Uncertainty and reliability analysis applied to slope stability

Uncertainty and reliability analysis applied to slope stability

Abdallah I. Husein Malkawi *, Waleed F. Hassan, Fayez A. Abdulla

Civil Engineering Department, Jordan University of Science and Technology, 221-10 Irbid, Jordan

Abstract

In this paper, reliability analysis of slope stability is presented using two methods of uncertainty ®rst-order second-moment method (FOSM) and Monte Carlo simulation method (MCSM). The results of thesemethods are compared using four recognized methods of slope stability. These are Ordinary method ofslices, simpli®ed Bishop's method, simpli®ed Janbu's method and Spencer's method. Two illustrativeexamples are presented in this paper: one is homogenous slope and the other is non-homogeneous layeredslope. The study shows that the reliability index (�) is independent of the seed random number generatorand a sample size of 700 or greater is a good choice for MCSM. In the case of homogeneous slope a goodagreement is observed between the calculated (�) using FOSM and MCSM for both the Ordinary and theBishop's method. However, slight di�erence in (�) is observed between the two uncertainty methods whe-ther Janbu's method or Spencer's method is used. In the case of the layered slope good agreement isobtained between the two uncertainty methods for Ordinary, Bishop and Janbu methods. Similar toexample 1, Spencer's method shows also slight di�erence in (�) between FOSM and MCSM methods.Model uncertainty is addressed by evaluating the relative performance of the three slope stability methodsi.e. Ordinary, Bishop and Janbu methods as compared to Spencer's method. # 2000 Elsevier Science Ltd.All rights reserved.

Keywords: Slope stability; Factor of safety; Model uncertainty; Reliability index; First-order second-moment; MonteCarlo simulation

1. Introduction

In geotechnical engineering analysis and design various sources of uncertainties are encoun-tered and well recognized. Several features usually contribute to such uncertainties, like: (1) thoseassociated with inherent randomness of natural processes; (2) Model uncertainty re¯ecting theinability of the simulation model, design technique or empirical formula to represent the system'strue physical behavior, such as calculating the safety factor of slopes using limiting equilibriummethods of slices; (3) Model parameter uncertainties resulting from inability to quantify accurately

0167-4730/00/$ - see front matter # 2000 Elsevier Science Ltd. All rights reserved.

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Structural Safety 22 (2000) 161±187www.elsevier.nl/locate/strusafe

* Corresponding author. Tel.: +962-2-295111, ext. 2112; fax: +962-2-710-2299.

E-mail address: [email protected] (A.I. Husein Malkawi).

the model input parameters and (4) Data uncertainties including (a) measurement errors, (b) datainconsistency and non-homogeneity and (c) data handling. In slope stability computations, var-ious sources of uncertainties are encountered, such as geological details missed in the explorationprogram, estimation of soil properties that are di�cult to quantify, i.e. the spatial variability inthe ®eld cannot be reproduced accurately, ¯uctuation in pore water pressure, testing errors andmany other relevant factors.In a deterministic analysis, the factor of safety (F) is de®ned as the ratio of resisting to driving

forces on a potential sliding surface. The slope is considered safe only if the calculated safetyfactor clearly exceeds unity. Whereas, in a probabilistic framework the factor of safety is expres-sed in terms of its mean value as well as its variance. Reliability analysis is therefore used to assessuncertainties in engineering variables such as the factor of safety of slope stability. The reliabilityindex (�) is often used to express the degree of uncertainty in the calculated factor of safety. Suchuncertainty is usually assessed by di�erent approaches such as the ®rst-order second-momentmethod, point estimate method, and Monte Carlo simulation method.Wu and Kraft [1], Cornell [2], Alonso [3], Tang et al., [4], Venmarcke [5], Wol� [6], Li and

