INTERNATIONAL JOURNAL FOR NUMERICAL AND ANALYTICAL METHODS IN GEOMECHANICS, VOL. 12, 533-550 (1988)

SLOPE STABILITY ANALYSES FOR MATERIALS WITH A NON-LINEAR FAILURE ENVELOPE

I.F. COLLINS, C.I.M. GUNN, M.J. PENDER AND WANG YAN

School of Engineering, University of Auckland, Private Bag, Auckland, New Zealand

SUMMARY

Upper bound techniques are used to analyse the stability of slopes of material failing according to a non- linear failure criterion, such as closely jointed rocks. The global, extremal nature of upper bound calculations is discussed, as is the inverse use of such procedures for obtaining safe lower bound estimates for material strength properties in back analyses. Specific results are presented for the material model proposed by Hoek and Brown.

1. INTRODUCTION

In many situations the failure of rock slopes is governed by pre-existing features such as faults, joints and bedding planes. However, for jointed rock masses, where the directions of the joint planes are quasi-randomly distributed and where the joint separation is small compared with the slope dimensions, the slip surface is free to adopt the shape of ‘least resistance’. Classical soil slope stability analyses assume that the material failure is governed by a linear Coulomb failure envelope. Such linear analyses have been adapted for application to closely jointed rock slopes by allowing for the presence of vertical tension cracks.’ However, Hoek and Brown’ and Hoek3 have conclusively demonstrated that the failure criterion for such rock masses is a non-linear relation between shear and normal stresses. This non-linearity is most marked at lower stress levels, the failure envelope being of quasi-parabolic shape.

Having analysed a large number of triaxial tests on a variety of rock types, with varying degrees of fracturing, Hoek and Brown’ developed an empirical failure criterion relating the major and minor principal stresses c1 and ~ 7 ~ :

(a, - a3)/cc = (ma3/~, + s)”’ (1 ) where a, is the uniaxial compressive strength of the intact rock, and m and s are dimensionless constants which depend on the rock type and degree of fracturing. It is assumed that the directions of the fractures are sufficiently random for the behaviour of the rock mass to be isotropic. This criterion is a generalization of Griffith’s classical model for the development of cracks in brittle materials. The value of s lies between unity for intact rock and zero for a completely granulated specimen. The value of the second parameter, m, also decreases as the degree of fracturing increases, over a range from around 28 for intact granite down to the order of lo-’ for very poor- quality carbonate rocks. Detailed tables of these parameters are given by H ~ e k . ~ The experi- mental scatter in these data is quite large, particularly for the softer rocks.’

As can be seen from equation (l), the parameter s is a measure of the decrease in uniaxial strength due to fracturing, since sl” a, is the compressive failure strength under uniaxial

0363-9061/88/050533-18$09.00 0 1988 by John Wiley & Sons, Ltd.

Received 23 March 1987 Revised 14 March 1988

534 I . F. COLLINS ET AL.

conditions (03 =O). The physical interpretation of the second parameter rn is not so straightfor- ward; it governs the shearing strength of the material and plays a role similar to that of the angle of internal friction for a linear material.

The majority of soil slope stability analyses are based on the limit equilibrium procedure. The critical slope height or factor of safety is estimated by firstly assuming that the failure criterion is satisfied on some slip surface, typically a circular arc, and secondly ensuring that the failing mass is in overall equilibrium, e.g. by using one of the methods of While this procedure is similar to those of plastic limit analysis, it does not have the same rational basis. A number of attempts have been made, using calculus-of-variations arguments, to show that the slip curve which gives the minimum value of the safety factor for a uniform slope is in fact a logarithmic However, as emphasized by Morgenstern* and Josselin de J ~ n g , ~ . ~ ~ these analyses yield only stationary values of the factor of safety, and that deeper analyses show that this problem is highly degenerate, and that the logarithmic spiral solution does not give even a local minimum solution. Moreover it is not possible, by limit equilibrium arguments alone, to find such a minimum solution. From the point of view of general continuum mechanics this is not surprising since one would not expect to be able to construct a rational theory of material failure, based on statics alone, without reference to the kinematics of the associated failure mechanism.

