91631_07

103

CHAPTER 7

Methods of Analyzing Slope Stability

Methods for analyzing stability of slopes include sim-ple equations, charts, spreadsheet software, and slopestability computer programs. In many cases more thanone method can be used to evaluate the stability for aparticular slope. For example, simple equations orcharts may be used to make a preliminary estimate ofslope stability, and later, a computer program may beused for detailed analyses. Also, if a computer programis used, another computer program, slope stabilitycharts, or a spreadsheet should be used to verify re-sults. The various methods used to compute a factorof safety are presented in this chapter.

SIMPLE METHODS OF ANALYSIS

The simplest methods of analysis employ a single sim-ple algebraic equation to compute the factor of safety.These equations require at most a hand calculator tosolve. Such simple equations exist for computing thestability of a vertical slope in purely cohesive soil, ofan embankment on a much weaker, deep foundation,and of an innite slope. Some of these methods, suchas the method for computing the stability of an inniteslope, may provide a rigorous solution, whereas others,such as the equations used to estimate the stability ofa vertical slope, represent some degree of approxima-tion. Several simple methods are described below.

Vertical Slope in Cohesive SoilFor a vertical slope in cohesive soil a simple expres-sion for the factor of safety is obtained based on aplanar slip surface like the one shown in Figure 7.1.The average shear stress, , along the slip plane is ex-pressed as

2W sin W sin W sin (7.1)l H /sin H

where is the inclination of the slip plane, H is theslope height, and W is the weight of the soil mass. Theweight, W, is expressed as

21 HW (7.2)2 tan

which when substituted into Eq. (7.2) and rearrangedgives

1 H sin cos (7.3)2

For a cohesive soil ( 0) the factor of safety isexpressed as

c 2cF (7.4)

H sin cos

To nd the minimum factor of safety, the inclinationof the slip plane is varied. The minimum factor ofsafety is found for 45. Substituting this value for (45) into Eq. (7.4) gives

4cF (7.5)

H

Equation (7.5) gives the factor of safety for a verticalslope in cohesive soil, assuming a plane slip surface.Circular slip surfaces give a slightly lower value forthe factor of safety (F 3.83c /h); however, the dif-ference between the factors of safety based on a planeand a circular slip surface is small for a vertical slopein cohesive soil and can be ignored.

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Copyright 2005 John Wiley & Sons Retrieved from: www.knovel.com

104 7 METHODS OF ANALYZING SLOPE STABILITY

H

W

Figure 7.1 Vertical slope and plane slip surface.

Equation (7.5) can also be rearranged to calculatethe critical height of a vertical slope (i.e., the heightof a slope that has a factor of safety of unity). Thecritical height of a vertical slope in cohesive soil is

4cH (7.6)critical

Bearing Capacity EquationsThe equations used to calculate the bearing capacity offoundations can also be used to estimate the stabilityof embankments on deep deposits of saturated clay.For a saturated clay and undrained loading ( 0),the ultimate bearing capacity, qult, based on a circularslip surface is1

q 5.53c (7.7)ultEquating the ultimate bearing capacity to the load,q H, produced by an embankment of height, H,gives

H 5.53c (7.8)

where is the unit weight of the soil in the embank-ment; h represents the maximum vertical stress pro-duced by the embankment. Equation (7.8) is anequilibrium equation corresponding to ultimate condi-tions (i.e., with the shear strength of the soil fully de-veloped). If, instead, only some fraction of the shearstrength is developed (i.e., the factor of safety is

1 Although Prandtls solution of qult 5.14c is commonly used forbearing capacity, it is more appropriate to use the solution based oncircles, which gives a somewhat higher bearing capacity and offsetssome of the inherent conservatism introduced when bearing capacityequations are applied to slope stability.

greater than unity), a factor of safety can be introducedinto the equilibrium equation (7.8) and we can write

cH 5.53 (7.9)

F

In this equation F is the factor of safety with respectto shear strength; the term c /F represents the devel-oped cohesion, cd. Equation (7.9) can be rearranged togive

cF 5.53 (7.10)

H

Equation (7.10) can be used to estimate the factor ofsafety against a deep-seated failure of an embankmenton soft clay.

Equation (7.10) gives a conservative estimate of thefactor of safety of an embankment because it ignoresthe strength of the embankment and the depth of thefoundation in comparison with the embankment width.Alternative bearing capacity equations that are appli-cable to reinforced embankments on thin clay foun-dations are presented in Chapter 8.

Innite SlopeIn Chapter 6 the equations for an innite slope werepresented. For these equations to be applicable, thedepth of the slip surface must be small compared tothe lateral extent of the slope. However, in the case ofcohesionless soils, the factor of safety does not dependon the depth of the slip surface. It is possible for a slipsurface to form at a small enough depth that the re-quirements for an innite slope are met, regardless ofthe extent of the slope. Therefore, an innite slopeanalysis is rigorous and valid for cohesionless slopes.The innite slope analysis procedure is also applicableto other cases where the slip surface is parallel to theface of the slope and the depth of the slip surface issmall compared to the lateral extent of the slope. Thiscondition may exist where there is a stronger layer ofsoil at shallow depth: for example, where a layer ofweathered soil exists near the surface of the slope andis underlain by stronger, unweathered material.

The general equation for the factor of safety for aninnite slope with the shear strength expressed in termsof total stresses is

cF cot tan (cot tan ) (7.11)

z

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SLOPE STABILITY CHARTS 105

where z is the vertical depth of the slip surface belowthe face of the slope. For shear strengths expressed byeffective stresses the equation for the factor of safetycan be written as

uF cot (cot tan ) tan zc

(cot tan ) (7.12)z

where u is the pore water pressure at the depth of theslip surface.

For effective stress analyses, Eq. (7.12) can also bewritten as

F [cot r (cot tan )] tan uc

(cot tan ) (7.13)z

where ru is the pore pressure ratio dened by Bishopand Morgenstern (1960) as

ur (7.14)u z

Values of ru can be determined for specic seepageconditions. For example, for seepage parallel to theslope, the pore pressure ratio, ru, is given by

hw w 2r cos (7.15)u z

where hw is the height of the free water surface verti-cally above the slip surface (Figure 7.2a). If the seep-age exits the slope face at an angle (Figure 7.2b), thevalue of ru is given by

1wr (7.16)u 1 tan tan

where is the angle between the direction of seepage(ow lines) and the horizontal. For the special case ofhorizontal seepage ( 0), the expression for ru re-duces to

wr (7.17)u

Recapitulation

Simple equations can be used to compute the fac-tor of safety for several slope and shear strengthconditions, including a vertical slope in cohesivesoil, an embankment on a deep deposit of satu-rated clay, and an innite slope.

Depending on the particular slope conditions andequations used, the accuracy ranges from excel-lent, (e.g., for a homogeneous slope in cohesion-less soil) to relatively crude (e.g., for bearingcapacity of an embankment on saturated clay).

SLOPE STABILITY CHARTS

The stability of many relatively homogeneous slopescan be calculated using slope stability charts based onone of the analysis procedures presented in Chapter 6.Fellenius (1936) was one of the rst to recognize thatfactors of safety could be expressed by charts. Hiswork was followed by the work of Taylor (1937) andJanbu (1954b). Since the pioneering work of these au-thors, numerous others have developed charts for com-puting the stability of slopes. However, the early chartsof Janbu are still some of the most useful for manyconditions, and these are described in further detail inthe Appendix. The charts cover a range in slope andsoil conditions and they are quite easy to use. In ad-dition, the charts provide the minimum factor of safetyand eliminate the need to search for a critical slip sur-face.

Stability charts rely on dimensionless relationshipsthat exist between the factor of safety and other pa-rameters that describe the slope geometry, soil shearstrengths, and pore water pressures. For example, theinnite slope equation for effective stresses presentedearlier [Eq. (7.13)] can be written as

tan c2 2F [1 r (1 tan )] (1 tan )u tan z(7.18)

or

tan cF A B (7.19)

tan z

where

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z

hw

(a)

(b)

Figure 7.2 Innite slope with seepage: (a) parallel to slope face; (b) exiting the slope face.

2A 1 r (1 tan ) (7.20)u2B 1 tan (7.21)

A and B are dimensionless parameters (stability num-bers) that depend only on the slope angle, and in thecase of A, the dimensionless pore water pressure co-efcient, ru. Simple charts for A and B as functions ofthe slope angle and pore water pressure coefcient, ru,are presented in the Appendix.

For purely cohesive ( 0) soils and homogeneousslopes, the factor of safety can be expressed as

cF N (7.22)0 H

where N0 is a stability number that depends on theslope angle, and in the case of slopes atter than about11, on the depth of the foundation below the slope.For vertical slopes the value of N0 according to theSwedish slip circle method is 3.83. This value (3.83)is slightly less than the value of 4 shown in Eq. (7.5)based on a plane slip surface. In general, circular slipsurfaces give a lower factor of safety than a plane,especially for at slopes. Therefore, circles are gener-

ally used for analysis of most slopes in cohesive soils.A complete set of charts for cohesive slopes of variousinclinations and foundation depths is presented in theAppendix. Procedures are also presented for using av-erage shear strengths with the charts when the shearstrength varies.

