Limit analysis solutions for three dimensional undrained ...€¦ · of the mobilised shear strength derived from the 2D back analysis for a 3D slope, will be unsafe. In order to

Computers and Geotechnics 36 (2009) 1330–1351

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Computers and Geotechnics

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Limit analysis solutions for three dimensional undrained slopes

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

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

a r t i c l e i n f o

Article history:Received 27 February 2009Received in revised form 27 May 2009Accepted 12 June 2009Available online 19 July 2009

Keywords:Safety factorStability chartCohesive clayInhomogeneousLandslide

0266-352X/$ - see front matter � 2009 Elsevier Ltd.doi:10.1016/j.compgeo.2009.06.002

* Corresponding author. Tel.: +61 2 4921 5735; faxE-mail addresses: [email protected] (A.J. Li),

edu.au (R.S. Merifield), [email protected]

a b s t r a c t

This paper uses numerical finite element upper and lower bound limit analysis to produce stabilitycharts for three dimensional (3D) homogeneous and inhomogeneous undrained slopes. Althoughthe conventional limit equilibrium method (LEM) is used more often in practice for evaluating slopestability, the accuracy of the method is often questioned due to the underlying assumptions that itmakes. Using the limit theorems can not only provide a simple and useful way of analysing the sta-bility of slopes, but also avoid the shortcomings and arbitrary assumptions under pinning the LEM.The rigorous limit analysis results in this paper were found to bracket the slope stability numberto within ±9% or better and therefore can be used to benchmark for solutions from other methods.In addition, it was found that using a two dimensional (2D) analysis to analyse a 3D problem willlead to a significant difference in the factors of safety depending on the slope geometries. This isof particular relevance to any back analyses of slope failure as it will lead to an unsafe estimationof material strengths.

� 2009 Elsevier Ltd. All rights reserved.

1. Introduction

Estimating the slope stability remains an important problem forgeotechnical engineers. This problem has drawn the attention ofmany investigators [1–7] in the past and continues to do so. Cur-rently, two dimensional (2D) limit equilibrium analysis, such asBishop’s simplified method [2] and Janbu’s simplified method [8],are two of the most popular approaches used to evaluate slope sta-bility. It is commonly believed that 2D solutions utilised in designwill obtain a conservative evaluation for a three dimensional (3D)slope failure. However, as pointed out by Gens et al. [9], estimatesof the mobilised shear strength derived from the 2D back analysisfor a 3D slope, will be unsafe. In order to account for the threedimensional effects on slope stability many 3D methods had beenproposed [10–13]. The majority of methods proposed in thesestudies are simply based on extensions of Bishop’s simplified [2],Spencer’s [14], or Morgenstern and Price’s [15] original 2D limitequilibrium slice methods. The differences between each studyare the arbitrary assumptions made regarding inter-column forces.It should be acknowledged that the inherent limitations of the lim-it equilibrium method (LEM) still remain in these 3D solutions.

Using the limit theorems can not only provide a simple and use-ful way of analysing the stability of geotechnical structures, butalso avoid the shortcomings of the arbitrary assumptions under

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: +61 2 4921 [email protected] (A.V. Lyamin).

pinning the LEM. Limit analysis studies of 3D slope stability to datehave generally been based on the upper bound method. Conse-quently, the full power of the limit theorems to bracket the truecollapse load has not been utilised, and there exists no true boundson the 3D stability of slopes.

Currently, there are no widely accepted three dimensional sta-bility analysis solutions for soil slopes available for practicing geo-technical engineers. In most cases it is not feasible to perform a fulldisplacement finite element analysis and as such the three dimen-sional effects of the slope in question are often ignored. However,ignoring the 3D effects when analysing slopes can lead to unsafeanswers. In the back analyses of shear strengths, for example,neglecting the 3D effects will lead to values that are too high,and therefore affect any further stability assessments at the samelocation. As stated previously, one aim of this study is to produce3D stability charts that can be used by practicing engineers,extending those currently used regularly for 2D slope stabilityevaluation.

Fortunately, the finite element upper and lower bound limitanalysis techniques developed by Lyamin and Sloan [16,17] andKrabbenhoft et al. [18] provide a useful method for dealing withthe problems of slope stability. These numerical upper and lowerbound methods have been used to provide chart solutions by Yuet al. [19] for 2D purely cohesive and cohesive-frictional soil slopes,Loukidis et al. [20] for seismic cohesive-frictional soil slopes, and Liet al. [21,22] for static and seismic rock slopes. The purpose of thispaper is to provide sets of 3D stability charts for undrainedhomogeneous and inhomogeneous soil slopes by using the finite

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A.J. Li et al. / Computers and Geotechnics 36 (2009) 1330–1351 1331

element bounding methods of Lyamin and Sloan [17] and Krab-benhoft et al. [18] which can bracket the actual stability numbersfrom above and below. The chart solutions in this study can beseen as convenient tools to be used by practising engineers to esti-mate the initial stability for excavated or man-made slopes.

2. Previous studies

Three dimensional methods of analysis of slopes available todate can be broadly divided into three categories. Those basedon: (1) the limit equilibrium method (LEM) [9–11,23]; (2) thelimit analysis method [6,24] and; (3) the finite element method(FEM) [25,26]. Other methods of analysis, like the finite differ-ence method [27,28], distinct element method [29], probabilityassessment [30,31] are also used in current slope stability anal-yses. Duncan [32] provides a comprehensive review for twodimensional (2D) and three dimensional (3D) LEM and FEMestimates of slope stability, and therefore the review of litera-ture herein will be referring to more recent publications (post1996).

Fixed face

u = v = w = 0 (Upper bound)


σn = τ = 0 (Lower boun

σn = τ = 0 (Lower

σn = τ = 0 (Lower bound)

Fig. 1. Problem configuration for homoge

2.1. Limit equilibrium analysis

Chang [33] considered force equilibrium for individual blocksand an overall system of blocks in a 3D limit equilibrium analysis.Huang and Tsai [34] and Huang et al. [35] took into account a forceand/or moment limit equilibrium in two orthogonal directions toanalyse the 3D stability of a potential failure mass. In addition,Zhu [36] employed numerical schemes in limit equilibrium analy-sis to approximate the critical slip surfaces where initial trial sur-faces are not required and no restrictions are imposed on theshape of slip surfaces. Although the above studies based on theLEM are appropriate for analysing practical cases, chart solutionswere not provided.