The techniques of plastic limit analysis offer a more rational approach to slope stability problems, However, as is very well known, the normal flow rule assumption, which is necessary for the validity of the limit theorems (at least as they are traditionally stated) predicts dilatation rates for a linear Coulomb material which are far too large. The application of the limit theorems to materials with non-normal flow rules seems still to be a source of confusion in the geomechanics literature. l 1

An upper bound to the collapse load for a ‘standard’ material with an associated normal flow rule is obtained from a kinematically admissible velocity field, while lower bounds can be calculated from statically admissible stress fields. Since the stress field at collapse of a correspond- ing ‘non-standard’ material which has the same failure envelope but a non-normal flow rule satisfies all the conditions of a statically admissible stress field, it follows that the collapse load for the non-standard material cannot exceed that for the standard material. In other words, the standard material is ‘maximal’ in the sense that a failure load associated with it is greater than or equal to that associated with any other model material which has the same failure envelope but shows different kinematic behaviour (i.e. obeys a different flow rule). Furthermore, this means that an ‘upper bound’ calculated from a kinematically admissible velocity field is an upper bound to the failure load, irrespective of the flow rule. However , a ‘lower bound’ calculated from a statically admissible stress field is not necessarily a lower bound for a non-standard material. (For further discussion of this theorem see References 12-15.) This result is, of course, unfortunate for most design purposes; where lower bounds would provide safe design estimates.

In this paper systematic upper-bound estimates of the slope stability factor are obtained for various non-linear materials which model the behaviour of closely jointed rocks. In this approach we are generalizing the work on linear materials by Chen.6 Although these upper bounds provide ‘unsafe’ estimates, there are a number of factors in favour of such an approach.

1. As explained above, an upper-bound solution still overestimates the failure load, even when the material behaviour is non-standard, and when a number of competing solutions are available the one giving the lowest failure load is definitely the best. This is in contrast to estimates of load factors obtained from limit equilibrium procedures, which may be either upper or lower- bounds and which cannot be systematically optimized.

2. As will be shown, it is no more difficult to obtain upper-bound estimates when the non-linear

NON-LiNEAR SLOPE STABILITY 535

failure criterion is expressed in terms of principal stresses, as in Hoek and Brown’s criterion (equation (1)) than when it is expressed in terms of the local shear and normal stresses. A limit equilibrium procedure requires the latter type of formulation. Chen6 has compared stability factor estimates for linear materials obtained from optimum upper-bound calculations and limit equilibrium techniques, using either Bishop’s method of slices or the 4-circle method. The difference between the estimates is not large, and in particular the agreement between the upper-bound and 4-circle values is within 1-6 per cent for slope angles of 45” or greater (see Table 9.3 in Reference 6). In view of this and the uncertainties in the material constitutive parameters, it is argued that for practical purposes values obtained from upper bounds are at least as ‘useful’ as those obtained from limit equilibrium arguments. Here we are limiting attention to the simplest situation of isotropic, homogeneous slopes with no water table. The relative merits of the two procedures for more complex situations remain to be determined. Although the upper-bound procedure provides unsafe overestimates of the slope height when applied directly to slope stability problems, it will provide conservative lower-bound estimates of the material strength parameters when the results are used inversely in the form of a ‘back- analysis’. This aspect is discussed in Section 5, in which is presented a back-analysis procedure that is an application of the extremal stress field concept introduced by Hi11.I6 From this viewpoint the problem is one of pure stress analysis, and the ‘normal flow rule’ is introduced as a way of characterizing the extremal conditions; it is not a consequence of any assumption of a plastic constitutive law.

2. NON-LINEAR FAILURE ENVELOPES FOR CLOSELY JOINTED ROCKS

Kingston and Spencer” have developed a general theory of the statics of plane strain deformations of rigid/plastic materials with non-linear failure envelopes. The failure envelope can be expressed in terms of stress invariants as

P =f (4) (2) where p=4(a, +a3) is the mean principal stress and q=$(o, -a3) the maximum in-plane shear stress. Alternatively, the envelope can be expressed in the Mohr diagram as

7 = s(4 (3) where z and a are the shear and direct stresses on the slip curve.

Hoek and Brown’s criterion, equation (l), is readily expressed in form of equation (2):

but leads to a very complicated, implicit relation when expressed in terms of z and a, as shown by Hoek3 and Ucar.’ * Kingston and SpencerI7 have shown that the general transformation between forms (2) and (3) is best accomplished by introducing the local angle of internal friction 4 as an intermediate variable, since

tan4 = dz/da = g’(o) and cosec4 = dp/dq =p(q) ( 5 )

The angle 4 is the slope of the failure envelope in the Mohr (a,z) plane and is, of course, constant for a linear material. The second of equations (5) is readily established by using the transformation

z = q cos4, a = p - q sin4 (6)

536 I. F. COLLINS ET AL.

(see Figure 1). For a Hoek-Brown material the angle of internal friction is given by

sin@= - + I ( 2, )- (7)

In addition to using (4), stability factors have been computed'for a non-linear material, whose envelope is written as

where zo is the unconfined shear strength, and n and $ are material parameters. Ucar" has shown that the Hoek-Brown model can be approximated by an equation of the form of (8), with n =* for sufficiently small direct stresses. However, the present authors have found that this approximation is not valid at the level of stresses encountered in slope stability problems. Hoek and Brown' have also used laws of this type to describe the behaviour of a range of rock types, the values of the parameters n and t,h depending on the degree of fracturing.

