Kitch Wright and Gilbert (1995)

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Probabilistic Analysis of Reinforced Soil SlopesWilliam A. Kitch l , Stephen G. Wright2 and Robert B. Gilbert3

Abstract Submitted to:10th ASCE Engineering Mechanics Specialty ConferenceProfessor Stein StureDepartment of Civil, Environmental andArchitectural EngineeringUniversity of ColoradoBoulder, CO 80309-0428Tel: (303) 492-7651Fax: (303) 492-7317

12 May 1994

Pub, May 1995, 10th ASCE EMC, Boulder CO, Vol 1. pp 325-328


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Abs t rac t

Geosynthetic reinforcement IS being used increasingly to improve thestability and performance of soil slopes and embankments. Thecurrent methods of design for such reinforcement are deterministic,based on the same limit equilibrium procedures that are used fordesign of conventional, unreinforced slopes. Stability is evaluatedbased on a factor of safety that is typically defined with respect tosoil shear strength. The most critical potential failure surface withthe minimum factor of safety is found by trial and error. For typicalreinforced slope designs, the most critical potential failure surface islocated primarily outside of the reinforced zone; thus, the factor ofsafety essentially reflects the potential for failure in the unreinforcedsoil. In reality there are two potential modes of failure. Oneprimarily through the reinforced soil mass, the other primarilythrough the unreinforced soil mass. The conventional factor of safetydoes not reflect the likelihood of failure by each mechanism. In thispaper, we present a design approach for reinforced slopes that is


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Probabilistic Analysis of Reinforced Soil SlopesWilliamA. Kitch\ Stephen G. Wrighe, and Robert B. Gilbert3, Members ASCE

AbstractThis paper presents an analysis of a geosynthetically reinforced slope that is based onstructural reliability theory. This probabilistic analysis is compared to currentdeterministic analysis methods. Current deterministic methods use limit equilibriumtechniques to locate the failure mode with the minimum factor of safety. The analysispresented considers many possible failure modes. I t is shown that the failure modewith the highest probability of failure does not necessarily correspond to the failuremode with the minimum factor of safety.IntroductionCurrent design methods for geosynthetically reinforced slopes are deterministic, basedon the same limit equilibrium procedures that are used for design of conventional,unreinforced slopes. Stability is evaluated based on a minimum factor of safety alonga potential failure surface. There are numerous potential failure surfaces in reinforcedslopes, some lying within the reinforced zone of the slope and some lying outside of


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Deterministic AnalysisTraditional limit equilibrium analysis techniques, adapted for reinforced slopes, wereused in the deterministic analysis. The stability method used was Spencer's procedureof slices, including reinforcement forces following the technique developed by Wrightand Cuenca (1986). The allowable reinforcement forces were input to the stabilitycomputation and the traditional factor of safety was then computed. The factor ofsafety was defined as the ratio of the available soil shear strength to the applied shearstress. In this case, for cohesionless soil, the factor of safety, F, wasF = tan (1)

' fwhere r is the applied shear stress and is the angle of internal friction of the soil.The analytical procedure, then, is to search for the potential shear surface that has theminimum factor of safety as defined in Equation 1. The computer programUTEXAS3 (Wright 1991) was used to compute the factor of safety and to aid in thesearch for the shear surface with the minimum factor of safety. Shear surfaces wererestricted to circular arcs. An extensive search was performed to locate the circlewith the minimum factor of safety, and included circles lying both within and outsideof the reinforced zone of the slope.Probabilistic AnalysisThe probabilistic analysis used the first-order reliability method. The limit statefunction, g, was defined as

g=F-1 (2)where F is the factor of safety, as defined in Equation 1. Two randomvariables wereconsidered: the soil shear strength, tan, and the allowable force in the primaryreinforcement, R. Both were considered to be normally distributed. The mean value


