.10 0.15 0.20 0.25 0.30

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0



h=0.1)

Lower Bound Fitted yield surfaceσ

1 / σ

ci=6.0417 (σ

3 / σ

ci) + 0.8826

Fig. 13. Comparisons of yield surfaces between numerical lower boundand tangential method (b = 75�).

Table 1Comparisons of tangential Mohr–Coulomb parameters for various qualityrocks (b = 75� and kh = 0.1)

GSI mi Tangential method Lower bound linear regression

/t ct/rci /t ct/rci

10 10 60.7 0.000314 61.74 0.00031550 10 48.51 0.021347 59.31 0.00701

100 10 20.038 1.56842 45.72 0.1796


purposes, linear regression and Eq. (5) are also employedhere to obtain the magnitudes of /t and ct/rci from thelower bound solutions.

Table 1 displays optimised /t and ct/rci values from thetangential approach. Compared with the lower bound lin-ear regression values, the differences in /t and ct/rci are sig-nificant, except for the case of GSI = 10. Due to the factthat /t and ct/rci are almost the same for GSI = 10 and

mi = 10, the difference in stability numbers between thesetwo approaches is small, as shown in Fig. 4d.

Referring to Fig. 13, based on Eq. (5) and Table 1, theyield surface adopted in tangential method can be plottedin the r1/rci � r3/rci plane. The difference between the tan-gential method and native Hoek–Brown yield surfaces isfound to increase significantly with increasing GSI. There-fore, the difference in stability numbers between the aver-age bounding solutions and results of tangential method,as mentioned previously, can be explained. For the caseof GSI = 50 and 100, it is apparent in Fig. 13 that the yieldsurface from the lower bound solutions plots below thatfrom the results using the tangential technique. In particu-lar, for GSI = 100, the yield surface of the tangentialmethod is significantly above the Hoek–Brown failure sur-face which will lead to a larger shear strength and thus alarger factor of safety. This explains the large differencesobserved in Figs. 4 and 5 and the unconservative natureof the tangential technique for these cases.

The predicted slip surfaces by using the tangentialmethod (bold line) are compared with the upper boundplastic zones in Figs. 10 and 11. It can be observed thatthe difference between the obtained slip surfaces increasessignificantly with increasing GSI, which agrees with the dif-ference in the yield surfaces shown in Fig. 13. In Fig. 11, byusing the tangential method, the variation of slip surface isalso found to be insignificant for various kh.

5.3. Limit equilibrium method (LEM) solutions

In this study, the commercial limit equilibrium softwareSLIDE [47] and Bishop’s simplified method [48] have beenemployed to make comparisons with the average lowerand upper bound solutions. A limit equilibrium analysiswhich adopts the Hoek–Brown strength parameters can beperformed when using SLIDE. More information in howSLIDE calculates and determines the shear strength of eachindividual slice can be found in Li et al. [19] and Hoek [23].

Referring to Figs. 4–6, the displayed triangular pointsare the stability numbers obtained using the Hoek–Brownstrength parameters based on the LEM. It can be seen thatmost of the results from SLIDE are remarkably close to theaverage lines of the upper and lower bound limit analysissolutions. However, the exception to this observation canbe found in the case of b = 75�, shown in Figs. 4d and6d. In these figures, for lower GSI values (GSI 6 50), thestability numbers obtained using the LEM have beenunderestimated by 18–30%, compared to the average

5 10 15 20 25 30 35

1

10

100

GSI=70

GSI=30

SLIDE-Hoek-Brown Model

mi

N=

σ ci/γ

HF

β = 75, kh

= 0.4

GSI=10

GSI=50

GSI=100

Hβ

Fig. 14. Stability numbers from the limit equilibrium analyses (kh = 0.4).

-0.2 -0.1 0.0 0.1 0.2 0.30.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

σ 1/ σ

ci

σ3 / σ

ci

GSI=100 mi=35, Lower bound (k

h= 0.0)


h= 0.3)


h= 0.0)


h= 0.3)

GSI=100 mi=35

GSI=100 mi=5

GSI=50 mi=5

Tensile Compressive

Fig. 15. The Hoek–Brown failure criterion for variable GSI and mi valuesand observed lower bound results.


bound solutions. Although comparisons between the upperbound and LEM show an underestimation of between 10–22%, the difference in stability numbers between the limitanalysis and the LEM is still quite significant. Due to thetrue stability numbers being bounded by the upper andlower bound limit analysis solutions, results obtained fromthe LEM tend to be unconservative. This means that, forrock slope of high slope inclination (b P 75�), unsafe fac-tors will be obtained by using the PS limit equilibriumanalyses.

5.4. Investigation of stability numbers increasing with

increasing mi

As mentioned above, Fig. 6d indicated that for highslope angles (b P 75�) and high lateral coefficients(kh P 0.3), the stability number N was found to actuallyincrease slightly with increasing mi. This implies that thestability of the slope is essentially independent of the mate-rial shear strength. This observation requires a more thor-ough investigation.

