Continuous slip surface method for stability analysis of …scientiairanica.sharif.edu/article_21227_fcb5b9b6bbd60f... · 2021. 4. 21. · more suitable, at least for the selected

Scientia Iranica A (2020) 27(6), 2657{2668

Sharif University of TechnologyScientia Iranica

Transactions A: Civil Engineeringhttp://scientiairanica.sharif.edu

Continuous slip surface method for stability analysis ofheterogeneous vertical trenches

R. Jamshidi Chenari�, H. Kamyab Farahbakhsh, and A. Izadi

Department of Civil Engineering, Faculty of Engineering, University of Guilan, Rasht, P.O. Box 3756, Guilan, Iran.

Received 5 July 2016; received in revised form 21 May 2018; accepted 17 December 2018

KEYWORDSCritical excavationdepth;Random �eld theory;Continuous slipsurface;Vertical trench;Un-drained shearstrength.

Abstract. Evaluating the reliability of trenches against sliding failure is complicatedsince most alluvial deposits are heterogeneous and spatially variable. This means thatrather than perfectly circular or linear failure surfaces, trench failure tends to be morecomplex, following the weakest path or zones through the material; thereby, a new methodcalled Continuous Slip-Surface (CSS) was adopted to calculate the critical excavation depth.CSS runs an algorithm to seek a continuous slip surface. The Finite Di�erence Method(FDM) coupled with random �eld theory and CSS method is well suited for calculatingslope stability since it allows the failure surface to seek out the weakest path through thesoil. For an unsupported vertical cut, it was shown that the critical excavation depthacquired from CSS method was indeed an upper bound solution. In terms of numericaland analytical terms, an idealized variation model was used to illustrate that increasingthe un-drained shear strength density attenuated the e�ect of shear strength variability.Correlation structure of the input variable was shown to in uence the results, although itsbehavior varied on low and high scales of uctuation.

© 2020 Sharif University of Technology. All rights reserved.

1. Introduction

In geotechnical engineering, traditionally, the criticalexcavation depth of vertical unsupported trenches isthe point at which safety factor reaches one. It ispossible to calculate the safety factor using two generalmethods: analytical and numerical solutions. Chen(1975) proposed the upper and lower bound approachesbased on theory of plasticity to investigate at whichpoint a trench reaches its critical depth and causesa slip surface [1]. Numerical methods proposed byZienkiewicz et al. [2], Naylor [3], Matsui and San [4],

*. Corresponding author. Tel.: +98 13 33690485;Fax: +98 13 33505866E-mail addresses: Jamshidi [email protected] (R. JamshidiChenari); [email protected] (H. KamyabFarahbakhsh); [email protected] (A. Izadi)

doi: 10.24200/sci.2019.21227

Ugai and Leshchinsky [5], Lane and Gri�ths [6],Dawson et al. [7], and Rachez et al. [8] have longbeen used to evaluate the critical excavation depth ofunsupported trenches. However, the second approach,utilizing one of the numerical methods such as FiniteDi�erence Method (FDM) or Finite Element Method(FEM), is more reliable, especially in complicatedgeometrical and geotechnical conditions. Nevertheless,such methods are subject to some weaknesses andmay result in failure by using the strength reductiontechnique based on iterative analyses. Application ofstrength reduction technique requires advanced numer-ical modeling skills and a rather long calculation timeand in case of complicated models, it can last as longas several hours [9].

In the strength reduction technique, failure isintroduced based on one or more points of velocitythat did not converge to a constant value regardlessof such requirements as continuity of slip surface and

2658 R. Jamshidi Chenari et al./Scientia Iranica, Transactions A: Civil Engineering 27 (2020) 2657{2668

magnitude of the velocity value. This probably leadsto underestimation of the critical excavation depth.With an understanding of the di�erences between thesemethods, a comprehensive study has been performedto compare di�erent approaches; meanwhile, a newapproach is introduced to investigate the critical depthof excavation based on velocity convergence and con-tinuity of slip surface requirements using the FDM.It should be noted that this study does not considerany retaining structure provisions for vertical cuts andindeed takes unsupported vertical cuts into account toinitiate the idea and they do not overlook or neglectthe necessity of support system for vertical cuts. Theaim of this study is to show the e�ciency of the newmethod in estimating the critical excavation depth ofunsupported vertical cuts.

