Important Note on Slope stability Analysis

Theoretical Background

All limit equilibrium methods of slope stability analysis have four characteristics in common

(Duncan and Wright, 1980):

1. All use the following definition of the factor of safety (F):

(1)

2. Placing a factor on shear strength is appropriate because evaluation of the shear strength

typically involves the greatest uncertainty in practical applications of slope stability

analyses. Note, however, that by definition the factor of safety is the same at all points

along the potential slip surface. This is reasonable only at failure; that is, when the factor

of safety equals unity. Because the factor of safety is taken to be the same at all points

along the potential slip surface even when the factor of safety is greater than unity, limit

equilibrium methods of analysis cannot model the mechanism of progressive failure.

3. All assume that the strength parameters are independent of stress-strain behaviour.

4. All use some or all of the equations of equilibrium to calculate the average values of

and on each slice, where is the normal stress on the base of the slice. is

required to determine the shear strength using the following equation:

(2)

in which c and are Mohr-Coulomb strength parameters. Since the forces involved in

equilibrium methods are statically indeterminate, all methods employ assumptions to

make up the balance between the number of equilibrium equations and the number of

unknowns in the problem.

The most commonly used slope stability analysis methods divide the mass above an assumed slip

surface into vertical slices. This is to accommodate conditions where the soil properties and pore

pressures vary with location throughout the slope. The forces acting on a typical slice are shown

in Figure 1.

W= weight of slice

kW= seismic force applied at center of slice

S/F= mobilized shear forces at base of slice

P'= effective normal forces on base

U= water pressure force on base

B= resultant top boundary forces

X=vertical side force

E = horizontal side force

Figure 1 Forces acting on a typical slice.

As stated previously, equilibrium methods employ assumptions to make the problem statically

determinate. The most critical of these assumptions typically deals with the side forces X and E.

Figure 2 shows the assumption made concerning side forces for several of the more common

methods.

Figure 2 Differences in assumptions regarding side forces in common methods of slope stability

analysis.

Ordinary Method of Slices

The ordinary method of slices assumes that the resultant of the side forces acting on a slice act

parallel to the base of the slice. By resolving forces normal to the base of the slice, the side

forces are eliminated and the following equation results:

(3)

Moment equilibrium around the center of the circular slip surface is the only condition of

equilibrium satisfied by this method. The value of P given in Equation 3 is a low approximation,

which leads to a conservative value of F. In cases of flat slopes with high pore pressures, the

error in the value of F may be as much as 50%. In total stress analyses the error is typically not

more than 10%.

Bishop's Simplified Method

In Bishop's Simplified Method the effect of the inter-slice force is eliminated by assuming that

the vertical component of the inter-slice forces is zero. The forces on a typical slice are shown in

Figure 3 and consist of:

a. the slice weight, W

b. the pseudo-static seismic force, kW, in which k is the seismic coefficient

c. the pore pressure force, U ( = u . l )

d. the effective normal force on base, P'

e. the mobilized shear force,

f. the resultant of boundary forces perpendicular and parallel to the top of the slice, N and

M

g. the horizontal slide forces, En and En+1

Figure 3 Force polygon for Bishop's Simplified Method.

Solving force equilibrium in the vertical direction, thereby eliminating the side forces, yields;

(4)

or rearranging and solving for P':

(5)

Once the normal force at the base of each slice is found, overall moment equilibrium yields and

implicit expression for the factor of safety:

(6)

where yk, aN and aM are appropriate moment arms and is given by

(7)

Whitman and Bailey (1967) point out a rare numerical difficulty with this method. When is

negative the possibility exists that the denominator in Equation 5 could be negative, or worse,

zero. When this occurs, a warning is printed and the resisting moment is set to zero.

Spencer's Method

Spencer's Method assumes that the inter-slice forces are parallel (Spencer, 1967). A typical slice

and the corresponding force polygon are shown in Figure 4. The forces on the slice are:

a. the slice weight, W

b. the pseudo-static seismic force, kW, in which k is the seismic coefficient

c. the pore pressure force, U ( = u . l )

d. the effective normal force on base, P'

e. the mobilized shear force,

f. the resultant of boundary forces perpendicular and parallel to the top of the slice, N and

M respectively

g. the resultant of the parallel side forces, Q

Figure 4 Force polygon for Spencer's Method.

