Structural Safety, 9 (1990) 1-20 1 Elsevier

PROBABILISTIC THREE-DIMENSIONAL STABILITY ANALYSIS OF SLOPES

M. Semih YOcemen

Department of Statistics, Middle East Technical University, Ankara 06531, Turkey

and Azm S. AI-Homoud

Department of Civil Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139, U.S.A.

(Received July 8, 1988; accepted in revised form March 24, 1989)

Key words: failure; probability theory; random variables; reliability; safety; slope stability; soil mechanics; statistical analysis.

ABSTRACT

A probabilistic model is described to evaluate the three-dimensional (3-D) stability of earth slopes under long-term conditions. The model takes into consideration the spatial variabilities and correla- tions of soil properties, as well as the uncertainties stemming from the discrepancies between laboratory-measured and in-situ values of shear strength parameters. It also accounts for the effects of modeling errors and progressive failure. In the 3-D analysis, the critical and total slope widths become two new and important parameters.

An actual failure case is analysed in detail taking into consideration all sources of uncertainties. Based on the best estimates of the different soil parameters, the failure width is predicted to be 63 m versus 55 m of observed failure width and the probability of slope failure is computed to be O. 11. The results agree well with those actually observed supporting the predictive ability of the PTDSSA (Probabilistic Three-Dimensional Slope Stability Analysis) model The numerical evaluation of the model is carried out by the PTDSSA computer program prepared for this purpose.

1. INTRODUCTION

The stability of slopes in natural or man-made earth structures is a classical problem in geotechnical engineering. Like most of the geotechnical engineering problems the stability analysis of earth slopes involves not only spatially varying soil properties but also numerous

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uncertainties resulting from insufficient information and inadequate knowledge of the random characteristics of soil profiles. In the conventional deterministic design or safety checking an overall factor of safety together with conservative values of soil strength and load parameters are used to account implicitly for these uncertainties. In the deterministic analysis the safety criterion requires that the computed factor of safety be greater than a prescribed allowable value.

On the other hand, the probabilistic approach to geotechnical problems permits a rational consideration of various sources of uncertainties that significantly influence the failure of a slope. In fact, for purposes of analysis and design, probabilistic concepts are the proper tools for the modeling and analysis of these uncertainties. In this case, the probability of survival a n d / o r the reliability index are used as measures of safety.

Three-dimensional probabilistic models of slope stability have been investigated in a number of studies [1-7] confirming the importance of 3-D effects. However, in most of the studies to date, all important aspects of the problem (such as all sources of uncertainties, 3-D formulation, short and long-term stabilities) were not treated at the same time. The aim of this paper is to provide a comprehensive probabilistic 3-D slope stability model with all its aspects. The proposed model takes into consideration the spatial variation of soil parameters related to the shear strength (i.e., cohesion, angle of friction and pore pressure) in three perpendicular directions in a soil volume. In this model all sources of uncertainties are modeled and systematically analysed.

The model is applicable to any slope under short and long-term conditions. However, the proposed probabilistic three-dimensional slope stability analysis (PTDSSA) model is presented here with respect to the long-term conditions, and its implementation is also illustrated considering an actual failure case under long-term conditions. An application of the PTDSSA model concerning the short-term conditions is given elsewhere [8]. Numerical computations are carried out by using the computer program (PTDSSA-program) coded for this purpose [1].

Throughout the paper, - (bar) and - (tilde) over a random quantity denote the mean and standard deviation of that random variable, respectively.

2. PTDSSA MODEL

2.1 3-D formulation of slope stability

In this paper the PTDSSA model is developed by following the general framework outlined by Vanmarcke [2,9] but with certain extensions and modifications. The details of the PTDSSA model is given in Ref. [1]. Here, we shall only present an outline of this model considering long-term conditions.

In the probabilistic formulation, similar to the deterministic models, the reliability of a slope can be analysed by comparing the restoring moment, R, and the disturbing moment, S, about the center of rotation of a slip circle. In the two-dimensional (2-D) model, the cross-sectional values R(xo) and S(xo) are compared and the 2-D safety factor at a location x = Xo, F(xo), becomes

F(xo) = R ( x o ) / S ( x o ) (1)

The 3-D formulation deviates from the 2-D formulation basically with respect to the consideration of the end effects in the former one and the total length, B, and the failure width,

X

0

Fig. 1. View of failing soil mass and cross-section within the failing soil mass (Ref. [2]).

b, enter into the picture as additional variables. Our model for the probabilistic 3-D analysis of slope stability will rest on the following basic assumptions (see Fig. 1): (i) Failure surfaces are cylindrical; (ii) the location and width of the sliding soil mass are constrained to their critical values; (iii) the soil properties are statistically homogeneous over the soil volume under consideration, so that the correlation functions of the soil properties do not depend on location but they depend only on the distance between two points; (iv) cross sections along the axis of the slope are the same. These assumptions can be relaxed whenever necessary as explained in Refs. [1] and [2].

In view of the above assumptions, for a cylindrical potential failure surface centered at x = x 0 and bounded by vertical end sections at xl = x0 - b / 2 and x 2 = x 0 + b / 2 (see Fig. 1), the 3-D safety factor Fb(xo) is expressed as follows [1,2]:

/

" " ~ ] / " x I \ x l

where, R(x) and S(x) are cross-sectional resisting and driving moments, respectively, and R e is the contribution of the end sections of the failure surface to the resisting moment.

