Monte Carlo simulation of rock slope reliability

Compurers

& S~rucrures

Vol.

33. No. 6. PP. 1495-1505,

1989

Printed in Great Britain.

MONTE

0

CARLO

SIMULATION RELIABILITY

004%7949/89 $3.00 + 0.00 1989 Pergamon Press plc

OF ROCK SLOPE

SAMERTAMIMI,BERNARDAMADEIand DAN M. FRANGOP~L Department of Civil Engineering, University of Colorado, Boulder, CO 80309-0428,U.S.A. (Received 29 November 1988)

Abstract--One of the most important problems encountered in probabilistic rock slope stability analysis is the correlation that exists among the random variables involved in the design equation. Some existing probabilistic models assume independence between the random variables by ignoring all possible correlations. Other models realize the importance of including the correlations in the mathematical formulation; however, the associated mathematical complications require inclusion of some simplifying assumptions, such as Gaussian distribution of all random variables involved in the design equation. Therefore, it is important to develop techniques to deal with the correlations while maintaining the desired accuracy. This paper is concerned with reliability analysis of rock slopes against single plane sliding under the influence of water. A modified Monte Carlo Simulation is used to determine the reliability of rock slopes including possible correlations between the variables entering into the design equation. These variables do not have to be normally distributed and can have different probability density functions. Finally, a computer program has been developed to perform all the necessary calculations.

INTRODUCTION Uncertainty in geotechnical engineering is inevitable. In nature, most variables governing geotechnical problems are random rather than deterministic. Properties of soil and rock are inherently heterogeneous, and natural earth deposits are characterized by irregular layers of various materials with wide ranges of material properties [l]. Uncertainty also exists in material testing where no single value can be representative. Since geotechnical engineering problems are characterized by uncertain variables, design is always subjected to uncertainties. A common engineering practice is to ignore the probabilistic nature of the variables of interest and treat them as if they were single valued quantities. The worst case is generally considered and a factor of safety is used to account for the uncertainties. In many cases, this approach results in conservative designs that can add substantially to the cost of a project. On the other hand, an inexpensive design may not ensure the desired level of performance or safety. Therefore, it is necessary to make decisions based on trade-off between cost and safety [2,3]. Deterministic analyses have been widely used in rock slope stability analysis. Based on the assumption of limit equilibrium, deterministic methods assess the safety of rock slopes through a factor of safety defined as the ratio of strength available along a potential failure surface to that required for failure to occur. Deterministic methods of rock slope stability use point estimates of the variables, and hence, no consideration is given to randomness. Therefore, deterministic analyses fail to provide an accurate estimate of slope reliability.

Furthermore, there is an ambiguity as to the exact interpretation of the factor of safety. There is full agreement that a factor of safety less than 1.O implies inadequate capacity and therefore ‘failure’. However, the full meaning of the factor of safety is not very clear. For instance, as pointed out by Baecher [4], a slope with a factor of safety of 2.0 is not twice as stable as one with a value of 1.0. To overcome the deficiency of a single value estimate, researchers have used probability theory to help express the uncertainties in geotechnical engineering problems. Some investigators have used analytical techniques to evaluate design uncertainty [3]. Others have proposed the use of the Monte Carlo simulation rather than closed-form solutions [S]. Whatever method is used to evaluate the uncertainty associated with variables to be accounted for within a design equation, the results must reveal the mean value, the standard deviation and the probability density function of the final distribution. Existing probabilistic methods of rock slope stability cannot be considered as having met the requirements to handle rock slope stability analysis effectively. The main purpose of this paper is to present an approach for determining the reliability of rock slopes including the influence of correlations. Some examples are presented for both correlated and independent random variables, using analytical and simulation techniques, and a comparison of the results is given. Other examples are provided to demonspate the capabilities of the proposed method when analytical solutions cannot be obtained. Finally, a summary and a concluding discussion are given.

1495

SAMER TAMIMIet al.

