StabilityChartsforPseudostaticStabilityAnalysisofRockSlopes … · 2020. 8. 6. · slope angle β...

Research ArticleStability Charts for Pseudostatic Stability Analysis of Rock SlopesUsing the Nonlinear Hoek–Brown Strength Reduction Technique

Chaowei Sun ,1 Junrui Chai,1,2 Tao Luo,2 Zengguang Xu,2 Yuan Qin,2

Xiaosa Yuan,1 and Bin Ma1

1Shaanxi Key Laboratory of Safety and Durability of Concrete Structures, Xijing University, Xi’an 710123, China2State Key Laboratory Base of Ecohydraulics Engineering in Northwest Arid Area, Xi’an University of Technology,Xi’an 710048, China

Correspondence should be addressed to Chaowei Sun; [email protected]

Received 9 April 2020; Revised 1 July 2020; Accepted 8 July 2020; Published 6 August 2020

Academic Editor: Chunshun Zhang

Copyright © 2020 Chaowei Sun et al. (is is an open access article distributed under the Creative Commons AttributionLicense, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work isproperly cited.

(is paper presents a set of stability charts for the stability assessment of rock slopes that satisfy the Hoek–Brown (HB) criterionunder various seismic loading conditions. (e nonlinear Hoek–Brown strength reduction technique is used to conductpseudostatic stability analysis of rock slopes subjected to horizontal seismic excitation. Based on an extensive parametric study,first, a set of stability charts with a slope angle of β� 45° under static and pseudostatic conditions are proposed by using ABAQUS6.10 software. Second, the slope angle weighting factor (fβ) and the seismic weighting factor (fkh) are adopted to characterize theinfluence of slope angle (β) and horizontal seismic acceleration coefficient (kh) on the rock slope stability. Finally, the reliability ofthe proposed charts was validated by three typical examples and two case studies, and the results show that the values of the factorof safety (FOS) obtained from the proposed charts are consistent with the values from other methods.(e proposed charts providean efficient and convenient way to determine the FOS of rock slopes directly from the rock mass properties (c and σci), the HBparameters (mi and GSI), the slope geometry (H and β), and the horizontal seismic coefficients (kh).

1. Introduction

In regions of high seismic intensity, earthquakes are a majorcause of man-made and natural slope failures. (erefore,conducting stability analyses of rock slopes subjected to theseismic conditions has been regarded as an important anddifficult issue in civil and mining engineering. Stabilitycharts provide an efficient and convenient way for pre-liminary and rapid slope stability evaluation, and they havebeen routinely applied in practical application to calculatethe factor of safety (FOS) of a slope. Since Taylor [1] putforward a set of stability charts for soil slopes for the firsttime, many attempts have been made to develop such chartsfor rock or soil slopes, e.g., Gens et al. [2], Baker [3], Li et al.[4, 5], Michalowski [6, 7], Steward et al. [8], Gao et al. [9], Eid[10], and Sun et al. [11]. However, these charts are based onthe commonly used Mohr-Coulomb (MC) failure criterion

and need the shear strength parameters of cohesion c andinternal friction angle φ for slope stability analysis. However,the rock masses are discontinuous and inhomogeneousmedia characterized by intact rock and artificially or nat-urally occurring discontinuities such as joints, faults, bed-ding planes, and fractures. (erefore, the linear MC failurecriterion is generally not applicable to describe the failureenvelope of the rock mass [12–15].

Since the HB failure criterion, originally presented byHoek and Brown [16], reflects the nonlinear nature of therock mass strength, it is currently one of the most com-monly used failure criteria to predict the strength of intactrock (Group I) and rock masses with heavy joints ordiscontinuities (Group III), as shown in Figure 1. (elatest version is the generalized Hoek–Brown (GHB)criterion proposed by Hoek et al. [17] and is expressed asfollows:

HindawiAdvances in Civil EngineeringVolume 2020, Article ID 8841090, 16 pageshttps://doi.org/10.1155/2020/8841090

mailto:[email protected]://orcid.org/0000-0001-7970-3304https://creativecommons.org/licenses/by/4.0/https://creativecommons.org/licenses/by/4.0/https://creativecommons.org/licenses/by/4.0/https://creativecommons.org/licenses/by/4.0/https://doi.org/10.1155/2020/8841090

σ1 � σ3 + σci mbσ3σci

+ s

α

, (1)

where σci is the uniaxial compressive strength of the intactrock and

mb � mi expGSI − 10028 − 14D

, (2)

s � expGSI − 1009 − 3D

, (3)

α �12

+16

exp −GSI

15 − exp −

203

. (4)

Based on (2) to (4), the input properties of s, α, and mbrely on the geological strength index (GSI) depending on thedegree of fracturing of the rock mass. (e GSI values rangefrom 5 for extremely fractured rock masses to 100 for intactrock masses.mi is an index related to the type of rock, and itsvalue varies from 1 to 35, representing different magnitudesof rock hardness. D is the disturbance factor, ranging from 0for undisturbed in situ rock masses to 1.0 for disturbed rockmasses. Guidelines on how to obtain the values of GSI andmi can be found in Hoek and Bray [18], Marinos and Hoek[19], and Hoek et al. [20]. In this research, the value of D� 0is used in the analyses of rock slope stability.

