UNIFIED FORMULATION FOR ANALYSISOF SLOPES WITH GENERAL

SLIP SURFACEs

Discussion by Sunil Sharma,4 Member, ASCE

The limit equilibrium procedures presented within the uni­fied formulation offer an attractive approach for accessing thestability of slopes and are similar to the methodology previ­ously suggested by Fredlund and Krahn (1977). Typically,force equilibrium is satisfied for all slices and initially onlyoverall moment equilibrium is satisfied for the slide mass.Once a common factor of safety (i.e., F =Fff =Fmm) has beendetermined for this force and overall moment equilibrium con­ditions, (18) is used to determine the location of the thrustline, or the inclination of the interslice forces for the rigorousJanbu method (cases 7 and 8 in Table 3). This discussion isdirected towards three items: (1) Eq. (18); (2) reasonablenessof solution; and (3) implementation of Janbu's method.

In the opinion of the discusser, (18) (and its subsequentderivatives) is an important link between the force and mo­ment equilibrium equation and must be implemented correctly.In view of this, the following comments concerning (18) areoffered.

First, I believe that (18) is incorrect as it presumes a thrustline that is parallel to the shear surface. The slope of the thrustline will typically vary as a function of x, and within the au­thors' formulation, it could be rewritten as

stated by Leshchinsky (1990) and Sharma and Moudud(1992), the engineer must consider (1) the location of thethrust line; (2) the distribution of interslice forces; (3) the di­rection of the interslice forces; and (4) the distribution of nor­mal and shear stresses along the failure surface before accept­ing the calculated safety factor value as an accurate assessmentof the stability of the slope. A discussion about the validity ofthe safety factors reported in Tables 5-9 would have provideda most useful service to the readers who are struggling withthe many commercial slope-stability software packages thatoffer some of the advanced analytical procedures mentionedin the paper.

The last comment concerns the attempt to generalize Jan­bu's method with an arbitrary function for a "moment-arm"[Eq. (20b)] , conceptually similar to the assumed intersliceforce angle distribution employed by the Morgenstern-Pricemethod. This approach is tempting and does fit in with theproposed unified formulation but is likely to present seriousdifficulties with convergence (Li and White 1987; Wright1970). There are usually many angle functions that will suc­cessfully provide a solution. However, the authors will findthat the selection of a suitably shaped moment-arm functionis very difficult and the solution is considerably more sensitivethan the Morgenstern-Price analysis. Often, this function mustbe assembled interactively for the successful calculation of asensible value of the safety factor. Once a solution is gener­ated, the four conditions for reasonableness must again be sat­isfied before accepting the value of the safety factor. The re­ported results indicated very small moment-arms, and if thesubsequent distribution of the interslice force angles had beenconsidered, the authors would have discovered a very unrea­sonable distribution.

dM(x) dQT(x) =E(x)tan ex, - --;;;- + dx hQ (22) APPENDIX. REFERENCES

where the variable ex, represents the slope of the thrust-lineand will also be a function of (x). This equation only has adirect impact on the solutions provided for cases 7 and 8 forthe Janbu formulation. In programming the Janbu method, theauthors would have used the discrete form of the equation,and it is possible that such an error was not accidentally in­corporated into SLOPAS. However, if such an error was made,the paper should be corrected accordingly.

Second, the continuous form of (18), or (22), is usuallyderived by taking moments about the midpoint of the base ofa typical slice and then assuming that the relative magnitudeof the higher-order terms (e.g., AE;!i.Xj) are negligible in com­parison with the first-order terms such as E;!i.Xj. This simpli­fication is acceptable for internal slices, but is not appropriatefor the first and last slices due to their triangular shape (Chenand Morgenstern 1983). To avoid such problems, it is rec­ommended that all terms be used to satisfy moment equilib­rium, especially for the rigorous Janbu method. A completeformulation for the different methods is presented by Li andWhite (1987).