2. Analytical solutions

Analytical solutions to stability problems such as thebearing capacity of foundations and thrust behindthe retaining structures can be found widely amongdi�erent approaches [10{14]. Vertical cuts as anotherexample of stability issues in soil mechanics are be-lieved to remain stable up to several meters in heightwithout any supporting system. The upper and lowerbound solutions of limit analysis bracket any possiblevalue of critical excavation depth of vertical cut. Stressdiscontinuity should be considered to achieve lowerbound estimation. It should be mentioned that thereare many stress �elds satisfying equilibrium and yieldcondition simultaneously. The maximum value of thelower bound solution is safe excavation height. A lowerbound solution to determining the critical excavationdepth of vertical cuts in the un-drained condition is:

Hcr =2Cu0

� �; (1)

where Cu0 is the un-drained shear strength on thesurface, is the unit weight of the soil, and � is theun-drained shear strength density or its increasing ratewith depth. An upper bound solution of Hcr may befound by equating external rate of work to the energydissipation on the sliding surface. The upper boundsolution is thus given below:

Hcr =4Cu0

� 2�: (2)

3. Strength reduction-based method

The strength reduction method is one of the limit stateforms of solution to analyzing stability of foundationsand slopes. This method has been widely studied bymany researchers such as Song [15], Lian et al. [16],Eberhardt [17], Sai et al. [18], Zheng et al. [19], Huang

and Cang [20], Wei and Cheng [21], and Eser et al. [22].Calculation of safety factor using strength reductiontechnique can be considered as a progressive procedureby reducing the shear strength of the material to reachequilibrium in limit state.

The partial safety factor F is de�ned according toEqs. (3) and (4):

CTrial =1

FTrialC; (3)

�trial = arctan�

1F trial

tan��: (4)

Non-linear �nite di�erence program FLAC [23] is usedfor the bracketing approach, which is similar to themethod proposed by Dawson et al. [7]. This procedurecomprises some steps and criteria that help identify thefailure state. The unbalanced force ratio was de�nedby the ratio of the unbalancing force to the meanabsolute value of the force exerted by adjacent zones.According to the investigation by Rachez et al. [8], thesystem is in equilibrium condition if the unbalancedforce ratio is less than 10�3 for a given factor of safety.Otherwise, another analytic step should be run. Themean force ratio is compared with that in the previousanalyzed steps. If the di�erence between the previouslyobtained ratios is less than 10%, the system is in a non-equilibrium condition.

A drawback of this procedure is that when localfailure occurs, this criterion will alarm failure, whereasthe reality might be totally di�erent. Figure 1 showsthe �nite di�erence mesh for a soil stratum in theunexcavated state. The unsupported vertical cut issupposed to be excavated following the initial in-situ state equilibrium analysis. The model size anddimensions have been examined to eliminate boundarye�ects. A linear increase in the un-drained shearstrength with depth is considered. In addition, theun-drained Young's modulus is assumed to be fullycorrelated with the un-drained shear strength de�nedas 500 Cu, increasing with depth accordingly [24,25]. In

Figure 1. Con�guration of the model adopted fornumerical analyses.

R. Jamshidi Chenari et al./Scientia Iranica, Transactions A: Civil Engineering 27 (2020) 2657{2668 2659

the strength reduction technique, excavation proceedsstep by step until a safety factor of 1.0 is reached.The critical depth of excavation is introduced when thesafety factor reaches one. Jamshidi Chenari and Za-manzadeh [26] compared the critical excavation depthsestimated using C-Phi reduction technique with upperand lower bounds predictions. They showed that theestimated critical excavation depth for di�erent surfacecohesion values fell between these two bounds.

As clearly shown in Figure 2, a continuous slipsurface has not been formed, while the safety factorobtained based on the strength reduction techniquerenders a value of 1.0 at depth 13 m.

