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Journal of Rock Mechanics and Geotechnical Engineering 7 (2015) 659e667

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Journal of Rock Mechanics andGeotechnical Engineering

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Effects of spatial variation in cohesion over the concrete-rock interfaceon dam sliding stability

Alexandra Krounis*, Fredrik Johansson, Stefan LarssonDivision of Soil and Rock Mechanics, KTH Royal Institute of Technology, Stockholm, SE-10044, Sweden

a r t i c l e i n f o

Article history:Received 18 May 2015Received in revised form20 July 2015Accepted 3 August 2015Available online 8 October 2015

Keywords:Concrete gravity damSliding stabilityCohesionBrittle failureSpatial variation

* Corresponding author. Tel.: þ46 87908060.E-mail address: alexandra.krounis@byv.kth.se (A. KPeer review under responsibility of Institute o

Chinese Academy of Sciences.1674-7755 � 2015 Institute of Rock and Soil Mof Sciences. Production and hosting by Elsevier B.V. Ahttp://dx.doi.org/10.1016/j.jrmge.2015.08.005

a b s t r a c t

The limit equilibrium method (LEM) is widely used for sliding stability evaluation of concrete gravitydams. Failure is then commonly assumed to occur along the entire sliding surface simultaneously.However, the brittle behaviour of bonded concrete-rock contacts, in combination with the varying stressover the interface, implies that the failure of bonded dam-foundation interfaces occurs progressively. Inaddition, the spatial variation in cohesion may introduce weak spots where failure can be initiated.Nonetheless, the combined effect of brittle failure and spatial variation in cohesion on the overall shearstrength of the interface has not been studied previously. In this paper, numerical analyses are used toinvestigate the effect of brittle failure in combination with spatial variation in cohesion that is taken intoaccount by random fields with different correlation lengths. The study concludes that a possible exis-tence of weak spots along the interface has to be considered since it significantly reduces the overallshear strength of the interface, and implications for doing so are discussed.� 2015 Institute of Rock and Soil Mechanics, Chinese Academy of Sciences. Production and hosting by

Elsevier B.V. All rights reserved.

1. Introduction

In Sweden andmany other regions/countries of the world, thereare an increasing number of older dams in need of safety re-assessments to evaluate their compliance with modern safetyregulations. One important failure mode considered is sliding.Several techniques for safety assessment with regard to sliding areavailable, with the traditional limit equilibrium method (LEM) be-ing the most popular, accepted and widely used approach. In LEM,the dam is modelled as a rigid body allowed to slide along its base,and the safety is evaluated by the ratio between the driving forcesand the resisting forces. In general, the available shear strength ofthe dam-foundation contact is expressed by the MohreCoulombfailure criterion, and the resisting forces are determined by inte-grating the normal stresses over the potential sliding plane. Thetechnique is based on the assumption that the shear strength issimultaneously mobilised along the entire sliding surface at thetime of failure (Ruggeri et al., 2004). In order for that simplificationto be valid, the interface must behave as an elastic-perfectly plasticmaterial. However, tests conducted on concrete-rock cores taken

rounis).f Rock and Soil Mechanics,

echanics, Chinese Academyll rights reserved.

from dams show that an elastic-brittle response is to be expectedfor cores with bonded interfaces (Rocha, 1964; Link, 1969; Lo et al.,1990; EPRI, 1992). The elastic-brittle response, in combinationwiththe varying stress conditions along the interface, means that aprogressive mechanism of failure would be a better description ofthe interface behaviour. In addition, it is likely that parts with highand low values of cohesion could be expected to appear in clusterswith a certain correlation distance. According to Westberg Wildeand Johansson (2013), the reason for this is that the bondstrength depends on factors such as the results from cleaning therock surface prior to the concrete casting, the local rock massquality and the location of leakage and other degradation pro-cesses. Possible spatial variation in cohesion over the interface mayintroduce weak areas where the failure process can be initiated andcontributes further to the uncertainties regarding the failurebehaviour of the bonded contact. Since progressive failure can leadto the failure of interfaces which appear to be stable when only themean value of the peak strength is considered, the simplifiedMohreCoulomb model commonly used in LEM will result in anoverestimation of the interface shear resistance and thus damsafety.

