Research ArticleModified Limiting Equilibrium Method for StabilityAnalysis of Stratified Rock Slopes

Rui Yong1 Chang-Dong Li2 Jun Ye1 Man Huang1 and Shigui Du12

1Key Laboratory of Rock Mechanics and Geohazards Shaoxing University Zhejiang Shaoxing 312000 China2Engineering Faculty China University of Geosciences Wuhan Hubei 430074 China

Correspondence should be addressed to Shigui Du dsgusxeducn

Received 15 May 2016 Revised 24 July 2016 Accepted 30 August 2016

Academic Editor Jose J Munoz

Copyright copy 2016 Rui Yong et alThis is an open access article distributed under the Creative CommonsAttribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The stratified rock of Jurassic strata is widely distributed in Three Gorges Reservoir Region The limit equilibrium method isgenerally utilized in the stability analysis of rock slope with single failure plane However the stratified rock slope cannot beaccurately estimated by this method because of different bedding planes and their variable shear strength parameters Based onthe idealized model of rock slope with bedding planes a modified limiting equilibrium method is presented to determine thepotential sliding surface and the factor of safety for the stratified rock slope In this method the S-curve model is established todefine the spatial variations of the shear strength parameters 119888 and 120593 of bedding plane and the tensile strength of rock mass Thismethod was applied in the stability evaluation of typical stratified rock slope in Three Gorges Reservoir Region China The resultshows that the factor of safety of the case study is 0973 the critical sliding surface for the potential slip surface appears at beddingplane C and the tension-controlled failure occurs at 105m to the slope face

1 Introduction

The analysis for the estimation of rock slope stability hasbeen a challenging task for engineers in civil and miningengineering [1 2] Rock masses constituting the slopes oftenhave discontinuities (jointing and bedding patterns) andfree spaceexcavation surfaces in various forms resulting indifferent types of slope failures [3 4] Planar failure is one ofthe most common rock slope failure modes that happened instratified rock masses [5ndash7] The stratified structure of a rockmass is typically acquired during sedimentary depositionand planar structures such as foliation or schistosity canalso be produced by metamorphism This process can causedifferences in material composition particle size and fabricor mineral orientation that result in rock stratification [8 9]The dips and mechanical parameters of the bedding planesin naturally stratified rock masses have stronger influenceson rock mass strength and stability Therefore the appraisalof slope stability in stratified rock masses is complicatedbecause dominant discontinuities lead to a highly anisotropicbehavior [10] The potential sliding surfaces in stratified rock

masses should be considered in determining the criticalsliding surface and the stability of rock slope However theprevious studies on planar failure were mainly focused on thestability of rock slopes with a single failure plane and they arenot capable of analyzing the rock slope with various beddingplanes

The limit equilibrium method for the estimation of thefactor of safety (FS) of the rock slope against plane failurehas been widely accepted by the engineers mainly becauseof its validity and simplicity Hoek and Bray [11] presented ananalytical solution for the plane failure mode in rock slopesin which the upper slope surface and tension crack wereassumed to be horizontal and vertical respectively Based onthis research Sharma et al [12] extended the plane failureanalysis to incorporate an inclined upper slope and a nonver-tical tension crack Kroeger [13] illustrated the plane failureanalyses of compound slopes by limit equilibrium approachand proposed new equations for determining tension cracksJiang et al [14] proposed different limiting equilibriumapproaches for computing the stability of the planar failurerock slope Ahmadi and Eslami [15] explored the effects of

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2016 Article ID 8381021 9 pageshttpdxdoiorg10115520168381021

2 Mathematical Problems in Engineering

the forces due to water pressure on discontinuity surfacein plane failure rock slope However these methods arerestricted by the requirement that single valued parametersdescribe the slope characteristics [16] In actuality the shearstrength of a rock mass is spatially variable due to thefactors like climate carbonation geology relief vegetationcover or human activities For instance some slopes may becomposed of rocks which have been extensively and deeplyweathered in situ over long periods [17] The strength ofrock mass near the slope surface and upper slope surfaceis easily affected by these factors but less influence on thedeeper inside of the rock mass Obviously we cannot neglectthe strength variations along the sliding surface which mayaffect the accuracy of slope stability calculations by the limitequilibrium method However this inherent variability ofshear strength has often been portrayed in an oversimplifiedway in earlier researches on rock slope instability

The objectives of this paper are (1) to present amathemat-ical model for describing the spatial variations of the shearstrength rock discontinuities and tension strength of rockmass (2) to propose calculation procedure for determiningthe critical sliding surface and tension crack and (3) tointroduce amethod for evaluating the stability of the stratifiedrock slope Furthermore this modified limiting equilibriummethod was applied to the stability assessment of the casestudy Jiaxiao Slope in Three Gorges Reservoir RegionChina

2 Calculation Method and Model

21 Geomechanical Model Planar rock slope failure occurswhen a rockmass in a slope slides down and along a relativelyplanar failure surface These failure surfaces are usuallystructural discontinuities such as bedding planes faultsjoints or the interface between bedrock and an overlyinglayer of weathered rock Stability condition occurs if thecritical joint dip is less than the slope angle and mobilizedjoint shear strength is not enough to assure the stability ofthe slope

Theplanar failuremode provides insightful knowledge onthe behavior of rock slopes in which the material above thissurface is regard as a ldquofree bodyrdquoThe disturbing and resistingforces are estimated enabling the formulation of equationsconcerning force or moment equilibrium (or both) of thepotential sliding mass FS is defined as the ratio of resistingforces to driving forces

Let us consider a slope of inclination 119894 height 119867 and apotential failure plane of inclination 120573 (Figure 1) Accordingto the geometric properties in Figure 1 the weight of the rigidblock ABC is calculated by

ℎ =

119867

sin 119894sin (119894 minus 120573)

119871 =

119867

sin120573

119866 =

1

2

120588119892ℎ119871 =

1205881198921198672 sin (119894 minus 120573)

2 sin 119894 sdot sin120573

(1)

H

x

yUpper slope surface

Slope faceSliding mass

Sliding surface

A

CB

Gh

L

i 120573

Gsin120573

Gcos120573

Figure 1 A rock slope in plane failure

where 120588 is the average density of the rock mass 119892 is thegravitational acceleration (98ms2) 119871 is the length of thefailure plane AC It is easy to obtain the driving force 119865

119903and

resisting force 119865119904by the force analysis of the rigid block ABC

119865119904= 119866 cos120573 tan120593 + 119888119871

119865119903= 119866 sin120573

(2)

where 120593 and 119888 are the effective internal friction angle andcohesion of failure plane The factor of safety (FS) can beobtained by the force equilibrium equation as follows

FS =119865119904

119865119903

=

119866 cos120573 tan120593 + 119888119871119866 sin120573

=

2119888 sin 119894120588119892119867 sin (119894 minus 120573) sin120573

+

tan120593tan120573

(3)

This model is widely used in planar rock slope failureanalysis which offers considerable insights into the behaviorof real slopes However it failed to be utilized for the stabilityanalyzing of the stratified rock slope with series of beddingplanes Then the shear strength parameters 120593 and 119888 alongthe failure plane were treated as constant values which failedto reflect the strength variations along the sliding surfaceFurthermore the potential failure rock mass was commonlyassumed as a rigid block so the potential tensile failurecannot be considered in the planar rock slope failure analysis

The sliding mass of stratified rock slope can be dividedinto several plane segments by rock bedding surfaces (Fig-ure 2) However it is hard to find the slip surface from thesebedding surfaces geological discontinuities and surfacesof weakness The limit equilibrium analysis method wasextended for determining the critical position in all thepotential slip surfaces Here the exposed length of each layer119894 on the upper ground surface is labeled by 119871

119886119894 The exposed

length on the slope face is denoted as 119871119887119894 The potential

tension crack appears on the upper slope surface and itshorizontal distance to slope face is 119871

119888 The tensile strength

of sliding mass is considered in this idealized model becausethe rock mass may have a tensile failure and create the tensilecrack in some conditions (eg the shear strength of most ofthe sliding surface is very low but reaches high value nearupper slope surface) Furthermore the field investigations

Mathematical Problems in Engineering 3

Potential cracking surface

A

C

Upper slope surface

Bedding surface i

Slope face

B

120572

Bedding surface i minus 1

Lc

hc

La1 La2 La3 Lai

120573iLbi

Lb1

Lb3Lb2

middot middot middot

middot middotmiddot

Figure 2 Idealized model of rock slope with bedding planes

indicate that the cohesion 119888 and friction angle 120593 of beddingsurfaces are variable at different slip surfaces As commonsense the shear strength of the rock discontinuities near toslope surface is usually lower than that in the deep place Dueto the anisotropy of shear strength it is no longer reasonableto simplify the bedding rock slope failure by the planar rockslope mode in Figure 1

Furthermore in the idealized model of stratified rockslope the locations of the bedding planes are determinedbased on the exposed length 119871

119886119894of each layer on the upper

ground surface and the exposed length 119871119887119894on the slope face

However there is a challenge in the pairing up of the daylightlocations for a potential slip plane on the top boundary andslope face because sometimes the exposed length 119871

119887119894on

the slope face is difficult to measure especially for thoseslopes with large slope angles In rock engineering practicesthe geotechnical drilling is suggested to be implemented atdifferent positions on the upper surface of rock slope Basedon the core samples from drilling tubes the dip angles ofthe bedding planes can be obtained via measuring the dipangles of the bedding planes Generally the potential failuresurface is assumed to be planar in planar rock slope failureanalysis Then the corresponding exposed length 119871

119887119894on the

slope surface can be calculated through daylight location onthe top boundary and dip angle of the bedding plane

22 Basic Assumptions Comparing the single planar failuremodel with the idealized model of bedding rock slope it isnecessary to make a few assumptions for the stability assess-ment The main assumptions are summarized as follows

(1) It is assumed that the slope face upper slope surfacefailure plane and the tension crack are parallel ornearly parallel The rock slope has a vertical tensioncrack and the upper surface of the slope is horizontal

(2) The assumed failure mode is translational-crack-slipand the sliding block is assumed to have a unitthickness in the planar failure analysis

(3) Release or free surfaces are present parallel to thecross section of the analysis which provides negligible

Distance to slope surface

Stre

ngth

par

amet

ers

a b c

dA1 A2

b1 b2H1

H2

Figure 3 Relationship between strength parameters and distance toslope surface

or no resistance to sliding at the lateral boundaries ofthe failure

(4) The tensile strength 120590 of the rock mass and shearstrength parameters 119888 and 120593 of rock discontinuitiesare assumed to be anisotropic and the spatial vari-ation model of strength parameters is provided inSection 23

(5) The above assumptions represent rock block slid-ing along a single joint which is the potential slipsurface in the rock mass and there is no relativesliding between different bedding planes above theslip surface The shear force over the slip surface isdetermined by Mohr-Coulomb criterion here

23 Spatial Variation Model for Rock Mass and Rock Dis-continuities The spatial variability of rock discontinuitiesshear strength parameters 119888 and 120593 and the rock mass tensilestrength 120590 is considered in this idealized model of beddingrock slope S-curve model belongs to multiple variable anal-ysis methods and mainly appears in logit model or logisticregression model It is widely used in the statistical andempirical analysis in fields like sociology biostatistics clinicalmedicine marketing management and so forth S-curvemodel has the basic features for describing the progressivetransition from one value to another which increases non-linearly and continuously whose variation trend is shapedlike ldquoSrdquo (Figure 3) In previous studies the S-curve model hasbeen used successfully in describing the spatial variations ofthe shear strength parameters of the bedding plane [8 18 19]There are real data supporting the validity of the S-curvemodel and this model has been successfully used to describethe spatial variations of the shear strength parameters causedby the weakening effects Based on the S-curve model somesuggested methods were proposed for analyzing the stabilityof rock slope with bedding surfaces Here it is appropriateto illustrate the variations of the strength including shearstrength parameters 119888 and120593 of rock discontinuities and therock mass tensile strength 120590

4 Mathematical Problems in Engineering

According to S-curve model the strength can beexpressed as follows

119878 (119909) =

119860

1 + 119890minus1198881015840(119909minus119887)

+ 1198671015840 (4)

where 119878(119909) denotes the strength of rock mass or rockdiscontinuities 119909 is the distance to the slope surface 1198671015840 isthe lowest strength of rock mass or rock discontinuities Thestrength of rock mass or rock discontinuities deep insidethe rock slope is barely affected by the factors mentionedabove so it gets the largest value 119860 is the difference betweenthe lowest and the largest strength The comparison betweencurves (a) and (d) in Figure 3 shows that the variation oflowest and largest value in S-curve model is determined bycoefficients 119860 and 1198671015840 119887 and 1198881015840 are the shape controllingfactors of S-curve model Then 119887 represents the distance tothe slope surface when the strength is equal to themean valueof the lowest and the highest strength and 1198881015840 is positivelycorrelated with the inverse of 119887

In the present study the S-curve model is established todefine the spatial variations of the shear strength parametersof bedding plane and the tensile strength of rock mass Forthe shear strength parameters 1198671015840 is the shear strength ofhighly weathered rock discontinuities near slope surface119860 is the difference between the shear strength value ofthe highly weathered rock discontinuities and the highestshear strength value of discontinuities which are the leastaffected by weathering effect 119887 and 1198881015840 are obtained by curve-fitting of the shear strength of the rock discontinuities andthe locations of rock joint specimens For tensile strengthparameters1198671015840 is the tensile strength of rock specimens nearslope surface 119860 is acquired by calculating the differencebetween1198671015840 and the tensile strength of intact rock specimenbased on the variable tensile strength and the depth of rockspecimens 119887 and 1198881015840 values are acquired through the S-curvemodel fitting

It is assumed that the shear strength parameters areassociated with the distance to slope surface The distancesbetween upper slope surface and slope face have the sameinfluence on the shear strength parameters A set of equationsfor describing shear strength parameters 119888 and 120593 along thebedding surface are illustrated as follows

120593 (119909) =

119860120593

1 + 119890minus119888120593(119909minus119887120593)+ 119867120593 (119909 le

119871119894

2

)

119860120593

1 + 119890minus119888120593(119871119894minus119909minus119887120593)+ 119867120593 (

119871119894

2

le 119909 le 119904119894)

119888 (119909) =

119860119888

1 + 119890minus119888119888(119909minus119887119888)+ 119867119888 (119909 le

119871119894

2

)

119860119888

1 + 119890minus119888119888(119871119894minus119909minus119887119888)+ 119867119888 (

119871119894

2

le 119909 le 119904119894)

(5)

where 119867120593and 119867

119888are the shear strength of rock discon-

tinuities near slope surface and they are the lowest shearstrength parameters the shear strengths of new and completerock discontinuities reach the largest value 119860

120593and 119860

119888are

the differences between the lowest and largest shear strengthparameters 119887

120593 119887119888 119888120593 and 119888

119888are the shape controlling coeffi-

cients of equation set These coefficients can be obtained bydirect shear tests on the specimens taken from different partsof bedding surface In Figure 4 119871

119894is the length of bedding

surface and 119904119894is the length of the potential sliding surface

which is controlled by the cracking surfaceSimilarly tensile strength based on the S-curvemodel can

be expressed as

120590 (ℎ) =

119860120590

1 + 119890minus119888120590(ℎminus119887120590)+ 119867120590 (6)

where 119867120590is the lowest tensile strength of rock mass on the

slope surface 119860120590is the difference between the lowest and

largest tensile strength 119887120590and 119888120590are the shape controlling

coefficients of tensile strength change function ℎ is the depthfrom the upper slope surface to the intersection of beddingsurface and cracking surface (Figure 5) In this study thepotential vertical tensile cracks are assumed to appear whenthe sliding happens The influence on the distributions of thecohesion and frictional angle in tensile crack developing wasnot considered here

24 Modified Limiting Equilibrium Equations The rock massabove the failure surface is regarded as an entire sliding bodyand the back geometric boundary is the potential crackingsurface (Figure 6) Here the bedding surface 119894 is the failuresurface

The length of bedding surface 119894 can be described by

119871119894

= radic[(

119894

sum

119896=1

119897119887119896) sdot cos120572 + (

119894

sum

119896=1

119897119886119896)]

2

+ [(

119894

sum

119896=1

119897119887119896) sdot sin120572]

2

(7)

The failure plane dip is

120573119894= arcsin(

sum119894

119896=1119897119887119896sdot sin120572

radic[(sum119894

119896=1119897119887119896) sdot cos120572 + (sum119894

119896=1119897119886119896)]

2

+ [(sum119894

119896=1119897119887119896) sdot sin120572]

2

) (8)

Mathematical Problems in Engineering 5

Upper slope surface

Slope face

Bedding surface

0

S

x

L i

(a)

Upper slope surface

Slope face

Bedding surface

0

S

x

Cracking surface

s i

(b)

Upper slope surface

Slope face

Bedding surface

0

S xL i

(c)

Upper slope surface

Slope face

Bedding surface

0

S x

Cracking surface

s i

(d)

Figure 4 Relationships between shear strength 119878 of bedding surface and distance 119909 to slope surface (a) long bedding plane without crackingsurface (b) long bedding plane with cracking surface (c) small bedding plane without cracking surface and (d) short bedding plane with thecracking surface

The length of potential cracking surface can be expressedas

ℎ119888=

[(sum119896

119894=0119897119886119894) minus 119897119888] sdot (sum

119896

119894=0119897119887119894) sdot sin120572

(sum119896

119894=0119897119886119894) + (sum

119896

119894=0119897119887119894) sdot cos120572

(9)

The length of failure surface is

119904119894= (1 minus

(sum119894

119896=1119897119886119896) minus 119897119888

(sum119894

119896=1119897119886119896) + (sum

119894

119896=1119897119887119896) sdot cos120572

)

sdot radic[(

119894

sum

119896=1

119897119887119896) sdot cos120572 + (

119894

sum

119896=1

119897119886119896)]

2

+ [(

119894

sum

119896=1

119897119887119896) sdot sin120572]

2

(10)

The weight of sliding mass is

119866119894= 120588 sdot 119892 sdot [

1

2

(

119894

sum

119896=1

119897119886119896) sdot (

119894

sum

119896=1

119897119887119896) sdot sin (120572) minus 1

2

ℎ119888

sdot ((

119894

sum

119896=1

119897119886119896) minus 119871

119888)]

(11)

The resistance supplied by the tensile strength of rockmass can be written as

119879119894= int

ℎ119888

0

119908 (120590119905) 119889ℎ (12)

The normal force on the failure surface can be obtainedby

119873119894= 119866119894sdot cos120573

119894+ 119879119894sdot sin120573

119894 (13)

The resistance supplied by the cohesion of the beddingsurface is

119875 (119909) = int

1198711198942

0

(

119860119888

1 + 119890minus119888119888(119909minus119887119888)+ 119867119888)119889119909

+ int

119904119894

1198711198942

(

119860119888

1 + 119890minus119888119888(119871119894minus119909minus119887119888)+ 119867119888)119889119909

(14)

The average friction coefficient of the bedding surface is

119891 (119909) =

1

119904119894

[int

1198711198942

0

tan(119860120593

1 + 119890minus119888120593(119909minus119887120593)+ 119867120593)119889119909

+ int

119904119894

1198711198942

tan(119860120593

1 + 119890minus119888120593(119871119894minus119909minus119887120593)+ 119867120593)119889119909]

(15)

The factor of safety 120578 can be obtained by

120578 =

119865119904119894

119865119903119894

=

119873119894sdot 119891 (119909) + 119875 (119909)

119866119894sdot sin120573

119894minus 119879119894sdot cos120573

119894

(16)

where 119865119904119894is the resisting force and 119865

119903119894is the driving force of

this bedding plane

6 Mathematical Problems in Engineering

0h

Upper slope surface

Slope face

Bedding surface

Cracking surface120590

Figure 5 Relationships between tensile strength 120590 of rockmass anddistance ℎ to upper slope surface

Upper slope surface

AG

N

T

Potential cracking surface

CB

Bedding surface i

Ff

Slope face

120572

Lc

hc

La1 La2 La3 Lai

120573iLbi

Lb1

Lb3Lb2

middot middot middot

middot middotmiddot

Figure 6 Computing model of bedding rock slope

The critical sliding surface can be determined by findingthe potential sliding surface which has theminimum factor ofsafety The stability analysis was taken by performing seriesof iterations on various combinations of potential slidingsurface and tensile crack The flow chart for evaluating thestability of the stratified rock slope is shown in Figure 7

3 A Case Study by Modified LimitingEquilibrium Method

31 Introduction of the Jiaxiao Slope Jiaxiao Slope is a typicalbedding rock slope in Triassic Strata which is situated atJingmen City Hubei Province China The slope is mainlycomposed of weathered sandstone There are three beddingsurfaces namely bedding plane A bedding plane B andbedding plane C (Figure 8) The dip of bedding plane isapproximately 22∘ The upper slope surface is nearly hori-zontal and the slope height is about 12m The sandstonecrushing degree is controlled by the depth to slope surfacewhich severely influences the strength of rock mass anddiscontinuities