4. Continuous Slip Surface Method (CSSM)

Evaluating the reliability of excavations against failureis complicated due to the spatially variable nature ofmost soils and uncertainty in their properties [27,28].This means that instead of perfect circular failuresurfaces, excavation failure tends to be more complex,following the weakest path or zones through the mate-rial. Furthermore, the drawback of the strength reduc-tion technique makes the results even more unreliable.Basically, a slope or an excavation is set to fail, whilea continuous slip surface is formed. For this purpose, amethod has been employed to search for a continuousslip surface starting from the surface and extending tothe excavation boundary.

In Continuous Slip-Surface Method (CSSM), thealgorithm seeks to �nd continuously generated plasticzones. Figure 3 illustrates bounded shear zone formedin unsupported vertical cut excavation. The programautomatically ignores outliers, as indicated in Figure 3.Further to the slip surface continuity criterion, somelimiting conditions on the failure state are also checked.The excavation is conceived to fail when the unbalancedforce ratio does not converge (less than 10�3) after30000 steps of time marching analysis.

However, a failure might occur while the unbal-anced force ratio is below the set limit. For thisreason, a velocity criterion has also been considered. If

Figure 2. Generated failure state for F:S = 1:0 using theconventional strength reduction scheme.

Figure 3. Continuity of slip surface by identifying lowerand upper plastic boundaries.

grid points velocity located in the excavation boundaryexceeds 5 � 10�5 m/s, the excavation is said to havereached a failure state [23]. This means that in CSSM,three criteria are jointly checked to alarm failure:continuity of failure slip surface, unbalanced forceratio, and grid points velocity convergence. Figure 4demonstrates di�erent controls adopted using the newCSSM.

As seen in Figure 5, for all values of surface cohe-sion, the critical excavation depth estimated using thenew CSSM coincides with the upper bound solution.This is expected as CSSM allows full mobilization ofthe plastic zones in order to seek continuous failureenvelope.

5. Random �eld generation

Spatial variability of soil properties can be quanti�edby several parameters such as deterministic trend,Coe�cient Of Variation (COV), correlation length, andanisotropy among others, and they have been widelystudied in the literature. Further detailed analyses ofspatial variability of soil parameters for geotechnicalstructures can be found in the literature [29{38].According to Phoon and Kulhawy [30], the inherentvariability of soil properties is usually decomposed intotwo components:

1. A deterministic trend;2. Fluctuations around the deterministic trend given

below:

� (z) = t (z) + w (z) ; (5)

where �(z) is the in-situ geotechnical property, t(z) thetrend function, and w(z) the uctuating component,also known as \o� the trend" variation. Figure 6schematically represents di�erent components of inher-ent variability.

Deterministic trend can be estimated with areasonable amount of in situ soil data (for example,using least-square �t method). Eslami Kenarsari etal. [39] suggested that quadratic trend removal wasmore suitable, at least for the selected Cone Pene-tration Test (CPT) data soundings. The uctuating


Figure 4. Illustration of the Continuous Slip Surface Method (CSSM).

velocity components can be characterized as a randomvariable with zero mean and non-zero variances.

The log-normal distribution of stochastic varia-tion in the un-drained shear strength appears to bereasonable for probabilistic studies given that shearstrength is strictly non-negative. Application of severalprobability distribution functions, namely normal, log-normal, beta, etc., is found in literature and practice.A more detailed description of proper distributionfunctions for geo-materials can be found in the studiesof Lee et al. [40], Harr [41], and more recently inSeyedein et al. [42]. In practice, it is more common

to use dimensionless COV rather than standard devi-ation, which can be de�ned as the standard deviationdivided by the mean. A number of investigators havedetermined the typical values of the COV of the un-drained shear strength [40,43]. Jamshidi Chenari etal. [37] recommended that the range of the COVsfor the un-drained shear strength based on in situ orlaboratory tests varied from 10% {50%. The thirdimportant feature of a random �eld is its correlationstructure. If two samples are close together, theyare usually more correlated than the case when theyare widely separated. In literature, it is common to


Figure 5. Continuous Slip-Surface Method (CSSM)versus conventional strength reduction technique with� = 1 kPa/m and = 20 kN/m3.