Lumb [7], Barabosa et al., [8], among others, applied a probabilistic approach in analyzing slopestability using the ®rst-order second-moment method. Recently, a number of applications ofprobabilistic slope stability studies using other numerical approaches, such as Monte Carlosimulation or point estimate method, have been reported in the literature [9,10]Christian et al. [11] used the mean-®rst order reliability method, which is a simpli®cation of the

more general ®rst order reliability method. They found that the reliability analysis is especiallyuseful in establishing design values of safety factor representing consistent risks for di�erent typesof failure. Tobutt [12] used the Monte Carlo method as a sensitivity-testing tool for slope stabilityand also as a method for calculating the probability of failure of a given earth slope.This paper presents a probabilistic based approach by which the relevant sources of uncer-

tainty involved in slope stability analysis can be modeled and analyzed. Two di�erent methodsare used to quantify the uncertainty in the calculated safety factor. These are the ®rst-order second-moment method (FOSM) and Monte Carlo simulation method (MCSM). Two examples, repre-senting homogeneous and layered slopes, were selected to investigate how the probability of failureand the reliability index may vary among those two methods. The results were obtained using fourwell-recognized methods of slope stability analysis, i.e. Ordinary method [13], Bishop's method [14],Janbu's method [15,16] and Spencer's method [17]. Model uncertainty was addressed by evaluatingthe relative performance of the simpli®ed slope stability methods compared to Spencer's method.

2. Limiting equilibrium methods

Most problems in slope stability are statically indeterminate, and as a result, some simplifyingassumptions are made in order to determine a unique factor of safety. Due to the di�erences inassumptions, various methods have been developed. Among the most popular methods are pro-cedures proposed by Fellenius, Bishop, Janbu and Spencer referred to before [13±17]. Some ofthese methods satisfy only overall moment, like the Ordinary and simpli®ed Bishop methods andare applicable to a circular slip surface, while Janbu's method satis®es only force equilibrium andis applicable to any shape. Spencer's method, however, satis®es both moment and force equilibrium

162 A.I. Husein Malkawi et al. / Structural Safety 22 (2000) 161±187

and it is applicable to failure surfaces of any shape. It is considered as one of the rigorous andaccurate methods for solving stability problems. Table 1 presents a summary of static equilibriumconditions in limit equilibrium methods of slices considered in this study. In order to formulatethe algorithm to solve for the factor of safety based on the above-mentioned methods, one shouldconsider the forces acting on a typical slice as shown in Fig. 1.For the Ordinary method of slices, which is considered the simplest method of slices, the factor

of safety is directly obtained. The method assumes that the inter-slice forces are parallel to thebase of each slice, thus they can be neglected and the factor of safety is given as follows:

Table 1Summary of static equilibrium conditions in di�erent limit equilibrium methods of slices

Method Force equilibrium Moment equilibrium

1st Direction 2nd Direction

Ordinary or Fellenius Yes No YesBishop's simpli®ed Yes No YesJanbu's simpli®ed Yes Yes No

Spencer Yes Yes Yes

Fig. 1. Depicts forces acting on a typical slice.

A.I. Husein Malkawi et al. / Structural Safety 22 (2000) 161±187 163

F �Pni�1�c0b sec �� W cos��Q cos��ÿ �� WW cos��ÿ �� khW sin n�ÿU:b� tan�0�

Pni�1�W�WW cos��Q cos�� sin �ÿPn

i�1�WW sin��Q sin��cos�ÿ h

R��Pni�1

khW�cos�ÿ ha

R�

�1�

All terms and symbols appearing in Eqs. (1)±(5) are de®ned in Fig. 1.In Bishop's method the factor of safety is determined by trial and errors, using an iterative

process, since the factor of safety (F) appears in both sides of Eq. (2). The inter-slice shear forcesare neglected, and only the normal forces are used to de®ne the inter-slice forces. The factor ofsafety is given as follows:

F �

Pni�1�c0b sec �� 1

cos�� sin� tan�0F

�Wÿ c0b tan�F ÿU:b�WW cos��Q cos�� tan�0��

Pni�1�W�WW cos��Q cos�� sin �ÿPn

i�1�WW sin ��Q sin��cos�ÿ h

R�

�Xni�1

khW�cos�ÿ ha

R�

�2�

Similarly, for Janbu's method the factor of safety is determined also by an iterative procedurethrough varying the e�ective normal stress on the failure surface. The inter-slice shear forces areignored and the normal forces are derived from the summation of vertical forces. The resultingfactor of safety is given below:

F �

Pni�1�c0b sec�

�� 1


�Wÿ c0b tan�F ÿ ub cos��WW cos��Q cos�� tan�0� cos��

Pni�1�U:b sin��Wkh ÿWW sin �ÿQ sin��

�Xni�1

1


�Wÿ c0b sin�F ÿ ub cos��WW cos��Q cos�� sin�

�3�

In Spencer's method, the e�ect of inter-slice forces is included and both moment and force equi-librium are explicitly satis®ed. This eventually will lead to an accurate calculation of the factor ofsafety. The factor of safety is determined through an iterative procedure, slice by slice, by varyingF and � until force and moment equilibrium are satis®ed.The force equilibrium equation is as follows:


