research article application of probabilistic method to ... · instability probability of gravity...

Research ArticleApplication of Probabilistic Method to Stability Analysis ofGravity Dam Foundation over Multiple Sliding Planes

Gang Wang and Zhenyue Ma

School of Hydraulic Engineering Dalian University of Technology No 2 Linggong Road Dalian 116024 China

Correspondence should be addressed to Gang Wang ydwanggang163com

Received 9 March 2016 Accepted 5 June 2016

Academic Editor Egidijus R Vaidogas

Copyright copy 2016 G Wang and Z MaThis is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The current challenge to the engineering profession is to carry out probabilistic methods in practice The design point methodin generalized random space (DPG method) associated with the method of divided difference can be utilized to deal with thecomplex problem of probability calculation of implicit performance function with nonnormal and correlated variables For apractical concrete gravity dam the suggested method is performed to calculate the instability probability of the dam foundationover multiple sliding places The general conclusions drawn in the paper are identical to those in other research and the methodis proved to be feasible accurate and efficient As the same analysis principle the method can also be used in other similar fieldssuch as in fields of slopes earth-rock dams levees and embankments

1 Introduction

It is a trend to use the probabilistic method for evaluation ofthe risk of failure in almost all engineering fields includingsome in geotechnical or structural engineering [1ndash3] Thecurrent challenge to the geotechnical and structural engi-neering profession is to carry out probabilistic methods inpractice [4] Many researchers have focused on the researchtopics of reliability-based design and risk analysis and madeprogress in resolving the problem about the instability failureof slopes dams levees embankments and other geotechnicalor structural engineering fields in recent years [1 4 5]

TheMonteCarlomethod (MC) is usually used to estimatethe reliability index120573 However thismethod is rarely adopteddue to its huge calculation time [5 6] Besides MC manyother methods have been proposed for reliability analysissuch as the first-order reliability method (FORM) [1 7]second-order reliability method (SORM) [8 9] and someimproved methods In order to obtain 120573 the partial deriva-tives of performance function 119892(X) are needed in thesemethods But in geotechnical or structural engineering119892(X)is usually implicit and its partial derivatives are complex ordifficult to be derived from implicit to explicit Thereforethese conventional reliability methods only can be used toanalyze small structures

Always the response surfacemethods (RSMs) are utilizedto obtain the solution of 120573 for reliability problem withimplicit 119892(X) for complex structures Wong applied RSM toevaluate the reliability of a homogeneous slope [10] Mooreand Sa constructed confidence intervals about the differencein mean responses at the stationary point and alternatepoints based on the proposed delta method and F-projectionmethod and compared coverage probabilities and intervalwidths [11] Zheng and Das proposed an improved responsesurface method and applied that to the reliability analysis ofa stiffened plate structure [12] Guan and Melchers evaluatedthe effect of response surface parameter variation on struc-tural reliability [13] Gupta and Manohar used the responsesurface method to study the extremes of Von Mises stress innonlinear structures under Gaussian excitations [14] Wonget al proposed an adaptive design approach to overcomethe problem which was that the solution of the reliabilityanalysis initially diverged when the loading was applied insequence in the nonlinear finite element (NLFE) analysisand made several suggestions to improve the robustnessof RSM [15] Xu and Low used RSM to approximate theperformance function of slope stability in slope reliabilityanalysis in which the response surface is taken as a bridgebetween standalone numerical packages and spreadsheet-based reliability analysis [16] Cheng et al presented a new

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2016 Article ID 4264627 12 pageshttpdxdoiorg10115520164264627

2 Mathematical Problems in Engineering

artificial neural network (ANN) based response surfacemethod in conjunction with the uniform design methodfor predicting failure probability of structures [17] Gavinand Yau described the use of higher order polynomials inorder to approximate the true limit state more accurately incontrast to recently proposed algorithms which focused onthe positions of sample points to improve the accuracy of thequadratic stochastic response surface method (SRSM) [18]Zou et al presented an accurate and efficient MC for limitstate-based reliability analysis at both component and systemlevels using a response surface approximation of the failureindicator function [19] Nguyen et al proposed an adaptiveconstruction of the numerical design in which the responsesurface was fitted by the weighted regression techniquewhich allowed the fitting points to be weighted according totheir distance from the true failure surface and their distancefrom the estimated design point [20] Similar to supportvector machine- (SVM-) based RSM Samui et al adoptedrelevance vector machine- (RVM-) based first-order second-moment method (FOSM) to build a RVM model to predictthe implicit performance function and evaluate the partialderivatives with sufficient accuracy [21] Tan et al discussedsimilarities and differences between radial basis functionnetworks (RBFN) based RSMs and SVM-based RSMs whichindicated that there is no significant difference between themand then proposed two new sampling methods and a hybridRSM to reduce the number of evaluations of the actualperformance function [22 23]

However RSM and its improved methods mentionedabove are relatively complicated and need much more com-puting cost since they have a lot of iterating calculationsat different design points associated with numerical method(ie finite element method) to fit limit state curved surfacewhich is represented by 119892(X) = 0 In this paper a noveldesign point approach in generalized random space (DPGmethod) associated with the method of divided differenceis proposed and applied to analyze the probability of gravitydam foundation instability overmultiple sliding planes In themethod implicit performance function with nonnormal andcorrelated variables is considered and iterating calculationfor 120573 is performed in generalized random space directly Thewhole procedure for reliability analysis does not need muchpreparation mathematically and is relatively simple for thecomplex engineering problems

2 Basic Principle of Computing InstabilityProbability of Gravity Dam Foundation

Due to the complexity of foundation stability against slidingof gravity dam for simplification the instability probabilityis commonly evaluated using models of single or dual slidingplane(s) for the dam foundation [24] However instabilityof foundation over multiple sliding planes is a general caseoccurring in bedrock under most engineering geologicalconditions for a gravity dam Therefore it is necessary toapply a model of multiple sliding planes to analysis oninstability probability of gravity dam foundation but there

is limited existing research regarding this case In tradi-tional deterministic analysis equal safety coefficient method(namely equal-119870 method) is used to analyze and assess thestability failure of dam foundation In the method factor ofsafety 119870 is regarded as an unknown number in nonlinearequations and needs to be solved by iterative calculatingnumerically similar to the application of the methods ofBishop Spencer and Janbu as well which are widely usedfor stability analysis on slope The reliability index 120573 orfailure probability 119901119891 is calculated by implicit performancefunction 119892(X) = 119870 minus 1 associated with the factor ofsafety 119870 Accounting for this the conventional FORM andits improved methods (such as JC method which combinesthe method of equivalent normalization transformation withFORM and is recommended by the Joint Committee ofStructural Safety) cannot be directly applied since the partialderivatives of 119892(X) with respect to the variable X cannot beevaluated directly [1 25] Hence if a partial derivative of 119892(X)is calculated by divided difference mathematically the valueof the derivative at any design point xlowast will be worked outeasily and the limit state curved surface will not be fittedcompletely as RSM [1 25] Usually the JC method associatedwith the method of divided difference could calculate designpoints by iteration on the actual limit state curved surfaceso that it could avoid fitting the overall curved surfaceThe method can provide sufficient accuracy and efficiencyto calculate probability of failure as prone to dealing withcomplex problems for large projects which has been verifiedby many examples in some literatures So it is a good idea touse the similar method to calculate instability probability ofgravity dam foundation over multiple sliding planes

21 Uncertainty in the Analysis Probabilistic methods canreveal the contributions of different components to theuncertainty in the analysis on the probability of failure ofdam foundation for gravity dam The uncertainty related toinstability of dam foundation could be classified as follows(1) the uncertainty of hydrographic andhydraulic parameters(2) the uncertainty of mechanical parameters of rocks andsoils (3) the uncertainty of seismic excitation (4) otheruncertainty underlying the process of dam design construc-tion operation management and so on

The second type of uncertainty above is studied mainlyin the paper and others are ignored by deterministic engi-neering applications just like in conventional analysis fordam stability against sliding The uncertainty of mechanicalparameters of rocks and soils might be enslaved to analysismethods due to subjective or objective conditions derivedfrom the lack of knowledge and so on Always the laboratoryand in situ measured data of the properties of rocks or soilsare inadequate So there is demonstrable distinction betweenthe measured values from the statistics of the small samplingand the true values of the actual physical and mechanicalproperties existing in real world

Many of studies have shown that friction coefficient 1198911015840and cohesion force 1198881015840 against shear fracture of rocks ordiscontinuities are two important uncertainty parameters

Mathematical Problems in Engineering 3

influencing the foundation stability of gravity dam Firstly bya great number of data of field measurements and their sta-tistical back-analyses there are obvious scatter in the spatialvariation of the two shear strength parameters not only forrockmass and faults of different projects but also for differenttypes of rockmass and faults at different locations in the sameproject Chen et al indicated that the variability of1198911015840 is nearlyat the same level but the variability of 1198881015840 is not so by thestudy on the variability of the shear strength parameters ofthe foundation rock mass and faults at several dam sites inChina [26 27] Secondly the probability distributions andnegative correlation of the two parameters cannot be ignoredThe random parameters might be normally distributed or ln-normally distributed

In our probabilistic analysis the two parameters againstshear fracture 1198911015840 and 1198881015840 are to be regarded as randomvariables following some distributions such as normal orln-normal distribution and considering their variability andcorrelation

22 Performance Function for ComputingInstability Probability

221 Equivalence Safety Factor Method for Foundation Insta-bility over Multiple Sliding Planes The sliding patterns ofgravity dam foundation could be divided by slip paths oversingle dual and multiple plane(s) The last one is the generalcase for analysis of foundation instability of gravity dam byassuming the potential failure surface as shown in Figure 1There are 119899 slip planes of 119899 sliding wedges from upstream todownstream in the foundation The 119894th wedge is subjected to(a) the vertical normal stress 120590 and the horizontal shear stress120591 induced by action of dam or other upside loads along thetopside of the wedge (b) dead load 119877119894 (c) force119873119894 and uplift119880119894 normal to the 119894th slip plane (d) resistance 119876119894 providedby the (119894 + 1)th wedge (e) angle of inclination 120572119894 of theinterface between the 119894th and (119894+1)th wedges and (f) angle ofinclination120573119894 the length 119897119894 and the shear strength parameters1198911015840119894 and 119888

1015840119894 of the 119894th slip plane

Based on the limit equilibriummethod the normal force119873119894 and applied shear 119879119894 of the 119894th slip plane respectively areexpressed as

