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Harmonising Rock Engineering and the Environment–Qian&Zhou(eds)©2012Taylor&Francis Group,London,ISBN978-0-415-80444-8 Probabilistic assessment of stability of underground rock caverns and cavern shapeoptimizationW.G.Zhang&A.T.C.GohSchool of Civil and Environment Engineering,Nanyang Technological University,SingaporeJ.Y.K.WongDefence Science&Technology Agency,SingaporeABSTRACT:The global stability of underground rock cavern excavated in a Mohr-Coulomb material is investigated by means of Universal Distinct Element Code(UDEC)in this paper.The following stochastic variables are considered:the friction angle, the cohesion,the deformation modulus of the rock mass and the in-situ stress ratio.The overburden thickness,the unit weight of rock materials,the Poisson’s ratio and the joint strength are assumed as deterministic.The cavern width and height are also assumed as uncertain variables in order to optimize the shape of the rock cavern.Most cavern shapes are horse-shoe or bullet-head shaped.It is widely argued that the use of a flat-arch cavern would make the best use of cavern space.Thus for this study,the initial cavern width and height are set as30m and18m,respectively.The influences of the flattening process on cavern stability can be investigated through incremental increases in the cavern width and reduction in the cavern height through six design levels to assess the changes of safety factor and probability of failure.For each configuration of the six levels, the probability of failure is determined by Monte Carlo simulation incorporated by neural network results.The configuration satisfying the critical safety factor and the expected performance level with the flattest cavern roof can be termed as the optimal design.It is also suggested that the critical factor of safety and the targeted performance level be used together,as complementary measures of acceptable design.Subject:Analysis techniques and Design methodsKeywords:neural network;numerical modeling;risks and hazards1INTRODUCTIONOne of the major considerations in the design of an under-ground rock cavern is the evaluation of its stability since the rock excavation causes a redistribution of the stresses in the proximity of the underground mon numerical methods used to evaluate cavern stability can be categorized as continuum methods such as the Finite Element Method (FEM)(Ren et al.,2005,Meguid and Rowe,2006,Xia et al., 2007,Cai,2008)and Finite Difference Method(FDM)(Cai, 2008,Roth et al.,2001,Hatzor et al.,2002,Alonso et al., 2006,Wang and Zhu,2006,Stavropoulou et al.,2007),and discontinuum methods such as the Distinct Element Method (DEM)(Cundall,1971,1976)and the Discontinuous Defor-mation Analysis(DDA)(Shi,1988).There are no universal quantitative guidelines to determine when one method should be used instead of the other(Bobet et al.,2009). Conventional deterministic evaluation of stability of geotechnical structures and underground openings involves the use of a factor of safety which cannot predict the state of system with absolute certainty.The alternative is to assess the reliability indexβor the probability of‘failure’P f, in which process the determination of the joint probability distribution of R and S and the integration of the proba-bility density function(PDF)over the failure domain are involved.Various methods have been proposed to estimate βand(or)P f.Among the most widely used methods are the First Order Reliability Method(FORM),First Order Second Moment(FOSM)method,and the direct Monte Carlo Sim-ulation(MCS)method.For reference,typicalβand P f for Table1.Target reliability indices(U.S.Army Corps of Engineers, 1997).Reliability Probability of failure,Expected index,βP f=1− (−β)performance level 1.00.16Hazardous1.50.07Unsatisfactory2.00.023Poor2.50.006Below average3.00.001Above average4.00.00003Good5.00.0000003High representative geotechnical components and systems and their expected performance levels are listed in Table1.For stability considerations,most cavern shapes are designed as horse-shoe or bullet-head shaped.However,it is widely argued that the use of a flat-arch cavern will make the best use of cavern space.Consequently,the influences of the flattening process on cavern stability can be investigated through incremental increases in the cavern width and reduc-tion in the cavern height to assess the changes of safety factor and probability of failure.This paper presents a practical approach to underground rock cavern stability reliability analysis that combines the Back Propagation Neural Network(BPNN)with MCS.The global stability of underground rock cavern excavated in a Mohr-Coulomb material is deterministically investigated for6different design levels by means of numericalFigure 1.Cavern configuration and boundaryfixity.Figure 2.Configurations for the cavern flattening process.experimentation.Based