Computers and Geotechnics 63 (2015) 291–298

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Computers and Geotechnics

journal homepage: www.elsevier .com/locate /compgeo

Research Paper

Slope stability analysis using the limit equilibrium methodand two finite element methods

http://dx.doi.org/10.1016/j.compgeo.2014.10.0080266-352X/� 2014 Elsevier Ltd. All rights reserved.

Abbreviations: LEM, limit equilibrium method; SRM, strength reductionmethod; ELSM, enhanced limit strength method; FOS, factor of safety; SRF, strengthreduction factor; PSO, particle swarm optimisation.⇑ Corresponding author.

E-mail addresses: liu-shiyi@hotmail.com (S.Y. Liu), shaolt@dlut.edu.cn (L.T.Shao), lijunli1995@163.com (H.J. Li).

S.Y. Liu a, L.T. Shao a,⇑, H.J. Li b

a State Key Laboratory of Structural Analysis for Industrial Equipment, Dalian University of Technology, 116024 Dalian, Chinab State Key Laboratory of Simulation and Regulation of Water Cycle in River Basin, China Institute of Water Resources and Hydropower Research, 100048 Beijing, China

a r t i c l e i n f o

Article history:Received 21 July 2014Received in revised form 29 September2014Accepted 15 October 2014Available online 5 November 2014

Keywords:Slope stabilityLimit equilibrium methodStrength reduction methodEnhanced limit strength methodCritical slip surface

a b s t r a c t

In this paper, the factors of safety and critical slip surfaces obtained by the limit equilibrium method(LEM) and two finite element methods (the enhanced limit strength method (ELSM) and strength reduc-tion method (SRM)) are compared. Several representative two-dimensional slope examples are analysed.Using the associated flow rule, the results showed that the two finite element methods were generally ingood agreement and that the LEM yielded a slightly lower factor of safety than the two finite elementmethods did. Moreover, a key condition regarding the stress field is shown to be necessary for ELSManalysis.

� 2014 Elsevier Ltd. All rights reserved.

1. Introduction

The limit equilibrium method (LEM) is widely used by research-ers and engineers conducting slope stability analysis. The mostcommon limit equilibrium techniques are methods of slices, suchas the ordinary method of slices (Fellenius) and the Bishop simpli-fied, Spencer, and Morgenstern-Price methods. The slices tech-nique is well known to be a statically indeterminate problem andis solved by assuming a distribution of internal forces. Conse-quently, the results obtained from particular methods can varybased on the different assumptions used.

Slope stability analysis using the finite element method hasbeen widely accepted in the literature for many years. The SRMand ELSM are the main finite element slope stability methods cur-rently employed. Comparisons between the LEM and finite ele-ment analyses of slope stability illustrate the advantages andlimitations of these methods for practical engineering problems.

The SRM was used for slope stability analysis as early as 1975by Zienkiewicz et al. [1]. This method was later termed the ‘‘shear

strength reduction technique’’ by Matusi and San [2], but the nameis now typically shortened to the ‘‘strength reduction method’’.During the last ten years, many researchers have applied theSRM to analyse slope stability problems and have compared theSRM and LEM [3–5]. The primary advantage of the SRM is thatthe critical slip surface is found automatically from the shearstrain, which increases as the shear strength decreases. However,the SRM suffers from the important limitation [3] of being unableto locate other ‘‘slip’’ surfaces (i.e., local minima).

The enhanced limit slope stability method calculates stressesusing the finite element method and searches for the critical slip sur-face with the minimum FOS. Brown and King [6] applied this methodto analyse the slope stability with a linear elastic soil model. Later,the method was named the ‘‘enhanced limit’’ slope stability methodby Nalyor [7]. The definitions of different FOSs in this method havebeen summarised, and the formulation by Kulhawy was termedthe ‘‘enhanced limit strength’’ method (ELSM) by Fredlund [8]. Theprimary task of the ELSM is to locate the critical slip surface usingmathematical optimisation. Many methods have been proposed toidentify the critical slip surface based on different optimisationmethods, such as the dynamic programming method [9–11], theBroyden–Fletcher–Goldfarb–Shanno (BFGS) algorithm [12], patternsearch [13], and particle swarm optimisation [14].