Currently, the strength reduction method (SRM) is veryattractive for solution of slope stability problem. Since thismethod was first introduced by Zienkiewicz et al. [21], it hasbeen commonly accepted and employed in practice by manygeotechnical engineers and researchers because there are twomajor advantages when SRM is applied to analyze the slopestability: (i) the SRM can not only consider the stress-strainbehavior but also simulate the progressive failure mode ofthe slope with complicated geometric shapes and loadingconditions; and (ii) the critical failure surface, including thelocation and shape, can be located automatically while theFOS is obtained simultaneously [22]. (erefore, the purpose

of the present research is to conduct a pseudostatic stabilityanalysis of rock slopes based on the nonlinear HB strengthreduction technique presented in our previous study [23]and to propose the stability charts for evaluating the stabilityof rock slopes that satisfy the HB criterion under the seismiccondition. A simple method consisting of locating thetangent of the HB envelope and introducing the instanta-neous MC shear strength parameters is used to implementthe nonlinear HB strength reduction technique. (e PSstability analysis in this paper only considers the effect ofhorizontal seismic loads, and the vertical seismic loads arenot considered. However, the proposed charts provide aconvenient way to determine the FOS of rock slopes directlyfrom rock mass properties (c and σci), the HB parameters(mi and GSI), slope geometry (H and β), and horizontalseismic coefficients (kh).

2. Review of Existing Stability Charts for RockSlopes Based on the HB Criterion

It remains difficult to develop stability charts on the basis ofthe HB failure criterion since at least six parameters (σci, mi,c, H, β, and GSI) are included in stability analysis for a rockslope under the condition of D� 0. Based on our literaturereview, only the stability charts proposed by Li et al. [24–26],Carranza-Torres [27], Jiang et al. [28], and Shen et al. [29]can be applied to obtain the FOS directly from the HB failurecriterion. Carranza-Torres [27] used the simplified Bishopmethod to analyze the rock slope stability and found that, fora given slope with α� 0.5, the FOS is related only to the threeindependent parameters of cH, s/m2b, and β. However, thesecharts are limited to the use of stability estimation of a rockslope with specified values of α� 0.5 and β� 45°. Recently,based on the limit equilibrium method (LEM), Shen et al.[29] first proposed a new chart-based stability analysismethod for rock slopes using the HB criterion (see (5)) byintroducing two weighting factors of fD and fβ (see (6)) toexamine the influence of the disturbance factor D and the

Group II

Single discontinuities Two discontinuities

Group III

Several discontinuities Jointed rock massIntact rock

Joint rockσciGSI, mi, γ

Group I

Figure 1: Applicability of the HB failure criteria for the rock slope [17].

2 Advances in Civil Engineering

slope angle β on the stability of rock slopes. However, thesechart solutions were suitable for stability analysis of rockslopes under static condition, and the seismic effects werenot considered in their studies:

FOS � FOS45° × fβ × fD, (5)

fβ � 2.66e− 0.022β

(0< FOS< 4). (6)

(e pseudostatic (PS) approach is a widely adopted andaccepted technique in engineering practice and has beenemployed to investigate the earthquake effects on rock slopestability [30–37]. Based on the PS method, Li et al. [25] firstused the limit analysis method (LAM) to develop the sta-bility charts for seismic stability assessment of the rockslopes directly based on the HB failure criterion. (e defi-nition of the dimensionless stability number (N) used intheir study is shown in

N �σci

cHFOSLAM, (7)

where FOSLAM is the FOS calculated by the LAM. (erepresentative seismic stability charts for a rock slope withβ� 45° presented by Li et al. [25] are shown in Figure 2. Anarrow range of the dimensionless stability numbers hasbeen bounded, so the average values ofN are used to developsuch charts for simplicity. Based on these charts, the stabilitynumber N is obtained when the parameters of GSI and miare given under the condition of D� 0. (en, the value of Nis adopted to calculate the FOS using (7). (e values ofFOSLAM are proved not to be equal to FOSLEM because thedefinition of FOSLAM differs from the definition of FOSLEMin slope limit equilibrium analysis [23, 29, 38].

3. Method of Analysis

3.1.Nonlinear Strength ReductionTechnique forHBCriterion.In the present study, to apply the HB failure criterion inconjunction with the strength reduction technique topseudostatic stability analysis of rock slopes, the instanta-neous MC shear strength parameters (the instantaneousinternal friction angle φ and instantaneous cohesion c) areobtained by locating the tangent of the HB envelope under anormal stress σn, as illustrated in Figure 3. (en, the slope ofthe tangent to the HB failure envelope yields the value of theinstantaneous φ, and the intercept with τ axis gives the valueof the instantaneous c. As shown in Figure 3, if the elementstress state (σ1, σ3) is given and described by the MC failurecriterion, the corresponding values of the instantaneous φand c can be obtained by the following equation [15]:

σnσci

�σ3iσci

+12

mbσ3σci

+ s

α

1 −αmb mb σ3/σci( + s(

α−1

2 + αmb mb σ3/σci( + s( α−1

⎡⎣ ⎤⎦,

(8)

τsσci

� mbσ3σci

+ s

α��

1 + αmb mb σ3/σci( + s( α−1

2 + αmb mb σ3/σci( + s( α−1 ,

(9)

φi � arcsin 1 −2

2 + αmb mb σ3i/σci( + s( α−1

⎡⎣ ⎤⎦, (10)

ci � τs − σn tanφi. (11)