For slopes with both cohesion and friction, addi-tional dimensionless parameters are introduced. Janbu(1954) showed that the factor of safety could be ex-pressed as

cF N (7.23)cf H

where Ncf is a dimensionless stability number. The sta-bility number depends on the slope angle, , the porewater pressures, u, and the dimensionless parameter,c, which is dened as

H tan (7.24)c c

Stability charts employing c and Eq. (7.23) to cal-culate the factor of safety are presented in the Appen-

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COMPUTER PROGRAMS 107

dix. These charts can be used for soils with cohesionand friction as well as a variety of pore water pressureand external surcharge conditions.

Although all slope stability charts are based on theassumption of constant shear strength (c, c and , are constant) or else a simple variation in undrainedshear strength (e.g., c varies linearly with depth), thecharts can be used for many cases where the shearstrength varies. Procedures for using the charts forcases where the shear strength varies are described inthe Appendix. Examples for using the charts are alsopresented in the Appendix.

Recapitulation

Slope stability charts exist for computing the fac-tor of safety for a variety of slopes and soil con-ditions.

SPREADSHEET SOFTWARE

Detailed computations for the procedures of slices canbe performed in tabular form using a table where eachrow represents a particular slice and each column rep-resents the variables and terms in the equations pre-sented in Chapter 6. For example, for the case where 0 and the slip surface is a circle, the factor ofsafety is expressed as

c lF (7.25)W sin

A simple table for computing the factor of safety usingEq. (7.25) is shown in Figure 7.3. For the OrdinaryMethod of Slices with the shear strength expressed interms of effective stresses, the preferred equation forcomputing the factor of safety is

2[c l (W cos u l cos )tan ]F W sin

(7.26)

A table for computing the factor of safety using thisform of the Ordinary Method of Slices equation is il-lustrated in Figure 7.4. Tables such as the ones shownin Figures 7.3 and 7.4 are easily represented and im-plemented in computer spreadsheet software. In fact,more sophisticated tables and spreadsheets can be de-veloped for computing the factor of safety using pro-cedures of slices such as the Simplied Bishop, forceequilibrium, and even Chen and Morgensterns proce-dures (Low et al., 1998).

The number of different computer spreadsheets thathave been developed and used to compute factors ofsafety is undoubtedly very large. This attests to theusefulness of spreadsheets for slope stability analyses,but at the same time presents several important prob-lems: First, because such a large number of differentspreadsheets are used and because each spreadsheet isoften used only once or twice, it is difcult to validatespreadsheets for correctness. Also, because one personmay write a spreadsheet, use it for some computationsand then discard the spreadsheet, results are oftenpoorly archived and difcult for someone else to in-terpret or to understand later. Electronic copies of thespreadsheet may have been discarded. Even if an elec-tronic copy is maintained, the software that was usedto create the spreadsheet may no longer be availableor the software may have been updated such that theold spreadsheet cannot be accessed. Hard copies ofnumerical tabulations from the spreadsheet may havebeen saved, but unless the underlying equations, for-mulas, and logic that were used to create the numericalvalues are also clearly documented, it may be difcultto resolve inconsistencies or check for errors.

Recapitulation

Spreadsheets provide a useful way of performingcalculations by the procedures of slices.

Spreadsheet calculations can be difcult to checkand archive.

COMPUTER PROGRAMS

For more sophisticated analyses and complex slope,soil, and loading conditions, computer programs aregenerally used to perform the computations. Computerprograms are available that can handle a wide varietyof slope geometries, soil stratigraphies, soil shearstrength, pore water pressure conditions, externalloads, and internal soil reinforcement. Most programsalso have capabilities for automatically searching forthe most critical slip surface with the lowest factor ofsafety and can handle slip surfaces of both circular andnoncircular shapes. Most programs also have graphicscapabilities for displaying the input data and the resultsof the slope stability computations.

Types of Computer ProgramsTwo types of computer programs are available forslope stability analyses: The rst type of computer pro-gram allows the user to specify as input data the slopegeometry, soil properties, pore water pressure condi-

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Summation:

b h1 1

1

W sinW(1)

2

3

4

5

6

7

8

9

10

Slice

No.

h2 2 h3 3

(1) W = b x (h11 + h22 + h33)

(2) = b / cos

(2) c c

b

1

2

3

h1

h2

h3

c Wsin

F =

Figure 7.3 Sample table for manual calculations using the Swedish circle ( 0) proce-dure.

tions, external loads, and soil reinforcement, and com-putes a factor of safety for the prescribed set ofconditions. These programs are referred to as analysisprograms. They represent the more general type ofslope stability computer program and are almost al-ways based on one or more of the procedures of slices.

The second type of computer program is the designprogram. These programs are intended to determinewhat slope conditions are required to provide one ormore factors of safety that the user species. Many ofthe computer programs used for reinforced slopes andother types of reinforced soil structures such as soilnailed walls are of this type. These programs allow the

user to specify as input data general information aboutthe slope geometry, such as slope height and externalloads, along with the soil properties. The programsmay also receive input on candidate reinforcement ma-terials such as either the tensile strength of the rein-forcement or even a particular manufacturers productnumber along with various factors of safety to beachieved. The computer programs then determine whattype and extent of reinforcement are required to pro-duce suitable factors of safety. The design programsmay be based on either procedures of slices or single-free-body procedures. For example, the logarithmicspiral procedure has been used in several computer

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COMPUTER PROGRAMS 109

Summation:

Slice

No.

W c

os

- u

co

s2

h1b 1 W c' ' u u W cos

(W co

s - u

co

s2

)tan

'

c' W sinh2 2 h3 3

1

2

3

4

5

6

7

8

9

10

Note: 1. W = b x (h11 + h22 + h33)2. = b / cos

[(W cos ucos2) tan ' c']W sin

F =

Figure 7.4 Sample table for manual calculations using the Ordinary Method of Slices andeffective stresses.

programs for both geogrid and soil nail design (Lesh-chinsky, 1997; Byrne, 20032). The logarithmic spiralprocedure is very well suited for such applicationswhere only one soil type may be considered in thecross section.

Design programs are especially useful for design ofreinforced slopes using a specic type of reinforcement(e.g., geogrids or soil nails) and can eliminate much ofthe manual trial-and-error effort required. However, thedesign programs are usually restricted in the range ofconditions that can be handled and they often makesimplifying assumptions about the potential failuremechanisms. Most analysis program can handle amuch wider range of slope and soil conditions.

Automatic Searches for Critical Slip SurfaceAlmost all computer programs employ one or moreschemes for searching for a critical slip surface withthe minimum factor of safety. Searches can be per-formed using both circular and noncircular slip sur-faces. Usually, different schemes are used depending

2 Byrne has utilized the log spiral procedure in an unreleased versionof the GoldNail software. One of the authors (Wright) has also usedthe log spiral successfully for this purpose in unreleased software foranalyzing soil nail walls.

on the shape (circular vs. noncircular) of slip surfaceused. Many different search schemes have been used,and it is beyond the scope of this chapter to discussthese in detail. Nevertheless, several recommendationsand guidelines can be offered for searching for a crit-ical slip surface:

1. Start with circles. It is almost always preferableto begin searching for a critical slip surface usingcircles. Very robust schemes exist for searchingwith circles, and it is possible to examine a largenumber of possible locations for a slip surfacewith relatively little effort on the part of the user.

2. Let stratigraphy guide the search. For both cir-cular and noncircular slip surfaces, the stratigra-phy often suggests where the critical slip surfacewill be located. In particular, if a relatively weakzone exists, the critical slip surface is likely topass through it. Similarly, if the weak zone isrelatively thin and linear, the slip surface mayfollow the weak layer and is more likely to benoncircular than circular.

3. Try multiple starting locations. Almost all au-tomatic searches begin with a slip surface that theuser species in some way. Multiple starting lo-cations should be tried to determine if one loca-tion leads to a lower factor of safety than another.

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4. Be aware of multiple minima. Many searchschemes are essentially optimization schemesthat seek to nd a single slip surface with thelowest factor of safety. However, there may bemore than one local minimum and the searchscheme may not necessarily nd the local mini-mum that produces the lowest factor of safetyoverall. This is one of the reasons why it is im-portant to use multiple starting locations for thesearch.

5. Vary the search constraints and other parame-ters. Most search schemes require one or moreparameters that control how the search is per-formed. For example, some of the parameters thatmay be specied include: The incremental distances that the slip surface

is moved during the search The maximum depth for the slip surface The maximum lateral extent of the slip surface

or search The minimum depth or weight of soil mass

above the slip surface The maximum steepness of the slip surface

where it exits the slope The lowest coordinate allowed for the center of

a circle (e.g., to prevent inversion of the circle)Input data should be varied to determine howthese parameters affect the outcome of the searchand the minimum factor of safety.