Finally, if one examines the unified formUlation, it shouldbe apparent that the term E(x) is directly related to the safetyfactor, as determined using (9a). As E(x) is also featured prom­inently in (18), this equation cannot be considered to be "in­dependent of the safety factor," as stated by the authors.

The authors also provided a comparison of the differentmethods of analysis using the five example slopes. However,in the opinion of discusser, the authors did not adequatelydiscuss the reasonableness of the reported safety factors. As

·July, 1994, Vol. 120, No.7, by R. D. Espinoza, P. L. Bourdeau, andB. Muhunthan (Paper 3982).

4 Assoc. Prof., Univ. of Idaho, Dept. of Civ. Engrg., Moscow, 1083844-1022.

1601 JOURNAL OF GEOTECHNICAL ENGINEERING 1 FEBRUARY 1996

Sharma, S., and Moudud, A. (1992). "Interactive slope analysis usingSpencer's method." Proc.• ASCE Spec. Con! on Stability and Perf. ofSlopes and Embankments, Vol. I, ASCE, New York, N.Y., 506-520.

Wright, S. G. (1969). "A study of stability and undrained shear strengthof clay shales," PhD dissertation, University of California, Berkeley,Calif.

Discussion byStephen G. Wright,s Member, ASCE,

and James Michael Duncan,6 Fellow, ASCE

The authors have presented an interesting generalization oflimit equilibrium analyses of slope stability, and have com­pared factors of safety for five different examples calculatedusing 12 different assumptions regarding side forces, each ofwhich satisfies all conditions of equilibrium. Their work servesto create an important perspective for this type of analysis,which is widely used in geotechnical engineering practice.

For four of the five examples examined by the authors, thevariations in factors of safety calculated using 12 different sideforce assumptions were small. These variations are shown inTable 10.

The variations for cases I, 3,4, and 5 are no more than 7%from the average values. This is sufficiently accurate for prac­tical purposes. Uncertainties in calculated values of factor ofsafety due to imperfect knowledge of shear strengths, pore­water pressures, and geometry are nearly always larger. Thus,

, Ashley H. Priddy Centennial Prof. of Civ. Engrg., Univ. of Texas,Austin, TX.

·Univ. Distinguished Prof., Virginia Polytechnic Inst. and State Univ.,Blacksburg, VA.

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TABLE 11. Values of F Calculated for Example 2 and PercentDifferences from Average Value of F

TABLE 10. Variations In Factors of Safety for Four of FiveExamples

0.50

!1.00~

~

1.50Case 6

Case 3 (Spence(s Method)

l!JF =1.75 -1.46=0.29 (20%)

l!JF =1.34 -1.27 = 0.07 (6 %)

2.00

I-"-~·~Last point on ~ •• •slip surface moves • 50 m

This point and others to 'scaieOfsketch Ithe right do not move

-3%-5%-7%-4%

Low"(3)

Differencefrom average

(3)

+2%+6%+5%+6%

High"(2)

Value of F(2)

1345

Case(1 )

Example(1 )

'Difference from average value.

'Case 1 does not satisfy force and moment equilibrium in the sameanalysis.

if the uncertainties that arise from assumptions made in for­mulating the method of analysis are as small as those shown,they are acceptable for almost all practical purposes. This isan important finding.

The magnitudes of the variations for the authors' cases 1,3, 4, and 5 are about the same as the discussers have foundfrom analyses of a wide variety of different conditions ana­lyzed by many different methods, all of which satisfy all con­ditions of equilibrium. In addition to limit equilibrium methodsof slices, the writers have used the finite element method andthe log spiral method for comparison, with the same result­as long as a method satisfies all conditions of equilibrium, thefactor of safety for the critical slip surface differed by no morethan ±6% from the most accurate value that could be calcu­lated using any method.

Eleven of the 12 values of F calculated by the authors forexample 2 also fall within the same small deviation from theaverage that they found for cases 1, 3, 4, and 5, as shown inTable 11.

