A softening block approach to simulate excavation in jointed rocks

Transcription

1 Bull Eng Gol Environ (2012) 71: DOI /s ORIGINAL PAPER A softning block approach to simulat xcavation in jointd rocks Qinghui Jiang Chuangbing Zhou Dianqing Li Man-chu Ronald Yung Rcivd: 5 July 2011 / Accptd: 7 Jun 2012 / Publishd onlin: 25 Sptmbr 2012 Ó Springr-Vrlag 2012 Abstract Th instability of rock masss as a rsult of xcavation is on of th most important rock mchanics problms. This papr proposs a nw softning block approach for th simulation of stag-wis squntial xcavations in jointd rock masss. Th concpt of a softning block is prsntd and th corrsponding algorithm dvlopd and implmntd into th original Discontinuous Dformation Analysis program proposd by Shi. Th proposd softning block approach obviats th nd to rmov th xcavatd blocks from th calculation modl, which furthr simplifis th work of pr-procssing for squntial xcavation simulations. Finally, two numrical xampls ar prsntd to dmonstrat th validity and capability of th proposd approach. Th rsults indicat that it can b usd to simulat th xcavation in jointd rock mass fficintly and accuratly. Kywords Jointd rock masss Discontinuous dformation analysis Softning block Excavation Résumé L instabilité d masss rochuss résultant d travaux d xcavation constitu l un ds plus importants problèms d mécaniqu ds rochs. L articl propos un nouvll approch d bloc affaibli pour la simulation d procssus d xcavation progrssiv dans ds masss rochuss fissurés. L concpt du bloc affaibli st présnté, ls algorithms corrspondants sont dévloppés t Q. Jiang (&) C. Zhou D. Li School of Civil Enginring, Wuhan Univrsity, Wuhan , Popl s Rpublic of China -mail: jqh1972@yahoo.com.cn M. R. Yung Civil Enginring Dpartmnt, California Stat Polytchnic Univrsity, Pomona, USA implémntés dans l programm original d analys ds déformations n miliu discontinu proposé par Shi. L approch proposé évit d avoir à nlvr ls blocs xcavés du modèl d calcul, c qui nsuit simplifi l travail d pré-traitmnt pour ls simulations d xcavations succssivs. Finalmnt, dux xmpls numériqus sont présntés pour démontrr la validité t la capacité d l approch proposé. Ls résultats indiqunt qu ll put êtr utilisé pour simulr l xcavation d masss rochuss fissurés d manièr fficac t précis. Mots clés Masss rochuss fissurés Analys d déformation n miliu discontinu Bloc affaibli Excavation Introduction It has long bn apprciatd that th mchanical/dformation bhavior of rock masss is significantly influncd by th containd discontinuitis. In th light of this, Shi (1988) and Shi and Goodman (1989) dvlopd th Discontinuous Dformation Analysis (DDA) mthod to simulat th bhavior of rock masss. Sinc thn, th original 2-D mthod has bn xtndd in trms of both thory and computr coding (Jing 1998; Kim t al. 1999; Parc t al. 2000; Doolin and Sitar 2001; Hatzor t al. 2004; Zhng and Jiang 2009). Maclaughlin and Doolin (2006) hav mad a thorough rviw of th xisting 2-D DDA validation litratur, citing 100 articls. In rcnt yars, prliminary fforts hav bn mad to xtnd DDA to thr-dimnsions. Shi (2001) discussd th contact submatrics including thos for normal and shar springs. Byabanaki t al. (2008) prsntd an algorithm to sarch and calculat gomtrical contacts in th 3-D DDA. Jiang

2 748 Q. Jiang t al. and Yung (2004) and Yung t al. (2007) prsntd modls for point-to-fac contact and dg-to-dg contact, including th dtction of contact typ, stimation of th first ntranc facs, and th critria for intr-pntration. Wu t al. (2005) proposd a contact-sarching algorithm for frictionlss vrtx-to-fac contact problms. Th instability of rock masss inducd by xcavation is on of th most important problms in rock mchanics. Convntional Finit Elmnt Analysis (FEM) is frquntly usd as a computation and dsign tool for xcavation problms. Th rock mass bfor xcavation is in a stat of quilibrium undr gravity and tctonic