The pseudo-static seismic force and the resultants of boundary pressures will not be included in

the following discussion; these forces are of known magnitude and direction and therefore do not

contribute to the understanding of the theory but do tend to lengthen the equations.

Summation of forces, normal and tangent to the base of each slice, provides two equations of

force equilibrium and two unknowns, P' and Q:

(8)

(9)

Solving Equation 8 for P' and substituting the expression into Equation 9 and then solving

Equation 9 for Q yields:

(10)

If the external forces on the slope are in equilibrium, the vectorial sum of the inter-slice forces

must be zero to assure overall force equilibrium. Since the inter-slice forces are all parallel, this

requirement reduces to:

(11)

Furthermore, the normal force and the weight of each slice are assumed to be coincidental at a

point on the slip surface with the same X-coordinate as the slice's center of gravity. For each

slice to be in moment equilibrium the resultant Q of the inter-slice forces must be concurrent

with the remaining forces acting on the slice. In other words, Q must act through the point on the

base of each slice where the normal and weight forces act, with sufficient modifications made to

account for top-of-slice boundary forces or pseudo-static seismic forces.

If the sum of the moments of the external forces about an arbitrary point, say the origin, is zero,

then the sum of the moments of the inter-slice forces about this point must also be zero:

(12)

where x and y are the coordinates of the point on the base of the slice where the forces are acting.

Satisfaction of Equations 11 and 12 assures that equilibrium is fully satisfied for each slice. Once

a solution is found to these two equations, the line of thrust can be calculated for each slice.

A solution to Equations 11 and 12 is obtained by simultaneously varying F and until the two

equations are satisfied. For the initial assumed values of F and , the equations may be in error

by the amounts R1 and R2 respectively, that is:

(13)

(14)

where Q is based on the assumed values of F and , and R1 and R2 are the horizontal force and

moment imbalances respectively. Note that Equation 11 has been modified slightly to give the

horizontal component of the inter-slice forces.

By the Newton-Raphson method for convergence, F and are varied until R1 and R2 are within

acceptable limits. This convergence process is discussed in detail in Wright's dissertation (1969).

Morgenstern and Price's Method

Morgenstern and Price's Method as originally formulated took a somewhat different approach to

the solution of complete slice equilibrium (Morgenstern and Price, 1965). While Spencer

considered overall moment equilibrium, Morgenstern and Price have considered only the

moment equations of individual slices. Each method satisfies all conditions of equilibrium but

Spencer's Method requires about half the computer time. For this reason TSLOPE's version of

the Morgenstern and Price Method is actually just an extension of Spencer's Method to allow

side forces that are not necessarily parallel (Spencer, 1973). Morgenstern and Price assume that

the ratio of the side forces is given by:

(15)

where f(x) represents a user-defined variational relationship between X and E. The parameter is

an unknown scaling factor determined by the program to yield complete equilibrium.

Spencer's Method assumes that the side forces are inclined at angle with respect to horizontal:

(16)

The angle is determined in the calculation process. To extend Spencer's Method to allow non-

parallel side forces we let:

(17)

where f(x) is as defined previously. Note that f(x) = 1 is equivalent to Spencer's method. The

angle of each side force becomes:

(18)

The forces on a typical slice and the force polygon are shown in Figure 5. The side forces are

calculated using:

(19)

where Q is calculated using Equation 10 in which is replaced by . The horizontal force and

moment imbalance are calculated and F and are varied until the imbalances are within

acceptable limits.

.

Figure 5 Force polygon for Morgenstern and Price Method.

Use of Limit Equilibrium Slope Stability Analysis

It is desirable that the user of any computer program for limit equilibrium slope stability analyses

should be aware of the mechanics of the method that is being used but it is even more important

that he or she have a thorough understanding of the principles involved in choosing the right

geometry to be analyzed and in specifying the soil properties and pore pressures used in the

analyses. It is beyond the scope of this manual to discuss these subjects at length but some

guidance and some of the more pertinent references are offered in the following paragraphs.

Choice of Geometry

Slopes often appear to fail on circular slip surfaces and it is often reasonable to analyze slope

stability using circular slip surfaces. However, there are also many instances when this is not the

case. Non-circular slip surfaces may be more critical than circular slip surfaces when:

1. There is a weak layer present in the foundation. The weak layer could be a soft clay

(Leonards, 1982; Fredlund et al., 1981) or a liquefiable sand (Seed and Wilson, 1967).