1496 THEORETICAL

BACKGROUND

Engineering problems often involve the evaluation of functional relations between a dependent variable and one or more basic independent variables. If any of the basic variables are random, the dependent variable will likewise be random; its probability dist~bution, as well as its moments, will be functionally related to and be derived from the basic random variables [l]. As previously indicated, our main concern in reliability analysis is to determine the main descriptors (the mean value p and standard deviation Q) as well as the probability density of the function of interest. The main descriptors of a function of random variables can only be derived analytically for multiplicative and additive random variables. The problem becomes even more complex if the variables are correlated. On the other hand, the problem of defining the probability density of a function of random variables is very difficult in many cases, especially when the function is nonlinear. Instead, approximate methods are normally used to arrive at the probability density of a function of random variables. One popular approach to determine the probability density function of a function of random variables is to use the Central Limit Theorem when dealing with the sum or multiplication of random variables. The Central Limit Theorem states [6]: Under very general conditions, as the number of variables in the sum becomes large, the distribution of the sum of random variables will approach a normal (Gaussian) distribution. In other words, the sum of a large number of variables will be normally distributed irrespective of the probability density function of the individual variables. The general conditions required by the Central Limit Theorem can be found in El] and [6]. Also, by the Central Limit Theorem, the product of a large number of random variables will tend to be log-normally distributed, irrespective of the distribution of the individual variables. If the general assumptions of the Central Limit Theorem are satisfied, and the mean Q.+)and standard deviation (CT!)of each variable i are known, with the mathematics of probability theory (e.g. [l]), the mean, standard deviation and the probability density function of the final distribution may be obtained. Another method of determining the probability density function of a function of random variables is by using Convolution Integrals. For example, given that 2 = X + Y and knowing the density functions of the random variables X and Y,_&(x) andf,(y), we want to determine the density function of Z. The derivation of the probability density function of the sum of two independent random variables requires,

according to [l], the evaluation

of

or

The form of this solution is mathematically intractable in all but a limited number of cases [7]. Therefore, approximate solutions must be determined by numerical integration techniques. This method requires the identification of the marginal dist~butions of all independent variables, as well as the probability distribution of the correlated variables conditioned on all other variables. This requirement can be tedious for continuous random variables and, therefore, the method will not be discussed in this paper. SIMU~TION

TECHNIQUE

Simulation techniques are becoming increasingly popular because computer development has decreased the computer cost needed to run the simulations. The Monte Carlo simulation provides an effective way of determining the final probability density of a function of random variables as well as its mean and standard deviation from any kind of input-distributions. In basic terms, Monte Carlo simulation involves repeating the simulation using a particular set of values in accordance with the probability density function of each individual variable. The mechanism is quite simple. The computer generates a random number between zero and one from a uniform distribution. By knowing the cumulative distribution of the probability density function for each variable entering into the design equation, the computer can pick up a unique value for each variable. Three methods are generally used to determine the random values from the cumulative density function (CDF): the inverse transform method, the composition method, and the function of random variables method [8,9]. This process is repeated many times independently to arrive at the probability density of the function of random variables. One advantage of the Monte Carlo simulation is that any form of input probability density function (binomial, skewed, etc.) may be used for each random variable. By definition, simulation involves certain errors which can be reduced by using a large number of iterations (simulation runs). In theory, Monte Carlo simulation resembles sampling theory in statistics. We can imagine that there is an infinite number of possible outcomes of the simulation, yet we use only a limited number to represent the final distribution. In Monte Carlo simulation, the sample mean 2 and

1497

Monte Carlo simulation of rock slope reliability variance s2 are used to estimate the population mean p and variance a’; therefore, the accuracy of this estimate is of concern. For a number of simulation runs (sample space) n, the sample mean

is also a random variable. Its expected value is given by E(x) = /l

(3)

The algorithm rearranges a random number sequence with reference to another so as to obtain the desired correlation between the two. The algorithm for simulating two correlated random variables X and Y can be summarized as follows. (1) Generate two independent random number sequences [X,] and [Z,] having the required distribution characteristics with the help of the techniques described in [8] and [9]. [Z,] is to be used as a parent population for generating [Y,]. (2) For every xi, calculate the best linear predictor of Y:

and its variance is given by Y,=E(YIX=xi)=py-p+-xi)

Var(n) = c.