119873119894 = (119877119894 + int120590d119860 119894) cos120573119894 minus lfloor119876119894 cos (120572119894 + 120573119894)

minus 119876119894minus1 cos (120572119894minus1 + 120573119894) + int 120591d119860 119894 sin120573119894rfloor minus 119880119894

(1)

119879119894 = 119876119894minus1 sin (120572119894minus1 + 120573119894) minus 119876119894 sin (120572119894 + 120573119894) + (119877119894

+ int120590d119860 119894) sin120573119894 + int 120591d119860 119894 cos120573119894(2)

The stability factor of safety for the 119894th wedge is defined as

119870119894 =1198731198941198911015840119894 + 1198881015840119894 119897119894

119879119894 (3)

So substituting the expressions for 119873119894 and 119879119894 into (3)for the typical wedge and according to the definition of

120590 120591Ai

Qiminus1

120572iminus1

Ri

Qi

120572iUi Ni

120573if998400i c

998400i li

Figure 1 Illustration of multiple sliding planes for stability calcula-tion of dam foundation

equivalence safety factor method 1198701 = 1198702 = sdot sdot sdot = 119870119899 = 119870the general wedge and wedge interaction equation can bewritten as

119870 =119876119894minus1 cos (120572119894minus1 + 120573119894) 119891

1015840119894 minus 119876119894 cos (120572119894 + 120573119894) 119891

1015840119894 + 119901119894

119876119894minus1 sin (120572119894minus1 + 120573119894) minus 119876119894 sin (120572119894 + 120573119894) + 119902119894 (4)

where 119901119894 = lfloor(119877119894 +int120590d119860 119894) cos120573119894 minusint 120591d119860 119894 sin120573119894 minus119880119894rfloor1198911015840119894 +1198881015840119894 119897119894

and 119902119894 = (119877119894 + int120590d119860 119894) sin120573119894 + int 120591d119860 119894 cos120573119894The nonlinear equations (4) are in terms of 119870 and 119876119894

(119894 = 0 1 119899) and 1198760 = 119876119899 = 0 So there are eventually119899 unknown numbers for 119899 sliding wedges of dam foundationAssuming 120572119894 = 90∘ according to conventional vertical slicemethod [26 28ndash30]119870 and119876119894 (119894 = 1 119899minus1) can be workedout by iterative calculation numerically

222 Performance Function for Probability Calculation Thebasic expression of the performance function for reliabilityanalysis on dam foundation stability could be given as

119885 = 119870 minus 1 (5)

where119870 is the factor of safety mentioned in (4) which is thefunction in terms of variables 1198831 1198832 119883119897 So (5) can berewritten as

119885 = 119870 minus 1 = 119892 (1198831 1198832 119883119897) (6)

where 119892(1198831 1198832 119883119897) is the function associated with therandom variables1198831 1198832 119883119897

In the analysis of instability of foundation over multiplesliding planes119885 = 119892(1198831 1198832 119883119897) is implicit function andlimit state equation 119885 = 0 can be represented by nonlinearequation (4) above Correspondingly the random variables1198831 1198832 119883119897 (119897 = 2119899) are actually the strength parametersagainst shear fracture 1198911015840119894 and 119888

1015840119894 where 119894 = 1 2 119899

23 ComputingMethod of Failure Probability with NonnormalandCorrelative RandomVariables It is a problem to calculateinstability probability with nonnormal and correlated ran-dom variables of dam rock base over multiple sliding planesWhat is more the random characteristics of the variables willinfluence the accuracy of the value of instability probability

A well known method for calculating reliability index orprobability of failure of a structureMC simulation is derived


from sampling principle In the method each continuousvariable is replaced by a large number of discrete valuesgenerated from the underlying distribution These values areused to compute a large number of values of function 119885and its distribution Finally the probability of failure can beestimated to be equal to the number of failures which faileddivided by the total number of calculations statistically MChas been widely applied for reliability calculation due to itsclear definition in spite of the low calculating efficiency andunder some circumstances it is always the unique method toverify the accuracy of any other probabilistic method

Besides MC method JC method is also a conventionalmethod for calculating reliability index or probability offailure of a structureWhen the randomvariables are nonnor-mally distributed the treatment of equivalent normalizationtransformation [31] for the variables is performed rationallyhowever when the random variables are correlative foreach other the problem becomes complicated because thecalculationwith nonnormal and correlative variables involvescomplex combining processes of equivalent normalizationtransformation and orthogonal transformation Calculationof the eigenvalues of matrices must be conducted in the pro-cedure in particular Other approaches such as the methodof Rosenblatt orNataf transformation are also utilized to dealwith the calculation with nonnormal and correlated variables[32 33] However in the former method the joint cumulativedistribution function of variables must be known and in thelatter the ratio of coefficients of correlation between pre-and posttransformation of variables needs to be determinedpreviously The preconditions in the two methods are hardlyensured usually

So a new reliability-based method which is called designpoint method in generalized random space (DPG method)can be applied since iterative calculation for 120573 is performedin generalized random space directly by assuming that thevalue of coefficient of correlation of any two variables isnot changed before and after variable transformation ingeneralized space based on the principle of isoprobabilistictransformation [34ndash37] Obviously the DPG method can beused in rather more general cases than JC method When thevariables are independent of each other the former becomesthe latter

A set of random variables 1198831 1198832 119883119899 describing therandomness in the geometry material loading and so oncan form an 119899-dimensional generalized random space inwhich there is a performance function119885 = 119892(1198831 1198832 119883119899)for the structure Similar to the Euclidean random space inconventional reliability theory the limit state of the structureis represented by 119885 = 0 119885 gt 0 then the structure is onthe reliable state (ldquosaferdquo) 119885 lt 0 then the structure is on thefailure state (ldquofailrdquo) Figure 2 shows the principle of the DPGmethod

To performDPGmethod firstly the angle of any two axesis defined as

120579119883119894119883119895 = 120587 minus arccos (120588119883119894119883119895) (7)

where 120588119883119894119883119895 is the coefficient of correlation for variables 119883119894and 119883119895 The coefficient of correlation for the two variables

X2

0 X1

plowast

120573

120579x1x2

Z lt 0

Z gt 0

Failure state

Reliable state

Z = 0 Limit state

Figure 2 A conceptual illustration of the design point and reliabilityindex in 2-dimensional generalized random space

after transformation in generalized random space based onthe principle of isoprobabilistic transformation is

12058811988310158401198941198831015840119895

asymp 120588119883119894119883119895 (8)

where 1198831015840119894 and 1198831015840119895 are arbitrary two variables by transforma-

tion in generalized space from119883119894 and119883119895 respectivelyThe mean and standard deviation of any variable 1198831015840119894 are

defined as

1205831198831015840119894

= 119909lowast119894 minus 1205901198831015840119894

Φminus1 [119865119894 (119909lowast119894 )]

1205901198831015840119894

=120593 Φminus1 [119865119894 (119909

lowast119894 )]

119891119894 (119909lowast119894 )

(9)

where 119909lowast119894 is any component of the vector xlowast =(119909lowast1 119909

lowast2 119909

lowast119894 119909

lowast119899 ) which is the value of the vector

X = (1198831 1198832 119883119899) at design point 119901lowast shown in Figure 2119891119894(sdot) denotes the probability density function (PDF) ofthe random variables and 119865119894(sdot) denotes the cumulativedistribution function (CDF) of the random variables

The reliability index at xlowast is

120573

=sum119899119894=1 (120597119892120597119883119894)

1003816100381610038161003816119901lowast (1205831198831015840119894 minus 119909lowast119894 )

(sum119899119894=1sum119899119895=1 1205881198831015840

1198941198831015840119895

((120597119892120597119883119894) (120597119892120597119883119895))100381610038161003816100381610038161003816119901lowast1205901198831015840119894

1205901198831015840119895

)12

(10)

and the sensitivity coefficient of any variable 119883119894 is calculatedas

1205721015840119894

=

sum119899119895=1 1205881198831015840

1198941198831015840119895

(120597119892120597119883119895)100381610038161003816100381610038161003816119901lowast1205901198831015840119895

sum119899119895=1sum119899119896=1 1205881198831015840

1198951198831015840119896

((120597119892120597119883119895) (120597119892120597119883119896))100381610038161003816100381610038161003816119901lowast1205901198831015840119895

1205901198831015840119896

(11)

119909lowast119894 for next iterative calculation is

119909lowast119894 = 1205831198831015840119894

+ 12057312057210158401198941205901198831015840119894

(12)

The initial xlowast are always the mean of random variableWhen xlowast converges by repeated calculation from (9) to (12)the resultant reliability index 120573 and probability of failure119901119891 =1 minus Φ(120573) are obtained finally


24 Method of Divided Difference to Computing PartialDerivative of Implicit Performance Function According to(10) and (11) the value of partial derivative of performancefunction at design point 119901lowast (120597119892120597119883119894)|119901lowast needs to be calcu-lated When the performance function 119892(1198831 1198832 119883119899) isimplicit the method of one order divided difference can beutilized as

120597119885

120597119883119894

10038161003816100381610038161003816100381610038161003816119901lowast=120597119892

120597119883119894

10038161003816100381610038161003816100381610038161003816119901lowastasympΔ119892

Δ119909119894

=119892 (1199091 1199092 119909119894 + Δ119909119894 119909119899)

Δ119909119894

minus119892 (1199091 1199092 119909119894 119909119899)

Δ119909119894

(13)

From the practical experience of calculation the com-puting efficiency and accuracy can be ensured while Δ119909119894 =(001sim005)119909119894

25 Computing Steps and Procedures DPG method asso-ciated with method of divided difference to calculate thestability reliability index 120573 or instability probability 119901119891 ofgravity dam foundation over multiple sliding planes hasbeen proposed above and can be implemented easily withinMatlab There are six steps of the solution process

(1) Regard all of the strength parameters against shearfracture 1198911015840119894 and 119888

1015840119894 (119894 = 1 2 119899) of the 119899 slip

planes under 119899 slidingwedges as the randomvariablesX that affect the foundation stability of gravity damgenerally assume the means of 1198911015840119894 and 119888

1015840119894 as the initial

value xlowast at current design point 119901lowast(2) Use the current xlowast to calculate the means and stan-

dard deviations of the variables by (9)(3) Calculate 120573 at the current design point 119901lowast by (10)

and then perform sensitivity analyses according to(11) The critical calculation of partial derivative ofperformance function 119885 = 119892(X) = 119870(X) minus 1 (see(6)) is performed by divided difference calculationaccording to (13) in which the factor of safety 119870solved by nonlinear equations (4) at the limit state(119885 = 0) is considered