2. There is a heavily overconsolidated, stiff fissured-clay or clay-shale foundation for an

embankment. These materials tend to have highly anisotropic shear strength in which the

strength may be as little as twenty percent along fissures as compared to other directions

(Wright and Duncan, 1972; Duncan and Dunlop, 1969).

3. A dam's core is sloping and is significantly weaker than its shell. (For illustration, see

Example 1 for TSLOPE.)

There are some slopes where three dimensional effects make an important contribution to

stability. For these cases TSLOPE3 should be used.

Specification of Shear Strengths and Pore Pressures

Slope stability analyses may be performed using either total stresses or effective stresses. The

use of total stress as opposed to effective stress analyses and the various ways in which design

shear strengths can be selected can produce a wide range of safety factors. In general, these

questions are more important than the choice of the method used for analyzing stability

(Johnson, 1974). Bishop and Bjerrum (1960) set forth the following basic guidelines on the

specification of shear strength for use in limit equilibrium slope stability analyses:

1. "Effective stress analysis is a generally valid method for analyzing any stability problem

and is particularly valuable in revealing trends in stability which would not be apparent

from total stress methods. Its application in practice is limited to cases where the pore

pressures are measured or can be estimated with reasonable accuracy, such as long-term

stability where the pore pressure is controlled either by the static water table or by a

steady-state flow pattern."

2. "Where a saturated clay is loaded or unloaded at such a rate that there is no significant

dissipation of the excess pore pressures set up, the stability can be determined by the

u =  0 analysis, using the undrained strength obtained in the laboratory or from in-situ

tests. This is essentially an end of construction method, and in the majority of foundation

problems, where the factor of safety increases with time, it provides a sufficient check on

stability. For cuts, on the other hand, where the factor of safety generally decreases with

time, the long term stability must be calculated by the effective stress method."

3. "For saturated soils the values of c' and ' are obtained from drained [triaxial] tests or

consolidated undrained tests with pore pressure measurements, carried out on undisturbed

samples. The range in stresses at failure should be chosen to correspond to those in the

field. Values measured in the laboratory appear to be in satisfactory agreement with field

records with two exceptions. In stiff fissured clays the field value of c' is lower than the

value given by standard laboratory tests; in some very sensitive clays the field value of '

is lower than the laboratory value."

These 1960 guidelines are still generally valid but increases in our understanding, particularly of

undrained strengths, since that time now allow us to do more accurate analyses albeit at the

expense of some complications!

Problems in slope stability can be broadly grouped in two classes: short-term problems and long-

term problems. When a saturated or partially saturated soil with a low permeability undergoes a

change in stress there will generally be a corresponding change in pore pressure. The stage at

which the excess pore pressures (positive or negative) resulting from the change in stress are

fully developed is referred to as the short-term condition. With the passage of time these out-of-

balance pore pressures are redistributed until eventually they are everywhere in equilibrium with

the steady state pore pressures appropriate for the new stress conditions. This final stage is

referred to as the long-term condition and the continuing stability of the slope under gravity or

applied loads is a problem with drained loading conditions.

Long-term or drained stability problems are usually simpler than short-term or undrained

stability problems since they always involve drained or effective stress strength parameters and

for a given soil these do not vary very much with the type of test that is used to determine them.

However, it should be noted that even the effective stress Mohr-Coulomb envelope is curved,

rather than straight, for most soils and that the values of ' are thus lower at higher confining

pressures. The curvature of the Mohr-Coulomb envelope should normally be taken into account

for slopes higher than about 100 feet since use of values of ' determined at the usual confining

pressures of about 1 t.s.f. will then lead to errors on the unsafe side.

In general, short-term stability problems involve undrained loading and they can be addressed

using total stresses and undrained strengths or effective stresses, drained strengths and pore water

pressures. It is commonly believed that both approaches should give the same answer but this is

not necessarily so. As noted by Bishop and Bjerrum (1960):

"for factors of safety other than 1 the two methods will not in general give numerically equal

values of F. In the effective stress method the pore pressure is predicted for the stresses in the

soil, under the actual loading conditions, and the value of F expresses the proportion of c' and

tan ' then necessary for equilibrium. The total stress method on the other hand implicitly uses a

value of pore pressure related to the pore pressure at failure in the undrained test."