(4)

Therefore, according to the Central Limit Theorem, the sample mean (as a random variable) has a normal distribution with a mean value p and a standard deviation a/& [i.e. f N N(p, cr/,,&)]. This indicates that the error generated in the mean value using Monte Carlo simulation decreases with the square root of the number of samples (simulation runs). Hence, the error decreases at a decreasing rate with the number of iterations. Therefore, there should be an optimal number, where the incremental decrease in error is not justified by the additional required number of iterations. To estimate the number of simulation runs needed, the confidence interval method can be used [l, 61; or, the error in the mean value p for a number of simulation runs n can be approximated by E,,=&

(5) J;;

Therefore, knowing the maximum desired error, one can calculate the number of simulation runs that are needed. SIMULATION

OF CORRELATED

RANDOM

VARIABLES

Monte Carlo simulation provides a convenient means for the probabilistic analysis of many engineering problems. However, a major difficulty is the correlation existing between the random variables. The Monte Carlo simulation is based on generating random numbers independently to resemble the sampling process. Therefore, the basic problem to be considered is the generation of two random number sequences [X,] and [Y,] of sample N, to represent two jointly distributed random variables X and Y. A number of algorithms have been developed in the literature to generate correlated random numbers [lO-123. A convenient approach to generate correlated random numbers was proposed by Nawathe and Rao [l l] and is based on optimal linear prediction.

where p is the correlation coefficient between X and Y. (3) For every xi, search through [Z,] until a zj is found such that qj=[z,-

Y,]‘-
(for q, > 0)

or qf < D:cJ:

(for qj < 0)

where u: is the error variance given by g: = E(q2) = .;(l

- p2)

and D, and D, are as indicated in Fig. 1. (4) Choose the first zj which satisfies the condition of the previous step, to be paired with xi as the corresponding yi. (5) If none of the zj s satisfies the condition, choose for zj to be paired with xi, the one that gives the minimum squared error. PROBABILISTIC

ROCK SLOPE STABILITY

ANALYSIS

Safety margin

In probabilistic reliability analysis, the safety margin M of a rock slope is defined as the force resisting sliding down the plane R less the force causing the sliding to occur Q. Failure is defined by the event M < 0. The probability of this event is J=,=P[M=(R-Q)
(6)

in which P stands for probability, and P, is the probability of failure or sliding. Figure 2 illustrates the concept of the safety margin. Note that the expected value of the safety margin is given as E(M) = E(R) -E(Q), and the probability of failure (sliding) equals the shaded area on the left-hand side of the probability density function. In order to assess the reliability of a rock slope, it is necessary to know the probability density function

SAMER TAMIMI et al.

1498

can be found as *

x

/3*= - cp_‘(P,) = Q,-‘(I - Pf)

v

Q-9

Ax f Ax,

jE

where @-’ is the value of the standard normal variate at the probability level (I - P,).

X

Shortcomings of existing probabilistic methods

I

-Ax,Ax = 1.96

Ax,

Ax, For positive valuesof error D, = ‘2

= 2.69

For negative values of error x

DI=

$

= 0.975

Fig. 1. Values of D, and D, [ll].

of the safety margin. The probability of failure of the slope can be found by calculating the probability of the event M < 0, and conversely, the reliability of the slope can be assessed by calculating the probability of the event M > 0. Another important parameter in reliability analysis is the reliability index with respect to sliding (/I,), defined as

in which p,,, [also denoted as E(M)] and u, are respectively the mean and standard deviation of the safety margin M. The larger the reliability index & the smaller the probability of sliding Pr. By assuming that the safety margin has a normal distribution, 8,

f(m)

Probability density function of the safety margin M = R-Q

m

Fig. 2. Probability density function of the safety margin.