For the implementation of the nonlinear HB strengthreduction technique, first, the grid elements are generatedfor the numerical slope model. When the basic parameters,boundary conditions, and loading are input and set accu-rately, the stress state of each element in the slope model canbe obtained by conducting the elastoplastic analysis. (en,each element within the slope model is considered to satisfythe instantaneous MC failure envelope, and the instanta-neous φ and c corresponding to different element stressstates can be determined based on (8) to (11). In the presentstudy, the estimation of the instantaneous MC strengthparameters from the HB failure criterion is combined withthe shear strength reduction technique for stability analysisof a rock slope using the finite element software ABAQUS6.10.More details about the implementation of the nonlinearHB strength reduction technique are given in the latestresearch [23].

3.2. Pseudostatic Approach. In the PS approach, the earth-quake load is replaced by an equivalent static force, and itsmagnitude is expressed as a product of horizontal or verticalseismic coefficients (kh and kv) and the weight of the soil orrock mass. Although it is generally recognized that the PSapproach is conservative, the method is commonlyemployed in research as a result of its simplicity and ef-fectiveness. In addition, currently, the slope design is oftenmore concerned about the horizontal seismic effect in slopePS stability analysis. (erefore, this research emphasizes theanalysis of the earthquake effects on rock slope stabilitybased on various horizontal seismic coefficients. For thepurpose of obtaining a reliable horizontal seismic coefficientfor a given site, Figure 4 presented in the California Divisionof Mines and Geology [39] gives the design suggestions forpseudostatic analysis. As Figure 4 shows, the values of khrange from 0 to 0.375 within the range of a recommendedpseudostatic safety factor. (erefore, the horizontal seismiccoefficients varying between kh � 0 and kh � 0.3 are used inthe present paper.

3.3. Problem Definition. Figure 5 shows the sketch of thedefinition for the pseudostatic stability analysis of rockslopes based on the HB failure criterion. In the present study,all the slope cases are assumed to have heavily jointed orfractured rock masses such as Group III in Figure 1, whichare thus considered as isotropic and homogeneousthroughout the slope. In this section, we carry out a para-metric study on the problem presented and discuss therelationship between the FOS and the related seven inputparameters such as the HB parameters (mi and GSI), rockmass properties (c and σci), slope geometry (β and H), andhorizontal seismic coefficients (kh).

Advances in Civil Engineering 3

Based on the investigation of 20000 landslide historiescaused by the Wenchuan Earthquake [40], the static analysisresults reveal that most of the landslides have a slope angleexceeding 35°.(erefore, the slope angles are taken to be 30°,45°, 60°, and 75° in our research. Slopes with the widely usedvalues of the horizontal seismic acceleration coefficient of0.1, 0.2, and 0.3 are analyzed.

3.4. 9eoretical Relationship between the FOS and RelatedParameters. For a rock slope with the given values of inputparameters s, α,mb, and σci, (8) and (9) can be used to calculatethe minimum principal stress σ3 and the shear stress τs, re-spectively, and the simplified equations are simply described as

σ3σci

� f1σnσci

, mb, s, α , (12)

τsσci

� f2σnσci

, mb, s, α . (13)

When the dimensionless horizontal seismic coefficientkh is introduced to conduct the pseudostatic stability analysisfor rock slopes under different horizontal seismic loadingconditions, the FOS is defined as a function of the resistingshear force Ts divided by the mobilized shear force Tm, which

100

10

1

0.1

0.01

Stab

ility

num

ber, N

5 10 15 20 25 30 35mi

AverageSlide HB model

GSI = 10

GSI = 50

GSI = 100

(a)

Stab

ility

num

ber, N

100

10

1

0.1

0.01

0.0015 10 15 20 25 30 35

mi

Average

Slide HB modelTangential method

GSI = 10

GSI = 50

GSI = 100Hβ

(b)

Stab

ility

num

ber, N

10

1

0.1

0.015 10 15 20 25 30 35

mi

Average

Slide HB modelTangential method

GSI = 10

GSI = 50

GSI = 100

(c)

Figure 2: Seismic stability charts for rock slopes using the limit analysis method [22]. (a) β � 45°, kh � 0.1. (b) β � 45°, kh � 0.2. (c) β � 45°, kh �0.3.

φi

ci

HB envelope

Instantaneous MC envelope

σn Normal stress

τs

σ1

σ3σn

τsShea

r str

ess

Figure 3: Relationship of instantaneous MC envelope and HBenvelope in the normal and shear stress plane.

Hynes andfranklin (1984)

0.40

0.35

0.30

0.25

0.20

0.15

0.10

0.05

0.00

Pseu

dost

atic

coeffi

cien

t, k h

Recommended pseudostatic safety factor1.00 1.05 1.10 1.15 1.20 1.25 1.30

Seed (1979)M 8.25

M 6.5

Figure 4: Design recommendations for pseudostatic analysis [39].


can also be described in terms of the slice weight cH and theseismic inertia force khcH, given by

Ts � cH cos β − khcH sin β,

Tm � cH sin β + khcH cos β,

⎧⎪⎨

⎪⎩(14)

FOS �Ts

Tm�

τsdAτmdA

�τsτm

�τs

cH sin β + kh cos β(

�τsσci∗

σcicH sin β + kh cos β(

�σci

cH sin β + kh cos β( f2

σnσci

, mb, s, α

�σcicH

f3σnσci

, mb, s, α, kh, β .