Case 6 is clearly different from the others. The calculatedfactor of safety differs by 27% from the average. Thus, theauthors' conclusion that "the choice of internal force hypoth­esis may have a significant influence on the computed safetyfactor" is based on one result out of 60 (Case 6 for example2), which is clearly different from the other 59 cases. The other59 of the 60 cases examined are in agreement with the pre­vious findings of the discussers and others who have studiedthe problem. It is thus of considerable interest to determinethe reason for this exceptional result, so that it can be viewedin perspective.

The discussers have performed additional analyses that in­dicate that there are two reasons for the unusual result for case6 of example 2.

The first reason for the unusual result is the fact that theauthors did not search for the critical slip surface for the casesthey compared. Their conclusions are based on values of

FIG. 8. Forces on Slice at Toe of Slope for Example 2

~W=47tons

R",=181 tons

Case 3 (Spence(s Melhod) forcepolygon for slice at toe of slope

Rss=2ff1~W=47tons

~Case 6 force polygon for slice at toe of slope

Resuttant 01 shearand normallorceon slip surface = Rss

Slice at toe of slope

FIG. 7. Variation of Factor of Safety with Distance from Toe ofSlope to Last Point on Slip Surface for Example 2

1':-5------~10':-------~5':----------!0 0.00

Distance from toe of slope to last point on slip surface - meters

safety factor calculated for arbitrary, noncritical slip surfaces.While this approach is easier than locating the critical slipsurface and calculating the minimum factor of safety for eachcase, it can lead to erroneous conclusions regarding accuracy(Duncan 1992).

This is illustrated in Fig. 7. This figure shows the results ofseveral analyses for the example 2 slope. In these analyses theexit point at the lower end of the slip surface was moved tovarious positions to the left of the toe of the slope. With theexit point at the toe of the slope, as for the analyses performedby the authors, the difference between the factor of safety cal­culated for case 6 and case 3 (Spencer's method) was 0.29, or20% of the Spencer's method value. With the exit point 10 mto the left of the toe of the slope, the difference between thefactor of safety calculated for the case 6 assumption and Spen­cer's method was 0.07, or 6% of the Spencer's method value,a value that is in agreement with previous experience.

The second reason that the factor of safety for case 6 is sohigh is a result of the interaction between the side force onthe last slice and the force on the steeply inclined slip surfaceat the base of this slice. The effect of this steep slip surfacesegment can be seen in Fig. 8. Because the base of the lastslice at the toe rises at an angle of about 52 degrees, the re­sultant of the side forces (Rs! in Fig. 8) is close to parallel tothe resultant of the shear and normal force on the slip surface(Rss in Fig. 8). The force polygon for the slice consists of thesetwo forces and the weight force. When these two forces areclose to parallel, they must be very large to balance the weightforce. When the force Rss is large, the last slice at the toe of

-4%5%

-5%4%

27%-5%-5%-4%

2%-4%-4%-6%

I'2 1.3263 1.4604 1.3155 1.4386 1.7567 1.3118 1.3119 1.326

10 1.41411 1.33712 1.32613 1.309

[Average] 1.386

JOURNAL OF GEOTECHNICAL ENGINEERING / FEBRUARY 1996/161

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represent positive real numbers smaller than 0.5, meaning thatthe forces P and W must lie on the base of the slice. Dividingeach term by Ax and rearranging leads to

.!. (T + T. ) + (Ei+lhl+ l - E;hi) _ (E + E) tan a2 I 1+1 Ax 1+1 I 2

(24)

(25)