Existing probabilistic rock slope stability analyses provide a better and more accurate understanding of the reliability of rock slopes than deterministic methods. Yet, there are some shortcomings and drawbacks in using these methods since they require many assumptions in their formulations. Existing methods can be broadly divided into two main groups: analytical methods (which are based on closed-form expressions for the main descriptors of a function of random variables) and simulation techniques. In general, pure analytical methods do not provide solutions to the probabilistic analysis without including some simplifying assumptions. Therefore, the accuracy of the analytical methods depends on the validity of the assumptions used. Difficulties arise in the application of pure analytical methods when the variables in the design equation are correlated. The mathematics required for a combination of uncorrelated variables cannot be generalized for correlated variables. In many appli~tions, the main descriptors of the safety margin (the mean and standard deviation) are calculated by using analytical methods. In general, these methods assume that some or all of the random variables within the equation of the safety margin are independent. However, there is a major shortcoming in this assumption since correlations between random variables can be an important consideration in reliability calculations for rock slopes. The effect of correlations was studied by many researchers. Preliminary studies conducted by Glynn and Ghosh [13] indicate that the probability of failure determined by neglecting the effect of correlations is greatly overestimated. They proposed other correlations between the random variables as shown in Fig. 3. Frangopol and Hong[3] found that neglecting the inherent positive correlations between the total force that resists sliding and the total force that induces sliding greatly underestimates the reliability of the slope. Neglecting these correlations underestimates the safety margin that results in understating the reliability of the rock slope. Therefore, in assessing rock slope stability, it is necessary to include the correlations in the mathematical formulation. In assessing rock slope reliability, we are not only interested in knowing the mean and standard deviation of the safety margin, but also in knowing its probability density function. In the context of rock slope stability, the Central Limit Theorem cannot be applied since all conditions required by this theorem are not satisfied; for instance, the number of variables involved in the analysis is not sufficiently large.

1499

Monte Carlo simulation of rock slope reliability

where:

UC&LPSAB

Dipof joint JRC = joint roughness coefficient,

Strike of joint Joint spaciq Joint persistence Joint Joint Joint Cleft

length friction angle cohesion water pressure

JCS = joint wall compressive strength,

4, = residual friction angle, and

C: Probably correlated P: Possibly correlated

0; = effective normal stress across the plane.

U: Probably uncorrelated

Fig. 3. Possible correlations between random variables as proposed by Glynn and Ghosh [13].

Because of that, some analytical models assume that all the random variables included in the analysis are normally distributed. Based on this assumption, the probability density function of the safety margin has a normal distribution. This assumption makes the analysis easier since the normal distribution of the safety margin requires only two statistical parameters (the mean and standard deviation) for its description. However, there is ample evidence that the variations of rock properties cannot always be characterized by the normal distribution. Examples of non-normal probability density functions of some rock properties can be found in [14], [15] and [16], among others. Some researchers have proposed to use the Monte Carlo simulation to calculate the probability of failure of a rock slope (e.g. Glynn and Ghosh [13]; Nguyen and Chowdhury [5]). However, no consideration has been given to the correlations existing between the basic random variables.

The joint wall compressive strength (JCS) can be determined using the Schmidt Hammer test, whereas the joint roughness coefficient (JRC) can be determined from a tilt test using the following equation [4, 191:

where GI,is the tilt angle, and a& is the corresponding value of the effective normal stress when sliding

occurs during the tilt test. Also, a: can be calculated from 0;=

+ 4,

1

(11)

A

R=[Tsinfl+lVcos$-U] .tan[,,log,,(

zP = a; tan JRC log,, F

+ WCOSI,!I - U

in which A is the base area of the block, W is the weight of the block, and U is the uplift force. Based on eqns (9)--(11), the resisting force R and the driving force Q can be written as

Proposed method

In this section, a new approach to reliability analysis is presented and applied to rock slope stability analysis. The proposed methodology uses the Monte Carlo simulation to perform the reliability analysis taking into account the inherent correlations that exist between the basic random variables involved in the analysis. An increased understanding of the influence of probabilistic dependencies between the random variables should provide a more accurate reliability analysis. The present method is only concerned with the reliability analysis of rock slopes against single plane sliding under the influence of water. Figure 4 shows the problem geometry where sliding of a block of weight W occurs along a plane of weakness with dip angle $. Rock reinforcement (bolts or cables) is installed at an angle /I to the plane of weakness in order to resist sliding. The peak shear strength of the plane of weakness, zP, is assumed to be defined by the criterion proposed by Barton [17, 181:

Tsinfi

““‘4

Wcos$+Tsmfi-CJ

(12) and Q=Wsin$-Tcosp+V where V is the water force in the tension Water filled tension crack

(9) Fig. 4. Plane sliding [22].