(4) Use (12) to obtain the new xlowast for next iterativecalculation

(5) Perform repeated calculation from Step (2) to Step (4)until the values of vector xlowast are converged

(6) Obtain the ultimate 120573 and calculate the instabilityprobability 119901119891 = 1 minus Φ(120573) finally

3 Assessment Standard

Dam failure is different from general safety accident Onthe one hand the consequence is very serious sometimesit even causes destructive disasters on the other hand damfailure is affected by many natural factors and cannot be

controlled by human Therefore risk standard of dam failureis a comprehensive index involving many factors such aspolitics economy engineering technology society naturalenvironment and cultural background If the risk standardis selected relatively low the safety reliability resisting damfailure will be insufficient and once failure takes place theconsequencewill be serious on the contrary if the standard isrelatively high the project cannot bring anticipated economicbenefit and may cause water waste and investment lossFrom this point many experts and researchers have studiedthe establishment of appropriate design standards Based onthe related data of different projects in the worldwide theallowable probability of failure119875119910 is 11times10

minus4a for large andmiddle dams and the failure probability is unacceptable whenit is larger than 10 times 10minus3a Among all the failure modesstability failure accounts for about 10 of the total of damfailure according to ICOLD (the International Commissionon Large Dams) Bulletin 99 (1995) ldquoDam FailuresStatisti-cal Analysisrdquo and Chinese ldquounified standard for reliabilitydesign of hydraulic engineering structuresrdquo Therefore it isconsidered that the acceptable probability of stability failureis less than 11 times 10minus5a and the instability probability isunacceptable when it is larger than 10 times 10minus4a

In contrast the allowable safety factor [119870] is 30 undernormal loading condition according to deterministic limitequilibriummethod in Chinese Design Specification of Con-crete Gravity Dam (DL 5108-1999) and the target reliabilityindex 120573119879 is 42 for the hydraulic structures at Grade 1 and37 for the hydraulic structures at Grade 2 in the relativereliability-based design codes

4 Application

41 Basic Data A hydropower station located in southeastChina is for power generation singly without other compre-hensive benefits Its main body is roller compacted concretegravity dam which has a maximum height of 1190m thecrest length of 2840m and the elevation of 256400m Thisdam consists of 14 blocks and is numbered from number1 to number 14 from left bank to right bank Number 5 tonumber 8 are overflow structures others are water retainingstructures The regulating storage of the reservoir is 0254billion m3 The usual pool elevation under normal operationis 25600m and the corresponding tailwater elevation is247161m The silt elevation is 249770m The uplift intensityfactor is 025 The structure grade of the dam is Grade 2so the design reference period 119873119889 is 50 a according to thedesign specification The allowable instability probability is[119875] = 119875119910times119873119889 = 55times10

minus4 and the probability is unacceptablewhen it is larger than 5 times 10minus3

411 Engineering Geological Conditions The geologicalexploration in the dam site has shown that there are agreat number of lowly inclined weak structural planes andseveral fault-fractured zones within the foundation whichare adverse to the dam stability The geologic parametersof the bed base rock and the structural planes are listed inTables 1 and 2 respectively


Table 1 Geologic parameters of bed base rock in the dam site

Class of rockmass

Drydensity120588gcm3

Saturatedcompressivestrength119877119887MPa

Bedrockbedrock Concretebedrock

Deformationmodule1198640GPa

Poissonrsquosratio 120583

Strength against shearfracture


1198911015840 1198881015840MPa 1198911015840 1198881015840MPaII 265ndash270 80ndash100 15sim21 020ndash025 120sim135 150sim160 110sim120 110sim115III1 260ndash265 60ndash80 8sim12 025ndash030 110sim120 100sim130 100sim107 090sim100

Table 2 Geologic parameters of structural planes in the dam site

Classification Strength against shearfracture

Category Subcategory 1198911015840 1198881015840MPaHard structural plane 060sim070 010sim015

Weak structural plane or fault-fracturedzones

Clastic and fragment 045sim050 010sim015Clastic with mud 035sim040 005sim010

Mud with silty grain clastic 025sim030 001sim005Clayey mud 018sim025 0002sim0010

Figure 3 shows the geologic section of typical slidingmodes of number 6 overflow sectionThere are lowly inclinedstructural planes (mainly fh09 fh10 and fh11) and joints(fh152-2 etc) below the dam base and steeply inclinedfaults (F16 F17 etc) near to the dam heel The geologicalcharacteristics of lowly inclined weak structural planes arelisted in Table 3 Table 4 lists the slip paths and the values ofthe material properties of the sliding planes used for stabilitycalculation

412 Analyses of Slip Paths and Wedges The slip paths ofdam foundation are consisted of the faults weak structuralplanes joints and rock bridges by their probable combina-tions (shown in Figure 3) For stability analysis of the damfoundation the bed base rock will be divided in severalwedges according to any different slip path Figure 4 givesthe divided wedges on Slip Path B for example There areone tensile fractured plane (B1-B2) below dam heal and threesliding planes (B2-B3 B3-B4 and B4-B5) at the bottom of thethree wedges (Wedges I II and III)

413 Results from Deterministic Analyses Under the normalloading condition the safety factors are respectively 256245 and 1071 for the three slip paths mentioned aboveby deterministic equivalence safety factor method withouttaking the variation of parameters into consideration Thefirst two factors of safety are less than the allowance safetyfactor [119870] = 30 So it is necessary to meet the stability ofdam foundation against sliding by some effective engineeringtreatments

In the next sections the influence of randomness ofstrength parameters against shear fracture of sliding planeson foundation stability will be researched by DPG methodassociated with the method of divided difference proposedby the paper That is the foundation instability probability

affected by the probability distribution variability and nega-tive correlation of the two parameters1198911015840 and 1198881015840 is consideredin detail

42 Limit State Equations and Random Parameters For thepractical dam project the limit state equations for the mostadverse slip path Slip Path B of number 6 overflow sectionare expressed by (4) as follows

1198701198760 sin (1205720 + 1205731) minus 1198701198761 sin (1205721 + 1205731) + 1198701199021

minus 1198760 cos (1205720 + 1205731) 11989110158401 + 1198761 cos (1205721 + 1205731) 119891

10158401

minus 1199011 = 0

1198701198761sin (1205721 + 1205732) minus 1198701198762sin (1205722 + 1205732) + 1198701199022

minus 1198761cos (1205721 + 1205732) 11989110158402 + 1198762cos (1205722 + 1205732) 119891

10158402

minus 1199012 = 0

1198701198762sin (1205722 + 1205733) minus 1198701198763sin (1205723 + 1205733) + 1198701199023

minus 1198762cos (1205722 + 120573119894) 11989110158403 + 1198763cos (1205723 + 1205733) 119891

10158403

minus 1199013 = 0

(14)

where

1205720 = 1205721 = 1205722 = 90∘

1205731 = minus14∘

1205732 = minus16∘

1205733 = minus14∘


Table 3 Geologic characteristics of lowly inclined weak structural planes

Name of structural planes OccurrenceWidth of fracturedzone for weak

structural planescmExtension lengthm Type of structural planes

fh09 N40ndash60∘WNE ang 20ndash25∘ 2ndash30 210 Clastic with mudfh10 N40ndash60∘WNE ang 20ndash25∘ 2ndash20 200 Clastic with mudfh11 N40ndash60∘WNE ang 20ndash25∘ 2ndash8 125 Clastic and fragmentfh152-5 N80∘ENW ang 20∘ 2sim5 Mud with silty grain clastic

Table 4 Description of the slip paths and the value of the material properties of the sliding planes

Number ofslip paths

Name of sliding planescomprising the paths

Angle ofapparent dip∘ Characteristics Strength against shear fracture

1198911015840 1198881015840MPa

A

A1-A2 90 Tensile fractured plane at dam heal 0000 0000A2-A3 minus16 Clastic with mud 0375 0075A3-A4 minus16 Clastic and fragment 0475 0125A4-A5 minus16 Clastic with mud 0375 0075A5-A6 0 Dam base 1150 1125

B

B1-B2 90 Tensile fractured plane at dam heal 0000 0000B2-B3 minus14 Clastic with mud 0375 0075B3-B4 minus16 Clastic and fragment 0475 0125B4-B5 minus14 Clastic with mud 0375 0075

C

C1-C2 90 Tensile fractured plane at dam heal 0000 0000C2-C3 minus15 Clastic and fragment 0475 0125C3-C4 minus19 Rock bridge (II) 1275 1550C4-C5 minus16 Mud with silty grain clastic 0275 0030C5-C6 minus16 Rock bridge (III1) 1050 0850

119901119894

= [(119877119894 + int120590d119860 119894) cos120573119894 minus int120591d119860 119894 sin120573119894 minus 119880119894]1198911015840119894

+ 1198881015840119894 119897119894 119894 = 1 2 3

119902119894 = (119877119894 + int120590d119860 119894) sin120573119894 + int 120591d119860 119894 cos120573119894

119894 = 1 2 3

int 120590d119860 119894 = 119881119894 119894 = 1 2 3

int 120591d119860 119894 = 119867119894 119894 = 1 2 3

1198971 = 499m

1198972 = 335m

1198973 = 1174m

1198760 = 1198763 = 0

(15)

According to the practical engineering experience and thespecial geological andmechanical parameters of the bed base

rock of the example dam in the paper the degree of scatter ofthe variation coefficients of the strength parameters againstshear fracture is obvious so the coefficients of variation of 1198911015840and 1198881015840 are assumed to change in certain rangesThe statisticalcharacteristics of 1198911015840 and 1198881015840 of Slip Path B used for calculationare listed in Table 5 According to Chinese ldquounified standardfor reliability design of hydraulic engineering structuresrdquo(1994) the parameter distribution type of slip planes can beassumed in normal or ln-normal distributions for reliabilityanalysis when the statistical data are not sufficient

43 Resultant Analyses

431 Feasibility ofDPGMethod TwomethodsDPGandMCmethods are used to calculate the instability probabilities ofSlip Path B during design reference period Table 6 gives thecalculated results obtained by the two methods when 1198911015840 and1198881015840 are both normally distributed their variation coefficients1205751198911015840 and 1205751198881015840 are 02 and 03 respectively and their coefficientof correlation 12058811989110158401198881015840 is 0

The instability probabilities of Slip PathB are respectively1310 times 10minus4 and 1244 times 10minus4 calculated by MC and DPGmethods and the absolute and relative errors between the twomethods are respectively 00007 and 53 Thus the twomethods have nearly the same accuracy but the latter requiresless iterating calculation for convergence Meanwhile by


