Applications of rock failure process analysis (RFPA) method

Journal of Rock Mechanics and Geotechnical EngineeringJournal online: www.rockgeotech.orgApplications of rock failure process analysis (RFPA) methodChun' an Tang , Shibin TangInstitute. for Rock Instability and Seismicity Research, Dalian University of Technology, Dalian, 116024, ChinaReceived 29 March 2011; accepted in revised form I September 2011; accepted 10 September 2011Abstract: Brittle failure of rocks is a classical rock mechanical problem. Rock failure not only involves initiation andpropagation of single crack, but also is associated with initiation, propagation and coalescence of many cracks. The rock failureprocess analysis (RFPA) tool has been proposed since 1995. The heterogeneity of rocks at a mesoscopic level is considered byassuming that the material properties follow the W eibull distribution. Elastic damage mechanics is used for describing theconstitutive law of the meso-level element. The finite element method (FEM) is employed as the basic stress analysis tool. Themaximum tensile strain criterion and the Mohr-Coulomb criterion are utilized as the damage threshold. In order to solve thestability problem related to rock engineering structures, fundamental principles of strength reduction method (SRM) and gravityincrease method (GIM) are integrated into the RFPA. And the acoustic emission (AE) event rate is employed as the criterion forrock engineering failure. The prominent feature of the RFPA-SRM and RFPA GIM for stability analysis of rock engineering isthat the factor of safety can be obtained without any presumption for the shape and location of the failure surface. In this paper,several geotechnical engineering applications that use the RFPA method to analyze their stability are presented to provide somereferences for relevant researches. The principles of the RFPA method in engineering are introduced firstly, and then the stabilityanalysis of tunnel, slope and dam is focused on. The results indicate that the RFPA method is capable of capturing themechanism of rock engineering stability and has the potential for application in a larger range of geo-engineering.Key words: case studies; rock slopes and foundations; stability analysis; rock failureand material sciences would strive for a very long time1 Introduction[1].Rock failure is induced by damage evolution of initialWith the development of deep mining of resources,defects. The mechanical behaviors of rocks arehydropower, underground storage of nuclear waste,determined by the internal mesoscopic structures, theunderground heat extraction from rocks, undergroundmesoscopic damage and its evolution, and thecoal gasification, geological storage of CO2 and otherdevelopment of cracks. Krajcinovic [2] indicated thatunderground engineering proects, studies on rockthe phenomenological model could not effectively dealfailure mechanism have attracted more attentions.with the mesoscopic damage process, which could onlyBrittle failure of rocks is a complex process, whichbe overcome by using the mesoscopic mechanicalinvolves nitition and propagation of muli-crack. In models. Dougill et al. [3- 9] and other researchers usedfact, rock failure and instability have been difficultdamage mechanics to study the failure of rocks fromproblems in solid mechanics. Earlier studies by seniordifferent points of view. Furthermore, many researchesscientists in China indicate that, for researches on solidestablished damage mechanics based method to studymaterials under external loads and environmentalthe behaviors of rocks at mesoscopic level. The relevantconditions, the failure or damage process induced byresults are further extended to general brittle damageevolution of defects in material is an interdisciplinaryproblems, and the researches on mesoscopic damagescientific problem, for which scientists in mechanicsmechanics are continuously enriched.It is . generally recognized that the most importantfeature of rock material properties is the heterogeneity.Doi: 10.3724/5P.J125.211.00352Coreponding author. Tel: +86- 189558; E-mai: catang@mechsof.cn Crack initiati中国煤化工冰s are differentSupported by the Stale Key Development Program for Basic Research offrom thoseCNMHGEven rocks areChina (2007CB209400), Projects of Intermational Cooperaion and Exchangessimplified ihto nomogeneous meaia (mesoscopicNSFC (50820125405), and the National Natural Science Foundation of Chinahomogeneity) under some circumstances, various(51004020)Chun'an Tang et al. 1 J Rock Mech Geotech Eng. 2011, 3 (4): 352-372353macroscopic defects or impurities (macroscopic material could be realized. This numerical simulationheterogeneity) in rocks lead to complex laws of crack tool has been applied to numerical studies on the failurepropagation. With the increase in stress, these defects process of heterogeneous rocks and their acousticor impurities become the inducing sources of stress emission (AE) characteristics [13-15], engineeringconcentration. Crack initiation and propagation lead to problems such as movement of rock strata [16],increasing crack density and stronger interaction development mode of seismic sources [17], and crackbetween cracks. The cracks eventually coalesce and propagation in brittle and heterogeneous materials [18].result in macroscopic failure. Although the britleness The simuation results are in good agreement with theof rocks determines their failure characteristics, experimental results.compared to homogeneous materials, such as glass, oneCurrently, the RFPA system has been widely used inof the most important differences is that rocks behave geotechnical engineering. In this paper, applications ofnonlinearly due to their heterogeneity. The britle the RFPA system to geotechnical engineering arecharacteristics of rocks lead to catastrophic failure. On introduced.the other hand, the heterogeneity results in aprogressive and evolutionary process before failure. To 2 Implement of engineering method inbe more accurate, the failure process of rocks isthe RFPAprogressive and catastrophic. The complexity of thisprocess results in the complexity of rock damageIn order to understand the failure mechanism of rockmechanics. In view of rock heterogeneity， manyengineering structures, numerical methods, such as theresearchers considered this feature by statisticalFEM, boundary element method (BEM), and discretedistribution. Weibull [10] proposed a statistical theoryelement method (DEM), have been developed, and theyof strength for brittle materials based on“the weakest-have become increasingly popular for the stabilitylink model" in 1939, i.e. the Weibull distribution,analysis of the structures. It has shown that thewhich has been widely applied. At present, a greatnumerical methods have a number of advantages ovenumber of researches on rock damage and failure basedthe traditional limit equilibrium approaches for stabilityon the Weibull distribution bave been reported. Sinceanalysis of rock engineering. Most importantly, the1995, the authors and their research group have beencritical failure surface can be found automatically.committed to study the rock failure process analysisNevertheless, the currently widely accepted numerical(RFPA). Based on the statistical distribution method inmethods do not take into account the heterogeneity ofrock material properties, mesoscopic damage theoryrock masses at macroscopic levels under complicatedand numerical computation method, a RFPA systemgeological conditions. The heterogeneity plays anwas developed and applied to a great number ofimportant role in determining the fracture paths andfundamental researches. The RFPA system is 8fracture patterns of rock masses. The influence ofnumerical tool based on the elastic damage model and heterogeneity is pronounced on the progressive failureits theoretical foundation is the academic idea that the process [13, 19]. In the RFPA method, the Weibullauthors have been thinking for many years. In 1991,the distribution is used to consider the heterogeneity ofauthors proposed the assumption on the normalrocks.distribution of element strength at meso-scale for rocksIn order to solve the stability problem related to rock[11, 12]. The mesoscopic heterogeneity was consideredengineering structures, the fundamental principle ofto be the root cause of the macroscopic nonlinearity of strength reduction method (SRM) and gravity increasequasi-rittle materials. The heterogeneity of rockmethod (GIM) are introduced into the RFPA.material and stochastic distribution of defects can be Mathematically, both of these two methods in the RFPAreflected by the statistical constitutive damage model. .(RFPA-SRM andRFPA-GIM)arecompletelyThereafter, for the convenience of solution, the Weibull continuum-based methods, processing nonlinear andistribution was adopted to reflect the stochastic discontinuous failure mechanism problems. The codedistribution instead of the normal distribution. The considers the deformation of a heterogeneous materialstatistical distribution of material properties wascontaining randomly distributed micro-fractures. Asintegrated into the numerical methods such as finite loads are app中国煤化工w, interact andelement method (FEM). Elements that satisfy the givencoalesce, resuYHCNMHGbehavior anstrength criterion were considered to be failed, and thus formation of hacluscuplr liavtuics. ine RFPA-SRMthe numerical simulations of heterogeneous rock and RFPA-GIM not only satisfy the global equilibrium,.354Chun'an Tang etal/J Rock Mech Geotech Eng 2011, 3 (4) 352- 372strain-consistent and nonlinear constitutive relationship the release of elastic energy. Therefore, as anof rock and soil materials, but also take into account the approximation, it is reasonable to assume that the AEheterogeneous characteristics of materials at counts are proportional to the number of damagedmicroscopic and macroscopic levels. A tensile cutoff elements and that all the strain energy released bycriterion is also incorporated to model tensile failure. damaged elements is in the form of AEs [14].The code has been successfully applied in failureIn the RFPA-SRM model, the AE counts areprocess analysis of rock material.determined by the number of damaged elements and the(1) For the RFPA-SRM, as an altermative approach energy release is calculated from the strain energyto the failure analysis problem related to geological or released by the damaged elements. Based on the aboverock engineering, the fundamental principle of strength assumptions, the cumulative AE counts and thereduction is