on numerical results,the performance function can be obtained through the use of BPNN.Perfor-mance level can be determined from rock cavern stability reliability assessment by means of MCS.The effects of the flattening process on cavern stability will be explored by com-paring the calculated safety factor and probability of failure of these 6levels.2NUMERICAL METHOD FOR ROCK CA VERN The Universal Distinct Element Code (UDEC)was adopted to carry out the stability analyses of the underground rock cavern using the shear strength reduction (SSR)technique (Dawson et al.,1999,Griffiths and Lane,1999,Hammah et al.,2004,2005a,2005b,2006,Matsui and San,1992).In conventional SSR analysis,a factor of safety is obtained by systemati-cally reducing (or increasing)the shear strengths of soil or rock materials until stress-induced failure occurs.It has been demonstrated that the SSR method works for a wide range of problems,including stability problems in blocky rock masses (Hammah et al.,2007a)and cavern problems (Hammah et al.,2007b).Both stress-controlled and structure-controlled instabilities were considered in this paper.For simplicity,two sets of joints angled 45◦and −80◦respectively,were considered as the only structural features in the model.The cross-section of the underground cavern configuration and the boundary fixity conditions are shown in Figure 1.The cavern roof flattening process is shown in Figure 2.The width is increased from 30m to 40m while the height is reduced from 18m to 15m,making the cavern wider and flatter.The cavern roof arcs are ellipti-cal and the heights of roof arcs reduce from 6m to 3m with a constant side wall height 12m.The thickness D from the ground surface to the top of arc crown is fixed at 60m and the overburden T is fixed at T =40m.Full-face excavation wasTable 2.Values and distributions of input variables.VariablesDistributions Statistics Intact rock properties Unit weight γ(kN /m 3)D27Y oung’s modulus E (GPa)Normal Mean =8,Cov =25%Poisson’s ratio υD0.25Cohesion c (MPa)Normal Mean =0.4,Cov =30%Friction angle φ(◦)Normal Mean =31,Cov =20%Tensile strength σt (MPa)D 11.37Dilation angle ϕ(◦)D 0Joint propertiesOrient.of joint sets 1/2(◦)D 45/−80Spacing of joint sets 1/2(m)D 8/12Normal stiffness k n (GPa/m)D 10Shear stiffness k s (GPa/m)D 10Cohesion c j (MPa)D 0.2Friction angle φj (◦)D 35Other inputsIn situ stress ratio k 0Normal Mean =2.5,Cov =20%Overburden pressureD1.422MPaconsidered and the limiting values for over-break and under-break are assumed as 500mm.Table 2lists the values and distributions of input variables (deterministic or probabilistic)considered.To investigate the influences of the flattening process on cavern stability,and also to take into account the uncertainties of some input variables,in each configuration of the six levels,two design combinations were considered and denoted by the “+”and “−”notations to represent the low and high values of each uncertain input variable.A high value is x h =µ+1.645σ(σ=µ×Cov)where µis the mean value and a low value is x l =µ−1.645σ.Factor combinations for the 4uncertain input variables are assumed as in the order of cohesion c ,friction angle φ,Y oung’s modulus E and in situ stress ratio k 0.To illustrate,a FS +−−+value means a safety factor value obtained at a high cohesion value,a low friction angle,a low Y oung’s modulus value and a high in situ stress ratio together with other deterministic inputs for each one of the 6config-urations.Thus for each configuration,24DEM experiments are needed.3DETERMINATIONS OF LIMIT STATE FUNCTION A total of 96cases considering different cavern configura-tions and different parameters were numerically simulated and the calculated FS values are shown in Figure 3.Gen-eral speaking,for the same input parameters,a larger cavern span means a smaller safety factor.A few exceptions can be observed from some design parameter combinations such as FS ++++,FS ++−+,FS −+++,FS −+−+,FS −++−,FS −+−−and FS −−−−.The influence from individual parameter on the safety factor can be quite large,for example,from 2.94of FS ++−+to 1.42of FS −+−+for configuration 1.The influ-ence can also be very small,for instance,from 1.75of FS +−++to 1.75of FS +−−+for configuration 5.But for the prob-lem analyzed in this paper,the safety factor is a function ofFigure3.DEM Factor of safety results.these4uncertain input variables,although no closed form limit state function of FS exists.Consequently,some numer-ical approaches should be proposed to obtain the implicit performance function.Among the most widely used methods are the response sur-face method(RSM)and the back propagation neural network (BPNN).For the BPNN used in limit state function approx-imation in geotechnical engineering,refer to(Goh,1994, Goh and Kulhawy,2003).Here a three-layered BPNN with 2hidden neurons,“tansig”and“purelin”transfer functions is adopted for limit state function approximation.A multiple