For the computat ion of R(x) and S(x) the ordinary method of slices is adopted. Under the long-term conditions the shear strength depends on the cohesion, c, angle of friction, ~, and the pore pressure, u. The spatial averages of these shear strength parameters are taken as the basic random variables, whereas the unit weight of soil and the geometric parameters are treated as deterministic basic variables, since the associated uncertainties are relatively small. In view of these assumptions, R(x) becomes the dominant random function in eqn. (2).

The PTDSSA model takes into consideration all sources of uncertainties, and systematically incorporates them into the assessment of the reliability of an earth slope. Besides the spatial variability of shear strength parameters, inaccuracies in the mechanics of the deterministic slope stability model as well as discrepancies between the in-situ and laboratory-measured values of soil properties are taken into consideration. Random correction factors denoted by N are introduced to adjust for these inaccuracies and discrepancies.

At a certain generic section, x = x 0, the cross-sectional overturning and resisting moments, with appropriate correction factors and consistent with the method of slices are expressed as follows [1]:

m t~7

S ( x ) = r Y'~ V,y i sin a i = r Y'~ W~ sin ai (3) i = 1 t = l

m

R ( x ) = NmNpr Y'~ {Nc c~l ~ + (l/V, cos a , - u~l~ cos 2 a~)tan(N~,qai)} (4) i = 1

where, r = radius of the circular failure surface, m -- number of slices, V, and W, = volume and weight of the i th slice, 7~ = average unit weight of soil volume within the ith volume, a~-- inclination of the base of the ith slice to the horizontal axis, l, = length of the ith segment of the failure surface, u~ = pore pressure acting on the ith slice computed from the average pore pressure distribution along the failure surface. Throughout the text the spatial averages of the shear strength parameters are denoted by lower case letters, whereas their "poin t" values are denoted by capital letters.

In eqn. (4), N m and Np are the correction factors for the modeling error and the effect of progressive failure, respectively. On the other hand, N, and Ne~ are the correction factors accounting for the systematic errors resulting from the discrepancies between laboratory and in-situ conditions in the estimation of the values of c and qa. Both of these correction factors can be written as the product of component correction factors to accomodate for errors resulting from limited number of samples, No; change in the stress system because of the removal of the sample from the ground, N1; mechanical disturbance during sampling, N2; size of specimen, N3; rate of shearing, N4; anisotropy, N 5. In other words we can express, for example, the correction factor for cohesion as follows:

5

No= l--I Nj(c) (5) j = 0

The mean value of a correction factor is denoted by N and is assumed to correct for the mean bias. The coefficient of variation (c.o.v.) of N is denoted by A and it accounts for the prediction and the test discrepancy errors.

The statistical parameters of the correction factors depend on the soil properties, site conditions and types of laboratory tests performed. General guidelines for the quantification of N, and A~ for different soil types are given in Refs. [10-12]. In the same references, methods for combining several sources of information in estimating the statistical parameters of the correc- tion factors are also presented, together with examples illustrating the assessment of these parameters.

The uncertainty in the driving moment S is neglected here, since the basic variables involved in the computation of S have comparatively small uncertainties. Thus, it is treated as a deterministic variable in eqn. (2). As a result, our interest lies now on the total (3-D) resisting moment Rb(x) and the assessment of its statistical parameters. Rb(x ) consists of two terms (see eqn. (2)). The first term is the integral of the random function R(x) . The exp._ected value and the standard deviation of this term is bR and bRFR(b), respectively [2]. Here, R and /~ denote the mean and standard deviation of R, respectively. FR(b ) is the dimensionless standard deviation reduction factor reflecting the smoothing effect of the integration operation on the spatial variability of R(x) .

The second term of Rb(X ) reflects the effect of end sections. Here, we shall assume the end sections of the failing soil mass to be vertical planes. Vanmarcke [2], incorporated the effect of end sections into the analysis by treating each end section as the equivalent of an extension by an amount d/2 of the width of failure area. Since the end surfaces are assumed to be vertical, this hypothetical extension of the failure width will not necessitate any increase in the weight___of the sliding soil mass. Here, the same procedure will be applied. Denoting the mean of R e by R e and defining d = R e / R (where d has the dimension of length and could be obtained by plane strain deterministic analysis) the mean of 3-D resisting moment becomes

Rb = bR --}- Re = R(b -'}- d) (6)

The variance of R b will be the sum of the variances associated with the resisting moment developed over the cylindrical surface and the vertical end sections expressed as follows [2]:

/ ~ = bZkZF~(b)[1 + e l (7)

where, e equals the ratio of the variance of the "end section" resisting moment to (bR)2. It was shown by Vanmarcke [2] that for failure widths of practical interest, the coefficient e is close to zero.

In effect, the contribution of the end sections is taken as an increase in the mean value of the resistance, but the contribution to the uncertainty is neglected. However, any uncertainty resulting from this is implicitly covered by the modeling uncertainty.

Uncertainties in the driving moment can be treated in a similar way except that there will be no end effect contribution if vertical end sections are assumed. As discussed earlier the variability in the driving moment is neglected and it is considered as a deterministic variable with its value equal to its true mean, S. Accordingly, based on eqn. (2) the mean of the 3-D safety factor F b becomes:

fib= R(b_+ d) _ if(1 + d/b) (8) Sb

where F = R / S is the "plane strain" (2-D) mean safety factor. In eqn. (8) and in other equations, the extension length, d, will be computed approximately from the following equation [21

2Ar' 2A d - L ~ - L (9)

where, L is the total arc length, A and r ' are the cross-sectional area and moment arm of each end section, respectively. It is assumed that r ' = r.