Gens et al. [9] provided stability charts for 3D purely cohesivesoil slopes. The results presented in their study showed that thedifference in the slope stability assessment between two and threedimensional analysis can ranges from 3–30% and average 13.9%.This difference is comparable in importance with the correctionscommonly made with regard to undrained shear strength (cu),and in back analysis, may be unsafe. Jiang and Yamagami [37] pro-


d

Symmetric face

τ = 0 (Lower bound)

v = 0 (Upper bound)


x, u y, v

z, w

L/2

H

d)

bound)

Mode of Failure:

F = Face failure

T = Toe failure

neous slopes in purely cohesive soil.

1332 A.J. Li et al. / Computers and Geotechnics 36 (2009) 1330–1351

posed chart solutions for cohesive-frictional slopes. In their study,both simple slopes and infinite slopes were accounted for. It wasfound by Jiang and Yamagami [37] that an infinite slope has a lar-ger factor of safety than a simple slope for a given cohesion (c0),friction angle (/0) and height of slip surface.

Baker et al. [38] adopted the pseudo static (PS) method in theirlimit equilibrium analyses and proposed 2D seismic chart solutions

Toe

Rigid Base

β

Fig. 3. The analysed strength profile for

u = v = w = 0(Upper bound)u = v = w = 0(Upper bound)

u = v = w = 0(Upper bound)

u = v = w = 0(Upper bound)

y, vx, u

z, w

v = 0Symmetric face(Upper bound)

(a) Upper bound

σn = τ = 0

(Lower bound)

y, vx, u

z, w

τ = 0Symmetric face(Lower bound)

(b) Lower bound

Fig. 2. Typical three dimensional finite element limit analysis meshes andboundary conditions.

for cohesive-frictional soil slopes. Their investigation focused onthe effects of the critical PS coefficient on the slope stability for arange of geometries, friction angle (/0) and stability numberðN ¼ c0=cHÞ. This form of stability number is the same as thatadopted by Gens et al. [9] which was proposed by Taylor [39].

Recently, Chen and Chameau [40] developed a 3D limit equilib-rium method and found the factor of safety from a 3D analysis issmaller than that from a 2D analysis. Later, Cavounidis [41] provedthe statement made by Chen and Chameau [40] is incorrect. Cavo-unidis [41] also highlighted that the 3D factor of safety of a slope isalways greater than the 2D factor of safety for the same slope.

Stark and Eid [42] reviewed three commercially available limitequilibrium based computer programs in their attempts to analyseseveral landslide case histories and concluded that the factor ofsafety is poorly estimated by these software packages because ofinbuilt limitations in describing geometry, material propertiesand ignoring the shear resistance along the vertical sides of slidingmass etc.

2.2. Finite element analysis

As highlighted by Duncan [32], the FEM is a general-purposemethod which can be used to calculate stresses, movements, porepressures and other characteristics of earth masses during con-struction [43,44] without a priori assumed sliding surface. Pottset al. [45] used the FEM to examine the failure mechanism forthe delayed collapse of a cut slope in stiff clays. Troncone [46]incorporated the soil stain-softening behaviour into the elasto-viscoplastic constitutive model and found that the strain-softeningbehaviour plays an important role in the progressive failure of theslope.

In order to estimate the slope stability and obtain its factor ofsafety by using the finite element analysis, the strength reductionmethod (SRM) is widely employed [5,47–50]. Using the SRM,Manzari and Nour [48] investigated the soil dilatancy effect onthe slope stability analysis. Griffiths and Lane [5] and Griffithsand Marquez [25] examined 2D and 3D slope stability respectively.They demonstrated that utilising the SRM in finite element analy-sis can obtain rational safety factors.

Hwang et al. [51] observed that the critical slip surface deter-mined by the simplified Bishop’s analysis compare well with thefailure surface obtained by using the mobilised friction angle con-tours from the finite element analysis of an excavated slope. Inaddition, the difference in the factors of safety between SRM andLEM was found to be insignificant by Baker et al. [38] and Psarro-poulos and Tsompanakis [52].

z

cu(z) = c

u0 + ρ z

ρ

cu0

d

1

H

inhomogeneous undrained slopes.


Reviewing the slope stability assessment based on the FEM, itshould be noted that only Griffiths and Marquez [25] and Yuet al. [53] have investigated the 3D slope stability under effect ofsuch general factors as the geometrical characteristics of the damand the topography of the canyon site. However, stability chartsfor 3D inhomogeneous soil slopes still do not exist.

2.3. Limit analysis

Although the limit theorems provide a simple and useful way ofanalysing the stability of geotechnical structures, they have notbeen widely applied to the 3D slope stability problem. Currently,most slope stability evaluations based on the limit theorems haveused the upper bound method alone, such as Chen et al. [54,55],Donald and Chen [56], Farzaneh and Askari [24], De Buhan andGarnier [57], Michalowski [6,58], and Viratjandr and Michalowski[59]. Major contributions for soil slope stability analysis were pre-sented by Michalowski and his co-worker who investigated localfooting load effects on the 3D slope stability [6] and provided setsof stability charts for cohesive-frictional slopes which took seismicloadings and pore pressure into account [58,59]. In addition,Michalowski [60] employed the limit analysis technique to esti-mate the stability of uniformly reinforced slopes.

1 2 3 4 5

0.06

0.08

0.10

0.12

0.14

0.16

0.18

0.20

0.22

0.24 β = 75°

β = 60°

Depth factor (d / H)

β = 7.5°

β = 15°β = 22.5°

β = 30°

Nh =

cu/γ

HF

β = 45°

Hβ

Less stable

(a) Lower bound

1 2 3 4 5

0.06

0.08

0.10

0.12

0.14

0.16

0.18

0.20

0.22

β = 75°

β = 60°


β = 7.5°

β = 15°

β = 22.5°β = 30°

Nh =

cu/γ

HF

β = 45°

Hβ

Less stable

(b) Upper bound

Fig. 4. Two dimensional limit analysis solutions of stability numbers for homoge-neous undrained slopes (L/H =1).