Numerical methods, which allow for the incorporation of thedeformability of the materials and different sliding and openingcriteria for the interfaces, have been implemented for analyses ofconcrete dams. The constitutive models usually adopted for po-tential sliding surfaces in the dam body at the dam-foundationinterface and in the foundation are of the MohreCoulomb type,ruled by the friction angle and cohesion (Foster and Jones, 1994;

A. Krounis et al. / Journal of Rock Mechanics and Geotechnical Engineering 7 (2015) 659e667660

Dawson et al., 1998; Liu et al., 2003; Zhou et al., 2008; Chen and Du,2011; Jia et al., 2011; Sun et al., 2011). Linear elastic and nonlinearfracture mechanics models, governed by fracture toughness andfracture energy, are also used (Ebeling et al., 1997; Kishen, 2005;Saouma, 2006). When fracture mechanics models are used, thesemi-brittle behaviour of bonded interfaces is taken into account.However, when MohreCoulomb type constitutive models are usedfor the concrete-rock interface, the semi-brittle behaviour ofbonded contacts is rarely considered. Dawson et al. (1998) and Jiaet al. (2011) constituted the rare exceptions. Yet in none of thecases published has the spatial variation in cohesion been includedin the analysis, which means that its impact on the assessed damsafety is still uncertain. There is thus a need to generically study andquantitatively determine the impact of brittle failure in combina-tion with a possible variation in the bond strength on the assesseddam safety.

In this paper, the influence of a brittle material model in com-bination with spatial variation in cohesion on the assessed slidingstability of a hypothetical dam monolith, with dimensions similarto those of a typical Swedish concrete gravity dam, is investigatedusing numerical experiments. The reason put forward for usingnumerical methods is that physical observations of dam failure dueto sliding are almost impossible to be realized for economic, tech-nical and environmental reasons. The numerical experiments areperformed using the three-dimensional (3D) finite difference pro-gramme FLAC3D (Itasca Consulting Group Inc, 2011). MohreCoulomb constitutive material models are used for the interface toprovide a straightforward comparison with the factor of safety (FS)obtained using LEM. The distribution of cohesion is estimated fromresults of direct tensile tests on concrete-rock cores extracted froma concrete dam located in Sweden, and the spatial variation overthe sliding surface is taken into account through the use of randomfields. Comparisons are conducted between the numerical resultsobtained using ductile and brittle material models for the interfaceand values of FS determined analytically by LEM to analyse thediscrepancy between the different models. Finally, there is a dis-cussion on the results and how possible spatial variation in bondstrength could be incorporated into re-assessments of existingconcrete dams.

2. Factor of safety

In this section, the definition of FS used in this paper is pre-sented. The limit equilibrium and strength reduction techniques,used to determine FS in the analytical and numerical analyses,respectively, are also described. A detailed review of the twomethods for use in rock engineering and a comparison betweenthem are provided in Ureel and Momayez (2014). Previous studies,within the fields of geotechnical and rock engineering, wherecomparisons between FS obtained using LEM and the strengthreduction technique are conducted, can be found in Matsui and San(1992), Cala and Flisiak (2001), Cheng et al. (2007), Chen et al.(2014), etc.

2.1. Limit equilibrium method (LEM)

According to LEM, a structure is stable with regard to slidingwhen, for any potential sliding surface, the resultant shear stressrequired for equilibrium (s) is lower than the available shearstrength (sF). FS is thus determined as the ratio between thesequantities, i.e. FS ¼ sF/s.

Defining themaximumshear strength that canbemobilisedusingthe MohreCoulomb failure criterion, the shear strength availablelocally for each point of the concrete-rock interface is given by

sF ¼ cþ sN tan f (1)

where sN is the effective normal stress; and c and f are the cohe-sion and internal friction angle of the bonded interface, respec-tively. In order to estimate the shear force of the total interface (TF),it is assumed that the ultimate capacity is simultaneously achievedalong the entire sliding surface and the normal stress is integratedover the potential sliding plane. This gives

TF ¼ cAþ N0 tan f (2)

where N0is the resultant of the effective forces normal to the

assumed sliding plane including the effects of uplift, and A is thearea of the sliding surface. By also integrating the resultant shearstresses over the sliding plane, the global FS against sliding at theconcrete-rock interface, FSLEM, can be determined according to

FSLEM ¼ cAþ N0 tan f

H(3)

where H is the resultant of the horizontal loads acting on thestructure.