linear regression is also shown for comparison in Figure4. The results show the BPNN performs better than the multiple regression approach.The multiple linear regression equation with a coefficient of correlation R2=0.95was expressedas:Using the optimal trained connection weights,it is possi-ble to develop a mathematical expression relating the input variables and the output variable(predicted FS)(Goh and Kulhawy,2003).After obtaining the implicit limit state function,it can be used for stability reliability analysis.4RELIABILITY ANAL YSISAssuming the limiting factor of safety FS lim=1,the limit state function is defined as G(x)=FS NN−1and incorporated into an@risk added-in excel sheet for Monte Carlo simulation, as shown in Figure5.The columns C,D,E and F are filled in with BPNN inter-layered weights,bias values and transfer functions.Figure6shows the P f values for various mean m c and Cov c values with different levels.The P f increases for increasing Cov c values and decreasing m c values.For the same cavern geometry,the P f value calculated from Cov c=0.3is larger than that from Cov c=0.2as a large Cov value means more uncertainty.This difference becomes more obvious as m c increases.It can be concluded that both m c and Cov c values significantly influence the P f values.Figure7shows the P f values for various m tan(φ)and Cov tan(φ)values with different levels.The P f increasesfor Figure4.The BPNN and regression results.increasing Cov tan(φ)values and decreasing m tan(φ)values.Compared with cohesion c,m tan(φ)significantly influences theP f value.But the effect from Cov tan(φ)is not so considerable.Figure8shows the P f values for various m E and Cov E val-ues with different levels.The P f increases for increasing m Evalues.It is interesting to note that the Cov E values have littleinfluence on the P f value.But theoretically,P f value shouldincrease as the Cov E increases from0.15to0.25.The effect is,however,so minimal that it is dominated by the MCS“noise”.Figure9shows the P f values for various m k0and Cov k0val-ues with different levels.It is observed that the uncertainty ofk0contributes the least to the P f value.For m k0,an interestingthing is that for the B=30m case,the P f decreases slightly forincreasing m k0values;while for the B=36m,B=38m andB=40m cases,the P f increases slightly for increasing m k0values;for the B=32m and B=34m cases,P f decreases atm k0=2.33and then slightly increases.The influence from m k0and Cov k0may be sensitive to the joint orientation andspacing,which will be further investigated in later research.Figures6–9show that a sudden increase of P f values hap-pens from design level2(B=32,H=18)to design level3(B=34,H=18),by one order of magnitude.It is generalpractice that design level2should be adopted as the pre-ferred choice.However,it fails to take into account the targetedFigure 5.P f calculation using BPNN incorporatedMCS.Figure 6.Effect of m c and Cov c on P f.Figure 7.Effect of m tan(φ)and Cov tan(φ)on P f.Figure 8.Effect of m E and Cov E on P f.Figure 9.Effect of m k0and Cov k0on P f.Figure 10.The choice of the optimal design level.performance levels.Consequently,Figure 10gives suggested procedure for rock cavern shape determination.It is based on the expected performance level (the calculated probability of failure)and the factor of safety.For the case study in this paper,if a targeted performance level of “Good”(P f =0.00003)isto be met,the only possible choice is design level1(B=30mand H=18m).If a targeted performance level of“Belowaverage”(P f=0.006)is to be met,design level1(B=30m and H=18m),level2(B=32m and H=18m)and level3(B=34m and H=18m)are potential options;the choice ofa certain design depends on the determination of the criticalfactor of safety and how conservative one expects the designto be.5SUMMAR Y AND CONCLUSIONSNumerical analyses have been carried out using the Dis-tinct Element Method to assess the stress-induced stabilityof underground rock caverns.Analyses using neural net-works were used to model the relationship between the cavernsafety factor FS and c,φ,E and k0.Monte Carlo simula-tion is combined with the optimal trained neural networkmodel to calculate the probability of failure and to conductthe parametric study.The reliability analyses indicated that the probability of fail-ure is significantly influenced by the mean values and thecoefficients of variation of c andφ.This confirms the point thata single global safety factor cannot provide the complete infor-mation regarding the cavern stability as the same safety factormay indicate completely different probabilities of failure fordifferent coefficient of variations of c andφ.In addition,it isfound that the P f value is considerably influenced by the meanvalue of Y oung’s modulus.However,there remains a need forfurther investigation on whether influence on P f values fromin situ stress ratio are sensitive to the joint orientation andspacing.It is suggested that the critical factor of safety and thetargeted performance level(probability of failure)be usedtogether,as complementary measures of acceptable design.This study will be extended to consider other relevantexcavation factors such as excavation sequence and rockbolt system.Serviceability limit state(SLS)will also beconsidered.ACKNOWLEDGEMENTSThe authors thank the Defence Science and TechnologyAgency Singapore for providing the research funding for thisproject.REFERENCESAlonso,E.,Alejano,L.R.