P AQ- 131 -- - 132W - - h = 0

cos a Ax Q

When the width of slice dx becomes very small, Ti = Ti + 1 =T(x) , Ei+1 = Ei = E(x) , and the second term in (6) becomes(dMldx) where M(x) is the internal moment distribution de­fined as the product E(x)h (x). It is also noted that h (x) = y ­y" where y is the vertical coordinate of the failure surface andy, represents the location of the line of thrust. The contribu­tions of forces P and W to the sum of moments can be ne­glected in the limit (Lh ~ dx), thus yielding the moment equi­librium equation given by (18). Replacing hex) = y - y, in(18) an alternative expression may be obtained in terms of theslope failure surface (tan a, =dy,ldx) as

dE dQT(x) =E tan a, - hex) dx + dx

which is identical to the equation presented by Janbu (1954).Hence, (18) does not presume that the line of thrust is parallelto the slip failure surface, as suggested by Sharma. Eitherequation, (18) or (25), may be used indistinctively in the nu­merical solution of the system of equations. The use of (18)was more convenient for the numerical implementation. Fur­thermore, comparing (25) with (22), suggested by the dis­cusser, it is clear that the second term of the right-hand sideof (22) is incorrect.

Another concern raised by the discusser refers to the effectof the second-order terms neglected in the derivation of (18).As shown in the preceding, in the derivation of (18) or (25),the only second-order terms are the contribution of the forcesP and W, and they can be sensibly neglected from the momentequilibrium equation. This assumption is equivalent to assum­ing that the weight of the slice and the normal force appliedat the base of the slice pass through point I (Fig. 2). Althoughsuch assumptions are not true for the extreme slices where atriangular shape is usually found (Chen and Morgenstern1983), they do not drastically affect the computed factor ofsafety and the corresponding internal forces. Nonetheless, con­sidering the locations of the forces P and W at the extremeslices gives a more reasonable distribution of the location ofthe line of thrust at the extreme slices.

Sharma further suggests that since E(x) is related to thefactor of safety, it makes (18) dependent on the factor ofsafety. As shown here, (18), which has been derived based onmoment equilibrium of a single slice, provides a natural re­lationship between the interslice forces T and E. Thus, from atheoretical standpoint, (18) is independent of any assumptionson the factor of safety. From a practical standpoint, however,solutions based on limit equilibrium hypotheses imply a con­stant factor of safety. Thus, when (18) is used in the solutionalgorithm it is indeed related to the factor of safety. Otherpopular relationships between the interslice forces are onlyvalid within the limit equilibrium concept, and have been pro­posed as means to render the system of equilibrium equationsdetermined.

The writers disagree with the discusser's suggestion that thedistribution of internal forces obtained with a small momentarm are necessarily ,unreasonable. The numerical scheme hasbeen recently implemented and tested with several exampleswith no serious difficulties, including the general moment-armfunction, [(20b)]. Analysis of the internal force distribution forseveral cases that yielded a small moment arm did not indicate

where h = height of the line of thrust with respect to the slipsurface, I31A1 = distance between point I and the line of forceP; and 132Ax = distance between point I and the line of actionof the vertical resultant force Ws' The parameters 13, and 132

the slope is providing a large part of the resistance to slidingfor the entire slope, and the factor of safety is large. As Rss

and Rst become closer and closer to parallel, the factor of safetyapproaches infinity. This can lead to numerical problems thatcan be solved by making the slip surface flatter at the toe ofthe slope.

Two lessons can be learned from these examples. First, dif­ferent methods of slope-stability analysis and different as­sumptions regarding internal forces should be compared bycalculating the minimum factor of safety for each method orassumption, and comparing these minimum values. Comparingvalues of safety factor for arbitrarily selected slip surfaces canproduce misleading results, because the arbitrary slip surfacemay be further from critical for one method than for another.This is illustrated by the authors' case 6 analysis of example2. Second, the calculated factor of safety increases very rapidlyas the slip surface segment at the exit point becomes steep,and the resultant of forces on the side of the slice comes closeto parallel to the resultant of forces on the slip surface. Steepinclinations of the slip surface at the toe are far from critical,and they thus amplify the apparent difference between meth­ods when comparisons are made on the basis of arbitrarilyselected slip surfaces.