(13) crack.

SAMERTAMIMIet al.

1500

Utilizing the deterministic approach, safety for plane sliding is given by

the factor of

4, and KS. In addition, there is a strong positive correlation between the uplift force U and the water I

In probabilistic analysis, the reliability of a rock slope is assessed through the safety margin M. The first step in reliability analysis is to define the equation for the safety margin. For plane sliding, the safety margin can be expressed as

force in the tension crack V. Neglecting these correlations understates the safety margin and the reliability of the rock slope. A similar expression for the safety margin can be derived if eqn (9) is replaced by a more simple Coulomb criterion such as

M=R-Q r,=C+uitan4 =[Tsin/?+

rtan[JRClog,,( -(Wsin$

(16)

Wcost+k-U]

where C and 4 are the plane of weakness cohesion and friction angle, respectively. Then, the safety margin is again defined by eqn (15) with R now being equal to

“““:

Wcost,b+Tsm~-U

- Tcosp

+ V).

(15)

R=CA+(WcosI(/+Tsin/l-U)tan4. Note that the joint properties in eqn (15) are highly random in nature. They are determined by repeated field observations (JRC and I,G)or by repeated field or laboratory tests (JCS, 4, and tit). The weight of the sliding block W is known with a certain accuracy. Similarly, the water forces U, Vdeveloping along the critical joints may vary with drainage, rain, etc. Finally, the bolts or cables acting as the reinforcing forces T may be different to those applied due to possible time-dependent strength reduction associated with corrosion or movement of the anchor points. The next step is to identify any probabilistic dependencies that may exist between the basic random variables involved in the equation for the safety margin. With this regard, the influence diagram methodology is effective to identify probabilistic dependencies between random variables [20,21]. In the influence diagrams arcs or arrows are used to show the interrelationships between the random variables involved in the analysis. An arc connecting two random variables indicates probabilistic dependency between the two. In other words, a random variable (node) is modeled as being probabilistically dependent on all the random variables that directly precede it. In this paper, the influence diagram proposed in Fig. 5 is used to identify these probabilistic dependencies. Since the driving force Q and the resisting force R are based on the same state of information (T, j?, +, W), inherent correlations exist between R and Q even though there is no direct link (correlation) between them. Correlations exist between the joint roughness coefficient JRC and the joint compressive strength JCS according to eqn (10). Possible correlation may also exist between the friction angle

(17)

The influence diagram shown in Fig. 5 must now contain C and 4 instead of JCS, JRC and 4,. The last step in the proposed methodology is to evaluate the influence diagram of Fig. 5 taking into account all the dependencies (arcs) among the random variables. Evaluating the influence diagram means solving the probabilistic inference problem which results in determining the main descriptors of the safety margin as well as its probability density function. In the present analysis, however, the equation for the safety margin includes multiplicative correlated random variables, and involves operations other than additions and multiplications. Hence, evaluating the influence diagram and therefore determining the main descriptors and the probability density function of the safety margin is not possible using classical analytical methods without including

‘-*T Fig. 5. Proposed influence diagram.

1501

Monte Carlo simulation of rock slope reliability some simplifying assumptions. Instead, we propose a method that utilizes the Monte Carlo simulation to evaluate the influence diagram. The method takes into account all possible correlations between the random variables, and different probability density functions can be attached to these variables. In order to solve the probabilistic inference, the proposed method makes use of the sample space of each random variable (node) in Fig. 5. The sample space of each random variable is based on the conditioning variables; the sample space of an independent variable is its marginal probability density function, and that of a dependent variable is its marginal density function along with all correlation coefficients with all its conditioning variables (predecessors). To start, a unique value is selected for each independent variable using a random number generator in accordance with the probability density function of the variable of interest. Once a value has been selected, the variable (node) is said to be released. The next cycle involves selecting a unique value for each first-order conditional variable if all its predecessors have been released. The selection is made using a random number generator in accordance with the first-order conditional probability density function (using the method proposed by Nawathe to generate correlated random numbers). The cycle repeats itself until all the nodes in the diagram have been released. Since the model is characterized in probabilistic terms, the procedure has to be repeated many times to ensure the randomness in selecting the values. Because the method deals with the sample space of each random variable, all kinds of operations can be included in the mathematical formulation, correlations can be effectively taken into account, and different types of probability density functions can be assigned to the variables. A computer program, written in Fortran, has been developed to perform the simulation calculations. The program considers different types of probability density functions to describe random variables. The choice can be made from: Normal, Lognormal, Uniform, Triangular, Beta and Gamma distributions. NUMERICAL