A number of author^'^-^' have worked with failure envelopes of the simpler form

z = Aab (9)

with values of b in the range 0.65485. Such failure envelopes pass through the origin and so have zero unconfined shear strength. As shown in Appendix 1 , this means that for b > 0.5 the minimum slope stability factor for a vertical slope is actually zero! For this reason we have preferred to work with the slightly more complicated form (8).

We return now to the Hoek-Brown formulation (1) or (4). Since the values of m and s both decrease as the degree of fracturing increases, it is possible that, for a given rock type, these two parameters are not independent. A log-log plot of the values of these two parameters given by Hoek3 is shown in Figure 2. It is found that the two-coefficient formula

rn=m,exp[8.27(1 (10)

f i ts these data with a maximum error for rn of 'N 13 per cent, which occurs in the middle of the range of values of s z 4 x and less than 5 per cent for intact rock (s = 1.0). In view of the scatter in the original data, this approximation is held to be reasonable. The parameter m, is the value of m for intact rock, and can in principle be obtained from the measured inclination of the fracture planes in a simple compression test using equation (7). Typical values3 of m, vary from about 7 for carbonate rocks, through 10 for lithified

The error is zero for poor-quality rock ( s z

Figure 1 . Non-linear failure envelope-notation

NON-LINEAR SLOPE STABILITY 537

Figure 2. Variation of the material parameter rn with s in the Hoek-Brown model

argillaceous rocks such as mudstone, to 15 for sandstone, 17 for andesite and up to about 25 for coarse-grained igneous rocks such as granite.

Thus for a given rock type, with known values of oC and m, the degree of fracturing can be represented by the single parameter s. The corresponding value of rn needed in the failure criterion (1) or (4) is calculated from the formula (10). It has been suggested that the value of s can be related to the CSIR or NGI standard fracture ratings3

3. AN UPPER-BOUND PROCEDURE FOR NON-LINEAR FAILURE ENVELOPES

An upper bound to the collapse height of a slope can be found by postulating a collapse mechanism and then equating the rate of working by gravity to the rate ofenergy dissipation in the deforming regions. To keep the calculation of the latter simple it is usual, in slope stability analyses, to assume the slope collapses by sliding or rotating as a rigid block along one or more discrete slip lines. More complicated deformation modes involving deforming fan-shaped regions are sometimes considered, but these lead to much more complex computations which, in the present context, are not justified.

The energy dissipation rates must be calculated using the normal flow rule assumption, so the ratio of the compressive normal strain rate E to the shear strain rate y must satisfy (Figure 1) the relation

E / y = - tan4 (1 1)

538 I . F. COLLINS E T AL.

A slip curve can be regarded as a vanishingly thin layer across which the velocity components vary rapidly. If 6u and 6v are the jumps in the tangential and normal velocities across such a curve, then

where t is the layer thickness. It follows from equations (11) and (12) that y = 8u/t and E = - sv/t (12)

&/6u = tan 4 (13) so that the velocity discontinuity vector across a slip line makes an angle q5 with the slip line.

For a linear material 4 is a material constant, so that the angle between the velocity vector and the slip line must be constant along its length. Hence, as is well known, if the rigid block is translating the slip line must be a straight line, whereas if the block is rotating the slip line is a logarithmic spiral.6 For a non-linear material +and hence the traction components-will in general vary along the slip line, and no such simple conclusions about its shape can be drawn. A possible way of computing the shape of the slip line is to require the stresses on this line to satisfy the equilibrium equations,23 i.e. Kotter’s equations. This approach was adopted by Zhang and CheqZ4 who used a non-linear power envelope of type (8) and a calculus-of-variations argument to demonstrate that the stability factor (factor of safety) is stationary when Kotter’s equilibrium equations are satisfied along the slip surface. Essentially the same result had earlier been established by Baker and Frydman,’$ who also claimed that this stationary value was in fact a minimum. These authors did not investigate the nature of the second variation but instead appealed to the upper-bound extremum prinkiple of limit analysis. Although this conjecture may well be true, and the computations presented here support it, it is not sufficient in the present author’s view simply to study stationary values, since the minima encountered in limit analysis are nearly always non-analytic, so that the first variation does not v a n i ~ h . ~ ~ , ~ ’

Our main concern here, however, is to find a procedure which is sufficiently simple and general to be of practical value and which may be able to compete with limit equilibrium procedures. The computational procedure using Kotter’s equation to determine the critical slip surface is complex, as pointed out by Zhang and Chen,24,28 who were forced to use an inverse procedure to determine the critical stability factor. Here we use instead a much simpler procedure which as described in Section 4, yields results which differ typically by less than 2.5 per cent from those obtained by using Kotter’s equation.