A relatively large number of examples and bench-marks can be found in the literature for the factor ofsafety for a particular slip surface. However, manyfewer examples can be found to conrm the locationof the most critical slip surface (lowest factor ofsafety), even though this may be the more importantaspect of verication. For complex slopes, much moreeffort is usually spent in a slope stability analysis toverify that the most critical slip surface is found thanis spent to verify that the factor of safety for a givenslip surface has been computed correctly.

Restricting the Critical Slip Surfaces of InterestIn general, all areas of a slope should be searched tond the critical slip surface with the minimum factorof safety. However, is some cases it may be desirableto search only a certain area of the slope by restrictingthe location of trial slip surfaces. There are two com-mon cases where this is appropriate. One case is wherethere are insignicant modes of failure that lead to lowfactors of safety, but the consequences of failure aresmall. The other case is where the slope geometry is

such that a circle with a given center point and radiusdoes not dene a unique slip surface and slide mass.These two cases are described and discussed furtherbelow.

Insignicant modes of failure. For cohesionlessslopes it has been shown that the critical slip surfaceis a very shallow plane, essentially coincident with theface of the slope. However, the consequences of a slidewhere only a thin layer of soil is involved may be verylow and of little signicance. This is particularly thecase for some mine tailings disposal dams. In suchcases it is desirable to investigate only slip surfacesthat have some minimum size and extent. This can bedone in several ways, depending on the particular com-puter program being used:

The slip surfaces investigated can be required tohave a minimum depth.

The slip surfaces investigated can be forced topass through a specic point at some depth belowthe surface of the slope.

The soil mass above the slip surface can be re-quired to have a minimum weight.

An articially high shear strength, typically ex-pressed by a high value of cohesion, can be as-signed to a zone of soil near the face of the slopeso that shallow slip surfaces are prevented. In do-ing so, care must be exercised to ensure that slipsurfaces are not unduly restricted from exiting inthe toe area of the slope.

Ambiguities in slip surface location. In some casesit is possible to have a circle where more than onesegment of the circle intersects the slope (Figure 7.5).In such cases there is not just a single soil mass abovethe slip surface, but rather there are multiple, disasso-ciated soil masses, probably with different factors ofsafety. To avoid ambiguities in this case, it is necessaryto be able to designate that only a particular portion ofthe slope is to be analyzed.

Recapitulation

Computer programs can be categorized as designprograms and analysis programs. Design pro-grams are useful for design of simple reinforcedslopes, while analysis programs generally canhandle a much wider range of slope and soil con-ditions.

Searches to locate a critical slip surface with aminimum factor of safety should begin with cir-cles.

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VERIFICATION OF ANALYSES 111

? ?

?

?

?

Figure 7.5 Cases where the slide mass dened by a circularslip surface is ambiguous and may require selective restric-tion.

Multiple searches with different starting pointsand different values for the other parameters thataffect the search should be performed to ensurethat the most critical slip surface is found.

In some case it is appropriate to restrict the regionwhere a search is conducted; however, care mustbe taken to ensure that an important slip surfaceis not overlooked.

VERIFICATION OF ANALYSES

Most slope stability analyses are performed usinggeneral-purpose computer programs. The computerprograms offer a number of features and may involvetens of thousands, and sometimes millions, of lines ofcomputer code with many possible paths through thelogic, depending on the problem being solved. Foresterand Morrison (1994) point out the difculty of check-ing even simple computer programs with multiple

combinations of paths through the software. Consider,for example, a comprehensive computer program forslope stability analysis that contains the features listedin Table 7.1. Most of the more sophisticated computerprograms probably contain at least the number of op-tions or features listed in this table. Although someprograms will not contain all of the options listed, theymay contain others. A total of 40 different features andoptions is listed in Table 7.1. If we consider just twodifferent possibilities for the input values for each op-tion or feature, there will be a total of over 1 1012( 240) possible combinations and paths through thesoftware. If we could create, run, and verify problemsto test each possible combination at the rate of one testproblem every 10 minutes, over 20 million years wouldbe required to test all possible combinations, working24 hours a day, 7 days a week. Clearly, it is not pos-sible to test sophisticated computer programs for allpossible combinations of data, or even a reasonablysmall fraction, say 1 of 1000, of the possible combi-nations. Consequently, there is a signicant possibilitythat any computer program being used has not beentested for the precise combination of paths involved ina particular problem.

Because it is very possible that any computer pro-gram has not been veried for the particular combi-nation of conditions the program is being used for,some form of independent check should be made ofthe results. This is also true for other methods of cal-culation. For example, spreadsheets are just anotherform of computer program, and the difculty of veri-fying spreadsheet programs was discussed earlier. It isalso possible to make errors in using slope stabilitycharts and even in using simple equations. Further-more, the simple equations generally are based on ap-proximations that can lead to important errors for someapplications. Consequently, regardless of how slopestability computations are performed, some indepen-dent check should be made of the results. A numberof examples of slope stability analyses and checks thatcan be made are presented in the next section.

Recapitulation

Because of the large number of possible pathsthrough most computer programs, it is likely mostprograms have not been tested for the precisecombination of paths involved in any particularanalysis.

Some check should be made of the results ofslope stability calculations, regardless of how thecalculations are performed.

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Table 7.1 Possible Options and Features for a Comprehensive Slope Stability Computer Program

Soil prole linesstratigraphySoil shear strength c soiltotal stresses

c soileffective stressesCurved Mohr failure envelopetotal stressesCurved Mohr failure envelopeeffective stressesUndrained shear strength varies with depth below horizontal datumUndrained shear strength dened by contour lines or interpolationShear strength dened by a c /p ratioAnisotropic strength variationundrained strength and total stressesAnisotropic strength variationdrained strength and effective stressesConsolidatedundrained shear strength (e.g., for rapid drawdownlinear strength envelopes)Consolidatedundrained shear strength (e.g., for rapid drawdowncurved strength envelopes)Structural materials (e.g., steel, concrete, timber)

Pore water pressure Constant pore water pressureConstant pore pressure coefcient, ruPiezometric linePhreatic surfaceInterpolated values of pore water pressure (e.g., from nite element analyses)Interpolated values of pore water pressure coefcient, ruSlope geometryLeft vs. right face of slope analyzedDistributed surface loads (e.g., water)Line loads

Reinforcement GeotextilesGeogridsSoil nailsTieback anchorsPilesPiers

Slip surface(s) Individual circleIndividual noncircular slip surfaceSystematic search with circlesRandom search with circlesSystematic search with noncircular slip surfacesRandom search with noncircular slip surfaces

Procedure ofanalysis

Simplied Bishop procedureSpencers procedureCorps of Engineers Modied Swedish procedureSimplied Janbu procedureChen and Morgensterns procedure

EXAMPLES FOR VERIFICATION OF STABILITYCOMPUTATIONS

Ten example slopes were selected for the slope stabil-ity analyses presented in this section. These exampleswere selected with two purposes in mind: First, to il-lustrate the different methods for computing the factorof safety that were discussed in the preceding sectionsof this chapter, and second, to illustrate several impor-

tant details and features of slope stability analyses. Forexample, one problem addresses the use of submergedunit weights. Several other problems illustrate the dif-ferences among various procedures of slices. Some ofthese and other examples illustrate the importance oflocating the critical slip surface. Most of the examplesare presented with enough detail that they can be usedas benchmarks for verifying results of calculations us-ing other means (e.g., with other computer programs).

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EXAMPLES FOR VERIFICATION OF STABILITY COMPUTATIONS 113

The 10 example problems selected for analysis aresummarized in Table 7.2. Each example is describedbriey and the methods of calculation (simple equa-tions, charts, spreadsheets, and computer programs) areindicated. Any calculations presented using computerprograms were performed with the UTEXAS4 soft-ware (Wright, 1999) unless otherwise stated. The sum-mary also indicates whether analyses were performedfor short- or long-term stability conditions. Additionalfeatures illustrated by each example are indicated inthe last column of Table 7.2. The 10 cases listed inthis table provide a useful collection of problems forcomputer program verication.

Example 1: Unbraced Vertical Cut in ClayTschebotarioff (1973) describes the failure of a verticalexcavated slope that was made for a two-story base-ment in varved clay. The excavation was made, withoutbracing, to a depth of 22 ft on one side and 31.5 ft onthe other side. The average unconned compressivestrength of the clay from an investigation nearby wasreported to be 1.05 tons/ft2 and the unit weight of theclay was 120 lb/ft3. Factors of safety were calculatedfor the deeper of the two cuts (Figure 7.6) using theequation for a vertical slope with a plane slip surface,and using the slope stability charts presented in theAppendix. Calculations were also performed using acomputer program. For an undrained shear strength, Suof 1050 psf ( qu /2), the factor of safety for a planeslip surface is calculated as

4c (4)(1050)F 1.11 (7.27)

H (120)(31.5)

Using Janbus charts for 0 presented in the Ap-pendix, the factor of safety is calculated as

c 1050F N (3.83) 1.06 (7.28)0 H (120)(31.5)

Calculations with circles using the computer programresulted in a factor of safety of 1.06. The calculationswith the charts conrm the results with the computerprogram, and both show that circular slip surfaces givea slightly lower factor of safety than plane slip sur-faces.