(15)

(e normal stress σn on the slip surface depends on theparameters of cH and kh. (erefore, (15) can be described asfollows:

FOS � f4σcicH

f3cH

σci, mb, s, α, kh, β . (16)

(e input parameters mb, s, and α in (16) can be de-termined from (2)–(4), respectively. (en, the final FOS canthen be given by

FOS � f5σcicH

, mb, s, α, kh, β � f6 mi, GSI, D, SR, kh, β( .

(17)

Equation (17) reveals that, for a rock slope with the givenvalues of mi, GSI, D, β, and kh, the FOS depends only on thenondimensional parameter of strength ratio SR (SR� σci/€H) despite the magnitude of σci, c and H. (erefore, thenumber of independent parameters for determining the FOSof a rock slope can be reduced to five (SR, mi, GSI, D, β, andkh) under the condition of D� 0, as shown in

FOS � f6 SR,GSI, mi, kh, β( . (18)

To verify this theoretical relationship as shown in (18),Table 1 shows three slope cases with different values of c, σci,and H, which have the same values of SR, GSI, mi, kh, and β.(en, the FOS values for these cases are calculated based onfive different LEMs using Slide 6.0 and the nonlinear HB

S = 1.5H

u = 0 v = 0

u =

0

u =

0

H

L = 2.5H

β

Joint rockσci, GSI, mi, γ, D

khW

W

(a)

(b)

Figure 5: Sketch of definition for pseudostatic stability analysis of rock slope in this paper: (a) typical slope model and the boundaryconditions; (b) mesh of the slope model.


strength reduction method by the software ABAQUS 6.10,and the results are listed in Table 1. Values of the FOS for allthe three cases are exactly the same. For further study, threeadditional groups that have the same values of SR, kh, and βover a range of GSI and mi were analyzed, and the FOSresults are given in Table 2. (e FOS depends only on themagnitude of SR for slopes with the same values of GSI, mi,β, and kh. Based on this observation, stability charts for rockslopes can easily be developed under different seismicloading conditions.

4. Results and Analysis

4.1. Stability Charts for Rock Slopes with β� 45° under StaticandPseudostaticConditions. Using the previously expressedparameters for stability analysis in previous section, first, aset of charts for rock slope stability with a specific slope angleof β� 45° under static conditions (kh � 0) is developed in thissection, as shown in Figure 6. Figure 6 reveals that the FOSclearly increases with the increase of SR and GSI. For in-stance, as illustrated in Figure 6(a), when SR� 1.0, increasingthe geological strength index from GSI� 10 to 100 can in-crease the FOS by more than 6 times. (e values of SR alsohave a remarkable impact on rock slope stability, especiallyunder the conditions of high GSI values, but, for low valuesof GSI≤ 60, the FOS can be seen to increase slightly with theincrease of SR. For instance, in Figure 6(a), when GSI� 100,the value of FOS is equal to 4.94 for SR� 2.0 and increases to17.88 for SR� 10. When GSI� 10, the FOS is 0.58 for SR� 2and increases to 0.95 for SR� 10. (e charts presented inFigure 6 provide a good benchmark and foundation forcomparison in subsequent parts.

Figure 7 illustrates the relationship between the FOS andmi, GSI, and SR for a slope with kh � 0.1, 0.2, and 0.3 andβ� 45°. (e FOS increases slightly with the increase of SRwhen the GSI values are at low levels (GSI≤ 60), and themaximum value ofmi � 35 and the minimum value ofmi � 5yield a narrow range of FOS values. Nevertheless, the FOScan be seen to increase dramatically as SR increases whenGSI＞ 60. In addition, as Figure 7 shows, the FOS for

different values of mi meets at one point at high GSI values,so, for the small SR values (smaller than the intersectionpoints), the FOS increases with the increase in mi, but whilethe SR values exceed the intersection points, the FOS de-creases when mi increases. As expected, when the charts inFigure 7 are compared with the charts in Figure 6, weconclude that kh has a remarkable influence on the rockslope stability. For instance, when mi � 15, GSI� 50, andSR� 10 (Figure 6(c)), the FOS values decrease from 3.05 to2.55, 2.17, and 1.87 with kh increasing from 0 to 0.3.

4.2. Charts for the Seismic Weighting Factor fkh and the SlopeAngle Weighting Factor fβ. In previous section, both theslope angle (β) and the horizontal seismic acceleration co-efficient (kh) have significant effects on rock slope stability.(e main purpose of this section is to produce the charts forthe seismic weighting factor (fkh) and the slope angleweighting factor (fβ) for use in rock slope stability analysis.

4.2.1. Seismic Weighting Factor fkh. (e seismic weightingfactor fkh is defined as the ratio of the FOS under the seismiccondition to the FOS under static conditions and adopted toexamine the influence of kh on stability of the rock slope.(e

Table 1: Comparison of the FOS for a given slope with the same values of SR.

