StaticandSeismicStabilityChartsforThree-DimensionalCut ...the limit equilibrium method, the limit analysis method, andtheﬁniteelementmethod. For homogeneous slopes, stability charts

Research ArticleStatic and Seismic Stability Charts for Three-Dimensional CutSlopes and Natural Slopes under Short-TermUndrained Conditions

Yuan Zhou,1,2 Fei Zhang ,1 and Bing Li3

1Key Laboratory of Ministry of Education for Geomechanics and Embankment Engineering, Hohai University,Nanjing 210098, China2'e Science & Technology Department of Hohai University, Hohai University, Nanjing 210098, China3Key Laboratory of Concrete and Prestressed Concrete Structures of the Ministry of Education, Southeast University,Nanjing 210096, China

Correspondence should be addressed to Fei Zhang; [email protected]

Received 19 September 2018; Revised 25 November 2018; Accepted 19 December 2018; Published 22 January 2019

Academic Editor: Pier Paolo Rossi

Copyright © 2019 Yuan Zhou et al. ,is is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

In the strict framework of limit analysis, an analytical approach is derived to obtain the upper bound solutions for three-dimensional inhomogeneous slopes in clays under undrained conditions. Undrained strength profiles increasing linearly withdepth below the crest of the slope and below the outline surface of the slope are assumed representative of cut slopes and naturalslopes, respectively. Stability charts are produced for the cut slopes and the natural slopes under both static and pseudostaticseismic loading conditions.,e presented charts are convenient to assess the preliminary and short-term stability for 3D slopes inpractical applications, such as rapid excavation or buildup of embankments and slopes subjected to earthquakes. Compared withthe available results from the finite element limit analysis method, a better estimate of the slope safety is obtained from theanalytical approach.

1. Introduction

Stability charts for slopes provide an efficient tool for thepreliminary assessment of slope safety and for the cali-bration of any sophisticated numerical models that areultimately used for solving more complex slope stabilityproblems. ,e development of stability charts has been thesubject of many investigations since the pioneering work ofTaylor [1]. One of the most intensively studied issues is thestability charts for slopes in clays under short-term un-drained conditions. ,ese charts are usually used to obtainthe stability for rapid excavation or buildup of embank-ments and slopes subjected to earthquakes. Based on dif-ferent assumptions about the undrained strength profiles,failure mechanisms, and loading conditions, a series ofstability charts has been derived using the traditionalmethods for total stress analysis of slope stability such as

the limit equilibrium method, the limit analysis method,and the finite element method.

For homogeneous slopes, stability charts have beenderived for two-dimensional (2D) slope failures [1–6] andthree-dimensional (3D) slope failures [7–12]. For in-homogeneous slopes, as encountered in most practical sit-uations, an undrained strength profile increasing linearlywith depth below the crest of the slope, as shown inFigure 1(a), is usually assumed representative of cut slopes,as generally expressed by

cu(z) � cu0 + ρz, (1)

where cu is the undrained strength of soil at a given depth; zis the depth below the crest of the slope; cu0 is the undrainedstrength at the crest of the slope; and ρ is the gradient ofundrained strength with respect to the depth. ,e limitequilibrium (LE) method [13–15] is employed to stability

HindawiAdvances in Civil EngineeringVolume 2019, Article ID 1914674, 18 pageshttps://doi.org/10.1155/2019/1914674

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analyze of cuttings in normally consolidated clays. Koppula[16] conducted seismic stability analysis of undrainedslopes using the pseudostatic approach. Yu et al. [17] alsoadopted the �nite element limit analysis method to obtainthe least upper bound solutions, which are compared withtraditional LE results. Based on the presented methods, thestability charts for such de�ned cut slopes have been de-rived for 2D slope failures. For more realistic 3D failures ofcut slopes, Li et al. [18] derived a set of stability charts byusing the �nite element methods for upper bound andlower bound limit analyses developed by Lyamin and Sloan[19, 20] and Krabbenhoft et al. [21]. Furthermore, by usingthese methods, Li et al. [22] derived another set of stabilitycharts for 3D failures of natural slopes by assuming anundrained strength pro�le increasing linearly with depthbelow the outline surface instead of the crest of slope, asshown in Figure 1(b), where z is the depth below the surfaceof the slope and cu0 is the undrained strength at the surfaceof the slope. ese two sets of the stability charts providethe upper and lower bounds to the safety factors for cutslopes and natural slopes. Nevertheless, their accuracy maybe further re�ned by using a rigorous analytical method forslope stability analysis to avoid being aected by the ar-ti�cial boundary conditions and mesh sizes speci�ed in theadopted numerical methods.

In the strict framework of limit analysis, Michalowskiand Drescher [23] proposed a class of kinematicallyadmissible 3D rotational failure mechanisms for slopes inclays under both undrained and drained conditions. Also,

the proposed is the corresponding analytical approachfor the upper bound limit analysis of stability of ho-mogeneous slopes. Recently, their work has been ex-tended by Gao et al. [24] to include face-failure and base-failure mechanisms. And an optimization techniqueproposed by Chen [25] had been adopted and shownmore eective in �nding the critical failure surfaces andthe least upper bounds to the critical heights of homo-geneous slopes.

In the present paper, the analytical approach originated byMichalowski and Drescher [23] and extended by Gao et al.[26] is further exploited to derive the stability charts forinhomogeneous cut slopes and natural slopes (as shown inFigure 1) under both static and pseudostatic seismic loadingconditions [27–29].e stability charts are plotted for a rangeof parameters wider than those presented by Li et al. [18, 22].A comparison is made between the calculated results fromthis study and from the study of Li et al. [18, 22] to illustratethe eectiveness of the adopted failure mechanisms in �ndingthe least upper bounds to the slope safety factors.

2. Analytical Procedures for Limit Analysis ofSlope Stability

2.1. 3D Failure Mechanism. e 3D rotational failuremechanism proposed byMichalowski and Drescher [23] wasextended to involve the failure surface passing below the toeby Gao et al. [24]. Figure 2(a) shows the extended failuremechanism for slopes in clays under undrained conditions.