A geotechnical investigation was carried out in June 2013which included the drilling of six geotechnical drill holes onthe upper surface of Jiaxiao Slope Then based on the drillcores we obtained six rock samples for the tensile strengthtest and nine rock joint samples for the shear strength test

The tensile specimens are disc-shaped with a diameter 119863 =

50mm and a thickness 119905 = 25mm and the sizes of the rockjoint samples are about 80mm times 80mm Here the Braziliantest was performed to determine the indirect tensile strengthaccording to ISRM (1978) [20] The test result shows thatthe tensile strength of highly weathered rock specimens nearslope surface is 03MPa and the tensile strength of intact rockspecimen is 48MPa Thus it is obvious that 119867

120590= 03MPa

and 119860120590= 45MPa in the S-curve model (Figure 9) and the

S-curve model for the tensile strength can be written as

120590 (ℎ) =

45

1 + 119890minus119888120590(ℎminus119887120590)+ 03 (17)

Similarly based on the direct shear test for joint spec-imens the friction angle and cohesion of highly weath-ered rock discontinuities near slope surface are 1100∘ and019MPaThe friction angle and cohesion of the unweatheredrock discontinuities are 1870∘ and 028MPaAccording to theS-curve model fitting result (Figure 10) the S-curve modelsfor shear strength in (5) can be written as

120593 (119909)

=

770

1 + 119890minus1440(119909minus010)

+ 11 (119909 le

119871119894

2

)

770

1 + 119890minus1440(119871

119894minus119909minus010)

+ 11 (

119871119894

2

le 119909 le 119904119894)

119888 (119909)

=

009

1 + 119890minus2300(119909minus01)

+ 019 (119909 le

119871119894

2

)

009

1 + 119890minus2300(119871

119894minus119909minus01)

+ 019 (

119871119894

2

le 119909 le 119904119894)

(18)

The strength coefficients of bedding plane and rock massare tabulated in Table 1 Moreover the shear strength ofbedding rock joints was obtained by the direct shear test Theaverage unit weight 120574 is 28 kNm3

32 Stability Analyses on the Case Study The stability of thisbedding rock slope was calculated by the modified limitingequilibrium method presented above The distributions ofsafety factor 120578 of three bedding planes are illustrated inFigure 11 The 120578 values of each bedding plane decline firstand tend to be stable later However in the stable sectioneach bedding planersquos 120578 firstly decrease and then increase alittle bit Most of the safety factors of bedding plane B arelarger than those of bedding plane A and bedding plane CMoreover the safety factor of bedding A is larger than thesafety factor of bedding C when 119871

119888is shorter than 3m The

minimum 120578 of bedding plane A is 1012 the minimum 120578 ofbedding plane B is 1071 and theminimum 120578 of bedding planeC is 0973 Therefore the critical sliding surface is beddingplane C which has the minimum 120578 of the bedding rockslope Moreover the result shows that the potential tension-controlled failure would be most likely to occur at 105m tothe slope face

Mathematical Problems in Engineering 7

Table 1 Strength coefficients of bedding plane and rock mass (MPa)

Coefficient of strength function Bedding plane Rock massFriction angle (∘) Cohesion (MPa) Tensile strength

119867 1100 019 030119860 770 009 450119888 1440 2300 418119887 010 010 330

Start

Assume the potential slidinghappens in layer i

No

No

End

Input the analysis data① Density 120588② Geometric parameters 120572 La Lb

③ Strength function 120593(x) c(x) 120590(h)

Assume the distance betweenthe slope surface and crack

potential crack is Lc

Failure planedip 120573i

Failure planelength Li

Failure surfacelength si

WeightGi

Tensile force Ti

slip surfaceNormal force Ni on the

Resisting force Fsi

Driving force Fri

Safety factor 120578

If 120578 = 120578min

120578min

= 120578i

If 120578i = 120578min

120578 = 120578min

Yes

Figure 7 Flow chart of stability calculation for bedding rock slope

4 Discussion

The spatial variationmodel of the rockmass and rock discon-tinuities discussed in this paper is based on S-curve modelIt is appropriate for indicating the spatial variability of rockdiscontinuities shear strength parameters 119888 and 120593 and therock mass tensile strength 120590 under the high weathering andexcavation unloading The strength of rock discontinuitiesnear the upper slope surface is lower than deep inside partIf the resistance supplied by the tensile strength of rock massis constant the most unstable sliding surface is the beddingplane However the tensile strength and the length of thepotential cracking surface are constantly changing which

makes the stability of rock slope more complicated If thisresistance force is lower than the sliding force the safetyfactor along the bedding plane is consistently decreasingWith the tensile strength and the length of potential crackingsurface increasing the resistance force may make the safetyfactor decrease first and then increase

In the modified limiting equilibrium equations the spa-tial variations of the strength of rock discontinuities androck mass are introduced in this method If we set resistancesupplied by the tensile strength of rock mass to zero thiscalculation model will degenerate into an idealized modelof planar failure Note that this strength parameters modelof the rock mass and rock discontinuities sometimes cannot

8 Mathematical Problems in Engineering

263m312m 945m

464m

113m200m

72∘

Bedding plane ABedding plane C

Bedding plane B

Figure 8 Analysis model of the case study

Tensile strength data pointFitting curve

Depth h (m)545435325215105

0

1

2

3

4

5

Tens

ile st

reng

th 120590

(MPa

)

120590(h) =45

1 + eminusc120590 (hminusb120590)+ 03

c120590 = 418b120590 = 330

R2 = 098

Figure 9 S-curve model for the tensile strength

describe the spatial variability of strength parameters exactlyNevertheless the main idea for calculating the safety factorof stratified rock slope is still valid and needs to change theexpression of strength parameters

The sliding surface in a slopemay consist of a single planecontinuous over the full area of the surface or a complex sur-face made up of both discontinuities and fractures throughintact rock [21] Here the sliding surface is composed ofthe bedding surface and vertical potential cracking surfaceIt is different from the failure mode caused by propagatingcracks which are extended and coalesced with neighboringjoints The suggested method in this paper is appropriate foranalyzing the stability of stratified rock slope whose jointcrack is undeveloped

5 Conclusions

The plane failure is one of the rock slope failure modes infield situations and the limit equilibrium approach for theestimation of the factor of safety of the rock slope againstplane failure has been well accepted by the engineers Insedimentary rock mass the stratified rock slope usually

Friction angle data pointFitting curve

120593(x) =770

1 + eminusc120593(xminusb120593)+ 11

R2 = 087

b120593 = 010

01 02 03 04 050

Length x (m)

10

12

14

16

18

20

Fric

tion

angl

e (de

gree

)

c120593 = 144

(a)

Fitting curveCohesion data point

R2 = 094

cc = 2300bc = 010

c(x) =009

1 + eminusc119888(xminusb119888)+ 019

01 02 03 04 050

Length x (m)

018

02

022

024

026

028

03

Coh

esio

n (M

Pa)

(b)

Figure 10 S-curve model for the friction angle and cohesion of therock discontinuities

has several bedding planes whose dips and mechanicalparameters have significant effects on rock mass strengthand stability However previous limit equilibrium approacheswere mainly about the plane failure mode with single slidingsurface and failed to predict the stability of the rock slopewith various bedding planesThis paper presented amodifiedlimit equilibrium method for predicting the stability of thesedimentary rock slope with flat bedding planes In thismethod the spatial variations of shear strength parameters119888 and 120593 of bedding plane and the tensile strength 120590 of rockmass were considered and they are expressed by the S-curve model The potential tensile at the top of rock masscan also be considered in plane failure mode incorporatingvarious combinations of potential sliding surface and tensilecrack The critical sliding surface and the tensile crack weredetermined by finding the potential sliding surfacewhich had

Mathematical Problems in Engineering 9

(65 1012)

(85 1071)

(105 0973)

090

095

100

105

110

115

120

125

130

135

140

Safe

ty fa

ctor

2 4 6 80 12 14 1610Lc

Bedding plane ABedding plane B

Bedding plane C

Figure 11 Relations between factor of safety and 119871119888

theminimum factor of safetyThismethod was utilized in thestability analysis of a case study in Three Gorges ReservoirRegion China The factors of safety were different betweenthe bedding planes and they were affected by the resistancesupplied by the tensile strength of rock mass The factor ofsafety of the case study was 0973 the critical sliding surfaceappears at bedding planeC and the tension-controlled failureoccurs at 105m to the slope face On the basis of generalizedanalytical expressions for the factor of safety the examplewas provided to show the effectiveness of the modified limitequilibrium method in the stability analysis of stratified rockslopes

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

This study was funded by National Natural Science Foun-dation of China (nos 41427802 41172292 41302257 and41502300) Zhejiang Provincial Natural Science Foundation(no Q16D020005) and Zhejiang Provincial Natural ScienceFoundation (no Q13D020001)

References

[1] Y Zhao Z-Y Tong and Q Lu ldquoSlope stability analysis usingslice-wise factor of safetyrdquoMathematical Problems in Engineer-ing vol 2014 Article ID 712145 6 pages 2014

[2] SGDu Y JHu andX FHu ldquoMeasurement of joint roughnesscoefficient by using profilograph and roughness rulerrdquo Journalof Earth Science vol 20 no 5 pp 890ndash896 2009

[3] S K Shukla and M M Hossain ldquoAnalytical expression forfactor of safety of an anchored rock slope against plane failurerdquoInternational Journal of Geotechnical Engineering vol 5 no 2pp 181ndash187 2011

[4] N Sivakugan S K Shukla and B M Das Rock Mechanics AnIntroduction CRC Press New York NY USA 2013

[5] M Gencer ldquoProgressive failure in stratified and jointed rockmassrdquo Rock Mechanics and Rock Engineering vol 18 no 4 pp267ndash292 1985

[6] J-J Dong C-H Tu W-R Lee and Y-J Jheng ldquoEffectsof hydraulic conductivitystrength anisotropy on the stabilityof stratified poorly cemented rock slopesrdquo Computers andGeotechnics vol 40 pp 147ndash159 2012

[7] J Ching Z-Y Yang J-Q Shiau and C-J Chen ldquoEstimationof rock pressure during an excavationcut in sedimentary rockswith inclined bedding planesrdquo Structural Safety vol 41 pp 11ndash19 2013

[8] H Tang R Yong and M A M Ez Eldin ldquoStability analysis ofstratified rock slopes with spatially variable strength parame-ters the case of Qianjiangping landsliderdquoBulletin of EngineeringGeology and the Environment 2016

[9] X-L Yang and J-H Yin ldquoLinear Mohr-Coulomb strengthparameters from the non-linear Hoek-Brown rock massesrdquoInternational Journal of Non-LinearMechanics vol 41 no 8 pp1000ndash1005 2006

[10] P Fortsakis K Nikas V Marinos and P Marinos ldquoAnisotropicbehaviour of stratified rock masses in tunnellingrdquo EngineeringGeology vol 141-142 pp 74ndash83 2012

[11] E Hoek and J D Bray Rock Slope Engineering CRC Press 1981[12] S Sharma T K Raghuvanshi and R Anbalagan ldquoPlane

failure analysis of rock slopesrdquo Geotechnical and GeologicalEngineering vol 13 no 2 pp 105ndash111 1995

[13] E B Kroeger ldquoAnalysis of plane failures in compound slopesrdquoInternational Journal of Surface Mining Reclamation and Envi-ronment vol 14 no 3 pp 215ndash222 2000

[14] B S Jiang M F Cai and H Du ldquoAnalytical calculation onstability of slope with planar failure surfacerdquo Chinese Journal ofRock Mechanics and Engineering vol 23 no 1 pp 91ndash94 2004

[15] M Ahmadi and M Eslami ldquoA new approach to plane failure ofrock slope stability based on water flow velocity in discontinu-ities for the Latian dam reservoir landsliderdquo Journal ofMountainScience vol 8 no 2 pp 124ndash130 2011

[16] A Johari A Fazeli and A A Javadi ldquoAn investigation intoapplication of jointly distributed random variables method inreliability assessment of rock slope stabilityrdquo Computers andGeotechnics vol 47 pp 42ndash47 2013

[17] HAViles ldquoLinkingweathering and rock slope instability non-linear perspectivesrdquo Earth Surface Processes and Landforms vol38 no 1 pp 62ndash70 2013

[18] HM Tang Z X Zou C R Xiong et al ldquoAn evolutionmodel oflarge consequent bedding rockslides with particular referenceto the Jiweishan rockslide in Southwest Chinardquo EngineeringGeology vol 186 pp 17ndash27 2015

[19] Z X Zou H M Tang C R Xiong Y P Wu X Liu and SB Liao ldquoGeomechanical model of progressive failure for largeconsequent bedding rockslide and its stability analysisrdquoChineseJournal of Rock Mechanics and Engineering vol 31 no 11 pp2222ndash2230 2012 (Chinese)

[20] Z T Bieniawski and I Hawkes ldquoSuggested methods for deter-mining tensile strength of rock materialsrdquo International Journalof RockMechanics andMining Sciences vol 15 no 3 pp 99ndash1031978

[21] C W Duncan and W M Christopher Rock Slope Engineering-Civil and Mining Taylor amp Francis New York NY USA 4thedition 2005

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Mathematical Problems in Engineering

the forces due to water pressure on discontinuity surfacein plane failure rock slope However these methods arerestricted by the requirement that single valued parametersdescribe the slope characteristics [16] In actuality the shearstrength of a rock mass is spatially variable due to thefactors like climate carbonation geology relief vegetationcover or human activities For instance some slopes may becomposed of rocks which have been extensively and deeplyweathered in situ over long periods [17] The strength ofrock mass near the slope surface and upper slope surfaceis easily affected by these factors but less influence on thedeeper inside of the rock mass Obviously we cannot neglectthe strength variations along the sliding surface which mayaffect the accuracy of slope stability calculations by the limitequilibrium method However this inherent variability ofshear strength has often been portrayed in an oversimplifiedway in earlier researches on rock slope instability

The objectives of this paper are (1) to present amathemat-ical model for describing the spatial variations of the shearstrength rock discontinuities and tension strength of rockmass (2) to propose calculation procedure for determiningthe critical sliding surface and tension crack and (3) tointroduce amethod for evaluating the stability of the stratifiedrock slope Furthermore this modified limiting equilibriummethod was applied to the stability assessment of the casestudy Jiaxiao Slope in Three Gorges Reservoir RegionChina

2 Calculation Method and Model

21 Geomechanical Model Planar rock slope failure occurswhen a rockmass in a slope slides down and along a relativelyplanar failure surface These failure surfaces are usuallystructural discontinuities such as bedding planes faultsjoints or the interface between bedrock and an overlyinglayer of weathered rock Stability condition occurs if thecritical joint dip is less than the slope angle and mobilizedjoint shear strength is not enough to assure the stability ofthe slope

Theplanar failuremode provides insightful knowledge onthe behavior of rock slopes in which the material above thissurface is regard as a ldquofree bodyrdquoThe disturbing and resistingforces are estimated enabling the formulation of equationsconcerning force or moment equilibrium (or both) of thepotential sliding mass FS is defined as the ratio of resistingforces to driving forces

Let us consider a slope of inclination 119894 height 119867 and apotential failure plane of inclination 120573 (Figure 1) Accordingto the geometric properties in Figure 1 the weight of the rigidblock ABC is calculated by

ℎ =

119867

sin 119894sin (119894 minus 120573)

119871 =

119867

sin120573

119866 =

1

2

120588119892ℎ119871 =

1205881198921198672 sin (119894 minus 120573)

2 sin 119894 sdot sin120573

(1)

H

x

yUpper slope surface

Slope faceSliding mass

Sliding surface

A

CB

Gh

L

i 120573

Gsin120573

Gcos120573

Figure 1 A rock slope in plane failure

where 120588 is the average density of the rock mass 119892 is thegravitational acceleration (98ms2) 119871 is the length of thefailure plane AC It is easy to obtain the driving force 119865

119903and

resisting force 119865119904by the force analysis of the rigid block ABC

119865119904= 119866 cos120573 tan120593 + 119888119871

119865119903= 119866 sin120573

(2)

where 120593 and 119888 are the effective internal friction angle andcohesion of failure plane The factor of safety (FS) can beobtained by the force equilibrium equation as follows

FS =119865119904

119865119903

=

119866 cos120573 tan120593 + 119888119871119866 sin120573

=

2119888 sin 119894120588119892119867 sin (119894 minus 120573) sin120573

+

tan120593tan120573

(3)

This model is widely used in planar rock slope failureanalysis which offers considerable insights into the behaviorof real slopes However it failed to be utilized for the stabilityanalyzing of the stratified rock slope with series of beddingplanes Then the shear strength parameters 120593 and 119888 alongthe failure plane were treated as constant values which failedto reflect the strength variations along the sliding surfaceFurthermore the potential failure rock mass was commonlyassumed as a rigid block so the potential tensile failurecannot be considered in the planar rock slope failure analysis

The sliding mass of stratified rock slope can be dividedinto several plane segments by rock bedding surfaces (Fig-ure 2) However it is hard to find the slip surface from thesebedding surfaces geological discontinuities and surfacesof weakness The limit equilibrium analysis method wasextended for determining the critical position in all thepotential slip surfaces Here the exposed length of each layer119894 on the upper ground surface is labeled by 119871

119886119894 The exposed

length on the slope face is denoted as 119871119887119894 The potential

tension crack appears on the upper slope surface and itshorizontal distance to slope face is 119871

119888 The tensile strength

of sliding mass is considered in this idealized model becausethe rock mass may have a tensile failure and create the tensilecrack in some conditions (eg the shear strength of most ofthe sliding surface is very low but reaches high value nearupper slope surface) Furthermore the field investigations

Mathematical Problems in Engineering 3

Potential cracking surface

A

C

Upper slope surface

Bedding surface i

Slope face

B

120572

Bedding surface i minus 1

Lc

hc

La1 La2 La3 Lai

120573iLbi

Lb1

Lb3Lb2

middot middot middot

middot middotmiddot

Figure 2 Idealized model of rock slope with bedding planes

indicate that the cohesion 119888 and friction angle 120593 of beddingsurfaces are variable at different slip surfaces As commonsense the shear strength of the rock discontinuities near toslope surface is usually lower than that in the deep place Dueto the anisotropy of shear strength it is no longer reasonableto simplify the bedding rock slope failure by the planar rockslope mode in Figure 1

Furthermore in the idealized model of stratified rockslope the locations of the bedding planes are determinedbased on the exposed length 119871

119886119894of each layer on the upper

ground surface and the exposed length 119871119887119894on the slope face

However there is a challenge in the pairing up of the daylightlocations for a potential slip plane on the top boundary andslope face because sometimes the exposed length 119871

119887119894on

the slope face is difficult to measure especially for thoseslopes with large slope angles In rock engineering practicesthe geotechnical drilling is suggested to be implemented atdifferent positions on the upper surface of rock slope Basedon the core samples from drilling tubes the dip angles ofthe bedding planes can be obtained via measuring the dipangles of the bedding planes Generally the potential failuresurface is assumed to be planar in planar rock slope failureanalysis Then the corresponding exposed length 119871

119887119894on the

slope surface can be calculated through daylight location onthe top boundary and dip angle of the bedding plane

22 Basic Assumptions Comparing the single planar failuremodel with the idealized model of bedding rock slope it isnecessary to make a few assumptions for the stability assess-ment The main assumptions are summarized as follows

(1) It is assumed that the slope face upper slope surfacefailure plane and the tension crack are parallel ornearly parallel The rock slope has a vertical tensioncrack and the upper surface of the slope is horizontal

(2) The assumed failure mode is translational-crack-slipand the sliding block is assumed to have a unitthickness in the planar failure analysis

(3) Release or free surfaces are present parallel to thecross section of the analysis which provides negligible

Distance to slope surface

Stre

ngth

par

amet

ers

a b c

dA1 A2

b1 b2H1

H2

Figure 3 Relationship between strength parameters and distance toslope surface

or no resistance to sliding at the lateral boundaries ofthe failure

(4) The tensile strength 120590 of the rock mass and shearstrength parameters 119888 and 120593 of rock discontinuitiesare assumed to be anisotropic and the spatial vari-ation model of strength parameters is provided inSection 23

(5) The above assumptions represent rock block slid-ing along a single joint which is the potential slipsurface in the rock mass and there is no relativesliding between different bedding planes above theslip surface The shear force over the slip surface isdetermined by Mohr-Coulomb criterion here

23 Spatial Variation Model for Rock Mass and Rock Dis-continuities The spatial variability of rock discontinuitiesshear strength parameters 119888 and 120593 and the rock mass tensilestrength 120590 is considered in this idealized model of beddingrock slope S-curve model belongs to multiple variable anal-ysis methods and mainly appears in logit model or logisticregression model It is widely used in the statistical andempirical analysis in fields like sociology biostatistics clinicalmedicine marketing management and so forth S-curvemodel has the basic features for describing the progressivetransition from one value to another which increases non-linearly and continuously whose variation trend is shapedlike ldquoSrdquo (Figure 3) In previous studies the S-curve model hasbeen used successfully in describing the spatial variations ofthe shear strength parameters of the bedding plane [8 18 19]There are real data supporting the validity of the S-curvemodel and this model has been successfully used to describethe spatial variations of the shear strength parameters causedby the weakening effects Based on the S-curve model somesuggested methods were proposed for analyzing the stabilityof rock slope with bedding surfaces Here it is appropriateto illustrate the variations of the strength including shearstrength parameters 119888 and120593 of rock discontinuities and therock mass tensile strength 120590