Instead of considering slip surfaces of general shapes we limit our attention to log-spiral surfaces. It follows therefore that the friction angle 4 will be constant along the length of the slip surface, and for a non-linear material the normal and tangential stress components 0 and z must also be constant (Figures 3 and 4). The upper bound to the stability factor obtained for such a slip surface is of course an upper bound for the given non-linear material. However, since 4 is constant, the upper bound can be calculated just as for a linear material with this constant value of friction angle. The best upper bounds that can be obtained for fixed 4 and fixed slope angle /3 by varying the position of the spiral centre 0 (Figure 3) have been tabulated by Chen.6 We can thus obtain upper bounds for non-linear materials by using previously calculated linear stability factors. If now we let the (0, z) point traverse around the non-linear failure envelope (Figure 4), so that we now consider all possible values that 4 can take, we generate a family of upper-bound estimates for the non-linear material. Clearly we will now pick the value of 4 which gives the lowest such estimate, and will be the best estimate that can be obtained by using log-spiral slip surfaces. This argument is expressed more formally below.

The linear stability factor for a given slope angle /3 is defined by6

NON-LINEAR SLOPE STABILITY 539

Figure 3. Log-spiral failure surface

Figure 4. Generalized cohesion for non-linear failure envelope

where y is the specific weight, h the slope height and c the cohesion. For fixed 8, NL is an increasing function of 4. However, the effective local cohesion

c(4) = z-otan4 = q sec4 - p tan4 (15)

is a decreasing function of 4 when o>O, but an increasing function of 4 when o<O (Figure 4). There is hence a critical value of = say, for CY 2 0, at which the stability factor for the non- linear material is a minimum. The particular log-spiral failure mechanism thus selected is the best of the class of all such log-spiral mechanisms. It cannot be the exact solution, since the equilibrium equation along the slip line can never be satisfied with constant o and z values.

There is some arbitrariness in the choice of reference stress used to define the dimensionless stability factor for a non-linear material. For the power-law model (8) the natural choice is zo, the unconfined shear strength, while for the Hoek-Brown model (1) the uniaxial compression strength of intact rock oc was chosen initially. Hence, as appropriate, we define the non-linear stability factor as

N = y h / z , or N = y h / o c (16)

N ( 4 ) = NL(4) c(4) lzo or N ( 4 ) = NL(4) c(4)/ac (17)

so that in terms of the local linear stability factor

The values of NL(4) were taken from the table given by Chen.6 The local cohesion was calculated using equation (15), together with the appropriate failure envelope for each value of 4. For the

540 I. F. COLLINS ET AL.

power-law material obeying equation (S),

44) __ = [ ( 1 - a) (n/A)' - + A]

70

where A = tan $/tan I), while for the Hoek-Brown model (4)

+-tan c($) - m(1-sin $)2 s

C J ~ 16 sin $ cos $ m

As explained briefly in Appendices 1 and 2, single- and double-block sliding failure mechanisms have also been considered, but in general the optimum log-spiral solution was found to give the best estimates.

4. SLOPE STABILITY FACTORS

4.1. Power-law materials

A typical variation of the non-linear stability factor with the local angle of internal friction, calculated from equations (17) and (lS), is shown in Figure 5. To facilitate this calculation the values of the linear stability factors N , given by Chen6 were fitted to a sixth-order polynomial and the minimum value of N , namely Nmin, was found using a standard library optimization routine employing a quadratic interpolation procedure. The 'effective' angle of internal friction $min at which this minimum occurs is found to increase with slope angle /I. The optimum stability factor Nmin increased with $ (see equation (8)) but decreases with increasing slope angle /I, as illustrated in Figure 6. In order to gauge the effect of considering only log-spiral slip surfaces, it is necessary to compare the stability factor estimates of the present technique with the more 'exact' procedure of Zhang and Chen.24 The two procedures will clearly give the same answer for a linear material, since for such a material the stresses are in equilibrium along a log-spiral, so the discrepancy

I I I I 1 I 0.0 15.0 30.0 45.0 60.0 75.0 90.0

LOCRL FRICTION ANGLE

Figure 5. Variation of stability factor N with local friction angle $, for Power-law material Z / T ~ =(u/z0 tan $ + ly, with n -_ - $ = 85" and slope angle = 75"