Z

1

Cu0

Cu=Cu0 + ρZ

ρH

Z

d

Rigid base

β

Cu0

(a)

Z

1

Cu0

Cu=Cu0 + ρZ

ρH Z

Z

Z

d

Rigid base

β

Cu0

Cu0

Cu0

(b)

Figure 1: Depth contours and undrained strength pro�le assumed representative of (a) a cut slope and (b) a natural slope (adopted from thestudy of Li et al., 2010).

2 Advances in Civil Engineering

e failure mechanism is generated by rotating a circle ofdiameter R (shaded area in Figure 2(a)) about an axis rmpassing through point O outside the circle:

R �r− r′2,

rm �r + r′2

,

(2)

where the radii r and r′ are shown in Figure 2(a). e failuresurface passes through the toe when angle β′ equals theslope angle β (where angle β′ can be found in Figure 2(a)and H is the slope height). Figure 2(b) shows 3D drawingsof the failure surface for slopes with �nite width Bmodi�ed with a plane insert of width b, to allow the

transition to the 2D failure mechanism of Chen [30] as bapproaches in�nity. For details of the construction of the3D admissible rotational failure mechanism, see thesource references [23, 24]. e failure mechanism is ki-nematically admissible for both homogeneous slopes andinhomogeneous slopes under undrained conditions.

2.2. Stability Analysis. Based on the abovementioned 3Dmechanism, an upper bound to the stability number Nρ �cHF/cu0 (where F is the factor of safety and c is the unitweight of soil), which has been analyzed numerically by Liet al. [18, 22], can be determined by equating the rate of wokWcdone by soil weight to the rate of internal energy dissipationD.To account for the eect of horizontal seismic forces on slope

N

θ0θMθM

θN

θCθh

M

H

x

x

xe

yy

yda

v

R B

ββ′

A

O

r′rm

r

D C

D′A′

(a)

Plane insert

H

Bb

β

(b)

Figure 2: 3D failure mechanisms for undrained slopes: (a) torus-type failure surface; (b) failure surface with limited width B (adopted fromthe study of Michalowski and Drescher, 2009).

Advances in Civil Engineering 3

stability, an additional rate of workWs done by the pseudostaticseismic forces is counted into the energy balance equation. Ingeneral, the balance equation is given as follows:

Wcurvec + W

planec + kh W

curves + W

planes � D

curve+ D

plane,

(3)

where kh is the horizontal seismic acceleration coefficient;the superscript “curve” denotes the work rates for asection of the curvilinear cone at the two ends of thefailure mechanism and the superscript “plane”’ relates tothe plane insert in the center of the mechanism. ,eexpressions of Wplanec , W

planes , and D

plane for the plane

insert can be found elsewhere [30, 31]. ,e expressions ofWplanec , W

planes , and D

curve for cut slopes and naturalslopes can be derived following the same procedures andsimilar symbols for obtaining the expressions for ho-mogeneous slopes as presented by Michalowski andDrescher [23].

Since the work rate done by soil weight is independentof the undrained strength of soil, the expressions of Wplanecand Wplanes for a cut slope are identical to those for a naturalslope with the same failure mechanism. For the extendedfailure mechanism, the work rates Wplanec and W

planes are

derived as

Wcurvec � 2ωc⎡⎣

θB

θ0

��R2−a2

√

0

��R2−x2

√

arm + y(

2 cos θ dy dx dθ + θC

θB

��R2−d2

√

0

��R2−x2

√

drm + y(

2 cos θ dy dx dθ

+ θh

θC

��R2−e2

√

0

��R2−x2

√

erm + y(

2 cos θ dy dx dθ⎤⎦,

Wcurves � 2ωkhc⎡⎣

θB

θ0

��R2−a2

√

0

��R2−x2

√

arm + y(

2 sin θ dy dx dθ + θC

θB

��R2−d2

√

0

��R2−x2

√

drm + y(

2 sin θ dy dx dθ

+ θh

θC

��R2−e2

√

0

��R2−x2

√

erm + y(

2 sin θ dy dx dθ⎤⎦,

(4)

where ω is the angular velocity and variables a, d, e, θB, andθC are obtained from the geometrical and trigonometricrelations in Figure 2(a) as

a �sin θ0sin θ

r− rm,

d �sin θC + β( sin θhsin(θ + β)sin θC

r− rm,

e �sin θhsin θ

r− rm,

θB � arctansin θ0

cos θ0 −A′,

θC � arctansin θh

cos θ0 −A′ − sin θh − sin θ0( /tan β,

A′ �sin θh − θ0(

sin θh−sin θh + β′( sin θh sin β′

sin θh − sin θ0( .

(5)

Unlike the work rates Wplanec and Wplanes , the rate of

internal energy dissipation Dcurve is, however, dependent onthe undrained strength profile of soil. To account for the

effect of the assumed linearly increasing undrained strengthwith depth, a dimensionless parameter λcρ, called cohesionratio and defined by Koppula [15], is used here, as it gives

λcρ �ρHFcu0

. (6)

For a cut slope, the energy dissipation rate Dcurve isderived as

Dcurve

� 2ωcu0⎡⎣ θB

θ0

R

arm + y(

2 R��R2 −y2

η dy dθ

+ θC

θB

R

drm + y(

2 R��R2 −y2

η dy dθ

+ θh

θC

R

erm + y(

2 R��R2 −y2

η dy dθ⎤⎦,

(7)

where

η � 1 +λcρ

sin θh − sin θ0rm + y

rsin θ − sin θ0 . (8)

For a natural slope, it becomes


Dcurve

� 2ωcu0⎡⎣ θB

θ0

R

arm + y(

2 R��R2 −y2

η dy dθ + θM

θB

R

drm + y(

2 R��R2 −y2

η dy dθ

+ θN

θM

R

drm + y(

2 R��R2 −y2

1 +λcρ

sin θh − sin θ0sin(θ + β)

cos βy− d

r0 dy dθ

+ θC

θN

R

drm + y(

2 R��R2 −y2

ξ dy dθ + θh

θC

R

erm + y(

2 R��R2 −y2

ξ dy dθ⎤⎦,

(9)

where

ξ � 1 +λcρ

sin θh − sin θ0rm + y

rsin θ− sin θh , (10)

and variables θM and θN are obtained from the trigonometricrelations in Figure 2(a) as

θM � arccos cos θ0 −sin θh − θ0(

sin θh+sin θh + β′( sin θh sin β′

sin θh − sin θ0( ,

θN � arccos cos θh +sin β− β′( sin θh − sin θ0(

sin β sin β′ .