F18

F17

A2

B2

C2

F16

f52

A1(B1 C1)fh06

fh07

fh09

fh10

fh11

A3

B3

A4

B4

C3

A5 A6B5

C5C6fh152-2

fh152-4

fh152-7fh152-8

fh152-9C4

Figure 3 Geological profile and assuming slip paths of the dam foundation

Table 5 Statistical characteristics of random variables of slip path B

Slip plane Random variable Mean value Variation coefficient Probability distribution

B2-B3 11989110158401 0375 014sim026 Normalln-normal distribution11988810158401MPa 0075 015sim045 Normalln-normal distribution



Table 6 Comparison of resultant instability probabilities calculated by MC and DPG method

MethodDesign points (sliding plane

B2-B3) 120573 119901119891 Notes11989110158401lowast

11988810158401lowastMPa

MC 3650 1310 times 10minus4The numbers of calculating performance function119899 = 1 000 000 and the variation coefficient for 119901119891119881119901119891 = 00873 when 95 confidence level is specified

DPG 0107 0039 3664 1244 times 10minus4The numbers of calculating performance function 119899 = 36with the step Δ119909119894 = 003119909119894 for divided differencecalculation

DPGmethod the design points can be obtained easily and theprobability distributions variability and negative correlationof 1198911015840 and 1198881015840 can be taken into consideration for specialresearch So the DPG method suggested in the paper hasmuchmerit and is rational and feasible for themajor researchgoal of the paper

432 Analyses of the Effects of Correlation of 1198911015840 and 1198881015840 onInstability Probability Table 7 lists the calculated results ofthe instability probabilities of Slip Path B by DPG methodwhen 1198911015840 and 1198881015840 are both normally distributed 1205751198911015840 and 1205751198881015840 are02 and 03 respectively and the coefficient of correlation12058811989110158401198881015840 is within the range of minus07sim00 The negative value


B1

B2B3

B4

B5

V

H

III III

Figure 4Wedges on Slip Path B119881 vertical load due to dam action119867 horizontal load due to dam action

Table 7 Calculated results under the conditions of different 12058811989110158401198881015840 byDPG method

Number 12058811989110158401198881015840Design points (sliding

plane B2-B3) 120573 119901119891

11989110158401lowast

11988810158401lowastMPa

1 00 0107 0039 3664 1244 times 10minus4

2 minus01 0100 0044 3789 7554 times 10minus5







Note if 12058811989110158401198881015840 is less than minus08 11989110158401

lowast will be less than 0 and the simulation willbe not converged at design points

of 12058811989110158401198881015840 indicates the degree of negative correlation of 1198911015840

and 1198881015840 It is clear from Table 7 that the resultant instabilityprobability calculated decreases with the increasing of theabsolute value of negatively correlated coefficient 119901119891 =

1244 times 10minus4 when 12058811989110158401198881015840 = 00 (independent case) and 119901119891 =3106 times 10minus7 when 12058811989110158401198881015840 = minus07 (strongly negatively correlatedcase) By comparison the values of the instability probabilityunder the two cases are different in 3 orders of magnitudeCorrespondingly the values of the reliability index under thetwo cases are 3664 and 4984 respectively Obviously theindependent case is on the conservative side for simulation

433 Analyses of the Effects inDifferentDistributions of1198911015840 and1198881015840 on Instability Probability Figure 5 illustrates the resultantinstability probability under four different cases of combina-tions of normal or ln-normal distributions of 1198911015840 and 1198881015840 for1205751198911015840 = 02 and 1205751198881015840 = 03 by DPGmethod In all cases 119901119891 tendsto increase as 12058811989110158401198881015840 increases fromminus07 (negatively correlated)to 0 (independent) When 1198911015840 is normally distributed theinstability probabilities are approximately identical whether1198881015840 is normally or ln-normally distributed When 1198911015840 is ln-normally distributed the instability probabilities are smaller

100E minus 03

100E minus 05

100E minus 07

100E minus 09

100E minus 11

100E minus 1300minus01minus02minus03minus04minus05minus06minus07minus08

pf

120588f998400c998400

ln-normal distributions for both f998400 and c998400

Normal distributions for both f998400 and c998400

ln-normal distribution for f998400 and normal for c998400Normal distribution for f998400 and ln-normal for c998400

Figure 5 Relation of 119901119891 and 12058811989110158401198881015840 assuming that 1198911015840 and 1198881015840 followdifferent distributions

010 015 020 025 030

100E minus 02

100E minus 03

100E minus 04

100E minus 05

100E minus 06

100E minus 07

100E minus 08

120575f998400

120575c998400 = 045

120575c998400 = 030

120575c998400 = 015

pf

Figure 6 Relation of 119901119891 and 1205751198911015840

and the values calculated under the case that 1198881015840 is in ln-normaldistribution are less than in normal distribution at the samelevel of correlation of the two parameters From the figure 119901119891= 124 times 10minus4 if 1198911015840 and 1198881015840 are both normally distributed and119901119891 = 464 times 10

minus9 if the two parameters are both ln-normallydistributed for 1205751198911015840 = 02 1205751198881015840 = 03 and 12058811989110158401198881015840 = 00

434 Analyses of the Effects of Variability of 1198911015840 and 1198881015840 onInstability Probability On the basis of the above analyses itis reasonable simple and conservative to assume that 1198911015840 and1198881015840 are normally distributed and independent of each otherunless otherwise demonstrated Considering the range of thevalues of coefficients of variation 1205751198911015840 = 014sim026 1205751198881015840 =015sim045 the calculations are made and the curves about119901119891 varying with 1205751198911015840 and 1205751198881015840 are displayed in Figures 6 and7 According to the resultant curves 119901119891 starts to apparentlyclimb as 1205751198911015840 increases for different 1205751198881015840 (shown in Figure 6)and in addition 119901119891 also tends to increase as 1205751198881015840 increases fordifferent 1205751198911015840 (shown in Figure 7) However 119901119891 = 192 times 10

minus3

if 1205751198911015840 = 026 and 119901119891 = 404times10minus7 if 1205751198911015840 = 014 for 1205751198881015840 = 030

meanwhile119901119891 = 318times10minus4 if 1205751198881015840 = 045 and119901119891 = 608times10

minus5

if 1205751198881015840 = 015 for 1205751198911015840 = 020 There is the difference of 119901119891in 4 orders of magnitude between 1205751198911015840 = 014 and 026 for


015 020 025 030 035 040 045

100E minus 02

100E minus 03

100E minus 04

100E minus 05

100E minus 06

100E minus 07

100E minus 08

120575c998400

120575f998400 = 026

120575f998400 = 020

120575f998400 = 014

pf


1205751198881015840 = 030 in contrast under the latter case there is thedeference of 119901119891 in one order of magnitude between 1205751198881015840 = 015and 045 for 1205751198911015840 = 020 So the influence of the variability of1198911015840 is more apparent than that of 1198881015840 on 119901119891 despite the fact thatthe varying range of coefficients of variation of 1198911015840 is less thanthat of 1198881015840

435 Stability Safety Analyses and Assessments of the PracticalDam The instability probability of themost adverse slip pathof the example dam section (number 6 overflow section)mentioned above is 1244 times 10minus4 for 1205751198911015840 = 020 1205751198881015840 =030 and 12058811989110158401198881015840 = 00 It is less than the allowable valueof the instability probability [119875] = 55 times 10minus4 but they arein the same order of magnitude Meanwhile it is 2861 times10minus3 for 1205751198911015840 = 026 1205751198881015840 = 045 and 12058811989110158401198881015840 = 00 It isgreater than [119875] but less than 5 times 10minus3 which is defined asunacceptable probability In contrast the factor of safety is245 by deterministic method without taking the variationof parameters into consideration The factor of safety is lessthan the allowable safety factor [119870] = 30 the reliabilityindex is 366 for 1205751198911015840 = 020 1205751198881015840 = 030 and 12058811989110158401198881015840 = 00 whichnearly equals the target reliability index 120573119879 = 37 Obviouslythe conclusions are more or less different following threedifferent safety standards Generally it has been verified thatthe deterministic method is most conservative in the threemethods by the analyses of the practical dam in the paperand the conclusions fromother literatures aswell Fromaboveanalyses the margins of safety of the foundation stability arenot adequate for the example dam section and the enhancingtreatments should be carried out especially for the fault-fractured zones and weak structural planes

Similarly the instability probabilities of the other two slippaths can be calculated by the DPG method Assume that 1198911015840and 1198881015840 of each sliding plane of each slip path are normallydistributed and independent and 1205751198911015840 = 020 1205751198881015840 = 030and 12058811989110158401198881015840 = 00 for any slip plane The resultant instabilityprobabilities are listed in Table 8 The three slip paths can beseen as three main failure modes for the rock foundation ofthe example dam and each failure mode is independent dueto the independence of each shear strength parameter of any

Table 8 Resultant instability probabilities of three slip pathscalculated by DPG method

Slip pathsReliability method

(DPG) Deterministic method

120573 119901119891 119870

A 3772 8094 times 10minus5 256B 3664 1244 times 10minus4 245C 5612 9993 times 10minus9 1071Note the deterministic method is equivalence safety factor method withoutconsidering the variation of the parameters against shear fracture

sliding plane Three failure modes comprise a series failuresystemof dam foundation So the overall probability of failurefor the overall dam foundation system can be calculated bysumming over the instability probabilities of the three slippaths The final instability probability of the overall rockfoundation is 2053 times 10minus4 for a conservative consideration

It should also be noted that for the example dam thecoefficients of variation and negative correlation of the twokey mechanical parameters of sliding planes were selected inthe relatively wider ranges which are summarized from thefield and experimental data In the calculation for Slip PathB four combinations of distributions of the two parameterswere detailedly taken into consideration So the final resultsobtained for the example dam can reveal the performanceabout foundation stability of any similar gravity dam ingeneral We confirm that the randomness of 1198911015840 is moresignificant than 1198881015840 and what is more it is reasonable toassume a normal distribution and independence of eachparameter in the absence of further informationThe assump-tion will probably overestimate the probability of foundationinstability of gravity dam in some degree

5 Conclusions

In this paper the main goal has been to try applying prob-abilistic approach to calculate the instability probability ofgravity dam foundation over multiple sliding planes ratherthan single or dual sliding plane(s) in traditional analysisThe simple and efficient probabilistic method DPG methodcombined with a divided difference approximation to thepartial derivatives is proposed The effects of randomness ofthe shear strength parameters of sliding planes for a practicaltypical gravity dam are analyzed in detail The followingconclusions can be obtained