2.2. Strength reduction technique

There are two common techniques to determine failure due tosliding using numerical techniques: reducing the shear strengthof the interface until failure occurs, or increasing the appliedloads until failure occurs. Since increasing the load could lead toother types of failures, e.g. overturning, which are not consideredin this paper, the shear strength reduction technique (SRT) isapplied here. SRT is a popular technique when numericalmodelling for stability analyses of rock and soil slopes is usedand was employed as early as 1975 by Zienkiewicz et al. (1975).In the field of dam engineering, the technique has been appliedby Alonso et al. (1996), Liu et al. (2003), Zhou et al. (2008), Chenand Du (2011), and Jia et al. (2011) among others. In this study,the technique is implemented using the 3D explicit finite dif-ference code FLAC3D. FLAC3D and the two-dimensional (2D) codeFLAC have been widely used for numerical stability evaluationsof rock slopes (Sjöberg, 1999a,b; Latha and Garaga, 2010) and soilslopes (Zettler et al., 1999; Dawson and Roth, 1999), and havealso been applied in stability analyses of concrete dams byKieffer and Goodman (1999), Bu (2001), Léger and Javanmardi(2006), Gustafsson et al. (2010), and Yang et al. (2012) amongothers.

The analysis is initially carried out with the actual load andresistance parameters of the structure studied to establish theinitial stress conditions. After this, the resistance parameters of theinterface, f and c, are simultaneously and progressively decreasedby a factor RF according to Eqs. (4) and (5) while keeping theapplied load conditions unchanged. The computation continuesuntil failure occurs.

fRF ¼ arctan�tan f

RF

�(4)

cRF ¼ c=RF (5)

The interface strength just prior to failure can be considered asthe shear strength required for equilibrium. It then follows, fromthe definition of FS in Section 2.1, that the numerical FSSRT can bedetermined according to

A. Krounis et al. / Journal of Rock Mechanics and Geotechnical Engineering 7 (2015) 659e667 661

FSSRT ¼ cþ sN tan fc

RFeqþ sN

tan fRFeq

¼ RFeq (6)

where RFeq is the reduction factor at the limit bearing capacitystate.

Fig. 2. Schematic representations of the post-peak responses of various materialmodels.

3. Method for numerical analyses

3.1. Modelling procedure

Three stages of numerical analyses are performed (Fig. 1). Themain purpose of the first stage is to check the accuracy of theresult obtained by LEM compared to the numerical solution,where stresses and strains are included in the analysis. Theinterface is therefore assumed to follow the requirementsimposed by plastic-perfectly failure theory (Fig. 2). This is ach-ieved by using the default constitutive model provided in FLAC3D,defined by the linear Coulomb shear strength criterion and gov-erned by f, c, dilation angle (j), normal stiffness (Kn) and shearstiffness (Ks) of the interface.

In the second stage of the numerical analyses, the effect ofbrittle failure is studied. The elastic-brittle-plastic or strain-softening models (Fig. 2) are commonly adopted to simulate brit-tle rock failure (Hajiabdolmajid et al., 2002), a mechanism of thesame type as the brittle concrete-rock interface failure. In thisstudy, the elastic-brittle-plastic model is implemented for theinterface, and the peak shear strength is described by the MohreCoulomb failure criterion. In order to reproduce the behaviourdescribed, the “bond”model available in FLAC3D is used. The logic ofthe “bond” model means that the interface remains elastic beforethe bond strength and that the bond breaks if either the shear stressexceeds the shear strength or the tensile normal stress exceeds thenormal strength of the interface. The “bond” model is managed bythe tension, representing the normal bond strength, and the sbratio, i.e. the ratio between the shear and normal bond strengths.Because of the normal stress distribution and, in the followingstages, the spatial variation in c over the interface, the shear andtensile strengths of the different interface elements vary. The built-in programming language FISH is therefore used to calculate thetensile strength of each element (st,i) according to st,i ¼ ci/2 basedon Griffith’s failure criterion (Griffith, 1921). FISH is also used toestimate the sb ratio assigned to each element according tosbi ¼ sp,i/st,i, where sp,i is calculated using Eq. (1). The computationthen proceeds in the same way as described for the previous stage.

Fig. 1. Schematic of the

In stages 1 and 2, the concrete-rock interface is assumed to behomogeneous with regard to c, i.e. a single value, equal to thesample mean of c, is assigned to all interface elements. In order toanalyse the effect of a possible spatial variation in c on the peakshear strength, a third stage of calculations is performed. The“bond” model is used to represent the interface behaviour, and theanalysis is performed in the same way as described for stage 2.However, a random field is used in order to represent the fluctua-tion of c from point to point. Each point of the interface is thusassigned a different value of c, randomly chosen from a predefineddistribution function for the parameter. Since the location of ele-ments with high or low values of c in relation to the normal stressesalong the interface may affect the initiation and propagation offailure, stage 3 is carried out for several different realisations ofrandom fields generated from the same distribution function. Themore the realisations implemented, the greater the confidencethat can be placed in the estimated magnitude of the error.Nevertheless, due to extensive computational time, the number ofrealisations is limited to 10.

