uncertainty assessment of critical excavation depth of

Scientia Iranica A (2016) 23(3), 864{875

Sharif University of TechnologyScientia Iranica

Transactions A: Civil Engineeringwww.scientiairanica.com

Uncertainty assessment of critical excavation depth ofvertical unsupported cuts in undrained clay usingrandom �eld theorem

R. Jamshidi Chenari� and M. Zamanzadeh

Department of Civil Engineering, Faculty of Engineering, University of Guilan, Rasht, Guilan, Iran.

Received 14 December 2014; received in revised form 22 May 2015; accepted 19 October 2015

KEYWORDSCritical excavationdepth;Probability of failure;Spatial variability;Random �nitedi�erence method;Random limitequilibrium method.

Abstract. Classical methods such as limit equilibrium or limit analysis, dealing withthe stability analysis of open cuts and trenches and calculation of critical excavationdepth for them, do not fully satisfy the theoretical requirements for stability problemsleading to di�erent solutions depending on the adopted method. It is now appreciated thatgeomaterials exhibit considerable heterogeneity, caused by the lithological and inherentvariation, which cannot be fully covered by simple methods. This paper highlightsthe uncertainty embedded in critical excavation depth calculation, arising from spatialvariability of shear strength parameters, using Random Finite Di�erence Method (RFDM)and Random Limit Equilibrium Method (RLEM). In the present study, the lognormallydistributed undrained shear strength is considered spatially correlated throughout thedomain. Surface cohesion value and the shear strength density were introduced as thedeterministic parameters along with the coe�cient of variation of undrained shear strengthand its scale of uctuation as stochastic parameters; these parameters were studiedto see their e�ect on uncertainty in critical excavation depth estimation. The resultsclearly demonstrated the uncertainty in critical excavation depth arising from the inherentvariability of shear strength parameters using the RFDM results; however, RLEM did notprove to re ect such uncertainty e�ciently due to local averaging in prescribed failuresurfaces.© 2016 Sharif University of Technology. All rights reserved.

1. Introduction

Slope and trench stability analysis is a branch ofgeotechnical engineering that is highly adaptable toprobabilistic treatment and has received considerableattention in the literature. Almost all practical prob-abilistic methods have, at some points, been appliedto slope stability problems as a general earth struc-

*. Corresponding author. Tel.: +98 13-33690485;Fax: +98 13-33505866E-mail addresses: Jamshidi [email protected] (R. JamshidiChenari); Masoud [email protected] (M.Zamanzadeh)

ture analysis. A deterministic slope stability anal-ysis method such as the Limit Equilibrium Method(LEM), Limit Analysis Method (LAM), or Finite El-ement/Di�erence Method (FEM/FDM) is needed asthe basis of a probabilistic slope stability analysis. Thechoice of a deterministic slope stability analysis methodalso determines how spatial variability can be applied.

Since four decades ago, many probabilistic meth-ods have been contrived for analysis of the stabilityof earth structures. These methods can be groupedinto four categories: analytical methods, approximatemethods, Monte-Carlo simulation, and �nite elementmethod. In analytical methods, the probability densityfunctions of input variables are expressed mathemati-

R. Jamshidi Chenari and M. Zamanzadeh/Scientia Iranica, Transactions A: Civil Engineering 23 (2016) 864{875 865

cally. They are then integrated analytically into theadopted slope stability analysis model to derive amathematical expression for the density function of thefactor of safety. Limited attempts have been made toutilize analytical methods [1-3]. The jointly distributedrandom variables method belongs to this category.

Most approximate methods are modi�ed versionsof two methods, i.e. First Order Second Moment(FOSM) Method [4] and Point Estimate Method [5].Both methods require knowing the mean and varianceof all input variables as well as the performancefunction de�ning safety factor (e.g., Bishop's equation).Some research on slope stability by the FOSM methodis reported (e.g., [6]). Gri�ths et al. [7] used random�eld theory in probabilistic analysis of in�nite slopesand concluded that the �rst order methods may notproperly account for spatial variability which can leadto unconservative estimates of the probability of slopefailure. Some attempts have also been made to applythe Point Estimate Method [8-9].

A Monte-Carlo simulation is a procedure whichseeks to simulate stochastic processes by random selec-tion of input values to an analysis model in proportionto their joint probability density function. It is apowerful technique that is applicable to both linear andnon-linear problems, but requires a large number ofsimulations to provide a reliable distribution of the re-sponses. Many attempts have been made to analyze thestability of slopes using Monte-Carlo simulation [10-15].