Figure 6. Inherent variability of soil properties.

use a correlation function (�) in the following singleexponential form provided in Eq. (6), which is alsoknown as Markov spatial correlation function proposedby Fenton and Gri�ths [44]:

� = exp��2 j� j

�

�; (6)

where � is the correlation coe�cient between the loga-rithmic values of the un-drained shear strength at anytwo points separated by a distance � in a random �eldwith spatial correlation length �. Correlation degree ofa soil property was described by the correlation length.It is worth noting that the random �eld will be roughin variation if the correlation length is large, while ittends to be smooth if correlation length is small.

The random �eld was generated by matrix de-composition based on Cholesky method, which wasproposed by El-Kadi and Williams in 2000 [45]. More-over, a correlated log-normal distribution was used tode�ne characteristics of random �eld. The values of un-drained shear strength in the log-normally distributedform are estimated as follows:lncu = L"+ �lnCu ; (7)

where � is the mean of ln cu (un-drained shearstrength), " is a Gaussian vector �eld with zero meanand unit variance, and L is the lower triangular matrixde�ned by:

A = LLT ; (8)

where A is the covariance matrix formed by using thespeci�ed form of the covariance function. The covari-ance matrix was given by El-Kadi and Williams [45].

A = �2e� 2j�j� ; (9)

where �2 is the variance of ln cu, � is the autocorrelationlength, and matrix � is separation lag, which tends tobe the distant matrix. Figure 7 illustrates a samplerealization of the un-drained shear strength with theassumed stochastic properties.

As depicted in Figure 8, in di�erent analyzedcases, the mean and COV of critical excavation depthare well stable after 500 realizations of Monte-Carloprocess; thus, no additional bene�t is achieved by usinga large number of realizations.

As stated before, the critical excavation depth ofvertical unsupported trenches will be calculated usinga FISH program developed by authors adopting the

Figure 7. Realization of un-drained shear strength withCOV = 90%, Cu0 = 25 kPa, � = 1 kPa/m, and � = 12 m.

Figure 8. Validation of realizations for � = 12 m.


Figure 9. Flowchart for the purpose of critical excavation depth estimation of heterogeneous vertical trench usingRandom Finite Di�erence Method (RFDM) merged with Continuous Slip-Surface Method (CSSM).

�nite di�erence program FLAC, merged with RandomFinite Di�erence Method (RFDM). Figure 9 shows a owchart of how to calculate the critical excavationdepth through repeated �nite di�erence analyses byusing a modi�ed C-Phi reduction algorithm. As statedbefore, the factor of safety is calculated by using themodi�ed C-Phi reduction algorithm and this procedurewill be in progress until a unit safety factor is achieved.

6. Results and discussion

6.1. In uence of the variabilityIn this code, Poisson's ratio (�) is assumed to beconstant, while un-drained shear strength (Cu) andun-drained Young's modulus (Eu) were randomizedthroughout the domain. Then, 80 sets of runswere performed to investigate the e�ects of COV, �

(strength density), and Cu0 (the surface un-drainedshear strength) separately. The parameters vary ac-cording to Table 1.

Spatial correlation lengths (�) of both horizontaland vertical directions remained equal and varied ac-cording to Table 1 in the isotropic sense. For the given

Table 1. Adopted values of the parameters in this study.

Parameter Values considered

Cu0 (kPa) 25, 50

(kN/m3) 20

� (kPa/m) 1, 2

COV (%) 10, 30, 50, 70, 90

� (m) 0.2, 8, 24, 200


con�guration and element size, the � = 0:2 m givesno property correlation. On the contrary, the lengthof 200 m represents the strict correlation betweenzones. However, for each set of adopted parameters,Monte Carlo simulations were conducted involving 500realizations of the un-drained shear strength as arandom �eld and the subsequent numerical analysesof critical excavation depth were performed. A typicalhistogram of the critical excavation depth for Cu0 =25 kPa, as estimated by 500 realizations, is shownin Figure 10. Since the critical excavation depthcannot be negative, the shape of the histogram suggestsa lognormal distribution, which was adopted in thisstudy.