4 Mathematical Problems in Engineering

According to S-curve model the strength can beexpressed as follows

119878 (119909) =

119860

1 + 119890minus1198881015840(119909minus119887)

+ 1198671015840 (4)

where 119878(119909) denotes the strength of rock mass or rockdiscontinuities 119909 is the distance to the slope surface 1198671015840 isthe lowest strength of rock mass or rock discontinuities Thestrength of rock mass or rock discontinuities deep insidethe rock slope is barely affected by the factors mentionedabove so it gets the largest value 119860 is the difference betweenthe lowest and the largest strength The comparison betweencurves (a) and (d) in Figure 3 shows that the variation oflowest and largest value in S-curve model is determined bycoefficients 119860 and 1198671015840 119887 and 1198881015840 are the shape controllingfactors of S-curve model Then 119887 represents the distance tothe slope surface when the strength is equal to themean valueof the lowest and the highest strength and 1198881015840 is positivelycorrelated with the inverse of 119887

In the present study the S-curve model is established todefine the spatial variations of the shear strength parametersof bedding plane and the tensile strength of rock mass Forthe shear strength parameters 1198671015840 is the shear strength ofhighly weathered rock discontinuities near slope surface119860 is the difference between the shear strength value ofthe highly weathered rock discontinuities and the highestshear strength value of discontinuities which are the leastaffected by weathering effect 119887 and 1198881015840 are obtained by curve-fitting of the shear strength of the rock discontinuities andthe locations of rock joint specimens For tensile strengthparameters1198671015840 is the tensile strength of rock specimens nearslope surface 119860 is acquired by calculating the differencebetween1198671015840 and the tensile strength of intact rock specimenbased on the variable tensile strength and the depth of rockspecimens 119887 and 1198881015840 values are acquired through the S-curvemodel fitting

It is assumed that the shear strength parameters areassociated with the distance to slope surface The distancesbetween upper slope surface and slope face have the sameinfluence on the shear strength parameters A set of equationsfor describing shear strength parameters 119888 and 120593 along thebedding surface are illustrated as follows

120593 (119909) =

119860120593

1 + 119890minus119888120593(119909minus119887120593)+ 119867120593 (119909 le

119871119894

2

)

119860120593

1 + 119890minus119888120593(119871119894minus119909minus119887120593)+ 119867120593 (

119871119894

2

le 119909 le 119904119894)

119888 (119909) =

119860119888

1 + 119890minus119888119888(119909minus119887119888)+ 119867119888 (119909 le

119871119894

2

)

119860119888

1 + 119890minus119888119888(119871119894minus119909minus119887119888)+ 119867119888 (

119871119894

2

le 119909 le 119904119894)

(5)

where 119867120593and 119867

119888are the shear strength of rock discon-

tinuities near slope surface and they are the lowest shearstrength parameters the shear strengths of new and completerock discontinuities reach the largest value 119860

120593and 119860

119888are

the differences between the lowest and largest shear strengthparameters 119887

120593 119887119888 119888120593 and 119888

119888are the shape controlling coeffi-

cients of equation set These coefficients can be obtained bydirect shear tests on the specimens taken from different partsof bedding surface In Figure 4 119871

119894is the length of bedding

surface and 119904119894is the length of the potential sliding surface

which is controlled by the cracking surfaceSimilarly tensile strength based on the S-curvemodel can

be expressed as

120590 (ℎ) =

119860120590

1 + 119890minus119888120590(ℎminus119887120590)+ 119867120590 (6)

where 119867120590is the lowest tensile strength of rock mass on the

slope surface 119860120590is the difference between the lowest and

largest tensile strength 119887120590and 119888120590are the shape controlling

coefficients of tensile strength change function ℎ is the depthfrom the upper slope surface to the intersection of beddingsurface and cracking surface (Figure 5) In this study thepotential vertical tensile cracks are assumed to appear whenthe sliding happens The influence on the distributions of thecohesion and frictional angle in tensile crack developing wasnot considered here

24 Modified Limiting Equilibrium Equations The rock massabove the failure surface is regarded as an entire sliding bodyand the back geometric boundary is the potential crackingsurface (Figure 6) Here the bedding surface 119894 is the failuresurface

The length of bedding surface 119894 can be described by

119871119894

= radic[(

119894

sum

119896=1

119897119887119896) sdot cos120572 + (

119894

sum

119896=1

119897119886119896)]

2

+ [(

119894

sum

119896=1

119897119887119896) sdot sin120572]

2

(7)

The failure plane dip is

120573119894= arcsin(

sum119894

119896=1119897119887119896sdot sin120572

radic[(sum119894

119896=1119897119887119896) sdot cos120572 + (sum119894

119896=1119897119886119896)]

2

+ [(sum119894

119896=1119897119887119896) sdot sin120572]

2

) (8)

Mathematical Problems in Engineering 5

Upper slope surface

Slope face

Bedding surface

0

S

x

L i

(a)

Upper slope surface

Slope face

Bedding surface

0

S

x

Cracking surface

s i

(b)

Upper slope surface

Slope face

Bedding surface

0

S xL i

(c)

Upper slope surface

Slope face

Bedding surface

0

S x

Cracking surface

s i

(d)

Figure 4 Relationships between shear strength 119878 of bedding surface and distance 119909 to slope surface (a) long bedding plane without crackingsurface (b) long bedding plane with cracking surface (c) small bedding plane without cracking surface and (d) short bedding plane with thecracking surface

The length of potential cracking surface can be expressedas

ℎ119888=

[(sum119896

119894=0119897119886119894) minus 119897119888] sdot (sum

119896

119894=0119897119887119894) sdot sin120572

(sum119896

119894=0119897119886119894) + (sum

119896

119894=0119897119887119894) sdot cos120572

(9)

The length of failure surface is

119904119894= (1 minus

(sum119894

119896=1119897119886119896) minus 119897119888

(sum119894

119896=1119897119886119896) + (sum

119894

119896=1119897119887119896) sdot cos120572

)

sdot radic[(

119894

sum

119896=1

119897119887119896) sdot cos120572 + (

119894

sum

119896=1

119897119886119896)]

2

+ [(

119894

sum

119896=1

119897119887119896) sdot sin120572]

2

(10)

The weight of sliding mass is

119866119894= 120588 sdot 119892 sdot [

1

2

(

119894

sum

119896=1

119897119886119896) sdot (

119894

sum

119896=1

119897119887119896) sdot sin (120572) minus 1

2

ℎ119888

sdot ((

119894

sum

119896=1

119897119886119896) minus 119871

119888)]

(11)

The resistance supplied by the tensile strength of rockmass can be written as

119879119894= int

ℎ119888

0

119908 (120590119905) 119889ℎ (12)

The normal force on the failure surface can be obtainedby

119873119894= 119866119894sdot cos120573

119894+ 119879119894sdot sin120573

119894 (13)

The resistance supplied by the cohesion of the beddingsurface is

119875 (119909) = int

1198711198942

0

(

119860119888

1 + 119890minus119888119888(119909minus119887119888)+ 119867119888)119889119909

+ int

119904119894

1198711198942

(

119860119888

1 + 119890minus119888119888(119871119894minus119909minus119887119888)+ 119867119888)119889119909

(14)

The average friction coefficient of the bedding surface is

119891 (119909) =

1

119904119894

[int

1198711198942

0

tan(119860120593

1 + 119890minus119888120593(119909minus119887120593)+ 119867120593)119889119909

+ int

119904119894

1198711198942

tan(119860120593

1 + 119890minus119888120593(119871119894minus119909minus119887120593)+ 119867120593)119889119909]

(15)

The factor of safety 120578 can be obtained by

120578 =

119865119904119894

119865119903119894

=

119873119894sdot 119891 (119909) + 119875 (119909)

119866119894sdot sin120573

119894minus 119879119894sdot cos120573

119894

(16)

where 119865119904119894is the resisting force and 119865

119903119894is the driving force of

this bedding plane

6 Mathematical Problems in Engineering

0h

Upper slope surface

Slope face

Bedding surface

Cracking surface120590

Figure 5 Relationships between tensile strength 120590 of rockmass anddistance ℎ to upper slope surface

Upper slope surface

AG

N

T

Potential cracking surface

CB

Bedding surface i

Ff

Slope face

120572

Lc

hc

La1 La2 La3 Lai

120573iLbi

Lb1

Lb3Lb2

middot middot middot

middot middotmiddot

Figure 6 Computing model of bedding rock slope

The critical sliding surface can be determined by findingthe potential sliding surface which has theminimum factor ofsafety The stability analysis was taken by performing seriesof iterations on various combinations of potential slidingsurface and tensile crack The flow chart for evaluating thestability of the stratified rock slope is shown in Figure 7

3 A Case Study by Modified LimitingEquilibrium Method

31 Introduction of the Jiaxiao Slope Jiaxiao Slope is a typicalbedding rock slope in Triassic Strata which is situated atJingmen City Hubei Province China The slope is mainlycomposed of weathered sandstone There are three beddingsurfaces namely bedding plane A bedding plane B andbedding plane C (Figure 8) The dip of bedding plane isapproximately 22∘ The upper slope surface is nearly hori-zontal and the slope height is about 12m The sandstonecrushing degree is controlled by the depth to slope surfacewhich severely influences the strength of rock mass anddiscontinuities

A geotechnical investigation was carried out in June 2013which included the drilling of six geotechnical drill holes onthe upper surface of Jiaxiao Slope Then based on the drillcores we obtained six rock samples for the tensile strengthtest and nine rock joint samples for the shear strength test

The tensile specimens are disc-shaped with a diameter 119863 =

50mm and a thickness 119905 = 25mm and the sizes of the rockjoint samples are about 80mm times 80mm Here the Braziliantest was performed to determine the indirect tensile strengthaccording to ISRM (1978) [20] The test result shows thatthe tensile strength of highly weathered rock specimens nearslope surface is 03MPa and the tensile strength of intact rockspecimen is 48MPa Thus it is obvious that 119867

120590= 03MPa

and 119860120590= 45MPa in the S-curve model (Figure 9) and the

S-curve model for the tensile strength can be written as

120590 (ℎ) =

45

1 + 119890minus119888120590(ℎminus119887120590)+ 03 (17)

Similarly based on the direct shear test for joint spec-imens the friction angle and cohesion of highly weath-ered rock discontinuities near slope surface are 1100∘ and019MPaThe friction angle and cohesion of the unweatheredrock discontinuities are 1870∘ and 028MPaAccording to theS-curve model fitting result (Figure 10) the S-curve modelsfor shear strength in (5) can be written as

120593 (119909)

=

770

1 + 119890minus1440(119909minus010)

+ 11 (119909 le

119871119894

2

)

770

1 + 119890minus1440(119871

119894minus119909minus010)

+ 11 (

119871119894

2

le 119909 le 119904119894)

119888 (119909)

=

009

1 + 119890minus2300(119909minus01)

+ 019 (119909 le

119871119894

2

)

009

1 + 119890minus2300(119871

119894minus119909minus01)

+ 019 (

119871119894

2

le 119909 le 119904119894)

(18)

The strength coefficients of bedding plane and rock massare tabulated in Table 1 Moreover the shear strength ofbedding rock joints was obtained by the direct shear test Theaverage unit weight 120574 is 28 kNm3

32 Stability Analyses on the Case Study The stability of thisbedding rock slope was calculated by the modified limitingequilibrium method presented above The distributions ofsafety factor 120578 of three bedding planes are illustrated inFigure 11 The 120578 values of each bedding plane decline firstand tend to be stable later However in the stable sectioneach bedding planersquos 120578 firstly decrease and then increase alittle bit Most of the safety factors of bedding plane B arelarger than those of bedding plane A and bedding plane CMoreover the safety factor of bedding A is larger than thesafety factor of bedding C when 119871

119888is shorter than 3m The

minimum 120578 of bedding plane A is 1012 the minimum 120578 ofbedding plane B is 1071 and theminimum 120578 of bedding planeC is 0973 Therefore the critical sliding surface is beddingplane C which has the minimum 120578 of the bedding rockslope Moreover the result shows that the potential tension-controlled failure would be most likely to occur at 105m tothe slope face

Mathematical Problems in Engineering 7

Table 1 Strength coefficients of bedding plane and rock mass (MPa)

Coefficient of strength function Bedding plane Rock massFriction angle (∘) Cohesion (MPa) Tensile strength

119867 1100 019 030119860 770 009 450119888 1440 2300 418119887 010 010 330

Start

Assume the potential slidinghappens in layer i

No

No

End

Input the analysis data① Density 120588② Geometric parameters 120572 La Lb

③ Strength function 120593(x) c(x) 120590(h)

Assume the distance betweenthe slope surface and crack

potential crack is Lc

Failure planedip 120573i

Failure planelength Li

Failure surfacelength si

WeightGi

Tensile force Ti

slip surfaceNormal force Ni on the

Resisting force Fsi

Driving force Fri

Safety factor 120578

If 120578 = 120578min

120578min

= 120578i

If 120578i = 120578min

120578 = 120578min

Yes

Figure 7 Flow chart of stability calculation for bedding rock slope

4 Discussion

The spatial variationmodel of the rockmass and rock discon-tinuities discussed in this paper is based on S-curve modelIt is appropriate for indicating the spatial variability of rockdiscontinuities shear strength parameters 119888 and 120593 and therock mass tensile strength 120590 under the high weathering andexcavation unloading The strength of rock discontinuitiesnear the upper slope surface is lower than deep inside partIf the resistance supplied by the tensile strength of rock massis constant the most unstable sliding surface is the beddingplane However the tensile strength and the length of thepotential cracking surface are constantly changing which

makes the stability of rock slope more complicated If thisresistance force is lower than the sliding force the safetyfactor along the bedding plane is consistently decreasingWith the tensile strength and the length of potential crackingsurface increasing the resistance force may make the safetyfactor decrease first and then increase

In the modified limiting equilibrium equations the spa-tial variations of the strength of rock discontinuities androck mass are introduced in this method If we set resistancesupplied by the tensile strength of rock mass to zero thiscalculation model will degenerate into an idealized modelof planar failure Note that this strength parameters modelof the rock mass and rock discontinuities sometimes cannot

8 Mathematical Problems in Engineering

263m312m 945m

464m

113m200m

72∘

Bedding plane ABedding plane C

Bedding plane B

Figure 8 Analysis model of the case study

Tensile strength data pointFitting curve

Depth h (m)545435325215105

0

1

2

3

4

5

Tens

ile st

reng

th 120590

(MPa

)

120590(h) =45

1 + eminusc120590 (hminusb120590)+ 03

c120590 = 418b120590 = 330

R2 = 098

Figure 9 S-curve model for the tensile strength

describe the spatial variability of strength parameters exactlyNevertheless the main idea for calculating the safety factorof stratified rock slope is still valid and needs to change theexpression of strength parameters

The sliding surface in a slopemay consist of a single planecontinuous over the full area of the surface or a complex sur-face made up of both discontinuities and fractures throughintact rock [21] Here the sliding surface is composed ofthe bedding surface and vertical potential cracking surfaceIt is different from the failure mode caused by propagatingcracks which are extended and coalesced with neighboringjoints The suggested method in this paper is appropriate foranalyzing the stability of stratified rock slope whose jointcrack is undeveloped

5 Conclusions

The plane failure is one of the rock slope failure modes infield situations and the limit equilibrium approach for theestimation of the factor of safety of the rock slope againstplane failure has been well accepted by the engineers Insedimentary rock mass the stratified rock slope usually

Friction angle data pointFitting curve

120593(x) =770

1 + eminusc120593(xminusb120593)+ 11

R2 = 087

b120593 = 010

01 02 03 04 050

Length x (m)

10

12

14

16

18

20

Fric

tion

angl

e (de

gree

)

c120593 = 144

(a)

Fitting curveCohesion data point

R2 = 094

cc = 2300bc = 010

c(x) =009

1 + eminusc119888(xminusb119888)+ 019

01 02 03 04 050

Length x (m)

018

02

022

024

026

028

03

Coh

esio

n (M

Pa)

(b)

Figure 10 S-curve model for the friction angle and cohesion of therock discontinuities

has several bedding planes whose dips and mechanicalparameters have significant effects on rock mass strengthand stability However previous limit equilibrium approacheswere mainly about the plane failure mode with single slidingsurface and failed to predict the stability of the rock slopewith various bedding planesThis paper presented amodifiedlimit equilibrium method for predicting the stability of thesedimentary rock slope with flat bedding planes In thismethod the spatial variations of shear strength parameters119888 and 120593 of bedding plane and the tensile strength 120590 of rockmass were considered and they are expressed by the S-curve model The potential tensile at the top of rock masscan also be considered in plane failure mode incorporatingvarious combinations of potential sliding surface and tensilecrack The critical sliding surface and the tensile crack weredetermined by finding the potential sliding surfacewhich had

Mathematical Problems in Engineering 9

(65 1012)

(85 1071)

(105 0973)

090

095

100

105

110

115

120

125

130

135

140

Safe

ty fa

ctor

2 4 6 80 12 14 1610Lc

Bedding plane ABedding plane B

Bedding plane C

Figure 11 Relations between factor of safety and 119871119888

theminimum factor of safetyThismethod was utilized in thestability analysis of a case study in Three Gorges ReservoirRegion China The factors of safety were different betweenthe bedding planes and they were affected by the resistancesupplied by the tensile strength of rock mass The factor ofsafety of the case study was 0973 the critical sliding surfaceappears at bedding planeC and the tension-controlled failureoccurs at 105m to the slope face On the basis of generalizedanalytical expressions for the factor of safety the examplewas provided to show the effectiveness of the modified limitequilibrium method in the stability analysis of stratified rockslopes

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

This study was funded by National Natural Science Foun-dation of China (nos 41427802 41172292 41302257 and41502300) Zhejiang Provincial Natural Science Foundation(no Q16D020005) and Zhejiang Provincial Natural ScienceFoundation (no Q13D020001)

References

[1] Y Zhao Z-Y Tong and Q Lu ldquoSlope stability analysis usingslice-wise factor of safetyrdquoMathematical Problems in Engineer-ing vol 2014 Article ID 712145 6 pages 2014

[2] SGDu Y JHu andX FHu ldquoMeasurement of joint roughnesscoefficient by using profilograph and roughness rulerrdquo Journalof Earth Science vol 20 no 5 pp 890ndash896 2009

[3] S K Shukla and M M Hossain ldquoAnalytical expression forfactor of safety of an anchored rock slope against plane failurerdquoInternational Journal of Geotechnical Engineering vol 5 no 2pp 181ndash187 2011

[4] N Sivakugan S K Shukla and B M Das Rock Mechanics AnIntroduction CRC Press New York NY USA 2013

[5] M Gencer ldquoProgressive failure in stratified and jointed rockmassrdquo Rock Mechanics and Rock Engineering vol 18 no 4 pp267ndash292 1985

[6] J-J Dong C-H Tu W-R Lee and Y-J Jheng ldquoEffectsof hydraulic conductivitystrength anisotropy on the stabilityof stratified poorly cemented rock slopesrdquo Computers andGeotechnics vol 40 pp 147ndash159 2012

[7] J Ching Z-Y Yang J-Q Shiau and C-J Chen ldquoEstimationof rock pressure during an excavationcut in sedimentary rockswith inclined bedding planesrdquo Structural Safety vol 41 pp 11ndash19 2013

[8] H Tang R Yong and M A M Ez Eldin ldquoStability analysis ofstratified rock slopes with spatially variable strength parame-ters the case of Qianjiangping landsliderdquoBulletin of EngineeringGeology and the Environment 2016

[9] X-L Yang and J-H Yin ldquoLinear Mohr-Coulomb strengthparameters from the non-linear Hoek-Brown rock massesrdquoInternational Journal of Non-LinearMechanics vol 41 no 8 pp1000ndash1005 2006

[10] P Fortsakis K Nikas V Marinos and P Marinos ldquoAnisotropicbehaviour of stratified rock masses in tunnellingrdquo EngineeringGeology vol 141-142 pp 74ndash83 2012

[11] E Hoek and J D Bray Rock Slope Engineering CRC Press 1981[12] S Sharma T K Raghuvanshi and R Anbalagan ldquoPlane

failure analysis of rock slopesrdquo Geotechnical and GeologicalEngineering vol 13 no 2 pp 105ndash111 1995

[13] E B Kroeger ldquoAnalysis of plane failures in compound slopesrdquoInternational Journal of Surface Mining Reclamation and Envi-ronment vol 14 no 3 pp 215ndash222 2000

[14] B S Jiang M F Cai and H Du ldquoAnalytical calculation onstability of slope with planar failure surfacerdquo Chinese Journal ofRock Mechanics and Engineering vol 23 no 1 pp 91ndash94 2004

[15] M Ahmadi and M Eslami ldquoA new approach to plane failure ofrock slope stability based on water flow velocity in discontinu-ities for the Latian dam reservoir landsliderdquo Journal ofMountainScience vol 8 no 2 pp 124ndash130 2011