The mean and variance of the 3-D resisting moment, R b, are now expressed in terms of the mean and variance of the 2-D resisting moment, R, as given by eqns. (6) and (7), respectively. The mean and variance of R(x ) are obtained by using the first-order second-moment (FOSM) approximation. In this case the statistical parameters required are the means and "poin t" variances of the basic variables and of the correction factors. Also any type of existing correlations have to be quantified.

The degree of spatial correlation associated with the shear strength parameters is quantified by the scales of fluctuation in the x, y and z directions, denoted by ?~x, ?~u and ?~z, respectively. The scale of fluctuation, ?~, which is introduced by Vanmarcke [2,9,13] is a convenient measure of the degree of correlation in a soil medium. Physically interpreted, it represents the distance over which a soil property (in certain direction) shows a relatively strong correlation.

2.2 Computation of the statistical parameters of R(x)

A first-order approximation of the mean of R ( x ) is obtained as follows [1]:

R = N m N p r , (c)? i l i + (W~ cos a , - ~ / , cos 2 a,) tan ~,(q~)~, (10) i = 1 ' = " =

The variance of the resisting moment, k 2, will be composed of contributions from all of the component basic variables. Moreover, each variance component will itself consist of contribu- tions from the different segments into which the failure arc is subdivided. In this variance contribution the spatial and other sources of correlations among the basic variables will also be taken into consideration. Below the contribution of cohesion to the uncertainty in R is given by considering the most general case and then simplifications for common special cases are discussed.

A generic segment i on the failure arc is identified by its length l~ and its inclination a t. The characteristic correlation distance for cohesion along the failure arc within segment i is denoted by At. In general, the random variation of the cohesion may not be isotropic. One must expect the scale of fluctuation of the cohesion to depend on direction; specifically, it tends to be different in the horizontal and vertical directions. Here •,. is approximated as follows [2,9]:

1 i/2 I c, = A2% sin 2 a i -Jr 12ci, cos 2 o/i ] (11)

where ),c and Xc are the scales of fluctuation for cohesion within the segment i in the z and y directions, respeci{vely (see Fig. 2).

The contribution of the cohesion to the variance of the resisting moment consists of contributions from the different segments into which the failure arc is divided. The total contribution of uncertainty associated with cohesion will consist of two parts: The first part results from the variability within individual segments (note that a segment forms the lower

Z

~k Sliding ~k~o~. surface ~ .~

/ I( £i "(Yk ' Zk)

T ~£o z

Fig. 2. View of the i th and k th slices on the sliding surface and the corresponding 1-D spatial averages.

surface of a slice). The second part results from the spatial correlation between segments. To find these contributions, the total failure arc length L is divided into m segments with arc lengths ll . . . . ,lm where Ei%ll~ = L. The mean and scale of fluctuation of c, may differ from segment to segment. The standard deviation at the mid point of segment i is denoted by Ci and the scale of fluctuation of this segment, ?~c, is as given by eqn. (11).

Based on the FOSM method (mean point expansion), the contribution of the cohesion to the total variance of R, denoted b y / ~ , is given as follows:

R~=NmN~,r c F2(l,) ~21 i= l

)) ]/ + 2 Y'~ • ( ~ ( c ) ( ~ k ( c ) Fc,(l~)I'~,(lk)C,(~kl~lkoc, c, (12) i=1 k = i + l j=O

where Fci(li) and Fc,(lk) are the standard deviation reduction factors due to spatial averaging of c over the segments i and k, respectively, and Pc, c, is the coefficient of correlation between the spatial averages of cohesion for segments i and k. In eqn. (12), the first part results from the variability of c within a segment and the second part is due to the spatial correlation between the spatial averages of c at different segments.

The computation of the standard deviation reduction factor requires information on the correlation function. Vanmarcke [2,13] showed that a reasonable approximate expression for the standard deviation reduction factor, regardless of the form of the underlying correlation function, would be

1.0 for l~<)t (13) F ( I ) = (~k/l) 1/2 for l>~X

The expression for the evaluation of correlation between the spatial averages of a soil property for two segments along the failure arc is derived in the appendix. The O~i~, term is to be evaluated based on eqn. (A.3) given in the appendix.

Depending on the degree of spatial correlation, the following special cases may arise in practice: (1) All characteristic correlation distances (i.e., scales of fluctuation for all segments), )t~,, along the failure arc are larger than the total arc length L; then eqn. (12) reduces to

k~ ~ 2 ~ 2 2 (14) = ~v m JVp r i ( C l i i=1

(2) The soil is homogeneous, the spatial correlation is isotropic (i.e., C~ = C and ?~c, = ?~c) and 2,,. ~< L; then eqn. (12) reduces to

=,Vm,Vpr f i ~.2(c) d2X~L (15) j=O

For obtaining the contribution of the other component random variables to ~2 similar formulations are carried out. The formulation for the contribution of the correction factors follows the same procedure. It should be noted that in the case of correction factors the most common case is the perfectly correlated case, for which the associated variance reduction factors become one.

At this stage, taking into consideration all contributions, the variance of the resisting m o m e n t is given by

7

/~2= ~ h2 (16) i = 1

where, k 2, R22, k 2, /~2, k2, k2 and k 2 are respectively, the contr ibut ions of cohesion, angle of friction, pore pressure, correction factors N m and Np, correction factors Nji(c ) and Nji(q0, correlation between cohesion and angle of friction and correlation between correction factors Nai(c ) and Nji(q~ ) to k 2.