ZR � ZL � FW sin�ÿ c0b sec�ÿW cos� tan�0

sin��ÿ �� tan�0 ÿ F cos��ÿ ��U:b sec� tan�0 �Wkh�Fÿ tan�0 tan�� cos�

sin��ÿ �� tan�0 ÿ F cos��ÿ ��Q�F sin��ÿ �� ÿ cos��ÿ �� tan�0�

sin��ÿ �� tan�0 ÿ F cos��ÿ �� WW�F sin��ÿ �� ÿ cos��ÿ �� tan�0�

sin��ÿ �� tan�0 ÿ F cos��ÿ �� 4�

While the moment equilibrium equation is given by:

hR � ZL

ZRhL ÿ ZL

ZR

b

2tan�� ZL

ZRtan �

b

2� b

2tan �ÿ b

2tan �

� hWW sin �

ZR cos �� hQ sin�

ZR cos �ÿ hakhW

ZR cos ��5�

The iteration is terminated when the calculated values of ZR and hR di�er within an acceptabletolerance from the known values of ZR and hR at the boundary.The computer software called stability analysis of slopes using Monte Carlo technique (SAS-

MCT) developed by Husein Malkawi and Hassan [18] is used in this study to perform slope sta-bility analysis using the above mentioned method of slices. The program performs probabilisticapproach in analyzing slopes stability problem; reliability analysis is performed using MonteCarlo simulation technique. It generates a large number of di�erent soil parameter sets and cal-culates the safety factor for each random set. Then, the generated factors of safety are used toconstruct the associated probability distribution. The corresponding reliability index (�) and theprobability of failure (Pf) of the slope can be obtained.

3. Methods of reliability evaluation

3.1. First-order second-moment (FOSM) method

The FOSM method estimates the uncertainty of the factor of safety of a slope againstinstability as a function of the variances of the stochastic input variables, such as soil frictionangle, soil cohesion, and soil unit weight. It uses Taylor's series expansion to estimate the localuncertainty of the factor of safety at a selected expansion point.Consider that the factor of safety (F) can be expressed as a function of stochastic input para-

meters Xs as

F � g�XT� � g�x1; x2; . . . ;xn� �6�

in which X is an n-dimensional vector of stochastic input parameters; the superscript (T) is thematrix or vector transpose; and g( ) represents a functional relationship for the safety factor. Inthe context of the present study, g(XT) is the factor of safety formulation using the four


mentioned methods of slope stability analysis. The stochastic input vector X consists of the soilvariables. The FOSM method considers the ®rst order Taylor series expansion term of Eq. (6)

F � g�xTo � �

Xni�1

sio�Xi ÿ xio� � g�xTo � � sT

o �Xÿ xo� �7�

where so is an n-dimensional column vector of sensitivity coe�cients with element sio � �@g=@xi�xo

being the sensitivity coe�cient of the safety factor F with respect to the ith input parameter, Xi atthe expansion point xo.Applying the expectation and variance operators to Eq. (7) with xo � �, the mean and variance

of the safety factor (F ) can be estimated as

E�F� � �F � g��T� �8�

Var�F� � �2F � sTX s �9�

where, �F and �F are the mean and standard deviation of the model output, respectively. � andX are the vector of means and the covariance matrix of stochastic input parameters, respectively,and s is the sensitivity coe�cient vector evaluated at xo � �. If all stochastic input parameters areindependent, the variance of the safety factor reduces to

Var�F� � �2F �Xni�1

s2i �2i �

Xni�1� @g@xi�2j�xi

�2i �10�

As can be seen from Eqs. (9) and (10), the uncertainty of the safety factor Var(F), depends notonly on the uncertainty of individual stochastic input parameters as measured by �2i , but also onthe associated sensitivity coe�cients (si). The assumption of independent (uncorrelated) soilproperties (�, C and ) is reported by various researchers. For example, Matsuo and Kuroda [19]and later on Matsuo [20], showed that cohesion and tanf are uncorrelated. Lamb [21,22] showedalso that cohesion and tan� have negligible correlation. YuÈ cemen et. al. [23], Dettinger andWilson [24], Chowdhury and Xu [10] and Christian et al. [11] assumed in their studies such sta-tistical independence between soil properties.Usually, two approaches are used to estimate the variance of the safety factor as approximated

by the ®rst-order second-moment method. The ®rst approach is a direct evaluation of the di�er-ential equation given in Eq. (10). This is a close form solution for Var(F). However, for mostmethods of slope stability such evaluation is practically impossible and inconvenient. The secondapproach involves a numerical approximation of the partial derivatives. This is usually used andrecommended for computing the variance (uncertainty) of the safety factor.