solution as well as an analytical solution can be obtained, and therefore a comparison of results can be achieved. Assume that it is desired to analyze a rock slope using the Coulomb failure criterion without including the effect of water. The area of the critical joint (A), the angle of friction (4) and the dip angle (JI) are assumed to be deterministic (single valued quantities) and have the following values: A = 6000 ft2 (554.5 m2) 4 = 30” * = 45”. The cohesion is assumed to have a normal distribution with a mean value of 1.0 kip/ft2 (47.88 kN/m2) and standard deviation of 0.1 kip/ft2 (4.79 kN/m’). The block weight W has a normal distribution with a mean value of 10,000 kips (44,480 kN) and a standard deviation of 500 kips (2224 kN). Analytical solution. According to the Coulomb failure criterion, the resisting force R and the driving force Q can be expressed as follows: R=CA+

Q= WsinII/.

This example is designed such that a simulation

(19)

R and Q are based on the same state of information, and therefore are correlated. The coefficient of correlation between R and Q is given by oR,Q (20)

P=ORaQ

whereas the covariance of R and Q is given as 0 R.Q

=

EBQI - JWE[QI

= sin $ cos $ tan d[E( W’) - E*(W)] sin 295 =-tan4a2,. 2

The goals of this section are as follows:

Example 1: Comparison between analytical and simulation solutions

(18)

and

EXAMPLES

(1) to illustrate the effect of correlations on calculating the main descriptors of the safety margin, (2) to demonstrate the capability of the computer program to deal with situations where analytical solutions cannot be obtained, and (3) to visualize the advantages of using probabilistic analyses over classical deterministic analyses when assessing rock slope reliability.

Wcos$tan4

(21)

The mean values for R and Q are equal to tan4

pR=Apc+pwcos~

(22)

and PQ

and the corresponding

=

sin

+

PW

(23)

variances are such that

a’,=A2a~+(cos2JI

tan24)ok

(24)

SAMERTAMIMIet al.

1502

and 0; = sin’+ ok.

(25)

Therefore, the expression for the correlation efficient between R and Q can be written as

co-

simulation runs, and taking into account the correlations between R and Q. The following results were obtained: Pf = 0.0000 pM = 3021.71 kips (13440.57 kN)

p = cos * tan f$

Substituting the (22)-(25) gives

OW

JA’,cT$ + cos2 $ tan2 4 c&’

corresponding

values

in

(26) eqns

uM = 621.83 kips (2765.90 kN) /I, = 4.8594. The factor of safety was calculated as FS = R/Q with R and Q defined in eqns (17) and (13), respectively, using the mean values of these random variables. It was found to be equal to 1.43. Discussion. By comparing the results obtained from the simulation with those obtained using analytical methods, the following errors in the main descriptors of the safety margin were found:

pR = 10082.48 kips (44846.87 kN) uR = 633.77 kips (2819.01 kN) p’a = 7071.07 kips (31,452.12 kN) a, = 353.55 kips (1572.59 kN). The safety margin is equal to the resisting force less the driving force

error in pM = - 0.34% error in uM = -0.56%

M=R-Q.

(27) error in /I,= +0.22%.

Hence, the expected value of the safety margin is equal to PM

and the corresponding

=

PR

-

(28)

PQ

variance is such that

aZ,=a:,+a~-2po,fJa.

(29)

Substituting the values of pR, pQ, uR and up in eqns (26x29) gives p = 0.32

pM = 3011.41 kips (13,394.75 kN) = 618.36 kips (2750.46 kN). U,+,

uM=Jm (30)

The reliability index is equal to the mean over the standard deviation of the safety margin, and therefore /?,=-= 301 1’41 618.36

4.8700.