The random �nite element method combineselasto-plastic �nite element analysis with random �eldsgenerated using the local average subdivision method.Several new slope stability analyses have been doneusing this method and the stochastic �nite elementmethod [16-21].

The current study investigates the statistical andprobability issues involved in stability of vertical un-supported cut slopes in undrained condition employingtwo distinctively di�erent reliability analysis methods,namely, Random Limit Equilibrium Method (RLEM)and random �nite di�erence method (RFDM). Al-though necessity of support system for vertical cutscannot be neglected, it should be clari�ed that thisstudy does not consider any retaining system forvertical cuts. The aim of the study is only to showthe e�ect of variability in shear strength parameters onthe critical excavation depth estimation of unsupportedvertical cuts.

2. Spatial variability and reliability

The spatial variability of geotechnical pro�les can bequanti�ed by using several statistical parameters, suchas the central trend (or the mean), the coe�cient ofvariation, the correlation length, and the anisotropy

Figure 1. Inherent soil variability along with di�erentcomponents [22].

among others (e.g., [22-23]). For example, the spatialvariation in geotechnical property, �, with depth z canbe decomposed into a trend function t and a uctuatingcomponent w:

�(z) = t(z) + w(z): (1)

Figure 1 schematically represents di�erent componentsof inherent variability. Deterministic trend can beestimated by a reasonable amount of in situ soil data(using for example least-square �t method), wherethe uctuating component can be characterized asa random variable having zero mean and non-zerovariance.

The standard deviation, �(z), normalized by thelocal mean geotechnical property, �(z), obtained fromthe trend function provides a useful dimensionless ratioknown as the Coe�cient of Variation (CoV):

CoV(z) =�(z)�(z)

: (2)

The generic range of CoVs for soil properties is sum-marized in Table 1. More comprehensive discussionon the CoV ranges and other statistical parametersfor geotechnical properties are provided in Phoon andKulhawy [24].

The scale of geotechnical property uctuation isanother important spatial characteristic of the ground.It indicates the distance scales within the materialproperties that show strong spatial correlations. Thescale of uctuations in geotechnical random �elds canbe simulated by using correlation lengths in covariancefunctions. The correlation length, or autocorrelationlength, is the distance at which spatial autocorrelations

Table 1. Representative Coe�cients of Variation (CoV)for geotechnical parameters [27].

Property CoV (%)

Dry unit weight ( d) 2-13Undrained shear strength (Cu) 6-80E�ective friction angle ('0) 7-20Elastic modulus (Es) 15-70

866 R. Jamshidi Chenari and M. Zamanzadeh/Scientia Iranica, Transactions A: Civil Engineering 23 (2016) 864{875

decay by 1=e, which is about 37% (e.g., [25]). The scaleof uctuations is usually between 1.4 and 2.0 times thecorrelation length for exponential, squared exponential,and spherical autocorrelation functions (e.g., [26]). TheMarkovian spatial correlation function is commonlyused in geotechnical engineering to simulate the soilheterogeneity (e.g., [27]).

3. Random Limit Equilibrium Method(RLEM)

Although most traditional limit equilibrium methodsdo not consider spatial variability, some investigatorshave combined LEM with random �eld theory (e.g.,[28-37]). However, the inherent nature of LEM isthat it leads to a critical failure surface which in 2-Danalysis appears as a straight line or curvilinear shapethat could be noncircular. The in uence of random�eld is only taken into account along a prescribed lineor curvilinear path. Table 2 provides a list of 2-Dslope stability reliability studies found in the literatureutilizing LEM and random �eld theory in some cases.

The objective of this study is to show the e�ectof uncertainty embedded in geotechnical parameters(undrained cohesion in this study) on critical excava-tion depth estimation. It is quite impossible to �nd ananalytical solution for critical excavation depth (Hcr)while considering the log-spiral slip surface, and itdemands fully-numerical methods to render a solution;therefore, a planar slip surface is de�ned in orderto obtain critical excavation depth in a speci�c soilcondition.

Denoting the speci�c weight of the soil body by ,cohesion value at the ground surface by Cu0, and theshear strength rate or density value by �, accordingto the Limit Analysis Method (LAM), the equilibrium

satisfaction leads to a general solution for the factor ofsafety of unsupported vertical cuts adopting a planarslip surface in undrained condition as:

F:S: =�

�CuH( � ��)

�: (3)

� and � values in Eq. (3) vary between 2 to 4 and 1to 2, respectively, depending on the adoption of loweror upper bound schemes. As will be discussed later, �and � values can be precisely approximated by the factthat the value obtained by Eq. (3) should converge tothat obtained by strength reduction method.