3.2. Monte Carlo simulation

Monte Carlo simulation is a method used to obtain the probability distribution of dependentrandom variables given the probability distribution of a set of independent random variables.Thus, in Monte Carlo simulation studies three steps are usually required, namely: determining the


independent variable (input), transforming the input as independent variable (output), and thenanalyzing the output. Monte Carlo simulation o�ers a practical approach to reliability analysisbecause the stochastic nature of the system response (output) can be probabilistically duplicated.In this technique a large number of soil variables such as shear strength, angle of internal fric-

tion and unit weight of the soil can be sampled from their known (or assumed) probability dis-tribution. For this purpose, the probability density function for each of these soil variables mustbe speci®ed. Usually, a normal distribution is assumed for the soil properties see Matsuo andKuroda [19] and Tobutt [12]. Then, the corresponding safety factor of each set is calculated.These values of safety factors are plotted on a probability paper in order to determine the dis-tribution of the safety factor. The reliability index (�) and the probability of failure (Pf) are thencalculated using the safety factor probability distribution. This approach can be applied to anymethod of slices, that uses limit equilibrium in the analysis of slopes. In this paper, uncertainty inslope stability is quanti®ed by evaluating the reliability index, which is de®ned as:

� � E�F� ÿ 1:0

��F� �11�

where � is the reliability index, E(F) the expected value of the safety factor, and �(F) is the stan-dard deviation.

4. Uncertainty and reliability index calculation

Fig. 2 illustrates schematically the methodology used to evaluate the uncertainty and thereliability of slope stability factor of safety for both the FOSM and MCSM. The ®rst step in themethodology is to specify the slope geometry and the probability distribution for the soil prop-erties (�, C and ). The second step is to search for the critical slip surface and its associatedfactor of safety using limiting equilibrium methods (Ordinary, Bishop, Janbu and Spencer). Oncethe critical slip surface and the soil uncertainties are known (or assumed), the reliability analysiscan be performed. In the case of FOSM, the partial derivatives of the factor of safety with respectto each of the soil properties must be evaluated. Then the mean and the variance of the factor ofsafety and the associated probability distribution can be determined. Accordingly, the reliabilityindex (�) and the probability of failure (Pf) of the slope can be calculated. On the other hand, forthe case of Monte Carlo simulation method independent sets of soil properties (�, C and ) aregenerated from their assigned probability distributions. Then the factor of safety for each set canbe calculated using any limiting equilibrium method. Accordingly, the mean, the standarddeviation and the associated probability distribution of the factor of safety are determined.Finally, the reliability index (�) and the probability of failure (Pf) can be calculated.To identify the probability distribution of the safety factor three sets of soil properties (�, C

and ) are generated from their probability distributions, the sizes of which are 1000, 10 000 and15 000. It is assumed that �, C and are normally distributed random variables with mean of 10�,9.8 kN/m2 and 17.64 kN/m3 and standard deviation of 1�, 2.94 kN/m2 and 0.1764 kN/m3,respectively. Then the factor of safety is determined by any limiting equilibrium method. Forexample, the frequency histogram for the generated factor of safety using data of example 1 for


Fig. 2. Schematic representation of methodology used.


the ®rst set is shown in Fig. 3. The generated factor of safety and the assumed probability dis-tribution are also shown in Figs. 3 and 4. The Chi-square goodness of ®t test indicates that thenormal distribution adequately ®ts the generated factor of safety for all three sets.

5. Factors a�ecting reliability index

Reliability in slope stability analysis is a�ected by various factors such as uncertainty asso-ciated with soil properties and uncertainty associated with the methods (or models) used.Although the focus of this paper is to address the reliability of the safety factor due to soiluncertainty, model uncertainty has been addressed in a simple manner by evaluating the relativeperformance of the methods used.