By using both lower and upper bound analyses to estimateslope stability, Yu et al. [19], Kim et al. [61] and Loukidis et al.[20] proposed sets of stability charts for inhomogeneous soil slopesand cohesive-frictional soil slopes subjected to pore pressure andseismic loadings respectively. However their studies were only fo-cused on investigating the stability of 2D slopes. The purpose ofthis paper is to extend the chart solutions of Yu et al. [19] to 3Dslope stability problems. Both upper and lower bounds are em-ployed here, and thus true failure load can be bounded. These solu-tions are obtained from numerical techniques developed byLyamin and Sloan [17] and Krabbenhoft et al. [18].

3. Problem definition

The typical 3D slope geometry for the problem of this paper isshown in Fig. 1, where the 2D (x � z) slope profile is extended inthe y direction by a distance L/2 to obtain the symmetric 3D slopeboundary. In general, the slope failure mode can be divided intothree types which are face-failure, toe-failure and base-failure, asshown in Fig. 1 where n is used to describe the lateral extent ofthe base-failure. The strength of cohesive soil is determined bythe undrained shear strength (cu). In this study, cu is assumed con-stant throughout the slope or increasing with depth, referred to asthe homogeneous and inhomogeneous undrained slopes respec-

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0.16β = 75°

β = 60°

β = 7.5°

β = 15°

β = 22.5°

β = 30°

Nh =

cu/γ

HF


β = 45°

Hβ

Less stable

(a) Lower bound

1 2 3 4 5

1 2 3 4 50.02

0.04

0.06

0.08

0.10

0.12

0.14 β = 75°

β = 60°


β = 7.5°

β = 15°

β = 22.5°

β = 30°

Nh =

cu/γ

HF

β = 45°

Hβ

Less stable

(b) Upper bound

ig. 5. Three dimensional limit analysis solutions of stability numbers for homo-
F geneous undrained slopes (L/H = 1).


tively. A range of slope inclination (b), depth factor (d/H) and L/Hratios are considered. The soil is modelled by the Mohr–Coulombyield criterion with zero friction angle due to the purely cohesivesoil under undrained loading conditions. In order to decrease thetotal number of elements, symmetry is exploited for the 3D cases.As shown in Fig. 1, the applied stress and velocity boundary condi-tions are given to simulate the fixed and symmetric faces in theupper and lower bound analyses.

The typical upper and lower bound finite element meshes andboundary conditions used to analyse the 3D slope problem areillustrated in Fig. 2. The overall mesh dimensions were adjustedso that a statically admissible stress field for the lower bound anal-ysis and a kinematically admissible the velocity/plastic field for theupper bound analysis were contained.

The stability of homogeneous slopes in purely cohesive un-drained clay is usually expressed in terms of a dimensionless sta-bility number in the following form

Nh ¼ cu=cHF ð1Þ

where Nh is the stability number, c is the soil unit weight, H is theslope height and F is the safety factor of the slope. This form of sta-bility number is as proposed by Taylor [39].

The shear strength profile assumed for inhomogeneous un-drained slopes is displayed in Fig. 3. The shear strength cu is as-

1 2 3 4 5

0.04

0.06

0.08

0.10

0.12

0.14

0.16

0.18

0.20

β = 75°

β = 60°


β = 7.5°

β = 15°

β = 22.5°

β = 30°

Nh =

cu/γ

HF

β = 45°

Hβ

Less stable

(a) Lower bound

1 2 3 4 5

0.04

0.06

0.08

0.10

0.12

0.14

0.16


β = 7.5°

β = 15°

β = 22.5°

β = 30°

Nh =

cu/γ

HF

β = 45°

Hβ

β = 60°

β = 75°

Less stable

(b) Upper bound

Fig. 6. Three dimensional limit analysis solutions of stability numbers for homo-geneous undrained slopes (L/H = 2).

sumed to increase linearly with depth as is the case in normallyconsolidated clays [62]. Therefore, cu is given by

cuðzÞ ¼ cu0 þ qz ð2Þ

where cu0 is the undrained shear strength at the slope crest, q is therate of increase in the undrained shear strength with depth and z isthe depth from the crest of the slope.

To be able to compare the results obtained in this paper to theexisting 2D results of Yu et al. [19], two dimensionless parameters,Nq and kcq are introduced as follows:

Nq ¼ cHF=cu0 ð3Þkcq ¼ qHF=cu0 ð4Þ

These two equations were proposed by Yu et al. [19] to account forthe effect of increasing strength with depth. Eq. (3) can be seen asthe stability number for inhomogeneous undrained slopes. Suchdefinition of the stability number is inverse of what is adopted forhomogeneous slopes (Eq. (1)). As a consequence, it is expected thatthe upper bound stability number will be larger than the lowerbound stability number for inhomogeneous cases (the reverse ofwhat is observed for homogeneous slopes).

0.04

0.06

0.08

0.10

0.12

0.14

0.16

0.18

0.20


β = 7.5°

β = 15°

β = 22.5°

β = 30°

Nh =

cu/γ

HF

β = 45°

Hβ

β = 60°

β = 75°

Less stable

(a) Lower bound

1 2 3 4 5

1 2 3 4 5

0.04

0.06

0.08

0.10

0.12

0.14

0.16

0.18


β = 7.5°

β = 15°

β = 22.5°

β = 30°

Nh =

cu/γ

HF

β = 45°

Hβ

β = 60°

β = 75°

Less stable

(b) Upper bound



4. Results and discussions for homogeneous undrained slopes

4.1. Stability charts for homogeneous undrained slopes based on thenumerical limit analyses

The 2D and 3D chart solutions for homogeneous cohesive un-drained slopes obtained from the numerical upper and lowerbound analysis are displayed in Figs. 4–9 for a range of slope angles(b), depth factors (d/H), and L/H ratios. It can be noted that theupper and lower bound limit analysis solutions bracket a rangeof stability numbers (Nh) to within ±9% for 3D cases and ±2.5%for 2D cases. No particular trend of the greatest difference in theupper and lower bound solutions was observed.

For the larger d/H ratios and L=H 6 5, the line of the stabilitynumbers should be flat for b P 45�. However, the obtained resultsdo not plot exactly as a flat line due to mesh-dependency. There-fore, except when d/H = 1, Nh in Figs. 5–8 represent average valuesfor the limit analysis solutions when b P 45�. It should be empha-sized that the error in average Nh values for these cases is less than2.5%.