strsss. Th rmoval of th xcavatd zon changs ths initial quilibrium conditions. Th displacmnts and incrmntal strsss inducd by th xcavation can b calculatd by applying th rvrs initial tractions to th xcavatd surfac (Goodman and Brown 1963; Brown and King 1966; Mana and Clough 1981). Rcntly th so-calld softning lmnt approach has bn dvlopd to simulat xcavation, and adoptd by commrcial softwar such as ANSYS (Gârlanu t al. 2010). In rock masss, thr ar gnrally many discontinuitis of diffrnt typs and sizs whil discontinuous dformations (such as opning, sliding and falling of blocks) ar likly to occur du to unloading on xcavation. Although som fforts to simulat th xcavation of rock slops and undrground opnings using DDA hav bn rportd (Pi and Lu 1996; Dong t al. 1996; Wu t al. 2006) in most cass th assumption is that th xcavation is compltd in only on stp. As th stability of a rock mass dpnds on th xcavation squnc and th strss history, Kim t al. (1999) dvlopd an algorithm to simulat squntial xcavation. Th algorithm modifis th initial gomtry of th block systm as wll as th initial strss data continuously for squntial xcavations, which lads to a considrabl incras of data manipulation and computing tim. This papr proposs a nw softning block approach to simulat th xcavation procss in a jointd rock mass using th DDA mthod. Th proposd algorithm liminats th nd to continuously rmov th xcavatd blocks from th calculation modls, hnc th simulation fficincy was improvd significantly whilst th numbr of blocks and thir storag positions in th gnral stiffnss matrix rmaind unchangd. Consquntly, th pr-procssing is highly simplifid. Two tst cass ar prsntd to support th validity of th proposd approach. Equations of motion of a discrt block systm Th DDA mthod proposd by Shi and Goodman is in ssnc a numrical mthod to solv th motion and dformation of multipl rock blocks. Starting from th dfinitions of displacmnt functions, th quations of motion of th blocks ar first stablishd using th potntial nrgy minimization. Thn, th dirct tim intgration schm is adoptd to solv th quations of motion. Displacmnt functions of dformabl blocks DDA carris out calculations basd upon tim stps, in which larg displacmnts and dformations ar th accumulatd rsults of th small ons obtaind from stp-wis calculations through tim stps. Th solutions for ach tim stp ar th incrmnts of displacmnts and strains of blocks. An incrmntal xprssion for th unknown variabls of a typical block can b writtn as ½DD Š¼½Dd 1 Dd 2 Dd 3 Dd 4 Dd 5 Dd 6 Š T ¼½Du 0 Dv 0 Dc 0 D x D y Dc xy Š T ð1þ whr ðdu 0 ; Dv 0 Þ ar th x- and y-translation incrmnts; Dc 0 is th rotation angl incrmnt of th block around its mass cntr ðx 0 ; y 0 Þ; and ðd x ; D y ; Dc xy Þ ar th incrmnts of normal and shar strains of th block. Th displacmnt incrmnts ðdu; DvÞ of an arbitrary point ðx; yþ in th block can b xprssd in trms of ½DD Š as " P 6 Du j¼1 ¼ ½T Dv Š½DD Š ¼ t # 1jDd j P 6 j¼1 t ð2þ 2jDd j whr ½T Š¼ 1 0 ðy y 0Þ ðx x 0 Þ 0 ðy y 0 Þ=2 0 1 ðx x 0 Þ 0 ðy y 0 Þ ðx x 0 Þ=2 ð3þ ½T Š is calld th shap function matrix of th block. Error du to approximation using a linar displacmnt function causs block xpansion, spcially in problms involving larg rigid body rotations (Koo and Chrn 1998). To corrct th rror, post-corrction is usd and th x- and y-displacmnt incrmnts ðdu; DvÞ of all rlvant points including block vrtics ar rcalculatd using Eq. (4) aftr ach tim stp as follows: " P 6 Du j¼1 ¼ t # 1jDd j P Dv 6 j¼1 t 2jDd j j6¼3 þ cos Dc 0 1 sin Dc 0 sin Dc 0 cos Dc 0 1 x x 0 : ð4þ y y 0 Strain and strss matrics With displacmnt incrmnts known at any point within th block, th strain incrmnts at any point can b