3.2. Definition of failure

When numerical methods are used for stability analyses, failureis commonly determined based on either the obtained de-formations or the occurrence of divergence. In the present study,

analyses outline.

A. Krounis et al. / Journal of Rock Mechanics and Geotechnical Engineering 7 (2015) 659e667662

the occurrence of a divergence (a simulation is considered to haveconverged when the normalised unbalanced force of every node inthe mesh is less than 10�5 within 200,000 iterations) is regarded asthe loss of the static-force equilibrium state and is taken as thefailure condition. The reason is that the technique where failure isdefined based on the inflection point of the deformation curve ismore susceptible for human errors. However, since the choice of200,000 as the iteration ceiling is subjective (it can be mentionedthat at a balanced state, the number of iterations required to reachstable conditions, once the shear strength properties have beenreduced by 0.1, varies between approximately 35,000 at earlystages to 150,000 just prior to failure), bonding conditions anddeformations at the interface of the dam are also checked in orderto verify the results obtained.

3.3. Limitations

One limitation associated with the methodology used is thesimultaneous reduction of all strength parameters, both peak andresidual. Once the peak shear strength of an interface element isreached, the residual shear strength of the node has been reducedto a lower value than the initial residual shear strength of theelement. This means that the stress redistribution may not beentirely representative of the real stress redistribution and thathigher stresses than that in reality may be assigned to nearby el-ements and cause them to yield.

It should also be mentioned that, for most real dams, it isreasonable to assume that the concrete-rock interface is partiallybonded (Lo et al., 1990). Yet, in this paper, the entire area under thedam monolith is assumed to be bonded, which may deviate fromreal conditions.

4. Application to dam monolith

4.1. Input

4.1.1. Geometry and meshAn outline of the hypothetical dam body used in the analyses is

shown in Fig. 3. The dam monolith is 30 m high and 8 mwide. Thedimensions of the dam monolith are based on the geometry oftypical Swedish concrete dams and were chosen so that the entirebase of the monolith will be under compression. The numericalmodel of the dam also includes part of the foundation, 45 m in

Fig. 3. Illustration of the dam monolith geometry. The dam monolith is 8 m wide.

depth, 101 m in length and 8 m in width. The dam and foundationare modelled using solid elements, while the dam-foundationcontact is modelled using specific interface elements. Each inter-face element has a size of 0.1 m � 0.1 m in order to be represen-tative of the measurement data, i.e. drilled core size, to minimisepossible scale effects and transformation errors in the determina-tion of the material properties. The mesh adopted in the analyses isshown in Fig. 4.

4.1.2. Loads and boundary conditionsA fixed displacement constraint is applied to the bottom

boundary while normal displacement constraints are applied to theside boundaries of the foundation. Only one load case, with thewater level at the retention level, is considered. The loads includedin the analyses are a horizontal water load acting on the upstreamface of the dam monolith (W), uplift (U) and deadweight of theconcrete monolith (G). In FLAC3D, W and U are applied as externalpressures to the upstream face of the dam monolith and the fullarea of the base of the dam, respectively. The uplift pressure isassumed to vary linearly from a hydrostatic pressure based on theretention level at the upstream face to zero pressure at thedownstream face, while W varies linearly from a hydrostaticpressure based on the retention water at the bottom of the reser-voir to zero pressure at the water surface. The relation G ¼ Vcgc,where Vc is the volume of the monolith and gc is the unit weight ofthe concrete, is used to determine G.

The concrete elastic modulus (Ec), Poisson’s ratio (nc), and theunit weight (gc) are set to 30 GPa, 0.2, and 24 kN/m3, respectively.The dam monolith is assumed to be located on a strong rockfoundation. The rock elastic modulus (Er), Poisson’s ratio (nr), andthe unit weight (gr) are set to 50 GPa, 0.2, and 26 kN/m3,respectively.

The internal friction angle of the bonded concrete-rock interface(fi) is assumed to be equal to that reported by EPRI (1992),fi,peak ¼ 54�. Based on the results presented by Lo et al. (1990), theresidual friction angle is set to fi,residual ¼ 45�.