5.1. Uncertainty due to soil properties

The randomness and uncertainty in the soil property are the most important factors that maya�ect the reliability of the safety factor. Christian et al. [11] reported that uncertainties in soilproperties can rise from inherent spatial variability in the properties and random testing errors in

Fig. 3. Frequency histogram of the factor of safety using MCSM for example 1.


their measurements. Also, such uncertainty may come from systematical errors due to samplingprocess and bias in the measurement process itself.In addition to the randomness in the soil properties, random seed and sequence length may

a�ect the calculated reliability index (�). Therefore, the following computer experiments are per-formed to see how the reliability index may be a�ected by these two factors. Data of example 1was used in these experiments.

5.1.1. Computer experiment 1In this experiment, the e�ect of seed random number generator on the reliability index � is

investigated. For this purpose, several computer runs are conducted by which the seed randomnumber generator is allowed to vary from 2000 to 15 000. For each run the reliability index wasdetermined. Fig. 5 shows the e�ect of changing the seed random number generator on the relia-bility index using the four-mentioned slope stability methods. As can be seen from Fig. 5 thereliability index � is not sensitive to the selected seed random number generator. Therefore, inthis study, the seed of 10 000 was selected.

5.1.2. Computer experiment 2To perform the reliability analysis the appropriate sample size of soil properties should be

determined. For this purpose the following computer runs are conducted. Di�erent sample sizes

Fig. 4. Normal distribution ®t for the generated factor of safety using MCSM for example 1.


ranged from 50 to 2000 were generated and the associated reliability index � for each of the fourstability methods was calculated. Fig. 6 shows the relationship between the sample size (generatednumber) and the reliability index (�). It is clearly shown in this ®gure that almost there is nosigni®cant di�erence in the calculated (�) as the sample size exceeds 700. Therefore, the samplesize of 700 was chosen in this study.

5.2. Illustrative examples

In this study two examples were selected for studying the reliability and uncertainty of slope'sfactor of safety using the ®rst-order second-moment and Monte Carlo simulation methods. The®rst example illustrates the stability of a homogeneous slope and the second illustrates the stability

Fig. 5. Relationship between seed random number generator and reliability index.

Table 2Values of factor of safety for both examples

Limiting equilibrium methods Ordinary Bishop Janbu Spencer Critical Circlex-y-r coordinates (m)

Factor of safety

Example 1 1.278 1.358 1.314 1.293 8.64±13.04±9.56Example 2 0.947 1.012 0.946 1.199 9.22±11.98±9.38


of a non-homogeneous slope. Four well-known slope stability methods were applied to theseexamples. Table 2 presents the calculated factor of safety for both examples using those fourmethods.

5.2.1. Example 1: homogeneous slopeThis example is a simple homogeneous slope with geometry presented in Fig. 7. The soil

properties of this example are shown in Table 3. The means of soil properties as well as theircoe�cients of variation are assumed from their reported values in the literature [25]. In thispaper, di�erent values of coe�cient of variations are assumed according to their proposed rangeby Harr [25]. For example, the coe�cient of variation for soil cohesion C ranges from 0.1 to 0.8,for soil angle of internal friction (�) between 0.05 to 0.2, and 0.01 to 0.03 for soil unit weight ( ).In the following section, the uncertainty and reliability analyses of this example are performedaccording to the two mentioned methods of reliability analysis (see Fig. 2).

Fig. 6. Relationship between sample size and reliability index.

Table 3Soil properties of example 1

Soil parameter Mean CV1 CV2 CV3 CV4

Cohesion (kN/m2) 10.0 0.10 0.20 0.30 0.40

Friction angle (�) 10.0 0.05 0.10 0.15 0.20Unit weight (kN/m3) 17.64 0.01 0.02 0.03 0.04