Researchers have compared the results from the SRM and LEMand those from the ELSM and LEM. However, there are few com-

292 S.Y. Liu et al. / Computers and Geotechnics 63 (2015) 291–298

parisons of the results among all three methods. Furthermore,extremely few studies compare the critical slip surfaces found bythese methods; instead, the FOS is the primary parameter of inter-est. In this paper, the locations and shapes of the slip surfaces andthe factors of safety of the slope stability calculated using the LEM,SRM and ELSM based on the assumption that the soil satisfies theMohr–Coulomb failure criterion are compared. The elastic-per-fectly plastic Mohr–Coulomb model is used in both finite elementmethods. Using the same shear strength parameters, closer agree-ment can be expected between the finite element method predic-tions and LEM results. Moreover, the elastic-perfectly plastic modelcan predict the behaviour of actual soil better than the rigid plasticmodel used by the LEM. To locate critical slip surfaces in the ELSM,a search technique combining particle swarm optimisation withpattern search is proposed.

2. Comparisons between the definition of the factors of safety

In slope stability analysis, the FOS describes the structuralcapacity of an embankment or slope, either natural or excavated,beyond the expected or actual loads. In this work, comparisonsare performed to determine the correlation among the factors ofsafety of these three methods.

2.1. LEM

The LEM defines the factor of safety (FOS) as follows:

FOS ¼ shear strength of soilshear stress required for equilibrium

ð1Þ

Eq. (1) can also be expressed as follows:

si ¼sf i

FOS¼ c0 þ r0i tan u0

FOS¼ c0

FOSþ r0i

tan u0

FOSð2Þ

where si and r0i are the shear stress and effective normal stress atthe ith slice of the slip surface, respectively, and c0 and u0 are thecohesion and internal friction angle, respectively. In other words,the FOS is ‘‘the factor by which the shear strength of the soil wouldhave to be divided to bring the slope into a state of barely stableequilibrium’’ [15]. The critical slip surface is the surface correspond-ing to the minimum value of the FOS; this minimum value is the‘‘true’’ factor of safety.

2.2. SRM

In the SRM, the FOS is defined as the factor by which the originalshear strength parameters must be divided to bring the slope tothe point of failure [4]. The factored shear strength parameters c0fand u0f are given by

c0f ¼c0

SRFð3Þ

Fig. 1. Generation of the trial slip surface.

u0f ¼ arctantan u0

SRF

� �ð4Þ

where SRF is a strength reduction factor. The FOS is equal to thevalue of the SRF that causes the slope to fail. Griffiths and Lanenoted that this definition of the FOS is exactly the same as that usedin the LEM [4].

2.3. ELSM

In the ELSM, for an arbitrary slip surface L, the FOS can bedefined as

FOS ¼Pn

i¼1sf iDLiPni¼1siDLi

¼Rsf dLRsdL

ð5Þ

where n is the number of discrete segments along L and DLi is thelength of segment i. Based on the stresses calculated by the finiteelement method, the shear stress si and the effective normal stressr0i can be expressed in the form of Eq. (2):

si ¼sf i

FOS DLið Þ ¼c0 þ r0i tan u0

FOS DLið Þ ¼ c0

FOS DLið Þ þ r0itan u0

FOS DLið Þ ð6Þ

where FOS(DLi) is the function of the strength reduction factor forsegment i.

Substituting Eq. (6) into (5),Pn

i¼1sf iDLiPn

i¼1siDLi¼

Pni¼1 c0 þ r0i tan u0� �

DLiPni¼1

c0FOS DLið Þ þ r0i

tan u0FOS DLið Þ

� �DLi

¼Rsf dLR sf

FOS DLið Þ dLð7Þ

Based on the first mean value theorem for integration, the lowerright side of Eq. (7) can be transformed intoZ sf

FOS DLið Þ dL ¼Rsf dL

FOSð8Þ

According to Eqs. (7) and (8), the FOS in the ELSM can beexpressed in the same form as Eq. (5).

For a slip line that is neither straight nor circular, the physicalmeaning of the FOS in the ELSM has been questioned by severalresearchers because the integration in Eq. (5) is neither the sum-mation of force vectors in space nor the summation of the projec-tions of force vectors in a fixed direction [16]. However, severalresearchers have already proven that the definition can be consid-ered acceptable in practical application [8,9,12,13].