Input parameters Case 1 Case 2 Case 3Β (°) 45 45 45GSI 50 50 50mi 15 15 15kh 0.1 0.1 0.1σci (MPa) 27.6 50 100c (kN/m3) 23 25 25H (m) 30 50 100SR (σci/cH) 40 40 40Fellenius 4.26 4.29 4.42Simplified Bishop 4.38 4.41 4.45Simplified Janbu 4.39 4.43 4.46Spencer 4.42 4.45 4.48

FOSMorgenstern Price 4.49 4.55 4.52HB strength reduction technique (present study) 4.47 4.52 4.50

Table 2: Comparison of the FOS for a given rock slope with variousHB parameters.

Input parameters FOSGSI mi Group 1 Group 2 Group 3

β� 45°kh � 0.1SR� 40

10 5 1.2 1.2 1.210 15 1.71 1.71 1.7110 25 2.01 2.01 2.0310 35 2.23 2.23 2.2450 5 4.5 4.51 4.5250 15 4.52 4.53 4.5350 25 4.7 4.69 4.750 35 4.89 4.88 4.89100 5 46.2 46.21 4.19100 15 35.25 35.26 35.25100 25 30.81 30.79 30.81100 35 27 27.02 26.99


0.0 0.5 1.0 1.5 2.0

FOS

100

50

10

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

0 5 10 15 20 25 30 35 400

2

4

6

8

10

100

9080

70

60

5040

3020

10

SR = σci/γH SR = σci/γH

(a)

FOS

10

50

100

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

0

2

4

6

8

10

2010

3040

50

60

70

8090

100

0.0 0.5 1.0 1.5 2.0 0 5 10 15 20 25 30 35 40SR = σci/γH SR = σci/γH

(b)

FOS

10

50

100

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

0

2

4

6

8

10

1020

3040

50

60

7080

90100

0.0 0.5 1.0 1.5 2.0 0 5 10 15 20 25 30 35 40SR = σci/γH SR = σci/γH

(c)

Figure 6: Continued.


FOS

10

50

100

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

0

2

4

6

8

10

1020

3040

5060

70

8090

100

0.0 0.5 1.0 1.5 2.0 0 5 10 15 20 25 30 35 40SR = σci/γH SR = σci/γH

(d)

FOS

10

50

100

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

0

2

4

6

8

10

10090

8070

6050

4030

2010

0.0 0.5 1.0 1.5 2.0 0 5 10 15 20 25 30 35 40SR = σci/γH SR = σci/γH

(e)

FOS

10

50

100

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

0

2

4

6

8

10

10090

8070

6050

4030

2010

0.0 0.5 1.0 1.5 2.0 0 5 10 15 20 25 30 35 40SR = σci/γH SR = σci/γH

(f )

Figure 6: Continued.


horizontal seismic acceleration coefficient kh is assignedvalues ranging from 0.1 to 0.3 while the values of SR,mi, andGSI are the same as slope models with β� 45°. (en, the FOSfor the slope is calculated, and the factor of fkh is obtained.Figure 8 gives the statistical data analysis results of theseismic weighting factor for slopes with β� 30°, 45°, 60°, and75°. For a slope with β� 30° and kh � 0.1 (Figure 8(a)), theminimum and maximum values of the seismic weightingfactors are 0.745 and 0.825, respectively, the average value is

0.802, and a majority of the values of fkh are located aroundthe average values. In addition, as expected, the seismicweighting factor fkh obviously decreases as kh increases. Forexample, the average seismic weighting factor decreasesfrom 0.802 to 0.544 when kh increases from 0.1 to 0.3. (esame regularity of statistical distribution can also be foundfor slopes with β� 45°, 60°, and 75°, as illustrated inFigures 8(b)–8(d). Finally, the curve fitting methods are usedto obtain the exponential functions with a high fitting degree

FOS

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

0

2

4

6

8

10

0.0 0.5 1.0 1.5 2.0 0 5 10 15 20 25 30 35 40SR = σci/γH SR = σci/γH

10

50

100100

9080

7060

5040

3020

10

(g)

Figure 6: Stability charts for rock slopes with β� 45° under static conditions kh � 0: (a)mi � 5; (b)mi � 10; (c)mi � 15; (d)mi � 20; (e)mi � 25;(f ) mi � 30; (g) mi � 35.

0 5 10 15 20 25 30 35 400

5

10

15

20

25

30

35

4040 35 30 25 20 15 10 5 0

40

35

30

25

20

15

10

5

0

FOS

FOS

2040

6080

100

1030 50

70

90

SR = σci/γH

SR = σci/γH

mi = 5mi = 25

mi = 15mi = 35

(a)

FOS

FOS

0

5

10

15

20

25

30

35

40

45

50

SR = σci/γH40 35 30 25 20 15 10 5 0

0 5 10 15 20 25 30 35 40SR = σci/γH

10 30 5070

90

50

45

40

35

30

25

20

15

10

5

0

204060

80

100

mi = 5mi = 25

mi = 15mi = 35

(b)

FOS

FOS

40 35 30 25 20 15 10 5 0SR = σci/γH

SR = σci/γH0 5 10 15 20 25 30 35 40

0

5

10

15

20

25

30

3510

30 5070

90

100

80

6040

2035

30

25

20

15

10

5

0

mi = 5mi = 25

mi = 15mi = 35

(c)

Figure 7: Stability charts for rock slopes with β� 45° under pseudostatic condition: (a) kh � 0.1; (b) kh � 0.2; (c) kh � 0.3. Note. In this figure,the biaxial coordinate is adopted to differentiate the trends of curves with different GSI values in one chart.


for slopes with β� 30°, 45°, 60°, and 75°, and the results areshown in Table 3.