incorporated into the constitutive model of cumulative AE energy release can be realisticallythe element described above. The shear strength simulated using the RFPA-SRM model. Provided thatreduction technique [20] is applied to each element. the AE event rate reaches the maximum value, aThe strength of element, %o, is linearly degraded as macroscopic failure surface forms and slope filurefollows:occurs. Simultaneously, the corresponding Fyrial is theFri=fg1fam(1) factor of safety, F,, of the slope.where Frial is the trial factor of safety, and ferial is theIn the following sections, we present sometrial strength of the element. The trial strength foinlengineering applications of the RFPA method to showused in the RFPA-SRM is to investigate the strength of the convenience of this method to analyze the stabilitythe geological media (in this case, the rock masses).of rock engineering.In this study, rock slope failure is examined. Slopestabilit simultion in the RFPA-SRM is run with the 3 Variation of deformation and stresstrial strength frial until the critical failure surfaceat key places in the Baziling tunnelslopes is determined.(2) For the RFPA-GIM, the critical failure surface of3.1 Overview of geology and numerical modelslopes is obtained by gradually increasing the gravityThe Baziling tunnel is located in Changyang Countywhile keeping material properties constant. In theof Yichang City and Badong County of Enshi Tujia andRFPA-GIM, the gravitation of the elements increasesMiao Autonomous Prefecture. The tunnel entrance is atlinearly. For each loading step, there is a correspondingBaziling Village, Langping Town, Changyang County,trial gravitational acceleration giu (m/s'). Referring tothe definition of factor of safety in the finite elementand the exit is at the east bank of the Sidu River,strength reduction technique [21, 22]， the safetyLiziyuan Village, Yesanguan Town, Badong County.repertory factor Fyina is defined as the ratio of theThe tunnel is designed as a forked tunnel. The entranceelement gravitation in the failure state to the initial is designed as two separated tunnels and connected withelement gravitation, which can be written asthe west end of the Baziling extra-long bridge. The exitFi:lnl=goW /gois a twin-arch tunnel, only 20- -30 m away from thewhere go is the initial gravitational acceleration (m/s).bridge abutment of the Sidu extra-long bridge. TheIn the RFPA-GIM, slope stability analysis is runtunnel extends along WE in plan. As the stability in thewith the trial gravitational acceleration go"ie until theforked section is of great importance, the RFPA isadopted to simulate the variations of deformation andcritical failure surface in the slope is determined.Several possible techniques can be used to definestress at key places and the failure mechanism of theslope failure, including the formation of critical failuresurrounding rocks in the Baziling tunnel.surface, non-convergence of the finite element solution,In consideration of computational precision duringetc. In the RFPA-SRM, the maximum AE event rate is numerical simulations, hexahedral elements are adoptedused as the criterion of slope failure. Slope failure is in the model. The mesh is refined near the tunnel and iscommonly accompanied by a dramatic increase in the coarser at some distance away from the tunnel. In othernodal displacement within the elements. Accordingly,words, the mesh is denser and smaller near the tunnelthere is a dramatic increase in the number of damagedand coarser and larger at some distance away ftom theelements. Monitoring AE event rates seems to be a tunnel, which中国煤化工computationalgood way of identifying the initiation and propagation precision. TCNMH;lated by fourof cracks and fractures in rocks. In quasibrittle models according tu me suepwise excavation. The totalmaterials, such as rocks, AE is predominantly related to number of elements in each model is about one million..35Chun'an Tang et al./J Rock Mech Geotech Eng. 2011, 3 (4): 352- -372Excavation stepbetween the two portions is prominent. At the separated122436486072 8tunnel sections, the left and right tunnels are separatedThe 4th sectionby thick rock wall and the transversal distance betweenthe left and right tunnels is large. Hence, the interactionTihe rd scctionbetween them is weak. Therefore, the displacement atThe 2nd sctionThe Ist sectionthe twin-arch tunnel sections is greater than that at theseparated tunnel sections. This shows that during tunnel(a) Variation of displacement at the crown of the left turnel.design and excavation, the thickness of the wallbetween the left and right tunnels shall be optimized andensured. The wall cannot be either too thick or too thin.The 4th sctionThe wall should be thick enough to effectively reduceThe 2nd sectionthe interaction between the two tunnels. However, it canThe 3rd sectionnot be too thick, which will result in much higherThe 1st sectionconstruction cost.