The variance, k 2, expressed by eqn. (16) takes care only the spatial averaging along the arc length and does not account for the averaging along the axis of the slope. The effect of spatial averaging along the axis of the slope is taken care by F2(b) in eqn. (7). An indirect but a simple way to compute 1"2(b) is to evaluate a weighted average reduct ion in the variance of the resisting moment due to the variance reduction associated with each variable in the x direction. This can be expressed as follows:

r 2 ( b ) = g , U ( b ) + + 3U(b) + . . . +¢kr2(b) (17) where

Variance contr ibution (from all slices) for the k th variable ~'~ = k2 (18)

and k refers to c, O, u, N o, Nj (c ) , . . . , e t c , and ~1 + ~2 + ~3 + " '" +~k = 1.0. In eqn. (17), F~(b) equals to 1.0 for X k >~ b, and it equals to X J b for X k ~< b.

3. COMPUTATION OF FAILURE PROBABILITIES

The probability of failure, Pf(b), of a soil volume of width b and centered at a specified location along the slope axis is defined as follows:

Pf (b) = P( F b <~ 1.0) (19)

Note that in this case the 3-D safety factor, F b, is treated as a r andom variable, since the location of the failure mass is fixed.

The reliability index, r , is also a convenient measure for evaluating the safety of a slope. In terms of the mean and s tandard deviation of the 3-D factor of safety, the reliability index associated with a soil mass of width b is [1,2]:

d -1 ( f ib - 1)/Fb= (1 + d / b - 1/ff)~2~,1(1 + ~ ) (20)

where, ~R,, = / ~ b / R b = ~2RFR(b)(1 + d/b) 1, and denotes the c.o.v, of R e. In order to compute the probabili ty of failure we have to assume a probabil i ty distr ibution for

F e. The distribution of F b depends on the joint distr ibution of the shear strength parameters which is generally not available. However, if for convenience we assume F b to have a Gaussian distribution, then the probability of failure Pf (b) becomes:

Pf (b) = P ( F b <~ 1) = ,#(-13) (21)

where ~(-) is the cumulative distribution function of the s tandardized Gaussian distribution.

Equation (21) gives the risk of failure at a specific location over a width of b. Now, we shall evaluate the probability of a soil mass of width b failing at any location along the axis of the slope. In other words, we shall evaluate the safety of an earth slope along its total length, B. In this case, Fb(X ) forms a random process, since the location of the sliding soil mass is not fixed but random. A failure over a specified width of b will occur anywhere along the axis of the slope, whenever the random process, Fb(X ) crosses into the unsafe domain defined by { F b <~ 1.0}.

Equations to compute the probability that failure may take place at any location along the slope, PF, was derived by Vanmarcke [2] by utilizing the level crossing concepts of random functions. These equations are as follows:

1 - [ ( 1 - P f ( b ) } e x p { - v ( B - b ) } ] B>~b PF ----- (22)

Pf (B) B <~ b

where

v = (,rr 2 b ~ R ) - ' e x p ( - fl2/2) (23)

Here, ~k R is the scale of fluctuation of R and equals to bF2(b), assuming b >/~k R.

In computing Pf(b) from eqn. (21) it is assumed that the failure width b is fixed. Actually there will be a critical width which gives the highest failure probability. This critical or most likely failure width, denoted by b c, is taken to be the value which maximizes the probability of slope failure, PF-

4. PTDSSA COMPUTER PROGRAM

The PTDSSA computer program is prepared to carry out the numerical computations associated with the probabilistic model discussed in the previous sections. A flowchart of this program is presented in Fig. 3. The program can analyse slopes located in multilayered deposits and under short- and long-term conditions. The main input parameters are: the mean values and the uncertainties of the soil properties and the means and coefficients of variation of different correction factors. The correlation characteristics of the different soil parameters represented by the scales of fluctuation in the x, y and z directions are also required as an input to the program. The geometry of the slope should also be defined.

The main outputs of the program are: the geometric parameters of the most critical sliding surface (i.e., center of rotation, radius of rotation and critical width of failure), mean 2-D safety factor, c.o.v, of the resisting moment, probability of slope failure at a certain location and the probability of slope failure. A listing of the program together with the instructions for its use can be found in Ref. [1].

5. ANALYSIS OF THE SELSET LANDSLIDE

5.1 Description of the landslide

The Selset landslide was a rotational slide within a deposit of heavily overconsolidated boulder clay. The clay was remarkably uniform, without fissures or joints and the long-term conditions

10

S/lop Read e geometry

-Soil properties for layers

• Means, c.o.v.'s and scales of fluctuation of the soil properties

• Average location of water table and c.o.v, of pore pressure

• Mean correction factors and the corresponding c.o.v.'s

Ntunber of slices, trial centers, etc.

tD2 loop for d i f f e r e n t t r i a l c i r c l e s ~ . ~ . _ t . certain section along slope a x i s ) /

1 Establish coordinates of the trial circle

Define mean s l i c e p rope r t i e s including correction factors

I omp t . . . . . afety footor I

c o n ' t

Do loop for different failure widths 0 < b < total slope width (B)

1 Establish width of failure b

Evaluate var iance con t r ibu t ion from each soil parameter considering all slices

Compute total variance of 3-D resisting moment and the scale of fluctuation of resisting moment along the slope axis

Compute probability of total slope failure PF for all possible failure surfaces as

specified

i Select dimensions of slip surface (i.e• center and radius of circular surface and critical failure width) which maximizes PF

C-End )

Fig. 3. Flow chart for PTDSSA model•

0

.5 m

£h=2.0 m

I - 9 . X ~ ~ S - ]Boulder I

h : 12.8 1.0 / I ~ - - / Critical |- / i" ~ surface as |h

/ /" / / found in | w / I" / / this study

f ~_~_m~ ~y -~- f~A\w'2~'~/ __V_ __/ -" ~ Critical surface

| ~- as found by

7 9 ml ~ Skempton and • ~ Brown

/ ! ~~//----//~///

4.1 m

0 !