A population mean of mc ¼ 1.44 MPa and a population varianceof s2c ¼ 0:94 are estimated for c of the bonded interface from theresults of direct tensile tests on four bonded cores extracted from aconcrete gravity dam located in northern Sweden. Although basedon a small number of tests, the results seem reasonable comparedto those presented by EPRI (1992). A lognormal distribution ischosen to avoid unrealistic negative values.

In stages 1 and 2, c is assumed to be equal to mc. In stage 3, thedistribution function of c is used to generate the random fieldsutilized. A measure to describe the degree of spatial correlation of cis also necessary to generate the fields. Such a measure is the cor-relation length, also called the scale of fluctuation (e.g. Vanmarcke,2010). Unfortunately, the magnitude of the correlation length ofcohesion (qc) at existing dams is generally unknown. Therefore, twodifferent correlation distances, qc ¼ 0m and qc ¼ 10m, are assumedfor comparison. The correlation lengths are taken to be statisticallyisotropic.

Realisations of the cohesion fields are made in the computersoftware R! (Hornik, 2006) using the Random Fields package(Schlater, 2001) that simulates Gaussian random fields. Since c isassumed to follow a lognormal distribution, the first step in therealisation procedure is to determine the mean value (mlnc) andvariance ðs2lncÞ of lnc based on the relationship between the normaland lognormal random variables. Then random fields are generatedfor lnc and transformed back into c according to c ¼ elnc. Fig. 5ashows a realisation of a random field with correlation length of 0 mwhere high values of cohesion are randomly mixed with lowervalues. A realisation of a random field with correlation length of10 m is shown in Fig. 5b.

Fig. 4. Finite difference mesh adopted in the study.

Fig. 5. Realisation of c with (a) qc ¼ 0 m and (b) qc ¼ 10 m.

A. Krounis et al. / Journal of Rock Mechanics and Geotechnical Engineering 7 (2015) 659e667 663

The empirical relation proposed by Barton and Choubey (1977) isused to determine the initial value of Ks:

Ks ¼ 100sfL

(7)

where sf is the peak shear strength, and L is the critical joint length.However, since sF of the concrete-rock interface is not homoge-neous, Ks cannot be defined by a single value. Eq. (4) is thereforeapplied separately to each element along the interface in the nu-merical analyses. This is done by first deciding an initial value forthe entire interface based on the mean values of c and fi, where theestimated average s0N of the interface, based on the assumption ofequal loading conditions, is obtained by dividing the sum of allvertical forces (G and U) by the total area of the interface. Theestimated initial value is then used as input in the numericalanalysis where the stress conditions over the interface are esti-mated. Once the initial stress distribution over the interface hasbeen calculated, Ks of each element is updated based on c and thecalculated value of s0N for the specific element. This is an iterativeprocess, repeated for every step, which may cause stress

redistribution along the interface. The initial value is set toKs ¼ 0.19 MPa/mm based on L ¼ 1 m and s0N ¼ 0.35 MPa. Since Eq.(4) is based on the behaviour of rock joints, the reasonableness ofthe estimated value of Ks is checked by comparing it with the re-sults presented by Saiang et al. (2005) for shotcrete-trachyte andshotcrete-magnetite cores. For the current range of normal stresses,the estimated initial value of Ks is within the limits of the valuespresented by Saiang et al. (2005).

The normal stiffness to shear stiffness ratio R ¼ Kn/Ks is used todetermine Kn. In this paper, a ratio of 10, which lies within theinterval of values presented by Bandis et al. (1983), is assumed, andthis leads to an initial value of Kn ¼ 1.92 MPa/mm.

4.2. Results

For the monolith studied, FSLEM ¼ 9.6 is obtained. According tothe normal stress distribution at the concrete-rock interface,determined using Navier’s formula, the entire base of the dam isunder compression so there is no need to update the uplift pressuredue to a potential tensile crack at the heel of the dam.

In order for the dam monolith to reach failure in stage 1, theinitial shear strength parameters of the interface are reduced by afactor of 10.4 under current loading conditions. Thus, the factors ofsafety against sliding obtained using LEM and SRT with a homo-geneous interface and a ductile constitutive model for the interfaceagree reasonably well, with FSSRT being slightly higher than FSLEM.The numerical results also show that, under normal loading con-ditions and initial shear strength values, the dammonolith is stablewith regard to all relevant failure modes, i.e. sliding, overturningand crushing of the foundation or concrete, and the entire base ofthe dam is under compression, in agreement with the analyticalcalculations.