5.2.2. Results of example 1To study the relationship between the factor of safety and the variation of soil properties, the

factor of safety was calculated as function of each of soil variables. Each variable (soil property)allowed varying independently about its respective mean value while the other properties are kept®xed at their mean values. Figs. 8±10 show the relationships between the factor of safety and theangle of internal friction, soil cohesion, and unit weight, respectively.The relationship between the factor of safety and changes in the soil properties are almost lin-

ear. These ®gures are used in calculating the partial derivatives of the safety factor with respect toeach of these soil properties. Such derivatives are required for evaluating the uncertainty of thefactor of safety [Var(F)] as expressed in Eq. (10).Three experiments have been conducted to illustrate how the uncertainty in soil properties may

a�ect the calculated reliability index of the safety factor. These experiments represent three casesof soil uncertainties (low, moderate and high). In the ®rst experiment, the CV of the angle ofinternal friction allowed to change from 0.05 to 0.2, where low uncertainties are assumed for soilcohesion and soil unit weight, the CVs were assumed 0.1 and 0.01, respectively. In the secondexperiment, moderate uncertainties are assumed for both the soil cohesion and the soil unitweight with the CVs 0.2 and 0.02, respectively. In the third experiment high uncertainties areassumed for the soil cohesion and the soil unit weight with the CVs 0.3 and 0.03, respectively. Inthese experiments, the reliability index is estimated using the FOSM method for the four slopestability methods, Fig. 11 shows the results of these experiments. It is clear from Fig. 11 how thereliability index may be a�ected with the soil uncertainties, for example, in the case of lowuncertainties in soil properties high reliability index is obtained and vice versa.A comparative study between the FOSM method and the MCSM is presented using the soil

properties of example one. The e�ect of variation in CV for the angle of internal friction on thecalculated reliability index using the four method of slope stability is shown in Fig. 12. The results

Fig. 7. Cross-section and soil properties for example 1.


indicate that no variations were observed between the FOSM and MCSM for the Ordinary, andBishop methods. However, slight variation is observed between the two uncertainty methods inthe case of Janbu and Spencer methods. Similar trends between the two approaches are observedfor the case of the variation in the CV of soil cohesion and soil unit weight, see Figs. 13 and 14.

Fig. 9. Variation of factor of safety with soil cohesion for example 1 (other soil variables are taken equal to theirexpected values).

Fig. 8. Variation of factor of safety with angle of internal friction for example 1 (other soil variables are taken equal to

their expected values).


5.2.3. Example 2: non-homogeneous slopeThis example is a layered slope with the soil properties presented in Table 4. The slope cross-

section is shown in Fig. 15. This example is used to compare the reliability index calculated usingFOSM and MCSM. Figs. 16±18 present the calculated reliability index verses the variation in thecoe�cient of variation of the angle of internal friction, soil cohesion and soil unit weight. Theresult indicated that no variation was observed between the two approaches when the ordinary,Bishop and Janbu methods are used. However, slight variation is observed in case of Spencermethod. This variation in � between the two reliability methods is due to the approximation fol-lowed in evaluating the partial derivative of the safety factor as required by the FOSM method.While in case of MCSM such derivative is not required.

5.3. Model uncertainty

In engineering design and analysis, one frequently uses models involving parameters that aresubject to uncertainty. Therefore, model outputs on which engineering design and analysis arebased are subject to uncertainty. However, even when the input is exactly speci®ed, the predic-tions of an engineering model can be expected to deviate from reality due to model uncertainty.In slope stability analysis, various sources of model errors are recognized. According to Christianet al. [11] three sources of model uncertainties were de®ned, these are: three-dimensional failurecompared to two-dimensional approximation, failure to ®nd the most critical failure surface, andnumerical and rounding error. In addition to what Christian et al. [11] reported in their study,model uncertainty in slope stability analysis arises from the approximation made when repre-senting physical phenomena by equations. So model uncertainty is mainly attributable to the

Fig. 10. Variation of factor of safety with soil unit weight for example 1 (other soil variables are taken equal to theirexpected values).


inappropriate selection of the method to evaluate the safety factor, which causes some inherenterror in the estimation.Estimating the model uncertainty can be troublesome; however, the most e�ective approach is

to rely on empirical observations when they exist. Usually, model selection can be made by con-ducting Monte Carlo simulation experiments, and then comparing the performance of the com-peting models in reproducing observational statistics. In the case of slope stability studies suchprocedure is not possible because it is di�cult to have historical observations to compare withresults of slope stability methods. Therefore, in this paper, model uncertainty is evaluated bycomparing the relative performance of the simpli®ed slope stability methods to the one considered

Fig. 11. E�ect of soil uncertainty on reliability index for example 1.


Fig. 12. Comparison of reliability index calculated by FOSM and MCSM based on the four slope stability methods forexample 1 (CV for C=0.1 and for =0.01).