3 Excavation in jointd rock 749 dtrmind according to th strain displacmnt rlationship ½D Š¼½LŠ½T Š½DD Š¼½BŠ½DD Š ð5þ whr ½LŠ is th diffrntial oprator matrix for twodimnsional problms and th diffrntiation of th matrix ½T Š lads to th matrix ½BŠ givn by ½BŠ ¼½LŠ½T Š¼ : ð6þ Assuming homognous, linar and isotropic lastic bhavior for th rock blocks, th rlation btwn strsss and strains can b givn by ½Dr Š¼½EŠ½BŠ½DD Š ð7þ whr ½EŠ is th lastic constant matrix. Equations of motion of th block systm Th potntial nrgy of th block, P p, includs lastic strain nrgy P s, initial strss potntial nrgy P r0, surfac load potntial nrgy P a, body forc potntial nrgy P w and inrtia forc potntial nrgy P i in this papr, bing xprssd as P p ¼ P s þ P r0 þ P a þ P w þ P i Z 1 ¼ V 2 ½D Š T ½EŠ½D Šþ½D Š T ½r 0 Š dxdy Z ½DD Š T ½T Š T ð½b Š q½t Š½ D ŠÞdxdy V Z ½DD Š T ½T Š T ½p Šds ð8þ S r whr q is th mass dnsity of th matrial concrnd, ½b Š th body forc vctor of th block, ½r 0 Š th initial strss vctor of th block and ½p Š th surfac forc vctor actd on th block s boundary, rspctivly. For a discrt modl, th total potntial nrgy of th block systm, P p, is th sum of potntial nrgis for all individual blocks. Substituting Eq. (5) into Eq. (8) givs th following potntial functional nrgy of th block systm P p ¼ X ½DD Š T ½M Š½ D Šþ X X ½DD Š T ½f Š 1 2 ½DD Š T ½K Š½DD Š ð9þ whr ½M Š¼ R V q½t Š T ½T Šdxdy is th mass matrix of th block; ½K Š¼ R V ½BŠ T ½EŠ½BŠdxdy is th stiffnss matrix of th block; ½f Š¼ R V ð½t Š T ½b Š ½BŠ T ½r 0 ŠÞdxdy þ R S ½T Š T r ½p Šds is th load matrix of th block. According to th variation principl, th minimization procss dp p ¼ 0 lads to quations of motion of th block systm in a matrix form ½MŠ½ DŠþ½KŠ½DDŠ ¼½FŠ ð10þ whr ½MŠ, ½KŠ and ½FŠ ar th global mass matrix, stiffnss matrix and load matrix, rspctivly, and thy ar assmbld block-wis, i.., ½MŠ ¼ X ½M Š ½KŠ ¼ X ½K Š ½FŠ ¼ X ½f Š: ð11þ A dirct tim intgration schm is usd for solving Eq. (10), in which th acclration within ach tim stp is assumd to b constant. Dnoting th initial vlocity vctor of th block systm as ½V 0 Š¼½_Dð0ÞŠ, thn ½ DŠ ¼ 2 ðdtþ 2 ½DDŠ 2 Dt ½V 0Š ð12þ whr Dt is tim incrmnt of th currnt tim stp. Substituting Eq. (12) into Eq. (10), th global quilibrium quations of th block systm at th currnt tim stp bcom ½KŠþ 2½MŠ Dt 2 ½DDŠ ¼½FŠþ 2 Dt ½MŠ½V 0Š: ð13þ Kinmatic conditions Within ach tim stp, th solution of global quilibrium quations (Eq. 13) must satisfy th constraint conditions,.g., no pntration and no tnsion btwn th block intrfacs. Th kinmatic constraints on a givn block systm ar imposd using th pnalty mthod. Numrical pnaltis analogous to stiff springs ar applid at th contacts to prvnt intrpntration of th blocks. As shown in Fig. 1, th contact typs for th problms in two dimnsions includ vrtx-to-vrtx, vrtx-to-dg and dg-to-dg contacts. Th dg-to-dg contact can usually b convrtd to two vrtx-to-dg contacts (s Fig. 1). For ach contact, thr ar thr possibl stats: opn, sliding, and lockd. Onc th contact stat is dtrmind, th sub-matrics of contact springs and frictional forcs ar calculatd and addd to th matrics of th global quations at th corrsponding positions. Within ach tim stp, th global systm of quilibrium quations (Eq. 13) is solvd rpatdly with variations of th typs, locations and stats of contacts. Th procdur of adding and rmoving stiff springs dpnding on th changs in contact stats is known as opn clos itration (Shi 1988). Whn th normal componnt of th contact forc is tnsil, th contact is opn. In this cas, no springs or frictional forcs ar applid at th contact location. Whn th normal componnt of th contact forc is comprssiv, th