In stage 2, when the brittle material model is used to describethe failure behaviour of the homogeneous interface, a somewhatlower value of FSSRT ¼ 8.1, which supports the previous findingsthat numerical models incorporating the FLAC “bond” model castlower safety factors than LEM models for gravity dams (Altarejos-Garcia et al., 2012), is obtained. The reason for this is that due tothe brittle material model, the post-peak load that an element ofthe interface can carry is lower than the estimated peak shearstrength. Thus, once the peak shear strength of an interfaceelement is exceeded, the stresses around the element will increase.This may cause nearby points to yield, resulting in further stress

Table 2A summary of the mean values of c for random fields, used as input in the numericalanalyses in stage 3, the corresponding FSLEM and FSSRT and the deviations betweenFSLEM and FSSRT for realisations with qc ¼ 10 m.

Realisation mc,random field (MPa) FSLEM FSSRT j(FSSRT � FSLEM)/FSSRTj (%)1 1.17 8.3 5.3 56.62 1.28 8.8 6 54.43 1.72 11 7.2 52.84 1.46 9.7 6.3 545 3.07 17.8 11.4 56.16 1.35 9.2 6 53.37 1.16 8.2 5.8 41.48 1.26 8.7 5.5 58.29 0.76 6.2 4.1 51.2

10 0.77 6.3 4.2 50

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 μc,random field (MPa)

100908070605040302010

0|(FS S

RT−FS

LEM

)/FS S

RT|

(%) Stage 2 Stage 3, θc=0 m Stage 3, θc=10 m

Fig. 6. Deviations between FSLEM and FSSRT plotted against the mean value of interfacecohesion for each realisation.

A. Krounis et al. / Journal of Rock Mechanics and Geotechnical Engineering 7 (2015) 659e667664

redistributions and so the process continues in a consecutivemanner until the surface of failure extends to the point where ki-nematic release becomes possible as shown in previous studies ofdam stability (Bolzon, 2010; Jia et al., 2011). A comparison betweenFSLEM and FSSRT obtained in stage 2 shows that for a homogeneousinterface, the assumption of ductile failure results, for the analysedmonolith, in an overestimation of FSLEM by approximately 20%.

FSSRT values calculated in stage 3, when the spatial variation incohesion over the interface is included in the analyses, togetherwith FSLEM for each realisation are given in Tables 1 and 2 forqc ¼ 0 m and qc ¼ 10 m, respectively. The deviations between FSLEMand FSSRT for each realisation, representative of the error associatedwith Eq. (4), are also presented. When qc ¼ 0 m, the deviation be-tween FSLEM and FSSRT varies between 46.8% and 66.1%, with anaverage of 56.7% and standard deviation of 5.9%. When qc ¼ 10 m isassumed for cohesion, the deviation between FSLEM and FSSRT variesbetween 41.4% and 58.2%, with an average of 52% and standarddeviation of 5%.

In Fig. 6, the deviations between FSLEM and FSSRT are plottedagainst mc,random field for each realisation. The deviation betweenFSLEM and FSSRT obtained in stage 2 is also shown in the figure. It canbe seen that no overall trend can be detected for the different cases.The deviations obtained when qc ¼ 0m follow a linear trend, wherethe deviation increases with themean value of c. This is due to FSSRTbeing nearly constant for all realisations (Table 1) while FSLEM iscorrelated to mc,random field. When qc ¼ 10 m is assumed, the de-viations calculated exhibit a more random scatter. From the valuespresented in Table 2, it seems that, although FSSRT in general in-creases with the mean value of c, the degree of correlation isslightly lower than that for FSLEM, which results in the patternshown in Fig. 6.

In Fig. 7 the debonding of the concrete-rock interface due todecreased shear strength is plotted against RF for stage 2 and twotypical cases of stage 3. It can be seen here that, for a homogeneousinterface, the debonding process initiates at the upstream part ofthe interface, where sN is at its lowest, for RF close to RFeq and thenalmost immediately propagates to the rest of the interface. Whenthe spatial variation in c is taken into account, the debondingprocess is initiated much earlier. Furthermore, from the start, thedebonding is scattered over the entire interface. Nonetheless, whenqc ¼ 0 m is assumed, the debonding is completely randomlydistributed; while for qc ¼ 10 m, a pattern where nearby elementsdebond at the same time can be detected. This indicates that thelocation and size of the clusters of elements with related values of chave an impact on the failure process of the interface, which couldexplain the somewhat lower degree of association between FSSRTand mc,random field than that between FSLEM and mc,random field,resulting in the scatter observed in Fig. 6.

To check the reasonableness of the obtained results, the defor-mation of the monolith was also controlled for the different stages.