Fig. 4. Average cross section of slope and the position of the most critical potential failure surfaces at Selset (Ref. [14]).

11

existed [14]. The landslide took place in the south slope of the River Lune valley. An average cross section through the slope was given by Skempton and Brown [14] with an average height of 12.8 m and an average inclination of 28 ° (Fig. 4). The failure width was about 55 m and the exact location of the slip surface was not determined. However, Skempton and Brown [14] examined various possible slip circles and the most critical surface was identified as shown in Fig. 4. A 2-D probabilistic analysis of the Selset landslide have been carried out by Yucemen et al. [10], which included an assessment of the statistics of the correction factors.

5.2 Estimation of the input parameters

From each of the eight samples which have been taken at different locations of the slope [14], at least three specimens (37.5 mm diameter × 75.0 mm height) were prepared, and the peak effective stress parameters were estimated from consolidated drained triaxial tests. Based on these eight samples, the average soil properties and their coefficients of variation were computed to be [10]: ~ = 8.67 kPa, ~ = 32 °, 6(C) = 0.33 and 6(q~) = 0.05. The average unit weight of the boulder clay is computed to be 2230 k g / m 3.

For the intact non-fissured clay of this site and based on the studies of Skempton and Brown [14] and Skempton and Hutchinson [15], Yucemen et al. [10] estimated the mean correction factors for c and q, and the corresponding c.o.v.'s due to the different factors causing discrepancies between laboratory and in-situ strengths. These mean and c.o.v, values are:

N, (c ) = N~(q)) = 1.0, N: (c ) = 1.15, N2(q)) = 1.10,

-N3(c)=0.93, N3(q ) )= l . 0 , N 4 ( c ) = N 4 ( d p ) = 0 . 8 7 , N 5 ( c ) = ~ / 5 ( , ) = 0 . 9 8

and

Al(C) =Ai ( (~ ) ----0 , A2(C) -~0 .08 , A2(~ ) ---- 0.05 ,

A3(C) = 0 . 0 5 , A3(~/)) = 0 , A4(C) = A4(~/~) = 0 . 1 1 ,

As(c) = As(q~) = 0.04.

The mean value of Np (for this first-time slide) and the c.o.v, were given by Yucemen et al. [10] as 0.95 and 0.04, respectively. For the modeling error N m = 1.16 and the c.o.v, is taken as 0.11 as suggested in Ref. [10]. Finally, the uncertainties associated with insufficient sampling are computed to be

A0(c ) = 0.33/f8- = 0.12 and A0(q~ ) = 0 . 0 5 / f 8 = 0.02.

Since no local data was available to estimate the scales of fluctuation for cohesion and angle of friction, their best estimates are taken as hc. = ?%: = 1.0 m and hc, = ?~c, = ?~,, = h , , = 5.0 m based on the values suggested in Ref. [1].

As indicated by Skempton and Hutchinson [15], at the time of failure the ground water level was expected to be at the ground surface, except at the top of the slope. In the PTDSSA model we are interested in the average water table level and the associated c.o.v, No data was available about the fluctuations in the water table level to evaluate these values. Nevertheless, for purposes of analysis, a reasonable mean condition of water pressure within the slope could be represented by a static ground water table with its level parallel to the slope surface and whose top level is Ah meters below the surface. Here, a best estimate of Ah is taken to be 2.0 m. The c.o.v, for pore pressure, ~2(u), is taken the same as that of the water table level, and is computed from the

12

specified value of the probability of the water table level exceeding the slope surface level. Assuming a normal distribution for water table level, f~(u) is found to be 0.35 for a 0.001 probability of exceedance [1].

The total slope width, B, was not available. Hence, a value of 200 m is assumed. The sensitivity of results to the total slope width will be discussed in Section 5.4.

5.3 Estimation of the failure probability

Using the best estimates of the input parameters summarized above and assuming the resisting moment to have a Gaussian distribution, the computations are performed by utilizing the PTDSSA computer program. The assumption of the total resisting moment to be Gaussian is justified by the central limit theorem, since this quantity results from the summation (integra- tion) of the resisting moments due to the cross-sectional elements lying within the failure width (eqn. (2)). The program identifies the most critical failure surface and the associated failure probability, by computing failure probabilities for different potential slip surfaces and finally selects the one with the maximum failure probability. The failure probability computed in this way can be taken as a close lower-bound to the true slope failure probability. The factors which support this statement are [2]: the existence of a strong correlation among the failure events associated with adjacent failure surfaces and the decay of the failure probability for surfaces different from the critial failure surface. The flowchart for the PTDSSA computer program is given in Fig. 3 to show the sequence and details of computations.