The common definitions of the LEM and SRM can be interpretedas corresponding to two different methods used to obtain a set ofreduction strength parameters that cause the slope to reach its crit-ical limit equilibrium state. Based on the derivations in the ELSM, itcan be shown that the definition of the FOS is also established basedon the strength reduction, as in the LEM and SRM. Furthermore, theFOS in the ELSM is the ‘‘average’’ strength reduction factor along theslip surface based on the first mean value theorem for integration

Fig. 2. Discretisation of the slip surface.

Fig. 3. Flow chart of the pattern search.

Fig. 4. Example 1: Undrained clay slope with a weak foundation layer, cu1/cH = 0.25,H = 5 m. The global size of the seeds for the mesh is H/25.

Table 1Geotechnical parameters for example 1.

Case c (kN/m3) cu2/cu1 (–) E0 (kN/m2) m0 (–)

1 20.0 1.0 105 0.492 20.0 0.8 105 0.49

Fig. 5. Comparisons of the critical slip surfaces obtained from (a) the LEM (solidlines) and ELSM (dashed line) and (b) the SRM (solid line and contour of equivalentplastic strain) for case 1.

S.Y. Liu et al. / Computers and Geotechnics 63 (2015) 291–298 293

[13]. The critical slip surface is the surface corresponding to theminimum value of the FOS.

3. Slip surface location process

3.1. LEM

In the plane strain problem, the slip surface is assumed to be acircular arc in many existing LEMs. However, numerical analysishas revealed that the shape of the slip surface might be a combina-tion of a straight line and a circular arc or a combination of astraight line and a logarithmic spiral. Optimisation techniqueshave been shown to be the most efficient means of locating non-circular slip surfaces [17–21].

3.2. SRM

In the SRM, the critical slip surface is found automatically fromthe increase in the shear strain due to the reduction of the shearstrength. Typically, the critical slip surface is approximated using

technical measures, mesh deformation plots [4], velocity fields[22], etc. Cheng et al. [3] defined the critical slip surface as a wavyline that connects the maximum shear strain increment. Zhenget al. [23] similarly defined the critical slip surface as a wavy linethat connects the maximum equivalent plastic strain, where thewavy line is smoothed using the least squares method.

In several cases, the location of the critical slip surface definedby the SRM (e.g., the contour of the equivalent plastic strain) isunclear. In this paper, two technical measures are used to deter-mine the critical slip surface. First, the equivalent plastic strainband in the slope from the toe to the top is defined as the criticalslip surface in the SRM. The equivalent plastic strain up

eq isobtained by integrating the equivalent plastic strain rate:

epeq ¼

Z t

0

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2 _ep

ij_ep

ij=3q

dt ð9Þ

Second, once a slope is led to the limit equilibrium state by SRManalysis using the associated flow rule, the ELSM can be used as atechnical measure to search for the critical slip surface of theslopes. In such a process, the stress field is calculated with a

reduced shear strength tan u0SRF and c0

SRF

� �, and the critical slip surface

is defined by the minimum value of the FOS from the ELSM. Addi-tionally, the minimum value of the FOS of the ELSM should satisfyEq. (10):

FOS ¼Rsf dLRsdL

� 1 ð10Þ

3.3. ELSM

In the ELSM, an optimisation technique is used to locate thecritical slip surface. In the present study, a two-step search tech-nique combining particle swarm optimisation (PSO) with a patternsearch is proposed. PSO, a heuristic global optimisation algorithmdeveloped by Kennedy and Eberhart [24], has been broadly appliedin optimisation problems. In PSO, a group of particles is referred toas a candidate, and each ‘‘particle’’ is represented by one vector V.The specific procedures [21,24] are not described in this paper.

In the search technique, the trial slip surface generation algo-rithm is similar to that used by Greco [18], Malkawi et al. [19],Cheng et al. [21] and Li et al. [14]. Consider the trial surface of foursegments with five vertices, as shown in Fig. 1. Eight control vari-ables (i.e., x1, x5, b1, b5, d5, d6, d7 and d8) represent the slip surface,where d5, d6, d7 and d8 are random numbers in the range of (�0.5,0.5). A trial surface is implemented according to a three-step pro-

Table 2Summary of the factors of safety for example 1.