[16] A Johari A Fazeli and A A Javadi ldquoAn investigation intoapplication of jointly distributed random variables method inreliability assessment of rock slope stabilityrdquo Computers andGeotechnics vol 47 pp 42ndash47 2013

[17] HAViles ldquoLinkingweathering and rock slope instability non-linear perspectivesrdquo Earth Surface Processes and Landforms vol38 no 1 pp 62ndash70 2013

[18] HM Tang Z X Zou C R Xiong et al ldquoAn evolutionmodel oflarge consequent bedding rockslides with particular referenceto the Jiweishan rockslide in Southwest Chinardquo EngineeringGeology vol 186 pp 17ndash27 2015

[19] Z X Zou H M Tang C R Xiong Y P Wu X Liu and SB Liao ldquoGeomechanical model of progressive failure for largeconsequent bedding rockslide and its stability analysisrdquoChineseJournal of Rock Mechanics and Engineering vol 31 no 11 pp2222ndash2230 2012 (Chinese)

[20] Z T Bieniawski and I Hawkes ldquoSuggested methods for deter-mining tensile strength of rock materialsrdquo International Journalof RockMechanics andMining Sciences vol 15 no 3 pp 99ndash1031978

[21] C W Duncan and W M Christopher Rock Slope Engineering-Civil and Mining Taylor amp Francis New York NY USA 4thedition 2005

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 3

Potential cracking surface

A

C

Upper slope surface

Bedding surface i

Slope face

B

120572

Bedding surface i minus 1

Lc

hc

La1 La2 La3 Lai

120573iLbi

Lb1

Lb3Lb2

middot middot middot

middot middotmiddot

Figure 2 Idealized model of rock slope with bedding planes

indicate that the cohesion 119888 and friction angle 120593 of beddingsurfaces are variable at different slip surfaces As commonsense the shear strength of the rock discontinuities near toslope surface is usually lower than that in the deep place Dueto the anisotropy of shear strength it is no longer reasonableto simplify the bedding rock slope failure by the planar rockslope mode in Figure 1

Furthermore in the idealized model of stratified rockslope the locations of the bedding planes are determinedbased on the exposed length 119871

119886119894of each layer on the upper

ground surface and the exposed length 119871119887119894on the slope face

However there is a challenge in the pairing up of the daylightlocations for a potential slip plane on the top boundary andslope face because sometimes the exposed length 119871

119887119894on

the slope face is difficult to measure especially for thoseslopes with large slope angles In rock engineering practicesthe geotechnical drilling is suggested to be implemented atdifferent positions on the upper surface of rock slope Basedon the core samples from drilling tubes the dip angles ofthe bedding planes can be obtained via measuring the dipangles of the bedding planes Generally the potential failuresurface is assumed to be planar in planar rock slope failureanalysis Then the corresponding exposed length 119871

119887119894on the

slope surface can be calculated through daylight location onthe top boundary and dip angle of the bedding plane

22 Basic Assumptions Comparing the single planar failuremodel with the idealized model of bedding rock slope it isnecessary to make a few assumptions for the stability assess-ment The main assumptions are summarized as follows

(1) It is assumed that the slope face upper slope surfacefailure plane and the tension crack are parallel ornearly parallel The rock slope has a vertical tensioncrack and the upper surface of the slope is horizontal

(2) The assumed failure mode is translational-crack-slipand the sliding block is assumed to have a unitthickness in the planar failure analysis

(3) Release or free surfaces are present parallel to thecross section of the analysis which provides negligible

Distance to slope surface

Stre

ngth

par

amet

ers

a b c

dA1 A2

b1 b2H1

H2

Figure 3 Relationship between strength parameters and distance toslope surface

or no resistance to sliding at the lateral boundaries ofthe failure

(4) The tensile strength 120590 of the rock mass and shearstrength parameters 119888 and 120593 of rock discontinuitiesare assumed to be anisotropic and the spatial vari-ation model of strength parameters is provided inSection 23

(5) The above assumptions represent rock block slid-ing along a single joint which is the potential slipsurface in the rock mass and there is no relativesliding between different bedding planes above theslip surface The shear force over the slip surface isdetermined by Mohr-Coulomb criterion here

23 Spatial Variation Model for Rock Mass and Rock Dis-continuities The spatial variability of rock discontinuitiesshear strength parameters 119888 and 120593 and the rock mass tensilestrength 120590 is considered in this idealized model of beddingrock slope S-curve model belongs to multiple variable anal-ysis methods and mainly appears in logit model or logisticregression model It is widely used in the statistical andempirical analysis in fields like sociology biostatistics clinicalmedicine marketing management and so forth S-curvemodel has the basic features for describing the progressivetransition from one value to another which increases non-linearly and continuously whose variation trend is shapedlike ldquoSrdquo (Figure 3) In previous studies the S-curve model hasbeen used successfully in describing the spatial variations ofthe shear strength parameters of the bedding plane [8 18 19]There are real data supporting the validity of the S-curvemodel and this model has been successfully used to describethe spatial variations of the shear strength parameters causedby the weakening effects Based on the S-curve model somesuggested methods were proposed for analyzing the stabilityof rock slope with bedding surfaces Here it is appropriateto illustrate the variations of the strength including shearstrength parameters 119888 and120593 of rock discontinuities and therock mass tensile strength 120590

4 Mathematical Problems in Engineering

According to S-curve model the strength can beexpressed as follows

119878 (119909) =

119860

1 + 119890minus1198881015840(119909minus119887)

+ 1198671015840 (4)

where 119878(119909) denotes the strength of rock mass or rockdiscontinuities 119909 is the distance to the slope surface 1198671015840 isthe lowest strength of rock mass or rock discontinuities Thestrength of rock mass or rock discontinuities deep insidethe rock slope is barely affected by the factors mentionedabove so it gets the largest value 119860 is the difference betweenthe lowest and the largest strength The comparison betweencurves (a) and (d) in Figure 3 shows that the variation oflowest and largest value in S-curve model is determined bycoefficients 119860 and 1198671015840 119887 and 1198881015840 are the shape controllingfactors of S-curve model Then 119887 represents the distance tothe slope surface when the strength is equal to themean valueof the lowest and the highest strength and 1198881015840 is positivelycorrelated with the inverse of 119887

In the present study the S-curve model is established todefine the spatial variations of the shear strength parametersof bedding plane and the tensile strength of rock mass Forthe shear strength parameters 1198671015840 is the shear strength ofhighly weathered rock discontinuities near slope surface119860 is the difference between the shear strength value ofthe highly weathered rock discontinuities and the highestshear strength value of discontinuities which are the leastaffected by weathering effect 119887 and 1198881015840 are obtained by curve-fitting of the shear strength of the rock discontinuities andthe locations of rock joint specimens For tensile strengthparameters1198671015840 is the tensile strength of rock specimens nearslope surface 119860 is acquired by calculating the differencebetween1198671015840 and the tensile strength of intact rock specimenbased on the variable tensile strength and the depth of rockspecimens 119887 and 1198881015840 values are acquired through the S-curvemodel fitting

It is assumed that the shear strength parameters areassociated with the distance to slope surface The distancesbetween upper slope surface and slope face have the sameinfluence on the shear strength parameters A set of equationsfor describing shear strength parameters 119888 and 120593 along thebedding surface are illustrated as follows

120593 (119909) =

119860120593

1 + 119890minus119888120593(119909minus119887120593)+ 119867120593 (119909 le

119871119894

2

)

119860120593

1 + 119890minus119888120593(119871119894minus119909minus119887120593)+ 119867120593 (

119871119894

2

le 119909 le 119904119894)

119888 (119909) =

119860119888

1 + 119890minus119888119888(119909minus119887119888)+ 119867119888 (119909 le

119871119894

2

)

119860119888

1 + 119890minus119888119888(119871119894minus119909minus119887119888)+ 119867119888 (

119871119894

2

le 119909 le 119904119894)

(5)

where 119867120593and 119867

119888are the shear strength of rock discon-

tinuities near slope surface and they are the lowest shearstrength parameters the shear strengths of new and completerock discontinuities reach the largest value 119860

120593and 119860

119888are

the differences between the lowest and largest shear strengthparameters 119887

120593 119887119888 119888120593 and 119888

119888are the shape controlling coeffi-

cients of equation set These coefficients can be obtained bydirect shear tests on the specimens taken from different partsof bedding surface In Figure 4 119871

119894is the length of bedding

surface and 119904119894is the length of the potential sliding surface

which is controlled by the cracking surfaceSimilarly tensile strength based on the S-curvemodel can

be expressed as

120590 (ℎ) =

119860120590

1 + 119890minus119888120590(ℎminus119887120590)+ 119867120590 (6)

where 119867120590is the lowest tensile strength of rock mass on the

slope surface 119860120590is the difference between the lowest and

largest tensile strength 119887120590and 119888120590are the shape controlling

coefficients of tensile strength change function ℎ is the depthfrom the upper slope surface to the intersection of beddingsurface and cracking surface (Figure 5) In this study thepotential vertical tensile cracks are assumed to appear whenthe sliding happens The influence on the distributions of thecohesion and frictional angle in tensile crack developing wasnot considered here

24 Modified Limiting Equilibrium Equations The rock massabove the failure surface is regarded as an entire sliding bodyand the back geometric boundary is the potential crackingsurface (Figure 6) Here the bedding surface 119894 is the failuresurface

The length of bedding surface 119894 can be described by

119871119894

= radic[(

119894

sum

119896=1

119897119887119896) sdot cos120572 + (

119894

sum

119896=1

119897119886119896)]

2

+ [(

119894

sum

119896=1

119897119887119896) sdot sin120572]

2

(7)

The failure plane dip is

120573119894= arcsin(

sum119894

119896=1119897119887119896sdot sin120572

radic[(sum119894

119896=1119897119887119896) sdot cos120572 + (sum119894

119896=1119897119886119896)]

2

+ [(sum119894

119896=1119897119887119896) sdot sin120572]

2

) (8)

Mathematical Problems in Engineering 5

Upper slope surface

Slope face

Bedding surface

0

S

x

L i

(a)

Upper slope surface

Slope face

Bedding surface

0

S

x

Cracking surface

s i

(b)

Upper slope surface

Slope face

Bedding surface

0

S xL i

(c)

Upper slope surface

Slope face

Bedding surface

0

S x

Cracking surface

s i

(d)

Figure 4 Relationships between shear strength 119878 of bedding surface and distance 119909 to slope surface (a) long bedding plane without crackingsurface (b) long bedding plane with cracking surface (c) small bedding plane without cracking surface and (d) short bedding plane with thecracking surface

The length of potential cracking surface can be expressedas

ℎ119888=

[(sum119896

119894=0119897119886119894) minus 119897119888] sdot (sum

119896

119894=0119897119887119894) sdot sin120572

(sum119896

119894=0119897119886119894) + (sum

119896

119894=0119897119887119894) sdot cos120572

(9)

The length of failure surface is

119904119894= (1 minus

(sum119894

119896=1119897119886119896) minus 119897119888

(sum119894

119896=1119897119886119896) + (sum

119894

119896=1119897119887119896) sdot cos120572

)

sdot radic[(

119894

sum

119896=1

119897119887119896) sdot cos120572 + (

119894

sum

119896=1

119897119886119896)]

2

+ [(

119894

sum

119896=1

119897119887119896) sdot sin120572]

2

(10)

The weight of sliding mass is

119866119894= 120588 sdot 119892 sdot [

1

2

(

119894

sum

119896=1

119897119886119896) sdot (

119894

sum

119896=1

119897119887119896) sdot sin (120572) minus 1

2

ℎ119888

sdot ((

119894

sum

119896=1

119897119886119896) minus 119871

119888)]

(11)

The resistance supplied by the tensile strength of rockmass can be written as

119879119894= int

ℎ119888

0

119908 (120590119905) 119889ℎ (12)

The normal force on the failure surface can be obtainedby

119873119894= 119866119894sdot cos120573

119894+ 119879119894sdot sin120573

119894 (13)

The resistance supplied by the cohesion of the beddingsurface is

119875 (119909) = int

1198711198942

0

(

119860119888

1 + 119890minus119888119888(119909minus119887119888)+ 119867119888)119889119909

+ int

119904119894

1198711198942

(

119860119888

1 + 119890minus119888119888(119871119894minus119909minus119887119888)+ 119867119888)119889119909

(14)

The average friction coefficient of the bedding surface is

119891 (119909) =

1

119904119894

[int

1198711198942

0

tan(119860120593

1 + 119890minus119888120593(119909minus119887120593)+ 119867120593)119889119909

+ int

119904119894

1198711198942

tan(119860120593

1 + 119890minus119888120593(119871119894minus119909minus119887120593)+ 119867120593)119889119909]

(15)

The factor of safety 120578 can be obtained by

120578 =

119865119904119894

119865119903119894

=

119873119894sdot 119891 (119909) + 119875 (119909)

119866119894sdot sin120573

119894minus 119879119894sdot cos120573

119894

(16)

where 119865119904119894is the resisting force and 119865

119903119894is the driving force of

this bedding plane

6 Mathematical Problems in Engineering

0h

Upper slope surface

Slope face

Bedding surface

Cracking surface120590

Figure 5 Relationships between tensile strength 120590 of rockmass anddistance ℎ to upper slope surface

Upper slope surface

AG

N

T

Potential cracking surface

CB

Bedding surface i

Ff

Slope face

120572

Lc

hc

La1 La2 La3 Lai

120573iLbi

Lb1

Lb3Lb2

middot middot middot

middot middotmiddot

Figure 6 Computing model of bedding rock slope

The critical sliding surface can be determined by findingthe potential sliding surface which has theminimum factor ofsafety The stability analysis was taken by performing seriesof iterations on various combinations of potential slidingsurface and tensile crack The flow chart for evaluating thestability of the stratified rock slope is shown in Figure 7

3 A Case Study by Modified LimitingEquilibrium Method

31 Introduction of the Jiaxiao Slope Jiaxiao Slope is a typicalbedding rock slope in Triassic Strata which is situated atJingmen City Hubei Province China The slope is mainlycomposed of weathered sandstone There are three beddingsurfaces namely bedding plane A bedding plane B andbedding plane C (Figure 8) The dip of bedding plane isapproximately 22∘ The upper slope surface is nearly hori-zontal and the slope height is about 12m The sandstonecrushing degree is controlled by the depth to slope surfacewhich severely influences the strength of rock mass anddiscontinuities

A geotechnical investigation was carried out in June 2013which included the drilling of six geotechnical drill holes onthe upper surface of Jiaxiao Slope Then based on the drillcores we obtained six rock samples for the tensile strengthtest and nine rock joint samples for the shear strength test

The tensile specimens are disc-shaped with a diameter 119863 =

50mm and a thickness 119905 = 25mm and the sizes of the rockjoint samples are about 80mm times 80mm Here the Braziliantest was performed to determine the indirect tensile strengthaccording to ISRM (1978) [20] The test result shows thatthe tensile strength of highly weathered rock specimens nearslope surface is 03MPa and the tensile strength of intact rockspecimen is 48MPa Thus it is obvious that 119867

120590= 03MPa

and 119860120590= 45MPa in the S-curve model (Figure 9) and the

S-curve model for the tensile strength can be written as

120590 (ℎ) =

45

1 + 119890minus119888120590(ℎminus119887120590)+ 03 (17)

Similarly based on the direct shear test for joint spec-imens the friction angle and cohesion of highly weath-ered rock discontinuities near slope surface are 1100∘ and019MPaThe friction angle and cohesion of the unweatheredrock discontinuities are 1870∘ and 028MPaAccording to theS-curve model fitting result (Figure 10) the S-curve modelsfor shear strength in (5) can be written as

120593 (119909)

=

770

1 + 119890minus1440(119909minus010)

+ 11 (119909 le

119871119894

2

)

770

1 + 119890minus1440(119871

119894minus119909minus010)

+ 11 (

119871119894

2

le 119909 le 119904119894)

119888 (119909)

=

009

1 + 119890minus2300(119909minus01)

+ 019 (119909 le

119871119894

2

)

009

1 + 119890minus2300(119871

119894minus119909minus01)

+ 019 (

119871119894

2

le 119909 le 119904119894)

(18)

The strength coefficients of bedding plane and rock massare tabulated in Table 1 Moreover the shear strength ofbedding rock joints was obtained by the direct shear test Theaverage unit weight 120574 is 28 kNm3

32 Stability Analyses on the Case Study The stability of thisbedding rock slope was calculated by the modified limitingequilibrium method presented above The distributions ofsafety factor 120578 of three bedding planes are illustrated inFigure 11 The 120578 values of each bedding plane decline firstand tend to be stable later However in the stable sectioneach bedding planersquos 120578 firstly decrease and then increase alittle bit Most of the safety factors of bedding plane B arelarger than those of bedding plane A and bedding plane CMoreover the safety factor of bedding A is larger than thesafety factor of bedding C when 119871

119888is shorter than 3m The

minimum 120578 of bedding plane A is 1012 the minimum 120578 ofbedding plane B is 1071 and theminimum 120578 of bedding planeC is 0973 Therefore the critical sliding surface is beddingplane C which has the minimum 120578 of the bedding rockslope Moreover the result shows that the potential tension-controlled failure would be most likely to occur at 105m tothe slope face

Mathematical Problems in Engineering 7

Table 1 Strength coefficients of bedding plane and rock mass (MPa)

Coefficient of strength function Bedding plane Rock massFriction angle (∘) Cohesion (MPa) Tensile strength

119867 1100 019 030119860 770 009 450119888 1440 2300 418119887 010 010 330

Start

Assume the potential slidinghappens in layer i

No

No

End

Input the analysis data① Density 120588② Geometric parameters 120572 La Lb

③ Strength function 120593(x) c(x) 120590(h)

Assume the distance betweenthe slope surface and crack

potential crack is Lc

Failure planedip 120573i

Failure planelength Li

Failure surfacelength si

WeightGi

Tensile force Ti

slip surfaceNormal force Ni on the

Resisting force Fsi

Driving force Fri

Safety factor 120578

If 120578 = 120578min

120578min

= 120578i

If 120578i = 120578min

120578 = 120578min

Yes

Figure 7 Flow chart of stability calculation for bedding rock slope

4 Discussion

The spatial variationmodel of the rockmass and rock discon-tinuities discussed in this paper is based on S-curve modelIt is appropriate for indicating the spatial variability of rockdiscontinuities shear strength parameters 119888 and 120593 and therock mass tensile strength 120590 under the high weathering andexcavation unloading The strength of rock discontinuitiesnear the upper slope surface is lower than deep inside partIf the resistance supplied by the tensile strength of rock massis constant the most unstable sliding surface is the beddingplane However the tensile strength and the length of thepotential cracking surface are constantly changing which

makes the stability of rock slope more complicated If thisresistance force is lower than the sliding force the safetyfactor along the bedding plane is consistently decreasingWith the tensile strength and the length of potential crackingsurface increasing the resistance force may make the safetyfactor decrease first and then increase

In the modified limiting equilibrium equations the spa-tial variations of the strength of rock discontinuities androck mass are introduced in this method If we set resistancesupplied by the tensile strength of rock mass to zero thiscalculation model will degenerate into an idealized modelof planar failure Note that this strength parameters modelof the rock mass and rock discontinuities sometimes cannot

8 Mathematical Problems in Engineering

263m312m 945m

464m

113m200m

72∘

Bedding plane ABedding plane C

Bedding plane B

Figure 8 Analysis model of the case study

Tensile strength data pointFitting curve

Depth h (m)545435325215105

0

1

2

3

4

5

Tens

ile st

reng

th 120590

(MPa

)

120590(h) =45

1 + eminusc120590 (hminusb120590)+ 03

c120590 = 418b120590 = 330

R2 = 098

Figure 9 S-curve model for the tensile strength

describe the spatial variability of strength parameters exactlyNevertheless the main idea for calculating the safety factorof stratified rock slope is still valid and needs to change theexpression of strength parameters

The sliding surface in a slopemay consist of a single planecontinuous over the full area of the surface or a complex sur-face made up of both discontinuities and fractures throughintact rock [21] Here the sliding surface is composed ofthe bedding surface and vertical potential cracking surfaceIt is different from the failure mode caused by propagatingcracks which are extended and coalesced with neighboringjoints The suggested method in this paper is appropriate foranalyzing the stability of stratified rock slope whose jointcrack is undeveloped

5 Conclusions

The plane failure is one of the rock slope failure modes infield situations and the limit equilibrium approach for theestimation of the factor of safety of the rock slope againstplane failure has been well accepted by the engineers Insedimentary rock mass the stratified rock slope usually

Friction angle data pointFitting curve

120593(x) =770

1 + eminusc120593(xminusb120593)+ 11

R2 = 087

b120593 = 010

01 02 03 04 050

Length x (m)

10

12

14

16

18

20

Fric

tion

angl

e (de

gree

)

c120593 = 144

(a)

Fitting curveCohesion data point

R2 = 094

cc = 2300bc = 010

c(x) =009

1 + eminusc119888(xminusb119888)+ 019

01 02 03 04 050

Length x (m)