Practically, the geotechnical engineers use theplanar or more advanced log-spiral slip surface toinvestigate the factor of safety of a vertical trench andconsequently its critical excavation depth. In LEManalyses, it is not required to consider realizations inthe whole body of the problem and thus the very time-consuming factor of safety calculation process for eachrealization is averted; also, it seems very simple to takeaccount of just the slip surface over which realizationcan be done. To do this, it is �rst needed to identify theslip surface and then the slip surface can be discretizedand divided into elements having correlated cohesionvalues. Two approaches are available here: the planarslip surface and the log-spiral one. In the undrainedcondition for the planar slip surface, slip angle wouldbe �=4 with respect to the vertical plane. As statedin literature [43], the slip surface is not planar; hence,for the undrained condition, the log-spiral slip surfaceformula is also employed, which would be transformedto the circular one for undrained condition.

Before conducting any stochastic analysis, modelgeometry was adopted corresponding to unit safetyfactor for each set of geotechnical parameters. Al-though the retaining structures are mandatory to

Table 2. Slope stability reliability studies combining LEM with random �eld theory in some cases.

Deterministic method Probabilistic method Reference

Level-crossing method First Order Second Moment (FOSM) Catalan and Cornell [38]Bishop First Order Second Moment (FOSM) Alonso [39]Morgenstern-Price First Order Reliability Method (FORM) Li and Lumb [28]Morgenstern-Price First Order Reliability Method (FORM) Mostyn and Soo [29]Bishop Monte-Carlo Simulation (MCS) El-Ramly et al. [32]Spence First Order Reliability Method (FORM) Low [33]Bishop First Order Second Moment (FOSM) Sivakumar et al. [34]LEM Monte-Carlo Simulation (MCS) Cho [36]Spencer First Order Reliability Method (FORM) Low et al. [35]Chen and Morgenstern Monte-Carlo Simulation (MCS) Hong and Roh [6]LEM Importance sampling technique Ching et al. [40]LEM Monte-Carlo Simulation (MCS) Ching et al. [41]LEM Kriging-based response surface method Zhang J. et al. [42]


be utilized in such cases, the aim of this study isonly to show the e�ect of spatial variability of shearstrength parameters on stability of vertical cuts whena small variability probably leads to more probabilityof failure in comparison with the classic deterministiccases. Uncertainty was applied to the undrained shearstrength so as to assess the reliability of calculations byrendering a safety factor for each stochastic realization.For this reason, two approaches were adopted. The�rst approach considers planar slip surface and dividesit into 1000 elements. The second approach considerscircular slip surface and seeks the most critical circularslip surface as traditionally adopted in slope stabilityanalysis literature. In the �rst approach, for the planarslip surface, the plane of slip is divided into 1000elements and an auto-correlated cohesion �eld usingrandom �eld theory is generated to assign the relatedcohesion to each element. In the second approach, amesh grid has been generated while each grid pointrepresents the center of a circular slip surface witha speci�ed radius, as shown in Figure 2. Severalslip surfaces can be found; however, the best of themcorresponds to the minimum deterministic safety factorsought based on a simple optimization scheme found inconventional slope stability analyses. After selection ofthe critical slip surface, as schematically illustrated inFigure 2, 1000 elements can be created along it andrealizations can be performed to set a combination ofauto-correlated cohesion values afterwards.

Using Bishop's formula to calculate the factor ofsafety of a circular slip surface and assuming that eachslice has its own cohesion value obtained by random�eld theory, the safety factor can be calculated by:

Safety factor =Pni=1 CRealized:bi= cos �iPn

i=1 wi: sin �i; (4)

in which bi is the horizontal width of each slice (projectlength of each element on the horizon), �i is theinclination angle for each slice, and wi is the weightof the respective slice. CRealized can be obtained by

Figure 2. Finding the best estimated circular slip surfacebased on Bishop's formulation.

Figure 3. Best �tted curve on the result of RLEM withCu0 = 50 kPa, � = 1:0 kPa/m, CoVCu = 50%, and� = 24 m: (a) Planar slip surface; and (b) circular slipsurface.

the result of Monte-Carlo realizations as discussedearlier. For both planar and circular slip surface, 1000realizations have been performed. It is observed fromFigure 3 that the results of safety factor calculationfor both failure slip assumptions follow a lognormalProbability Distribution Function (PDF).