&Varas,F.2006.Observations on bifurca-tion and localization phenomena around an excavation with FLAC. In FLAC and Numerical Modeling in Geomechanics,Madrid; Paper No.08-08.Bobet,A.et al.2009.Numerical models in discontinuous media:A review of advances for rock mechanics applications.J Geotech Geoviron;135(11):1547–1561.Cai,M.2008.Influence of stress path on tunnel excavation response-Numerical tools election and modeling strategy.Tunnelling Under-ground Space Technol;23:618–628.Cundall,P.A.1971.A computer model for simulating progressive large scale movements in blocky rock systems.In Proceedings of the Symposium of the International Society of Rock Mechanics, Nancy;Paper No.II-8.Cundall,P.A.1976.Explicit finite difference methods in geomechan-ics,numerical methods in engineering.In Proceedings of the EF Conference on Numerical Methods in Geomechanics,Blacksburg, Virginia;1:132–150.Dawson,E.M.,Roth,W.H.&Drescher,A.1999.Slope stability analysis by strength reduction,Geotechnique;49(6):835–840. Goh,A.T.C.1994.Back-propagation neural networks for model-ing complex systems.Artificial Intelligence in Engineering;9: 143–151.Goh,A.T.C&Kulhawy,F.H.2003.Neural network approach to model the limit state surface for reliability analysis.Can Geotech J;40:1235–1244.Griffiths,D.V.&Lane,P.A.1999.Slope Stability Analysis by Finite Elements.Geotechnique;49(3):387–403.Hammah,R.E.et al.2004.Stability Analysis of Rock Slopes using the Finite Element Method.In Proceedings of the ISRM Regional Symposium EUROCK2004and the53rd Geomechanics Colloquy, Salzburg,Austria.Hammah,R.E.et al.2005a.The Shear Strength Reduction Method for the Generalized Hoek-Brown Criterion.In Proceedings of the40th U.S.Symposium on Rock Mechanics,Alaska Rocks, Anchorage,Alaska.Hammah,R.E.et al.2005b.A comparison of finite element slope stability analysis with conventional limit-equilibrium investiga-tion.In Proceedings of the58th Canadian Geotechnical and6th Joint IAH-CNC and CGS Groundwater Specialty Conferences–GeoSask,Saskatoon,Canada.Hammah,R.E.,Y acoub,T.E.&Curran,J.H.2006.Investigating the performance of the shear strength reduction(SSR)method on the analysis of reinforced slopes.In Proceedings of the59th Cana-dian Geotechnical and7th Joint IAH-CNC and CGS Groundwater Specialty Conferences–Sea to Sky Geotechnique.Vancouver, Canada.Hammah R.E.et al.2007a.Analysis of Blocky Rock Slopes with Finite Element Shear Strength ReductionAnalysis.In Proceedings of the1st Canada-U.S.Rock Mechanics Symposium.Vancouver, Canada.Hammah,R.E.,Y acoub,T.E.&Curran,J.H.2007b.Serviceability-based slope factor of safety using the shear strength reduction (SSR)method.The Second Half Century of Rock Mechanics11th Congress of the International Society for Rock Mechanics,1137–1140,Taylor&Francis.Hatzor,Y.H.,Talesnick,M.&Tsesarsky,M.2002.Continuous and discontinuous stability analysis of the bell-shaped caverns at Bet Guvrin,Israel.Int J Rock Mech Min Sci;39:867–886. Matsui,T.&San,K.C.1992.Finite element slope stability analy-sis by shear strength reduction technique.Soils and Foundations; 32(1):59–70.Meguid,M.A.&Rowe,R.K.2006.Stability of D-shaped tunnels in a Mohr-Coulomb material under anisotropic stress conditions.Can Geotech J;43:273–281.Ren,G.et al.2005.Underground excavation shape optimization using an evolutionary put Geotech;32:122–132. Roth,W.,Poisel R.&Zettler,parison between analytical and numerical stability analyses using FLAC for shal-low tunnels in soft rock.In FLAC and Numerical Modeling in Geomechanics,Lyon;399–402.Shi,G.H.1988.Discontinuous deformation analysis-a new model for the statics and dynamics of block systems.PhD Thesis,University of California,Berkeley.Stavropoulou,M.Exadaktylos,G.&Saratsis,G.2007.A combined three-dimensional geological-geostatistical-numerical model of underground excavations in rock.Rock Mech Rock Eng;40(3): 213–243.U.S.Army Corps of Engineers.1997.Engineering and design: introduction to probability and reliability methods for use in geotechnical engineering.Department of the Army,Washington, D.C.Engineer Technical Letter1110-2-547.Wang,T.&Zhu,H.2006.The application of FLAC and FLAC3D to the support of an underground cavern.In FLAC and Numerical Modeling in Geomechanics,Madrid;Paper No.02-01.Xia,Y.L.et al.2007.Stability analysis of an underground power cavern in a bedded rock formation.Tunnelling Underground Space Technol;22(2):161–165.。