(6) With the advance of excavation, the tunnel(b) Variation of displacement at the crown of the right tunnel.deformation gradually becomes smaller from the twin-arch section to the separated tunnel section. The reason12243648607284is that the interaction between the left and right tunnelsThe 4th se:ctionbecomes weaker as the thickness of the wall in theforked tunnel increases.According to the above simulation results, in combinationwith the distribution and development of stress,(C) Variation of displacement at the left sidewall of the let tunnel,displacement and AEs, the factors of safety after eachexcavation step are shown in Table 1. As seen from thesimulation results, the factor of safety after the 1stexcavation step is the highest, which indicates that theexcavation of the half left tunnel has no significantoneffect on the stability of the surrounding rock. The3Efactor of safety drops from 50 after the lst excavationstep to 10 after the 2nd excavation step, which indicates(d) Variation of displacement at the right sidewall of the right tunel.lower stability. The factors of safety after the 3rd andFig3 Variations of dsplacement in the surounding rocks of the 4th excavation steps are close to that afer the 2nd step.forked tunnel with excavation steps.Table 1 Factors of safety of the tunnel.observed. For instance, when the excavation of the leftFactor of safetytunnel stops at a certain section and the excavation50continues for the right tunnel, the deformation of the10left tunnel still increases. Similarly, when the7.69excavation of the right tunnel stops at a certain section6.67and the excavation continues for the left tunnel, thedeformation of the right tunnel increases as well.The calculation results indicate that:Therefore, the distance between the excavation faces of(1) Due to unloading after tunnel excavation, thethe left and right tunnels shall be controlld properly so surrounding rocks deform towards the tunnel. Under theas to reduce the effect of interaction between the left action of gravity, the deformation is predominantly theand right tunnels due to altemative excavation.subsidence of the crown and heave of the floor. The(5) The displacement at the twin arch tunnel sections vertical deformation is larger than the horizontal(i.e. the lst and 2nd sections) is generally larger thandeformation. The displacement of the tunnel wall isthat in the separated tunnel sections (i.e. the 3rd and 4th relatively small after excavation, with the maximumsections). As the left and right portions of the twin-archdeformation中国煤化工:rall deformationtunnel are separated by a thin layer of concrete partition satisfies theMHC N M H Gl construction.wall, the transversal distance between the left and right(2) The large arch section of the forked tunnel isportions is relatively small. Therefore, the interaction excavated in upper and lower benches. The twin-arch.h Geotech Eng 2011, 3(4): 352-372357section (including the section with integrated partionmouth. The Jiaozhou Bay tunnel is an express roadwall and the section with sandwiched partitin wal) istunnel with six lanes in two diretions and the designexcavated by alternative half-section excavation of thevehicle velocity is 80 km/b. Completion of the tunnelleft and right tunnels with central heading, and thecan fundamentally solve the temporary shortage,separated tunnels are excavated by alternativeimprove the investment environment in the west, speedexcavation of the left and right tunnels. Adoption ofup the development of new economic zone, realize thevarious excavation methods leads to higher excavationefficiency, less repeated diturbances to thecomplementary advantages of the new and old portsurrounding rocks, and enhanced overall stability of thedistricts, and enhance the overall eficiency. It is asurrounding rocks.major engineering measure to provide strong support for(3) With tunnel excavation, the radial stress in the Qingdao City to develop into a moderm international city.surrounding rocks is released and the shear stressThe RFPA system is employed to analyze theincreases. The stress in the surrounding rocksminimum rock cover depth for the Qingdao Jiaozhoupredominantly compressive stress. The maximumBay tunnel. Relevant theoretical and numerical studiescompressive stress is far less than the compressiveare carried out to study the minimum rock cover depthstrength of the rock mass. At the same time, due tofor the Qingdao Jiaozhou Bay tunnel. The studies areexcavation unloading, stress concentration occurs at the performed according to the variation of geological strataarch comers and crown, and the shoulder of thealong the longitudinal profile, especially those of somepartition wall, However, the magnitude of stressweak strata and structural zones. The minimum coverconcentration is small.depth under different combinations of engineering(4) During excavation of the forked tunnel, the geology, hydrogeology and construction method isinteraction between the left and right tunnels is proposed and the measures to reduce the minimumobvious, i.e. excavation of the left tunnel affects thecover depth are suggested, which can provide someright tunnel and vice versa. Therefore, the distance references for the tunnel alignment and design.between the excavation faces of the left and rightThe plane strain model is adopted. In the model, thetunnels shall be properly controlled so as to reduce the x-axis is perpendicular to the tunnel axis in theinteraction between the left and right tunnels during horizontal plane and the y-axis is along the verticalaltemative excavation.direction. The range ofx-coordinateis - 60m