(11)

According to the balance equation (3), the least upperbounds to the stability number Nρ � cHF/cu0 can be derivedin terms of λcρ from the optimization scheme of Chen [25].For a slope of given values of β, λcρ, kh, and relative width B/H, independent variables in the optimization process include(c.f. Figure 2(a)) angles θ0, θh, and β′, ratio r′/r, and relativewidth of the plane insert b/H. Similar results for a homo-geneous slope (i.e., ρ � 0) can be also derived by applying λcρ� 0 to the above expressions.

3. Results and Discussions

Figures 3 and 4 show the upper bounds to the stabilitynumber Nρ for cut slopes and natural slopes under staticconditions, respectively. ,ey are plotted against the co-hesion ratio λcρ for ratios of B/H ranging from 1.5 to 10.0 andfor the 2D case. According to the practical experiences ofHunter and Schuster [14], Koppula [15], and Zhang et al.[32, 33], the value of λcρ is selected in the range of 0.0–5.0.Each chart in Figures 3 and 4 illustrates the results for oneinclination angle of the slope.

It can be seen from Figures 3 and 4 that the stabilitynumber Nρ increases almost linearly with the cohesion ratioλcρ. For a given value of λcρ, the stability number Nρ de-creases with the increasing ratio of B/H, and the value of Nρfor 3D failure is greater than that for 2D failure. Obviously,the constraint B on the width of the slope has a significanteffect on the stability number.

In the numerical limit analysis of Li et al. [18, 22],collapses of 3D cut slopes and natural slopes are limited bya rigid base at a depth d below the crest of the slope(Figure 1). When the slope is gentle and the depth d issmall, the rigid base makes the slope more stable, as

expected. ,e static solutions in Figures 3 and 4 do not takeinto account the effect of rigid base on the slope stabilitydue to a conservative estimate on the slope stability. Be-sides, the maximum depths of slope failure surfaces are lessthan 2H for 3D slopes with β ≥ 15°, and then, the rigid baseat the depth d � 2H has no effect on the critical values. Forthis reason, a comparison can be made between the criticalvalues of Nρ calculated from this study and from the nu-merical limit analysis method by Li et al. [18, 22] for cutslopes and natural slopes with depth factor d/H � 2.0 and aratio of B/H � 5.0, as shown in Figure 5. It can be seen thatthe stability number Nρ of this study is always less than thatin the numerical results. Figure 6 shows the comparisons ofthe analytical upper bound results derived from this studyand the numerical upper and lower bound solutions pre-sented by Li et al. [18, 22] for various slope angles. It can beseen that the analytical upper bound is closely bracketed bythe numerical upper and lower bounds. ,e upper boundresult of this study is obviously close to the numerical lowerbound solution rather than the numerical upper bound.,erefore, the best estimate of the upper bound to thecritical value of Nρ has been obtained by the analyticalapproach performed in this study.

For cut slopes and natural slopes subjected to seismicexcitation, a set of stability charts are presented inFigures 7–12 for horizontal acceleration coefficients kh of 0.1,0.2, and 0.3. It should be noted that, when the slopes with thesmall value of slope angle β are subjected to stronger seismicexcitation, the critical value of Nρ will tend to zero as the 2Dhomogeneous slopes in Michalowski [3]. A more rationalvalue is obtained by limiting the depth of the failuremechanism to a realistic value d below the crest. ,e depthfactor of d/H � 2.0 is adopted in the mechanism for gentleslopes under seismic conditions. As expected, the stabilitynumber Nρ reduces with increasing magnitude of the hor-izontal acceleration.

3.1. Applications. According to the above-derived stabilitycharts (Figures 3 and 4 for static conditions and Figures 7 to12 for seismic conditions), the factor of safety F can be easilyobtained for a given 3D slope.

For comparison purposes, the same example slope asanalyzed numerically by Li et al. [18, 22] is adopted here.,ecalculation parameters for the example slope are as follows:the slope height H � 12m, the slope angle β � 60°, the unitweight of soil c � 18.5 kN/m3, the undrained strength at thecrest of slope cu0 � 40 kPa, and the gradient of undrained


80

70

60

50

40

Nρ =

γH

F/c u

0

λcρ = ρHF/cu0

30

B/H = 1.5

2D10.0

5.03.0

2.020

10

00 1 2 3 4 5

(a)

λcρ = ρHF/cu0

Nρ =

γH

F/c u

0 B/H = 1.5

2D10.0

5.03.0

2.0

50

40

30

20

10

00 1 2 3 4 5

(b)

λcρ = ρHF/cu0

Nρ =

γH

F/c u

0

B/H = 1.5

2D10.0

5.03.0

2.0

35

30

25

20

15

10

5

00 1 2 3 4 5

(c)

λcρ = ρHF/cu0

Nρ =

γH

F/c u

0

30

B/H = 1.5

2D10.0

5.03.0

2.0

25

20

15

10

5

00 1 2 3 4 5

γ/ρ= 12.33

(d)

λcρ = ρHF/cu0

Nρ =

γH

F/c u

0 B/H = 1.5

2D10.0

5.03.0

2.0

25

20

15

10

5

00 1 2 3 4 5

(e)

λcρ = ρHF/cu0

Nρ =

γH

F/c u

0 B/H = 1.5

2D10.0

5.03.0

2.0

20

15

10

5

00 1 2 3 4 5

(f )

Figure 3: Static stability charts for cut slopes. (a) β � 15°. (b) β � 30°. (c) β � 45°. (d) β � 60°. (e) β � 75°. (f ) β � 90°.


