Sarma, S. K. (1987). GCotechnique 37, No. 1, 107-l I1

TECHNICAL NOTES

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A note on the stability analysis of slopes

S. K. SARMA*

KEYWORDS: analysis; earthquakes; slopes; stability.

In the stability analysis of slopes, Sarma (1973, 1979) has shown that for a given slip surface the determination of the critical acceleration factor K, involves a straightforward and easy procedure and that K, is a measure of the static factor of safety. He suggests the iterative technique of determining the factor of safety by first reducing the shear strength parameters by a known factor F and then by computing the corresponding K, value for the reduced strength. The appropriate choice of three or four values of F is generally suflicient to allow the determination of the shape of the grafih of factor of safety F versus K as shown in Fig. 1. Sarma & Bhave (1974) suggest a starting value of F, given by

F = 1 + 3.3K, (1)

The subsequent values of F can be obtained by linear interpolation.

However, the determination of the factor of safety involves overcoming some computational difficulties which arise from the method of slices.

Whether the slip surface is a circular or a non- circular arc and whether the method of solution is a simplified or a rigorous method, except for the simplified solutions of Fellenius (1936) or Skemp- ton & Hutchinson (1969), the determination of the factor of safety requires the solution of an equation of the kind

Discussion on this Technical Note closes on 1 July 1987. For further details see p. ii. * Imperial College of Science and Technology.

s=qL F + Bi

in which A, B and S depend on the method of solution. For example, for a circular arc slip surface of radius R, Bishops (1955) simplified sol- ution is

F = x{[cb + w(1 - r,) tan 41 set a

x [l + (tan z tan @)/F]-}/xw sin a (3)

This equation takes the form given in equation (2) where

A = [cb + w(1 - r,) tan 4) set a (4)

B = tan CI tan 4 (5)

s=c w sin a (6)

If a horizontal load Kw which acts at the centre of gravity of each slice is included in the problem, the resulting equation will be similar to equation (2) except that

S=zwsincr+Kw(cosor-i) (7)

where h is the height of the centre of gravity of the slice from the slip surface.

With Sarmas (1973) method rewritten to be solved for F rather than for K, or with Janbus (1957) equation, the form of the equation for F will remain the same as equation (2), but these methods now involve

A = [cb + w(1 - r,) tan 4 - DX tan 41 set a

B = tan a tan 4 (8)

(9)

107

108 TECHNICAL NOTES

KC

/(-- ---

0 1

F Fs

Fig. 1. Variation in factor of safety F with horizontal acceleration factor K

S = c (w - DX) tan CI + Kw (10)

The presence of the interslice shear force differ- ential DX does not make any difference to the form of the equation but the final solution will depend on its values. The following discussion is therefore valid for all methods of stability analysis that use the method of slices which involves an equation of similar form to equation (2).

The solution to equation (2) can be represent- ed, graphically, as the points of intersection of two curves, plotting F versus Y, where Y is given by

Y=S (11)

Y=4L F + Bi (12)

The first curve is a straight line and the second is a discontinuous curve of several segments as shown in Fig. 2. This has singularities at

F= -B, (13)

The singularity corresponding to the maximum value of F occurs in conjunction with the largest value of -tan CI tan c#/, i.e. F, is the largest value of -tan a tan 4.

The artificial cases such as when 1 AJ(F + Bi) is a negative quantity, which implies that the slipped mass will move uphill, or when S < 0 which implies a negative driving moment, may be disregarded. Therefore, the sections of the curve below the Y = 0 line in Fig. 2 are to be com- pletely discarded.

For values of F > F, the curve defined by equation (12) is entirely positive and asymptotic to Y = 0. This part of the curve is a higher order rectangular hyperbola. For values of F that are smaller than the smallest value of -B, the curve is entirely negative and is again a higher order rectangular hyperbola which is asymptotic to Y = 0. The line Y = S therefore intersects the curve for equation (12) at n different points which correspond to the n different solutions of F in equation (2), n being the number of distinctly dif- ferent values of tan a tan 4 on the slip surface.

It can be seen that solutions of F which are smaller than F, correspond to shear stresses at the base of the slip surface which are in the same direction as the slip movement in one or more segments. This condition is kinematically inad- missible because it violates the principle of limit

I\ i\

i\

Fig. 2. Graphical solution for the factor of safety

TECHNICAL NOTES 109

equilibrium and is therefore unacceptable. The only solution which is acceptable is the largest value of F which exceeds F,. The value of S changes with the applied horizontal load Kw; the critical acceleration factor K, is obtained from the value of S corresponding to a factor of safety of unity.

For realistic failure surfaces, the value of F, is usually significantly less than unity and therefore the solution of K,, i.e. the critical acceleration factor, is physically meaningful. This implies that, if a horizontal load larger than K, w is applied, a slide should occur. However, when F, > 1, the critical acceleration factor for F = 1, as obtained from Sarma (1973) or by using any other method of slices, is physically meaningless. The solution may correspond to the discarded portion of Fig. 2, i.e. Y < 0 or, if the solution corresponds to Y > 0, an acceleration factor greater than K, on these assumed slip surfaces will not produce failure. The reason for this is that for the same acceleration factor there is another solution of F which is greater than F, and therefore greater than unity. The surface is therefore safe. Slip sur- faces which have F, very close to but less than unity will yield large values of K, but will be found to produce solutions which are unaccept- able from other criteria. From the stress condi- tions, the exit angle at the toe end of a slip surface close to the free horizontal ground surface should be equal to x = -(45 - &/2) which gives a value of F, = tan 4 tan (45 - 412) which is very much smaller than unity.