4. Random Finite Di�erence Method (RFDM)

In recent years, a more rigorous method of probabilis-tic geotechnical analysis has been pursued in whichnonlinear �nite element/di�erence methods are com-bined with random �eld generation techniques (e.g.,[16-17]). This method, called the \Random FiniteElement/Di�erence Method" (RFEM/RFDM), fullyaccounts for spatial correlation and averaging and isalso a powerful slope stability analysis tool that doesnot require a priori assumptions related to the shapeor location of the failure mechanism. To calculatethe factor of safety, the C-Phi reduction approachwas employed in which the strength characteristics ofsoil are reduced to where a failure occurs. The non-linear �nite di�erence program FLAC [44], which takesthe mechanical behavior of excavations into accountand is able to calculate the factor of safety of slopes,


Figure 4. Con�guration of the model utilized in theanalysis.

was utilized to analyze the stability of unsupportedtrenches. Typical geometry of the model analyzed inthis study is shown in Figure 4.

The �nite di�erence mesh has been examined toeliminate the in uence of size and boundary e�ect onthe accuracy of results. Analyses with deterministicvalues of undrained shear strength increasing withdepth were �rst performed to see the accuracy ofthe proposed method and compare it with the exist-ing approximate solutions. The undrained Young'smodulus is de�ned as 500 Cu increasing with depth,accordingly. In the C-Phi reduction-based approach,the step by step excavation proceeds until a safetyfactor of 1.0 is approached. Based on this method,the deterministic critical excavation depth for di�erentsurface cohesion values is seen to be higher thanthe lower bound solution values and lower than theupper bound solution values as shown in Figure 5.The calculated deterministic critical excavation depthcorresponding to zero coe�cient of variation is thenemployed in subsequent stochastic analyses so as toapply uncertainty by introducing stochastic variationto the deterministic trend. The objective of thisstudy is to show the e�ect of uncertainty embeddedin geotechnical parameters (undrained cohesion inthis study) on critical excavation depth estimation.

Figure 5. The critical excavation depth calculated fromdeterministic strength reduction analysis for di�erentsurface cohesion values with � = 1:0 kPa/m.

Obviously, the starting point was chosen based on adeterministic C-Phi reduction analysis assuming \zero"coe�cient of variation corresponding to a \unit" safetyfactor. Spatial variability of shear strength parameterswas then applied to the domain under study to seeits e�ect on the reliability of deterministic analysisschemes. Even a small variation in shear strength(CoV = 10%) might lead to a great probability offailure, which is expected while the deterministic factorof safety is taken as \unity". The aim of the study isonly to compare the probabilities of failure (Pf ) corre-sponding to di�erent Coe�cients of Variation (CoVs).Therefore, the absolute values of Pf are not of value.Although retaining structures like cantilver retainingwalls are mandatory to utilize in such cases, the studyemphasizes the necessity of such provisions more thanthe deterministic cases when a small variability willlead to more probability of failure in comparison withthe classic deterministic cases.

In the probabilistic studies, the stochastic vari-ation of undrained shear strength is modeled byadopting a lognormal distribution with the aid of themost recognized representative statistical parametersincluding the mean value, the standard deviation, andthe scale of uctuation. There are several distributionslike normal, lognormal, beta, etc. used in practice.However, the use of lognormal distribution lies in thefact that shear strength is strictly non-negative and inlognormal distribution, there is no possibility of theexistence of negative values. For a detailed descriptionof proper distributions in geomaterials, Lee et al. [45]and Harr [46] can be conferred. In practice, it ismore common to use the dimensionless coe�cient ofvariation instead of standard deviation, which can bede�ned as the standard deviation divided by the mean.Typical values for the CoV of the undrained shearstrength have been suggested by several investigatorsbased on in situ or laboratory tests as pointed outearlier. The third important feature of a random�eld is its correlation structure. It is obvious that iftwo samples are close together, they will usually bemore correlated than the case where they are widelyseparated. It is common in literature to use a singleexponential form of correlation function, known asMarkovain spatial-correlation function:

� = e(�2j�j� ); (5)

in which � is the scale of uctuation, which is twice thecorrelation length for Markovian correlation functionde�nition, and � is the separation or lag distance. Thecorrelation length is the parameter which describes thedegree of correlation of a soil property and is de�nedas the distance beyond which the random values willbe no more correlated at all. It should be noted thatin the case of a small correlation length, random �eld


tends to be rough, and, oppositely, when it is large,random �eld tends to be smooth [47].