Superimposed on the histogram is a lognormaldistribution with parameters given by �lnHcr, �lnHcrin the line key. The �t appears reasonable, at leastvisually. In fact, this is one of the worst cases, out ofwhich the 80 parameter sets are given in Table 1. Byaccepting the lognormal distribution as a reasonable �tto the simulation results, the next task is to estimatethe parameters of the �tted lognormal distributionas a function of the input parameters (Cu0, COVCu,�, and �lnCu). The lognormal distribution has twoparameters: �lnHcr and COVlnHcr.

The COV of the un-drained shear strength variedfrom 10% to 90% and it was used to investigate thee�ects of un-drained shear strength variability on thecritical excavation depth statistics. The parameter ofthe transformed lnCu Gaussian random �eld may bedetermined using the following relations:

�2lnCu = ln

�1 +

�2Cu�2Cu

�; (10)

�lnCu = ln (�Cu)� 12�2

ln Cu; (11)

Figure 10. Typical frequency plot and �tted log-normaldistribution of critical depth for Cu = 25 kPa, � = 200 m,and COV = 70%.

Figure 11. Estimated mean of log-critical excavationdepth for � = 1:0 kPa/m.

where �cu and �cu represent mean and standard devi-ation of the un-drained shear strength, respectively.

The variance of lnCu varies from 0.01 to 0.59in this study. Figure 11 shows how the estimator of�lnHcr varies with �2

lnCu for Cu0 = 25 kPa and 50 kPa.All scales of uctuation are drawn in the plot, but theyare not individually labelled. This observation impliesthat the mean log-critical excavation depth is largelyindependent of the scale of uctuation, �lnCu. This isexpected since the scale of uctuation does not a�ectthe local average of a normally distributed process.Figure 11 suggests that the mean log-critical excavationdepth can be closely estimated by a straight line.

To investigate the e�ect of the uctuation scale,�lnCu, on the critical excavation depth statistics, thisparameter varied from 0.2 m (i.e., very much smallerthan the mesh size) to 200 m (i.e., substantially largerthan the soil model size). Broadly, at the limit�lnCu ! 0, the cohesion �eld becomes a white noise�eld. Because of the averaging e�ect of the un-drainedshear strength �eld, the critical excavation depth inthe limiting case �lnCu ! 0 is expected to approachthe depth obtained in the deterministic case, withCu = �Cu everywhere, and it has a negligible variancefor �nite �2

lnCu.At the other extreme end, as �lnCu ! 1, the

stochastic component of the cohesion �eld becomes thesame everywhere. In this case, the critical excavationdepth statistics were obtained by using a linearly vary-ing log-normally distributed variable, Cu(x) = C + �z.The critical excavation depth, Hcr, on a soil layer withlinear, yet random, un-drained shear strength, Cu, isgiven as follows:

Hcr =HdetCu�Cu

: (12)

For Hdet, the deterministic critical excavation depthwhen Cu = Cu0 + �z was obtained from a single �nitedi�erence set of analyses; then, as �lnCu ! 1,


the critical excavation depth assumes a lognormaldistribution with parameters:

�lnHcr = ln (Hdet) + �lnCu � ln(�Cu)! �lnHcr

= ln (Hdet)� 12�2

lnCu; (13)

�lnHcr = �lnCu: (14)

6.2. In uence of the correlationIt can easily be proven that the variance of criticalexcavation depth deviates from the un-drained shearstrength variance for small-scale uctuations, as shownin Figure 12. In all cases, �lnHcr approaches �lnCuas �lnCu increases. Reduction in variance of criticalexcavation depth as �lnCu decreases results from thelocal averaging variance reduction of the log-cohesion�eld around the excavation. Following this reasoningand assuming the local averaging of the area around theexcavation, the standard deviation of log-excavationcritical depth is calculated as follows:

�lnHcr = 4p (B;H)�lnCu ; (15)

where (B;H) is the so-called variance function [27],which depends on the averaging region, 3B�H, as wellas on the scale of uctuation, �lnCu. The variance func-tion corresponding to the isotropic Markov correlationfunction in Eq. (6) is approximated by:

(d1; d2) =12

[ (d1) (d2jd1)+ (d2) (d1jd2)] ; (16)

where:

(di) =

"1 +

�di

�lnCu

� 32#� 2

3

; (17)

Figure 12. Variation of simulated sample standarddeviation of log-excavation critical depth with the scale of uctuation of cohesion �eld.