Method Slip surfaces FOS FOS difference with LEM2 (%)

Case 1 Case 2 Case 1 Case 2

LEM (Bishop/Geoslope) 1 Slip surface A (circular) 1.474 1.235 1.3 2.72 Slip surface B (fully specified) 1.455 1.2023 Slip surface C (optimised) 1.362 1.120 �6.4 �6.8

SRM [4] (Coarse mesh) 1.460 1.210 0.3 0.7SRM (Fine mesh) 1.451 1.215 �0.3 0.7ELSM Slip surface B 1.448 1.211 �0.5 0.7

Fig. 6. Comparisons of the critical slip surfaces and FOS obtained from the ELSM fordifferent Poisson’s ratios in example 1.

Fig. 8. Stress state of a body within the Cartesian coordinate system.

294 S.Y. Liu et al. / Computers and Geotechnics 63 (2015) 291–298

cedure. First, a temporary intersection point V 06 is obtained afterthe first four control variables x1, x5, b1 and b5 are given; these fourvariables are random values in the given range. Second, based onthe random numbers d5 and d6, the position of vertex V2 and thetemporary vertex V 07 is given by the following equations:

x2 ¼ d5 x1 � x60ð Þ þ 0:5 x1 þ x60ð Þy2 ¼ d5 y1 � y60ð Þ þ 0:5 y1 þ y60ð Þx70 ¼ d6 x5 � x60ð Þ þ 0:5 x5 þ x60ð Þy70 ¼ d6 y5 � y60ð Þ þ 0:5 y5 þ y60ð Þ

8>>><>>>:

ð11Þ

Third, by using similar equations, V3 and V4 can be located in thetwo adjacent segments whose total length in the horizontal direc-tion is the largest. In general, the trial surface with m segments canbe obtained using a step-by-step process based on the vectorV = (x1, xm+1, b1, bm+1, d5,. . ., d2m).

The slip surface is assumed to be a curve that is discretised intoseveral segments. The more vertices are used, the ‘‘smoother’’ thecurve and the higher the calculation precision. However, comput-ing with a large number of control variables is time-consuming.Thus, the search strategy in this paper discretises each segment

Fig. 7. Contour of the displacement w

between control points (i.e., a filled circle) according to a givenmesh size, as shown in Fig. 2. New intersection points are called‘‘secondary control points’’ (i.e., a hollow circle). The number ofvertices is the number of control points and secondary controlpoints combined. However, the number of control variables inone particle only depends on the number of control points. There-fore, the purpose of the search strategy is to detect the ‘‘smooth-ness’’ of the slip surface while minimising the number of controlvariables. To improve the calculation precision, the search tech-nique combines particle swarm optimisation with pattern searchand is conducted in the following steps: (i) in the POS step, the fac-tors of safety, whose objective function (F) is calculated using thestresses of the vertices, are obtained; and (ii) in the pattern searchstep, the position of the critical slip surface that corresponds to theminimum value of the FOS from the POS step is used as the initial

ith (a) v0 = 0.3 and (b) v0 = 0.49.

Fig. 9. Direction of shear stresses at several points in case 1 of example 1, (a) v0 = 0.49 and (b) v0 = 0.3. In the polar coordinate system (enlarged partial view), angularcoordinate: h ¼ aþ 90

�, radial coordinate: r ¼ smax=sj j.

Fig. 10. Example 2: A homogeneous slope with a slope angle of 25.67� (2:1), c0/cH = 0.05, H = 5 m. The global size of the seed for the mesh is H/50.

Table 3Geotechnical parameters for example 2.

Case c (kN/m3) u0 (�) E0 (kN/m2) m0 (–) w (�)

1 20.0 20 105 0.4 202 20.0 20 105 0.4 0

Fig. 11. Comparisons of the critical slip surfaces obtained from (a) the LEM (solidline) and ELSM (dashed line) (w ¼ u0 or 0

�) and (b) the SRM (solid line and contour

of equivalent plastic strain) (w ¼ u0). (c) Comparisons of the critical slip surfacesobtained from the SRM (contour of equivalent plastic strain) and the ELSM (dashedline) (w = 0�).

S.Y. Liu et al. / Computers and Geotechnics 63 (2015) 291–298 295

slip surface. In the pattern search step, the search step size (Dd)should be given first. The flow chart for the pattern search whensearching for the critical slip surface is shown in Fig. 3. The vectorU0 is the initial control variable. Along the slip surface, U(1) andU(n) of the control variable (vector U) are the x-coordinates ofthe first and last control points, respectively; other elements ofvector U include the y-coordinates of the control points and sec-ondary control points.