For the Selset landslide the most critical surface is found to be that cylindrical surface with centre of rotation having coordinates of y = 4.0 m and z = 26.5 m and radius of 26.5 m (toe type of failure) as shown in Fig. 4 along with the critical surface estimated by Skempton and Brown [14]. The 2-D factor of safety for the critical surface equals to 1.35. The total slope failure probability Pv equals to 0.11 which is relatively high and consistent with the fact that the slope has actually failed. The most critical width of failure b c is found to be 63.0 m, which is in error of 15% of the actual failure width of 55.0 m.

The total uncertainty in the resisting moment, denoted by f~Rh' is computed to be 0.194, for the most critical failure surface. The contribution of different parameters to ~22 R,, are shown in

TABLE 1

Participation coefficients

Contributing Participation Participation parameter to ~2~b coefficient (%)

c 0.0001 0.27 ~b 0.0001 0.27 u 0.0071 18.83 N(~) 0.0148 39.25 N(c) 0.0017 4.51 Pc.q, 0.0000 0.00 ON(c), U(~) 0.0002 0.53 N m 0.0121 32.10 Np 0.0016 4.24

Total 0.0377 100.00%

13

Table 1 along with the corresponding participation coefficients. A participation coefficient is ~22 " divided by ~22 defined as the ratio of the "contr ibution of a certain parameter to ,% Rh"

Analysis of the results given in Table 1 shows that significant contributions to the total uncertainty come from pore pressure, and the different correction factors• The largest contribu- tion comes from the correction factor for q~ in this case. This is not due to high prediction uncertainty in ~ but due to the fact that the resisting moment developed is largely due to the frictional component of the shearing strength•

5 . 4 S e n s i t i v i t y a n a l y s e s

In the previous section, the best estimates of the different parameters involved in the stability analysis were used to calculate the mean value of the 2-D safety factor, the probability of slope failure PF and the critical failure width b c. Because of lack of data our estimates of pore pressure, scales of fluctuation and total slope width were highly subjective and could be in error• It is therefore desirable to study the sensitivity of our results to the variations in these parameters. For this purpose, one parameter will be perturbed within a certain range while other parameters are fixed to their best estimates and the variation in the basic results of the PTDSSA model with respect to critical failure width and probability of slope failure will be examined. Figures 5-15 show the results of these sensitivity analyses and the relevant discussion of the results are itemized below:

(1) An increase in mean water table level (or pore pressure) increases the probability of slope failure and reduces the expected critical failure width (Figs. 5 and 6).

(2) Variation in the pore pressure is one of the dominant parameters which affect the total uncertainty in the resisting moment• A change in the pore pressure uncertainty (i.e., ~2(u)) from 0 to 0.45 doubles the probability of failure (Fig. 7) and reduces the critical failure width from 72 m to 58 m in this case (Fig. 8).

(3) For purposes of analysis, the scales of fluctuation for c and ¢ are assumed to be the same. Furthermore the scales of fluctuation on the horizontal plane, X x a n d by, are also assumed to be the same. It is observed that if the horizontal scale of fluctuation is increased from 1 m to 30 m, b c decreases from 64 m to 55 m and PF increases from 0.11 to 0.13 (Figs. 9 and 10).

.~ o.6o

0.40 >i

-~ 0 . 2 0

.Q o 0.00

0.50 0.70 0.90 -- h / h

w

Fig. 5. Effect of average water table on slope failure probabi l i ty (hw = mean water table level, h = height of

slope).

v o cJ

z::

8o

,--t

4O

• 5o o.~o o.bo ' " ~w/h

Fig. 6. Effect of average water table on critical width of failure (hw = mean water table level, h = height of

slope).

14

O. 14

0.13

n

,-4 "~ O . i 1

4J • ~ 0.09

.Q m

Cu 0.07 ' ' J -'- ~(u)

0.0 0.2 0.4

Fig. 7. Effect of pore pressure uncertainty on probabil- ity of slope failure.

o 75

~ 7o

~ 6o

4J -,-t ~4 0 50 ' ' ' ., ~ (u)

0.0 0.2 0.4

Fig. 8. Effect of pore pressure uncertainty on critical width of failure.

0

~Z 4~ q3

@

0 ..4 4J

0

Fig. 9.

66

62

58

54

1 L I, = l.Om

. . . . 'o ' 2' 0 4 8 12 16 2 24 8 (m) x "y

Effect of horizontal scales of fluctuation on critical width of failure (kc, = Ac, = A~, = k , , ) .

[.q

,.-4 .,.4

4-4 O

43 -,.4

Fig. 10.

,-4 .,-4

0 )4

ICz =l~z = 1.0 m

O. 126

0.122

0.118

0.114

0.100 , , . . . . . . . k :I - x y

0 4 8 12 16 20 24 28 (m)

Effect of horizontal scales of fluctuation on probability of slope failure ( A , , = k , , = k~, = k,~, ).

0 60

zZ

~: 5O

O)

,-..-I -,-4 r~

"~ 30 u

-,-I 4..1

"~ 20 O

i0

70}

%

i i i i i

0 4 8 12 16 20 24 28 (m) X (m)

Z

Fig. 11. Effect of vertical scales of fluctuation on critical width of failure ( A c = ~ : ) .

]5

(4) The sensitivity to vertical scale of fluctuation, ?~z is examined for different values of Ax/B , where X x is normalized with respect to the slope width B. The influence of ?~z on the results are more significant compared to the horizontal scales of fluctuation. For example, if ?,z is increased from 1 m to 15 m the predicted failure width decreases from 63 m to 18 m and the probability of slope failure increases from 0.11 to 0.55, for Ax/B = 0.025 (Figs. 11 and 12).