Since the limit equilibrium method is most applicable at failure, in effective stress analyses one

should in fact use the pore pressures "at failure," rather than the "actual" pore pressures for the

short-term loading condition, and then both approaches will give similar answers. In this

connection one should note that the pore pressures specified in effective stress analyses affect

only the resisting forces that are computed and not the driving forces. This occurs because the

total normal force at the base of each slice is an unknown and an increase in the specified pore

pressure decreases the effective normal force but has no effect on the total normal force.

Similarly, changes in the pore pressures created by shearing under undrained loading conditions

are not included as driving forces in total stress analyses.

The traditional argument for using effective stress analyses for short-term, undrained problems is

that it is the effective stresses which really count in determining deformations and therefore

effective stress analyses provide greater insight into the problem at hand. However, effective

stress analyses require determination of either the "actual" pore pressures or the pore pressures

"at failure" and this is no easy task. Indeed, it is about as easy as it is to determine the undrained

strength that should be used in a total stress analysis since the reason that undrained strengths

vary with sample orientation, the type of test and the details of the loading conditions is largely

that the excess pore pressures are sensitive to these factors. In other words, it is about equally as

difficult to predict excess pore pressures for use in effective stress analyses as it is to determine

the appropriate undrained strengths for use in total stress analyses.

In practice various methods may be used to either predict excess pore pressures or to determine

undrained shear strengths for use in short-term stability analyses. Excess pore pressures in fully

saturated soils are most commonly predicted using the pore pressure coefficients A and B

(Skempton, 1954). Prediction of excess pore pressures in partly saturated soils is extremely

difficult. Fredlund and Morgenstern (1977) and Fredlund (1979) discuss various approaches.

Special methods have been developed for particular problems such as the end-of-construction

condition for embankments, as discussed subsequently.

The preferred method for determining undrained strengths has changed over the years. In 1960

Bishop and Bjerrum recommended the use of UU triaxial tests and cautioned against the use of

CU triaxial tests. For a time, use of the vane shear test was popular but it is now recognized that

correction factors should normally be applied to the measured strengths (Bjerrum, 1972; Duncan

and Buchignani, 1973; Larsson, 1980). Deficiencies in the UU test were also subsequently

recognized (e.g. Bjerrum, 1973) and it is now generally agreed that use of UU triaxial tests

should normally be restricted to those cases where local experience has shown that use of UU

strengths leads to safe and economical construction. More generally, Su should be obtained from

tests on reconsolidated samples using the test equipment and rate of loading which best

reproduces the in-situ stress and deformation conditions, using anisotropic consolidation if

necessary to represent the initial shear stresses on potential failure planes in the field and using

procedures such as the SHANSEP method (Ladd and Foott, 1974) to minimize the effects of

sample disturbance.

Some of the issues involved in selecting appropriate pore pressures and/or strengths for slope

stability analyses are discussed further in the following sections in terms of the common classes

of slope stability problems.

Long-Term Stability Problems

1. The simplest slope stability problem is a dry embankment as shown in Figure 6(a). The

pore pressures are equal to zero and the effective stress strength parameters, c' and ',

should be used. Consolidated-drained (CD) tests should be performed to determine c' and

'.

2. A partially submerged slope is shown in Figure 6(b). In this case, the water table is static

and the pore pressures are easily determined by taking the depth below phreatic surface

and multiplying by the unit weight of water. Effective stress strength parameters should

be used as determined by CD or consolidated-undrained (CU) tests with pore pressure

measurements. This problem may be solved two ways:

a. Use total unit weights throughout, apply the boundary water pressure and specify

the pore pressures in the slope.

b. Use buoyant unit weight below the water table and neglect the boundary water

pressure and pore pressures. This type of analysis is demonstrated in Example 3i

for TSLOPE.

Note that if a pseudo-static seismic loading is subsequently applied, method (a) must be

used because the correct inertia forces are obtained only by using total unit weights.