Since all the variables are assumed to be normally distributed, the probability density function of the safety margin is normal, and therefore the probability of failure is equal to P,= @(--p,)=

@(-4.87)=0.56

The results indicate that the error in using the Monte Carlo simulation is negligible. The deterministic analysis shows that the factor of safety is equal to 1.43. On the other hand, the probability of failure is equal to zero. In this case, the results are in agreement since both indicate a stable slope. However, this is not always the case as it will be shown in the next example. Now, to see the effect of correlations, we will calculate the standard deviation of the safety margin assuming independency between R and Q. Hence, using eqn (29) with p = 0 gives

x 10-6.

Simulation solution. Using the same values, the Monte Carlo simulation was performed using 10,000

= J(633.77)’ + (353.55)2 = 725.72 kips (3228.00 kN) instead of 618.36 kips (2750.46 kN). This leads to an error of - 17%. In other words, the standard deviation of the safety margin is overestimated by 17% when the correlation between R and Q is ignored. Note that according to eqn (29), the error caused by assuming p = 0 increases with the variances of R and Q. As a result of this misestimation, the probability of failure will not be properly estimated, and will thus provide an incorrect assessment of the reliability of the rock slope of interest. Example 2: Probability of failure us safety factor

The previous example was intended to compare an analytical solution with that of a simulation output

Monte Carlo simulation of rock slope reliability to show that the Monte Carlo simulation is indeed satisfactory. In addition, the Monte Carlo simulation has the advantage that all the variables in the design equation can have separate probability density functions. This example serves to demonstrate further this characteristic of the Monte Carlo simulation when calculating the probability of failure of a rock slope. Again, assume that the Coulomb failure criterion is used to assess the reliability of a rock slope for which the following information is given: and

A = 7500 ft2 (693.12 m2) * = 45”. In addition, 4 has a triangular distribution with a minimum value of 25”, a mode of 30” and a maximum value of 45”, W has a normal distribution with a mean value of 10,000 kips (44,480 kN) and a standard deviation of 1700 kips (7561.6 kN), and C has a normal distribution with a mean value of 0.85 kip/ft’ (40.7 kN/m2), and a standard deviation of 0.45 kip/ft2 (21.55 kN/m’). Simulation solution. The simulation was performed using these data, and the computer output indicates the following results: p,,, = 3441.84 kips (15,309.30 kN) u,,, = 735.46 kips (3271.33 kN) P,= 15.78%

/?, = 0.9982 FS = pR/pe = 1.48. Discussion. According to Hoek and Bray [22], a factor of safety between 1.0 and 1.3 is adequate for mine slopes, whereas a minimum factor of safety of 1.5 is required for critical slopes adjacent to roads or important installations. Therefore, the slope is stable since it has a factor of safety equal to 1.48. However, this factor of safety is misleading since there is a substantial probability of failure of 15.78%. Example 3: InfIuence of water pressure and reinforcement on rock slope reliability

This example demonstrates how to account for uncertainty in water pressure and reinforcement when assessing rock slope reliability. In this example, Barton’s failure criterion is used, and all the correlations proposed in Fig. 5 are taken into consideration. Assume that a rock slope is to be analyzed, with the following information: A = 4500 ft2 (415.87 m2)

$=45”

1503

CI= 66” aio = 0.001 MPa. In addition, 4: has a triangular distribution that is characterized by three values: a minimum of 20”, a mode of 25” and a maximum of 35”; JCS has a uniform distribution with a maximum value of 100 MPa and a minimum value of 50 MPa; and W has a normal distribution with a mean value of 10,000 kips (44,480 kN) and a standard deviation of 3000 kips (13,344 kN). Simulation solution. Assuming, first, that the slope is dry, the simulation was performed and the following results were obtained: pM= 1574 kips (7001.15 kN) bM = 439.5 kips (1954.9 kN) P, = 0.0002%

/$ = 3.5811 FS=p,/p,=

1.23.

Discussion. The results indicate that the slope is stable. Now, suppose that the effect of water is to be included in the analysis. To do that, assume the following values for the water forces: U has a normal distribution with a mean value of 1500 kips (6672 kN) and a standard deviation of 500 kips (224 kN), V has a normal distribution with a mean value of 600 kips (2668.8 kN) and a standard deviation of 300 kips (1334.4 kN). The simulation was performed assuming perfect correlations (p = 1.0) between U and V, and the following results were obtained:

pM = -665.2 kips (2958.8 kN) rsM= 857.4 kips (3813.7 kN) Pf = 77.1%

B, = -0.7759 FS = pR/pLe= 0.92.