0 1 2 3 4 5

50

40

30

20

10

0

λcρ = ρHF/cu0

Nρ

= γH

F/c u

0

B/H = 1.5

2.03.0

5.010.02D

(a)

0 1 2 3 4 5

30

25

20

15

10

5

0

λcρ = ρHF/cu0

Nρ

= γH

F/c u

0

B/H = 1.5

2.03.0

5.010.02D

(b)

0 1 2 3 4 5

25

20

15

10

5

0

λcρ = ρHF/cu0

Nρ

= γH

F/c u

0

B/H = 1.5

2.03.0

5.010.02D

(c)

0 1 2 3 4 5

25

20

15

10

5

0

λcρ = ρHF/cu0

Nρ

= γH

F/c u

0 B/H = 1.5

2.03.0 5.0

10.02D

(d)

0 1 2 3 4 5

20

15

10

5

0

λcρ = ρHF/cu0

Nρ

= γH

F/c u

0

B/H = 1.5

2.03.0

5.010.02D

(e)

0 1 2 3 4 5

20

15

10

5

0

λcρ = ρHF/cu0

Nρ

= γH

F/c u

0

B/H = 1.5

2.03.0

5.010.0

2D

(f )

Figure 4: Static stability charts for natural slopes. (a) β � 15°. (b) β � 30°. (c) β � 45°. (d) β � 60°. (e) β � 75°. (f ) β � 90°.


25

This studyLi et al. (2009, 2010)

Nρ =

γHF/c u

0

λcρ = 1.0 (cut slopes)

λcρ = 1.0 (natural slopes)

λcρ = 0.0 (homogeneous slopes)

Slope angle β (°)

20

15

10

5

015 30 45 60 75

Figure 5: Comparisons between upper bounds to the stability number derived from this study and those presented by Li et al. (2009, 2010)for undrained slopes with B/H � 5.0 and d/H � 2.0.

20

15

10

5

0

Nρ =

γHF/c u

0

λcρ = ρHF/cu00.0 0.2 0.4 0.6 0.8 1.0

β = 30°

60°

90°

Analytical upper bound (this study)Numerical upper bound (Li et al., 2009)Numerical lower bound (Li et al., 2009)

(a)

20

15

10

5

0

Nρ =

γHF/c u

0

λcρ = ρHF/cu00.0 0.2 0.4 0.6 0.8 1.0

β = 30°

60°

90°

Analytical upper bound (this study)Numerical upper bound (Li et al., 2010)Numerical lower bound (Li et al., 2010)

(b)

Figure 6: Comparisons between analytical upper bound results obtained from this study and the numerical upper and lower boundsolutions presented by Li et al. (2009, 2010) for (a) cut slopes and (b) natural slopes with B/H � 5.0.


60

50

40

30

20

10

00 1 2

2.0

B/H = 1.5

3.05.0

10.02D

3 4 5λcρ = ρHF/cu0

Nρ

=γH

F/c u

0

(a)

40

30

20

10

00 1 2

2.0

B/H = 1.5

3.05.0

10.02D


Nρ

=γH

F/c u

0

(b)

35

20

30

15

25

10

5

00 1 2

2.0

B/H = 1.5

3.05.0

10.02D


Nρ

=γH

F/c u

0

(c)

30

25

20

15

10

5

00 1 2

2.0

B/H = 1.5

3.05.0

10.02D


Nρ

=γH

F/c u

0

(d)

λcρ = ρHF/cu0

Nρ

=γH

F/c u

0

25

20

15

10

5

00 1 2

2.0

B/H = 1.5

3.05.0

10.02D

3 4 5

(e)

20

15

10

5

00 1 2

2.0

B/H = 1.5

3.05.0

10.02D


Nρ

=γH

F/c u

0

(f )

Figure 7: Seismic stability charts for cut slopes with kh � 0.1. (a) β � 15°. (b) β � 30°. (c) β � 45°. (d) β � 60°. (e) β � 75°. (f ) β � 90°.


50

40

30

20

10

00 1 2

2.0

B/H = 1.5

3.05.0

10.02D


Nρ

=γH

F/c u

0

(a)

35

30

25

20

15

10

5

00 1 2

2.0

B/H = 1.5

3.05.0

10.02D


Nρ

=γH

F/c u

0

(b)

35

20

30

15

25

10

5

00 1 2

2.0

B/H = 1.5

3.05.0

10.02D


Nρ

=γH

F/c u

0

(c)

30

25

20

15

10

5

00 1 2

2.0

B/H = 1.5

3.0 5.010.02D


Nρ

=γH

F/c u

0

(d)

λcρ = ρHF/cu0

Nρ

=γH

F/c u

0

25

20

15

10

5

00 1 2

2.0

B/H = 1.5

3.05.0

10.02D

3 4 5

(e)

20

15

10

5

00 1 2

2.0

B/H = 1.5

3.05.0

10.02D


Nρ

=γH

F/c u

0

(f )



40

30

20

10

00 1 2

2.0

B/H = 1.5

3.0 5.010.0

2D


Nρ

=γH

F/c u

0

(a)

30

25

20

15

10

5

00 1 2

2.0

B/H = 1.5

3.05.0

10.02D


Nρ

=γH

F/c u

0

(b)

35

20

30

15

25

10

5

00 1 2

2.0

B/H = 1.5

3.0 5.010.0

2D


Nρ

=γH

F/c u

0

(c)

30

25

20

15

10

5

00 1 2

2.0

B/H = 1.5

3.05.0

10.02D


Nρ

=γH

F/c u

0

(d)

λcρ = ρHF/cu0

Nρ

=γH

F/c u

0

25

20

15

10

5

00 1 2

2.0

B/H = 1.5

3.0 5.010.0

2D

3 4 5

(e)

20

15

10

5

00 1 2

2.0

B/H = 1.5

3.05.0

10.02D


Nρ

=γH

F/c u

0

(f)