Although the foregoing calculations are in closeagreement, they may not correctly reect the true fac-tor of safety of the slope. Terzaghi (1943) pointed outthat the upper part of the soil adjacent to a verticalslope is in tension. If the soil cannot withstand tension,cracks will form and the factor of safety will be re-duced. Terzaghi showed that if one conservatively es-

timates that a crack will form to a depth equal toone-half the slope height, the equation for the factorof safety (assuming a planar slip surface) becomes

cF 2.67 (7.29)

H

Thus, for the slope described above,

(2.67)(1050)F 0.74 (7.30)(120)(31.5)

which would clearly indicate that the slope was notstable. A computed factor of safety less than 1.0 forthis case seems reasonable, because the slope failedand the unconned compression tests that were usedto measure the shear strength would be expected tounderestimate strength due to sample disturbance.

In the rst calculations with the computer program,tension was observed on the bottoms of several of theslices near the upper part of the slope. Subsequently,a series of slope stability calculations was performedin which vertical tension cracks were introduced, be-ginning with a crack depth of 1 ft, and successivelyincreasing the crack depth in 1-ft increments until therewas no longer tension. The assumed crack depths, cor-responding factors of safety, and minimum normalstresses on the base of slices are summarized in Table7.3. If we take the factor of safety as being the valuewhere the tensile stresses are rst eliminated, wewould conclude that the factor of safety is less than 1(between 0.96 and 0.99).

For this example the stability calculations supportthe behavior observed quite well. However, the close-ness of the factor of safety to unity may be due in partto compensating errors caused by factors that were notconsidered. The shear strengths used were based onunconned compression tests, which typically under-estimate the shear strength. Thus, it is likely that theundrained shear strength of the clay was actuallygreater than what was assumed. At the same time, be-cause the slope was excavated, the unloading due toexcavation would cause the soil to swell gradually andlose strength with time. Also, it is possible that verticalcracks may have opened to substantial depths. It ispossible to imagine that the undrained strength mea-sured in more appropriate UU tests would have beenconsiderably higher than the shear strength used, whilelosses of strength due to swell and the development ofdeep tension cracks could have reduced the stability bya substantial amount. These offsetting factors couldhave affected the stability of the slope signicantly,and it can be seen that the failure may have taken place

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114

Table 7.2 Summary of Example Problems for Verication of Slope Stability Analyses

No. DescriptionShort or

long term

Methods of analysis and verication of results

Simple equations

Verticalslopeplane

Bearingcapacity

Inniteslope

Charts

0:Janbu

0:Hunter

andSchuster

c, Soil:Janbu

Spreadsheets

0 OMSSimplied

BishopForce

equilibrium

Computer program

UTEXAS4 Other Additional features

1 Unbraced verticalcut in saturatedclay (afterTschebotarioff);includingeffects oftension crack

S Y Y Y Effects of tension and atension crack.

2 LASH terminal:submergedslope excavatedin saturated,nearly normallyconsolidatedclay

S Y Y Use of total unitweights and porewater pressures vs.submerged unitweights.

3 Bradwell slipexcavated slopein stiff-ssuredclay

S Y Y Application of Janbucorrection factor insimplied Janbuprocedure. Slopemay fail even withhigh factor of safety.

4 Hypotheticalexample ofcohesionlessslope (c 0)on saturatedclay ( 0)foundation

S Y Y Y Application of Janbucorrection factor insimplied Janbuprocedure. Relativelylarge differences inF by variousprocedures.

5 Oroville Damhigh rocklldam

L Y Y Stability computationswith a curved Mohrshear strengthenvelope.Co

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115

6 James Bay dikeembankmentsconstructed onsoft clayfoundation

S Y Y Importance of ndingcritical slip surface.

7 Homogeneousearth dam withsteady-stateseepage

L Y Y Effects of how porewater pressures arerepresented (by ownet, piezometric line,phreatic surface).Illustrates effects ofpore pressure inOrdinary Method ofSlices.

8 Zoned (or claycore) earth damwith steady-state seepage

L Y Y Effects of how porewater pressures arerepresented (by ownet, piezometric line,phreatic surface).

9 Reinforced slope(1):embankment ona soft clayfoundation

S Y Y Reinforced slopeanalysis; inuence oflocation of criticalcircle.

10 STABGMreinforced slope(2): steepreinforced slope

L Y Y Reinforced slopeanalysis; inuence oflocation of criticalcircle.

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Varved Clay:qu = 1.05 tons/ft2 = 120 lb/ft3

31.5 ft

Figure 7.6 Unbraced vertical cut in clay described byTschebotarioff (1973).

Table 7.3 Variation in the Factor of Safety andMinimum Normal Stress on the Slip Surface with theAssumed Depth of Tension Crack

Assumed crackdepth(ft)

Minimum normalstress on slip

surface (base ofslices)(psf)

Calculated factorof safety

0 241 1.061 160 1.042 67 1.013 62 0.994 40 0.96

under conditions quite different from what was as-sumed in the stability calculations.

Recapitulation

Slope stability charts, the computer program, andthe simple equation for stability of a vertical cutbased on plane slip surfaces all gave nearly iden-tical values for the factor of safety.

Plane slip surfaces, compared to circles, give sim-ilar but slightly higher values for the factor ofsafety of a vertical slope.

Tensile stresses may develop behind the crest ofsteep slopes in clay and may lead to cracking thatwill substantially reduce the stability of the slope.

Close agreement between computed and actualfactors of safety may be fortuitous and a result ofmultiple large errors that compensate.

Example 2: Underwater Slope in Soft ClayDuncan and Buchignani (1973) described the failureof a slope excavated underwater in San Francisco Bay.The slope was part of a temporary excavation and wasdesigned with an unusually low factor of safety to min-imize construction costs. During construction a portionof the excavated slope failed. A drawing of the slopecross section is shown in Figure 7.7. The undrainedshear strength prole is presented in Figure 7.8. Theoriginal design factor of safety based on undrainedshear strengths was reported by Duncan and Buchig-nani to be 1.17.

Recently (2003), new slope stability calculationswere performed by the writers, rst using a computerprogram with Spencers procedure of slices. The min-imum factor of safety calculated was 1.17. Because theundrained shear strength for the clay in the slope in-creases linearly with depth, Hunter and Schusters(1968) slope stability charts described in the Appendixcan also be used to compute the factor of safety. Thefactor of safety computed using these charts is 1.18.

The slope stability calculations described abovewere performed using submerged (buoyant) unitweights to account for the slope being fully sub-merged. Submerged unit weights are convenient to usewhen the computations are being performed with eitherslope stability charts or by hand using a spreadsheet.Submerged unit weights can be used for this examplebecause there was no seepage force (no ow of water).However, in general when using computer programs itis preferable to use total unit weights and to specifyexternal and internal water pressures. Computer cal-culations were repeated for this slope using total unitweights and distributed loads on the surface of theslope to represent the water pressures. The factor ofsafety was again found to be 1.17. This not only con-rms what is expected but provides a useful check onthe calculations of the weights of slices and the forcesdue to external distributed loads calculated by the com-puter program.

A simple and useful check of any computer programis to perform separate sets of slope stability calcula-tions for a submerged slope (with no ow) using (1)submerged unit weights and (2) total unit weights withwater pressures. If the computer program is workingproperly and being used properly, it should give thesame result for both sets of calculations.3

3 This may not be true with force equilibrium procedures with in-clined interslice forces. Similar results may not be obtained withsubmerged unit weights and total unit weights plus water pressureswhen the interslice forces are total forces, due to both earth and waterpressures, as described in Chapter 6. In this case the differences infactors of safety calculated using submerged unit weights and totalunit weights plus water pressures may be large.

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Debris dike

San Francisco Bay mud

Firm soil

0.875 0.8751 1

Elev

atio

n - f

t (MLL

W)

-120

-80

-40

0

40

Figure 7.7 Underwater slope in San Francisco Bay mud described by Duncan and Buchig-nani (1973) and Duncan (2000).

12008004000

Undrained Shear Strength - psf

Dep

th B

elow

Mud

line

- fe

et

-100

-80

-60

-40

-20

0

Figure 7.8 Undrained shear strength prole for underwaterslope in San Francisco Bay mud. (From Duncan, 2000.)

Although the calculations presented above conrmthe factor of safety calculated by Duncan and Buchig-nani (1973) and indicate that the slope would be ex-pected to be stable, a portion of the slope failed, asnoted earlier. Duncan and Buchignani (1973) showedthat the effects of sustained loading (creep) under un-drained conditions was probably sufcient to reducethe shear strength and cause the failure. More recentreliability analyses by Duncan (2000) have shown that

the probability of failure was almost 20%. This prob-ability of failure is consistent with the fact that about20% of the length of the slope actually failed. Giventhe accuracy with which such analyses can be made,the close agreement between the probability of failureand the fraction of the slope that failed is probablyfortuitous.