NON-LINEAR SLOPE STABILITY 54 1

: 85.0

: 82.5

: 80.0

: 50.0

I 1 1 I rn d l

90 80 70 60 SLOPE INCLINATION

Figure 6. Variation of minimum stability factor Nmi, with slope angle for a power-law material with n=+ and various friction angles +

Table I. Comparison of stability factors calculated from Kotter's equation24 with those calculated using an optimum log-spiral for a power-law material (equation (8)) with n =+ and

$ = 20"

P 90" 75" 60" 45" Reference 28 4.52 5.66 6.97 8.78 Optimal log-spiral 4.61 5.80 7.19 8-99

between the two can be expected to increase with the degree of non-linearity. Table I gives such a comparison for a highly non-linear material with n =+. It is seen that the results obtained from the optimum log-spiral solution are consistently greater than those obtained using Kotter's equations, thus lending support to Baker and Frydman's result, but that this difference is always less than 2.5 per cent. We thus conclude that the errors introduced by considering only spiral slip surfaces are not likely to be significant in practice. It is common experience in metal-forming technology, where upper-bound techniques are used extensively, that very good load estimates may be obtained from fairly crude bounds. The load functional usually has a rather shallow minimum. More detailed results for the Hoek-Brown model are presented below since these are more readily related to particular rock types.

4.2. Hoek-Brown materials

For a vertical cut ( p = 900) it is found that the minimum stability factor Nmin varies markedly with s, but to all intents and purposes is independent of the rock type parameter m,. This finding can be understood by considering the single-block sliding solution, which is known to give reasonable estimates for steep slopes for linear materials. The theory of the single-block sliding model for a vertical cut can be developed in complete generality for an arbitrary non-linear failure

542 1. F. COLLINS ET AL.

criterion, as described in Appendix 1. For a Hoek-Brown material the stability factor is predicted to be 2s1/’ (see equation (A 6)). The optimum spiral solution is found to fit the formula

Nmi,=(1.935 +0.005)s1/‘

for all degrees of fracturing (s=O-l.O) and all rock types (m,=7-26). These are the non-linear analogues of the well-known results N,=4.0 and 3.83 for a vertical cut consisting of a linear, perfectly cohesive material, predicted by the single-block and log-spiral solutions respectively.6

Although the minimum stability factor is found to be effectively independent of rock type, this is not true of the local friction angle , at which this minimum occurs, as shown in Table 11. For the simple sliding mode, is given by the simple formula sin #min=(4s1’2/m+ 1)- l , where the dependence of rock type occurs explicitly through the parameter m. In turn this means, of course, that the position of the straight or log-spiral slip lines will also vary with rock type. The reason why the stability factor itself is independent of rock type is that the failure of a vertical cut is governed predominantly by the uniaxial strength of the material, sl/’ 0,.

The above considerations suggest that instead of using the compressive strength 0, of the intact rock in the definition of the stability factor, it is preferable to use s1iZ o,, the compressive strength of the fractured specimen. Accordingly we define a ‘reduced stability factor’

N, = yh/s’/’ c, = Nmi,/sl/’ (20) With this definition the variation in the values of the stability factor is now in the range 1-50 instead of varying over several decades. Computed values of N, for the five basic rock types and for the complete range of fracturing are given in Table 11. For non-vertical slopes the value of N , is seen to depend on both rock type and the degree of fracturing. For a given rock type the reduced stability factor first increases with the degree of fracturing to a maximum for s = and then decreases again. This behaviour reflects the analogous variation of the ‘effective’ angle of internal friction q5min (Table III), although this effect is less marked for shallower slopes (p=45”).

It should also be noted that for these shallow slopes the ‘effective’ angle of internal friction is largely independent of rock type. This implies that the unreduced stability factor Nmin depends only on the parameter m and is effectively independent of m, and s, as illustrated in Figure 7. The parameter m of course depends on both the rock type and the degree of fracturing, as in equation (lo), but essentially governs the ‘frictional behaviour’ of the material-see equation (7). For shallow slopes the slope stability factor is governed by these frictional properties and is more or less independent of the compressive strength.

5. BACK-ANALYSES

In a back-analysis, information about the ‘strength parameters’ of the material is deduced from the observed behaviour of a slope. The technique used here provides upper bounds to the collapse height of a slope, provided the material properties are known. Conversely, lower-bound estimates of the values of the material strength parameters can be deduced from observations of the heights of slopes which are not failing. Designs based on such conservative estimates will be ‘safe’, since the material strength is being underestimated.