018

02

022

024

026

028

03

Coh

esio

n (M

Pa)

(b)

Figure 10 S-curve model for the friction angle and cohesion of therock discontinuities

has several bedding planes whose dips and mechanicalparameters have significant effects on rock mass strengthand stability However previous limit equilibrium approacheswere mainly about the plane failure mode with single slidingsurface and failed to predict the stability of the rock slopewith various bedding planesThis paper presented amodifiedlimit equilibrium method for predicting the stability of thesedimentary rock slope with flat bedding planes In thismethod the spatial variations of shear strength parameters119888 and 120593 of bedding plane and the tensile strength 120590 of rockmass were considered and they are expressed by the S-curve model The potential tensile at the top of rock masscan also be considered in plane failure mode incorporatingvarious combinations of potential sliding surface and tensilecrack The critical sliding surface and the tensile crack weredetermined by finding the potential sliding surfacewhich had

Mathematical Problems in Engineering 9

(65 1012)

(85 1071)

(105 0973)

090

095

100

105

110

115

120

125

130

135

140

Safe

ty fa

ctor

2 4 6 80 12 14 1610Lc

Bedding plane ABedding plane B

Bedding plane C

Figure 11 Relations between factor of safety and 119871119888

theminimum factor of safetyThismethod was utilized in thestability analysis of a case study in Three Gorges ReservoirRegion China The factors of safety were different betweenthe bedding planes and they were affected by the resistancesupplied by the tensile strength of rock mass The factor ofsafety of the case study was 0973 the critical sliding surfaceappears at bedding planeC and the tension-controlled failureoccurs at 105m to the slope face On the basis of generalizedanalytical expressions for the factor of safety the examplewas provided to show the effectiveness of the modified limitequilibrium method in the stability analysis of stratified rockslopes

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

This study was funded by National Natural Science Foun-dation of China (nos 41427802 41172292 41302257 and41502300) Zhejiang Provincial Natural Science Foundation(no Q16D020005) and Zhejiang Provincial Natural ScienceFoundation (no Q13D020001)

References

[1] Y Zhao Z-Y Tong and Q Lu ldquoSlope stability analysis usingslice-wise factor of safetyrdquoMathematical Problems in Engineer-ing vol 2014 Article ID 712145 6 pages 2014

[2] SGDu Y JHu andX FHu ldquoMeasurement of joint roughnesscoefficient by using profilograph and roughness rulerrdquo Journalof Earth Science vol 20 no 5 pp 890ndash896 2009

[3] S K Shukla and M M Hossain ldquoAnalytical expression forfactor of safety of an anchored rock slope against plane failurerdquoInternational Journal of Geotechnical Engineering vol 5 no 2pp 181ndash187 2011

[4] N Sivakugan S K Shukla and B M Das Rock Mechanics AnIntroduction CRC Press New York NY USA 2013

[5] M Gencer ldquoProgressive failure in stratified and jointed rockmassrdquo Rock Mechanics and Rock Engineering vol 18 no 4 pp267ndash292 1985

[6] J-J Dong C-H Tu W-R Lee and Y-J Jheng ldquoEffectsof hydraulic conductivitystrength anisotropy on the stabilityof stratified poorly cemented rock slopesrdquo Computers andGeotechnics vol 40 pp 147ndash159 2012

[7] J Ching Z-Y Yang J-Q Shiau and C-J Chen ldquoEstimationof rock pressure during an excavationcut in sedimentary rockswith inclined bedding planesrdquo Structural Safety vol 41 pp 11ndash19 2013

[8] H Tang R Yong and M A M Ez Eldin ldquoStability analysis ofstratified rock slopes with spatially variable strength parame-ters the case of Qianjiangping landsliderdquoBulletin of EngineeringGeology and the Environment 2016

[9] X-L Yang and J-H Yin ldquoLinear Mohr-Coulomb strengthparameters from the non-linear Hoek-Brown rock massesrdquoInternational Journal of Non-LinearMechanics vol 41 no 8 pp1000ndash1005 2006

[10] P Fortsakis K Nikas V Marinos and P Marinos ldquoAnisotropicbehaviour of stratified rock masses in tunnellingrdquo EngineeringGeology vol 141-142 pp 74ndash83 2012

[11] E Hoek and J D Bray Rock Slope Engineering CRC Press 1981[12] S Sharma T K Raghuvanshi and R Anbalagan ldquoPlane

failure analysis of rock slopesrdquo Geotechnical and GeologicalEngineering vol 13 no 2 pp 105ndash111 1995

[13] E B Kroeger ldquoAnalysis of plane failures in compound slopesrdquoInternational Journal of Surface Mining Reclamation and Envi-ronment vol 14 no 3 pp 215ndash222 2000

[14] B S Jiang M F Cai and H Du ldquoAnalytical calculation onstability of slope with planar failure surfacerdquo Chinese Journal ofRock Mechanics and Engineering vol 23 no 1 pp 91ndash94 2004

[15] M Ahmadi and M Eslami ldquoA new approach to plane failure ofrock slope stability based on water flow velocity in discontinu-ities for the Latian dam reservoir landsliderdquo Journal ofMountainScience vol 8 no 2 pp 124ndash130 2011

[16] A Johari A Fazeli and A A Javadi ldquoAn investigation intoapplication of jointly distributed random variables method inreliability assessment of rock slope stabilityrdquo Computers andGeotechnics vol 47 pp 42ndash47 2013

[17] HAViles ldquoLinkingweathering and rock slope instability non-linear perspectivesrdquo Earth Surface Processes and Landforms vol38 no 1 pp 62ndash70 2013

[18] HM Tang Z X Zou C R Xiong et al ldquoAn evolutionmodel oflarge consequent bedding rockslides with particular referenceto the Jiweishan rockslide in Southwest Chinardquo EngineeringGeology vol 186 pp 17ndash27 2015

[19] Z X Zou H M Tang C R Xiong Y P Wu X Liu and SB Liao ldquoGeomechanical model of progressive failure for largeconsequent bedding rockslide and its stability analysisrdquoChineseJournal of Rock Mechanics and Engineering vol 31 no 11 pp2222ndash2230 2012 (Chinese)

[20] Z T Bieniawski and I Hawkes ldquoSuggested methods for deter-mining tensile strength of rock materialsrdquo International Journalof RockMechanics andMining Sciences vol 15 no 3 pp 99ndash1031978

[21] C W Duncan and W M Christopher Rock Slope Engineering-Civil and Mining Taylor amp Francis New York NY USA 4thedition 2005

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Mathematical Problems in Engineering

According to S-curve model the strength can beexpressed as follows

119878 (119909) =

119860

1 + 119890minus1198881015840(119909minus119887)

+ 1198671015840 (4)

where 119878(119909) denotes the strength of rock mass or rockdiscontinuities 119909 is the distance to the slope surface 1198671015840 isthe lowest strength of rock mass or rock discontinuities Thestrength of rock mass or rock discontinuities deep insidethe rock slope is barely affected by the factors mentionedabove so it gets the largest value 119860 is the difference betweenthe lowest and the largest strength The comparison betweencurves (a) and (d) in Figure 3 shows that the variation oflowest and largest value in S-curve model is determined bycoefficients 119860 and 1198671015840 119887 and 1198881015840 are the shape controllingfactors of S-curve model Then 119887 represents the distance tothe slope surface when the strength is equal to themean valueof the lowest and the highest strength and 1198881015840 is positivelycorrelated with the inverse of 119887

In the present study the S-curve model is established todefine the spatial variations of the shear strength parametersof bedding plane and the tensile strength of rock mass Forthe shear strength parameters 1198671015840 is the shear strength ofhighly weathered rock discontinuities near slope surface119860 is the difference between the shear strength value ofthe highly weathered rock discontinuities and the highestshear strength value of discontinuities which are the leastaffected by weathering effect 119887 and 1198881015840 are obtained by curve-fitting of the shear strength of the rock discontinuities andthe locations of rock joint specimens For tensile strengthparameters1198671015840 is the tensile strength of rock specimens nearslope surface 119860 is acquired by calculating the differencebetween1198671015840 and the tensile strength of intact rock specimenbased on the variable tensile strength and the depth of rockspecimens 119887 and 1198881015840 values are acquired through the S-curvemodel fitting

It is assumed that the shear strength parameters areassociated with the distance to slope surface The distancesbetween upper slope surface and slope face have the sameinfluence on the shear strength parameters A set of equationsfor describing shear strength parameters 119888 and 120593 along thebedding surface are illustrated as follows

120593 (119909) =

119860120593

1 + 119890minus119888120593(119909minus119887120593)+ 119867120593 (119909 le

119871119894

2

)

119860120593

1 + 119890minus119888120593(119871119894minus119909minus119887120593)+ 119867120593 (

119871119894

2

le 119909 le 119904119894)

119888 (119909) =

119860119888

1 + 119890minus119888119888(119909minus119887119888)+ 119867119888 (119909 le

119871119894

2

)

119860119888

1 + 119890minus119888119888(119871119894minus119909minus119887119888)+ 119867119888 (

119871119894

2

le 119909 le 119904119894)

(5)

where 119867120593and 119867

119888are the shear strength of rock discon-

tinuities near slope surface and they are the lowest shearstrength parameters the shear strengths of new and completerock discontinuities reach the largest value 119860

120593and 119860

119888are

the differences between the lowest and largest shear strengthparameters 119887

120593 119887119888 119888120593 and 119888

119888are the shape controlling coeffi-

cients of equation set These coefficients can be obtained bydirect shear tests on the specimens taken from different partsof bedding surface In Figure 4 119871

119894is the length of bedding

surface and 119904119894is the length of the potential sliding surface

which is controlled by the cracking surfaceSimilarly tensile strength based on the S-curvemodel can

be expressed as

120590 (ℎ) =

119860120590

1 + 119890minus119888120590(ℎminus119887120590)+ 119867120590 (6)

where 119867120590is the lowest tensile strength of rock mass on the

slope surface 119860120590is the difference between the lowest and

largest tensile strength 119887120590and 119888120590are the shape controlling

coefficients of tensile strength change function ℎ is the depthfrom the upper slope surface to the intersection of beddingsurface and cracking surface (Figure 5) In this study thepotential vertical tensile cracks are assumed to appear whenthe sliding happens The influence on the distributions of thecohesion and frictional angle in tensile crack developing wasnot considered here

24 Modified Limiting Equilibrium Equations The rock massabove the failure surface is regarded as an entire sliding bodyand the back geometric boundary is the potential crackingsurface (Figure 6) Here the bedding surface 119894 is the failuresurface

The length of bedding surface 119894 can be described by

119871119894

= radic[(

119894

sum

119896=1

119897119887119896) sdot cos120572 + (

119894

sum

119896=1

119897119886119896)]

2

+ [(

119894

sum

119896=1

119897119887119896) sdot sin120572]

2

(7)

The failure plane dip is

120573119894= arcsin(

sum119894

119896=1119897119887119896sdot sin120572

radic[(sum119894

119896=1119897119887119896) sdot cos120572 + (sum119894

119896=1119897119886119896)]

2

+ [(sum119894

119896=1119897119887119896) sdot sin120572]

2

) (8)

Mathematical Problems in Engineering 5

Upper slope surface

Slope face

Bedding surface

0

S

x

L i

(a)

Upper slope surface

Slope face

Bedding surface

0

S

x

Cracking surface

s i

(b)

Upper slope surface

Slope face

Bedding surface

0

S xL i

(c)

Upper slope surface

Slope face

Bedding surface

0

S x

Cracking surface

s i

(d)

Figure 4 Relationships between shear strength 119878 of bedding surface and distance 119909 to slope surface (a) long bedding plane without crackingsurface (b) long bedding plane with cracking surface (c) small bedding plane without cracking surface and (d) short bedding plane with thecracking surface

The length of potential cracking surface can be expressedas

ℎ119888=

[(sum119896

119894=0119897119886119894) minus 119897119888] sdot (sum

119896

119894=0119897119887119894) sdot sin120572

(sum119896

119894=0119897119886119894) + (sum

119896

119894=0119897119887119894) sdot cos120572

(9)

The length of failure surface is

119904119894= (1 minus

(sum119894

119896=1119897119886119896) minus 119897119888

(sum119894

119896=1119897119886119896) + (sum

119894

119896=1119897119887119896) sdot cos120572

)

sdot radic[(

119894

sum

119896=1

119897119887119896) sdot cos120572 + (

119894

sum

119896=1

119897119886119896)]

2

+ [(

119894

sum

119896=1

119897119887119896) sdot sin120572]

2

(10)

The weight of sliding mass is

119866119894= 120588 sdot 119892 sdot [

1

2

(

119894

sum

119896=1

119897119886119896) sdot (

119894

sum

119896=1

119897119887119896) sdot sin (120572) minus 1

2

ℎ119888

sdot ((

119894

sum

119896=1

119897119886119896) minus 119871

119888)]

(11)

The resistance supplied by the tensile strength of rockmass can be written as

119879119894= int

ℎ119888

0

119908 (120590119905) 119889ℎ (12)

The normal force on the failure surface can be obtainedby

119873119894= 119866119894sdot cos120573

119894+ 119879119894sdot sin120573

119894 (13)

The resistance supplied by the cohesion of the beddingsurface is

119875 (119909) = int

1198711198942

0

(

119860119888

1 + 119890minus119888119888(119909minus119887119888)+ 119867119888)119889119909

+ int

119904119894

1198711198942

(

119860119888

1 + 119890minus119888119888(119871119894minus119909minus119887119888)+ 119867119888)119889119909

(14)

The average friction coefficient of the bedding surface is

119891 (119909) =

1

119904119894

[int

1198711198942

0

tan(119860120593

1 + 119890minus119888120593(119909minus119887120593)+ 119867120593)119889119909

+ int

119904119894

1198711198942

tan(119860120593

1 + 119890minus119888120593(119871119894minus119909minus119887120593)+ 119867120593)119889119909]

(15)

The factor of safety 120578 can be obtained by

120578 =

119865119904119894

119865119903119894

=

119873119894sdot 119891 (119909) + 119875 (119909)

119866119894sdot sin120573

119894minus 119879119894sdot cos120573

119894

(16)

where 119865119904119894is the resisting force and 119865

119903119894is the driving force of

this bedding plane

6 Mathematical Problems in Engineering

0h

Upper slope surface

Slope face

Bedding surface

Cracking surface120590

Figure 5 Relationships between tensile strength 120590 of rockmass anddistance ℎ to upper slope surface

Upper slope surface

AG

N

T

Potential cracking surface

CB

Bedding surface i

Ff

Slope face

120572

Lc

hc

La1 La2 La3 Lai

120573iLbi

Lb1

Lb3Lb2

middot middot middot

middot middotmiddot

Figure 6 Computing model of bedding rock slope

The critical sliding surface can be determined by findingthe potential sliding surface which has theminimum factor ofsafety The stability analysis was taken by performing seriesof iterations on various combinations of potential slidingsurface and tensile crack The flow chart for evaluating thestability of the stratified rock slope is shown in Figure 7

3 A Case Study by Modified LimitingEquilibrium Method

31 Introduction of the Jiaxiao Slope Jiaxiao Slope is a typicalbedding rock slope in Triassic Strata which is situated atJingmen City Hubei Province China The slope is mainlycomposed of weathered sandstone There are three beddingsurfaces namely bedding plane A bedding plane B andbedding plane C (Figure 8) The dip of bedding plane isapproximately 22∘ The upper slope surface is nearly hori-zontal and the slope height is about 12m The sandstonecrushing degree is controlled by the depth to slope surfacewhich severely influences the strength of rock mass anddiscontinuities

A geotechnical investigation was carried out in June 2013which included the drilling of six geotechnical drill holes onthe upper surface of Jiaxiao Slope Then based on the drillcores we obtained six rock samples for the tensile strengthtest and nine rock joint samples for the shear strength test

The tensile specimens are disc-shaped with a diameter 119863 =

50mm and a thickness 119905 = 25mm and the sizes of the rockjoint samples are about 80mm times 80mm Here the Braziliantest was performed to determine the indirect tensile strengthaccording to ISRM (1978) [20] The test result shows thatthe tensile strength of highly weathered rock specimens nearslope surface is 03MPa and the tensile strength of intact rockspecimen is 48MPa Thus it is obvious that 119867

120590= 03MPa

and 119860120590= 45MPa in the S-curve model (Figure 9) and the

S-curve model for the tensile strength can be written as

120590 (ℎ) =

45

1 + 119890minus119888120590(ℎminus119887120590)+ 03 (17)

Similarly based on the direct shear test for joint spec-imens the friction angle and cohesion of highly weath-ered rock discontinuities near slope surface are 1100∘ and019MPaThe friction angle and cohesion of the unweatheredrock discontinuities are 1870∘ and 028MPaAccording to theS-curve model fitting result (Figure 10) the S-curve modelsfor shear strength in (5) can be written as

120593 (119909)

=

770

1 + 119890minus1440(119909minus010)

+ 11 (119909 le

119871119894

2

)

770

1 + 119890minus1440(119871

119894minus119909minus010)

+ 11 (

119871119894

2

le 119909 le 119904119894)

119888 (119909)

=

009

1 + 119890minus2300(119909minus01)

+ 019 (119909 le

119871119894

2

)

009

1 + 119890minus2300(119871

119894minus119909minus01)

+ 019 (

119871119894

2

le 119909 le 119904119894)

(18)

The strength coefficients of bedding plane and rock massare tabulated in Table 1 Moreover the shear strength ofbedding rock joints was obtained by the direct shear test Theaverage unit weight 120574 is 28 kNm3

32 Stability Analyses on the Case Study The stability of thisbedding rock slope was calculated by the modified limitingequilibrium method presented above The distributions ofsafety factor 120578 of three bedding planes are illustrated inFigure 11 The 120578 values of each bedding plane decline firstand tend to be stable later However in the stable sectioneach bedding planersquos 120578 firstly decrease and then increase alittle bit Most of the safety factors of bedding plane B arelarger than those of bedding plane A and bedding plane CMoreover the safety factor of bedding A is larger than thesafety factor of bedding C when 119871

119888is shorter than 3m The

minimum 120578 of bedding plane A is 1012 the minimum 120578 ofbedding plane B is 1071 and theminimum 120578 of bedding planeC is 0973 Therefore the critical sliding surface is beddingplane C which has the minimum 120578 of the bedding rockslope Moreover the result shows that the potential tension-controlled failure would be most likely to occur at 105m tothe slope face

Mathematical Problems in Engineering 7

Table 1 Strength coefficients of bedding plane and rock mass (MPa)

Coefficient of strength function Bedding plane Rock massFriction angle (∘) Cohesion (MPa) Tensile strength

119867 1100 019 030119860 770 009 450119888 1440 2300 418119887 010 010 330

Start

Assume the potential slidinghappens in layer i

No

No

End

Input the analysis data① Density 120588② Geometric parameters 120572 La Lb

③ Strength function 120593(x) c(x) 120590(h)

Assume the distance betweenthe slope surface and crack

potential crack is Lc

Failure planedip 120573i

Failure planelength Li

Failure surfacelength si

WeightGi

Tensile force Ti

slip surfaceNormal force Ni on the

Resisting force Fsi

Driving force Fri

Safety factor 120578

If 120578 = 120578min

120578min

= 120578i

If 120578i = 120578min

120578 = 120578min

Yes

Figure 7 Flow chart of stability calculation for bedding rock slope

4 Discussion

The spatial variationmodel of the rockmass and rock discon-tinuities discussed in this paper is based on S-curve modelIt is appropriate for indicating the spatial variability of rockdiscontinuities shear strength parameters 119888 and 120593 and therock mass tensile strength 120590 under the high weathering andexcavation unloading The strength of rock discontinuitiesnear the upper slope surface is lower than deep inside partIf the resistance supplied by the tensile strength of rock massis constant the most unstable sliding surface is the beddingplane However the tensile strength and the length of thepotential cracking surface are constantly changing which

makes the stability of rock slope more complicated If thisresistance force is lower than the sliding force the safetyfactor along the bedding plane is consistently decreasingWith the tensile strength and the length of potential crackingsurface increasing the resistance force may make the safetyfactor decrease first and then increase

In the modified limiting equilibrium equations the spa-tial variations of the strength of rock discontinuities androck mass are introduced in this method If we set resistancesupplied by the tensile strength of rock mass to zero thiscalculation model will degenerate into an idealized modelof planar failure Note that this strength parameters modelof the rock mass and rock discontinuities sometimes cannot

8 Mathematical Problems in Engineering

263m312m 945m

464m

113m200m

72∘

Bedding plane ABedding plane C

Bedding plane B

Figure 8 Analysis model of the case study

Tensile strength data pointFitting curve

Depth h (m)545435325215105

0

1

2

3

4

5

Tens

ile st

reng

th 120590

(MPa

)

120590(h) =45

1 + eminusc120590 (hminusb120590)+ 03

c120590 = 418b120590 = 330

R2 = 098

Figure 9 S-curve model for the tensile strength

describe the spatial variability of strength parameters exactlyNevertheless the main idea for calculating the safety factorof stratified rock slope is still valid and needs to change theexpression of strength parameters