As expected, the stability number Nh increases when b and theL/H ratio increase. For a given b and d/H, Nh achieves the maximumvalue when L/H =1. This implies that the factor of safety will re-duce with increasing L/H ratio. As is known, the plain strain anal-

0.04

0.06

0.08

0.10

0.12

0.14

0.16

0.18

0.20

0.22


β = 7.5°

β = 15°

β = 22.5°

β = 30°

Nh =

cu/γ

HF

β = 45°

Hβ

β = 60°

β = 75°

Less stable

(a) Lower bound

1 2 3 4 5

1 2 3 4 50.04

0.06

0.08

0.10

0.12

0.14

0.16

0.18

0.20


β = 7.5°

β = 15°

β = 22.5°

β = 30°

Nh =

cu/γ

HF

β = 45°

Hβ

β = 60°

β = 75°

Less stable

(b) Upper bound


ysis does not consider the resistance provided by the two curvedends of the slip surface. The boundary resistance from these twocurved ends can be seen as 3D end boundary effect which makesthe slope more stable. While increasing the L/H ratio, the relativecontributions of resistances provided by these two curved ends de-crease which means that 3D end boundary effect reduces. There-fore, using 2D stability numbers will lead to a more conservativeslope design.

It should be noted that the magnitude of n is not shown in Figs.4–8. This is due to the fact that the plastic zones obtained from theupper bound analyses are not exactly the single slip surfaces asused in LEM. Using a finer mesh in the 3D analyses would enablen to be defined more accurately, but would make computationsmore time-consuming. Therefore an accurate value of n was notfound. In addition, Gens et al. [9] observed that there are diver-gences between the actual and predicted n values, although noexplanation was provided.

Fig. 9 presents the stability numbers (Nh) obtained from theupper and lower bound limit analyses for vertical slopes(b ¼ 90�). These numbers can be used for estimating the stabilityof the shallow vertical cuts without retaining walls and props.Due to the fact that the vertical slope is shallow, the soil proper-ties can be assumed as uniform. As shown in Fig. 9, the stability

420.10

0.12

0.14

0.16

0.18

0.20

0.22

0.24

0.26

0.28

L / H = 5

L / H = 2

L / H = 3

Hβ

L / H = ∞

L / H = 1

Less stableN

h = c

u/ γH

F


(a) Lower bound

1 2 3 4 50.10

0.12

0.14

0.16

0.18

0.20

0.22

0.24

0.26

0.28

L / H = 5

L / H = 2

L / H = 3

Hβ

L / H = ∞

L / H = 1

Less stableN

h = c

u/ γH

F


(b) Upper bound

Fig. 9. Three dimensional limit analysis solutions of stability numbers for homo-geneous undrained slopes ðb ¼ 90�Þ.

(a) °= 30β

(b) °= 22 .5β

(c) °= 15β

Fig. 11. 2D upper bound plastic zones for d/H = 5.

(a) °= 30β

(b) °= 22 .5β

(c) °= 15β

Fig. 10. 2D plastic zones for d/H = 2.


number Nh increases with increasing L/H ratio, however it exhib-its no evident dependence on the depth factor (d/H) for verticalslopes.

Figs. 10 and 11 show the 2D upper bound plastic zones for d/H = 2 and d/H = 5, respectively. It can be seen that the major fail-ure mode is base-failure. The transition of the failure modes isshown in Fig. 12 where the failure mode changes from base-fail-ure to toe-failure as b increases. In this part of the study, all anal-yses indicate that base-failure is the primary failure type forpurely cohesive homogeneous slopes when b 6 60�. This meansthat the slip surfaces occur from the slope crest and pass belowthe toe of the slope. On the other hand, from the 2D plastic zonesin Fig. 12, the d/H ratio is found to have almost no effect on sta-bility numbers for b P 75�. Therefore, the stability numbers plotas flat lines for b P 75� in Figs. 4–9. It should be stressed thatall of the propagated slip surfaces in Figs. 10 and 11 have reachedthe rigid bottom base. It can be concluded that a rigid layerunderneath the slope controls the slip surface for a 2D uniformundrained slope with b 6 60�.

For b = 15� and L/H = 5, the 3D plastic zones obtained fromthe upper bound limit analysis for various depth factors (d/H)are shown in Fig. 13. It can be observed that the majority of3D slope failure modes for homogeneous undrained slopes arestill of base-failure. In general, the depth of slip surface increaseswith depth factor (d/H). However, it should be noted that thefailure surface for L/H = 5 (Fig. 13c) does not touch the rigid baselayer. Compared with the 2D case (Fig. 11c), the depth of slipsurface is found to be shallower. Fig. 14 shows the failure sur-faces for b ¼ 22:5�, L/H = 5 and the various d/H values. Compared

(a) °= 60β

(b) °= 75β

(c) °= 90β

Fig. 12. 2D plastic zones for various slope angles (d/H = 2).


to Fig. 13, it is found that the slip surface does not touch therigid bottom layer when L/H = 4. It implies that the boundary ef-fect of the depth factor (d/H) reduces with increasing slope incli-nation for 3D homogeneous undrained slopes. On the otherhand, d/H plays a more important role for a slope with lowerslope angle.

Figs. 13b and 15 show that the depth of slip surface varies whenthe ratio of L/H is changed. As expected, the depth of failure surfacedecreases with a reduction of L/H ratio. Again this is due to the 3Dend boundary effect increasing. The depth of slip surface changessignificantly when the ratio of L/H varies between 1 and 5. A tran-sition of 3D failure mode is displayed in Figs. 13a, 14a and 16. The

(a) 2=

(b) 4=

(c) 5=

Hd

Hd

Hd

Symmetric face

Fig. 13. 3D plastic zones for various d/H (b = 15� and L/H = 5).

failure mode is observed to change from base-failure to toe-failurewith b increasing. Observations of plastic zones in Figs. 12a and16b demonstrate that 3D end boundary effects may influence thedepth of the slip surface.

A comparison of the equivalent 2D and 3D cases can be madeby investigating the factor of safety ratio F3D/F2D for the sameslope angle (b), depth factor (d/H), slope height (H), unit weight(c) and undrained shear strength (cu). The ratio F3D/F2D is alsosimply the inverse ratio of the stability numbers (Nh)2D/(Nh)3D.Fig. 17 shows the average of the upper and lower bound ratio

(a) 2=

(b) 4=

(c) 5=

Hd

Hd

Hd

Symmetric face

Fig. 14. 3D plastic zones for various d/H (b = 22.5� and L/H = 5).


of F3D/F2D for various depth factors (d/H) and slope angles (b). Inthis figure, the magnitude of F3D/F2D denotes the degree in whichthe 2D analysis underestimated the slope stability. It should beacknowledged that the true ratio of F3D/F2D has been bracketedby the numerical upper and lower bound analysis within a rangeof ±7.5%.