This result is an example of the general extremal stress field concept introduced by Hi11.16 Suppose a body is observed not to fail under some prescribed loads, ‘failure’ being determined by the magnitude of some scalar functionf(cij) of the stresses. The precise type of failure is irrelevant; it could be brittle or ductile. There is an infinity of possible self-equilibrated stress fields which will be in equilibrium with these given loads. For each of these fields the value off will attain a maximum value,fm,, say, at one or more points in the body. If, out of the class of all such stress

NON-LINEAR SLOPE STABILITY 543

Table 11. Reduced stability factors N , for five different rock types

S

10-1 4 x 1 0 - 3 10-4 10-5 7 x 1 0 - 7 B 1 .o m, = 7.3: limestone 90" 1.93 80" 2.68 70" 3.78 60" 5.50 50" 8.37 45" 10.51

m, = 10.5: mudstone

90" 1.93 80" 2.84 70" 4.28 60" 6.7 1 50" 10.92 45" 14.09

m, = 15.7: sandstone

90" 1.93 80" 3.09 70" 5.1 1 60" 8.80 50" 15.36 45" 20.28

m, = 17.8: andesite

90" 1.93 80" 3.18 70" 5.45 60" 9.66 50" 17.18 45" 22.82

m, = 26.2 granite

90" 1.93 80" 333 70" 6.79 60" 13.21 50" 24.59 45" 33.08

1.93 2.8 1 4.17 6.44

10.34 13.28

1.93 3.01 4.83 8.09

13.86 18.18

1.93 3.3 1 5.95

10.97 19.93 26.64

1.94 3.43 6.40

12.15 22.4 1 30.06

1.94 3.88 8.23

17.02 32.41 43% 1

1.93 2.94 4.6 1 1.52

12.64 16.48

1.93 3.18 5.46 9.70

17.26 22.93

1.94 3.56 6.90

13.49 25.18 33.89

1.94 3.71 7.49

15.04 28.38 38.29

1.94 4.26 9.9 1

21.41 41.31 55.99

1.93 2.96 4.68 7.69

13.00 16.99

1.93 3.2 1 5.57 9.95

17.78 23.67

1.94 3.60 7.05

13.89 26.00 35.01

1.94 3.75 7.66

1550 29.3 1 3957

1.94 4.3 1

10.17 22.10 42.70 57.88

1.93 2.89 4.42 7.04

1 1.63 15.08

1.93 3.10 5.18 9.00

15.77 20.85

1.94 3.45 6.48

12.38 22.89 30.73

1.94 339 7.0 1

13.79 25.77 34.71

1.94 4.09 9.17

19.50 37.43 50.68

1.94 2.80 3.92 5.83 9.05

11.47

1.94 2.90 4.48 7.18

11.94 1550

1 :94 3.17 5.40 9.55

16.94 22.49

1.94 3.27 5.78

10.52 18.99 25.34

1.94 3.66 7.29

14.52 27.31 36.82

fields, the lowest suchf,,, value is$*, corresponding to a stress field o*, then it follows that the critical value off at which an element of this material fails cannot be less thanf*.

A stress field 05 which possesses this minimal value offis termed an extremal stress field. Hilli6 shows that 05 has the property that, in the region of the body wheref(oij) attains the extremal valuef*, there exists a virtual strain-rate field ei j which is directed along the outward normal to the surface f (oi j ) = constant at the appropriate local stress point 05. Thus the familiar 'normal flow rule' concept is introduced as a way of characterizing an extremal stress field; it is not part of any

544 I . F. COLLINS ET AL.

Table 111. 'Effective' angles of internal friction &in for limestone and granite, for various slope angles and degrees of fracturing

S

B 1 .o lo-' 4 ~ 1 0 - ~ 10-5 7 x 10-7

m, = 7.3: limestone

90" 40" 80" 38" 70" 35" 60" 31" 50" 26" 45" 23"

m, = 26.2: granite

90" 60" 80" 56" 70" 49" 60" 40" 50" 31" 45" 26"

45" 42" 39" 34" 28" 24"

64" 59" 51" 41" 31" 26"

49" 46" 42" 36" 29" 25"

66" 62" 53" 42" 31" 26"

50" 46" 42" 36" 29" 25"

67" 62" 53" 42" 31" 26"

48" 45" 40" 35" 28" 25"

65" 61" 52" 42" 31" 26"

42" 40" 36" 32" 27! 24"

62" 51" 50" 41" 31" 26"

constitutive assumption. These considerations reaffirm that the lower bounds to the material strength parameters obtained from the application of the upper-bound theorem of limit analysis are global, and independent of the actual flow behaviour of the material.

A linear Coulomb material involves two strength parameters: the cohesion c and the angle of internal friction Cp. The above argument can be applied by first keeping Cp fixed and finding the extremal value of c for each such Cp. As demonstrated below, this procedure generates a convex region in Mohr (a, z) space, termed the 'forbidden region', bounded by a resistance The actual failure envelope of the material must lie outside this region and cannot cross this resistance envelope.