3. The classic long-term stability problem is the steady state seepage condition shown in

Figure 6(c). This represents, for instance, the most critical condition for the downstream

slope of a dam with a full pool and with steady seepage through the dam. Again, use c'

and ' as determined by CD or CU tests. Pore pressures should be determined by

drawing a flow net or by field measurement. Apply boundary water pressures on

upstream and downstream slopes where applicable.

Figure 6 Long-term stability problems.

Short-Term Stability Problems

The most common short-term stability problem is the end-of-construction condition for materials

which dissipate excess pore pressures slowly in comparison with the rate of construction. In

more permeable soils, such as sands and gravels, the period of pore pressure redistribution is

very short and, except under conditions of transient loading, stability problems typically will fall

into the long-term category. Clays, on the other hand, dissipate excess pore pressures so slowly

that the period of pore pressure adjustment may last for months or years after the completion of

construction.

A classic problem in short-term stability is the case of a fill constructed on a soft clay foundation

as shown in Figure 7(a). This problem is normally analyzed using total stresses and undrained

shear strengths, the procedure termed the u =  0 analysis by Bishop and Bjerrum. The

undrained strength  Su  used in such problems is commonly expressed in terms of the in situ

vertical effective stress   'v  and the overconsolidation ratio (OCR) (e.g. Ladd and Foott, 1974),

but in more sophisticated analyses the position of the element on the potential sliding surface

should also be taken into account (e.g. Ladd et al., 1977). Cuts in saturated clay, Figure 7 (b), can

also be analyzed for short-term stability using the u =  0 method; however, a long-term

effective stress analysis should also be performed as this is usually the more critical case.

The end-of-construction condition for a constructed embankment, Figure 7(c), is also a problem

in short-term stability. This problem may be analyzed using total or effective stress methods. The

total stress method normally involves determination of the undrained strengths using UU triaxial

tests and the effective stress method commonly relies on the use of the procedure for computing

pore pressures developed by Hilf (1961). The total stress method is shown in Example 1 for

TSLOPE. Johnson (1974) provided a detailed discussion of the relative merits of various

approaches to end-of-construction stability problems.

Figure 7 End-of-construction stability problems.

The rapid drawdown condition (Figure 8(a)) is another short-term stability problem. While

effective stress analyses could be used for this problem it is more common to use total stress

analyses, as illustrated in Example 3i and Example 3ii for TSLOPE. In this example the

undrained strength, now termed ff has been determined as a function of the normal stress at

consolidation fc and the anisotropic consolidation ratio, Kc, using ACU triaxial tests as

recommended by Lowe (1967) . While other procedures could also be used for the rapid

drawdown problem the Lowe procedure has been widely adopted as a standard procedure. As

described by Lowe, the procedure involves conduct of two slope stability analyses--an initial

effective stress analysis is conducted for the high water steady state condition in order to obtain

the normal and shear stresses on the potential sliding surface for the consolidated condition, and

a second total stress analysis is conducted for the low water condition using the appropriate

undrained strengths. Lowe used a graphical procedure to do the slope stability analyses and it

sometimes seems to be assumed that one has to use this graphical procedure in order to obtain

the initial normal and shear stresses. This is of course not correct and the normal and shear

stresses at the base of each slice can be obtained more easily using TSLOPE. As explained by

Johnson (1974), the total stress approach to rapid drawdown stability problems can lead to a

substantial component of the shear strength at low confining pressures resulting from negative

pore pressures. The practice of the Corps of Engineers, and others, is therefore to use a combined

envelope, normally referred to as an S-R envelope, to avoid reliance on the shear strengths

associated with negative pore pressures.

The final case of short-term stability analyses to be discussed are earthquake and post-

earthquake stability analyses. In past practice the stability of embankments for earthquake

loadings has often been checked by applying a "pseudo-static" horizontal force, specified in

terms of a seismic coefficient which is used as a multiplier on the weight of the potential sliding

mass. TSLOPE allows the user to specify seismic coefficients but it should be noted that pseudo-

static seismic analyses are, in general, so crude as to be worthless. A better procedure that is still

less complicated than a full dynamic analysis, is that suggested by Newmark (Newmark, 1965;

Pyke, 1982) in which it is necessary to determine the seismic coefficient that reduces the factor

of safety to unity. In order to facilitate use of the Newmark method TSLOPE provides options

for automatic calculation of this critical seismic coefficient as demonstrated in Example 2 for

TSLOPE. In such analyses total unit weights must be used in conjunction with either undrained

strengths or drained strengths and steady state plus excess pore pressures at failure. Ideally the

undrained strengths or the excess pore pressures at failure will be determined as a function of the

initial normal and shear stresses along potential sliding surfaces.