(1) By comparison between MC method and DPGmethod the two methods have nearly the sameaccuracy but the latter needs less cost for calculationMeanwhile the DPG method has much other meritFor example the design points can be obtainedeasily during the calculating process and probabilitydistributions variability and correlation of variablescan be taken into consideration rationally Hence theDPG method associated with the method of divideddifference is feasible for the calculation of probabilityof failure with implicit performance function just like


the case of instability of dam foundation overmultiplesliding planes

(2) The randomness of material properties of slidingplanes is a major contributor to the uncertainty of thefoundation stability of a gravity damThe randomnessof the twoparameters against shear fracture1198911015840 and 1198881015840of the sliding planes may be described by assumptionof their distributions variability and correlation Sothe effects of the randomness of the parameters onresultant instability probability of gravity dam foun-dation can be analyzed based on these aspects Thestudy of the practical dam shows that (1) when bothof the two parameters follow normal distribution theresultant value of instability probability is greater thanthat which is calculated when one or both of theparameters do not follownormal distribution and thedistributed type of 1198911015840 is dominant (2) the strongerthe negative correlation of the two parameters is thesmaller the instability probability calculated is and(3) the instability probability tends to increase as thecoefficients of variability of 1198911015840 and 1198881015840 increases andthe influence of the variability of 1198911015840 is more apparentthan that of 1198881015840 on the instability probability So itis reasonable to assume a normal distribution andindependence of each parameter in the absence offurther information The assumption will probablyoverestimate the instability probability of gravity damfoundationThe spatial variation of 1198911015840 is more signif-icant than 1198881015840 on the instability probability All of theconclusions drawn in the paper are identical to thoseof other research

(3) Currently there are three kinds of safety indicesmdashfactor of safety reliability index and probabilityof failuremdashfor engineering stability problem Thefactor of safety is used in deterministic method Itis so convenient for engineers to utilize and haveso much experience in designs of many existinggravity dam projects that has become a major indexto assess foundation stability of gravity dam for along time However the assessment method basedon this index is relatively more conservative andshould be improved upon undoubtedly The trendis application of probabilistic approach in whichthe reliability index or probability of failure will beutilized Now a fundamental problem is the lack ofdata Field exploration is expensive and laboratorytests are always based on small numbers of samples sothere are seldom enough data to support meaningfulstatistical analyses and broad conclusions about thestatistics of rock or soil properties In a word thecurrent challenges to the profession are to make useof appropriate probabilistic methods in practice andto sharpen the investigations and analyses so that eachadditional data point provides maximal information

(4) In the paper we applied a simple accurate andefficientmethod to calculate the instability probabilityof concrete gravity dam foundation over multiplesliding places It can also be utilized in other similar

analyses of slopes levees embankments earth-rockdams and so on

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

The research has been supported in part by National NaturalScience Foundation of China under Grant no 51309048

References

[1] G B Baecher and J T Christian Reliability and Statistics inGeotechnical Engineering John Wiley amp Sons Chichester UK2003

[2] K K Phoon Reliability-Based Design in Geotechnical Engi-neering Computations and Applications Taylor amp Francis NewYork NY USA 2008

[3] G A Fenton and D V Griffths Risk Assessment in GeotechnicalEngineering John Wiley amp Sons New York NY USA 2008

[4] J T Christian and G B Baecher ldquoUnresolved problems ingeotechnical risk and reliabilityrdquo in Proceedings of the GeoRisk2011 Geotechnical Risk Assessment and Management (GSP 224)ASCE Specialty Conference pp 50ndash63 Atlanta Ga USA June2011

[5] J T Christian C C Ladd andG B Baecher ldquoReliability appliedto slope stability analysisrdquo Journal of Geotechnical Engineeringvol 120 no 12 pp 2180ndash2207 1994

[6] X-S Tang D-Q Li C-B Zhou and K-K Phoon ldquoCopula-based approaches for evaluating slope reliability under incom-plete probability informationrdquo Structural Safety vol 52 pp 90ndash99 2015

[7] A Der Kiureghian and T Dakessian ldquoMultiple design points infirst and second-order reliabilityrdquo Structural Safety vol 20 no1 pp 37ndash49 1998

[8] Y J Hong J Xing and J B Wang ldquoA second-order third-moment method for calculating the reliability of fatiguerdquoInternational Journal of Pressure Vessels and Piping vol 76 no8 pp 567ndash570 1999

[9] H U Koyluoglu and S R K Nielsen ldquoNew approximations forSORM integralsrdquo Structural Safety vol 13 no 4 pp 235ndash2461994

[10] F S Wong ldquoSlope reliability and response surface methodrdquoJournal of Geotechnical Engineering vol 111 no 1 pp 32ndash531985

[11] L J Moore and P Sa ldquoComparisons with the best in responsesurface methodologyrdquo Statistics amp Probability Letters vol 44no 2 pp 189ndash194 1999

[12] Y Zheng andPKDas ldquoImproved response surfacemethod andits application to stiffened plate reliability analysisrdquo EngineeringStructures vol 22 no 5 pp 544ndash551 2000

[13] X L Guan and R E Melchers ldquoEffect of response surfaceparameter variation on structural reliability estimatesrdquo Struc-tural Safety vol 23 no 4 pp 429ndash444 2001

[14] S Gupta and C S Manohar ldquoImproved response surfacemethod for time-variant reliability analysis of nonlinear ran-dom structures under non-stationary excitationsrdquo NonlinearDynamics vol 36 no 2ndash4 pp 267ndash280 2004


[15] S M Wong R E Hobbs and C Onof ldquoAn adaptive responsesurface method for reliability analysis of structures with multi-ple loading sequencesrdquo Structural Safety vol 27 no 4 pp 287ndash308 2005

[16] B Xu and B K Low ldquoProbabilistic stability analyses of embank-ments based on finite-element methodrdquo Journal of Geotechnicaland Geoenvironmental Engineering vol 132 no 11 pp 1444ndash1454 2006

[17] J Cheng Q S Li and R-C Xiao ldquoA new artificial neuralnetwork-based response surface method for structural reliabil-ity analysisrdquo Probabilistic Engineering Mechanics vol 23 no 1pp 51ndash63 2008

[18] H P Gavin and S C Yau ldquoHigh-order limit state functions inthe response surface method for structural reliability analysisrdquoStructural Safety vol 30 no 2 pp 162ndash179 2008

[19] T Zou Z P Mourelatos S Mahadevan and J Tu ldquoAn indicatorresponse surface method for simulation-based reliability anal-ysisrdquo Journal of Mechanical Design Transactions of the ASMEvol 130 no 7 Article ID 071401 pp 1ndash11 2008

[20] X S Nguyen A Sellier F Duprat and G Pons ldquoAdaptiveresponse surfacemethod based on a doubleweighted regressiontechniquerdquo Probabilistic Engineering Mechanics vol 24 no 2pp 135ndash143 2009

[21] P Samui T Lansivaara and M R Bhatt ldquoLeast square supportvector machine applied to slope reliability analysisrdquo Geotechni-cal andGeological Engineering vol 31 no 4 pp 1329ndash1334 2013

[22] X-H Tan W-H Bi X-L Hou and W Wang ldquoReliabilityanalysis using radial basis function networks and support vectormachinesrdquo Computers and Geotechnics vol 38 no 2 pp 178ndash186 2011

[23] X-H Tan M-F Shen X-L Hou D Li and N Hu ldquoResponsesurfacemethod of reliability analysis and its application in slopestability analysisrdquo Geotechnical and Geological Engineering vol31 no 4 pp 1011ndash1025 2013

[24] X Chang C JiangW Zhou andDHuang ldquoEqual safety factormethod and its reliability analysis for rock foundation of damrdquoChinese Journal of Rock Mechanics and Engineering vol 26 no8 pp 1594ndash1602 2007

[25] J T Christian ldquoGeotechnical engineering reliability how welldo we know what we are doingrdquo Journal of Geotechnical andGeoenvironmental Engineering vol 130 no 10 pp 985ndash10032004

[26] Z Chen J Xu P Sun CWu YWang and L Chen ldquoReliabilityanalysis on sliding stability of gravity dams part I an approachusing criterion of safety margin ratiordquo Journal of HydroelectricEngineering vol 31 no 3 pp 148ndash159 2012

[27] Z Chen J Xu L Chen Y Wang P Sun and C WuldquoReliability analysis on sliding stability of gravity dams Part IIdetermination of shear strength parameters and partial factorsrdquoJournal of Hydroelectric Engineering vol 31 no 3 pp 160ndash1672012 (Chinese)

[28] A W Bishop ldquoThe use of the slip circle in the stability analysisof slopesrdquo Geotechnique vol 5 no 1 pp 7ndash17 1955

[29] E Spencer ldquoA method of analysis of the stability of embank-ments assuming parallel inter-slice forcesrdquoGeotechnique vol 17no 1 pp 11ndash26 1967

[30] N Janbu Slope Stability Computation in Embankment DamEngineering John Wiley amp Sons New York NY USA 1973

[31] R Rackwitz and B Flessler ldquoStructural reliability under com-bined random load sequencesrdquo Computers amp Structures vol 9no 5 pp 489ndash494 1978

[32] M Hohenbichler and R Rackwitz ldquoNon-normal dependentvectors in structural safetyrdquo ASCE Journal of EngineeringMechanics vol 107 no 6 pp 1227ndash1238 1981

[33] P-L Liu and A Der Kiureghian ldquoMultivariate distributionmodels with prescribed marginals and covariancesrdquo Probabilis-tic Engineering Mechanics vol 1 no 2 pp 105ndash112 1986

[34] O Ditlevsen and H O Madsen Structural Reliability MethodsJohn Wiley amp Sons New York NY USA 1996

[35] M Lemaire Structural Reliability John Wiley amp Sons NewYork NY USA 2009

[36] G F Zhao Theory and Applications of Engineering StructuralReliability Dalian Institute of Technology Press Dalian China1996 (Chinese)

[37] Y G Li G F Zhao and B H Zhang ldquoMethod for structuralreliability analysis in generalized random spacerdquo Journal ofDalian University of Technology vol 33 supplement 1 pp 1ndash51993 (Chinese)