If the undrained shear strength, cu, is considereda random variable and assumed to be lognormallydistributed throughout the domain of study, the pointstatistics parameters, i.e. mean, �; standard deviation,�; and the most representative spatial statistics pa-rameter, i.e. scale of uctuation �, are used to realize alog-normally distributed random �eld given by:

cu(~x) = exp�L:"ln cu(~x) + �ln cu(~x)

�+ �z; (6)

where cu(~x) is the undrained shear strength assignedto the zones, ~x is the spatial position at which cu isdesired, "ln cu(~x) is an independent normally distributedrandom �eld with zero mean and unit variance, �ln cu isthe mean of the logarithm of cu, � is the shear strengthrate or density value as de�ned earlier, and L is a lower-triangular matrix computed from decomposition of thecovariance matrix.

A = LLT ; (7)

where A is the covariance between the logarithms of theundrained shear strength values at any two points. Inthis study, lnCu values are assumed to be characterizedby an exponential Markovian covariance function givenby [48]:

A = �2ln cu � e(�

2j�j� ); (8)

where �lnCu is the standard deviation of the logarithmof cu, and � is the lag distance, as mentioned before,considered to be the center to center distance of theconsecutive zones.

The parameters �lnCu and �lnCu are obtainedfrom the speci�ed mean and variance of undrained co-hesion, using log-normal distribution transformationsgiven by:

�ln cu = ln (�cu)� 12�2

ln cu ; (9)

�2ln cu = ln

�1 + (�cu=�cu)2

�= ln

�1 + CoV2

cu

�: (10)

Once the covariance matrix is established, it is de-composed into lower and upper triangular matrices(Eq. (7)) using Cholesky decomposition technique.Then, realizations of lognormally distributed soil prop-erties at each zone center are obtained by the trans-formation presented in Eq. (6) for a speci�ed mean,standard deviation, and scale of uctuation of the givensoil property (cu).

Figure 6 demonstrates a owchart on how thecalculation proceeds and the step by step procedurefor safety factor prediction is demonstrated. Similarprocedure was adopted for approximate methods (LEMor LAM) with the sole di�erence on the method ofsafety factor calculation which is conceptually di�erentfrom FDM. Figure 7 illustrates a sample realization

Figure 6. Flowchart for the process of safety factor analysis using RFDM.


Figure 7. A typical realization of undrained shearstrength with Cu0 = 50 kPa, � = 1:0 kPa/m,CoVCu = 10%, and � = 200 m.

Figure 8. Calculated and �tted log-normal cumulativeprobability of the factor of safety for Cu0 = 50 kPa,� = 1:0 kPa/m, � = 24 m, and CoVCu = 70%.

of the undrained shear strength with the assumedstochastic properties.

A typical cumulative probability of the factor ofsafety, as estimated from 1000 realizations, is shownin Figure 8 for Cu0 = 50 kPa and � = 1 kPa/m.A lognormal distribution was adopted in this studywith the requirement that safety factor is non-negative.Superimposed on the cumulative probability is a �ttedlognormal function with parameters given by mFS =0:817, �FS = 0:0844. At least visually, the lognormalcumulative graph appears perfectly �tted.

5. Results and discussion

In RFDM analyses, the Poisson's ratio (v) is assumedto be constant while the undrained shear strength(Cu) and the Young's modulus (Eu) are randomizedthroughout the domain. Forty sets of Monte-Carlosimulations were performed to investigate the e�ectsof the surface cohesion, Cu0, and undrained shearstrength density, �, as deterministic parameters andcoe�cient of variation of undrained shear strength,CoVCu, and its scale of uctuation, �, representingstochastic parameters on reliability of an open cut exca-

Table 3. Di�erent parameter sets adopted in this study.

Parameter Considered values

Cu0 (kPa) 50� (kPa/m) 1, 2COVCu (%) 10, 30, 50, 70, 90

� (m) 0.2, 8, 24, 200

vation. The above-mentioned parameters are variablethrough di�erent sets of analyses according to Table 3.

Spatial correlation lengths in both the horizontaland vertical directions were held the same implyingisotropic correlation structure. For the given problemcon�guration and element size, � = 0:2 m gives noappreciable correlation for the parameter under studyand results in ragged property �eld. On the contrary,the length of 200 m represents the strict correlationbetween zones. For each set of adopted Cu0, �, COVCu,and �, Monte-Carlo simulations have been conductedinvolving 1000 realizations of the undrained shearstrength random �eld and the subsequent numericalprogress of calculating the factor of safety.