The sliding surface in a slopemay consist of a single planecontinuous over the full area of the surface or a complex sur-face made up of both discontinuities and fractures throughintact rock [21] Here the sliding surface is composed ofthe bedding surface and vertical potential cracking surfaceIt is different from the failure mode caused by propagatingcracks which are extended and coalesced with neighboringjoints The suggested method in this paper is appropriate foranalyzing the stability of stratified rock slope whose jointcrack is undeveloped

5 Conclusions

The plane failure is one of the rock slope failure modes infield situations and the limit equilibrium approach for theestimation of the factor of safety of the rock slope againstplane failure has been well accepted by the engineers Insedimentary rock mass the stratified rock slope usually

Friction angle data pointFitting curve

120593(x) =770

1 + eminusc120593(xminusb120593)+ 11

R2 = 087

b120593 = 010

01 02 03 04 050

Length x (m)

10

12

14

16

18

20

Fric

tion

angl

e (de

gree

)

c120593 = 144

(a)

Fitting curveCohesion data point

R2 = 094

cc = 2300bc = 010

c(x) =009

1 + eminusc119888(xminusb119888)+ 019

01 02 03 04 050

Length x (m)

018

02

022

024

026

028

03

Coh

esio

n (M

Pa)

(b)

Figure 10 S-curve model for the friction angle and cohesion of therock discontinuities

has several bedding planes whose dips and mechanicalparameters have significant effects on rock mass strengthand stability However previous limit equilibrium approacheswere mainly about the plane failure mode with single slidingsurface and failed to predict the stability of the rock slopewith various bedding planesThis paper presented amodifiedlimit equilibrium method for predicting the stability of thesedimentary rock slope with flat bedding planes In thismethod the spatial variations of shear strength parameters119888 and 120593 of bedding plane and the tensile strength 120590 of rockmass were considered and they are expressed by the S-curve model The potential tensile at the top of rock masscan also be considered in plane failure mode incorporatingvarious combinations of potential sliding surface and tensilecrack The critical sliding surface and the tensile crack weredetermined by finding the potential sliding surfacewhich had

Mathematical Problems in Engineering 9

(65 1012)

(85 1071)

(105 0973)

090

095

100

105

110

115

120

125

130

135

140

Safe

ty fa

ctor

2 4 6 80 12 14 1610Lc

Bedding plane ABedding plane B

Bedding plane C

Figure 11 Relations between factor of safety and 119871119888

theminimum factor of safetyThismethod was utilized in thestability analysis of a case study in Three Gorges ReservoirRegion China The factors of safety were different betweenthe bedding planes and they were affected by the resistancesupplied by the tensile strength of rock mass The factor ofsafety of the case study was 0973 the critical sliding surfaceappears at bedding planeC and the tension-controlled failureoccurs at 105m to the slope face On the basis of generalizedanalytical expressions for the factor of safety the examplewas provided to show the effectiveness of the modified limitequilibrium method in the stability analysis of stratified rockslopes

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

This study was funded by National Natural Science Foun-dation of China (nos 41427802 41172292 41302257 and41502300) Zhejiang Provincial Natural Science Foundation(no Q16D020005) and Zhejiang Provincial Natural ScienceFoundation (no Q13D020001)

References

[1] Y Zhao Z-Y Tong and Q Lu ldquoSlope stability analysis usingslice-wise factor of safetyrdquoMathematical Problems in Engineer-ing vol 2014 Article ID 712145 6 pages 2014

[2] SGDu Y JHu andX FHu ldquoMeasurement of joint roughnesscoefficient by using profilograph and roughness rulerrdquo Journalof Earth Science vol 20 no 5 pp 890ndash896 2009

[3] S K Shukla and M M Hossain ldquoAnalytical expression forfactor of safety of an anchored rock slope against plane failurerdquoInternational Journal of Geotechnical Engineering vol 5 no 2pp 181ndash187 2011

[4] N Sivakugan S K Shukla and B M Das Rock Mechanics AnIntroduction CRC Press New York NY USA 2013

[5] M Gencer ldquoProgressive failure in stratified and jointed rockmassrdquo Rock Mechanics and Rock Engineering vol 18 no 4 pp267ndash292 1985

[6] J-J Dong C-H Tu W-R Lee and Y-J Jheng ldquoEffectsof hydraulic conductivitystrength anisotropy on the stabilityof stratified poorly cemented rock slopesrdquo Computers andGeotechnics vol 40 pp 147ndash159 2012

[7] J Ching Z-Y Yang J-Q Shiau and C-J Chen ldquoEstimationof rock pressure during an excavationcut in sedimentary rockswith inclined bedding planesrdquo Structural Safety vol 41 pp 11ndash19 2013

[8] H Tang R Yong and M A M Ez Eldin ldquoStability analysis ofstratified rock slopes with spatially variable strength parame-ters the case of Qianjiangping landsliderdquoBulletin of EngineeringGeology and the Environment 2016

[9] X-L Yang and J-H Yin ldquoLinear Mohr-Coulomb strengthparameters from the non-linear Hoek-Brown rock massesrdquoInternational Journal of Non-LinearMechanics vol 41 no 8 pp1000ndash1005 2006

[10] P Fortsakis K Nikas V Marinos and P Marinos ldquoAnisotropicbehaviour of stratified rock masses in tunnellingrdquo EngineeringGeology vol 141-142 pp 74ndash83 2012

[11] E Hoek and J D Bray Rock Slope Engineering CRC Press 1981[12] S Sharma T K Raghuvanshi and R Anbalagan ldquoPlane

failure analysis of rock slopesrdquo Geotechnical and GeologicalEngineering vol 13 no 2 pp 105ndash111 1995

[13] E B Kroeger ldquoAnalysis of plane failures in compound slopesrdquoInternational Journal of Surface Mining Reclamation and Envi-ronment vol 14 no 3 pp 215ndash222 2000

[14] B S Jiang M F Cai and H Du ldquoAnalytical calculation onstability of slope with planar failure surfacerdquo Chinese Journal ofRock Mechanics and Engineering vol 23 no 1 pp 91ndash94 2004

[15] M Ahmadi and M Eslami ldquoA new approach to plane failure ofrock slope stability based on water flow velocity in discontinu-ities for the Latian dam reservoir landsliderdquo Journal ofMountainScience vol 8 no 2 pp 124ndash130 2011

[16] A Johari A Fazeli and A A Javadi ldquoAn investigation intoapplication of jointly distributed random variables method inreliability assessment of rock slope stabilityrdquo Computers andGeotechnics vol 47 pp 42ndash47 2013

[17] HAViles ldquoLinkingweathering and rock slope instability non-linear perspectivesrdquo Earth Surface Processes and Landforms vol38 no 1 pp 62ndash70 2013

[18] HM Tang Z X Zou C R Xiong et al ldquoAn evolutionmodel oflarge consequent bedding rockslides with particular referenceto the Jiweishan rockslide in Southwest Chinardquo EngineeringGeology vol 186 pp 17ndash27 2015

[19] Z X Zou H M Tang C R Xiong Y P Wu X Liu and SB Liao ldquoGeomechanical model of progressive failure for largeconsequent bedding rockslide and its stability analysisrdquoChineseJournal of Rock Mechanics and Engineering vol 31 no 11 pp2222ndash2230 2012 (Chinese)

[20] Z T Bieniawski and I Hawkes ldquoSuggested methods for deter-mining tensile strength of rock materialsrdquo International Journalof RockMechanics andMining Sciences vol 15 no 3 pp 99ndash1031978

[21] C W Duncan and W M Christopher Rock Slope Engineering-Civil and Mining Taylor amp Francis New York NY USA 4thedition 2005

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 5

Upper slope surface

Slope face

Bedding surface

0

S

x

L i

(a)

Upper slope surface

Slope face

Bedding surface

0

S

x

Cracking surface

s i

(b)

Upper slope surface

Slope face

Bedding surface

0

S xL i

(c)

Upper slope surface

Slope face

Bedding surface

0

S x

Cracking surface

s i

(d)

Figure 4 Relationships between shear strength 119878 of bedding surface and distance 119909 to slope surface (a) long bedding plane without crackingsurface (b) long bedding plane with cracking surface (c) small bedding plane without cracking surface and (d) short bedding plane with thecracking surface

The length of potential cracking surface can be expressedas

ℎ119888=

[(sum119896

119894=0119897119886119894) minus 119897119888] sdot (sum

119896

119894=0119897119887119894) sdot sin120572

(sum119896

119894=0119897119886119894) + (sum

119896

119894=0119897119887119894) sdot cos120572

(9)

The length of failure surface is

119904119894= (1 minus

(sum119894

119896=1119897119886119896) minus 119897119888

(sum119894

119896=1119897119886119896) + (sum

119894

119896=1119897119887119896) sdot cos120572

)

sdot radic[(

119894

sum

119896=1

119897119887119896) sdot cos120572 + (

119894

sum

119896=1

119897119886119896)]

2

+ [(

119894

sum

119896=1

119897119887119896) sdot sin120572]

2

(10)

The weight of sliding mass is

119866119894= 120588 sdot 119892 sdot [

1

2

(

119894

sum

119896=1

119897119886119896) sdot (

119894

sum

119896=1

119897119887119896) sdot sin (120572) minus 1

2

ℎ119888

sdot ((

119894

sum

119896=1

119897119886119896) minus 119871

119888)]

(11)

The resistance supplied by the tensile strength of rockmass can be written as

119879119894= int

ℎ119888

0

119908 (120590119905) 119889ℎ (12)

The normal force on the failure surface can be obtainedby

119873119894= 119866119894sdot cos120573

119894+ 119879119894sdot sin120573

119894 (13)

The resistance supplied by the cohesion of the beddingsurface is

119875 (119909) = int

1198711198942

0

(

119860119888

1 + 119890minus119888119888(119909minus119887119888)+ 119867119888)119889119909

+ int

119904119894

1198711198942

(

119860119888

1 + 119890minus119888119888(119871119894minus119909minus119887119888)+ 119867119888)119889119909

(14)

The average friction coefficient of the bedding surface is

119891 (119909) =

1

119904119894

[int

1198711198942

0

tan(119860120593

1 + 119890minus119888120593(119909minus119887120593)+ 119867120593)119889119909

+ int

119904119894

1198711198942

tan(119860120593

1 + 119890minus119888120593(119871119894minus119909minus119887120593)+ 119867120593)119889119909]

(15)

The factor of safety 120578 can be obtained by

120578 =

119865119904119894

119865119903119894

=

119873119894sdot 119891 (119909) + 119875 (119909)

119866119894sdot sin120573

119894minus 119879119894sdot cos120573

119894

(16)

where 119865119904119894is the resisting force and 119865

119903119894is the driving force of

this bedding plane

6 Mathematical Problems in Engineering

0h

Upper slope surface

Slope face

Bedding surface

Cracking surface120590

Figure 5 Relationships between tensile strength 120590 of rockmass anddistance ℎ to upper slope surface

Upper slope surface

AG

N

T

Potential cracking surface

CB

Bedding surface i

Ff

Slope face

120572

Lc

hc

La1 La2 La3 Lai

120573iLbi

Lb1

Lb3Lb2

middot middot middot

middot middotmiddot

Figure 6 Computing model of bedding rock slope

The critical sliding surface can be determined by findingthe potential sliding surface which has theminimum factor ofsafety The stability analysis was taken by performing seriesof iterations on various combinations of potential slidingsurface and tensile crack The flow chart for evaluating thestability of the stratified rock slope is shown in Figure 7

3 A Case Study by Modified LimitingEquilibrium Method

31 Introduction of the Jiaxiao Slope Jiaxiao Slope is a typicalbedding rock slope in Triassic Strata which is situated atJingmen City Hubei Province China The slope is mainlycomposed of weathered sandstone There are three beddingsurfaces namely bedding plane A bedding plane B andbedding plane C (Figure 8) The dip of bedding plane isapproximately 22∘ The upper slope surface is nearly hori-zontal and the slope height is about 12m The sandstonecrushing degree is controlled by the depth to slope surfacewhich severely influences the strength of rock mass anddiscontinuities

A geotechnical investigation was carried out in June 2013which included the drilling of six geotechnical drill holes onthe upper surface of Jiaxiao Slope Then based on the drillcores we obtained six rock samples for the tensile strengthtest and nine rock joint samples for the shear strength test

The tensile specimens are disc-shaped with a diameter 119863 =

50mm and a thickness 119905 = 25mm and the sizes of the rockjoint samples are about 80mm times 80mm Here the Braziliantest was performed to determine the indirect tensile strengthaccording to ISRM (1978) [20] The test result shows thatthe tensile strength of highly weathered rock specimens nearslope surface is 03MPa and the tensile strength of intact rockspecimen is 48MPa Thus it is obvious that 119867

120590= 03MPa

and 119860120590= 45MPa in the S-curve model (Figure 9) and the

S-curve model for the tensile strength can be written as

120590 (ℎ) =

45

1 + 119890minus119888120590(ℎminus119887120590)+ 03 (17)

Similarly based on the direct shear test for joint spec-imens the friction angle and cohesion of highly weath-ered rock discontinuities near slope surface are 1100∘ and019MPaThe friction angle and cohesion of the unweatheredrock discontinuities are 1870∘ and 028MPaAccording to theS-curve model fitting result (Figure 10) the S-curve modelsfor shear strength in (5) can be written as

120593 (119909)

=

770

1 + 119890minus1440(119909minus010)

+ 11 (119909 le

119871119894

2

)

770

1 + 119890minus1440(119871

119894minus119909minus010)

+ 11 (

119871119894

2

le 119909 le 119904119894)

119888 (119909)

=

009

1 + 119890minus2300(119909minus01)

+ 019 (119909 le

119871119894

2

)

009

1 + 119890minus2300(119871

119894minus119909minus01)

+ 019 (

119871119894

2

le 119909 le 119904119894)

(18)

The strength coefficients of bedding plane and rock massare tabulated in Table 1 Moreover the shear strength ofbedding rock joints was obtained by the direct shear test Theaverage unit weight 120574 is 28 kNm3

32 Stability Analyses on the Case Study The stability of thisbedding rock slope was calculated by the modified limitingequilibrium method presented above The distributions ofsafety factor 120578 of three bedding planes are illustrated inFigure 11 The 120578 values of each bedding plane decline firstand tend to be stable later However in the stable sectioneach bedding planersquos 120578 firstly decrease and then increase alittle bit Most of the safety factors of bedding plane B arelarger than those of bedding plane A and bedding plane CMoreover the safety factor of bedding A is larger than thesafety factor of bedding C when 119871

119888is shorter than 3m The

minimum 120578 of bedding plane A is 1012 the minimum 120578 ofbedding plane B is 1071 and theminimum 120578 of bedding planeC is 0973 Therefore the critical sliding surface is beddingplane C which has the minimum 120578 of the bedding rockslope Moreover the result shows that the potential tension-controlled failure would be most likely to occur at 105m tothe slope face

Mathematical Problems in Engineering 7

Table 1 Strength coefficients of bedding plane and rock mass (MPa)

Coefficient of strength function Bedding plane Rock massFriction angle (∘) Cohesion (MPa) Tensile strength

119867 1100 019 030119860 770 009 450119888 1440 2300 418119887 010 010 330

Start

Assume the potential slidinghappens in layer i

No

No

End

Input the analysis data① Density 120588② Geometric parameters 120572 La Lb

③ Strength function 120593(x) c(x) 120590(h)

Assume the distance betweenthe slope surface and crack

potential crack is Lc

Failure planedip 120573i

Failure planelength Li

Failure surfacelength si

WeightGi

Tensile force Ti

slip surfaceNormal force Ni on the

Resisting force Fsi

Driving force Fri

Safety factor 120578

If 120578 = 120578min

120578min

= 120578i

If 120578i = 120578min

120578 = 120578min

Yes

Figure 7 Flow chart of stability calculation for bedding rock slope

4 Discussion

The spatial variationmodel of the rockmass and rock discon-tinuities discussed in this paper is based on S-curve modelIt is appropriate for indicating the spatial variability of rockdiscontinuities shear strength parameters 119888 and 120593 and therock mass tensile strength 120590 under the high weathering andexcavation unloading The strength of rock discontinuitiesnear the upper slope surface is lower than deep inside partIf the resistance supplied by the tensile strength of rock massis constant the most unstable sliding surface is the beddingplane However the tensile strength and the length of thepotential cracking surface are constantly changing which

makes the stability of rock slope more complicated If thisresistance force is lower than the sliding force the safetyfactor along the bedding plane is consistently decreasingWith the tensile strength and the length of potential crackingsurface increasing the resistance force may make the safetyfactor decrease first and then increase

In the modified limiting equilibrium equations the spa-tial variations of the strength of rock discontinuities androck mass are introduced in this method If we set resistancesupplied by the tensile strength of rock mass to zero thiscalculation model will degenerate into an idealized modelof planar failure Note that this strength parameters modelof the rock mass and rock discontinuities sometimes cannot

8 Mathematical Problems in Engineering

263m312m 945m

464m

113m200m

72∘

Bedding plane ABedding plane C

Bedding plane B

Figure 8 Analysis model of the case study

Tensile strength data pointFitting curve

Depth h (m)545435325215105

0

1

2

3

4

5

Tens

ile st

reng

th 120590

(MPa

)

120590(h) =45

1 + eminusc120590 (hminusb120590)+ 03

c120590 = 418b120590 = 330

R2 = 098

Figure 9 S-curve model for the tensile strength

describe the spatial variability of strength parameters exactlyNevertheless the main idea for calculating the safety factorof stratified rock slope is still valid and needs to change theexpression of strength parameters

The sliding surface in a slopemay consist of a single planecontinuous over the full area of the surface or a complex sur-face made up of both discontinuities and fractures throughintact rock [21] Here the sliding surface is composed ofthe bedding surface and vertical potential cracking surfaceIt is different from the failure mode caused by propagatingcracks which are extended and coalesced with neighboringjoints The suggested method in this paper is appropriate foranalyzing the stability of stratified rock slope whose jointcrack is undeveloped

5 Conclusions

The plane failure is one of the rock slope failure modes infield situations and the limit equilibrium approach for theestimation of the factor of safety of the rock slope againstplane failure has been well accepted by the engineers Insedimentary rock mass the stratified rock slope usually

Friction angle data pointFitting curve

120593(x) =770

1 + eminusc120593(xminusb120593)+ 11

R2 = 087

b120593 = 010

01 02 03 04 050

Length x (m)

10

12

14

16

18

20

Fric

tion

angl

e (de

gree

)

c120593 = 144

(a)

Fitting curveCohesion data point

R2 = 094

cc = 2300bc = 010

c(x) =009

1 + eminusc119888(xminusb119888)+ 019

01 02 03 04 050

Length x (m)

018

02

022

024

026

028

03

Coh

esio

n (M

Pa)

(b)

Figure 10 S-curve model for the friction angle and cohesion of therock discontinuities

has several bedding planes whose dips and mechanicalparameters have significant effects on rock mass strengthand stability However previous limit equilibrium approacheswere mainly about the plane failure mode with single slidingsurface and failed to predict the stability of the rock slopewith various bedding planesThis paper presented amodifiedlimit equilibrium method for predicting the stability of thesedimentary rock slope with flat bedding planes In thismethod the spatial variations of shear strength parameters119888 and 120593 of bedding plane and the tensile strength 120590 of rockmass were considered and they are expressed by the S-curve model The potential tensile at the top of rock masscan also be considered in plane failure mode incorporatingvarious combinations of potential sliding surface and tensilecrack The critical sliding surface and the tensile crack weredetermined by finding the potential sliding surfacewhich had

Mathematical Problems in Engineering 9

(65 1012)

(85 1071)

(105 0973)

090

095

100

105

110

115

120

125

130

135

140

Safe

ty fa

ctor

2 4 6 80 12 14 1610Lc

Bedding plane ABedding plane B

Bedding plane C

Figure 11 Relations between factor of safety and 119871119888

theminimum factor of safetyThismethod was utilized in thestability analysis of a case study in Three Gorges ReservoirRegion China The factors of safety were different betweenthe bedding planes and they were affected by the resistancesupplied by the tensile strength of rock mass The factor ofsafety of the case study was 0973 the critical sliding surfaceappears at bedding planeC and the tension-controlled failureoccurs at 105m to the slope face On the basis of generalizedanalytical expressions for the factor of safety the examplewas provided to show the effectiveness of the modified limitequilibrium method in the stability analysis of stratified rockslopes

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

This study was funded by National Natural Science Foun-dation of China (nos 41427802 41172292 41302257 and41502300) Zhejiang Provincial Natural Science Foundation(no Q16D020005) and Zhejiang Provincial Natural ScienceFoundation (no Q13D020001)

References

[1] Y Zhao Z-Y Tong and Q Lu ldquoSlope stability analysis usingslice-wise factor of safetyrdquoMathematical Problems in Engineer-ing vol 2014 Article ID 712145 6 pages 2014

[2] SGDu Y JHu andX FHu ldquoMeasurement of joint roughnesscoefficient by using profilograph and roughness rulerrdquo Journalof Earth Science vol 20 no 5 pp 890ndash896 2009