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


artificial neural network (ANN) based response surfacemethod in conjunction with the uniform design methodfor predicting failure probability of structures [17] Gavinand Yau described the use of higher order polynomials inorder to approximate the true limit state more accurately incontrast to recently proposed algorithms which focused onthe positions of sample points to improve the accuracy of thequadratic stochastic response surface method (SRSM) [18]Zou et al presented an accurate and efficient MC for limitstate-based reliability analysis at both component and systemlevels using a response surface approximation of the failureindicator function [19] Nguyen et al proposed an adaptiveconstruction of the numerical design in which the responsesurface was fitted by the weighted regression techniquewhich allowed the fitting points to be weighted according totheir distance from the true failure surface and their distancefrom the estimated design point [20] Similar to supportvector machine- (SVM-) based RSM Samui et al adoptedrelevance vector machine- (RVM-) based first-order second-moment method (FOSM) to build a RVM model to predictthe implicit performance function and evaluate the partialderivatives with sufficient accuracy [21] Tan et al discussedsimilarities and differences between radial basis functionnetworks (RBFN) based RSMs and SVM-based RSMs whichindicated that there is no significant difference between themand then proposed two new sampling methods and a hybridRSM to reduce the number of evaluations of the actualperformance function [22 23]

However RSM and its improved methods mentionedabove are relatively complicated and need much more com-puting cost since they have a lot of iterating calculationsat different design points associated with numerical method(ie finite element method) to fit limit state curved surfacewhich is represented by 119892(X) = 0 In this paper a noveldesign point approach in generalized random space (DPGmethod) associated with the method of divided differenceis proposed and applied to analyze the probability of gravitydam foundation instability overmultiple sliding planes In themethod implicit performance function with nonnormal andcorrelated variables is considered and iterating calculationfor 120573 is performed in generalized random space directly Thewhole procedure for reliability analysis does not need muchpreparation mathematically and is relatively simple for thecomplex engineering problems

2 Basic Principle of Computing InstabilityProbability of Gravity Dam Foundation

Due to the complexity of foundation stability against slidingof gravity dam for simplification the instability probabilityis commonly evaluated using models of single or dual slidingplane(s) for the dam foundation [24] However instabilityof foundation over multiple sliding planes is a general caseoccurring in bedrock under most engineering geologicalconditions for a gravity dam Therefore it is necessary toapply a model of multiple sliding planes to analysis oninstability probability of gravity dam foundation but there

is limited existing research regarding this case In tradi-tional deterministic analysis equal safety coefficient method(namely equal-119870 method) is used to analyze and assess thestability failure of dam foundation In the method factor ofsafety 119870 is regarded as an unknown number in nonlinearequations and needs to be solved by iterative calculatingnumerically similar to the application of the methods ofBishop Spencer and Janbu as well which are widely usedfor stability analysis on slope The reliability index 120573 orfailure probability 119901119891 is calculated by implicit performancefunction 119892(X) = 119870 minus 1 associated with the factor ofsafety 119870 Accounting for this the conventional FORM andits improved methods (such as JC method which combinesthe method of equivalent normalization transformation withFORM and is recommended by the Joint Committee ofStructural Safety) cannot be directly applied since the partialderivatives of 119892(X) with respect to the variable X cannot beevaluated directly [1 25] Hence if a partial derivative of 119892(X)is calculated by divided difference mathematically the valueof the derivative at any design point xlowast will be worked outeasily and the limit state curved surface will not be fittedcompletely as RSM [1 25] Usually the JC method associatedwith the method of divided difference could calculate designpoints by iteration on the actual limit state curved surfaceso that it could avoid fitting the overall curved surfaceThe method can provide sufficient accuracy and efficiencyto calculate probability of failure as prone to dealing withcomplex problems for large projects which has been verifiedby many examples in some literatures So it is a good idea touse the similar method to calculate instability probability ofgravity dam foundation over multiple sliding planes

21 Uncertainty in the Analysis Probabilistic methods canreveal the contributions of different components to theuncertainty in the analysis on the probability of failure ofdam foundation for gravity dam The uncertainty related toinstability of dam foundation could be classified as follows(1) the uncertainty of hydrographic andhydraulic parameters(2) the uncertainty of mechanical parameters of rocks andsoils (3) the uncertainty of seismic excitation (4) otheruncertainty underlying the process of dam design construc-tion operation management and so on

The second type of uncertainty above is studied mainlyin the paper and others are ignored by deterministic engi-neering applications just like in conventional analysis fordam stability against sliding The uncertainty of mechanicalparameters of rocks and soils might be enslaved to analysismethods due to subjective or objective conditions derivedfrom the lack of knowledge and so on Always the laboratoryand in situ measured data of the properties of rocks or soilsare inadequate So there is demonstrable distinction betweenthe measured values from the statistics of the small samplingand the true values of the actual physical and mechanicalproperties existing in real world

Many of studies have shown that friction coefficient 1198911015840and cohesion force 1198881015840 against shear fracture of rocks ordiscontinuities are two important uncertainty parameters










(1)


+ int120590d119860 119894) sin120573119894 + int 120591d119860 119894 cos120573119894(2)


119870119894 =1198731198941198911015840119894 + 1198881015840119894 119897119894

119879119894 (3)


120590 120591Ai

Qiminus1

120572iminus1

Ri

Qi

120572iUi Ni

120573if998400i c

998400i li



119870 =119876119894minus1 cos (120572119894minus1 + 120573119894) 119891

1015840119894 minus 119876119894 cos (120572119894 + 120573119894) 119891

1015840119894 + 119901119894






119885 = 119870 minus 1 (5)


119885 = 119870 minus 1 = 119892 (1198831 1198832 119883119897) (6)



1015840119894 where 119894 = 1 2 119899









120579119883119894119883119895 = 120587 minus arccos (120588119883119894119883119895) (7)


X2

0 X1

plowast

120573

120579x1x2

Z lt 0

Z gt 0

Failure state

Reliable state

Z = 0 Limit state



12058811988310158401198941198831015840119895

asymp 120588119883119894119883119895 (8)



defined as

1205831198831015840119894

= 119909lowast119894 minus 1205901198831015840119894

Φminus1 [119865119894 (119909lowast119894 )]

1205901198831015840119894

=120593 Φminus1 [119865119894 (119909

lowast119894 )]

119891119894 (119909lowast119894 )

(9)


lowast2 119909

lowast119894 119909




120573

=sum119899119894=1 (120597119892120597119883119894)


(sum119899119894=1sum119899119895=1 1205881198831015840

1198941198831015840119895

((120597119892120597119883119894) (120597119892120597119883119895))100381610038161003816100381610038161003816119901lowast1205901198831015840119894

1205901198831015840119895

)12

(10)


1205721015840119894

=

sum119899119895=1 1205881198831015840

1198941198831015840119895

(120597119892120597119883119895)100381610038161003816100381610038161003816119901lowast1205901198831015840119895

sum119899119895=1sum119899119896=1 1205881198831015840

1198951198831015840119896

((120597119892120597119883119895) (120597119892120597119883119896))100381610038161003816100381610038161003816119901lowast1205901198831015840119895

1205901198831015840119896

(11)


119909lowast119894 = 1205831198831015840119894

+ 12057312057210158401198941205901198831015840119894

(12)




120597119885

120597119883119894

10038161003816100381610038161003816100381610038161003816119901lowast=120597119892

120597119883119894

10038161003816100381610038161003816100381610038161003816119901lowastasympΔ119892

Δ119909119894

=119892 (1199091 1199092 119909119894 + Δ119909119894 119909119899)

Δ119909119894

minus119892 (1199091 1199092 119909119894 119909119899)

Δ119909119894

(13)




1015840119894 (119894 = 1 2 119899) of the 119899 slip














4 Application






Class of rockmass





















1198701198760 sin (1205720 + 1205731) minus 1198701198761 sin (1205721 + 1205731) + 1198701199021

minus 1198760 cos (1205720 + 1205731) 11989110158401 + 1198761 cos (1205721 + 1205731) 119891

10158401

minus 1199011 = 0

1198701198761sin (1205721 + 1205732) minus 1198701198762sin (1205722 + 1205732) + 1198701199022

minus 1198761cos (1205721 + 1205732) 11989110158402 + 1198762cos (1205722 + 1205732) 119891

10158402

minus 1199012 = 0

1198701198762sin (1205722 + 1205733) minus 1198701198763sin (1205723 + 1205733) + 1198701199023

minus 1198762cos (1205722 + 120573119894) 11989110158403 + 1198763cos (1205723 + 1205733) 119891

10158403

minus 1199013 = 0

(14)

where

1205720 = 1205721 = 1205722 = 90∘

1205731 = minus14∘

1205732 = minus16∘

1205733 = minus14∘







Number ofslip paths



1198911015840 1198881015840MPa

A


B


C


119901119894


+ 1198881015840119894 119897119894 119894 = 1 2 3

119902119894 = (119877119894 + int120590d119860 119894) sin120573119894 + int 120591d119860 119894 cos120573119894

119894 = 1 2 3

int 120590d119860 119894 = 119881119894 119894 = 1 2 3

int 120591d119860 119894 = 119867119894 119894 = 1 2 3

1198971 = 499m

1198972 = 335m

1198973 = 1174m

1198760 = 1198763 = 0

(15)







F18

F17

A2

B2

C2

F16

f52

A1(B1 C1)fh06

fh07

fh09

fh10

fh11

A3

B3

A4

B4

C3

A5 A6B5

C5C6fh152-2

fh152-4

fh152-7fh152-8

fh152-9C4









B2-B3) 120573 119901119891 Notes11989110158401lowast

11988810158401lowastMPa






B1

B2B3

B4

B5

V

H

III III




plane B2-B3) 120573 119901119891

11989110158401lowast

11988810158401lowastMPa

1 00 0107 0039 3664 1244 times 10minus4














100E minus 03

100E minus 05

100E minus 07

100E minus 09

100E minus 11


pf

120588f998400c998400





010 015 020 025 030

100E minus 02

100E minus 03

100E minus 04

100E minus 05

100E minus 06

100E minus 07

100E minus 08

120575f998400

120575c998400 = 045

120575c998400 = 030

120575c998400 = 015

pf





minus3



minus5



015 020 025 030 035 040 045

100E minus 02

100E minus 03

100E minus 04

100E minus 05

100E minus 06

100E minus 07

100E minus 08

120575c998400

120575f998400 = 026

120575f998400 = 020

120575f998400 = 014

pf








120573 119901119891 119870




5 Conclusions









Competing Interests


Acknowledgments


References














































Volume 2014




Journal of











Journal of


Function Spaces






Algebra


















(1)


+ int120590d119860 119894) sin120573119894 + int 120591d119860 119894 cos120573119894(2)


119870119894 =1198731198941198911015840119894 + 1198881015840119894 119897119894

119879119894 (3)