It is reasonable to estimate the probability offailure based on lognormal probability distributionaccording to the goodness-of-�t test. For example, inFigure 8, the probability of factor of safety falling lessthan unity gives:

Pf =P [FS < 1]='Longnormal

�1� �FS

�FS

��= 0:7758:

(11)

Another approach to conclude about the probability offailure of the system is to estimate the number of casescorresponding to the factor of safety less than unity andthen to estimate the probability of failure from Eq. (14)which renders similar results with good precision:

Pf =NfN

= 0:7620; (12)

where Nf is the number of failed cases and N isthe total number of cases. This again con�rms thesuitability of lognormal distribution function for thefactor of safety data. The results show that thedi�erence between two methods is less than 2%.

The probability of failure shall mathematicallyincrease with the coe�cient of variation of cohesionvalues, as con�rmed by Figure 9. This can be explainedby the essence of strength reduction method in whichsome weak points may cause instability to occur.The number of weak points soars with the increasein the coe�cient of variation of cohesion. Figure 9shows the probability of failure (probability of thefactor of safety falling below the unity) for di�erentstrength densities computed by RFDM analyses. Therate of increase in the probability of failure with


Figure 9. The probability of failure computed by RFDManalyses versus the coe�cient of variation of undrainedshear strength for Cu0 = 50 kPa: (a) � = 1:0 kPa/m; and(b) � = 2:0 kPa/m.

CoVcu variation however decreases when increasing thestrength density (�). This is clear from the comparisonof di�erent parts in Figure 9 and it is mainly dueto the fact that when the undrained shear strengthdensity (�) increases, the deterministic portion of thecohesion �eld gets more highlighted than the stochasticcomponent and so the e�ect of stochastic variationbecomes less important. This is endorsed by thefact that the coe�cient of variation is de�ned basedon the surface cohesion value, Cu0, not the wholedeterministic trend, and indeed increasing the strengthdensity translates into decreased coe�cient of variationwith depth.

To investigate the e�ect of the scale of uctuation,�lnCu , on the safety factor statistics, �lnCu is variedfrom 0.2 (i.e., very much smaller than the soil modelsize) to 200 (i.e., very much bigger than the soilmodel size). In the limit, smaller values of �lnCuthan the element size result in a set of elements whichare largely independent (increasingly independent as�lnCu decreases) and the undrained shear strength �eldbecomes a white noise, with independent Cu values atany two distinct points. In such cases, the undrainedshear strength �eld becomes in�nitely \rough" and any

point at which the soil is weak will be surroundedby points where the soil is strong. A path throughthe weakest points in the soil might have very lowaverage strength, but at the same time, it will becomein�nitely tortuous and thus in�nitely long. If the failurecriterion was to seek a Critical Slip Line (CSL) bearingthe minimum shear strength or contouring plasticpoints, the above-mentioned e�ects, combined withshear interlocking dictated by stress �eld, would implythat the weakest path should return to the traditionallog-spiral (circular in undrained state) as explained byFenton and Gri�ths [49]. However, the routine safetyfactor calculation scheme adopted in FLAC is the so-called strength reduction method which seeks to �ndnonconvergent point(s) being an indicator of collapse.When the algorithm cannot converge within a user-speci�ed maximum number of iterations, together witha dramatic increase in nodal displacements within themesh, the implication is that no stress distribution canbe found that is able to simultaneously satisfy both thefailure criterion and global equilibrium, and failure issaid to have occurred. The aforesaid failure criterionimplies that weak point(s) formation may lead to theearly nonconvergence and the failure state in otherwords. Therefore, the scale of uctuation has a seriouse�ect on the factor of safety as any decrease in thescale of uctuation leads to a reduction in safety factoror an increase in probability of failure, as appears fromFigure 9.

As �lnCu ! 1, the stochastic component ofundrained shear strength �eld becomes the same ev-erywhere. In this case, the factors of safety statisticsare expected to approach those obtained by using alinearly varying lognormally distributed variable, Cu,to model the soil, Cu(x) = C+�z. That is because thesafety factor, FS, of a vertical trench in a soil layer withlinear (but random) undrained shear strength, Cu, isgiven by:

FS =FSdet�cu

cu; (13)

where FSdet is the \deterministic" safety factor whenCu = Cu0 + �z, and which is obtained from a single�nite di�erence analysis (or any other appropriateapproximate calculation); then, as �lnCu ! 1, thesafety factor assumes a lognormal distribution withparameters:

�ln FS =ln FSdet+ln�cu��ln�cu =�12�2

ln cu ; (14)

�ln FS = �ln cu ; (15)

where FSdet is taken as unity in Eq. (15), as discussedearlier, and the relationships for the parameters oftransformed lnCu Gaussian random �eld are employed.The probability of failure can be calculated assuming


a lognormal distribution for the safety factor.