(5) It is observed that for cases where the total slope width is greater than the critical width of failure, bc will have approximately a constant value. By increasing the total slope width from 70

1.0

toO.8

0.6

0.4

]~ 0.2

0

0.0 , , ' ' ~-- ;~ (m) 0 4 8 12 16 z

Fig. 12. Effect of vertical scales of fluctuation on probability of slope failure (Xc= = A,=).

16

0.50

m~0.40

.~ 0 . 3 0

0.20

4~

o . l o

,Q o 0.00 i I

0 20o 4o; 6oo a'00 ld00 ~-~--B (m)

Fig. 13. Effect of total slope width, B, on probability of slope failure.

m to 1000 m, the critical failure width is increased from 61 m to only 64 m in the case study considered here. On the other hand, the probability of slope failure is quite sensitive to the total slope width. For our case study, as B increases from 70 m to 1000 m, PF increases from 0.034 to 0.465 (Fig. 13). Therefore, slopes with smaller widths appear to be safer than those with larger widths, provided all other conditions are the same. These results justify our assumption of a rather arbitrary value of 200 m for the unknown total slope width in the case of the Selset landslide, since this assumption will affect significantly only the probability of slope failure, which may also be used as a relative measure of slope safety.

(6) One of the methodological questions in the implementation of the PTDSSA model is the selection of the number of slices. In our case study the critical width decreases from 70 m to 62 m as the number of slices is increased from 5 to 50 (Fig. 14). The probability of slope failure is

O c 3

"t3 .,--I

0)

,---I -,--I

,-.q

O

-,--I ka

72

68

64

6O i i I i i

0 i0 20 30 40 50

Number of slices

Fig. 14. Effect of number of slices on critical failure width.

,--4 -,4

q~ 0

-,-4 ,-4 -.-4 .Q

i 0.117

0.115

0.113

0.iii

0. 109 I I o I I

0 tO 20 30 40 50

Number o f s l i c e s

Fig. 15. Effect of number of slices on probability of slope failure.

17

quite insensitive to the number of slices (Fig. 15). As the number of slices exceeds 35, the change in all results is very small. Therefore, in our analysis the failure arc is divided into 35 equal slices.

6. C O N C L U S I O N S

A probabilistic model to analyse the 3-D stability of earth slopes under long-term conditions is presented. An actual landslide is studied in detail to illustrate the implementation of the proposed model. The numerical computations associated with the probabilistic model are carried out by using the PTDSSA computer program. In this study the description and the implementa- tion of the PTDSSA model are carried out considering the long-term conditions only. However, it should be noted that both the model and the accompanying computer program are applicable also for slopes under short-term conditions (i.e., total stress analysis).

On the basis of this study the following conclusions could be stated: (1) In the long-term stability, main sources of uncertainty are those associated with the method of analysis, pore pressure distribution, and the in-situ values of angle of friction and cohesion. (2) The degree of spatial correlation associated with shear strength parameters within a soil deposit also influences the probability of slope failure as well as the predicted failure width. This correlation is quantified by scales of fluctuation in the x, y, and z directions. It is found that a larger scale of fluctuation increases the uncertainty in the resisting moment, which causes a reduction in the critical failure width and an increase in the slope failure probability. The scale of fluctuation in the vertical direction (i.e. z direction) is found to affect the results more significantly than those associated with the horizontal direction (i.e. x and y directions). (3) The 3-D slope stability analysis requires the spatial averaging to be carried out over the soil volume which necessitates the consideration of 3-D correlation functions. However, in this study the (volumetric) spatial averaging is carried out first over the arc length, then over the slope axis. Such a procedure simplifies the computations, since the knowledge of the 1-D correlation functions (or scales of fluctuation) in the x, y, z directions becomes sufficient.

18

(4) Considerat ion of the third d imension and end effects in the slope stabil i ty analysis influences the slope failure probabil i ty, and the critical failure width becomes an impor tan t and new ou tpu t of such an analysis. (5) Total slope width has an impor tan t effect on the probabi l i ty of slope failure while it does not affect the predicted failure width significantly. It is found that as the slope width increases, the probabi l i ty of failure also increases in a relat ion which depends basically on the reliability index and the scale of f luctuat ion of the resisting mo me n t along the slope axis. (6) A decrease in the 2-D safety factor (due to effects such as a rise in the mean ground water level, lower shear strength parameters , lower values for the correct ion factors, etc.) would result in an increase in the slope failure probabi l i ty and a reduct ion in the critical width of failure. (7) An increase in the uncer ta in ty in the resisting m o m e n t (due to increases in the c.o.v.'s of pore pressure, cohesion, angle of friction, correction factors, etc.) would result in an increase in the probabi l i ty of slope failure and a decrease in the critical wid th of failure. (8) For the Selset landslide, based on the best est imates of the di f ferent input parameters , the proposed model predicted a failure width of 63.0 m which is in error of 15% of the observed failure width of 55.0 m. The probabi l i ty of slope failure is calculated to be 0.11 which is relatively high and consistent with the fact that the slope has actual ly failed.

REFERENCES

1 A.S. AI-Homoud, Probabilistic three-dimensional stability analysis of slopes, M.Sc. Thesis, Dept. Civil Eng., Yarmouk University, Irbid, Jordan, 1985.