120590 120591Ai

Qiminus1

120572iminus1

Ri

Qi

120572iUi Ni

120573if998400i c

998400i li



119870 =119876119894minus1 cos (120572119894minus1 + 120573119894) 119891

1015840119894 minus 119876119894 cos (120572119894 + 120573119894) 119891

1015840119894 + 119901119894






119885 = 119870 minus 1 (5)


119885 = 119870 minus 1 = 119892 (1198831 1198832 119883119897) (6)



1015840119894 where 119894 = 1 2 119899









120579119883119894119883119895 = 120587 minus arccos (120588119883119894119883119895) (7)


X2

0 X1

plowast

120573

120579x1x2

Z lt 0

Z gt 0

Failure state

Reliable state

Z = 0 Limit state



12058811988310158401198941198831015840119895

asymp 120588119883119894119883119895 (8)



defined as

1205831198831015840119894

= 119909lowast119894 minus 1205901198831015840119894

Φminus1 [119865119894 (119909lowast119894 )]

1205901198831015840119894

=120593 Φminus1 [119865119894 (119909

lowast119894 )]

119891119894 (119909lowast119894 )

(9)


lowast2 119909

lowast119894 119909




120573

=sum119899119894=1 (120597119892120597119883119894)


(sum119899119894=1sum119899119895=1 1205881198831015840

1198941198831015840119895

((120597119892120597119883119894) (120597119892120597119883119895))100381610038161003816100381610038161003816119901lowast1205901198831015840119894

1205901198831015840119895

)12

(10)


1205721015840119894

=

sum119899119895=1 1205881198831015840

1198941198831015840119895

(120597119892120597119883119895)100381610038161003816100381610038161003816119901lowast1205901198831015840119895

sum119899119895=1sum119899119896=1 1205881198831015840

1198951198831015840119896

((120597119892120597119883119895) (120597119892120597119883119896))100381610038161003816100381610038161003816119901lowast1205901198831015840119895

1205901198831015840119896

(11)


119909lowast119894 = 1205831198831015840119894

+ 12057312057210158401198941205901198831015840119894

(12)




120597119885

120597119883119894

10038161003816100381610038161003816100381610038161003816119901lowast=120597119892

120597119883119894

10038161003816100381610038161003816100381610038161003816119901lowastasympΔ119892

Δ119909119894

=119892 (1199091 1199092 119909119894 + Δ119909119894 119909119899)

Δ119909119894

minus119892 (1199091 1199092 119909119894 119909119899)

Δ119909119894

(13)




1015840119894 (119894 = 1 2 119899) of the 119899 slip














4 Application






Class of rockmass





















1198701198760 sin (1205720 + 1205731) minus 1198701198761 sin (1205721 + 1205731) + 1198701199021

minus 1198760 cos (1205720 + 1205731) 11989110158401 + 1198761 cos (1205721 + 1205731) 119891

10158401

minus 1199011 = 0

1198701198761sin (1205721 + 1205732) minus 1198701198762sin (1205722 + 1205732) + 1198701199022

minus 1198761cos (1205721 + 1205732) 11989110158402 + 1198762cos (1205722 + 1205732) 119891

10158402

minus 1199012 = 0

1198701198762sin (1205722 + 1205733) minus 1198701198763sin (1205723 + 1205733) + 1198701199023

minus 1198762cos (1205722 + 120573119894) 11989110158403 + 1198763cos (1205723 + 1205733) 119891

10158403

minus 1199013 = 0

(14)

where

1205720 = 1205721 = 1205722 = 90∘

1205731 = minus14∘

1205732 = minus16∘

1205733 = minus14∘







Number ofslip paths



1198911015840 1198881015840MPa

A


B


C


119901119894


+ 1198881015840119894 119897119894 119894 = 1 2 3

119902119894 = (119877119894 + int120590d119860 119894) sin120573119894 + int 120591d119860 119894 cos120573119894

119894 = 1 2 3

int 120590d119860 119894 = 119881119894 119894 = 1 2 3

int 120591d119860 119894 = 119867119894 119894 = 1 2 3

1198971 = 499m

1198972 = 335m

1198973 = 1174m

1198760 = 1198763 = 0

(15)







F18

F17

A2

B2

C2

F16

f52

A1(B1 C1)fh06

fh07

fh09

fh10

fh11

A3

B3

A4

B4

C3

A5 A6B5

C5C6fh152-2

fh152-4

fh152-7fh152-8

fh152-9C4









B2-B3) 120573 119901119891 Notes11989110158401lowast

11988810158401lowastMPa






B1

B2B3

B4

B5

V

H

III III




plane B2-B3) 120573 119901119891

11989110158401lowast

11988810158401lowastMPa

1 00 0107 0039 3664 1244 times 10minus4














100E minus 03

100E minus 05

100E minus 07

100E minus 09

100E minus 11


pf

120588f998400c998400





010 015 020 025 030

100E minus 02

100E minus 03

100E minus 04

100E minus 05

100E minus 06

100E minus 07

100E minus 08

120575f998400

120575c998400 = 045

120575c998400 = 030

120575c998400 = 015

pf





minus3



minus5



015 020 025 030 035 040 045

100E minus 02

100E minus 03

100E minus 04

100E minus 05

100E minus 06

100E minus 07

100E minus 08

120575c998400

120575f998400 = 026

120575f998400 = 020

120575f998400 = 014

pf








120573 119901119891 119870




5 Conclusions









Competing Interests


Acknowledgments


References














































Volume 2014




Journal of











Journal of


Function Spaces






Algebra















120579119883119894119883119895 = 120587 minus arccos (120588119883119894119883119895) (7)


X2

0 X1

plowast

120573

120579x1x2

Z lt 0

Z gt 0

Failure state

Reliable state

Z = 0 Limit state



12058811988310158401198941198831015840119895

asymp 120588119883119894119883119895 (8)



defined as

1205831198831015840119894

= 119909lowast119894 minus 1205901198831015840119894

Φminus1 [119865119894 (119909lowast119894 )]

1205901198831015840119894

=120593 Φminus1 [119865119894 (119909

lowast119894 )]

119891119894 (119909lowast119894 )

(9)


lowast2 119909

lowast119894 119909




120573

=sum119899119894=1 (120597119892120597119883119894)


(sum119899119894=1sum119899119895=1 1205881198831015840

1198941198831015840119895

((120597119892120597119883119894) (120597119892120597119883119895))100381610038161003816100381610038161003816119901lowast1205901198831015840119894

1205901198831015840119895

)12

(10)


1205721015840119894

=

sum119899119895=1 1205881198831015840

1198941198831015840119895

(120597119892120597119883119895)100381610038161003816100381610038161003816119901lowast1205901198831015840119895

sum119899119895=1sum119899119896=1 1205881198831015840

1198951198831015840119896

((120597119892120597119883119895) (120597119892120597119883119896))100381610038161003816100381610038161003816119901lowast1205901198831015840119895

1205901198831015840119896

(11)


119909lowast119894 = 1205831198831015840119894

+ 12057312057210158401198941205901198831015840119894

(12)




120597119885

120597119883119894

10038161003816100381610038161003816100381610038161003816119901lowast=120597119892

120597119883119894

10038161003816100381610038161003816100381610038161003816119901lowastasympΔ119892

Δ119909119894

=119892 (1199091 1199092 119909119894 + Δ119909119894 119909119899)

Δ119909119894

minus119892 (1199091 1199092 119909119894 119909119899)

Δ119909119894

(13)




1015840119894 (119894 = 1 2 119899) of the 119899 slip














4 Application






Class of rockmass





















1198701198760 sin (1205720 + 1205731) minus 1198701198761 sin (1205721 + 1205731) + 1198701199021

minus 1198760 cos (1205720 + 1205731) 11989110158401 + 1198761 cos (1205721 + 1205731) 119891

10158401

minus 1199011 = 0

1198701198761sin (1205721 + 1205732) minus 1198701198762sin (1205722 + 1205732) + 1198701199022

minus 1198761cos (1205721 + 1205732) 11989110158402 + 1198762cos (1205722 + 1205732) 119891

10158402

minus 1199012 = 0

1198701198762sin (1205722 + 1205733) minus 1198701198763sin (1205723 + 1205733) + 1198701199023

minus 1198762cos (1205722 + 120573119894) 11989110158403 + 1198763cos (1205723 + 1205733) 119891

10158403

minus 1199013 = 0

(14)

where

1205720 = 1205721 = 1205722 = 90∘

1205731 = minus14∘

1205732 = minus16∘

1205733 = minus14∘







Number ofslip paths



1198911015840 1198881015840MPa

A


B


C


119901119894


+ 1198881015840119894 119897119894 119894 = 1 2 3

119902119894 = (119877119894 + int120590d119860 119894) sin120573119894 + int 120591d119860 119894 cos120573119894

119894 = 1 2 3

int 120590d119860 119894 = 119881119894 119894 = 1 2 3

int 120591d119860 119894 = 119867119894 119894 = 1 2 3

1198971 = 499m

1198972 = 335m

1198973 = 1174m

1198760 = 1198763 = 0

(15)







F18

F17

A2

B2

C2

F16

f52

A1(B1 C1)fh06

fh07

fh09

fh10

fh11

A3

B3

A4

B4

C3

A5 A6B5

C5C6fh152-2

fh152-4

fh152-7fh152-8

fh152-9C4









B2-B3) 120573 119901119891 Notes11989110158401lowast

11988810158401lowastMPa






B1

B2B3

B4

B5

V

H

III III




plane B2-B3) 120573 119901119891

11989110158401lowast

11988810158401lowastMPa

1 00 0107 0039 3664 1244 times 10minus4














100E minus 03

100E minus 05

100E minus 07

100E minus 09

100E minus 11


pf

120588f998400c998400





010 015 020 025 030

100E minus 02

100E minus 03

100E minus 04

100E minus 05

100E minus 06

100E minus 07

100E minus 08

120575f998400

120575c998400 = 045

120575c998400 = 030

120575c998400 = 015

pf





minus3



minus5



015 020 025 030 035 040 045

100E minus 02

100E minus 03

100E minus 04

100E minus 05

100E minus 06

100E minus 07

100E minus 08

120575c998400

120575f998400 = 026

120575f998400 = 020

120575f998400 = 014

pf








120573 119901119891 119870




5 Conclusions









Competing Interests


Acknowledgments


References














































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











120597119885

120597119883119894

10038161003816100381610038161003816100381610038161003816119901lowast=120597119892

120597119883119894

10038161003816100381610038161003816100381610038161003816119901lowastasympΔ119892

Δ119909119894

=119892 (1199091 1199092 119909119894 + Δ119909119894 119909119899)