Pf = '��ln FS

�ln FS

�= '

�12�ln cu

�; (16)

Figure 9 suggests that the probability of failure of thevertical cut with largely correlated undrained shearstrength �eld can be approximately estimated by astraight line of the form suggested by Eq. (16). It canbe seen that the agreement is fairly acceptable. Thesmall deviation arises from the fact that the predictedprobability of failure computed from Eq. (18) is basedon a LEM assumption (Eq. (15)) which takes a planarslip surface into account for a model bearing linearlydepth-varying undrained shear strength property, andthe FDM solution which adopts the strength reductionscheme to seek a failure point is conceptually di�erentand thus a deviation might be expected.

Another important stochastic property of interestfor the factor of safety is the coe�cient of variation ofthe safety factor. Figure 10 illustrates the variation ofthe safety factor against the CoV of undrained shearstrength. It is noted that introducing variation incohesion �eld causes variability of the results and theCoVFS increases with increase in the COVCu. However,this behavior becomes less eminent when increasing thestrength density from � = 1 kPa/m to � = 2 kPa/m.

Figure 10. Variability of the factor of safety versus thecoe�cient of variation of undrained shear strengthcomputed by RFDM analyses for Cu0 = 50 kPa: (a)� = 1:0 kPa/m; and (b) � = 2:0 kPa/m.

This is clearly seen from comparing di�erent partsof Figure 10. It is again explained by resorting tothe fact that the increase in strength density leadsthe deterministic component of cohesion to have moreimportant role than the stochastic component, asdiscussed earlier.

The RFDM is more time-intensive than theRLEM analyses which seek failure only along a pre-scribed and plausible failure surface. As RLEM anal-yses take place along a prescribed surface, one maylook into the e�ects in a di�erent view. The a�ectingparameters, namely the coe�cient of variation andthe scale of uctuation of undrained shear strength,are studied while assuming a planar or circular slipsurface. Figure 11 demonstrates the variation of theprobability of failure of slope as de�ned earlier withthe variability of undrained shear strength for twodi�erent strength densities. All scales of uctuationare drawn in the plot and individually labeled. It isimplied in these plots that the probability of failure ofthe vertical cut is not largely dependent on the scaleof uctuation, �lnCu . This is as expected since thescale of uctuation does not a�ect the global averageof a normally distributed process along a prede�nedslip surface. The shear resistance along a plausible

Figure 11. The probability of failure computed by RLEManalyses versus the coe�cient of variation of undrainedshear strength assuming a circular slip surface forCu0 = 50 kPa: (a) � = 1:0 kPa/m; and (b) � = 2:0 kPa/m.


and prede�ned slip surface is computed by globalintegration of the undrained shear strength along acircular surface and it is expected not to be a�ectedby the correlation structure of the random process.Figure 11 suggests that the probability of failure ofvertical cut in an undrained condition computed byRLEM analyses can be approximately estimated by astraight line de�ned by Eq. (16).

6. Conclusion

The main purpose of the study was to evaluate thereliability of the calculated critical excavation depthconsidering spatially variable random shear strength�eld. The factor of safety of excavation in a spatiallyvariable layer of �nite depth overlying bedrock is wellrepresented by a lognormal distribution with param-eters �ln FS and �ln FS, while the a�ecting strengthparameter (`Cu' in this study) is also assumed to belognormally distributed.

Random Finite Di�erence Methods (RFDM)coupled with strength reduction method have beenadopted to calculate the factor of safety in a spatiallycorrelated random �eld. Two deterministic param-eters, i.e. surface cohesion (Cu0) and the undrainedshear strength rate (�), along with two stochasticparameters, i.e. the coe�cient of variation of shearstrength (COV ) and its scale of uctuation (�), havebeen considered to evaluate the e�ect of the spatialvariation of the input parameter on the uncertainty em-bedded in calculation of the critical excavation depth.