The resistance envelope for a particular material can be constructed by first constructing a series of universal bounding envelopes in dimensionless (8, ?) space, where

8 = o/yh, 7 = ~ / y h (21) as shown in Figure 8. Each envelope corresponds to a particular slope angle /J. In terms of these dimensionless stresses, the Coulomb yield criterion can be written as

f(?, 8,Cp) = ? - 8 tan Cp - l/NL(Cp) = 0 (22) where NL(Cp) = yh/c is the linear stability factor. As Cp varies, equation (22) describes a family of straight lines in (6, ?) space whose envelope can be obtained from equation (22) and the equation obtained by equating df/dCp to zero. The equation of this envelope is most conveniently expressed parametrically:

8 = N i cos2 Cp/Nt, ? = (N, + N , sin Cp cos Cp)/Nt (23) where N,=dN,/dCp. The lower bounds to these resistance curves shown in Figure 8 were

NON-LINEAR SLOPE STABILITY

Beta= 45.0

545

MC = 2623 Mc = 17.84 Mc = 15.74 Mc= 10.49 Mc= 7.35

la.! I 1 I 1 1 1 1 1 1 1 1 1 1 1 1 1 I I I I 1 1 1 1 1 1 I I C I I I I 1 I 1 1 1 1 1 I I I l l 1 1

LO+ ic* lie Ib-l id lid ;'d b€aterialconstantm

Figure 7. Variation of minimum stability factor Nmin with rn for a Hoek-Brown material for various values of rn,

calculated using Chen's6 upper-bound estimates of NL, and an appropriate finite difference formula was used to calculate the derivatives Nt. It was found necessary to model the infinity in NL as 4-p before forming these finite differences. The form of this singularity is not easy to find as the computation of the optimum log-spiral is complex, but in the simpler case of a single-block failure mode it is easy to show that NL-(fl-$)-* as 4-p.

The form of the resistance curve can be found analytically when f l = 90" since then N, = 3.85 tan (b +*@), which when substituted in equation (23) gives the equation of the quarter-circle centred at (1/3.85,0) and passing through the origin, together with the straight line T = 1/3.85. All the resistance curves have straight-line segments parallel to the 0 axis for 13 2 Nt(0) with 7 = l/NL(O), as shown in Figure 8.

A conservative 'inner bound' to the failure envelope of a given material can be deduced from the above dimensionless resistance curves by using field data of observations of maximum slope height for various slope angles. Such data have been recorded by M ~ M a h o n , ~ ' who fitted simple power-law relations to the observed relation between maximum slope height h and slope length 1 for various rock types. For example, for the strong granitic rock slopes in the Yosemite National Park, h = aZb where, if h and I are measured in feet, a = 139 and b = 028. The resulting conservative failure envelope is shown in Figure 9. Of course, conservative estimates of failure envelopes obtained by using the resistance envelopes are of practical value only if the observed slopes are close to failure, and it may well be that a particular estimate so obtained is too conservative to be useful.

546 1. F. COLLINS ET AL.

I I 0 . 0 0 . 1 0 . 2 0 . 3 0 . 4 0 . 5 0 . 6 0 . 7 0 . 8

Figure 8. Dimensionless resistance curves ( C , ? ) for various slope angles p from 45" to 90"

6. DISCUSSION AND CONCLUSIONS

The object of this paper has been to demonstrate that upper-bound limit analysis techniques can be used to estimate the slope stability factors of closely fractured rocks and rock fills, whose failure is governed by non-linear failure envelopes. A simple procedure has been developed which assumes a log-spiral failure mechanism but which finds the optimal spiral that minimizes the stability factor for the given non-linear failure criterion.

Where the results can be compared, the procedure developed here is found to give results which are very close (to within 2.5 per cent) to those obtained by Zhang and Chen,24 who used a more 'exact' but computationally much more complex procedure. Attention has been concentrated on producing results for the Hoek-Brown models of highly fractured rocks expressed in terms of principal stress components instead of shear and normal stress components, which makes an upper-bound approach particularly appropriate. The influence of the degree of fracturing parameters m and s on the slope stability have been investigated.

NON-LINEAR SLOPE STABILITY 547

1.0

- ; 0.5 z P

u

0.0

0.0 0.5 1,. 0 1.5 2.0 2.5

0 (MPa)

Figure 9. Inner bound on failure envelope deduced from observed slope heights for granitic rock slopes of the Yosemite National Parkz6 for various slope angles p from 45" to 90"

A major criticism of the upper-bound approach to slope stability, which we do not refute, is that such estimates are unsafe. However, as we point out in Section 5, such estimates when used inversely in a back-analysis actually yield conservative estimates of the material strength.