Table 1A summary of the mean values of c for random fields, used as input in the numericalanalyses in stage 3, the corresponding FSLEM and FSSRT and the deviations betweenFSLEM and FSSRT for realisations with qc ¼ 0 m.

Realisation mc,random field (MPa) FSLEM FSSRT j(FSSRT � FSLEM)/FSSRTj (%)1 1.34 9.1 6.2 46.82 1.39 9.4 6.1 54.13 1.38 9.3 6.1 52.54 1.32 9 6 505 1.43 9.6 6.1 57.46 1.57 10.3 6.2 66.17 1.49 9.9 6.1 62.38 1.45 9.7 6.1 599 1.52 10 6.2 61.3

10 1.43 9.6 6.1 57.4

Fig. 8 shows typical deformation curves. It can be seen from thefigure that the obtained deformations just prior to failure areapproximately 1.5 mm. These deformations are on the high end ofthe range measured and presented by Saiang et al. (2005), forlaboratory samples with bonded interfaces. However, since theyrepresented the deformation of a dam monolith with significantlylarger dimensions than regular laboratory test samples, they aredeemed as reasonable. It can also be seen that, as expected, definingfailure based on deformations generates a somewhat lower FSSRTthan that when the convergence criterion is used. Nevertheless, thediscrepancy is nearly negligible and the obtained values of FSSRT aretherefore considered acceptable.

5. Discussion

5.1. Application example

The results from the numerical analyses show that the analyticalLEM, commonly used in dam stability analyses, results in an over-estimation of the assessed dam stability with regard to sliding. Theresults also indicate that spatial variation in c along the interfaceresults in weak spots where failure can be initiated, in contrast to ahomogeneous interface where failure is initiated at the locationwith lowest normal stress, and further diminishs the validity of theductile model, leading to an overestimation of the factor of safetywith regard to sliding by approximately 40%e65%. From the spreadof the results presented in Section 3.2, both within and betweendifferent cases, it is evident that, for the monolith studied, theuncertainty associated with Eq. (4) cannot be represented by oneconstant value unless the exact composition of the interface isknown.

Although the results presented in this paper are only valid forthe specific dammonolith under the stated boundary conditions, itis reasonable to believe that a possible spatial variation in c may

100

80

60

R

θ

θc = 0 m, R5 Stage 3, θ = 10 m, R4

Fig. 7. Debonding of the contact area due to gradual shear strength reduction. Slip now marks the elements that have debonded during the current step of the numerical analysis,slipped in past marks elements that have debonded during previous steps with lower RF, and no slip marks elements where the bond is still intact. R4 and R5 mean realisations 4and 5, respectively.

A. Krounis et al. / Journal of Rock Mechanics and Geotechnical Engineering 7 (2015) 659e667 665

also have a significant influence on the behaviour of other dam-foundation systems with bonded or partly bonded concrete-rockinterfaces. It follows that a reliable framework for incorporatingthe uncertainties associated with the erroneous assumptions ofductile failure in combination with spatial variation in cohesion isrequired in order to avoid an overestimation of the sliding stabilityof gravity dams with bonded concrete-rock interfaces.

5.2. Incorporation of spatial variation in c in sliding stabilityanalyses of existing dams

A majority of the national regulations/guidelines for dam safetyassessment and reassessment available today are based on deter-ministic techniques and uncertainties are generally dealt with byusing target safety factors. However, in the Swedish guidelines(SwedEnergy, 2012), uncertainties related to the shear strength of

bonded contacts are regarded as so significant that cohesion isdisregardedwhen sliding stability assessments of existing dams areperformed. This approach is on the safe side but may lead toexpensive and unnecessary strengthening of dams. An alternativeapproach for dealing with the greater uncertainties associated withcohesion is to apply separate, higher partial safety factors tocohesion than to the frictional component. According to Ruggeriet al. (2004), this technique has been adopted in the Spanish, Por-tuguese, Chinese, Indian, French, and Swiss regulations. In otherregulations/guidelines, where a single global FS is used, it is insteadcommon to refer to higher target values for FS when the effect ofcohesion is included in the overall shear strength of the interfacethan that when only frictional strength is considered (FERC, 2002;CDA, 2007). Nonetheless, it is unclear to what extent the higherreduction factors applied to the cohesion component and thehigher target values of global FS reflect the error associatedwith the

Fig. 8. Horizontal displacements vs. reduction factor for typical cases.