Although large differences between factors of safety cal­culated by different methods may be calculated for differentarbitrarily selected slip surfaces, these are not significant. Theydo not indicate a fundamental effect of using different as­sumptions because the difference in the calculated values offactor of safety is a function of the slip surface as well as themethods compared. Only if minimum factors of safety arecompared does the comparison indicate meaningful differencesbetween methods that are independent of the choice of slipsurface.

Closure by R. D. Espinoza,'P. L. Bourdeau,S Member, ASCE, and

B. Muhunthan,9 Associate Member, ASCE

The writers wish to thank the discussers for their interest inour paper and for their invaluable discussion.

As suggested by Sharma, the moment equilibrium equationof an individual slice, (18), is important because it relates theinternal forces with the line of thrust. The line of thrust maybe used to assess the physical admissibility of the solution. Adetailed derivation of (18) is needed to address the concernsraised by the discusser. Considering moment equilibrium ofthe forces acting on a single slice with respect to the centerof the slice base [point I in Fig. l(b)], we have

Ax ( Ax )(T; + T1+ 1 ) 2 + E;+I h i+, - 2 tan a

- E; (hi + ~x tan a) - P13,Al - W132Ax - AQhQ = 0

(23)

7 Asst. Proj. Engr., GeoSyntec Consultants, Columbia, MD 21044.8 Assoc. Prof., School of Civ. Engrg., Purdue Univ., West Lafayette,

IN.•Asst. Prof., Dept. of Civ. and Envir. Engrg., Washington State Univ.,

Pullman, WA 99164.

162/ JOURNAL OF GEOTECHNICAL ENGINEERING / FEBRUARY 1996

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(11 )

an unreasonable interslice force distribution. Moreover, forpurely cohesive slopes with a circular failure surface, in whichthe factor of safety is independent of the internal force distri­bution and equal to Fmm' small values of moment arm wererequired to make FJf equal to Fmm (Espinoza et al. 1992).

The writers agree with Sharma that in slope-stability prob­lems, the factor of safety is only one of many variables to takeinto consideration. The engineer must also consider other im­portant variables such as (1) the line of thrust variation; (2)the interslice force inclination; and (3) the distribution of theinterstice and base forces. Based on the extensive analysis per­formed by the writers, the first two considerations are the mostimportant ones for helping assess the reasonableness of thesolution.

The first aspect of the discussion by Wright and Duncan iscentered around the variation of the computed factors of safetyand its practical implications. As an "exact" factor of safetyof a slope based on limit equilibrium theory cannot be ob­tained, except for cohesive soils with circular failure surfaces,an average factor of safety computed from different rigorousmethods may provide a good estimate of the degree of stabilityfor a slope. Unfortunately, average factors of safety may beinfluenced by the number and type of cases analyzed, espe­cially since a standard rigorous procedure is yet to be defined.Therefore, an average value may not necessarily represent the"most accurate" value of a slope factor of safety.

The selection of a particular limit equilibrium method in agiven slope-stability analysis is often based on personal ex­perience and preferences rather than on technical considera­tions such as assumptions on internal force distribution.Hence, comparisons with respect to a "representative" factorof safety should be more appropriate. Let us use the Spencer'smethod as the representative factor of safety. Variations ob­tained with respect to this method for the five examples ana­lyzed in the paper range from -10% to +7% (case 6 of ex­ample 2 is not considered). It is important to note that adifference in safety factor of up to 10% may be obtained whencomparing Spencer's method with other popular proceduressuch as Morgenstern and Price's and Janbu's. Such differencesmay be important for a slope on the verge of failure or whena mandate safety factor must be obtained (e.g., U.S. Environ­mental Protection Agency subtitle D regulations for solid­waste landfills). Thus, depending on what rigorous method isbeing used in the analysis, additional costs may be involvedto meet such regulations.