For further study, Figure 9 presents the relationshipbetween the seismic weighting factor and the horizontalseismic acceleration coefficient. (e maximum value ofβ� 75° and the minimum value of β� 30° obviously yield anarrow range of values of the seismic weighting factor fkh,which means that the effect of the slope angle on the seismicweighting factor is small. (erefore, the average values of fkhare used to establish the simplified fitting equation, and theresult is given as follows:

fkh � 1.006 kh( 2

− 1.594 kh( + 1.0. (19)

4.2.2. Slope Angle Weighting Factor fβ. (e slope angleweighting factor fβ, which is defined as the ratio of the FOSunder other various slope angles to the FOS under β� 45°, isadopted to describe the effect of slope angles on the rockslope stability. (e first procedure for determining theweighting factor of fβ is to calculate the FOS under slopeangles ranging from 30° to 75° with the same values of mi,GSI, SR, and kh. After hundreds of runs to conduct thestability analysis for various slopes with a wide range ofgeometries and rock mass parameters, the statistical resultsof the slope angle weighting factor under the seismic

conditions of kh � 0, 0.1, 0.2, and 0.3 are shown in Figure 10.(e results reveal that when β� 30° and kh � 0 (Figure 10(a)),the minimum and maximum values of the slope angleweighting factor are 1.413 and 1.106, respectively, and theaverage value is 1.330. In addition, most of the slope angleweighting factors are distributed within the scope of theaverage values. Furthermore, it is observed that fβ decreasesas β increases. For example, the average value of fβ decreasesfrom 1.274 to 0.629 when β increases from 30° to 75°. (esame statistical distribution regularity is also found forslopes with kh � 0.1, 0.2, and 0.3 in Figures 10(b)–10(d).

Figure 11 shows the relationship between the slope angleweighting factor and the slope angle. (e minimum value ofkh � 0 and maximum value of kh � 0.3 are easily observed togenerate a narrow scope of the slope angle weighting factorfβ, which means that kh has relatively little effect on fkh.(erefore, the average values of the seismic weighting factor

0 10 20 30 40 50 60 70 80 90 100 110 120No. of calculated cases

1.0

0.8

0.9

0.6

0.7

0.4

0.5Seism

ic w

eigh

ting

fact

or, fkh fave: the average values of seismic weighting factor

kh = 0.1 fave = 0.802fmin = 0.745 fmax = 0.825

kh = 0.2 fave = 0.653fmin = 0.575 fmax = 0.704

kh = 0.3 fave = 0.544fmin = 0.465 fmax = 0.593

β = 30°

(a)


1.0

0.8

0.9

0.6

0.7

0.4

0.5Seism

ic w

eigh

ting

fact

or, fkh

fave: the average values of seismic weighting factor

kh = 0.1 fave = 0.838fmin = 0.787 fmax = 0.867

kh = 0.2 fave = 0.713fmin = 0.624 fmax = 0.760

kh = 0.3 fave = 0.611fmin = 0.515 fmax = 0.667

β = 45°

(b)


1.0

0.8

0.9

0.6

0.7

0.4

0.5Seism

ic w

eigh

ting

fact

or, fkh

fave: the average values of seismic weightingfactor

kh = 0.1 fave = 0.883fmin = 0.750 fmax = 0.939

kh = 0.2 fave = 0.761fmin = 0.596 fmax = 0.882

kh = 0.3 fave = 0.643fmin = 0.510 fmax = 0.776

β = 60°

(c)


1.0

0.8

0.9

0.6

0.7

0.4

0.5Seism

ic w

eigh

ting

fact

or, fkh

fave: the average values of seismic weighting factor

kh = 0.1 fave = 0.882fmin = 0.778 fmax = 0.954

kh = 0.2 fave = 0.758fmin = 0.600 fmax = 0.806

kh = 0.3 fave = 0.653fmin = 0.514 fmax = 0.719

β = 75°

(d)

Figure 8: (e seismic weighting factors for rock slopes with (a) β� 30°, (b) β� 45°, (c) β� 60°, and (d) β� 75°.

Table 3: Regression equations for the seismic weighting factors.

Slope angle, β Regression equations, fkh Fitting degree, R2

30° fkh � 0.99e−2.0kh 0.99845° fkh � 0.99e−1.6kh 0.99960° fkh � 1.01e−1.47kh 0.99675° fkh � 1.01e−1.47kh 0.996


Seismic weightingfactor for β = 30°Seismic weightingfactor for β = 45°Seismic weightingfactor for β = 60°

Fitting equation for seimicweighting factors

Seismic weightingfactor for β = 75°Average values of seismicweighting factors

fkh = 1.006(kh)2 – 1.594(kh) + 1.0 (fitting degree, R

2 = 0.999)

Horizontal acceleration coefficient, kh

1.1

1.0

0.9

0.8

0.7

0.6

0.5Se

smic

wei

ghtin

g fa

ctor

, fkh

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35

Figure 9: (e chart for seismic weighting factor.