Because this slope was only temporary, it was ap-propriate to compute the stability using undrainedshear strengths. However, if the slope was permanent,much lower drained shear strengths would apply. Asthe soil swells due to unloading by excavation, theshear strength would gradually be reduced. Eventually,the fully drained shear strength would become appli-cable. Representative values of the drained (effectivestress) shear strength parameters for San Francisco Baymud are c 0, 34.5 (Duncan and Seed, 1966b).For a fully submerged slope and c 0, the factor ofsafety can be calculated using the equation for an in-nite slope as

tan tan 34.5F 0.60 (7.31)

tan 1/0.875

Clearly, this factor of safety (0.60) is much less thatthe factor of safety (1.17) based on undrained shearstrengths, indicating that a substantial reduction in fac-tor of safety would have occurred if the excavatedtrench had not been lled with sand.

Recapitulation

Identical values for the factor of safety were ob-tained using a computer program and a slope sta-bility chart.

Either submerged unit weights or total unitweights and water pressures may be used to com-pute the stability of a submerged slope when thereis no ow.

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+17.5'

+6'

-3'

-27'

Clay Fill

Marsh Clay

BrownLondon Clay

BlueLondon Clay

1:1

1:1

1/2:1

12'

6'

-31'

Original ground level

28'

48.5'

Figure 7.9 Cross section of excavated slope for reactor 1 at Bradwell. (From Skempton andLaRochelle, 1965.)

Figure 7.10 Undrained shear strength prole for reactor 1excavation slope at Bradwell. (From Skempton and La-Rochelle, 1965.) Table 7.4 Summary of Short-Term Slope Stability

Analyses for an Excavated Slope in Stiff-FissuredClay: The Bradwell Slip

Procedure of slicesFactor of

safety

Spencer 1.76Simplied Bishop 1.76Corps of Engineers Modied Swedish 1.80Simplied Janbuno correction 1.63Simplied Janbuwith correction, 0 1.74

Even though the calculated factor of safety wasgreater than unity (1.17), the slope failed due tocreep strength loss.

For an excavated slope, the short-term factor ofsafety based on undrained conditions may bemuch higher than the long-term factor of safetybased on drained conditions.

Example 3: Excavated Slope in Stiff-Fissured ClaySkempton and LaRochelle (1965) describe a deep ex-cavation in the London Clay at Bradwell. A cross sec-tion of the excavation for reactor 1 is shown in Figure7.9. The excavation is 48.5 ft deep. The lower 28 ft ofthe excavation is in London Clay and is inclined at(horizontal)1(vertical). The London Clay is overlain12

by 9 ft of Marsh clay where the excavation slope wasinclined at 11 (45). Approximately 11.5 ft of clayfrom the excavation was placed at the top of the ex-cavation, over the marsh clay. The clay ll was alsoinclined at 11.

Short-term stability analyses were performed for theslope using undrained shear strengths. The marsh claywas reported to have an average undrained shearstrength of 300 psf and a total unit weight of 105 pcf.The clay ll was assumed to crack to the full depth ofthe ll (11.5 ft), and thus its strength was ignored.Skempton and LaRochelle reported a total unit weightof 110 pcf for the ll. The undrained shear strengthprole for the London Clay is shown in Figure 7.10.The undrained shear strength increases at a decreasing

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Summations:

Slice

No.

b(ft)

10.8

hfill(ft)

11.5

fill(pcf)

110

hmarsh(ft)

marsh(pcf)

4.5 105

hclay(ft)

clay(pcf)

- -1

(deg)

39.8

(ft)

14.1

c(psf)

300

c

4,215

W sinW(pounds)

18,748 12,008

2

3

4

5

6

7

8

9.1

11.5

12.0

9.0

6.0

5.5

8.5

11.5

5.8

-

-

-

-

-

110

110

-

-

-

-

-

9.0

9.0

9.0

4.5

-

-

-

105

105

105

105

-

-

-

3.2

9.8

16.1

20.7

23.4

19.7

7.5

120

120

120

120

120

120

120

23,637

31,665

34,466

26,857

16,862

13,002

7,645

35.1

30.5

25.6

21.3

18.3

16.0

13.3

11.1

13.3

13.3

9.7

6.3

5.7

8.7

1069

1585

1968

2222

2349

2429

2503

11,908

21,157

26,183

21,461

14,841

13,900

21,861

13,602

16,082

14,873

9,636

5,284

3,590

1,759

135,525 76,835

F = 135,52576,835= 1.76

Figure 7.11 Manual calculations by the Ordinary Method of Slices for short-term stabilityof the slope at Bradwell.

rate with depth. A representative unit weight for theLondon Clay at the site is 120 pcf.

Stability computations were rst performed for thisexample using a computer program and several pro-cedures of slices. The resulting factors of safety aresummarized in Table 7.4. The values for the factor ofsafety are as expected: Spencers procedure and theSimplied Bishop procedure give identical values be-cause they both satisfy moment equilibrium; there isonly one value for the factor of safety that will satisfymoment equilibrium for a circular slip surface. TheCorps of Engineers Modied Swedish procedure, aforce equilibrium procedure, overestimates the factorof safety compared to procedures that satisfy completeequilibrium, as is commonly the case. The SimpliedJanbu procedure (force equilibrium with horizontal in-terslice forces) without Janbu et al.s (1956) correctionfactor underestimates the factor of safety, as is alsotypically the case. The correction factor, 0, for theSimplied Janbu procedure was calculated from thefollowing equation presented by Abramson et al.,(2002):

2d d 1 b 1.4 (7.32) 0 1 L Lwhere b1 is a factor that depends on the soil type (cand ) and d /L represents the slide depth-to-length

ratio. For 0, b1 is 0.69 and the depth-to-lengthratio for the critical circle found for the SimpliedJanbu procedure is 0.13. The resulting correction factorcalculated from Eq. (7.32) is 1.07 and the correctedfactor of safety is 1.74 ( 1.07 1.63). This correctedvalue (1.74) for the factor of safety by the SimpliedJanbu procedure agrees well with the value (1.76) cal-culated by procedures that satisfy moment equilibrium.

The factor of safety was also calculated manuallyusing a spreadsheet program based on the OrdinaryMethod of Slices. Because is zero for this problemand the Ordinary Method of Slices satises momentequilibrium, the Ordinary Method of Slices should givethe same value for the factor of safety as Spencersand the Simplied Bishop procedures. There is no needto use a more complex procedure than the OrdinaryMethod of Slices for this case. The calculations for theOrdinary Method of Slices are shown in Figure 7.11.As expected, the factor of safety is 1.76, which is thesame as the value shown previously for Spencers andthe Simplied Bishop procedures.

Although the factor of safety calculated for thisslope is almost 1.8, the slope failed approximately 5days after excavation was completed. Skempton andLaRochelle (1965) discuss the probable causes of fail-ure. These include overestimates of the shear strengthdue to testing of samples of small size, strength lossesdue to sustained loading (creep), and the presence ofssures. Skempton and LaRochelle concluded that the

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100 ft

50 ft

100 ft

21

21

Sand:c = 0 = 40 = 140 pcf

Saturated clay: su = 2500 ( = 0) = 140 pcf

r = 174 ftCritical circle -Spencersprocedure

Figure 7.12 Cohesionless ll slope on saturated clay foundation.

Table 7.5 Summary of Slope Stability Analyses fora Cohesionless Embankment Supported by aSaturated Clay Foundation


safety

Spencer 1.19Simplied Bishop 1.22Corps of Engineers Modied Swedish 1.54Simplied Janbuno correction 1.07Simplied Janbuwith correction a0 1.16

aCorrection based on Eq. (7.32) with b1 0.5 andd /L 0.34; 0 1.09.

opening of ssures and a lower, residual strength alongthe ssures were probable causes of failure of theslope, even though the factor of safety computed basedon undrained shear strengths was relatively high.

Recapitulation

Spencers, the Simplied Bishop, and OrdinaryMethod of Slices procedures all gave the samevalue for the factor of safety for circular slip sur-faces because 0, and all these proceduressatisfy moment equilibrium.

The computer solution and the manual solutionusing a spreadsheet gave the same value for thefactor of safety.

The Corps of Engineers Modied Swedish pro-cedure overestimated the factor of safety for thiscase by a small amount (2%).

The Simplied Janbu procedure without the cor-rection factor applied underestimated the factor ofsafety by about 7%.

The corrected factor of safety by the SimpliedJanbu procedure agrees within 1% with the valueof the factor of safety calculated using methodsthat satisfy moment equilibrium.

Although the factor of safety for short-term sta-bility was much greater than 1, the slope failedapproximately ve days after construction, due toseveral factors that inuenced the shear strength.

Example 4: Cohesionless Slope on Saturated ClayFoundationThe fourth example is for a hypothetical embankmentconstructed of cohesionless granular material restingon a saturated clay ( 0) foundation, as shown inFigure 7.12. The embankment is assumed to drain al-most instantaneously, and thus its strength will notchange over time. The clay in the foundation is ex-pected to consolidate with time and its strength is ex-

pected to increase with time. Therefore, the critical pe-riod (lowest factor of safety) for the embankmentshould be immediately after construction.