Therefore, as a result of water pressure, the probability of failure increases. To improve the stability of the rock slope, assume that rock reinforcement is to be installed with the following variability. T has a lognormal distribution with a mean value of 1500 kips (6672 kN) and a standard deviation of 500 kips (2224 kN), and p = 30”. Using this data, the simulation was performed again and the following results for the safety margin were obtained: pM = 1539 kips (6845.5 kN)

1504

SAMERTAMIMIetal. u M=

1109 kips (4932.8 kN)

Pr = 1.6% j?, =

1.4805

FS = /+./pa =

variables by using Monte Carlo simulations. Finally, note that if Monte Carlo simulations are limited by constraints of economy and/or computer capability, a combination of analytical and simulation (i.e. hybrid) approaches to rock slope reliability problems may become more effective.

1.24. REFERENCES

The reinforcement substantially decreases the probability of failure and increases the factor of safety. SUMMARY

AND CONCLUSIONS

The evaluation of rock slope stability can be made deterministically if the nature and magnitude of the variables included in the analysis are certain. Unfortunately, many of these variables are probabilistic rather than single valued quantities. Traditionally, the safety of a rock slope with respect to sliding is expressed by means of the so called factor of safety, FS, defined as the ratio of the resisting force to the driving force. In recent years, considerable attention has been given to the identification and description of the uncertainties that exist in soil and rock properties. Probabilistic reliability analyses provide an alternate approach in assessing the reliability of rock slopes, whereby the variables included in the analyses are expressed in probabilistic terms. This approach has the advantage over deterministic techniques in incorporating the uncertainties that inherently exist in soil and rock properties, thus providing a more realistic approach. Two types of probabilistic techniques are commonly used, the pure analytical methods and the simulation techniques. Although the pure analytical methods are more accurate, they often cannot be used without including some simplifying assumptions. Moreover, in some cases even if a formulation is possible, the required solution may be analytically intractable. In these instances, a probabilistic solution may be obtained through Monte Carlo simulation. Compared to other probabilistic methods, the Monte Carlo simulation has two major advantages: (1) it includes the influence of possible correlations between the variables entering into the expression for the safety margin, and (2) it takes into account different probability density functions of the random variables involved in the design equation. These advantages are of great interest when looking at the reliability of rock slopes where the rock mass properties, the weight of the sliding block, the water forces and the reinforcing force entering into the expression for the safety margin are uncertain. Furthermore, probabilistic dependencies exist between these variables. In the context of rock slope reliability evaluation, this paper stressed the importance of including these dependencies among random

1.

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Monte Carlo simulation of rock slope reliability 19. G. Barla and F. Forlati, Shear behavior of filled discontinuities. Proc. Int. Symp. on Fundamentals of Rock Joinrs, pp. 163473 (1985). 20. A. M. Agogino and A. Redge, A tutorial on influence diagrams. Working Paper 85-07-03, Department of Mechanical Engineering, University of California, Berkeley, CA (1986). 21. R. D. Shachter, Evaluating influence diagrams. In Reliability and Quality Conirol, pp. 321-344. Elsevier-North Holland, Amsterdam (1986). 22. E. Hoek and J. W. Bray, Rock Slope Engineering, Revised 2nd Edition. The Inst. of Mining and Metallurgy, London (1977). 23. N. R. Barton, S. Bandis and K. Bakhtar, Strength,

1505

deformation and conductivity coupling of rock joints.

Int. J. Rock Mech. Min. Sci. Geomech. Abstr. 22, 121-140 (1985). 24. R. N. Chowdbury and A. Grivas, Probabilistic model for progressive failure of slopes. J. Georech. Engng, ASCE 108, 803-819 (1982). 25. E. F. Glynn, Stability of kinematic instability in slopes-a numerical approach. 20th U.S. Symp. on Rock Mechanics, Austin, Texas, pp. 317-321 (1979).

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