(dijdj) =

"1 +

�diRj

� 32#� 2

3

; (18)

Rj = �lnCu

"�2

+�

1� �2

�exp

(��

dj�2 �lnCu

�2)#

;(19)

where dj is the dimension of the averaging region (inthis case, d1 = 2B and d2 = 3Hcr). Predictions of�lnHcr using Eqs. (15) and (16) are given in Figure 12using solid lines.

6.3. In uence of deterministic heterogeneityIt can be seen that increasing COVCu results in morevariability of critical excavation depth estimation. Fig-ure 13 represents the box plot of each dataset obtainedfrom �nite di�erence analyses. On each box, the centralmark is the median and the edges of the box are the25th and 75th percentiles. This plot proves that whenthe cohesion variation increases, each box breadth

Figure 13. Increasing the Coe�cient Of Variation (COV)of critical excavation depth with increasing the COV ofcohesion: (a) � = 1:0 kPa/m and (b) � = 2:0 kPa/m.


and the variation of individual points increase. Thiscon�rms the increase in variability of critical excavationdepth results, as stated above. On the contrary, thebreadth of each box plot will shrink in case of anincrease in the un-drained shear strength density, �.This can be explained by the fact that COVCu isde�ned based on the surface un-drained shear strengthand the realized random �eld for the cohesion valuesis super-positioned with linearly varying deterministicportions. This means that the adopted COV is indeedfake and not realistic. Higher values of un-drainedshear strength density, �, cause a reduction in thevariability of the cohesion values, which in turn causesless variability of critical excavation depth results.

A simple theoretical justi�cation of the e�ect ofun-drained shear strength density � on the variationof the critical excavation depth results can be per-formed by adopting the harmonic variation form forthe random and stochastic components of the wholepro�le. Eq. (20) demonstrates superposition of linearand harmonic functions that represent \deterministic"and \stochastic" components, respectively. In theharmonic component, the amplitude A is theoreticallyshown to be

p2 times the standard deviation � and

the wave length �2 represents the scale of uctuationof the random �eld.

Cu (z) =Linear +Harmonic = Cu0 + �1z

+A sin�

2�z�2

�; (20)

where �1 represents the un-drained shear strengthdensity (�), as de�ned earlier. Figure 14 schematicallyillustrates two parts and their superposition.

By using a simple limit equilibrium calculation,the critical excavation depth is calculated by �ndingthe minimum depth through Eq. (21). It can easily be

Figure 14. Cohesion variation with depth forCu0 = 25 kPa, �1 = 2 kPa/m, A = 17:5 kPa, and �2 = 5m.

Figure 15. Dimensionless critical excavation depth fordi�erent values of strength density, �1, Cu0 = 25 kPa, andCOV = 90%.

shown that � = 45� will satisfy the minimum solutionfor the excavation depth in Eq. (21):

Cu0H +�1H2

2+A�2

2�

�1� cos

�2�H�2

��=

14 H2 sin 2�: (21)

Results are given in Figure 15 against the un-drained shear strength correlation length for di�erentstrength densities. It is seen that the harmonic com-ponent e�ect representing the stochastic component isless highlighted when the deterministic part contributesmore. This is attributed to the fact that quantitatively,lower \o�-trend" values than the deterministic portionare encountered when higher strength densities arechosen. This is actually equivalent to considering lowerCOVs expected to render higher critical excavationdepth, as proven earlier in Figure 13.