Δ119909119894

minus119892 (1199091 1199092 119909119894 119909119899)

Δ119909119894

(13)




1015840119894 (119894 = 1 2 119899) of the 119899 slip














4 Application






Class of rockmass





















1198701198760 sin (1205720 + 1205731) minus 1198701198761 sin (1205721 + 1205731) + 1198701199021

minus 1198760 cos (1205720 + 1205731) 11989110158401 + 1198761 cos (1205721 + 1205731) 119891

10158401

minus 1199011 = 0

1198701198761sin (1205721 + 1205732) minus 1198701198762sin (1205722 + 1205732) + 1198701199022

minus 1198761cos (1205721 + 1205732) 11989110158402 + 1198762cos (1205722 + 1205732) 119891

10158402

minus 1199012 = 0

1198701198762sin (1205722 + 1205733) minus 1198701198763sin (1205723 + 1205733) + 1198701199023

minus 1198762cos (1205722 + 120573119894) 11989110158403 + 1198763cos (1205723 + 1205733) 119891

10158403

minus 1199013 = 0

(14)

where

1205720 = 1205721 = 1205722 = 90∘

1205731 = minus14∘

1205732 = minus16∘

1205733 = minus14∘







Number ofslip paths



1198911015840 1198881015840MPa

A


B


C


119901119894


+ 1198881015840119894 119897119894 119894 = 1 2 3

119902119894 = (119877119894 + int120590d119860 119894) sin120573119894 + int 120591d119860 119894 cos120573119894

119894 = 1 2 3

int 120590d119860 119894 = 119881119894 119894 = 1 2 3

int 120591d119860 119894 = 119867119894 119894 = 1 2 3

1198971 = 499m

1198972 = 335m

1198973 = 1174m

1198760 = 1198763 = 0

(15)







F18

F17

A2

B2

C2

F16

f52

A1(B1 C1)fh06

fh07

fh09

fh10

fh11

A3

B3

A4

B4

C3

A5 A6B5

C5C6fh152-2

fh152-4

fh152-7fh152-8

fh152-9C4









B2-B3) 120573 119901119891 Notes11989110158401lowast

11988810158401lowastMPa






B1

B2B3

B4

B5

V

H

III III




plane B2-B3) 120573 119901119891

11989110158401lowast

11988810158401lowastMPa

1 00 0107 0039 3664 1244 times 10minus4














100E minus 03

100E minus 05

100E minus 07

100E minus 09

100E minus 11


pf

120588f998400c998400





010 015 020 025 030

100E minus 02

100E minus 03

100E minus 04

100E minus 05

100E minus 06

100E minus 07

100E minus 08

120575f998400

120575c998400 = 045

120575c998400 = 030

120575c998400 = 015

pf





minus3



minus5



015 020 025 030 035 040 045

100E minus 02

100E minus 03

100E minus 04

100E minus 05

100E minus 06

100E minus 07

100E minus 08

120575c998400

120575f998400 = 026

120575f998400 = 020

120575f998400 = 014

pf








120573 119901119891 119870




5 Conclusions









Competing Interests


Acknowledgments


References














































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











Class of rockmass





















1198701198760 sin (1205720 + 1205731) minus 1198701198761 sin (1205721 + 1205731) + 1198701199021

minus 1198760 cos (1205720 + 1205731) 11989110158401 + 1198761 cos (1205721 + 1205731) 119891

10158401

minus 1199011 = 0

1198701198761sin (1205721 + 1205732) minus 1198701198762sin (1205722 + 1205732) + 1198701199022

minus 1198761cos (1205721 + 1205732) 11989110158402 + 1198762cos (1205722 + 1205732) 119891

10158402

minus 1199012 = 0

1198701198762sin (1205722 + 1205733) minus 1198701198763sin (1205723 + 1205733) + 1198701199023

minus 1198762cos (1205722 + 120573119894) 11989110158403 + 1198763cos (1205723 + 1205733) 119891

10158403

minus 1199013 = 0

(14)

where

1205720 = 1205721 = 1205722 = 90∘

1205731 = minus14∘

1205732 = minus16∘

1205733 = minus14∘







Number ofslip paths



1198911015840 1198881015840MPa

A


B


C


119901119894


+ 1198881015840119894 119897119894 119894 = 1 2 3

119902119894 = (119877119894 + int120590d119860 119894) sin120573119894 + int 120591d119860 119894 cos120573119894

119894 = 1 2 3

int 120590d119860 119894 = 119881119894 119894 = 1 2 3

int 120591d119860 119894 = 119867119894 119894 = 1 2 3

1198971 = 499m

1198972 = 335m

1198973 = 1174m

1198760 = 1198763 = 0

(15)







F18

F17

A2

B2

C2

F16

f52

A1(B1 C1)fh06

fh07

fh09

fh10

fh11

A3

B3

A4

B4

C3

A5 A6B5

C5C6fh152-2

fh152-4

fh152-7fh152-8

fh152-9C4









B2-B3) 120573 119901119891 Notes11989110158401lowast

11988810158401lowastMPa






B1

B2B3

B4

B5

V

H

III III




plane B2-B3) 120573 119901119891

11989110158401lowast

11988810158401lowastMPa

1 00 0107 0039 3664 1244 times 10minus4














100E minus 03

100E minus 05

100E minus 07

100E minus 09

100E minus 11


pf

120588f998400c998400





010 015 020 025 030

100E minus 02

100E minus 03

100E minus 04

100E minus 05

100E minus 06

100E minus 07

100E minus 08

120575f998400

120575c998400 = 045

120575c998400 = 030

120575c998400 = 015

pf





minus3



minus5



015 020 025 030 035 040 045

100E minus 02

100E minus 03

100E minus 04

100E minus 05

100E minus 06

100E minus 07

100E minus 08

120575c998400

120575f998400 = 026

120575f998400 = 020

120575f998400 = 014

pf








120573 119901119891 119870




5 Conclusions









Competing Interests


Acknowledgments


References














































Volume 2014




Journal of











Journal of


Function Spaces






Algebra















Number ofslip paths



1198911015840 1198881015840MPa

A


B


C


119901119894


+ 1198881015840119894 119897119894 119894 = 1 2 3

119902119894 = (119877119894 + int120590d119860 119894) sin120573119894 + int 120591d119860 119894 cos120573119894

119894 = 1 2 3

int 120590d119860 119894 = 119881119894 119894 = 1 2 3

int 120591d119860 119894 = 119867119894 119894 = 1 2 3

1198971 = 499m

1198972 = 335m

1198973 = 1174m

1198760 = 1198763 = 0

(15)







F18

F17

A2

B2

C2

F16

f52

A1(B1 C1)fh06

fh07

fh09

fh10

fh11

A3

B3

A4

B4

C3

A5 A6B5

C5C6fh152-2

fh152-4

fh152-7fh152-8

fh152-9C4









B2-B3) 120573 119901119891 Notes11989110158401lowast

11988810158401lowastMPa






B1

B2B3

B4

B5

V

H

III III




plane B2-B3) 120573 119901119891

11989110158401lowast

11988810158401lowastMPa

1 00 0107 0039 3664 1244 times 10minus4














100E minus 03

100E minus 05

100E minus 07

100E minus 09

100E minus 11


pf

120588f998400c998400





010 015 020 025 030

100E minus 02

100E minus 03

100E minus 04

100E minus 05

100E minus 06

100E minus 07

100E minus 08

120575f998400

120575c998400 = 045

120575c998400 = 030

120575c998400 = 015

pf





minus3



minus5



015 020 025 030 035 040 045

100E minus 02

100E minus 03

100E minus 04

100E minus 05

100E minus 06

100E minus 07

100E minus 08

120575c998400

120575f998400 = 026

120575f998400 = 020

120575f998400 = 014

pf








120573 119901119891 119870




5 Conclusions









Competing Interests


Acknowledgments


References














































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










F18

F17

A2

B2

C2

F16

f52

A1(B1 C1)fh06

fh07

fh09

fh10

fh11

A3

B3

A4

B4

C3

A5 A6B5

C5C6fh152-2

fh152-4

fh152-7fh152-8

fh152-9C4









B2-B3) 120573 119901119891 Notes11989110158401lowast

11988810158401lowastMPa






B1

B2B3

B4

B5

V

H

III III




plane B2-B3) 120573 119901119891

11989110158401lowast

11988810158401lowastMPa

1 00 0107 0039 3664 1244 times 10minus4














100E minus 03

100E minus 05

100E minus 07

100E minus 09

100E minus 11


pf

120588f998400c998400





010 015 020 025 030

100E minus 02

100E minus 03

100E minus 04

100E minus 05

100E minus 06

100E minus 07

100E minus 08

120575f998400

120575c998400 = 045

120575c998400 = 030

120575c998400 = 015

pf





minus3



minus5



015 020 025 030 035 040 045

100E minus 02

100E minus 03

100E minus 04

100E minus 05

100E minus 06

100E minus 07

100E minus 08

120575c998400

120575f998400 = 026

120575f998400 = 020

120575f998400 = 014

pf








120573 119901119891 119870




5 Conclusions









Competing Interests


Acknowledgments


References














































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










B1

B2B3

B4

B5

V

H

III III




plane B2-B3) 120573 119901119891

11989110158401lowast

11988810158401lowastMPa

1 00 0107 0039 3664 1244 times 10minus4














100E minus 03

100E minus 05

100E minus 07

100E minus 09

100E minus 11


pf

120588f998400c998400





010 015 020 025 030

100E minus 02

100E minus 03

100E minus 04

100E minus 05

100E minus 06

100E minus 07

100E minus 08

120575f998400

120575c998400 = 045

120575c998400 = 030

120575c998400 = 015

pf





minus3



minus5



015 020 025 030 035 040 045

100E minus 02

100E minus 03

100E minus 04

100E minus 05

100E minus 06

100E minus 07

100E minus 08

120575c998400

120575f998400 = 026

120575f998400 = 020

120575f998400 = 014

pf








120573 119901119891 119870




5 Conclusions









Competing Interests


Acknowledgments


References














































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










015 020 025 030 035 040 045

100E minus 02

100E minus 03

100E minus 04

100E minus 05

100E minus 06

100E minus 07

100E minus 08

120575c998400

120575f998400 = 026

120575f998400 = 020

120575f998400 = 014

pf








120573 119901119891 119870




5 Conclusions









Competing Interests


Acknowledgments


References














































Volume 2014




Journal of











Journal of


Function Spaces






Algebra















Competing Interests


Acknowledgments


References














































Volume 2014




Journal of











Journal of


Function Spaces






Algebra








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra









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