Figure 8 Other short-term stability problems.

Post-earthquake stability analyses are a special case of short-term stability in which no seismic

coefficient is applied but the undrained strengths may be reduced in order to account for the

effects of excess pore pressures developed during earthquake shaking. This kind of problem is

illustrated in TSLOPE Example 4. Note, however, that for dilatant materials, the undrained

strengths are largely independent of the pore pressures prior to shearing (e.g. Castro and

Christian, 1976) so that the undrained shear strengths after cyclic loading may be essentially the

same as those prior to shaking. In any case, one should again use total unit weights in

conjunction with undrained strengths or drained strengths and steady state plus excess pore

pressures at failure. Again, the undrained strengths or the excess pore pressures at failure should

ideally be determined as a function of the initial normal and shear stresses along potential sliding

surfaces as well as a cyclic loading which simulates the earthquake loading.

For both earthquake and post-earthquake stability analyses the initial normal and shear stresses

along potential sliding surfaces can be obtained by conducting an effective stress analysis of the

pre-earthquake condition using TSLOPE, as described previously for rapid drawdown analyses.

However, for both earthquake and rapid drawdown analyses the initial normal and shear stresses

can also be obtained by use of the finite element method and a comparison of the stresses

obtained from finite element and slope stability analyses for use in a post-earthquake stability

analysis is shown in Figure 9. Note that the ratio of the shear stress to the normal stress obtained

from the slope stability analysis is constant along the sliding surface as a result of the basic

assumption in this kind of analysis that the factor of safety is constant along the sliding surface.

Note also that this ratio is constant only for materials with no cohesion. Thus the finite element

method actually gives more reasonable results but the difference in initial stresses led to a

difference in the computed factors of safety of less than 0. 1 for the example shown, and it is

likely that the difference in the factors of safety in other cases will also be small.

A final point to note with regard to post-earthquake stability analyses is that while the excess

pore pressures induced by cyclic loading may be determined as part of the overall analysis

procedure, they cannot or should not be included in the slope stability analysis. For the same

reason as discussed previously, excess pore pressures which are specified in the analysis affect

only the computation of resisting forces and have no effect on driving forces. Further, if effective

stress analyses are conducted, it is generally more reasonable to estimate the excess pore

pressures at large strains and to use these in the analysis rather than the excess pore pressures

prior to shearing since the computation of the factor of safety is more valid as failure is

approached and, from a practical point of view, one wants to know whether large strains and

failure can develop, rather than the value of an arbitrarily defined factor of safety for a non-

failure condition.

.

Figure 9 Comparison of initial stresses obtained from finite element and slope stability analysis.

Summary

In addition to drawing on local experience, the engineer conducting slope stability analyses

should draw on the wider experience published in the literature. Particularly good references on

various classes of slope stability problems include:

Embankments on Soft Ground: Bjerrum (1972), Ladd and Foott (1974)

Constructed Embankments: Lowe (1967)

Natural Slopes: Johnson (1974), Skempton & Hutchinson (1969)

Cut Slopes: Vaughan & Walbancke (1973), Chandler and Skempton (1974)

Remember also that the choice of soil properties and pore pressures should always be given the

utmost consideration in stability analyses as they will affect results more than any other factor.

The acceptable factor of safety for a slope should take into account the uncertainty in shear

strengths and pore pressures used in the analysis.

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Colorado, June 1960.

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Duncan, J.M. and Wright, S.G., "The Accuracy of Equilibrium Methods of Slope Stability

Analysis." Engineering Geology, Vol. 16, Elsevier Scientific Publishing Company, Amsterdam,

1980, pp. 5-17.

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Stability Methods." Proc. 10th International Conference on Soil Mechanics and Foundations

Engineering, Stockholm, Vol. 3, June 1981, pp. 409-416.

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ASCE, Vol. 103, No. GT5, May 1977.

Johnson, S.J., "Analysis and Design Relating to Embankments." Proc. ASCE Specialty

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