Referring to Fig. 17, the ratio of F3D/F2D is found to increase withincreasing d/H, decreasing b and decreasing L/H. The comparisonsof F3D/F2D between b ¼ 7:5� and b ¼ 45� show that the stabilityanalyses of slopes with higher depth factors (d/H) and the lowerslope angle (b) result in more significant underestimates in the fac-tor of safety. In particular, for b ¼ 7:5� and d/H = 5, the ratio of F3D/F2D shows that 3D factor of safety is approximately 4.05 times tothe 2D factor of safety from the upper and lower bound solutions.Therefore, a very conservative design would be obtained by using2D solutions when the slope has a low slope angle (b). However,the F3D/F2D ratio of b ¼ 7:5� changes more significantly than thatof b ¼ 45� while d/H increases from 1 to 5. As discussed above,the difference in the depth of slip surface between various L/H ra-tios is quite insignificant for a slope with higher slope angle. Thechange of the F3D/F2D ratio should be influenced by the change ofslip surface depth. Therefore, the phenomenon that the F3D/F2D ra-tio of b ¼ 7:5� changes more significantly than that of b ¼ 45� forvarious d/H values could be due to the fact that the boundary effectof the depth factor (d/H) reduces with increasing slope inclinationfor 3D slopes.

It can be noticed from Fig. 17 for b ¼ 75� that the F3D/F2D ratio isalmost unchanged when d=H P 2. Again this shows the d/H ratio

(a) 3=

(b) 1=

HL

HL

Symmetric face

Fig. 15. 3D plastic zones for various L/H ratios (b = 15� and d/H = 4).

does not play an important role for the cases with b P 75�. Basedon the solutions presented in Fig. 17, it is obvious that the factorof safety from a 3D analysis will be greater than that from a 2Danalysis for homogeneous undrained slopes. Therefore, the state-ment made by Chen and Chameau [40] is not valid for homoge-neous undrained slopes.

Chugh [27] analysed a sample problem and pointed out that thedifference between 2D and 3D safety factors tends to lose signifi-cance when L/H > 5. However, results of current study give theF3D/F2D ratio as high as 1.76 for the undrained uniform slopes whenL/H = 5. The difference between the 2D and 3D factor of safety esti-mates is greater than 76% which would be important and is notnegligible for the back analysis of a failed slope in practise.

(a) ο45=β

(b) ο60=β

(c) ο75=β

Symmetric face

Fig. 16. 3D plastic zones for various slope angles (d/H = 2 and L/H = 5).

1 2 3 4 51 .0

1 .5

2 .0

2 .5

3 .0

3 .5

4 .0

4 .5

F3D

/ F

2D

d / H = 1d / H = 2d / H = 3d / H = 5

L / H

L / H

L / H

β = 7 .5 °

1 2 3 4 51 .0

1 .5

2 .0

2 .5

d / H = 1d / H = 2d / H = 3d / H = 5

β = 2 2 .5 °

1 2 3 4 51 .0

1 .5

2 .0

2 .5

F3D

/ F

2D

d / H = 1d / H = 2d / H = 3d / H = 5

β = 3 0°

1 2 3 4 51 .0

1 .5

2 .0

d / H = 1d / H = 2d / H = 3d / H = 5

β = 4 5°

1 2 3 4 51 .0

1 .2

1 .4

1 .6

1 .8

2 .0

F3D

/ F

2D

d / H = 1d / H = 2d / H = 3d / H = 5

β = 6 0°

1 2 3 4 51 .0

1 .5

2 .0

d / H = 1d / H = 2d / H = 3d / H = 5

β = 7 5°

Fig. 17. Factor of safety ratio of F3D/F2D (limit analysis).


Moreover, this range of difference is greater than one reported byGens et al. [9] where it was found to be 3–30% with the averagebeing 13.9% based on the case records.

The statement made by Chugh [27] was based on the results ob-tained for frictional soil slopes. The outcomes of the present studydemonstrate that this statement is not applicable to the purely

0.02

0.04

0.06

0.08

0.10

0.12

L / H = 1


β = 7.5°

β = 30°

Nh =

cu/γ

HF

Upper bound LEM (Gens et al.) Lower bound

Less stable

1 2 3 4 5

1 2 3 4 5

0.06

0.08

0.10

0.12

0.14

0.16

0.18

L / H = 3


Upper bound LEM (Gens et al.) Lower bound

β = 15°

β = 45°

Nh =

cu/γ

HF

Less stable

Fig. 19. Comparisons of 3D stability numbers between the numerical limit analysis,LEM and FEM.


cohesive slopes. Therefore, engineers need to apply 2D solutionswith caution.

4.2. Solutions based on limit equilibrium and analytical methods

Taylor [1] and Gens et al. [9] presented a set of two and threedimensional stability charts based on the conventional limit equi-librium analysis for homogeneous, isotropic purely cohesiveslopes. These provide a useful benchmark on the estimates ob-tained from the numerical limit analysis approach and thereforethe chart solutions of Taylor [1] and Gens et al. [9] are used herefor comparative purposes.

Figs. 18 and 19 show the comparison of stability numbers be-tween the solutions of the numerical limit analysis methods andLEM for homogeneous cohesive undrained slopes. It can be seenthat the majority of stability numbers from the LEM are boundedby the upper and lower bound solutions. Based on the upper boundtheorem, De Buhan and Garnier [57] found the 2D stability numberfor vertical cut slopes ðb ¼ 90�Þ is 0.261 which is close to the upperbound solution presented in this paper. Referring to Figs. 18 and19, the trends in stability numbers obtained from these methodsare the same in that the stability number (Nh) increases withincreasing d/H and decreasing b.

4.3. 3D numerical limit analysis solutions for inhomogeneousundrained slopes

Figs. 20, 21 and 23–25 display the limit analysis stability num-bers Nq for 3D cut slopes. They show Nq increasing with decreasingb and d/H and increasing kcq. The 3D stability number (Nq) in Figs.20–25 is bounded by the numerical upper and lower bound solu-tions within ±8%. It is interesting to note that Nq increases almostlinearly with the dimensionless parameter kcq.