With Sarmas (1979) solution, which cannot be written in the form of equation (2), the use of F = F, to reduce the shear strength properties on the slip surface may not produce a large value of K, but instead will yield results which will be unacceptable from considerations of the direc- tions of the interslice forces.

The function S varies linearly with the horizon- tal load factor K. Therefore, the curve of F versus K as shown in Fig. 1 is also a rectangular hyper- bola of higher order for values of F > F,. This hyperbola has an asymptote at F = F, and the rectangular asymptote at K = K,. The value of K, will depend on the kind of function S which in turn depends on the method of solution. In Sarmas and in Janbus method

(14)

while in Bishops method

c w sin a K = - 1 w(cos a - h/R)

(15)

The value of K, is a negative number. If it is not

negative, there is no stability problem. The higher order rectangular hyperbola can be

expressed as

M N F-F,,,=-

K - K, + (K - K,)

P + (K _ K,)3 + ... (16)

where M, N and P are defined later. It is found that only three terms on the right-hand side are usually sufficient to define the curve.

To use this solution in Sarmas (1973, 1979) methods to find the factor of safety for any value of K, including K = 0, F, and K, are first deter- mined. There are then two situations.

Case 1 In case 1, F, < 1. This is the usual situation. Step I. Assume F = F, = 1; derive K, = K, . Step 2. Compute

M = (F, - F,)(K, - K,) (17)

Assume N = 0 and P = 0 which gives

F, = F, + & 0

Step 3. Reduce the shear strength properties by the factor F, and determine K, = K, for the reduced strength.

Step 4. Compute new values of M and N

M = (F, - F,J(K, - K,)

(f, - Fn,MK, - Ko)* - KI - K,

(19)

N = (F, - FJK, - K,)*

- M(K, - Ko) (20)

Assume P = 0 and determine

M N F=F,=F,+-

K - K, + (K _ K,)* (21)

Step 5. Reduce the shear strength properties on the slip surface by the factor F = F, and determine K, = K, for the reduced strength.

Step 6. Compute new values of M, N and P

M, = (F, - F,)(K, - Ko)3

(F2 - F,XK, - Ko13 - K,--2

(22)

110 TECHNICAL NOTES

Fig. 3. Cross-section of a slope and rock toe with an assumed slip surface

M, = (F, - FJK, - Ko)3

(F, - F,,,XK, - Kd3 -

K, - K3 (23)

M= MI-M, KI - K3

(24)

N = M, - M(K, + K, - 2K,) (25)

P = (F, - F,J(K, - K,J3

- M(K, - K,) - N(K, - K,) (26)

and determine F given by

F=F,+&+ N

o (K - Kc,)

P + (K - K,J3

Case 2 In case 2, F, > 1. K, is not computed for

F = 1. In step 1, an arbitrary value of

F=F,=F,+l (28)

is assumed and the rest of the procedure is the same as in case 1.

1 .o

Y

0 L 1 1.5

F

(a)

EXAMPLE Figure 3 shows a slope with a rock toe. A slip

surface ABCDEF is also shown. The material properties are given in the table. The point B of the slip surface is chosen to give a value for F, which is very close to unity. The point B is moved slightly up or down to produce the following two cases. Sarmas (1973) method is used for the solu- tion.

Case 1 In case 1, F, = 0.998. For a factor of safety of

unity, the critical acceleration factor is 12.33. The factor of safety for zero acceleration is 1.47 and the variation in the factor of safety with other acceleration factors is shown in Fig. 4(a). A detailed examination of the solution shows that the interslice forces are unacceptable for the criti- cal acceleration case.

Case 2 In case 2, Fm = 1.002. For a factor of safety of

unity, the critical acceleration factor is - 10.08, which is meaningless. The factor of safety for zero acceleration level is 1.59 and the variation in the factors of safety for other acceleration factors are shown in Fig. 4(b). The differences in the factor of safety for the two cases are considerable even though point B is moved by an insignificant amount; this shows the approach to a singularity.

The solution to the problem was also obtained by the Sarma (1979) method which gave a critical acceleration of 1.00 with vertical slices but the resulting interslice forces were unacceptable. No acceptable solution was found with inclined slices.

In conclusion, the value of F, should always be checked before any stability analysis is performed on a slip surface and if this value is found to be greater than or equal to unity the slip surface should be rejected.

(b)

Fig. 4. Variation in factor of safety with horizontal acceleration factor for two insignificantly different positions of point B in Fig. 3

TECHNICAL NOTES 111

REFERENCES and slopes. Gtotechnique 23, No. 3,423-433. Bishop, A. W. (1955). The use of the slip circle in the Sarma, S. K. (1979). Stability analysis of embankments

stability analysis of slopes. Giotechnique 5, No. 1, 7-17.

and slopes. .I. Geotech. Engng Div. Am. Sot. Cio. Engrs 105, GT12, 1511-1524.

Fellenius, W. (1936). Calculation of the stability of earth Sarma, S. K. & Bhave, M. V. (1974). Critical acceler- dams. Trans. 2nd Congr. Large Dams, Washington DC 4,445-459.

ation versus static factor of safety in stability analysis of earth dams and embankments. Gkotech-

Janbu, N. (1957). Earth pressures and bearing capacity nique 24, No. 4,661-665. calculations by generalized procedure of slices. Proc. Skempton, A. W. & Hutchinson, J. N. (1969). Stability 4th Int. Conf: Soil Mech. Fdn Engng, London 2,207- 212.

of natural slopes and embankment foundations. Proc. 7th Int. Conf: Soil Mech. Fdn Engng, Mexico

Sarma, S. K. (1973). Stability analysis of embankments City, State of the art volume, pp. 291-340.