[3] S K Shukla and M M Hossain ldquoAnalytical expression forfactor of safety of an anchored rock slope against plane failurerdquoInternational Journal of Geotechnical Engineering vol 5 no 2pp 181ndash187 2011

[4] N Sivakugan S K Shukla and B M Das Rock Mechanics AnIntroduction CRC Press New York NY USA 2013

[5] M Gencer ldquoProgressive failure in stratified and jointed rockmassrdquo Rock Mechanics and Rock Engineering vol 18 no 4 pp267ndash292 1985

[6] J-J Dong C-H Tu W-R Lee and Y-J Jheng ldquoEffectsof hydraulic conductivitystrength anisotropy on the stabilityof stratified poorly cemented rock slopesrdquo Computers andGeotechnics vol 40 pp 147ndash159 2012

[7] J Ching Z-Y Yang J-Q Shiau and C-J Chen ldquoEstimationof rock pressure during an excavationcut in sedimentary rockswith inclined bedding planesrdquo Structural Safety vol 41 pp 11ndash19 2013

[8] H Tang R Yong and M A M Ez Eldin ldquoStability analysis ofstratified rock slopes with spatially variable strength parame-ters the case of Qianjiangping landsliderdquoBulletin of EngineeringGeology and the Environment 2016

[9] X-L Yang and J-H Yin ldquoLinear Mohr-Coulomb strengthparameters from the non-linear Hoek-Brown rock massesrdquoInternational Journal of Non-LinearMechanics vol 41 no 8 pp1000ndash1005 2006

[10] P Fortsakis K Nikas V Marinos and P Marinos ldquoAnisotropicbehaviour of stratified rock masses in tunnellingrdquo EngineeringGeology vol 141-142 pp 74ndash83 2012

[11] E Hoek and J D Bray Rock Slope Engineering CRC Press 1981[12] S Sharma T K Raghuvanshi and R Anbalagan ldquoPlane

failure analysis of rock slopesrdquo Geotechnical and GeologicalEngineering vol 13 no 2 pp 105ndash111 1995

[13] E B Kroeger ldquoAnalysis of plane failures in compound slopesrdquoInternational Journal of Surface Mining Reclamation and Envi-ronment vol 14 no 3 pp 215ndash222 2000

[14] B S Jiang M F Cai and H Du ldquoAnalytical calculation onstability of slope with planar failure surfacerdquo Chinese Journal ofRock Mechanics and Engineering vol 23 no 1 pp 91ndash94 2004

[15] M Ahmadi and M Eslami ldquoA new approach to plane failure ofrock slope stability based on water flow velocity in discontinu-ities for the Latian dam reservoir landsliderdquo Journal ofMountainScience vol 8 no 2 pp 124ndash130 2011

[16] A Johari A Fazeli and A A Javadi ldquoAn investigation intoapplication of jointly distributed random variables method inreliability assessment of rock slope stabilityrdquo Computers andGeotechnics vol 47 pp 42ndash47 2013

[17] HAViles ldquoLinkingweathering and rock slope instability non-linear perspectivesrdquo Earth Surface Processes and Landforms vol38 no 1 pp 62ndash70 2013

[18] HM Tang Z X Zou C R Xiong et al ldquoAn evolutionmodel oflarge consequent bedding rockslides with particular referenceto the Jiweishan rockslide in Southwest Chinardquo EngineeringGeology vol 186 pp 17ndash27 2015

[19] Z X Zou H M Tang C R Xiong Y P Wu X Liu and SB Liao ldquoGeomechanical model of progressive failure for largeconsequent bedding rockslide and its stability analysisrdquoChineseJournal of Rock Mechanics and Engineering vol 31 no 11 pp2222ndash2230 2012 (Chinese)

[20] Z T Bieniawski and I Hawkes ldquoSuggested methods for deter-mining tensile strength of rock materialsrdquo International Journalof RockMechanics andMining Sciences vol 15 no 3 pp 99ndash1031978

[21] C W Duncan and W M Christopher Rock Slope Engineering-Civil and Mining Taylor amp Francis New York NY USA 4thedition 2005

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Mathematical Problems in Engineering

0h

Upper slope surface

Slope face

Bedding surface

Cracking surface120590

Figure 5 Relationships between tensile strength 120590 of rockmass anddistance ℎ to upper slope surface

Upper slope surface

AG

N

T

Potential cracking surface

CB

Bedding surface i

Ff

Slope face

120572

Lc

hc

La1 La2 La3 Lai

120573iLbi

Lb1

Lb3Lb2

middot middot middot

middot middotmiddot

Figure 6 Computing model of bedding rock slope

The critical sliding surface can be determined by findingthe potential sliding surface which has theminimum factor ofsafety The stability analysis was taken by performing seriesof iterations on various combinations of potential slidingsurface and tensile crack The flow chart for evaluating thestability of the stratified rock slope is shown in Figure 7

3 A Case Study by Modified LimitingEquilibrium Method

31 Introduction of the Jiaxiao Slope Jiaxiao Slope is a typicalbedding rock slope in Triassic Strata which is situated atJingmen City Hubei Province China The slope is mainlycomposed of weathered sandstone There are three beddingsurfaces namely bedding plane A bedding plane B andbedding plane C (Figure 8) The dip of bedding plane isapproximately 22∘ The upper slope surface is nearly hori-zontal and the slope height is about 12m The sandstonecrushing degree is controlled by the depth to slope surfacewhich severely influences the strength of rock mass anddiscontinuities

A geotechnical investigation was carried out in June 2013which included the drilling of six geotechnical drill holes onthe upper surface of Jiaxiao Slope Then based on the drillcores we obtained six rock samples for the tensile strengthtest and nine rock joint samples for the shear strength test

The tensile specimens are disc-shaped with a diameter 119863 =

50mm and a thickness 119905 = 25mm and the sizes of the rockjoint samples are about 80mm times 80mm Here the Braziliantest was performed to determine the indirect tensile strengthaccording to ISRM (1978) [20] The test result shows thatthe tensile strength of highly weathered rock specimens nearslope surface is 03MPa and the tensile strength of intact rockspecimen is 48MPa Thus it is obvious that 119867

120590= 03MPa

and 119860120590= 45MPa in the S-curve model (Figure 9) and the

S-curve model for the tensile strength can be written as

120590 (ℎ) =

45

1 + 119890minus119888120590(ℎminus119887120590)+ 03 (17)

Similarly based on the direct shear test for joint spec-imens the friction angle and cohesion of highly weath-ered rock discontinuities near slope surface are 1100∘ and019MPaThe friction angle and cohesion of the unweatheredrock discontinuities are 1870∘ and 028MPaAccording to theS-curve model fitting result (Figure 10) the S-curve modelsfor shear strength in (5) can be written as

120593 (119909)

=

770

1 + 119890minus1440(119909minus010)

+ 11 (119909 le

119871119894

2

)

770

1 + 119890minus1440(119871

119894minus119909minus010)

+ 11 (

119871119894

2

le 119909 le 119904119894)

119888 (119909)

=

009

1 + 119890minus2300(119909minus01)

+ 019 (119909 le

119871119894

2

)

009

1 + 119890minus2300(119871

119894minus119909minus01)

+ 019 (

119871119894

2

le 119909 le 119904119894)

(18)

The strength coefficients of bedding plane and rock massare tabulated in Table 1 Moreover the shear strength ofbedding rock joints was obtained by the direct shear test Theaverage unit weight 120574 is 28 kNm3

32 Stability Analyses on the Case Study The stability of thisbedding rock slope was calculated by the modified limitingequilibrium method presented above The distributions ofsafety factor 120578 of three bedding planes are illustrated inFigure 11 The 120578 values of each bedding plane decline firstand tend to be stable later However in the stable sectioneach bedding planersquos 120578 firstly decrease and then increase alittle bit Most of the safety factors of bedding plane B arelarger than those of bedding plane A and bedding plane CMoreover the safety factor of bedding A is larger than thesafety factor of bedding C when 119871

119888is shorter than 3m The

minimum 120578 of bedding plane A is 1012 the minimum 120578 ofbedding plane B is 1071 and theminimum 120578 of bedding planeC is 0973 Therefore the critical sliding surface is beddingplane C which has the minimum 120578 of the bedding rockslope Moreover the result shows that the potential tension-controlled failure would be most likely to occur at 105m tothe slope face

Mathematical Problems in Engineering 7

Table 1 Strength coefficients of bedding plane and rock mass (MPa)

Coefficient of strength function Bedding plane Rock massFriction angle (∘) Cohesion (MPa) Tensile strength

119867 1100 019 030119860 770 009 450119888 1440 2300 418119887 010 010 330

Start

Assume the potential slidinghappens in layer i

No

No

End

Input the analysis data① Density 120588② Geometric parameters 120572 La Lb

③ Strength function 120593(x) c(x) 120590(h)

Assume the distance betweenthe slope surface and crack

potential crack is Lc

Failure planedip 120573i

Failure planelength Li

Failure surfacelength si

WeightGi

Tensile force Ti

slip surfaceNormal force Ni on the

Resisting force Fsi

Driving force Fri

Safety factor 120578

If 120578 = 120578min

120578min

= 120578i

If 120578i = 120578min

120578 = 120578min

Yes

Figure 7 Flow chart of stability calculation for bedding rock slope

4 Discussion

The spatial variationmodel of the rockmass and rock discon-tinuities discussed in this paper is based on S-curve modelIt is appropriate for indicating the spatial variability of rockdiscontinuities shear strength parameters 119888 and 120593 and therock mass tensile strength 120590 under the high weathering andexcavation unloading The strength of rock discontinuitiesnear the upper slope surface is lower than deep inside partIf the resistance supplied by the tensile strength of rock massis constant the most unstable sliding surface is the beddingplane However the tensile strength and the length of thepotential cracking surface are constantly changing which

makes the stability of rock slope more complicated If thisresistance force is lower than the sliding force the safetyfactor along the bedding plane is consistently decreasingWith the tensile strength and the length of potential crackingsurface increasing the resistance force may make the safetyfactor decrease first and then increase

In the modified limiting equilibrium equations the spa-tial variations of the strength of rock discontinuities androck mass are introduced in this method If we set resistancesupplied by the tensile strength of rock mass to zero thiscalculation model will degenerate into an idealized modelof planar failure Note that this strength parameters modelof the rock mass and rock discontinuities sometimes cannot

8 Mathematical Problems in Engineering

263m312m 945m

464m

113m200m

72∘

Bedding plane ABedding plane C

Bedding plane B

Figure 8 Analysis model of the case study

Tensile strength data pointFitting curve

Depth h (m)545435325215105

0

1

2

3

4

5

Tens

ile st

reng

th 120590

(MPa

)

120590(h) =45

1 + eminusc120590 (hminusb120590)+ 03

c120590 = 418b120590 = 330

R2 = 098

Figure 9 S-curve model for the tensile strength

describe the spatial variability of strength parameters exactlyNevertheless the main idea for calculating the safety factorof stratified rock slope is still valid and needs to change theexpression of strength parameters

The sliding surface in a slopemay consist of a single planecontinuous over the full area of the surface or a complex sur-face made up of both discontinuities and fractures throughintact rock [21] Here the sliding surface is composed ofthe bedding surface and vertical potential cracking surfaceIt is different from the failure mode caused by propagatingcracks which are extended and coalesced with neighboringjoints The suggested method in this paper is appropriate foranalyzing the stability of stratified rock slope whose jointcrack is undeveloped

5 Conclusions

The plane failure is one of the rock slope failure modes infield situations and the limit equilibrium approach for theestimation of the factor of safety of the rock slope againstplane failure has been well accepted by the engineers Insedimentary rock mass the stratified rock slope usually

Friction angle data pointFitting curve

120593(x) =770

1 + eminusc120593(xminusb120593)+ 11

R2 = 087

b120593 = 010

01 02 03 04 050

Length x (m)

10

12

14

16

18

20

Fric

tion

angl

e (de

gree

)

c120593 = 144

(a)

Fitting curveCohesion data point

R2 = 094

cc = 2300bc = 010

c(x) =009

1 + eminusc119888(xminusb119888)+ 019

01 02 03 04 050

Length x (m)

018

02

022

024

026

028

03

Coh

esio

n (M

Pa)

(b)

Figure 10 S-curve model for the friction angle and cohesion of therock discontinuities

has several bedding planes whose dips and mechanicalparameters have significant effects on rock mass strengthand stability However previous limit equilibrium approacheswere mainly about the plane failure mode with single slidingsurface and failed to predict the stability of the rock slopewith various bedding planesThis paper presented amodifiedlimit equilibrium method for predicting the stability of thesedimentary rock slope with flat bedding planes In thismethod the spatial variations of shear strength parameters119888 and 120593 of bedding plane and the tensile strength 120590 of rockmass were considered and they are expressed by the S-curve model The potential tensile at the top of rock masscan also be considered in plane failure mode incorporatingvarious combinations of potential sliding surface and tensilecrack The critical sliding surface and the tensile crack weredetermined by finding the potential sliding surfacewhich had

Mathematical Problems in Engineering 9

(65 1012)

(85 1071)

(105 0973)

090

095

100

105

110

115

120

125

130

135

140

Safe

ty fa

ctor

2 4 6 80 12 14 1610Lc

Bedding plane ABedding plane B

Bedding plane C

Figure 11 Relations between factor of safety and 119871119888

theminimum factor of safetyThismethod was utilized in thestability analysis of a case study in Three Gorges ReservoirRegion China The factors of safety were different betweenthe bedding planes and they were affected by the resistancesupplied by the tensile strength of rock mass The factor ofsafety of the case study was 0973 the critical sliding surfaceappears at bedding planeC and the tension-controlled failureoccurs at 105m to the slope face On the basis of generalizedanalytical expressions for the factor of safety the examplewas provided to show the effectiveness of the modified limitequilibrium method in the stability analysis of stratified rockslopes

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

This study was funded by National Natural Science Foun-dation of China (nos 41427802 41172292 41302257 and41502300) Zhejiang Provincial Natural Science Foundation(no Q16D020005) and Zhejiang Provincial Natural ScienceFoundation (no Q13D020001)

References

[1] Y Zhao Z-Y Tong and Q Lu ldquoSlope stability analysis usingslice-wise factor of safetyrdquoMathematical Problems in Engineer-ing vol 2014 Article ID 712145 6 pages 2014

[2] SGDu Y JHu andX FHu ldquoMeasurement of joint roughnesscoefficient by using profilograph and roughness rulerrdquo Journalof Earth Science vol 20 no 5 pp 890ndash896 2009

[3] S K Shukla and M M Hossain ldquoAnalytical expression forfactor of safety of an anchored rock slope against plane failurerdquoInternational Journal of Geotechnical Engineering vol 5 no 2pp 181ndash187 2011

[4] N Sivakugan S K Shukla and B M Das Rock Mechanics AnIntroduction CRC Press New York NY USA 2013

[5] M Gencer ldquoProgressive failure in stratified and jointed rockmassrdquo Rock Mechanics and Rock Engineering vol 18 no 4 pp267ndash292 1985

[6] J-J Dong C-H Tu W-R Lee and Y-J Jheng ldquoEffectsof hydraulic conductivitystrength anisotropy on the stabilityof stratified poorly cemented rock slopesrdquo Computers andGeotechnics vol 40 pp 147ndash159 2012

[7] J Ching Z-Y Yang J-Q Shiau and C-J Chen ldquoEstimationof rock pressure during an excavationcut in sedimentary rockswith inclined bedding planesrdquo Structural Safety vol 41 pp 11ndash19 2013

[8] H Tang R Yong and M A M Ez Eldin ldquoStability analysis ofstratified rock slopes with spatially variable strength parame-ters the case of Qianjiangping landsliderdquoBulletin of EngineeringGeology and the Environment 2016

[9] X-L Yang and J-H Yin ldquoLinear Mohr-Coulomb strengthparameters from the non-linear Hoek-Brown rock massesrdquoInternational Journal of Non-LinearMechanics vol 41 no 8 pp1000ndash1005 2006

[10] P Fortsakis K Nikas V Marinos and P Marinos ldquoAnisotropicbehaviour of stratified rock masses in tunnellingrdquo EngineeringGeology vol 141-142 pp 74ndash83 2012

[11] E Hoek and J D Bray Rock Slope Engineering CRC Press 1981[12] S Sharma T K Raghuvanshi and R Anbalagan ldquoPlane

failure analysis of rock slopesrdquo Geotechnical and GeologicalEngineering vol 13 no 2 pp 105ndash111 1995

[13] E B Kroeger ldquoAnalysis of plane failures in compound slopesrdquoInternational Journal of Surface Mining Reclamation and Envi-ronment vol 14 no 3 pp 215ndash222 2000

[14] B S Jiang M F Cai and H Du ldquoAnalytical calculation onstability of slope with planar failure surfacerdquo Chinese Journal ofRock Mechanics and Engineering vol 23 no 1 pp 91ndash94 2004

[15] M Ahmadi and M Eslami ldquoA new approach to plane failure ofrock slope stability based on water flow velocity in discontinu-ities for the Latian dam reservoir landsliderdquo Journal ofMountainScience vol 8 no 2 pp 124ndash130 2011

[16] A Johari A Fazeli and A A Javadi ldquoAn investigation intoapplication of jointly distributed random variables method inreliability assessment of rock slope stabilityrdquo Computers andGeotechnics vol 47 pp 42ndash47 2013

[17] HAViles ldquoLinkingweathering and rock slope instability non-linear perspectivesrdquo Earth Surface Processes and Landforms vol38 no 1 pp 62ndash70 2013

[18] HM Tang Z X Zou C R Xiong et al ldquoAn evolutionmodel oflarge consequent bedding rockslides with particular referenceto the Jiweishan rockslide in Southwest Chinardquo EngineeringGeology vol 186 pp 17ndash27 2015

[19] Z X Zou H M Tang C R Xiong Y P Wu X Liu and SB Liao ldquoGeomechanical model of progressive failure for largeconsequent bedding rockslide and its stability analysisrdquoChineseJournal of Rock Mechanics and Engineering vol 31 no 11 pp2222ndash2230 2012 (Chinese)

[20] Z T Bieniawski and I Hawkes ldquoSuggested methods for deter-mining tensile strength of rock materialsrdquo International Journalof RockMechanics andMining Sciences vol 15 no 3 pp 99ndash1031978

[21] C W Duncan and W M Christopher Rock Slope Engineering-Civil and Mining Taylor amp Francis New York NY USA 4thedition 2005

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 7

Table 1 Strength coefficients of bedding plane and rock mass (MPa)

Coefficient of strength function Bedding plane Rock massFriction angle (∘) Cohesion (MPa) Tensile strength

119867 1100 019 030119860 770 009 450119888 1440 2300 418119887 010 010 330

Start

Assume the potential slidinghappens in layer i

No

No

End

Input the analysis data① Density 120588② Geometric parameters 120572 La Lb

③ Strength function 120593(x) c(x) 120590(h)

Assume the distance betweenthe slope surface and crack

potential crack is Lc

Failure planedip 120573i

Failure planelength Li

Failure surfacelength si

WeightGi

Tensile force Ti

slip surfaceNormal force Ni on the

Resisting force Fsi

Driving force Fri

Safety factor 120578

If 120578 = 120578min

120578min

= 120578i

If 120578i = 120578min

120578 = 120578min

Yes

Figure 7 Flow chart of stability calculation for bedding rock slope

4 Discussion

The spatial variationmodel of the rockmass and rock discon-tinuities discussed in this paper is based on S-curve modelIt is appropriate for indicating the spatial variability of rockdiscontinuities shear strength parameters 119888 and 120593 and therock mass tensile strength 120590 under the high weathering andexcavation unloading The strength of rock discontinuitiesnear the upper slope surface is lower than deep inside partIf the resistance supplied by the tensile strength of rock massis constant the most unstable sliding surface is the beddingplane However the tensile strength and the length of thepotential cracking surface are constantly changing which

makes the stability of rock slope more complicated If thisresistance force is lower than the sliding force the safetyfactor along the bedding plane is consistently decreasingWith the tensile strength and the length of potential crackingsurface increasing the resistance force may make the safetyfactor decrease first and then increase

In the modified limiting equilibrium equations the spa-tial variations of the strength of rock discontinuities androck mass are introduced in this method If we set resistancesupplied by the tensile strength of rock mass to zero thiscalculation model will degenerate into an idealized modelof planar failure Note that this strength parameters modelof the rock mass and rock discontinuities sometimes cannot

8 Mathematical Problems in Engineering

263m312m 945m

464m

113m200m

72∘

Bedding plane ABedding plane C

Bedding plane B

Figure 8 Analysis model of the case study

Tensile strength data pointFitting curve

Depth h (m)545435325215105

0

1

2

3

4

5

Tens

ile st

reng

th 120590

(MPa

)

120590(h) =45

1 + eminusc120590 (hminusb120590)+ 03

c120590 = 418b120590 = 330

R2 = 098

Figure 9 S-curve model for the tensile strength

describe the spatial variability of strength parameters exactlyNevertheless the main idea for calculating the safety factorof stratified rock slope is still valid and needs to change theexpression of strength parameters