7. Conclusion

Conventional C-Phi reduction method employed forcalculating safety factor was revisited and it was shownthat in some cases, overconservative estimation of thecritical excavation depth of vertical unsupported cutswas reached. The method was modi�ed by consideringthe continuity of slip surface. The following conclusionswere made through Monte-Carlo simulations of criticalexcavation depth of vertical unsupported cuts in theun-drained condition in the form of random �nitedi�erence analyses:

1. Continuous Slip Surface Method (CSSM) couldreach solutions for the critical excavation depth,quantitatively similar to the upper bound solution.It means that the conventional strength reductiontechnique neglects the formation of slip surfaceand only monitors speci�c control points to checkunbalancing force or grid points' velocity;

2. The critical excavation depth on a spatially ran-dom �eld of �nite depth overlying bedrock was


well represented by a lognormal distribution ifthe un-drained shear strength was log-normallydistributed;

3. Increasing Coe�cient Of Variation (COV) of un-drained shear strength led to a decrease in meancritical excavation depth. This means that COVincrease potentially increases the chance of weakpoint formation and the slip surface formationprobability;

4. Un-drained shear strength density or its variationrate with depth attenuated the e�ect of stochasticvariation on critical excavation depth estimation.Indeed, variability of un-drained shear strengthbecame less signi�cant at higher strength densities;

5. The scale of uctuations in the un-drained shearstrength was shown to largely a�ect the result.For small values, the �nite element zones becomemore independent. This leads to the formationof a so-called rugged �eld. However, local aver-aging caused less variability of critical excavationresults when small scales of uctuation were en-countered. Increasing the scale of uctuation re-sulted in greater diversity and variability of criticalexcavation depth. At the limits, when the scale of uctuation becomes very large, the �nite elementzones bear uniform �eld of random variable. Al-though the �eld is generated within random numbergeneration scope, it is still uniform. However, caserealizations are di�erent, leading to a considerablevariability of critical excavation depth results. Atthe limits, it converges to the standard deviation ofthe input variable.

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Biographies

Reza Jamshidi Chenari (PhD, PEng) is an Asso-ciate Professor in the Faculty of Civil Engineering,University of Guilan. He received his BSc and MScdegrees from Sharif University of Technology and hisPhD from Iran University of Science and Technology,Tehran, Iran, where his research focused on dynamicbehavior of retaining walls back�lled with geotextile�ber-reinforced sand. He conducted large cyclic triaxialtests and also shaking table experiments. ProfessorJamshidi Chenari's current research focuses on riskand reliability in geoengineering and geohazards, ap-plication of Random Numerical Methods (RNM) todi�erent geotechnical problems including foundationsettlement, shallow foundation bearing capacity, slopestability, consolidation of natural alluvial deposits, andseismic ampli�cation of alluviums. In general, he nowfocuses on the e�ect of heterogeneity and anisotropy


on deformation and strength properties of naturalalluviums. Dr. Jamshidi Chenari has authored orcoauthored numerous scienti�c publications.

Hassan Kamyab Farahbakhsh received his MScdegree in Geotechnical Engineering from Universityof Guilan in 2011. He completed his MSc thesis onthe uncertainty assessment of critical excavation depthby numerical modeling and arti�cial neural network.As a geo-engineer, he is interested in computationalgeomechanics and the application of numerical mod-eling to geotechnical engineering practice. He haswritten scienti�c publications on risk and reliability ingeotechnical engineering.

Ardavan Izadi is a PhD student of GeotechnicalEngineering at University of Guilan. He received MScof Geotechnical Engineering from Isfahan Universityof Technology in 2013, where his research focused onthe ultimate bearing capacity of shallow foundationson heterogeneous soil by upper bound limit analysis.Currently, he focuses on the e�ect of heterogeneity andanisotropy on the stability of geotechnical structuresvia various numerical and analytical methods under thesupervision of Dr. Jamshidi Chenari. His latest contri-bution is \Pseudo-Static Bearing Capacity of ShallowFoundations on Heterogeneous Marine Deposits usingLimit Equilibrium Method" accepted for publication inMarine Georesources and Geotechnology Journal.

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Continuous slip surface method for stability analysis of …scientiairanica.sharif.edu/article_21227_fcb5b9b6bbd60f... · 2021. 4. 21. · more suitable, at least for the selected