Soil shear strength and unit weight properties areshown in Figure 7.12. Drained (effective stress) shearstrength parameters are shown for the embankment,and undrained shear strengths are shown for the clayfoundation. Stability computations were rst per-formed using a computer program and several proce-dures of slices. The minimum factors of safety forvarious procedures are summarized in Table 7.5, andthe critical slip surface by Spencers procedure isshown in Figure 7.12.

As expected, the Simplied Bishop procedure givesa value for the factor of safety that is very close to theone calculated by Spencers procedure. The SimpliedJanbu procedure without the correction factor appliedgives a factor of safety that is approximately 10%lower, but the corrected value (1.16) agrees closelywith the values by the Simplied Bishop and Spencersprocedures. The Corps of Engineers Modied Swed-ish procedure produced a factor of safety about 25%higher than the value by Spencers procedure. Themuch higher value clearly demonstrates the potentially

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unconservative nature of the Modied Swedish forceequilibrium procedure.

The factor of safety was also computed using theOrdinary Method of Slices with a spreadsheet programfor the critical circle found by Spencers procedure.The computations are shown in Figure 7.13. The com-puted factor of safety is 1.08, approximately 10% lessthan the value calculated using Spencers procedure.Differences of this order (10%) are typical for caseslike this one where c and vary signicantly alongthe slip surface.

As an additional, approximate check on the stabilityof the embankment, the bearing capacity equation [Eq.(7.10)] was used to calculate a factor of safety. Thisgave

2500F 5.53 0.99 (7.33)(140)(100)

Although the bearing capacity solution represented byEq. (7.33) underestimates the stability of the embank-ment in this example, it provides a simple and con-venient way of preliminary screening for potentialproblems. In general, if the factor of safety for bearingcapacity is near or below 1, the factor of safety is likelyto be marginal and additional, more detailed analysesare probably warranted.

Recapitulation

Spencers procedure and the Simplied Bishopprocedure give very similar values for the factorof safety.

The Corps of Engineers Modied Swedish pro-cedure can substantially overestimate the factor ofsafety.

The Simplied Janbu procedure without the cor-rection factor applied underestimated the factor ofsafety, but the value is improved by applying thecorrection.

The Ordinary Method of Slices underestimates thefactor of safety but provides a convenient way ofchecking a computer solution using more accuratemethods.

The simple equation for bearing capacity on a sat-urated clay foundation gives a conservative esti-mate of stability but provides a useful tool forscreening for stability problems.

Example 5: Cohesionless Embankment(Oroville Dam)Curved MohrCoulomb EnvelopeThe next example is of the Oroville Dam, in particular,the stability of the downstream slope (Figure 7.14).

The downstream slope is composed of rockll (am-phibolite gravel). As for most granular materials, theMohr failure envelope is curved. Due to the greatheight of the Oroville Dam (778 ft) and the large var-iation in the pressures from the top to the bottom ofthe embankment, the curved Mohr failure envelope re-quires special consideration for the slope stability com-putations.

Curved (nonlinear) Mohr failure envelope. For thisexample the shear strength of the downstream shellmaterial is characterized by a secant friction angle(i.e., tan / ), which represents the slope of aline drawn from the origin of the Mohr diagram to apoint on the Mohr failure envelope. As discussed inChapter 5, the secant friction angle varies with conn-ing pressure and can be related to the minor principalstress, by3

3 log (7.34)0 10 pa

where 0 is the friction angle for a conning pressure( ) of 1 atm, is the change in friction angle per3log-cycle (10-fold) change in conning pressure, andpa is atmospheric pressure. Duncan et al. (1989) sum-marize shear strength data for the Oroville dam andreport values of 0 51 and 6 for the shellmaterial.

For slope stability computations the shear strengthneeds to be dened by a Mohr failure envelope thatexpresses the shear strength, , as a function of thenormal stress, or , depending on whether total oreffective stress analyses are being performed. The nor-mal stress, , should be the normal stress on the fail-ure plane at failure, . The relationship between

and conning pressure, , depends on the shear 3strength parameters. For a cohesionless soil (c 0)the relationship is expressed as

2cos (7.35) 3 1 sin

Ratios of to are tabulated in Table 7.6 for a 3range in friction angles that are representative for gran-ular materials. The ratios shown in Table 7.6 vary withfriction angle; however, it is convenient to assume thatthe ratio, is constant and equal to 1.5 for any / / 3value of . This assumption has very little effect onthe Mohr failure envelope that is subsequently com-puted. To illustrate this, consider a material with avalue of 0 of 40 and of 10. Computation of thefriction angles, , for values of effective normalstress, ( ), of 100, 1000, and 10,000 psf areshown in Table 7.7. For each value of the effective

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Summation:

Slice

No.

hfill(ft)

b(ft)

fill(pcf)

W(lb)

c(psf)

(deg) W cos

(W co

s) t

an'

c' W sinhclay(ft)clay(pcf)

1

2

3

4

5

6

7

8

9

10

10.7 19.2 140 0.0 125

597628

19.0 56.0 0.0

28706

22.5 86.8 0.0

148726

22.7 100.0 9.3

273040

35.2 91.2 28.0

343995

38.5 72.8 42.6

572546

27.3 56.3 49.0

30754

39.7 39.6 47.8

37.7 20.3 38.9

21.7 5.4 26.1

40 74.5 0 40

32

61.6 0

35 49.6 0

29 39.3 2500

40 28.1

40 15.3

27 4.4

40 -6.4

40 -19.3

25 -29.8

7666 0 276646432

99796

130872

68427

207956

99781

217869

99789

269954

62405

157348

79325

29322

-51247

-95926

11 24.7 0.0

140

140

140

140

140

140

140

140

140

140 10.0

125

125

125

125

125

125

125

125

125

125

382166

456562

290072

87202

40

-38.9

2500

2500

2500

2500

2500

2500

2500

40

40

0

0

0

0

0

0

0

0

70653

176932

266206

504909

576542

381040

453677

273752

75702

23931

59285

148464

0

0

0

0

0

0

0

0

0

0

73301

99788

-43283

-19317

682612214181 831211

F = [(Wcos)tan + c]Wsin = =214181 + 682612

8312111.08

Figure 7.13 Manual calculations for stability of cohesionless ll slope on saturated clayfoundation using the Ordinary Method of Slices and the critical circular slip surface foundusing Spencers procedure.

2.75:1

2.6:12.2:1

2.0:11.1

:1

0.9:1

0.5:

10.

2:1

Crest El. 922900

700

500

300

100

36003200280024002000160012008004000Distance (ft)

Elev

atio

n (ft)

1100

El. 900

Figure 7.14 Cross section of Oroville Dam.

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Table 7.6 Relationship Between the Ratio / 3and the Friction Angle

(deg)

3

30 1.5040 1.6450 1.77

Table 7.7 Computed Secant Friction Angles for Different Conning Pressures and Various Assumed Values forthe Ratio / 3

(psf)

1.53

3(psf)

(deg)

1.653

3(psf)

(deg)

1.83

3(psf)

(deg)

100 67 55.0 61 55.4 56 55.81,000 667 45.0 606 45.4 556 45.8

10,000 6,667 35.0 6,061 35.4 5556 35.8

Table 7.8 Points Calculated to Dene the NonlinearMohr Failure Envelope for the Oroville Dam ShellMaterial

3

150 100 59.0 250300 200 57.1 465600 400 55.3 870

1,200 800 53.5 1,6252,400 1,600 51.7 3,0404,800 3,200 49.9 5,7059,600 6,400 48.1 10,705

19,200 12,800 46.3 20,10038,400 25,600 44.5 37,74076,800 51,200 42.7 70,865

153,600 102,400 40.9 133,015

normal stress, the conning pressure, , was com-3puted assuming values of 1.5, 1.65, and 1.8 for theratio . The resulting conning pressures were / 3then used to compute the friction angles from Eq.(7.34). The assumed value for the ratio, , can / 3be seen to have very little effect on the friction anglecomputed for a given conning pressure. The maxi-mum difference between the friction angles computedassuming 1.5 and 1.8 for the ratio, , was only / 30.8. The shear stress, , dening the Mohr failureenvelope for a given normal stress, , is computedby multiplying the normal stress by the tangent of thefriction angle (i.e., tan ). A 0.8 differencein the friction angle corresponds to a difference of nomore than 3% in the shear stress. If it is assumed thatthe ratio, , is 1.5 when the value may actually / 3be somewhat higher, the resulting value for the frictionangle will be estimated slightly conservatively.

Based on the preceding discussion, a nonlinearMohr failure envelope was determined for the OrovilleDam shell material using values of 0 51 and 6, and assuming that 1.5. The en- / 3velope was dened by a series of discrete points com-puted using the following steps:

1. A range in values of normal stress wasestablished to encompass the maximum range ex-pected for Oroville Dam. The minimum normal

stress of interest was considered to be the normalstress at a depth of 1 ft, which for a total unitweight of 150 pcf was taken to be 150 psf. Themaximum normal stress was estimated based onthe height of the dam. For a height of 770 ft anda unit weight of 150 pcf, the maximum stress isapproximately 115,000 psf.