The second aspect of the discussion is centered around case6 of example 2. The discussers attribute the unusually largevalue of factor of safety to the fact that an arbitrary slip surfacewith a steep slope at the toe was used in the analysis. It shouldbe noted that the shape of the failure surface in example 2was prescribed by a thin weak layer (Morgenstern and Price1965). The writers performed a similar analysis (see example3 of the paper) where the exit point was moved to the left ofits actual position. The factor of safety variation was reducedto ±6% from the average, which is in agreement with thediscussers' results. Hence, the writers agree with Wright andDuncan with respect to their second reason for the factor ofsafety being so high in this case; it is the relationship of theinternal forces and the geometry of the slope.

There are many practical situations in which failure surfacesare controlled by stratigraphic conditions or existing failureplanes, and their locations are not influenced by the particularhypothesis assumed. For such cases, the same failure surfacescan be used to compute factors of safety using different hy­potheses. Furthermore, the authors believe that a critical sur­face found using a particular hypothesis (let us say Spencer's)can be used for subsequent comparisons. In such a case, onlythose hypotheses that render factors of safety lower than thatobtained by Spencer's method may be used in the analysis.

As the discussers pointed out, numerical problems may beexperienced in cases in which the interslice force resultant andthe force at the base of the last slice may become close tobeing parallel. Such numerical difficulties were not observedin the present analysis in the solution of example 2.

Errata. The following correction should be made to theoriginal paper:

Page 1190, Eq. (11): Should read

[;:] = [~~~naa ;~:] [~]

VERTICAL UPLIFT CAPACITY OF

HORIZONTAL ANCHORSH

Discussion by Abdelmalek Bouazza3

The authors have performed a thorough theoretical investi­gation on the behavior of plate anchors embedded in clay andsand. They presented an interesting analysis of the problem.The discusser would like to compare some of his experimentalfindings (Bouazza 1990) and field test results reported bySutherland (1965) with the theoretical analysis (case III) ob­tained by the authors.

The discusser's model pullout tests were performed on cir­cular plate anchors (diameter = 37.5 mm) embedded in varioussands at loose and dense states. The testing was conducted atdensity indexes (10) of 13% and 81% ("I = 14.81 kN/m2 and"I = 17.15 kN/m2

) with <I> = 32° and 42.8°, respectively. Tomake the comparison possible, an equivalent diameter (B, =Bcir/V4f;) was used. Ovesen (1981) showed that the sameuplift capacity is obtained for a square and a circular anchorat the same depth, provided their areas are equal.

In Fig. 14, predictions from the authors' theory are plottedfor two densities of sand corresponding to the experimentalvalues tested in a subrounded sand. Also included are resultsreported by Sutherland (1965). The discusser used (14), (15),and Fig. 7 to make the predictions. The authors' theory givesa breakout factor at A = 3 for loose sand of the order of 86%of the value for dense sand, compared with the experimentalresults, which gave 41 %. For the field tests carried out atSizewell (U.K) by Sutherland, the breakout factor is under­estimated in both densities (loose-medium and dense). Thisimplies that the authors' theory is insensitive to changes in'P values and therefore sand density. This might be due to thefailure mechanism, during uplift in the sand, chosen by theauthors, where the effect of density change has not been takeninto account. It has been shown that the mode of failure andthe breakout factor are dependent on soil properties (Suther­land et al. 1983; Bouazza 1991, 1993). This further shown byFig. 15, where stereophotogrammetric technique is applied tomonitor the development of the displacement field. In the caseof loose sand, it can be seen that the displacements in thesurrounded sand have mainly occurred 2 X B above the plateanchor and 0.25 X B laterally. It can also be seen that thedisplacement field does not extend to the ground surface,

"July, 1994, Vol. 120, No.7, by Kanakapura S. Subba Rao and JyantKumar (paper 4876).

3 Res. Fellow, Ghent Univ., Soil Mech. Lab., Technologiepark 9, B­9052 Zwijnaarde, Belgium.

JOURNAL OF GEOTECHNICAL ENGINEERING / FEBRUARY 1996/163

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