40

30

20

10

00 1 2

2.0

B/H = 1.5

3.05.0

10.02D


Nρ

=γH

F/c u

0

(a)

25

20

15

10

5

00 1 2

2.0

B/H = 1.5

3.05.0

10.02D


Nρ

=γH

F/c u

0

(b)

25

20

15

10

5

00 1 2

2.0

B/H = 1.5

3.05.0

10.02D


Nρ

=γH

F/c u

0

(c)

25

20

15

10

5

00 1 2

2.0

B/H = 1.5

3.05.0

10.02D


Nρ

=γH

F/c u

0

(d)

λcρ = ρHF/cu0

Nρ

=γH

F/c u

0

20

15

10

5

00 1 2

2.0

B/H = 1.5

3.0 5.010.0

2D

3 4 5

(e)

20

15

10

5

00 1 2

2.0

B/H = 1.5

3.05.0

10.02D


Nρ

=γH

F/c u

0

(f )

Figure 10: Seismic stability charts for natural slopes with kh � 0.1. (a) β � 15°. (b) β � 30°. (c) β � 45°. (d) β � 60°. (e) β � 75°. (f ) β � 90°.


30

25

20

15

10

5

00 1 2

2.0

B/H = 1.5

3.05.0

10.02D


Nρ

=γH

F/c u

0

(a)

25

20

15

10

5

00 1 2

2.0

B/H = 1.5

3.05.0

10.02D


Nρ

=γH

F/c u

0

(b)

25

20

15

10

5

00 1 2

2.0

B/H = 1.5

3.05.0

10.02D


Nρ

=γH

F/c u

0

(c)

25

20

15

10

5

00 1 2

2.0

B/H = 1.5

3.05.0

10.02D


Nρ

=γH

F/c u

0

(d)

λcρ = ρHF/cu0

Nρ

=γH

F/c u

0

20

15

10

5

00 1 2

2.0

B/H = 1.5

3.05.0

10.02D

3 4 5

(e)

20

15

10

5

00 1 2

2.0

B/H = 1.5

3.05.0

10.02D


Nρ

=γH

F/c u

0

(f )



25

20

15

10

5

00 1 2

2.0

B/H = 1.5

3.05.0

10.02D


Nρ

=γH

F/c u

0

(a)

20

15

10

5

00 1 2

2.0

B/H = 1.5

3.05.0 10.0

2D


Nρ

=γH

F/c u

0

(b)

25

20

15

10

5

00 1 2

2.0

B/H = 1.5

3.05.0

10.02D


Nρ

=γH

F/c u

0

(c)

25

20

15

10

5

00 1 2

2.0

B/H = 1.5

3.05.0 10.0

2D

3 4 5

λcρ = ρHF/cu0

Nρ

=γH

F/c u

0

(d)

λcρ = ρHF/cu0

Nρ

=γH

F/c u

0

20

15

10

5

00 1 2

2.0

B/H = 1.5

3.05.0

10.02D

3 4 5

(e)

20

15

10

5

00 1 2

2.0

B/H = 1.5

3.0 5.010.0

2D

3 4 5

λcρ = ρHF/cu0

Nρ

=γH

F/c u

0

(f )



strength with respect to the depth ρ � 1.5 kPa/m. To de-termine the safety factor of the cut slopes and the naturalslopes,Nρ/λcρ � c/ρ � 18.5/1.5 � 12.33 is first obtained. Fromthe charts for β � 60° in Figures 3 and 4, a straight linepassing through the origin with a gradient Nρ/λcρ � 12.33 isplotted. ,is straight line intersects the curves for the resultsof 3D and 2D slope failures. ,e safety factor F can be easilycalculated by reading Nρ for a given ratio of B/H and

dividing the value of cH/cu0. Table 1 shows the calculatedresults for various ratios of B/H and for the 2D case, togetherwith the corresponding data retrieved from Table 1 of Liet al. [18, 22]. Compared with the results derived from thefinite element upper bound limit analysis, a better estimateof the upper bounds to safety factor is obtained from theanalytical limit analysis proposed in this study, with theunexpected exception of the 2D case in the natural slope.

Table 1: Comparisons between calculated results from the analytical approach of this study and the numerical method of Li et al. (2009,2010) for the example slope.

B/H � 2 B/H � 3 B/H � 5 2D,isstudy

Numericalresult

,isstudy

Numericalresult

,isstudy

Numericalresult

,isstudy

Numericalresult

Cut slopes Nρ 9.33 10.25∗ 8.62 9.3∗ 7.92 9.0∗ 7.19 7.8∗

Fcut 1.68 1.85∗ 1.55 1.68∗ 1.43 1.62∗ 1.30 1.41∗

Naturalslopes

Nρ 8.21 9.1† 7.59 8.25† 7.16 7.75† 6.51 6.3†(6.6‡)Fnatural 1.48 1.64† 1.37 1.49† 1.29 1.40† 1.17 1.14†(1.19‡)

Note. ∗Data retrieved from Table 1 of Li et al. (2009); †data retrieved from Table 1 of Li et al. (2009); ‡data reread from Figure 7(a) of Li et al. (2010) by theauthors.

Table 2: Coefficients A and B in equation (12) for cut slopes.