Yu et al. [19] presented a set of two dimensional stability chartsbased on the upper and lower bound limit analysis for simpleslopes relevant to excavations and man-made fills built on soil.In their studies, the shear strength profile is the same as illustratedin Fig. 3a. The 2D solutions of Yu et al. [19] are shown in Figs. 20–25. It can be observed that the difference between the 2D [19] and3D stability numbers (Nq) decreases with increasing L/H ratio, asthe 3D end boundary effect decreases. Based on the comprehensiveobservation of all the results, it can be concluded that this range ofdifference varies from 30–60% to 8–25% when the ratio of L/H in-creases from 1 to 5. It should be noted that the chart solutionsare not presented in Fig. 25 for various d/H ratios as its effect isinsignificant for vertical cut slopes ðb ¼ 90�Þ.

1 2 3 4 50.0

0.1

0.2

0.3

β = 90°

β = 60°


Upper bound LEM (Taylor) Lower bound

Nh =

cu/ γ

HF

β = 30°

β = 7.5°

Fig. 18. Comparisons of 2D stability numbers between the numerical limit analysis,LEM and FEM.

From Fig. 26a it can be noticed that the difference in Nq be-tween the 2D and 3D upper bound solutions increases slightly withincreasing kcq and decreasing L/H. A similar trend also occurs forthe lower bound solutions (Fig. 26b). This implies that, for inhomo-geneous undrained slopes, the increasing strength with depth has amore significant effect on the stability numbers for slopes with alower L/H ratio.

Based on the upper bound solutions, Fig. 27 presents the effectof slope angle (b) on the stability number for L/H = 5 and differentvalues of depth factor (d/H) and kcq. The comparisons betweenthe plots for kcq ¼ 1:0 and kcq ¼ 0:0 show that the effects of slopeangle on the stability number are more significant for slopes witha high value of kcq and a low depth factor (d/H). A similar trendwas observed by Yu et al. [19] for 2D slopes. Moreover, the differ-ence in stability numbers between kcq ¼ 0:0 and kcq ¼ 1:0 isfound to decrease with b increasing. This indicates that the effectof kcq on gentle slopes is more significant than that on the steepslopes.

Fig. 28 displays several of the upper bound plastic zones forkcq ¼ 1:0, d/H = 2, L/H = 5 and various slope inclinations. The depthof failure surface increases with a reduction of the slope angle. Inaddition, it can be observed that the failure mode transfers gradu-ally from base-failure to toe-failure when b increases from 30�(Fig. 28a) to 60� (Fig. 28c). As expected, for most considered in thisstudy cases, the depth of slip surface for the inhomogeneous un-drained slopes is found to be shallower than that for the homoge-neous undrained slopes.

0

4

8

12

16

20

24

28

32

360.0 0.2 0.4 0.6 0.8 1.0

L / H = 2L / H = 1

Νρ =

γH

F /

c u0

λcp

= ρHF / cu0

0.0 0.2 0.4 0.6 0.8 1.0

0

4

8

12

16

20

24

28

32

36

UB

LB UB

LB

0.0 0.2 0.4 0.6 0.8 1.00

4

8

12

16

20

24

28

32

36

L / H = 5L / H = 3

UB

LB

0.0 0.2 0.4 0.6 0.8 1.00

4

8

12

16

20

24

28

32

36

3D 2D (Yu et al.)

UBLB

UBLB

More stable

(a) 1=Hd

0

4

8

12

16

20

24

28

32

360.0 0.2 0.4 0.6 0.8 1.0

L / H = 2L / H = 1

Νρ =

γH

F /

c u0

λcp

= ρHF / cu0

0.0 0.2 0.4 0.6 0.8 1.0

0

4

8

12

16

20

24

28

32

36

UB

LBUB

LB

0.0 0.2 0.4 0.6 0.8 1.00

4

8

12

16

20

24

28

32

36

L / H = 5L / H = 3

UB

LB

0.0 0.2 0.4 0.6 0.8 1.00

4

8

12

16

20

24

28

32

36

3D 2D (Yu et al.)

UBLB

UBLB

More stable

(b) 1.5=Hd

Fig. 20. Limit analysis solutions of stability numbers for inhomogeneous undrained cut slopes (b = 15�).


4.4. Application example for a cut slope

4.4.1. Case 1. Assumed case of Yu et al.In order to make comparisons of the factor of safety between

the newly proposed 3D chart solutions and the 2D chart solutions

presented by Yu et al. [19], the same example from Yu et al. [19] isused in this study. This example is a cut slope excavated in a nor-mally consolidated clay. The slope descriptions are as follows: theslope inclination b ¼ 60�, the height of the slope is H = 12 m, thedepth factor is d/H = 1.5, and the soil unit weight is c = 18.5 kN/

0

4

8

12

16

20

24

28

32

360.0 0.2 0.4 0.6 0.8 1.0

L / H = 2L / H = 1

Νρ =

γH

F /

c u0

λcp

= ρHF / cu0

0.0 0.2 0.4 0.6 0.8 1.0

0

4

8

12

16

20

24

28

32

36

UB

LBUB

LB

0.0 0.2 0.4 0.6 0.8 1.00

4

8

12

16

20

24

28

32

36

L / H = 5L / H = 3

UB

LB

0.0 0.2 0.4 0.6 0.8 1.00

4

8

12

16

20

24

28

32

36

3D 2D (Yu et al.)

UBLB

UBLB

More stable

(c) 2=Hd

Fig. 20 (continued)

0

4

8

12

16

20

240.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0

0

4

8

12

16

20

24

0.0 0.2 0.4 0.6 0.8 1.00

4

8

12

16

20

24

0.0 0.2 0.4 0.6 0.8 1.00

4

8

12

16

20

24

More stable

L / H = 2L / H = 1

Νρ =

γH

F /

c u0

λcp

= ρHF / cu0

UB

LBUB

LB

L / H = 5L / H = 3

UB

LB

3D 2D (Yu et al.)

UB

LB UBLB

(a) 1=Hd



0

4

8

12

16

20

240.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0

0

4

8

12

16

20

24

0.0 0.2 0.4 0.6 0.8 1.00

4

8

12

16

20

24

0.0 0.2 0.4 0.6 0.8 1.00

4

8

12

16

20

24

More stable

L / H = 2L / H = 1

Νρ =

γH

F /

c u0

λcp

= ρHF / cu0

UB

LB UB

LB

L / H = 5L / H = 3

UB

LB

3D 2D (Yu et al.)