The sliding surface in a slopemay consist of a single planecontinuous over the full area of the surface or a complex sur-face made up of both discontinuities and fractures throughintact rock [21] Here the sliding surface is composed ofthe bedding surface and vertical potential cracking surfaceIt is different from the failure mode caused by propagatingcracks which are extended and coalesced with neighboringjoints The suggested method in this paper is appropriate foranalyzing the stability of stratified rock slope whose jointcrack is undeveloped

5 Conclusions

The plane failure is one of the rock slope failure modes infield situations and the limit equilibrium approach for theestimation of the factor of safety of the rock slope againstplane failure has been well accepted by the engineers Insedimentary rock mass the stratified rock slope usually

Friction angle data pointFitting curve

120593(x) =770

1 + eminusc120593(xminusb120593)+ 11

R2 = 087

b120593 = 010

01 02 03 04 050

Length x (m)

10

12

14

16

18

20

Fric

tion

angl

e (de

gree

)

c120593 = 144

(a)

Fitting curveCohesion data point

R2 = 094

cc = 2300bc = 010

c(x) =009

1 + eminusc119888(xminusb119888)+ 019

01 02 03 04 050

Length x (m)

018

02

022

024

026

028

03

Coh

esio

n (M

Pa)

(b)

Figure 10 S-curve model for the friction angle and cohesion of therock discontinuities

has several bedding planes whose dips and mechanicalparameters have significant effects on rock mass strengthand stability However previous limit equilibrium approacheswere mainly about the plane failure mode with single slidingsurface and failed to predict the stability of the rock slopewith various bedding planesThis paper presented amodifiedlimit equilibrium method for predicting the stability of thesedimentary rock slope with flat bedding planes In thismethod the spatial variations of shear strength parameters119888 and 120593 of bedding plane and the tensile strength 120590 of rockmass were considered and they are expressed by the S-curve model The potential tensile at the top of rock masscan also be considered in plane failure mode incorporatingvarious combinations of potential sliding surface and tensilecrack The critical sliding surface and the tensile crack weredetermined by finding the potential sliding surfacewhich had

Mathematical Problems in Engineering 9

(65 1012)

(85 1071)

(105 0973)

090

095

100

105

110

115

120

125

130

135

140

Safe

ty fa

ctor

2 4 6 80 12 14 1610Lc

Bedding plane ABedding plane B

Bedding plane C

Figure 11 Relations between factor of safety and 119871119888

theminimum factor of safetyThismethod was utilized in thestability analysis of a case study in Three Gorges ReservoirRegion China The factors of safety were different betweenthe bedding planes and they were affected by the resistancesupplied by the tensile strength of rock mass The factor ofsafety of the case study was 0973 the critical sliding surfaceappears at bedding planeC and the tension-controlled failureoccurs at 105m to the slope face On the basis of generalizedanalytical expressions for the factor of safety the examplewas provided to show the effectiveness of the modified limitequilibrium method in the stability analysis of stratified rockslopes

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

This study was funded by National Natural Science Foun-dation of China (nos 41427802 41172292 41302257 and41502300) Zhejiang Provincial Natural Science Foundation(no Q16D020005) and Zhejiang Provincial Natural ScienceFoundation (no Q13D020001)

References

[1] Y Zhao Z-Y Tong and Q Lu ldquoSlope stability analysis usingslice-wise factor of safetyrdquoMathematical Problems in Engineer-ing vol 2014 Article ID 712145 6 pages 2014

[2] SGDu Y JHu andX FHu ldquoMeasurement of joint roughnesscoefficient by using profilograph and roughness rulerrdquo Journalof Earth Science vol 20 no 5 pp 890ndash896 2009

[3] S K Shukla and M M Hossain ldquoAnalytical expression forfactor of safety of an anchored rock slope against plane failurerdquoInternational Journal of Geotechnical Engineering vol 5 no 2pp 181ndash187 2011

[4] N Sivakugan S K Shukla and B M Das Rock Mechanics AnIntroduction CRC Press New York NY USA 2013

[5] M Gencer ldquoProgressive failure in stratified and jointed rockmassrdquo Rock Mechanics and Rock Engineering vol 18 no 4 pp267ndash292 1985

[6] J-J Dong C-H Tu W-R Lee and Y-J Jheng ldquoEffectsof hydraulic conductivitystrength anisotropy on the stabilityof stratified poorly cemented rock slopesrdquo Computers andGeotechnics vol 40 pp 147ndash159 2012

[7] J Ching Z-Y Yang J-Q Shiau and C-J Chen ldquoEstimationof rock pressure during an excavationcut in sedimentary rockswith inclined bedding planesrdquo Structural Safety vol 41 pp 11ndash19 2013

[8] H Tang R Yong and M A M Ez Eldin ldquoStability analysis ofstratified rock slopes with spatially variable strength parame-ters the case of Qianjiangping landsliderdquoBulletin of EngineeringGeology and the Environment 2016

[9] X-L Yang and J-H Yin ldquoLinear Mohr-Coulomb strengthparameters from the non-linear Hoek-Brown rock massesrdquoInternational Journal of Non-LinearMechanics vol 41 no 8 pp1000ndash1005 2006

[10] P Fortsakis K Nikas V Marinos and P Marinos ldquoAnisotropicbehaviour of stratified rock masses in tunnellingrdquo EngineeringGeology vol 141-142 pp 74ndash83 2012

[11] E Hoek and J D Bray Rock Slope Engineering CRC Press 1981[12] S Sharma T K Raghuvanshi and R Anbalagan ldquoPlane

failure analysis of rock slopesrdquo Geotechnical and GeologicalEngineering vol 13 no 2 pp 105ndash111 1995

[13] E B Kroeger ldquoAnalysis of plane failures in compound slopesrdquoInternational Journal of Surface Mining Reclamation and Envi-ronment vol 14 no 3 pp 215ndash222 2000

[14] B S Jiang M F Cai and H Du ldquoAnalytical calculation onstability of slope with planar failure surfacerdquo Chinese Journal ofRock Mechanics and Engineering vol 23 no 1 pp 91ndash94 2004

[15] M Ahmadi and M Eslami ldquoA new approach to plane failure ofrock slope stability based on water flow velocity in discontinu-ities for the Latian dam reservoir landsliderdquo Journal ofMountainScience vol 8 no 2 pp 124ndash130 2011

[16] A Johari A Fazeli and A A Javadi ldquoAn investigation intoapplication of jointly distributed random variables method inreliability assessment of rock slope stabilityrdquo Computers andGeotechnics vol 47 pp 42ndash47 2013

[17] HAViles ldquoLinkingweathering and rock slope instability non-linear perspectivesrdquo Earth Surface Processes and Landforms vol38 no 1 pp 62ndash70 2013

[18] HM Tang Z X Zou C R Xiong et al ldquoAn evolutionmodel oflarge consequent bedding rockslides with particular referenceto the Jiweishan rockslide in Southwest Chinardquo EngineeringGeology vol 186 pp 17ndash27 2015

[19] Z X Zou H M Tang C R Xiong Y P Wu X Liu and SB Liao ldquoGeomechanical model of progressive failure for largeconsequent bedding rockslide and its stability analysisrdquoChineseJournal of Rock Mechanics and Engineering vol 31 no 11 pp2222ndash2230 2012 (Chinese)

[20] Z T Bieniawski and I Hawkes ldquoSuggested methods for deter-mining tensile strength of rock materialsrdquo International Journalof RockMechanics andMining Sciences vol 15 no 3 pp 99ndash1031978

[21] C W Duncan and W M Christopher Rock Slope Engineering-Civil and Mining Taylor amp Francis New York NY USA 4thedition 2005

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Mathematical Problems in Engineering

263m312m 945m

464m

113m200m

72∘

Bedding plane ABedding plane C

Bedding plane B

Figure 8 Analysis model of the case study

Tensile strength data pointFitting curve

Depth h (m)545435325215105

0

1

2

3

4

5

Tens

ile st

reng

th 120590

(MPa

)

120590(h) =45

1 + eminusc120590 (hminusb120590)+ 03

c120590 = 418b120590 = 330

R2 = 098

Figure 9 S-curve model for the tensile strength

describe the spatial variability of strength parameters exactlyNevertheless the main idea for calculating the safety factorof stratified rock slope is still valid and needs to change theexpression of strength parameters

The sliding surface in a slopemay consist of a single planecontinuous over the full area of the surface or a complex sur-face made up of both discontinuities and fractures throughintact rock [21] Here the sliding surface is composed ofthe bedding surface and vertical potential cracking surfaceIt is different from the failure mode caused by propagatingcracks which are extended and coalesced with neighboringjoints The suggested method in this paper is appropriate foranalyzing the stability of stratified rock slope whose jointcrack is undeveloped

5 Conclusions

The plane failure is one of the rock slope failure modes infield situations and the limit equilibrium approach for theestimation of the factor of safety of the rock slope againstplane failure has been well accepted by the engineers Insedimentary rock mass the stratified rock slope usually

Friction angle data pointFitting curve

120593(x) =770

1 + eminusc120593(xminusb120593)+ 11

R2 = 087

b120593 = 010

01 02 03 04 050

Length x (m)

10

12

14

16

18

20

Fric

tion

angl

e (de

gree

)

c120593 = 144

(a)

Fitting curveCohesion data point

R2 = 094

cc = 2300bc = 010

c(x) =009

1 + eminusc119888(xminusb119888)+ 019

01 02 03 04 050

Length x (m)

018

02

022

024

026

028

03

Coh

esio

n (M

Pa)

(b)

Figure 10 S-curve model for the friction angle and cohesion of therock discontinuities

has several bedding planes whose dips and mechanicalparameters have significant effects on rock mass strengthand stability However previous limit equilibrium approacheswere mainly about the plane failure mode with single slidingsurface and failed to predict the stability of the rock slopewith various bedding planesThis paper presented amodifiedlimit equilibrium method for predicting the stability of thesedimentary rock slope with flat bedding planes In thismethod the spatial variations of shear strength parameters119888 and 120593 of bedding plane and the tensile strength 120590 of rockmass were considered and they are expressed by the S-curve model The potential tensile at the top of rock masscan also be considered in plane failure mode incorporatingvarious combinations of potential sliding surface and tensilecrack The critical sliding surface and the tensile crack weredetermined by finding the potential sliding surfacewhich had

Mathematical Problems in Engineering 9

(65 1012)

(85 1071)

(105 0973)

090

095

100

105

110

115

120

125

130

135

140

Safe

ty fa

ctor

2 4 6 80 12 14 1610Lc

Bedding plane ABedding plane B

Bedding plane C

Figure 11 Relations between factor of safety and 119871119888

theminimum factor of safetyThismethod was utilized in thestability analysis of a case study in Three Gorges ReservoirRegion China The factors of safety were different betweenthe bedding planes and they were affected by the resistancesupplied by the tensile strength of rock mass The factor ofsafety of the case study was 0973 the critical sliding surfaceappears at bedding planeC and the tension-controlled failureoccurs at 105m to the slope face On the basis of generalizedanalytical expressions for the factor of safety the examplewas provided to show the effectiveness of the modified limitequilibrium method in the stability analysis of stratified rockslopes

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

This study was funded by National Natural Science Foun-dation of China (nos 41427802 41172292 41302257 and41502300) Zhejiang Provincial Natural Science Foundation(no Q16D020005) and Zhejiang Provincial Natural ScienceFoundation (no Q13D020001)

References

[1] Y Zhao Z-Y Tong and Q Lu ldquoSlope stability analysis usingslice-wise factor of safetyrdquoMathematical Problems in Engineer-ing vol 2014 Article ID 712145 6 pages 2014

[2] SGDu Y JHu andX FHu ldquoMeasurement of joint roughnesscoefficient by using profilograph and roughness rulerrdquo Journalof Earth Science vol 20 no 5 pp 890ndash896 2009

[3] S K Shukla and M M Hossain ldquoAnalytical expression forfactor of safety of an anchored rock slope against plane failurerdquoInternational Journal of Geotechnical Engineering vol 5 no 2pp 181ndash187 2011

[4] N Sivakugan S K Shukla and B M Das Rock Mechanics AnIntroduction CRC Press New York NY USA 2013

[5] M Gencer ldquoProgressive failure in stratified and jointed rockmassrdquo Rock Mechanics and Rock Engineering vol 18 no 4 pp267ndash292 1985

[6] J-J Dong C-H Tu W-R Lee and Y-J Jheng ldquoEffectsof hydraulic conductivitystrength anisotropy on the stabilityof stratified poorly cemented rock slopesrdquo Computers andGeotechnics vol 40 pp 147ndash159 2012

[7] J Ching Z-Y Yang J-Q Shiau and C-J Chen ldquoEstimationof rock pressure during an excavationcut in sedimentary rockswith inclined bedding planesrdquo Structural Safety vol 41 pp 11ndash19 2013

[8] H Tang R Yong and M A M Ez Eldin ldquoStability analysis ofstratified rock slopes with spatially variable strength parame-ters the case of Qianjiangping landsliderdquoBulletin of EngineeringGeology and the Environment 2016

[9] X-L Yang and J-H Yin ldquoLinear Mohr-Coulomb strengthparameters from the non-linear Hoek-Brown rock massesrdquoInternational Journal of Non-LinearMechanics vol 41 no 8 pp1000ndash1005 2006

[10] P Fortsakis K Nikas V Marinos and P Marinos ldquoAnisotropicbehaviour of stratified rock masses in tunnellingrdquo EngineeringGeology vol 141-142 pp 74ndash83 2012

[11] E Hoek and J D Bray Rock Slope Engineering CRC Press 1981[12] S Sharma T K Raghuvanshi and R Anbalagan ldquoPlane

failure analysis of rock slopesrdquo Geotechnical and GeologicalEngineering vol 13 no 2 pp 105ndash111 1995

[13] E B Kroeger ldquoAnalysis of plane failures in compound slopesrdquoInternational Journal of Surface Mining Reclamation and Envi-ronment vol 14 no 3 pp 215ndash222 2000

[14] B S Jiang M F Cai and H Du ldquoAnalytical calculation onstability of slope with planar failure surfacerdquo Chinese Journal ofRock Mechanics and Engineering vol 23 no 1 pp 91ndash94 2004

[15] M Ahmadi and M Eslami ldquoA new approach to plane failure ofrock slope stability based on water flow velocity in discontinu-ities for the Latian dam reservoir landsliderdquo Journal ofMountainScience vol 8 no 2 pp 124ndash130 2011

[16] A Johari A Fazeli and A A Javadi ldquoAn investigation intoapplication of jointly distributed random variables method inreliability assessment of rock slope stabilityrdquo Computers andGeotechnics vol 47 pp 42ndash47 2013

[17] HAViles ldquoLinkingweathering and rock slope instability non-linear perspectivesrdquo Earth Surface Processes and Landforms vol38 no 1 pp 62ndash70 2013

[18] HM Tang Z X Zou C R Xiong et al ldquoAn evolutionmodel oflarge consequent bedding rockslides with particular referenceto the Jiweishan rockslide in Southwest Chinardquo EngineeringGeology vol 186 pp 17ndash27 2015

[19] Z X Zou H M Tang C R Xiong Y P Wu X Liu and SB Liao ldquoGeomechanical model of progressive failure for largeconsequent bedding rockslide and its stability analysisrdquoChineseJournal of Rock Mechanics and Engineering vol 31 no 11 pp2222ndash2230 2012 (Chinese)

[20] Z T Bieniawski and I Hawkes ldquoSuggested methods for deter-mining tensile strength of rock materialsrdquo International Journalof RockMechanics andMining Sciences vol 15 no 3 pp 99ndash1031978

[21] C W Duncan and W M Christopher Rock Slope Engineering-Civil and Mining Taylor amp Francis New York NY USA 4thedition 2005

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 9

(65 1012)

(85 1071)

(105 0973)

090

095

100

105

110

115

120

125

130

135

140

Safe

ty fa

ctor

2 4 6 80 12 14 1610Lc

Bedding plane ABedding plane B

Bedding plane C

Figure 11 Relations between factor of safety and 119871119888

theminimum factor of safetyThismethod was utilized in thestability analysis of a case study in Three Gorges ReservoirRegion China The factors of safety were different betweenthe bedding planes and they were affected by the resistancesupplied by the tensile strength of rock mass The factor ofsafety of the case study was 0973 the critical sliding surfaceappears at bedding planeC and the tension-controlled failureoccurs at 105m to the slope face On the basis of generalizedanalytical expressions for the factor of safety the examplewas provided to show the effectiveness of the modified limitequilibrium method in the stability analysis of stratified rockslopes

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

This study was funded by National Natural Science Foun-dation of China (nos 41427802 41172292 41302257 and41502300) Zhejiang Provincial Natural Science Foundation(no Q16D020005) and Zhejiang Provincial Natural ScienceFoundation (no Q13D020001)

References

[1] Y Zhao Z-Y Tong and Q Lu ldquoSlope stability analysis usingslice-wise factor of safetyrdquoMathematical Problems in Engineer-ing vol 2014 Article ID 712145 6 pages 2014

[2] SGDu Y JHu andX FHu ldquoMeasurement of joint roughnesscoefficient by using profilograph and roughness rulerrdquo Journalof Earth Science vol 20 no 5 pp 890ndash896 2009

[3] S K Shukla and M M Hossain ldquoAnalytical expression forfactor of safety of an anchored rock slope against plane failurerdquoInternational Journal of Geotechnical Engineering vol 5 no 2pp 181ndash187 2011

[4] N Sivakugan S K Shukla and B M Das Rock Mechanics AnIntroduction CRC Press New York NY USA 2013

[5] M Gencer ldquoProgressive failure in stratified and jointed rockmassrdquo Rock Mechanics and Rock Engineering vol 18 no 4 pp267ndash292 1985

[6] J-J Dong C-H Tu W-R Lee and Y-J Jheng ldquoEffectsof hydraulic conductivitystrength anisotropy on the stabilityof stratified poorly cemented rock slopesrdquo Computers andGeotechnics vol 40 pp 147ndash159 2012

[7] J Ching Z-Y Yang J-Q Shiau and C-J Chen ldquoEstimationof rock pressure during an excavationcut in sedimentary rockswith inclined bedding planesrdquo Structural Safety vol 41 pp 11ndash19 2013

[8] H Tang R Yong and M A M Ez Eldin ldquoStability analysis ofstratified rock slopes with spatially variable strength parame-ters the case of Qianjiangping landsliderdquoBulletin of EngineeringGeology and the Environment 2016

[9] X-L Yang and J-H Yin ldquoLinear Mohr-Coulomb strengthparameters from the non-linear Hoek-Brown rock massesrdquoInternational Journal of Non-LinearMechanics vol 41 no 8 pp1000ndash1005 2006

[10] P Fortsakis K Nikas V Marinos and P Marinos ldquoAnisotropicbehaviour of stratified rock masses in tunnellingrdquo EngineeringGeology vol 141-142 pp 74ndash83 2012

[11] E Hoek and J D Bray Rock Slope Engineering CRC Press 1981[12] S Sharma T K Raghuvanshi and R Anbalagan ldquoPlane

failure analysis of rock slopesrdquo Geotechnical and GeologicalEngineering vol 13 no 2 pp 105ndash111 1995

[13] E B Kroeger ldquoAnalysis of plane failures in compound slopesrdquoInternational Journal of Surface Mining Reclamation and Envi-ronment vol 14 no 3 pp 215ndash222 2000

[14] B S Jiang M F Cai and H Du ldquoAnalytical calculation onstability of slope with planar failure surfacerdquo Chinese Journal ofRock Mechanics and Engineering vol 23 no 1 pp 91ndash94 2004

[15] M Ahmadi and M Eslami ldquoA new approach to plane failure ofrock slope stability based on water flow velocity in discontinu-ities for the Latian dam reservoir landsliderdquo Journal ofMountainScience vol 8 no 2 pp 124ndash130 2011

[16] A Johari A Fazeli and A A Javadi ldquoAn investigation intoapplication of jointly distributed random variables method inreliability assessment of rock slope stabilityrdquo Computers andGeotechnics vol 47 pp 42ndash47 2013

[17] HAViles ldquoLinkingweathering and rock slope instability non-linear perspectivesrdquo Earth Surface Processes and Landforms vol38 no 1 pp 62ndash70 2013

[18] HM Tang Z X Zou C R Xiong et al ldquoAn evolutionmodel oflarge consequent bedding rockslides with particular referenceto the Jiweishan rockslide in Southwest Chinardquo EngineeringGeology vol 186 pp 17ndash27 2015

[19] Z X Zou H M Tang C R Xiong Y P Wu X Liu and SB Liao ldquoGeomechanical model of progressive failure for largeconsequent bedding rockslide and its stability analysisrdquoChineseJournal of Rock Mechanics and Engineering vol 31 no 11 pp2222ndash2230 2012 (Chinese)

[20] Z T Bieniawski and I Hawkes ldquoSuggested methods for deter-mining tensile strength of rock materialsrdquo International Journalof RockMechanics andMining Sciences vol 15 no 3 pp 99ndash1031978

[21] C W Duncan and W M Christopher Rock Slope Engineering-Civil and Mining Taylor amp Francis New York NY USA 4thedition 2005

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of