β B/Hkh � 0 kh � 0.1 kh � 0.2 kh � 0.3

A B A B A B A B

15°

1.5 14.3103 11.5429 10.7045 8.0734 8.5028 6.0669 6.7166 4.15102.0 11.9647 10.6815 8.6907 7.4181 6.8300 5.5594 5.6032 4.39823.0 9.9572 9.9457 6.9418 6.9535 5.3955 5.1557 4.3198 4.03435.0 8.6752 9.4006 5.8925 6.5406 4.4115 4.8650 3.4263 3.768810.0 7.8768 9.0236 5.2417 6.2664 3.7832 4.6572 2.8651 3.5592∞ (2D) 7.2769 8.7267 4.6914 6.0842 3.2662 4.5086 2.2893 3.3425

30°

1.5 9.6768 6.6362 8.0970 5.3987 6.8890 4.4735 6.0354 3.79672.0 8.7703 6.3038 7.0969 5.1453 5.9931 4.2410 5.2151 3.52793.0 7.7745 6.0632 6.2268 4.9168 5.1412 4.0708 4.3496 3.36355.0 7.1536 5.8459 6.2268 4.9168 4.4777 3.9215 3.4970 3.289710.0 6.7290 5.7188 5.5593 4.7623 4.0284 3.8400 2.9978 3.2242∞ (2D) 6.4288 5.5573 4.7424 4.5944 3.5700 3.8015 2.3854 3.2131

45°

1.5 8.1155 4.8325 7.1212 4.1478 6.3609 3.5965 5.7601 3.07942.0 7.5187 4.6165 6.5129 3.9617 5.7087 3.4071 5.0744 2.93033.0 6.9890 4.4338 5.8712 3.8236 5.0805 3.2883 4.3923 2.83725.0 6.5002 4.3138 5.3344 3.7402 4.4343 3.2333 3.5826 2.827310.0 6.1923 4.2501 4.9194 3.7136 4.0087 3.2056 3.1355 2.8042∞ (2D) 5.9501 4.1510 4.6053 3.6623 3.6720 3.1776 2.6632 2.8153

60°

1.5 7.0929 3.7568 6.4416 3.3219 6.0455 2.8950 5.5240 2.56192.0 6.5673 3.6151 5.8267 3.2500 5.3624 2.7979 4.8504 2.46103.0 6.1872 3.4545 5.4323 3.0604 4.9564 2.6698 4.1890 2.41295.0 5.7905 3.3836 5.0814 2.9724 4.2369 2.6748 3.6481 2.373010.0 5.5322 3.3267 4.7003 2.9604 3.9028 2.6499 3.2809 2.3613∞ (2D) 5.3239 3.2561 4.3973 2.9348 3.5675 2.6394 2.8129 2.3864

75°

1.5 6.1642 3.0140 5.7583 2.6924 5.4741 2.3745 4.9336 2.17582.0 5.7688 2.8644 5.2573 2.5894 4.7992 2.3368 4.4029 2.08693.0 5.3404 2.7583 4.8569 2.4881 4.3137 2.2430 3.9827 2.00935.0 5.0461 2.6674 4.5502 2.3975 4.0368 2.1564 3.7437 1.902810.0 4.8551 2.6220 4.3280 2.3516 3.7027 2.1488 3.4788 1.8779∞ (2D) 4.5987 2.5899 4.1375 2.3088 3.3854 2.1423 3.2900 1.8375

90°

1.5 5.4109 2.4119 5.1349 2.1369 4.7327 2.0014 4.4387 1.86642.0 4.9769 2.2792 4.6642 2.0652 4.3330 1.8852 4.0007 1.72503.0 4.5868 2.1895 4.1953 2.0087 3.8523 1.8380 3.5886 1.63325.0 4.2956 2.1136 3.9367 1.9290 3.6329 1.7508 3.2815 1.608510.0 4.0946 2.0605 3.7224 1.8910 3.3992 1.7088 3.0884 1.5669∞ (2D) 3.8672 2.0165 3.5075 1.8475 3.1401 1.6933 2.8899 1.5300


However, a careful check against Figure 7(b) of Li et al. [18]shows that the reading value ofNρ for the 2D natural slope isapproximately 6.6 rather than the number 6.3 presented inTable 1 of Li et al. [22]. ,erefore, the only exception isdoubted.

Alternatively, an analytical approximation of the curvesin the stability charts can be made as

cHF

cu0� A +

ρHFcu0

B, (12)

where coefficients A and B are determined by a linear fittingtechnique. It should be noted that the curves have a slightcurvature for natural slopes with small value of λcρ. Nev-ertheless, the goodness of fit measured by the statistical

Table 3: Coefficients A and B in equation (12) for natural slopes.

β B/Hkh � 0 kh � 0.1 kh � 0.2 kh � 0.3

A B A B A B A B

15°

1.5 14.0417 5.5213 10.7765 3.9044 8.3917 3.0683 6.8037 2.53672.0 12.1295 5.5304 8.8251 3.9860 6.8320 3.1272 5.5596 2.56553.0 10.2958 5.4513 7.1932 4.0178 5.5081 3.1519 4.3663 2.60455.0 8.9213 5.4354 6.1444 3.9802 4.5560 3.1486 3.5573 2.589110.0 8.1138 5.3815 5.4981 3.9474 3.9846 3.1224 3.0156 2.5757∞ (2D) 7.4668 5.3353 4.9707 3.9254 3.5103 3.1140 2.5252 2.5857

30°

1.5 9.8257 3.3207 8.2197 2.7972 6.9813 2.4186 6.0906 2.12912.0 9.0010 3.2809 7.2876 2.8136 6.0977 2.4466 5.2739 2.13433.0 7.9860 3.2971 6.4485 2.8280 5.2963 2.4598 4.3928 2.19595.0 7.3384 3.2848 5.7426 2.8412 4.5894 2.5033 3.6805 2.237310.0 6.8809 3.2821 5.3106 2.8429 4.1576 2.4996 3.2194 2.2420∞ (2D) 6.5250 3.2611 4.9572 2.8385 3.7456 2.5202 2.7498 2.2779

45°

1.5 8.2142 2.6961 7.1962 2.4184 6.3768 2.1509 5.7165 1.92962.0 7.6745 2.6371 6.6418 2.3650 5.7948 2.1230 5.1580 1.89643.0 7.0647 2.6149 6.0183 2.3402 5.1551 2.1331 4.4334 1.92625.0 6.5959 2.6024 5.4339 2.3723 4.5574 2.1465 3.7592 1.974510.0 6.2847 2.5785 5.0647 2.3742 4.1360 2.1727 3.3140 2.0039∞ (2D) 6.0276 2.5560 4.7538 2.3714 3.8115 2.1781 2.8561 2.0462