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(1.51) was found to be predominately outside the reinforced zone of the slope (Figure1). For circles contained within the reinforced zone of the slope, the minimum factorof safety was 1.85.Using the first-order reliability method, the reliability index, f3, was computed fordifferent potential failure modes (all circular). I t was found that the most probablefailure mode was always one of two critical modes: one consisted of a circle lyingcompletelywithin the reinforced zone of the slope and the second consisted of a circlelying predominately outside the reinforced zone. The locations of these two criticalfailure modes corresponded essentially to the two circles with the minimum factors ofsafety shown in Figure 1.The reliability index for the two critical failure modes was computed for coefficientsof variation of the shear strength of 0.1 and 0.2, and coefficients of variation of theallowable reinforcement force ranging from 0.05 to 0.5 (Kitch 1994). In Figure 2 thereliability indexes for these two critical failure modes are compared as a function ofthe coefficients of variation of shear strength and reinforcement force. It is apparentthat either failure mode may be the least reliable (have the lowest reliability index)depending on the magnitude of the coefficients of variation. In contrast, thedeterministic analysis indicated that the failure mode passing outside of the reinforcedzone of the slope is the most critical.ConclusionBoth the probabilistic and deterministic analyses show there are potential failuremodes within the reinforced zone of the slope and outside of the reinforced zone ofthe slope. The probabilistic analysis clearly shows that the most likely mode of failureis a function of the uncertainties in the reinforcement force and the soil shear strength.Because the deterministic factor of safety does not contain any information about the


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Critical Circle WithinReinforced Zone, F = 1.85

Soil Propertiesy = 18.8 kN/m3=320

1.22mt [email protected]

f----f------ - , . - - - 1.63 m

0.364 f - - r - - - ~ - - -

l O O . I I ~ [email protected] m Global Critical__L Circle, F = 1.51--- 'J

Primary Geogrids: SR2Allowable Load = 32 kN/m, Length = 8.5 m

Secondary Geogrids: SS2Allowable Load =5.6 kN/mVertical Spacing = 0.41 mLength = 1.5 i l l11.6 m

Figure 1: 70 Degree Slope Designed Using Tensar Guidelines, ShowingInternal and External Critical Circles.5-r-----------------------------,

a) Coefficient of Variation ofShear Strength =0.1

32 Location ofFailureMode

Within Reinforced Zone


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Probabilistic Analysis ofReinforced Soil Slopes5th ASCE Engineering Mechanics Conference22 May 1995

William A. KitchStephen G. WrightRobert B. Gilbert

United States Air Force AcademyUniversity of Texas at AustinUniversity of Texas at Austin


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Overview Problem Chosen for Analysis Deterministic Analysis Method Probabilistic Analysis Method Comparison ofDeterministic and ProbabilisticMethods Conclusions

Probabilistic Analysis ofReinforced Soil Slopes Slide 1


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Problem Chosen for Analysis Standard Design Procedures Used as Published

by Tensar Allowable Reinforcement Forces Specified by

Manufacturer Nominal Factor ofSafety Based on Soil ShearStrength was 1.3



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rr

=320

/

1f

---- A1.63 mllowable Load = 5.6 kN/m AVertical Spacing =0.41 m T'2mLength = 1.5 m 0.364 Soil Properties

I 0 370 [email protected] Y=18.8kN/m

Secondary Geogrids:SS2

38'

rr!!!/ 11 @0.41 m/!/ C Primary Geogrids: SR2

Allowable Load = 32 kN/mLength = 8.5 m



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Deterministic Analysis Methods Traditional Limit Equilibrium Methods Used Reinforcement Forces Incorporated as KnowForces Assume Circular Arc Shear Surfaces Search Performed to Find Shear Surface With

Minimum Factor of Safety Based on ShearStrength

F = tanr



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Critical Circle WithinReinforced Zone, F=1.85

Probabilistic Analysis ofReinforced Soil Slopes

Global Critical Circle, F=1.51

Slide 5


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Probabilistic Analysis Method Limit State Function, g, Defined as

g=F- Normally Distributed Random VariablesSoil Shear StrengthReinforcement Force

First-Order Reliability Method Used



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5 ... i

2 -4 Location of Failure Mode Within Reinforced Zone Outside of Reinforced Zone

43

Coefficient of Variation ofShear Strength = 0.1

1o I I i j t j j i i i t t l i I j I I4

"C..53>.

......

Coefficient of Variation ofShear Strength = 0.2

- - III-- - -c 2 _ o _ _,)0::: 10.5.1 0.2 0.3 0.4

Coefficient of Variation of Reinforcement Forceo f j I j I I I I I I I j I lit iii I I I t ro



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Conclusion Potential FailureModes Exist Both Within andOutside ofThe Reinforced Zone of the Slope The Factor of Safety Cannot Identify the Most

Probable Failure Mode The Probabilistic Analysis Clearly Shows the

Most ProbableMode is a Function ofUncertainties in Both Shear Strength andReinforcement Force

Probabilistic Analysis Can Compare DifferentFailureModes on a Rational Basis


Kitch Wright and Gilbert (1995)

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