4. Numerical examples

To compare the results of the three methods, four representa-tive examples were selected from the literature. The influence ofPoisson’s ratio and the dilation angle on the slope stability areinvestigated in the first two examples. The three methods are thencompared for a complicated geometry: a layered slope with a thin,weak layer and a multi-stage slope.

4.1. Effect of Poisson’s ratio

Example 1 is taken from the work of Griffiths and Lane [4]. Theslope geometry and geotechnical parameters are given in Fig. 4 and

Table 1. Three cases are studied in the LEM analysis: (1) the shapeof the slip surfaces is assumed to be a circular arc, (2) the slip sur-face is only assumed to be the critical slip surface by the ELSM, and(3) the critical slip surface obtained from the ELSM is assumed tobe a trial slip surface and SLOPE/W is used to locate the non-circu-lar critical slip surface by an optimisation procedure. In the SLOPE/W, a key element in the optimisation procedure is the techniqueused to move the end points of the line segments [25]; this tech-nique moves the points within an elliptical search area using a sta-tistical random walk procedure based on Monte Carlo methods.

As shown in Fig. 5 and Table 2, the critical slip surfaces from thetwo finite element methods are essentially the same (i.e., the dashedline is almost the same as the solid line within the equivalent plastic

Table 4Summary of factors of safety for example 2.

Method FOS FOS difference with LEM (%)

w ¼ u0 w ¼ 0� w not considered w ¼ u0 w ¼ 0

�

LEM (Bishop/Geoslope, Circular) 1.376SRM [4] (Coarse mesh) 1.400 1.7SRM (Fine mesh) 1.387 1.352 0.8 �1.7ELSM 1.384 1.384 0.6 0.6

296 S.Y. Liu et al. / Computers and Geotechnics 63 (2015) 291–298

strain band), and the FOS obtained from the ELSM is similar to theFOS from the SRM. The FOS from LEM1 is larger than those fromthe two finite element methods, which could be explained by theassumed circular arc slip surface. The FOS from LEM3 is the smallest;however, the corresponding non-circular critical slip surface isextremely different from those from the two finite elementmethods.

Griffiths and Lane [4] noted that elastic parameters have only aslight influence on the predicted FOS in slope stability analysisthrough the SRM. In most SRM analyses, the Poisson’s ratio is gen-erally assumed to be 0.3; however, Poisson’s ratio is 0.49 in thisexample. Fredlund et al. [8] found that the FOS calculated by theELSM is greater when Poisson’s ratio is near 0.5 than when Pois-son’s ratio is 0.3 in certain cases. A possible explanation for thisresult is that a Poisson’s ratio approaching 0.5 corresponds to zerovolume change (i.e., rigid body motion), yielding a FOS similar tothat obtained from the LEM [9]. In this paper, however, a moredetailed explanation is given.

To illustrate the effect of Poisson’s ratio on the slope stabilityfrom the ELSM, the critical slip surfaces are obtained for case 1 inTable 1 with two different Poisson’s ratios, as shown in Fig. 6.When m0 ¼ 0:49, the results of the ELSM are reasonable and reli-able. In this case, the soil material in most of the slope area is elas-tic, with the exception of the bottom boundary. Fig. 7(a) and (b)shows the contour of displacement for Poisson’s ratios of 0.3 and

Fig. 12. Comparison of the local FOS obtained from the ELSM using the originalstrength parameters and the reduction strength parameters for example 2,SRF = 1.387.

Fig. 13. Example 3: A slope in layered soil. The global size of the seeds for the meshis 0.15.

0.49, respectively. When Poisson’s ratio is 0.49, Fig. 7(b) revealsthat the deformation of the slope is more similar to rigid bodymotion. In addition, as shown in Fig. 8, the shear stress state atany point in the model can be expressed as

s ¼ 0:5 rx � ry� �

sin 2a� sxy cos 2að Þ2 ð12Þ

The shear stresses in different directions at several pointsaround the slip surface are plotted in Fig. 9. In Fig. 9(a), the direc-tions of the shear stresses at any point along the slip surface followthe same trend towards the slide. However, in Fig. 9(b), the direc-tions of the shear stresses at the right part of the slope begin totrend opposite to the slide. As a consequence, when the Poisson’sratio is 0.3, the critical slip surface is not kinematically acceptable.