Fig. 13. Comparison of reliability index calculated by FOSM and MCSM based on the four slope stability methods(CV for �=0.1 and for =0.01).


Fig. 14. Comparison of reliability index calculated by FOSM and MCSM based on the four slope stability methods(CV for �=0.1 and for C=0.01).


more rigorous. Since Spencer's method is considered as one of the rigorous method that leads to atrue factor of safety, the generated frequency distribution of the safety factor produced by thismethod is adopted as the real frequency distribution. Then, the frequency distributions of thesafety factor obtained by other slope stability methods for same slope geometry and same soilproperties are compared to that of the Spencer method to evaluate the relative performance ofthese methods. In all slope stability methods, Monte Carlo simulation was used to generate thefrequency distribution of the safety factor. Two criteria were used to compare the relative per-formance of the other slope stability methods to Spencer method: Relative mean absolute error(RMAE) and relative root mean squared error (RRMSE) given as follows:

RMAE � 1

M

XMi�1

eij j �12�

Fig. 15. Cross-section and soil properties for example 2.

Table 4

Soil properties of example 2

Soil parameter Mean CV1 CV2 CV3 CV4

Layer 1Cohesion (kN/m2) 10.0 0.10 0.20 0.30 0.40Friction angle (�) 10.0 0.05 0.10 0.15 0.20

Unit weight (kN/m3) 17.64 0.01 0.02 0.03 0.04Layer 2Cohesion (kN/m2) 8.0 0.10 0.20 0.30 0.40

Friction angle (�) 5.0 0.05 0.10 0.15 0.20Unit weight (kN/m3) 18.0 0.01 0.02 0.03 0.04


Fig. 16. Comparison of reliability index calculated by FOSM and MCSM based on the four slope stability methods forexample 2 (CV for C=0.1 and for =0.01).


Fig. 17. Comparison of reliability index calculated by FOSM and MCSM based on the four slope stability methods for

example 2 (CV for �=0.1 and for =0.01).


Fig. 18. Comparison of reliability index calculated by FOSM and MCSM based on the four slope stability methods for

example 2. (CV for �=0.1 and for C=0.01).


RRME �XMi�1

e2iM

" #1=2

�13�

ei � qsi ÿ qoi

qsi�14�

where qsi = value of the ith order quantile for the assumed true probability distribution obtainedusing Spencer method; qoi=estimated quantile value using other slope stability method; ei is therelative error; and M is the number of quantiles evaluated.Fifteen quantiles at selected probability levels were obtained to calculate the values of the two

performance measures. The discrete probability values associated with these quantiles are 0.01,0.025, 0.05, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 0.95, 0.975, and 0.99. The above procedure wasfollowed to evaluate the relative performance of the three slope stability methods (i.e. Orinary,Bishop and Janbu) as compared to Spencer method. Data of example 1 is used to quantify themeasures of the relative performance of these methods [Eqs. (12) and (13)]. From the perfor-mance criteria RMAE and RRMSE, given in Table 5, one can observe the relative performanceof the three methods as compared to Spencer method. It is found that the discrepancies betweenthese methods and the results of Spencer range from 0.04 to 0.1 as a relative mean absolute error.These discrepancies may be attributed to model uncertainty due to the simpli®cations and theunderlying assumptions used in formulating limiting equilibrium methods.

6. Discussions

It is shown in this paper that the selected seed random number generator does not a�ect MonteCarlo simulation results. It is also shown that sample sizes of 700 or greater are su�cient forMonte Carlo method to converge to the same reliability index (�) or probability of failure (Pf).Therefore, a sample size of 700 has been used in this paper in the application of Monte Carlosimulation method. It is also found that the probability distribution of the safety factor of suchsample size is normally distributed. Samples of sizes 1000, 10 000 and 15 000 also con®rmed thatthe distribution of the safety factor is normal.Due to the simplicity of the Ordinary method, in which the factor of safety is an explicit func-

tion of soil properties, an analytical solution for the FOSM can be obtained. In all the testedcases presented in examples 1 and 2 using Ordinary method, it is observed that there is no

Table 5Relative performance comparison between the three slope stability methods with Spencer's method

Method of slices REMEA RAMSE Mean S.D.