UBLB UB

LB

(b) 1.5=Hd

0

4

8

12

16

20

240.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0

0

4

8

12

16

20

24

0.0 0.2 0.4 0.6 0.8 1.0

0

4

8

12

16

20

24

0.0 0.2 0.4 0.6 0.8 1.0

0

4

8

12

16

20

24

More stable

L / H = 2L / H = 1

Νρ =

γH

F /

c u0

λcp

= ρHF / cu0

UB

LB UB

LB

L / H = 5L / H = 3

UB

LB

3D 2D (Yu et al.)

UBLB UB

LB

(c) 2=Hd

Fig. 21 (continued)


0

4

8

12

16

200.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0

0

4

8

12

16

20

0.0 0.2 0.4 0.6 0.8 1.0

0

4

8

12

16

20

0.0 0.2 0.4 0.6 0.8 1.0

0

4

8

12

16

20

More stable

L / H = 2L / H = 1

Νρ =

γH

F /

c u0

λcp

= ρHF / cu0

UB

LB UB

LB

L / H = 5L / H = 3

UB

LB

3D 2D (Yu et al.)

UBLB UB

LB

(a) 1=Hd

0

4

8

12

16

20

240.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0

0

4

8

12

16

20

24

0.0 0.2 0.4 0.6 0.8 1.0

0

4

8

12

16

20

24

0.0 0.2 0.4 0.6 0.8 1.0

0

4

8

12

16

20

24

More stable

L / H = 2L / H = 1

Νρ =

γH

F /

c u0

λcp

= ρHF / cu0

UB

LB UB

LB

L / H = 5L / H = 3

UB

LB

3D 2D (Yu et al.)

UBLB UB

LB

(b) 1.5=Hd



0

4

8

12

160.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0

0

4

8

12

16

0.0 0.2 0.4 0.6 0.8 1.00

4

8

12

16

0.0 0.2 0.4 0.6 0.8 1.00

4

8

12

16

More stable

L / H = 2L / H = 1

Νρ =

γH

F /

c u0

λcp

= ρHF / cu0

UB

LB UB

LB

L / H = 5L / H = 3

UB

LB

3D 2D (Yu et al.)

UB

LB

UB

LB

(a) 1=Hd


0

4

8

12

16

20

240.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0

0

4

8

12

16

20

24

0.0 0.2 0.4 0.6 0.8 1.00

4

8

12

16

20

24

0.0 0.2 0.4 0.6 0.8 1.00

4

8

12

16

20

24

More stable

L / H = 2L / H = 1

Νρ =

γH

F /

c u0

λcp

= ρHF / cu0

UB

LB UB

LB

L / H = 5L / H = 3

UB

LB

3D 2D (Yu et al.)

UBLB UB

LB

(c) 2=Hd

Fig. 22 (continued)


0

4

8

12

160.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0

0

4

8

12

16

γ / ρ =12.331

γ / ρ =12.331 γ / ρ =12.33

1

0.0 0.2 0.4 0.6 0.8 1.00

4

8

12

16

0.0 0.2 0.4 0.6 0.8 1.00

4

8

12

16

More stable

L / H = 2L / H = 1

Νρ =

γH

F /

c u0

λcp

= ρHF / cu0

UB

LB UB

LB

L / H = 5L / H = 3

UB

LB

3D 2D (Yu et al.)

UB

LB

UB

LB

(b) 1.5=Hd

0

4

8

12

160.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0

0

4

8

12

16

0.0 0.2 0.4 0.6 0.8 1.00

4

8

12

16

0.0 0.2 0.4 0.6 0.8 1.00

4

8

12

16

More stable

L / H = 2L / H = 1

Νρ =

γH

F /

c u0

λcp

= ρHF / cu0

UB

LB UB

LB

L / H = 5L / H = 3

UB

LB

3D 2D (Yu et al.)

UB

LB

UB

LB

(c) 2=Hd

Fig. 23 (continued)


m3. The undrained shear strength of the soil on the top of the slopesurface is cu0 = 40 kPa and the rate of the increasing undrainedshear strength with depth is estimated as q = 1.5 kPa/m.

A procedure for obtaining the factor of safety by using the chartsolutions presented in this study can be summarized in the follow-ing stages.

0

4

8

120.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0

0

4

8

12

0.0 0.2 0.4 0.6 0.8 1.0

0

4

8

12

0.0 0.2 0.4 0.6 0.8 1.0

0

4

8

12

More stable

L / H = 2L / H = 1

Νρ =

γH

F /

c u0

λcp

= ρHF / cu0

UB

LBUB

LB

L / H = 5L / H = 3

UB

LB

3D 2D (Yu et al.)

UB

LB

UB

LB

(a) 1=Hd

0

4

8

120.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0

0

4

8

12

0.0 0.2 0.4 0.6 0.8 1.0

0

4

8

12

0.0 0.2 0.4 0.6 0.8 1.0

0

4

8

12

More stable

L / H = 2L / H = 1

Νρ =

γH

F /

c u0

λcp

= ρHF / cu0

UB

LBUB

LB

L / H = 5L / H = 3

UB

LB

3D 2D (Yu et al.)

UB

LB

UB

LB

(b) 1.5=Hd



0

4

8

120.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0

0

4

8

12

0.0 0.2 0.4 0.6 0.8 1.0

0

4

8

12

0.0 0.2 0.4 0.6 0.8 1.0

0

4

8

12

More stable

L / H = 2L / H = 1

Νρ =

γH

F /

c u0

λcp

= ρHF / cu0

UB

LBUB

LB

L / H = 5L / H = 3

UB

LB

3D 2D (Yu et al.)

UB

LB

UB

LB

(c) 2=Hd

Fig. 24 (continued)

0

4

8

120.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0

0

4

8

12

0.0 0.2 0.4 0.6 0.8 1.00

4

8

12

0.0 0.2 0.4 0.6 0.8 1.00

4

8

12

More stable

L / H = 2L / H = 1

Νρ =

γH

F /

c u0

λcp

= ρHF / cu0

UB

LBUB

LB

L / H = 5L / H = 3

UB

LB

3D 2D (Yu et al.)

UB

LBUBLB



30


1. From the slope descriptions, the non-dimensional parameterscH/c � 12/40 = 5.55 and Nq=kcq ¼ ½cHF=cu0� ½qHF=cu0�= ¼ c=q ¼18:5=1:5 ¼ 12:33, respectively.