2. Specic values of normal stress, , ranging from150 psf to the maximum were selected for com-puting points dening the MohrCoulomb failureenvelope. Beginning with the minimum stress of150 psf, a geometric progression of values wasused (e.g., 150, 300, 600, 1200 psf). Particularattention was paid to selecting points at lowstresses because it was anticipated that the criticalslip surface would be relatively shallow due tothe cohesionless nature of the Oroville Dam shellmaterial.

3. For each value of normal stress the correspondingvalue of the conning pressure was computed as

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Figure 7.15 Critical circular slip surface for downstreamslope of Oroville Dam.

Fill: = 30, = 20 kN/m3Clay "crust": Su = 41 kN/m2, = 0, = 20 kN/m3

Marine clay: Su = 34.5 kN/m2, = 0, = 18.8 kN/m3

Lacustrine clay: Su = 31.2 kN/m2, = 0, = 20.3 kN/m3

Till (very strong)

12 ft

4 ft

8 ft

6.5 ft

6 ft3

13

1

56.3

6 ft

Figure 7.16 Cross section of James Bay dike.

(7.36)3 1.5

4. Secant values of the friction angle were com-puted for each value of from Eq. (7.34) as3

3 51 6 log (7.37)10 pa

where pa (atmospheric pressure) is 2116 psf forunits of pounds and feet that were used.

5. Shear stresses, , were calculated for each valueof normal stress from

tan (7.38)

The values calculated for the Oroville Dam shellmaterial are summarized in Table 7.8. The pairs of val-ues of and were used as points dening a nonlin-ear Mohr failure envelope for the slope stability

computations. Although not shown in this table, an ad-ditional point representing the origin ( 0, 0)was included in the data dening the envelope for theslope stability computations. Nonlinear Mohr failureenvelopes were also dened for the transition zone andthe core of Oroville Dam; however, because the criticalslip surface did not pass signicantly through thesezones, the shear strength data are not included here.

Slope stability computations. Slope stability com-putations were performed using the computer programand the nonlinear Mohr failure envelopes discussedearlier. Computations were performed using Spencersprocedure and circular slip surfaces. The critical slipsurface is shown in Figure 7.15, and the minimum fac-tor of safety is 2.28.

One way of checking the computer solution is tocalculate the factor of safety manually using a proce-dure of slices such as the Ordinary Method of Slicesor Simplied Bishop procedure. The friction anglecould be varied for each slice depending on the normalstress, . This is easiest to do with the OrdinaryMethod of Slices because the normal stress can be cal-culated independently of the shear strength using thefollowing equation

2W cos u l cos (7.39)

l

With the Simplied Bishop procedure, the normalstress depends on the friction angle (i.e., the normalstress is part of the solution for the unknowns). There-fore, the normal stress must rst be estimated to com-pute the friction angle, and then trial and error is useduntil the estimated and calculated values are in reason-able agreement. To estimate the friction angle initiallyfor the Simplied Bishop procedure, either the normalstress can be estimated from the vertical overburdenpressure or the normal stress can be calculated fromEq. (7.39) from the Ordinary Method of Slices.

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Fill

Clay "crust"

Marine clay

Lacustrine clay

Till (very strong)

Critical circle, F = 1.45

Critical noncircular slip surface, F = 1.17

Figure 7.17 Critical circular and noncircular slip surfaces for James Bay dike.

Because the critical slip surface for the downstreamslope was relatively shallow for this case, the inniteslope procedure was used to check the results of thecomputer solution. To do this the average normal stresswas calculated for the critical slip surface found fromthe computer solution. The average normal stress wascalculated using the equation

l i i (7.40)av l i

where the summations were performed for all slices.The average normal stress calculated for the criticalslip surface was 12,375 psf. From the nonlinear Mohrfailure envelope in Table 7.8, the corresponding shearstress, , is 13,421 psf and the equivalent secant fric-tion angle is 47.3. The factor of safety based on aninnite slope is then

tan tan 47.3F 2.17 (7.41)

tan 1/2.0

This value (2.17) from the innite slope analysis iswithin 5% of the value of 2.28 obtained from the com-puter solution with circular slip surfaces.

Recapitulation

When the friction angle depends on the conningstress, the friction angle is expressed convenientlyby a secant angle, which is a function of conningpressure, 3. This requires additional steps to de-termine an equivalent nonlinear Mohr failure en-velope for slope stability analyses.

To relate conning pressure, 3, to normal stress,, for a nonlinear Mohr failure envelope, the con-ning pressure can be assumed to be equal to two-thirds ( 1/1.5) the normal stress. This facilitatesdening points on the Mohr failure envelopewhen the friction angle is dened in terms of con-ning pressure, 3.

For shallow slides in cohesionless soils, stabilitycomputations can be checked with an inniteslope analysis, even when the Mohr failure en-velope is nonlinear. When the Mohr failureenvelope is nonlinear, the average normal stresson the slip surface from a computer solutioncan be used to dene an equivalent secant fric-tion angle that is then used in the innite slopeanalysis.

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Example 6: James Bay DikeThe James Bay hydroelectric project involved the de-sign of dikes that were to be constructed on soft andsensitive clays (Christian et al., 1994). A typical crosssection of one of the planned dikes is shown in Figure7.16. Soil properties for the materials in the dike andits foundation are summarized in this gure.

An analysis was rst performed using circular slipsurfaces, with a computer program and Spencers pro-cedure. The minimum factor of safety was calculatedto be 1.45 for the critical circle, which is shown inFigure 7.17. This value of 1.45 for the factor of safetyis the same as the value (1.453) that Christian et al.reported for the slope.

Additional analyses were performed using noncir-cular slip surfaces and an automatic search. The au-tomatic search was started with the critical circle fromthe previous analyses. Ten points were used to denethe slip surface. These points were shifted systemati-cally using the search routine implemented in the com-puter program until a minimum factor of safety wasfound. The corresponding critical noncircular slip sur-face was then adjusted by adding some points and re-moving others. Points were adjusted so that there wasa point located at the interfaces between soil layers,and these points were shifted in the horizontal direc-tion until the minimum factor of safety was againfound. The most critical noncircular slip surface foundafter searching is shown in Figure 7.17. The corre-sponding minimum factor of safety is 1.17, which isapproximately 20% less than the minimum value com-puted using circles.

Christian et al. (1994) discussed the effects of vari-ations and uncertainties in shear strength on the com-puted factors of safety for the James Bay dikes. Theyshowed that the variation in shear strength could havean important effect on the evaluation of stability. Theresults presented in the preceding paragraph show thatthe effect of using noncircular slip surfaces is of com-parable magnitude, thus illustrating the importance oflocating the critical slip surface accurately.

To verify the computations with noncircular slip sur-faces, additional computations were performed using aforce equilibrium procedure and a computer spread-sheet program. For these computations the intersliceforces were assumed to be parallel; the interslice forceinclination was assumed to be the same as the incli-nation (2.7 degrees) determined from the computer so-lution with Spencers procedure. The spreadsheetcomputations are presented in Figure 7.18. The com-puted factor of safety was 1.17, thus verifying thevalue calculated using the computer program.

Recapitulation

Noncircular slip surfaces may give signicantlylower factors of safety than circles.

The critical circle provides a good starting pointfor searching for the critical noncircular slip sur-face.

A force equilibrium solution using a spreadsheetwith the interslice force inclination from Spen-cers procedure provides a good method forchecking a computer solution with Spencers pro-cedure and is applicable to circular and noncir-cular slip surfaces.

Example 7: Homogeneous Earth Dam withSteady-State SeepageA homogeneous earth embankment resting on a rela-tively impervious foundation is illustrated in Figure7.19. The embankment impounds water on one side,and steady-state seepage is assumed to have developed.Stability computations were performed for this em-bankment to evaluate the long-term stability of thedownstream slope. Drained effective stress shearstrength parameters were used and are shown on thecross section of the embankment.

Pore water pressures. Finite element seepage anal-ysis was performed for the embankment to calculatepore water pressures. The GMS/SEEP2D software wasused for this purpose (Tracy, 1991; EMRL, 2001). Theentire cross section of the embankment was modeledwith nite elements, and appropriate saturated or un-saturated hydraulic conductivities were assigned de-pending on the pore water pressure. The hydraulicconductivity for the saturated soil (positive pressures)was 1 105 ft /min. The hydraulic conductivity forthe unsaturated soil was assumed to decrease sharplyto a residual hydraulic conductivity equal to 0.1% ofthe value for saturated conditions as the water pres-sures decreased below atmospheric. This essentially re-stricted almost all ow to the saturated (positive waterpressure) zone of the cross section. The nite elementmesh used for this problem contained 589 node pointsand 1044 elements.