2 E.H. Vanmarcke, Probabilistic stability analysis of earth slopes, Eng. Geol., 16 (1980) 29-50. 3 D.T. Bergado and L.R. Anderson, Stochastic analysis of pore pressure uncertainty for the probabilistic assessment

of the safety of earth slopes, Soils Foundations, 25 (1985) 87-105. 4 B. Peintinger and R. Rackwitz, Numerical uncertainty analysis of slopes, Rep. 52/1980, Technical University of

Munich, 1980. 5 E.H. Vanmarcke, Reliability of earth slopes, J.Geotech. Eng. Div., 103 (1977) 1247-1265. 6 D. Veneziano, D. Camacho and J. Antoniano, Three-dimensional models of slope reliability, Rep. R.77-17, Dept.

Civil. Eng. M.I.T., Cambridge, MA, 1977. 7 M.S. Yticemen and E.H. Vanmarcke, 3-D seismic reliability analysis of earth slopes, Proc. 4th ICASP, Florence,

Italy, 1 (1983) 195-208. 8 M.S. Yticemen and A.S. A1-Homoud, A probabilistic 3-D shortterm slope stability model, Proc. 5th ASCE

Speciality Conf. on Probabilistic Methods in Civil Eng., Blacksburg, VA, (1988) 140-143. 9 E.H. Vanmarcke, Probabilistic modeling of soil profiles, J. Geotech. Eng. Div., 103 (1977) 1227-1246.

10 M.S. Yt~cemen, W.H. Tang and A.H.-S. Ang, A probabilistic study of safety and design of earth slopes, Civil Eng. Studies, SRS No. 402, University of Illinois, Urbana, IL, 1973.

11 M.S. Yiicemen and W.H. Tang, Long-term stability of soil slopes: a reliability approach, Proc. 2nd ICASP, Aachen, Germany, (1975) 215-229.

12 W.H. Tang, M.S. Yticemen and A.H.-S. Ang, Probability-based short-term design of soil slopes, Can. Geotech. J., 13, (1976) 201-215.

13 E.H. Vanmarcke, Random Fields: Analysis and Synthesis, M.I.T. Press, Cambridge, MA, 1983. 14 A.W. Skempton and J.D. Brown, A land-slide in boulder clay at Selset, Yorkshire, Geotechnique, 11, (1961)

280-293. 15 A.W. Skempton and J.N. Hutchinson, Stability of natural slopes and embankment foundations, Proc. 7th Int.

Conf. on Soil Mech. and Foundation Eng., Mexico City (1969) 291-334.

19

APPENDIX

Correlation between two spatial averages

In the PTDSSA model it is necessary to evaluate the correlation between the spatial averages over the "areal" slices lying on the potential failure surface. Each of these areal slices has the dimensions shown in Fig. 2. These are the arc length l (existing on the y z plane) and the common width b (in the x direction).

Vanmarcke [9,13] has shown that for two rectangular areas A s and A k that lie on the same plane and have the same common dimension, such as b, then O(SAi, S & ) = O(Sli , Sl~ ) where, O(SA, , SA~) = correlation coefficient between the spatial average soil property s over areas A s and A k and O(sz, , sty)= correlation coefficient between the spatial average soil property s over lengths l s and l k. Here s denotes the spatial average of any soil property.

To evaluate the correlation coefficient between the spatial averages over A s and A~ (which are parts of the failure surface), the length l of each slice is decomposed into two components in the y and z directions, so that the areal slices A s and A k (Fig. 2) are each also decomposed and projected into areal slices on the x y and x z planes. By evaluating the correlation coefficient between spatial averages of areas projected to the x y and x z planes separately, and then combining the two results in an appropriate way, one obtains the correlation coefficient between the spatial averages of A i and A k. However, as discussed above, due to the common width b, the areal correlation reduces to the 1-D correlation.

With reference to Fig. 2, we can write: ls, = l s cos oti, Is. = I s sin as, lk, = l k COS Otk, lk_ = I k

sin a k, 1o, ' = l Yk - Yi l - ls cos a s, and lo_ = I Zk -- Zi l -- I s sin a,. Let X,, and Xs_ denote the scale of fluctuation of the soil property s in the y and z directions, respectively (if 1, and lk , lie in two different soil layers then to evaluate the coefficient of correlation, average correlation distances are to be used). The coefficient of correlation between the spatial averages of the soil property s over the two areal slices A s and A k becomes equal to the "combined" correlation coefficient between l s and l k in the y and z directions (due to common width), and can be written based on the general equation given by Vanmarcke [9,13]. For the correlation in the z direction [1]

2 2 _ to_) r,_(ts_+ lo~) lo.r,.(to:) ( i s + 2 2_ = =

2 2 2 2 -(to. + r;:(to + + (to,+ t , + < ) rs:(to + ts.+ + 21s l<Fs:( l,: ) F,:( l< ) (A.1)

where

1 if X,_>/lo_

)t, (A.2) 1'2(l°")= ~ if X~_~<Xo_

Expressions having the same format for F~(/i.), F~(lk: ) . . . . . etc, exist. In a similar way the coefficient of correlation between the spatial averages of the ith and k th

slices in the y direction O(,,sO, can be derived.

20

Finally, the "combined" coefficient of correlation between the i th and k th slices becomes:

lO(s,s;,) = [(p(sisk):) 2 sin20e + (p(s,,~),) 2 CoS2Oe ]11/2 (A.3)

where

0 e = tan-1 IZk--Z,I+ ( lk -- z , . ) / a

lYk-Yil+ (lk --li,)/2 (A.4)