In this section, a key condition of the stress field required forELSM analysis is revealed: the shear stresses at any point alongthe slip surface should follow the same trend towards the slidewhen the soil materials are elastic.

4.2. Effect of dilation angle

In the analysis of the SRM, it is evident that the dilation angleaffects the FOS and the shape of the critical slip surface. Griffithsand Line [4] noted that a value of w ¼ 0

�yields a reliable FOS in

RSM analysis. However, researchers are also interested in knowingthe shape of the critical slip line to understand the failure mecha-nism of slopes. When the flow rule is associated (w ¼ u0), the stressand velocity characteristics coincide, and closer agreement isexpected between the failure mechanisms predicted by the finiteelement methods and the critical slip surfaces generated by theLEM.

Example 2 is also taken from the work of Griffiths and Lane [4].The slope geometry and geotechnical parameters are given inFig. 10 and Table 3. In this example, both a non-associated flowrule (i.e., dilation angle = 0�) and an associated flow rule (i.e., dila-tion angle = friction angle) are applied in the two finite elementanalyses. The elastic parameters have only a weak influence onthe predicted FOS in the SRM [4]. Therefore, Poisson’s ratio m0

was adjusted to 0.4 from 0.3 (in the original paper) to satisfy theprevious key condition in the ELSM. Fig. 11 and Table 4 give theresults from the three methods. The critical slip surfaces obtainedby the LEM and ELSM almost coincide with each other for w ¼ 0

�

and w ¼ u0. When w ¼ u0, the shapes and locations of the criticalslip surface from the three methods and the FOS from the finiteelement methods are essentially the same. The FOS of the LEM isslightly less than those of the two finite element methods.

In the application of the SRM, Fig. 11(b) also compares the crit-ical slip surfaces determined by the ELSM (i.e., the green solid line)and the contour of the equivalent plastic strain. The comparisonreveals that the critical slip surface determined by the ELSM islocated in the centre of the band of equivalent plastic strain, andthe minimum value of the FOS is close to 1.

Fig. 12 presents the local FOS along the critical slip surfaceobtained by the ELSM using the original and reduced strengths.The local FOS, defined by Eq. (13), shows that almost all of theFLOCAL values are close to 1 when the stress field is calculated using

Table 5Geotechnical parameters for example 3.

Layer c (kN/m3) c0 (kPa) u0 (�) E0 (kN/m2) m0 (-) w (�)

1 18.62 15.0 20.0 105 0.35 20.02 18.62 17.0 21.0 105 0.35 21.03 18.62 5.0 10.0 105 0.35 10.04 18.62 35.0 28.0 105 0.35 28.0

(a)

(b)

Fig. 14. Comparisons of the critical slip surfaces obtained from (a) the LEM (solidline and dotted line) and ELSM (dashed line) and (b) the SRM (solid line and contourof equivalent plastic strain).

Fig. 15. Example 4: A multi-stage homogeneous slope. The global size of the seedsfor the mesh is 0.15. E0 = 105 kN/m2, m0 = 0.49, w = u0 , c = 20 kN/m3, u0 = 30�,c0 = 5 kPa.

Fig. 16. Comparisons of the slip surface and the FOS: (a) critical slip surfacesobtained from the SRM (solid line and contour of equivalent plastic strain) and (b)critical failure surface and local minima from the ELSM.

S.Y. Liu et al. / Computers and Geotechnics 63 (2015) 291–298 297

the reduced strength. Furthermore, the slope nearly reaches thelimit equilibrium state. In addition, for the case of the originalstrength, FLOCAL differs along the critical slip surface; however,the overall FOS = 1.384 (i.e., the ‘‘average’’ strength reduction fac-tor) from the ELSM is close to the FOS (SRF)=1.387 from the SRM:

FLOCAL ¼R DL

0 sf dLR DL0 sdL

ð13Þ

Fig. 17. Typical node displacement curve.

4.3. A slope with a thin, weak layer

The third example is a slope in layered soil. To analyse the sta-bility of this slope, the genetic algorithm using the Morgenstern-Price (M-P) method was used by Zolfaghari et al. [20], whereasCheng et al. [21] applied the particle swarm optimisation algo-rithm with the Spencer method. The slope geometry is shown inFig. 13 and Table 5 gives the geotechnical properties for layers1–4.