Spencer's method 0.0 0.0 1.4768 0.2155

Ordinary or Fellenius 0.1008 0.1111 1.3177 0.1410Bishop's simpli®ed 0.0819 0.0855 1.4385 0.1621Janbu's simpli®ed 0.0423 0.0449 1.3683 0.1469


di�erence in the calculated reliability index using FOSM and MCSM. Thus, Monte Carlo suc-cessfully gives similar results as the FOSM method.In the case of the homogeneous slope, the calculated values of reliability index based on the

FOSM method are very close to those determined on the basis of MCSM for both Ordinary andBishop methods. However, the reliability indexes calculated using MCSM are slightly di�erentthan those calculated using MOSM method whether Janbu's method or Spencer's method wasused. Similar trends in the results are observed in the case of the layered example (example 2). Inthis case, the Ordinary, Bishop and Janbu provide good agreement between the calculated (�)using FOSM and MCSM. However, the reliability index is slightly di�erent between the twouncertainty methods for Spencer's method. This is attributed to the fact that the factor of safety isnot an explicit function of the random variables (soil properties) in Spencer's method. Therefore,it is expected that FOSM will yield an approximate solution of the calculated reliability index.This is due to di�culty encountered in evaluating the analytical derivative of the factor of safetywith respect to each of these random variables. On the other hand, MCSM does not requireevaluation of such derivatives, which make it an appropriate scheme for reliability analysis in caseof rigorous stability method such as Spencer's method.Although the focus of this paper is to quantify the uncertainty of the factor of safety due to

uncertainty in soil properties, the model uncertainty is addressed by evaluating the relative per-formance of the simpli®ed slope stability methods as compared to Spencer's method. Such pro-cedure is proposed because it is not possible to measure the safety factor in ®eld and then tocompare it with the output of such stability method. The result reveals that discrepancies betweenthe three slope stability methods and Spencer are reasonably accepted. It is found that the dis-crepancies between these methods and Spencer ranges from 0.04 to 0.1 as a relative mean abso-lute error. Such discrepancies are mainly attributed to model uncertainty due to the underlyingassumption and simpli®cations associated with the formulation of these methods.

7. Conclusions

This paper outlines a procedure of probabilistic analysis of slope stability using ®rst-ordersecond-moment method and Monte Carlo simulation method. The results of these methods arecompared using four recognized methods of slope stability. These are Ordinary method, simpli-®ed Bishop's method, simpli®ed Janbu's method and Spencer's method. Two illustrative exam-ples are used in this paper: the ®rst example is for homogenous slope and the second is for slopein layered soil. The ®nding of this research warrant the following conclusions:

1. The study reveals that a sample size of 700 or greater is a good choice for MCSM, while forsample size less than 200 the reliability index shows a signi®cant variation. A sample size of700 was chosen in the MCSM.

2. For the homogeneous slope, a good agreement is observed in the calculated reliability index� for Ordinary and Bishop methods. For Janbu and Spencer methods slight variations existbetween FOSM and MCSM.

3. For the non-homogeneous slope the calculated reliability index (�) based on the two usedmethods is in good agreement for Ordinary, Bishop and Janbu methods. However, �obtained using the two uncertainty methods is di�erent in case of Spencer method.


4. The discrepancy in the calculated � between FOSM and MCSM in case of Spencer's methodis mainly attributed to the linearization procedure used in evaluating the ®rst derivative ofthe safety factor as required by FOSM. The highly non-linear formulation of the safetyfactor makes it impossible to evaluate such derivative; therefore, numerical approximationwas used. Whereas, MCSM does not require such derivative and this may explain this dis-crepancy between the two-reliability methods.

5. Model uncertainty was addressed by evaluating the relative performance of the simpli®edslope stability methods as compared to the more rigorous method. The discrepancies amongthese methods are mainly associated with model uncertainty due to the simpli®cation used inslope stability methods.

It is understood that FOSM method requires fewer calculations and computing time comparedto MCSM. However, with the advent of computers in data handling and speed Monte Carlosimulation o�ers a practical tool for geotechnical engineering problems. MCSM proved to bepowerful and e�ective scheme for more detailed reliability analysis of slope stability.

Acknowledgements

Financial support provided by Jordan University of Science and Technology is gratefullyacknowledged. The authors are grateful to the two anonymous referees for critically reviewing themanuscript, and improved the clarity of this paper as well.

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Uncertainty and reliability analysis applied to slope stability

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