Finite element seepage analyses produced values ofpore water pressure at each node point. These valuesof pore water pressure were then used to interpolatevalues of pore water pressure along each slip surfacein the slope stability computations. Interpolation wasperformed using the triangle-based interpolationscheme described by Wright (2002). This scheme isvery efcient and introduces negligible error caused by

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1234567

89

1011

7.7

4.710.211.212.128.228.211.9

9.99.34.2

Slice

No.

b h1 1

6.012.012.011.5

8.06.06.04.00.6

-

-

2020202020202020202020

h2 2

-

2.04.04.04.04.04.04.04.04.02.0

2020202020202020202020

h3 3 h4 4

- 18.8-

4.08.08.08.08.08.08.04.0

-

18.818.818.818.818.818.818.818.818.818.8

-

-

-

3.26.56.5

6.56.5

3.2-

-

20.320.320.320.320.320.320.320.320.320.320.3

W

926

13572

1319401958686333

13572

527030641446168

1234567

89

1011

Slice

No.

W

926

13572

1319401958686333

13572

527030641446

168

57.240.338.230.2

0.0

-40.6

0.00.00.0

-33.1

-43.7

14.36.2

12.912.912.128.228.211.911.912.3

5.8

c

0.041.034.531.531.531.531.531.531.534.5

41

3000000000000

u

000000000000

W s

in

779854

24852952

0000

-1675-942-116

- c

- (W

cos

- u

)ta

n'

-289-253-446-406-382-887-887-375-374-424-238

n

1.0500.7910.8140.8870.9990.9990.9990.9990.8110.7280.690

Zi+1

466122537306601

5331445340681541

6218

-336-847

n

0.972

Zi+1

554136639626909

5851511147982348

6591

569114

n

0.916

Zi+1

625147541377139

6232559753292934

6866

1224811

F1 = 1.0 F2 = 1.2 F3 = 1.4

0.7910.8140.8870.9990.9990.9990.9990.8110.7280.690

0.7910.8140.8870.9990.9990.9990.9990.8110.7280.690

Trial Factor of Safety1.0 1.2 1.4

Forc

e Im

bala

nce,

Zi+

1

-1000

1000

0

F = 1.17

Computation of slice weights

Computation of factor of safety

Zi+1 = Zi +W sin c + (W cos c) tan '

F

F

n

n = cos( ) + sin( )tan '

Figure 7.18 Manual calculations for James Bay dike using force equilibrium procedure.

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15 ft

40 ft48 ft2.5

2.5

11ksat = 10-5 ft./min; kunsat 0.001 ksat

c' = 100 psf, ' = 30, = 100 pcf

Figure 7.19 Cross section for homogeneous embankment with steady-state seepage.

Figure 7.20 Zero-pressure line (contour) used as phreatic surface and piezometric line forhomogeneous embankment.

Table 7.9 Summary of Factors of Safety from SlopeStability Computations for HomogeneousEmbankment Subjected to Steady-State Seepage(Spencers Procedure and Circular Clip Surfaces)


safety

Finite element seepage analysispore water pressures interpolated

1.19

Piezometric line 1.16Phreatic surface 1.24

interpolation. In fact, many fewer nodes and elementscould probably have been used for the nite elementseepage analysis with no loss in accuracy in the com-puted factor of safety (Wright, 2002).

The nite element seepage analysis was also usedto determine a position for a phreatic surface and apiezometric line (Figure 7.20). The phreatic surfaceand piezometric line were assumed to be the same asthe line of zero pore water pressure determined fromcontours of pore water pressure obtained from the -nite element seepage analysis. Experience with a num-ber of nite element seepage analyses where bothsaturated and unsaturated ow has been modeled hasshown that the contour of zero pore water pressurecorresponds very closely to the classical line of seep-age described by Casagrande (1937) for saturated ow.

For the slope stability computations, pore waterpressures were calculated from the phreatic surface andpiezometric line using the procedures discussed inChapter 6. The pore water pressures calculated fromthe phreatic surface were always as small as or smallerthan those based on the piezometric line and were usu-ally slightly larger than those based on the actual niteelement seepage analysis.

Although the pore water pressures calculated in thenite element analyses were negative in the uppermostpart of the ow region, above the piezometric line andphreatic surface, negative pore water pressures wereneglected in all of the slope stability calculations. Neg-ative pore water pressures probably would exist and

would contribute slightly to stability, but their effectwould be small for this problem and it seems reason-able to neglect them. Only when negative pore waterpressures can be sustained throughout the life of theslope should they be counted on for stability. Sustain-able negative pore water pressures seem unlikely formost slopes.

Stability analyses. Slope stability calculations wererst performed using a computer program and each ofthe three representations of pore water pressure dis-cussed above. Spencers procedure was used for all ofthe calculations. In each case an automatic search wasconducted to locate the most critical circle for eachrepresentation of pore water pressures. The minimumfactors of safety are summarized in Table 7.9. All three

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(1) hpiezometric = depth below piezometric line to slip surface.(2) u = w x hpiezometric

Summation:

Slice

No.

hsoil(ft)

b(ft)

soil(pcf)

W(lb)

(deg)

c'(psf)

(deg)

h pie

zom

etric

(1)

(W co

s

u

co

s2

)tan

c' W sin

1

2

3

4

5

6

7

8

9

7.3 2.6 125

11.1 7.6

2392

7.3 10.9

7.5 12.1

5.1 12.5

7.3 12.2

11.6 10.8

11.5 7.7

12.3 2.9

11.0 48.1 100 30

39.8

8.6 32.2

8.4 26.7

5.5 22.1

7.7 17.8

11.9 11.4

11.5 3.8

12.4 -4.0

0 17801096

14.4

0.0

1443

858

843

545

767

1186

1153

1235

912625087 28832

125

125

125

125

125

125

125

125

10580

9866

11383

7886

11153

15671

11098

4466

100

100

100

100

100

100

100

100

30

30

30

30

30

30

30

30

12.1

3.4

10.8

8.2

7.7

10.6

2.9

11.9

u(2)(psf)

213

515

663

745

756

673

482

181

u

co

s2

0

1818

3160

4462

3488

5253

7669

5529

2225

W c

os

1597

8128

8345

10174

7307

10617

15360

11073

4455

W c

os

- u

co

s2

1597

6310

5185

5711

3819

5364

7691

5545

2230

922

3643

2994

3297

2205

3097

4440

3201

1287

6772

5264

5107

2966

3418

3106

732

-315

[(W cos ucos2) tan c]W sin

= =F = 25087 + 912628832 1.19

Figure 7.21 Manual calculations for stability of embankment with steady-state seepage us-ing the Ordinary Method of Slices (the preferred method of representing pore water pres-sures).

representations of pore water pressure produced simi-lar values for the factor of safety, with the differencesbetween the rigorous nite element solution and theapproximate representations being 4% or less. As ex-pected, the piezometric line produced lower factors ofsafety than the phreatic surface, although the differ-ences were small. The phreatic surface did, however,result in a higher factor of safety than the rigorousnite element solution, and thus this approximationerrs on the unsafe side for these conditions.

A manual solution using a computer spreadsheetprogram and the Ordinary Method of Slices was per-formed as verication of the computer solutions. Cal-culations were performed using both the preferred andalternative methods of handling pore water pressuresin the Ordinary Method of Slices that were discussedin Chapter 6. For both sets of calculations the porewater pressures were calculated using the piezometricline.

The calculations for the Ordinary Method of Slicesare presented in Figures 7.21 and 7.22 for the preferredand alternative methods, respectively. Using the pre-

ferred method of handling pore water pressures, thefactor of safety was 1.19 (Figure 7.21). This value isidentical to the value calculated using the nite ele-ment seepage solution and slightly higher than thevalue with the piezometric line based on the computersolutions with Spencers procedure. The other, alter-native method of handling pore water pressures in theOrdinary Method of Slices resulted in a lower factorof safety of 1.08, which is approximately 10% lowerthan the other solutions. This is consistent with whatwas shown in Chapter 6 for the Ordinary Method ofSlices and illustrates why the form of the method il-lustrated in Figure 7.21 is preferred.

The nal set of calculations was performed usingthe charts in the Appendix and are summarized in Fig-ure 7.23. The factor of safety from the chart solutionis 1.08. This value is slightly lower than the valuesfrom the more accurate computer solutions. Theslightly lower value from the chart solution probablyreects use of the Ordinary Method of Slices to de-velop the charts as well as other approximations thatare made regarding seepage and pore water pressures.

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(1) hpiezometric = depth below piezometric line to slip surface.(2) u = w x hpiezometric

Summation:

Slice

No.

hsoil(ft)

b(ft)

soil(pcf)

W(lb)

(deg)

c'(psf)

(deg)

h pie

zom

etric

(1)

(W co

s

u

) tan

'

c' W sin

1

2

3

4

5

6

7

8

9

7.3 2.6 125

11.1 7.6

2392

7.3 10.9

7.5 12.1

5.1 12.5

7.3 12.2

11.6 10.8

11.5 7.7

12.3 2.9

11.0 48.1 100 30

39.8

8.6 32.2

8.4 26.7

5.5 22.1

7.7 17.8

11.9 11.4

11.5 3.8

12.4 -4.0

0 17801096

14.4

0.0

1443

858

843

545

767

1186

1153

1235

912622136 28832

125

125

125

125

125

125

125

125

10580

9866

11383

7886

11153

15671

11098

4466

100

100

100

100

100

100

100

100

30

30

30

30

30

30

30

30

12.1

91631_07

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