Fig. 14 compares the critical slip surfaces obtained using thethree methods. The critical slip surface and the FOS obtained fromthese methods are similar to one another with the exception ofthose from Zolfaghari et al., as shown in Table 6. To determine thisnon-circular critical slip surface, the LEM and ELSM using the non-circular search technique are both reliable; however, the SRM is

Table 6Summary of factors of safety for example 3.

Method FOS FOS differencewith LEM2 (%)

LEM1 (M-P, Non-circular) [20] (black dotted line) 1.240 12.6LEM2 (Spencer, Non-circular) [21] (black solid line) 1.101SRM 1.143 3.8ELSM 1.111 0.9

preferable to the other methods because it automatically locatesthe critical slip surface.

4.4. A multi-stage slope

This example is a multi-stage slope taken from Cheng et al. [3].Fig. 15 gives the geometry of the slope and the geotechnical param-eters. Such a multi-stage slope features not only a global minimum

Table 7Summary of factors of safety for example 4.

Method FOS Global FOS difference with LEM (%)

Global minimum Local minimum

LEM (Spencer, Non-circular) [21] 1.330 1.375 1.415 1.400 1.383SRM (w ¼ 0

�) [21] 1.327 �0.2

SRM (w ¼ u0) 1.370 3ELSM 1.366 1.412 1.432 1.402 1.387 2.7

298 S.Y. Liu et al. / Computers and Geotechnics 63 (2015) 291–298

but also many local minima. In the SRM, separate finite elementanalyses were performed, each with strength parameters (c andtan u0) of the material that were decreased by dividing by the SRF.The FOS can then be obtained when the SRF is associated with incip-ient failure. In addition, a sudden increase in the displacement oftypical nodes is assumed as the failure criterion of slopes.

Fig. 16 shows the location of the critical slip surface of this slopestability analysis. However, the critical slip surface (i.e., the con-tour of the equivalent plastic strain) from the SRM is not clearlyshown because the shear strains first build up at the toe of themain slope. Fig. 17 presents the displacement curves of two typicalnodes (nodes 28 and 32, as shown in Fig. 16(a)). It is evident thatonly the slip surface (solid line shown in Fig. 16(a)) is fully devel-oped. Furthermore, the critical slip surface (i.e., the solid linewithin the equivalent plastic strain band) obtained by the secondtechnical measure, as mentioned in Section 2.2, is the same asthe results from the LEM [3] and ELSM.

Fig. 16(b) and Table 7 summarise the local ‘‘slip surface’’ and thelocal minimum FOS. The locations and shapes of the local ‘‘slip sur-face’’ are similar to the results from Cheng et al. [3], and the localminimum FOS values are also slightly higher than those from theLEM. As mentioned before, the other local ‘‘slip surface’’ cannotbe located by the SRM; however, both the LEM and ELSM couldbe applied in the stability analysis of such a multi-stage slope toobtain the local minimum.

5. Conclusions

The FOS and the critical slip surfaces obtained by the LEM and twofinite element methods (ELSM and SRM) are compared. Several rep-resentative two-dimensional slope examples are analysed, and theresults obtained using the three methods are compared. Under theassumption of the associated flow rule, the critical slip surfaces fromthe two finite element methods and the limit equilibrium method aregenerally in good agreement; the FOS obtained from the limit equi-librium method is slightly less than those from the two finite elementmethods. Moreover, in the SRM, a number of separate finite elementanalyses must be performed to obtain the FOS and critical slip sur-face; thus, this is the most time-consuming solution.

When a slope model is in an elastic state, it was proven that thedirections of the shear stresses at any point along the slip surfacemust follow the same trend towards the slide for the applicationof the ELSM. Increasing the Poisson’s ratio was shown to be aneffective way to satisfy this condition.

In the application of the SRM, once a slope reaches the limitequilibrium state by the strength reduction analysis, in which theassociated flow rule is applied, the ELSM is shown to be useful indetermining the critical slip surface as a technical measure. Thecritical slip surface determined by the ELSM is located in the centreof the band of equivalent plastic strain.

The search technique combining PSO with the pattern searchtechnique to locate the slip surface of the slope was shown to befeasible and reliable in the stability analysis of these examples.This technique could also be applied in the LEM.

Acknowledgements

The first author gratefully acknowledges financial support fromthe China Scholarship Council (CSC). The authors are grateful to Dr.Chen Guangjing (SCK.CEN), whose suggestions and comments haveimproved the quality of the paper.

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