A. Krounis et al. / Journal of Rock Mechanics and Geotechnical Engineering 7 (2015) 659e667666

applied model for the shear strength since they are mainly deter-mined based on experience and not on actual knowledge about theimpact of the model error on the assessed safety of the dam.Furthermore, the uncertainty associated with the design values ofthe shear strength properties, arising not only from the naturalvariation of the parameters but also from the scarce amount of dataupon which they are commonly determined, has also to be takeninto account by the same FS. It is thus questionable whether auniform level of safety can be achieved by using a single, constantvalue of FS without regard to the degree of uncertainty involved inthe calculation.

A more robust framework for incorporating uncertainties in theanalyses is provided in probability-based methods such as struc-tural reliability analysis. Such methods have not been used exten-sively in dam safety assessments; however, in recent years, interestin using probability-based approaches for dam safety assessmentand reassessment has increased. Probability-based approacheshave been taken into account in various regulations or guidelines(ANCOLD, 2003; Hartford and Baecher, 2004; ICOLD, 2005; USBR,2011; SPANCOLD, 2013). Moreover, one of the main themes at the11th ICOLD Benchmark Workshop on Numerical Analysis of Damsdealt with estimation of the probability of failure of a gravity damfor the sliding failure mode (ICOLD, 2011). A review of publishedpapers in which probabilistic analysis of the sliding failure mode isperformed, including de Araújo and Awruch (1998), Ellingwoodand Tekie (2001), Saouma (2006), Altarejos et al. (2009), Lupoiand Callari (2010, 2012), ICOLD (2011), and Westberg Wilde andJohansson (2013), showed that the cohesion/adhesion of theconcrete-rock interface is commonly included in the evaluation ofthe peak shear strength without taking the progressive failuremechanism into consideration. The only exceptions in the afore-mentioned publications were Saouma (2006) and Westberg Wildeand Johansson (2013), who included the progressive failure

mechanism of bonded interfaces in their analyses. Saouma (2006),however, did not include the effect of a possible spatial variation incohesion, while Westberg Wilde and Johansson (2013) did not takeinto account the initial stress distribution, the stress redistributionsor the stiffness of the interface.

Nonetheless, probability-based methods are subjected to thesame uncertainty regarding the degree to which the appliedmodel mimics reality as the traditional deterministic LEM. Thus,for probability-based methods to be considered, a reliable optionfor sliding reassessment of dams with bonded concrete-rock in-terfaces, accurate modelling of the uncertainty associated withprogressive failure is required. JCSS (2001), among others, sug-gested that deviations between real behaviour and the behaviourpredicted by the model can be dealt with by introducing arandom variable representative of the uncertainty associatedwith a specific model. The numerical experiments presented inthis paper can be used as a starting point for quantifying such arandom variable representative of the uncertainty associatedwith the brittle material model in combination with spatialvariation in c.

6. Concluding remarks

Based on the results presented in this paper, it can be concludedthat it is not enough to only consider the brittle material model andmodel cohesion with a deterministic value; the influence of po-tential weak spots along the interface has to be taken into accountsince it further reduces the overall shear strength of the interfacesignificantly.

It can also be concluded that, if simple analytical methods suchas LEM are to be used for dam safety assessment and reassessment,further research is required to determine whether the targetvalues of FS commonly used today are representative of the un-certainty related to the interface properties and the simplifiedmodel used for defining the shear strength of bonded concrete-rock interfaces.

The limit state function commonly used in most probability-based methods is based on the same failure model as that used inanalytical deterministic analyses. Probability-based methods arethus also subjected to the same uncertainties as LEM is. Therefore, areliable and preferably simple methodology for including thisparticular uncertainty needs to be developed. The numerical ex-periments presented herein can be used as a starting point for sucha methodology.

Conflict of interest

The authors wish to confirm that there are no known conflicts ofinterest associated with this publication and there has been nosignificant financial support for this work that could have influ-enced its outcome.

Acknowledgements

The research presented was carried out as a part of “SwedishHydropower Centre e SVC”. SVC has been established by theSwedish Energy Agency, Elforsk and Svenska Kraftnät togetherwith Luleå University of Technology, KTH Royal Institute of Tech-nology, Chalmers University of Technology and Uppsala University.The authors also wish to acknowledge Diego Mas Ivar at ItascaConsultants AB for offering invaluable advice concerning FLAC3D

that greatly assisted the research, although any errors and the in-terpretations/conclusions of this paper are those of the authors.

A. Krounis et al. / Journal of Rock Mechanics and Geotechnical Engineering 7 (2015) 659e667 667

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