1.6

1.4

1.2

1.0

0.8

0.6

0.4

0.20 10 20 30 40 50 60 70 80 90 100

No. of calculated cases

Slop

e ang

le w

eigh

ting

fact

or, fβ

β = 30° fave = 1.330fmin = 1.106 fmax = 1.413

β = 60° fave = 0.766fmin = 0.679 fmax = 0.897

β = 75° fave = 0.601fmin = 0.466 fmax = 0.791

kh = 0 fave: the average values of slope angle weighting factor

(a)

0 10 20 30 40 50 60 70 80 90 100No. of calculated cases

1.6

1.4

1.2

1.0

0.8

0.6

0.4

0.2

Slop

e ang

le w

eigh

ting

fact

or, fβ

β = 30° fave = 1.274fmin = 1.038 fmax = 1.361

β = 60° fave = 0.807fmin = 0.701 fmax = 0.961

β = 75° fave = 0.629fmin = 0.462 fmax = 0.878

kh = 0.1 fave: the average values of slope angle weighting factor

(b)


1.6

1.4

1.2

1.0

0.8

0.6

0.4

0.2

Slop

e ang

le w

eigh

ting

fact

or, fβ

β = 30° fave = 1.219fmin = 1.002fmax = 1.314

β = 60° fave = 0.646fmin = 0.818 fmax = 1.026

β = 75° fave = 0.636fmin = 0.447 fmax = 0.968


(c)


1.6

1.4

1.2

1.0

0.8

0.6

0.4

0.2

Slop

e ang

le w

eigh

ting

fact

or, fβ

β = 30° fave = 1.186fmin = 0.984 fmax = 1.284

β = 60° fave = 0.772fmin = 0.438 fmax = 1.008

β = 75° fave = 0.640fmin = 0.404fmax = 1.005


(d)

Figure 10: (e slope angle weighting factors: (a) kh � 0; (b) kh � 0.1; (c) kh � 0.2; (d) kh � 0.3.


are used to establish the simplified fitting equation, which isgiven as follows:

fβ � −0.787 ln(β + 5.65) + 4.069. (20)

4.3. Use and Validation of the Proposed Seismic StabilityCharts. (e proposed seismic stability charts can easily beapplied to determine the FOS of a given rock slope. First,when the values of GSI, mi, and SR are given, the FOS forβ� 45° and kh � 0 can be obtained based on the charts inFigure 5. Second, the seismic weighting factor fkh for anygiven value of kh can be determined from Figure 8 or (19).(en, for a given value of slope angle β, Figure 10 or (20) canbe used to obtain the slope angle weighting factor fβ. Finally,the FOS can be obtained as follows:

F � fβ × fkh × FOSβ�45° . (21)

(e following three examples with different slope ge-ometries and rock mass parameters are adopted to elaboratethe use of the proposed charts. Example 1: a small slopeconsists of highly fractured rock masses with the followinginput parameters: σci � 2.7MPa, mi � 5, GSI� 10, β� 30°,H� 5m, c � 27 kN/m3, and kh � 0.1. Example 2: a mediumslope consists of good quality rock masses with the followinginput parameters: σci � 0.625MPa, mi � 15, GSI� 80, β� 75°,H� 25m, c � 25 kN/m3, and kh � 0.2. Example 3: a largeopen pit slope consists of blocky rock masses with thefollowing input parameters: σci � 46MPa, mi � 35, GSI� 50,β� 60°, H� 250m, c � 23 kN/m3, and kh � 0.3. According tothe procedures for application of the charts presented, Ta-ble 4 shows the calculation results of the three examples. (eresults of FOS obtained from the charts presented show goodagreement with the results obtained from the software ofSlide 6.0, which validates the feasibility and reliability of theproposed seismic stability charts.

5. Application of the Seismic Stability Charts toCase Studies

5.1. Case 1: A Rock Slope in Zipingpu Reservoir. (e casesanalyzed in this research are adopted from the studies ofJiang et al. [28]. (e rock slope analyzed is located 1.2 kmaway in the northern bank of the Zipingpu Reservoir ofSichuan, China. According to the field investigations of Jianget al. [28], the lithology of the rock mass within this slope ismainly dolomite.(e slope has a height of 100mwith a slopeangle of 50°. (e unit weight of the dolomite rock mass is28 kN/m3; the uniaxial compressive strength of the intactrock mass σci is 100MPa, and the HB parameter mi is 10.Based on the detailed mapping work, there are threedominant joint sets (J1, J2, and J3) and one damage zonewithin the slope. (e discontinuities are slightly weathered

Slope angle weightingfactor for kh = 0Slope angle weightingfactor for kh = 0.1Slope angle weightingfactor for kh = 0.2

Slope angle weightingfactor for kh = 0.3Average values of slopeangle weighting factorsFitting equation for slopeangle weighting factors

1.4

1.3

1.2

1.1

1.0

0.9

0.8

0.7

0.6

0.5

Slop

e ang

le w

eigh

ting

fact

or, fkh

Slope angle, β/°25 30 35 40 45 50 55 60 65 70 75 80

fβ = –0.7871n(β + 5.65) + 4.069 (0 ≤ FOS ≤ 3, fitting degree, R2 = 0.982)

Figure 11: (e chart for the slope angle weighting factor.

Table 4:(ree slope examples analyzed using the proposed seismicstability charts.