2. For b ¼ 60� and d/H = 1.5, the chart solutions shown in Fig. 23bare employed to determine the safety factor.

3. In Fig. 23b, a straight line passing through the origin with a gra-dient c/q = 12.33 is plotted. This straight line intersects withfour curves, which are the 2D and 3D chart solutions of thenumerical limit analysis.

4. The 2D upper and lower bound stability numbers of Yu et al.[19] are Nq = 6.8 and 7.8 for the lower bound and upper boundrespectively, and therefore the factors of safety are F ¼Nq=ðcH=cu0Þ ¼ 1:23 and 1.41. To account for the effect of L/Hratio on the safety factor, the factors of safety for 3D slopesare calculated, with details shown in Table 1.

In Table 1, the stability numbers (Nq) and the calculated safetyfactors from the upper and lower bound chart solutions for differ-ent L/H ratios are provided. The average of the upper and lowerbound safety factors are 1.7, 1.55 and 1.52 for L/H = 2, L/H = 3and L/H = 5, respectively. The safety factors for the 3D solutionsare around 1.15–1.30 times that of the safety factors of the 2Dsolutions of Yu et al. [19]. This demonstrates that the factor ofsafety obtained from 3D analysis will be always larger or equalto that obtained from 2D analysis, which is different from the re-sults of Chen and Chameau [40]. Therefore, using 2D solution isconservative for design and non-conservative when determiningstrength parameters from a back analysis of a failed slope. In addi-tion, from Table 1, the difference between the upper and lower

0.0 0.2 0.4 0.6 0.8 1.0

0

4

8

12

16

20

L / H = ∞

3D 2D (Yu et al.)

Νρ =

γH

F /

c u0

λcp

= ρHF / cu0

L / H = 1, 2, 3, 5

(a) Upper bound

0.0 0.2 0.4 0.6 0.8 1.00

4

8

12

16

20

L / H = ∞

3D 2D (Yu et al.)

Νρ =

γH

F /

c u0

λcp

= ρHF / cu0

L / H = 1, 2, 3, 5

(b) Lower bound

Fig. 26. Comparisons of stability numbers for different magnitudes of L/H (b = 45�and d/H = 1.5).

bound factors of safety for this example is found to be around18%. This difference decreases slightly when the ratio of L/Hincreases.

4.4.2. Case 2. An assumed collapse case using back analysisA similar case to the previous example is employed here to esti-

mate the soil strength from back analysis of a failed slope. The soilprofile for the failed slope is the same ðNq=kcq ¼ c=q ¼ 12:33Þ asabove, but the slope height H = 18 m and L/H = 5. The value of cu0

is unknown.By using the 2D solutions in Fig. 23b, an average Nq = 7.3 is ob-

tained, and thus cH/cu0 = 7.3, as the slope is at collapse (F = 1), andthus cu0 is calculated as cH/Nq = 18.5 � 18/7.3 = 45.6 kPa. Adoptingthe 3D solutions provided in Fig. 23b and Table 1, we find an aver-age Nq = 8.4, and thus cu0 = cH/Nq = 39.6 kPa. In the case cu0 is over-estimated by up to 15%. It is shown that 2D solutions are non-conservative for back analysis.

5. Summary and Conclusions

Several studies [10,63] have indicated that the factor of safetyfrom a 3D analysis will be greater than that from a 2D analysis.This has been proved based on the comparisons between 2D and3D limit analysis solutions for the factors of safety.

15 30 45 60 750

5

10

15

20

25

λcp

= 1.0

λcp

= 0.0

Νρ =

γH

F /

c u0

Slope angle (β )

(a) 1=Hd

15 30 45 60 750

5

10

15

20

25

λcp

= 1.0

λcp

= 0.0

Νρ =

γH

F /

c u0

Slope angle (β)

(b) 2=Hd

Fig. 27. Effect of slope angle on stability number based on the upper boundsolutions (L/H = 5).

(a) ο30=β

(b) ο45=β

(c) ο60=β

Symmetric face

Fig. 28. 3D plastic zones for various kcq ¼ 1:0, d/H = 2 and L/H = 5.

Table 1Safety factors for the cut slope example problem.

L/H = 2 L/H = 3 L/H = 5 2D

UB LB UB LB UB LB UB LB

Nq 10.25 8.6 9.3 7.9 9 7.8 7.8 6.8F = Nq/(cH/cu0) 1.85 1.55 1.68 1.42 1.62 1.41 1.41 1.23Average F 1.7 1.55 1.52 1.32


Three dimensional stability charts for homogeneous and inho-mogeneous purely cohesive slopes have been proposed in this pa-

per. Based on the results presented, the following conclusions canbe made:

1. Using the numerical upper and lower bound techniques, therange of stability numbers (Nh and Nq) have been boundedwithin ±9% or better for all cases considered. For homogeneousundrained slopes, the upper and lower bound limit analysissolutions bracket the ratio of F3D/F2D within ±7.5%. For theapplication example presented, the difference between theupper and lower bound factors of safety is found to be around17% for non-homogeneous undrained slopes.

2. For both the 3D homogeneous and inhomogeneous undrainedslopes with a slope angle ðb 6 30�Þ, the primary failure modeis that of base-failure which changes gradually to toe-failurewith increasing b.

3. The depth factor (d/H) boundary effect is found to reduce withincreasing slope inclination for the 3D homogeneous undrainedslopes due to the slip surfaces are not controlled by d/H ratio forsteeper slopes. In addition, the stability numbers (Nq) of the 3Dinhomogeneous undrained slopes are almost unchanged whend/H > 2.

4. Results of this study reveal the F3D/F2D ratio as high as 1.76 forthe undrained uniform slopes and 1.15 for the undrained cutslopes when L/H = 5. The difference between the 2D and 3D fac-tors of safety estimates is greater than 15% which would beimportant and is not negligible for the back analysis of a failedslope in practise.

5. For the 3D inhomogeneous undrained slopes, it is found that theeffect of kcq on the stability numbers is more significant for theslopes with a lower slope angle or L/H ratio, and the effect ofslope angle on the stability number is more significant for theslopes with a high value of kcq and a low depth factor (d/H).

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Limit analysis solutions for three dimensional undrained ...€¦ · of the mobilised shear strength derived from the 2D back analysis for a 3D slope, will be unsafe. In order to