It is evident from Fig. 6d that the LEM results do notexhibit the same response compared to the numericalbounds for b = 75� and kh = 0.3. Therefore, additionallimit equilibrium analyses for slopes with b = 75� andkh = 0.4 have been performed (using SLIDE) with theresults illustrated in Fig. 14. Now we can observe thatthe stability number N increases with mi increasing whenkh = 0.4 is adopted. This trend is similar to the results ofthe bounding methods shown in Fig. 6d where kh = 0.3.

With this being the case we can conclude that theobserved phenomenon is real and needs further investiga-tion. In doing so, the lower bound stress conditions inthe region of the failure plane were extracted and observedmore closely. The obtained information is displayed inFig. 15 along with the Hoek–Brown yield envelope for eachrock material. When GSI = 100, mi = 35 and kh = 0.0,most of stress points extracted along the slip surface, andtherefore much of the slip surface length itself, are in a stateof compression. In contrast, for the case of GSI = 100,mi = 35, and kh = 0.3 (square symbols), most of the stresspoints extracted along the slip surface fall in the regionwith small r1/rci � r3/rci and are in tension. This observa-tion is also true when GSI = 100, mi = 5, and kh = 0.3 (tri-angular symbols). This suggests that the tensile strength ofthe material will tend to dictate the overall stability. It isfound in Fig. 15 that when mi = 5 the tensile strength is lar-ger than that for mi = 35. This means that when collapse ofthe rock slope is due to tensile failure, rock masses with asmaller mi can provide high strength and therefore aremore stable. This would explain the decrease in stabilitynumber N with increasing mi seen in Fig. 6d. It should benoted that in Fig. 15, for the rock masses with lowerGSI, although the tensile strength is relative small, it stillplay an important role. Therefore, the phenomenon thatstability number N increases slightly with increasing mi

exists as well.

The observed stress conditions on the base of each slicein limit equilibrium analysis using SLIDE are shown inFig. 16 for b = 75�, mi = 35, GSI = 100 and kh = 0.4.The square points represent the stresses of each slice forthis case. It can be clearly seen that these stresses locateintensively in the region with the small rn/rci and compar-ing to the case with mi = 5, the shear stress is actuallylower. This means that, based on the Hoek–Brown analyt-ical model, the shear strength of rock masses with mi = 5

-0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.70.0

0.2

0.4

0.6

0.8

1.0

1.2

τ / σ

ci

σn / σ

ci

Tensile Compressive

GSI=100 mi=35 GSI=100 m

i=5

GSI=100 mi=35, SLIDE (k

h = 0.0)


h = 0.4)


h = 0.4)

Fig. 16. The observed stress conditions along the slip plane from the limitequilibrium analyses.


and mi = 35 will be similar for small ratios of rn/rci. Whileconsidering the case with kh = 0.0, larger rn/rci wasobserved on the base of the slices, where shear strengthfor mi = 5 is obviously less than that of mi = 35.

It was found that, under the strong earthquake loading,rock slopes tend to fail due to tensile stresses. It should benoted that this phenomenon only occurs for steep slopescombined with a high seismic coefficient. However, tensilestrength of rocks or rock masses can be small. Therefore,a rock slope stability based design is recommended toavoid this situation.

6. Conclusions

Using the numerical upper and lower bound techniques,rigorous bounds on the seismic rock slope stability havebeen presented as chart solutions. These stability charts,including the earthquake effects, are based on the Hoek–Brown failure criterion and can be used for estimating seis-mic rock slope stability in the initial design phase. Thisstudy follows the general consideration of the earthquakeeffects which only takes the horizontal seismic coefficient(kh) into account. A range of kh magnitudes is alsoincluded, which are consistent with most design codes.Although the vertical seismic coefficient (kv) is oftenignored in practice, it is still worth investigating its influ-ence on the stability of rock slopes in further studies. Basedon this study, the following conclusions can be made.

A narrow range of stability numbers (N) have beenbounded using pseudo-static (PS) upper bound and lowerbound solutions within ±9% or better for all considered cases.

The stability analysis of rock slopes using the limit equi-librium method was found to overestimate the factors ofsafety for the cases with higher slope angles and lowerGSI. Compared with the upper bound solutions alone, thisoverestimate can be as high as 22% for the steep slopes (i.eb = 75�) with GSI values less than approximately 50.

When the horizontal seismic coefficient (kh) increases bya factor of 0.1, the safety factor of a rock slopes maydecrease by more than 30%. But reducing the angle of slopeby 15� can increase the safety factor by at least 50%.

By using the tangential method proposed by Yang et al.[45], the overestimates of rock slope stability increase withGSI increasing and can be up to 80%. In addition, an inac-curate prediction of slip surface would be obtained.

It was found that for the cases with high slope angle andsignificant seismic coefficient the stability numbers for rockslopes obeying Hoek–Brown yield criterion will increasewith increasing mi. The reason for this was due to the ten-sile nature of overall failure for these specific cases.

Due to the fact that rocks and rock masses are not goodmaterials when it comes to providing tensile strength, thedesign for a rock slope should to be done to avoid tensilestresses development.

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