60°

1.5 7.1388 2.4670 6.4687 2.2474 5.9017 2.0296 5.4226 1.82602.0 6.6483 2.4103 5.9534 2.1969 5.3860 1.9818 4.8393 1.79183.0 6.1799 2.3493 5.5046 2.1364 4.9641 1.9215 4.2803 1.77375.0 5.8480 2.2981 5.1576 2.0883 4.3155 1.9593 3.6896 1.807510.0 5.6342 2.2414 4.7757 2.0956 3.9675 1.9572 3.3408 1.8128∞ (2D) 5.3836 2.2417 4.4778 2.0925 3.6433 1.9643 2.8754 1.8553

75°

1.5 6.2502 2.4269 5.7615 2.2425 5.4851 1.9719 4.9201 1.83962.0 5.7408 2.3540 5.2850 2.1560 4.8077 1.9557 4.3905 1.78733.0 5.3571 2.2359 4.8675 2.0673 4.3504 1.8950 3.9983 1.68945.0 5.0488 2.1735 4.5430 1.9894 4.0446 1.8197 3.7401 1.632710.0 4.8306 2.1313 4.3531 1.9368 3.7432 1.8107 3.4934 1.5955∞ (2D) 4.6297 2.0899 4.1505 1.8962 3.4157 1.7986 3.3170 1.5582

90°

1.5 5.4109 2.4119 5.1349 2.1369 4.7327 2.0014 4.4387 1.86642.0 4.9769 2.2792 4.6642 2.0652 4.3330 1.8852 4.0007 1.72503.0 4.5868 2.1895 4.1953 2.0087 3.8523 1.8380 3.5886 1.63325.0 4.2956 2.1136 3.9367 1.9290 3.6329 1.7508 3.2815 1.608510.0 4.0946 2.0605 3.7224 1.8910 3.3992 1.7088 3.0884 1.5669∞ (2D) 3.8672 2.0165 3.5075 1.8475 3.1401 1.6933 2.8899 1.5300

Table 4: Factors of safety for the example slope.

B/H � 1.5 B/H � 2.0 B/H � 3.0 B/H � 5.0 B/H � 10.0 2D

kh � 0Fcut 1.84 1.67 1.55 1.44 1.36 1.30

Fnatural 1.61 1.49 1.38 1.29 1.24 1.18

kh � 0.1Fcut 1.59 1.43 1.30 1.21 1.11 1.04

Fnatural 1.43 1.31 1.20 1.12 1.04 0.97

kh � 0.2Fcut 1.42 1.25 1.14 0.97 0.90 0.82

Fnatural 1.27 1.16 1.06 0.92 0.85 0.78

kh � 0.3Fcut 1.26 1.09 0.94 0.81 0.73 0.63

Fnatural 1.15 1.02 0.90 0.78 0.71 0.61


coefficient R2 can reach 0.99 for natural slopes and evenexceed 0.999 for cut slopes. ,e coefficients A and B aregiven in Tables 2 and 3 for cut and natural slopes, re-spectively. ,us, the safety factor F can be more easily de-rived from equation (12), and it can avoid the error byreading the stability charts.

For the above slope example, the factor of safety F isobtained from equation (12) and presented in Table 4. Notsurprisingly, the factor of safety for cut slope or natural slopedecreases with increasing magnitude of the horizontal ac-celeration kh and with increasing ratio of B/H. Moreover,using the results from the 2D analysis underestimates thestability of 3D cut slopes and natural slopes. It can be foundfrom Table 4 that the difference in the safety factors between3D and 2D analysis increases with increasing magnitude ofhorizontal acceleration kh and reducing ratio of B/H. Typ-ically, the difference can exceed 50% when the slope isconstrained to a narrow width of B/H � 1.5.

4. Conclusions

Based on the 3D kinematically admissible rotational failuremechanism, an analytical approach is derived for the upperbound limit analysis of the stability of cut slopes and naturalslopes under short-term undrained conditions. Comparedwith the finite element limit analysis method adopted by Liet al. [18, 22], the proposed analytical approach gives thebetter estimate of the upper bounds to the stability numberNρ. A set of stability charts is presented for both cut slopesand natural slopes under static and pseudostatic seismicloading conditions. ,e safety factor can be easily obtainedfrom the charts to evaluate the stability of cut slopes andnatural slopes. Furthermore, the results indicate that using2D solutions to evaluate the stability of 3D slopes will un-derestimate the factor of safety. ,e difference between 2Dand 3D factors of safety increases with the reducing ratio ofB/H and increasing horizontal acceleration coefficient kh.

Data Availability

,e data used to support the findings of this study are in-cluded within the article.

Conflicts of Interest

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

References

[1] D. W. Taylor, “Stability of earth slopes,” Journal of the BostonSociety of Civil Engineers, vol. 24, no. 3, pp. 197–246, 1937.

[2] N. Janbu, Slope Stability Computations, Soil Mechanics andFoundation Engineering Report, ,e Technical University ofNorway, Trondheim, Norway, 1968.

[3] R. L. Michalowski, “Stability charts for uniform slopes,”Journal of Geotechnical and Geoenvironmental Engineering,vol. 128, no. 4, pp. 351–355, 2002.

[4] R. Baker, “A second look at Taylor’s stability chart,” Journal ofGeotechnical and Geoenvironmental Engineering, vol. 129,no. 12, pp. 1102–1108, 2003.

[5] R. Baker, R. Shukha, V. Operstein, and S. Frydman, “Stabilitycharts for pseudo-static slope stability analysis,” Soil Dy-namics and Earthquake Engineering, vol. 26, no. 9, pp. 813–823, 2006.

[6] T. Steward, N. Sivakugan, S. K. Shukla, and B. M. Das,“Taylor’s slope stability charts revisited,” International Journalof Geomechanics, vol. 11, no. 4, pp. 348–352, 2011.

[7] A. Gens, J. N. Hutchinson, and S. Cavounidis, “,ree-dimensional analysis of slides in cohesive soils,”Géotechnique, vol. 38, no. 1, pp. 1–23, 1988.