Input parameters Example 1 Example 2 Example 3β (°) 30 75 60GSI 10 80 50mi 5 15 35kh 0.1 0.2 0.3σci (MPa) 2.7 0.625 46c (kN/m3) 27 25 23H (m) 5 25 250SR (σci/cH) 20 0.625 8

FOSβ�45° 1.170 3.085 3.434Calculated data

fβ 1.256 0.614 0.776fkh 0.851 0.721 0.612Proposed charts 1.251 1.366 1.631

FOSSlide 6.0 software 1.320 1.381 1.642


planar surfaces, and the persistence ranges from 0.2 to 3.0mwith the apertures

angle of slope increases from 30° to 90° as shown inFigure 13(b), respectively. It is known that the seismic forceand the angle of slope also exhibit nonnegligible negativeinfluences on slope stability, and thus it is necessary toinvestigate the effects of seismic forces on slope stability andto provide a convenient approach to estimate the seismicstability of a 3D slope.

(erefore, our further study will aim to conduct apseudostatic stability analysis of 3D rock slopes in Hoek–Brown media and develop a set of seismic stability chartsbased on the nonlinear Hoek–Brown strength reductiontechnique.

7. Conclusions

(is paper uses the nonlinear Hoek–Brown strength re-duction technique to conduct the pseudostatic stabilityanalysis of rock slopes and proposes the seismic stabilitycharts for rock slopes that satisfy the Hoek–Brown failurecriterion. (e proposed charts can be adopted for quicklyevaluating the seismic stability of rock slopes in the pre-liminary design phase. (e following main conclusions aredrawn from this study:

(1) Study of the theoretical relationship between the FOSand related parameters reveals that the FOS is de-pendent only on the dimensionless parameter ofstrength ratio SR (SR� σci/cH) despite the magni-tude of σci, c and H. (e number of independentparameters for determining the FOS is reduced tofive (SR,mi, GSI,D, β, and kh) under the condition ofD� 0, which make it possible to develop the seismicstability charts for rock slopes satisfying theHoek–Brown failure criterion. (is conclusion isvalidated by the results of the slope cases in Tables 1and 2.

(2) Utilizing a simple methodology to locate the tangentof the Hoek–Brown envelope and introducing theinstantaneous MC shear strength parameters make it

possible to implement the nonlinear Hoek–Brownstrength reduction technique by using ABAQUS 6.10software. Based on this method, a set of stabilitycharts is then developed as shown in Figures 6 and 7to determine the FOS of rock slopes with slope angleβ� 45° and horizontal seismic coefficient kh rangingfrom 0.1 to 0.3. (e FOS is observed to increaseobviously with the increasing values of GSI and SR.For GSI≤ 60, FOS increases slightly when SR in-creases, but the increase is remarkable for GSI> 60.In addition, the FOS decreases with increasingmi forhigh values of GSI (GSI> 60) and kh (kh≥ 0.3), andthe maximum value of mi � 35 and the minimumvalue of mi � 5 yield a narrow scope of FOS.

(3) (e seismic weighting factor fkh and slope angleweighting factor fβ are adopted to examine the effectof the horizontal seismic acceleration coefficient andthe slope angle on rock slope stability. (en, thecharts for determination of fkh (Figure 9) and fβ(Figure 11) are developed for 30°≤ β≤ 75° and0≤ kh≤ 0.3, respectively. Based on these charts,values of fkh and fβ are obtained for various values ofkh and β.(en, in conjunction with the static stabilitycharts for slope angle β� 45° (Figure 6), the final FOScan be described as F� fβ × fkh × FOSβ�45°. (e reli-ability of the proposed charts has been investigatedby three typical examples and two slope case studies,and the results of FOS obtained from the chartspresented show good agreement with those fromother methods, which validates the feasibility andreliability of the proposed seismic stability charts.

(4) In the present analysis for pseudostatic slope sta-bility, the seismic effect is approximately consideredas a steady loading, and only the horizontal seismicforce is considered in our research. Further work willpay attention to impacts of other factors such aspore-water pressure, the vertical seismic force, andslope geomorphology on the rock slope stability.

8

7

6

5

4

3

2

FOS

0 2 4 6 8 10W/H

γ = 20 kN/m3, H = 30m, β = 30°,GSI = 40, mi = 10, σci = 20 MPa

kh = 0

kh = 0.1

kh = 0.2

kh = 0.3

(a)

FOS

0 2 4 6 8 10W/H

9

8

7

6

5

4

3

2

1

γ = 20 kN/m3, H = 30 m, kh = 0.2,GSI = 40, mi = 10, σci = 20 MPa

β = 15°

β = 30°

β = 45°β = 60°

β = 75°β = 90°

(b)

Figure 13: Effects of seismic coefficient kh and slope angle β on 3D rock slope stability: (a) kh, the seismic coefficient; (b) β, the slope angle.


Data Availability

All data, models, or codes generated or used during the studyare available from the corresponding author upon request.

Conflicts of Interest

(e authors declare that they have no conflicts of interest.

Acknowledgments

(is research was financially supported by the NationalNatural Science Foundation of China (51909224 and51679197), the Natural Science Foundation Research Projectof Shaanxi Province (2020JQ-920 and 2017JZ013), theNatural Science Research Project of Education Departmentof Shaanxi Provincial Government (19JK0910), and Scien-tific Research Staring Foundation for Introduced Talents ofXijing University (XJ18T05). (is support is gratefullyacknowledged.

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