Results show that increasing the coe�cient ofvariation of undrained cohesion leads to more probabil-ity of failure or uncertainty in other words. It is seenthat even 10% variation in undrained shear strengthmay lead to unsatisfactory performance. Furthermore,it is observed that the more the undrained shearstrength varies, the more scattering in safety factorestimation is expected. Scale of uctuation of theundrained shear strength, on the other hand, showsa di�erent e�ect as it causes the probability of failureto rise when decreasing the correlation between shearstrength data. The explanation of this observation isinspired by the fact that in low values of the scale of uctuation, zones become highly independent and a so-called \Ragged" �eld forms. This causes an increase inthe number of weak points in the whole body, whichin e�ect induces a reduction in the factor of safety dueto the essence of strength reduction method. Strengthdensity of the cohesion value, �, on the other side of thespectrum, has a decreasing e�ect on the uncertainty ofresults owing to the fact that the coe�cient of variation(COV ) has been de�ned only based on the surfacecohesion value (Cu0), and the increase in shear strengthrate implies a reduction in coe�cient of variation ofundrained shear strength.

Random Limit Equilibrium Method (RLEM) wasalso adopted by choosing a planar and circular criti-cal slip surface using Bishop's formula to investigatethe e�ect of stochastic variation of undrained shearstrength properties. The slip surface was then dis-cretized into 1000 elements and an auto-correlatedcohesion �eld was generated afterwards. The factorof safety for each realization was calculated usingBishop's formula. The results consistently demon-strated that the RLEM might not properly re ectthe uncertainty in critical excavation depth estimationif the failure surface remained unchanged throughrealizations. The calculation of the safety factor in suchsimple and approximate methods was based mainlyon the assumption of a plausible and predeterminedslip surface. The shear resistance along the assumedfailure surface was calculated by integration of theshear strength increments along the planar or circularsurface and it was assumed not to be a�ected bythe correlation structure of the random �eld and hasvanishing variance for �nite �lnCu . An important pointis that in this study, RLEM �rst assumed the mostprobable slip surface corresponding to the deterministiccondition and then applied uncertainty and variabilityof parameters across the same surface and did notchange the surface location. However, it was possibleto seek surface variation through realizations; but, ifwe are to investigate the heterogeneity e�ect by RLEMwhen considering critical failure mechanism variation,which is time-intensive, it is better to resort to RFDMor RFEM which render the most probable failuresurface that passes through the weakest points and donot need any a priori assumptions regarding the shapeand mechanism of the failure. Also, the potential pitfallof the LEM, which is the inability to account for thestress-strain behavior of the soil, is overcome.

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Biographies

Reza Jamshidi Chenari, PhD, PEng is an AssociateProfessor in the Group of Civil Engineering, Universityof Guilan. He is also a Professional Engineer in Iran.He received his BS (�rst class) and MS degrees fromSharif University of Technology and his PhD from IranUniversity of Science and Technology, Tehran, Iran,where his research focused on dynamic behavior of re-taining walls back�lled with geotextile �ber reinforcedsand. He conducted large cyclic triaxial tests andalso shaking table experiments. Dr. Jamshidi's currentresearch focuses on risk and reliability in geoengineer-ing and geohazards, application of Random NumericalMethods (RNM) in di�erent geotechnical problemsincluding foundation settlement, shallow foundationbearing capacity, slope stability, consolidation of nat-ural alluvial deposits, seismic ampli�cation of alluvi-ums, and in general sense, he now focuses on thee�ect of heterogeneity and anisotropy in deformationand strength properties of natural alluviums on theaforesaid problems. Dr. Jamshidi has authored or co-authored more than 40 scienti�c publications.

Masoud Zamanzadeh received his BS degree in CivilEngineering from Islamic Azad University of Marand in2008 and MS degree in Geotechnical Engineering fromUniversity of Guilan in 2012. He ful�lled his masterdissertation, focused on the \Uncertainty Assessmentof Critical Excavation Depth of Vertical UnsupportedCuts in Undrained Clay Using Random Field The-orem", under the supervision of Dr. Reza JamshidiChenari in September 2012. As a geo-engineer, heis interested in computational geomechanics and theapplication of numerical modeling in geotechnical engi-neering practice. He has written scienti�c publicationson risk and reliability in geotechnical engineering. Hislast contribution is the \Investigation into the e�ectof the inherent heterogeneity of soil on the long-termstability of earth slopes" accepted for publication inSharif Civil Engineering Scienti�c Research Journal,Sharif University of Technology.

uncertainty assessment of critical excavation depth of

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