[8] R. L. Michalowski, “Limit analysis and stability charts for 3Dslope failures,” Journal of Geotechnical and GeoenvironmentalEngineering, vol. 136, no. 4, pp. 583–593, 2010.

[9] R. L. Michalowski and T. Martel, “Stability charts for 3Dfailures of steep slopes subjected to seismic excitation,”Journal of Geotechnical and Geoenvironmental Engineering,vol. 137, no. 2, pp. 183–189, 2011.

[10] F. Zhang, D. Leshchinsky, R. Baker, Y. Gao, andB. Leshchinsky, “Implications of variationally derived 3Dfailure mechanism,” International Journal for Numerical andAnalytical Methods in Geomechanics, vol. 40, no. 18,pp. 2514–2531, 2016.

[11] F. Zhang, D. Leshchinsky, Y. Gao, and S. Yang, “,ree-dimensional slope stability analysis of convex turning cor-ners,” Journal of Geotechnical and Geoenvironmental Engi-neering, vol. 144, no. 6, p. 06018003, 2018.

[12] F. Zhang, Y. Gao, D. Leshchinsky, S. Yang, and G. Dai, “3Deffects of turning corner on stability of geosynthetic-reinforced soil structures,” Geotextiles and Geomembranes,vol. 46, no. 4, pp. 367–376, 2018.

[13] R. E. Gibson and N. Morgenstern, “A note on the stability ofcuttings in normally consolidated clays,” Géotechnique,vol. 12, no. 3, pp. 212–216, 1962.

[14] J. H. Hunter and R. L. Schuster, “Stability of simple cuttings innormally consolidated clays,” Géotechnique, vol. 18, no. 3,pp. 372–378, 1968.

[15] S. D. Koppula, “On stability of slopes in clays with linearlyincreasing strength,” Canadian Geotechnical Journal, vol. 21,no. 3, pp. 577–581, 1984.

[16] S. D. Koppula, “Pseudo-static analysis of clay slopes subjectedto earthquakes,” Géotechnique, vol. 34, no. 1, pp. 71–79, 1984.

[17] H. S. Yu, R. Salgado, S. W. Sloan, and J. M. Kim, “Limitanalysis versus limit equilibrium for slope stability,” Journal ofGeotechnical and Geoenvironmental Engineering, vol. 124,no. 1, pp. 1–11, 1998.

[18] A. J. Li, R. S. Merifield, and A. V. Lyamin, “Limit analysissolutions for three dimensional undrained slopes,” Computersand Geotechnics, vol. 36, no. 8, pp. 1330–1351, 2009.

[19] A. V. Lyamin and S. W. Sloan, “Lower bound limit analysisusing non-linear programming,” International Journal forNumerical Methods in Engineering, vol. 55, no. 5, pp. 573–611,2002.

[20] A. V. Lyamin and S. W. Sloan, “Upper bound limit analysisusing linear finite elements and non-linear programming,”International Journal for Numerical and Analytical Methods inGeomechanics, vol. 26, no. 2, pp. 181–216, 2002.

[21] K. Krabbenhoft, A. V. Lyamin, M. Hjiaj, and S. W. Sloan, “Anew discontinuous upper bound limit analysis formulation,”International Journal for Numerical Methods in Engineering,vol. 63, no. 7, pp. 1069–1088, 2005.

[22] A. J. Li, R. S. Merifield, and A. V. Lyamin, “,ree-dimensionalstability charts for slopes based on limit analysis methods,”Canadian Geotechnical Journal, vol. 47, no. 12, pp. 1316–1334,2010.


[23] R. L. Michalowski and A. Drescher, “,ree-dimensionalstability of slopes and excavations,” Géotechnique, vol. 59,no. 10, pp. 839–850, 2009.

[24] Y. F. Gao, F. Zhang, G. H. Lei, and D. Y. Li, “An extended limitanalysis of three-dimensional slope stability,” Géotechnique,vol. 63, no. 6, pp. 518–524, 2013.

[25] Z.-Y. Chen, “Random trials used in determining globalminimum factors of safety of slopes,” Canadian GeotechnicalJournal, vol. 29, no. 2, pp. 225–233, 1992.

[26] Y. F. Gao, F. Zhang, G. H. Lei, D. Y. Li, Y. X. Wu, andN. Zhang, “Stability charts for 3D failures of homogeneousslopes,” Journal of Geotechnical and Geoenvironmental En-gineering, vol. 139, no. 9, pp. 1442–1454, 2013.

[27] S. L. Kramer, Geotechnical Earthquake Engineering, Prentice-Hall, Upper Saddle River, NJ, USA, 2003.

[28] M. Iskander, Z. Chen, M. Omidvar, I. Guzman, andO. Elsherif, “Active static and seismic earth pressure for c-φsoils,” Soils and Foundations, vol. 53, no. 5, pp. 639–652, 2013.

[29] E. Conte, A. Troncone, and M. Vena, “A method for thedesign of embedded cantilever retaining walls under static andseismic loading,” Géotechnique, vol. 67, no. 12, pp. 1081–1089,2017.

[30] W. F. Chen, Limit Analysis and Soil Plasticity, Elsevier Sci-ence, Rotterdam, Netherlands, 1975.

[31] R. L. Michalowski and L. You, “Displacements of reinforcedslopes subjected to seismic loads,” Journal of Geotechnical andGeoenvironmental Engineering, vol. 126, no. 8, pp. 685–694,2000.

[32] J. H. Zhang, J. Li, Y. Yao, J. Zheng, and F. Gu, “Geometricanisotropy modeling and shear behavior evaluation of gradedcrushed rocks,” Construction and Building Materials, vol. 183,pp. 346–355, 2018.

[33] J. H. Zhang, J. Peng, J. S. Zheng, and Y. Yao, “Characterisationof stress and moisture-dependent resilient behaviour forcompacted clays in South China,” Road Materials andPavement Design, pp. 1–14, 2018.


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StaticandSeismicStabilityChartsforThree-DimensionalCut ...the limit equilibrium method, the limit analysis method, andtheﬁniteelementmethod. For homogeneous slopes, stability charts