This treatment has been limited to idealized homogeneous slopes. However, there is no difficulty in principle in generalizing these upper-bound methods to deal with inhomogeneity, anisotropy and pore water pressures.

ACKNOWLEDGEMENTS

The authors are grateful to the New Zealand National Roads Board for financial support, and to Alison Broadbent for help with some computing.

APPENDIX 1: SINGLE-BLOCK SLIDING SOLUTIONS

Single-block sliding modes are known to give reasonable estimates of the slope stability factors for near-vertical slopes and linear materials. The optimal single-block sliding solution for a vertical cut can be found in complete generality for a material with an arbitrary non-linear failure envelope. The rate of working by gravity in the triangle CAB in Figure 10 sliding with speed V is

Wb =$h2 Vtane cos(8 + 4) ('41)

548 I. F. COLLINS ET AL

A 0

Figure 10. Single-block failure mechanism

while the rate of energy dissipation on BC is

W, = c(4)h V sec 8 cos 4 1.42)

where c(4) is the local cohesion. Equating these two expressions yields the following overestimate of h:

For a linear material c and 4 are constants, so h does not depend on the values of the stresses on the slip line BC, and the lowest overestimate is obtained from the slip line with 6 =in -*4. For a non-linear material, however, 4 and c are variables, so we now choose the stress components on BC, as well as its inclination, so as to minimize h. For fixed 4, the minimum h is again obtained when 8=$~-34, whence

from equations (5), (6) and ( 1 5). On differentiating h with respect to q and equating to zero, it is found that h has a minimum when q = q,, the solution to the equation q =f (q ) = p (see Figure 1). In other words the stress state on BC is one of vertical uniaxial compression, with g1 = 2q, and g3 = 0. (This result could also have been derived directly by considering the overall equilibrium of the sliding block, but such an argument would not have demonstrated that this solution is an optimum upper bound.) The corresponding optimum bound on h is

k i n = 4qOl~ (A51 For a Brown-Hoek material qo is easily shown to be +s’ /*r~~, so the corresponding stability

factor is

Nmin = 2s”2 (A6)

This value is independent of the second material parameter m, although the corresponding activated friction angle q5min and resulting slip line inclination depend on both s and m.

For a simple power-law material (z = soh), the value of q,-which, of course, is the radius of the critical Mohr circle that passes through the origin-is most easily found as the limit of z sec4 as 7-0. Since tan4=dz/do= bA1/bT1-(llhf, it is found that

0 for b>3 *AZ for b=;

qo=‘ co fo rb<+

NON-LINEAR SLOPE STABILITY 549

0

Figure 11. Two-block failure mechanism

It follows therefore from equation (A5) that for a material with 3< b< 1, which includes all those identified by Charles and Soares,” for example, a vertical cut has zero strength and will collapse at an arbitrarily small height. This result is ‘exact’ since equation (A5) gives a definite upper bound, irrespective of the particular flow rule.

APPENDIX 2: TWO-BLOCK MODELS

A number of authors have investigated two-block models (Figure 11). At collapse the blocks ACB and OAC translate with velocities of magnitude V , and V2 at angles 4, and 4z to BC and CO, respectively, where 4, and d 2 are the local angles of internal friction. Similarly, the velocity jump across CA must be inclined at an angle 43 to CA. The stress states on BC, CO and CA thus correspond to three different points on the failure envelope. An upper-bound estimate of the slope

Table IV. Comparison of optimum log-spiral and optimal two-block model predictions of the stability fac- tor N = yh/cr, for a 75” slope and m,

= 15.74

S Log-spiral Two-block

7 x 10-7 0.0026 0.0033 1 x 10-5 0.0157 00226 1 x 10-4 0.0360 0.0557 4 x 0.2252 0.3304

10-1 1.048 1 1.4783 1 3.0877 4.0050

550 I. F. COLLINS ET AL.

height (or stability factor) is obtained by equating the rate of working by gravity in the two moving blocks to the rate of energy dissipation on the three velocity discontinuities. The geometry of the mechanism and the three angles 4 can be varied to give the least upper bound obtainable from a two-block mechanism. This minimization was achieved using a standard NAG library routine and a quasi-Newton procedure. The minimum was found in all cases except when the slope was close to vertical, where the two-block model tries to reduce to the single-block model, which we know gives good results. However, in all cases the best upper bound on the slope stability factor obtained from the two-block model was inferior to that obtained from the optimum log-spiral solution. A typical comparison of stability factor prediction is given in Table IV.

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