4 750 Q. Jiang t al. Fig. 1 Diffrnt contact typs btwn two blocks contact is closd. If th shar componnt of th contact forc along th contact dg is larg nough to caus sliding, th closd contact occurs. In this cas, a normal spring and a pair of frictional forcs ar applid at th contact location. If th shar componnt of th contact forc along th contact dg is lss than that drivd from Coulomb s law, th closd contact occurs as a lockd stat. In this cas, a normal spring and a shar spring ar applid at th contact location. Convrgnc is vntually rachd whn ach of th contact stats and sliding dirctions rmains th sam aftr itration, judgd by a st of prscribd tolrancs. Simulation of xcavation in jointd rock mass In continuity-basd FEM analysis, th xcavation can b simulatd by rmoving th quivalnt nodal forcs on th xcavation surfac by rvrsing thir orintations but maintaining thir magnituds, which ar calculatd according to th initial strss condition or th strss condition at th prvious xcavation stp. In DDA analysis, blocks ar boundd by pr-xisting discontinuitis. Ths blocks ar linkd and transfr loads to ach othr through th normal and shar springs st btwn block intrfacs. If th discontinuitis in jointd rock masss cannot bar tnsil strsss, th rvrsd forcs on th xcavation surfac mak no ffct on th blocks byond th xcavation surfac. Th simulation approach basd upon th xcavation forcs can not b usd again. An xcavation stp implis unloading and strss-rlif, such that th baring capacity of th blocks within th xcavatd zon is rducd to zro; this ffct can b approximatd by assigning vry small stiffnss valus to th blocks to b rmovd by th xcavation stps. This simulation mthod is rfrrd to as th softning block approach. Rcntly th softning lmnt approach has bn dvlopd to simulat xcavation in convntional FEM analysis. Th softning block approach is diffrnt from th softning lmnt approach. As th FEM mthod is basd on th assumption of a continuum and uss only standard shap lmnts with pr-dtrmind topologis, it may rsult in difficult paramtrs, such as quivalnt lastic constants and an ovr-simplifid gomtry for a jointd rock mass. In addition, th softning finit lmnts may not b consistnt with th global stiffnss matrix for FEM analysis, spcially for multipl xcavations. Th softning block approach proposd in this study is suitabl for simulation of th squntial xcavation in a jointd rock mass. Th blocks usd by th DDA mthod ar physically isolatd blocks boundd by pr-xisting discontinuitis, which can b any convx or concav shap or vn polygons with hols. Furthrmor, DDA is stablishd basd on th block kintics, which adopts a tim intgral schm to solv th global quilibrium quations for static and dynamic problms. As th mass sub-matrics ar includd in th global stiffnss matrix for a discrt block systm, th inconsistncy causd by th softning blocks can b ovrcom ffctivly. Concpt of softning block Th rock mass bfor xcavation is in quilibrium undr th initial strsss. Excavation changs th modl gomtry and rmovs th rstraint action of th xcavatd rock blocks on th surrounding rock mass, subsquntly lading to strss r-distribution and displacmnts around th xcavatd ara. In rality, th xcavatd blocks ar rmovd from th calculation modl. For numrical simulation using DDA, such an ffct can b prformd by rducing th dformability of blocks to b xcavatd to such an xtnt that thir ffcts ar ngligibl as rgards th modl rsults, th so-calld block softning tchniqu. Manwhil, th gravity and th initial strsss of th softning blocks ar st as zro. Lt E b th actual lastic modulus of th xcavatd blocks. A softning factor, c, is dfind as th rduction cofficint of th lastic modulus. Thn th lastic modulus of th xcavatd blocks can b rprsntd as ke. Th xcavation can b simulatd simply by assigning small valus of k to all th blocks to b xcavatd at th rlvant xcavation stp. On major advantag of this tchniqu is that it is not ncssary to rmov th xcavatd blocks from th calculation domain, and th numbr of blocks and thir storag positions in th global stiffnss matrix rmain th sam so that th global stiffnss matrix maintains its total dgr of frdom. Th calculation procdur is considrably simplifid and th work of pr-procssing for squntial xcavation simulations much rducd. Squntial xcavation algorithm It is assumd that th calculation domain, X, is in quilibrium and th initial strss fild is known bfor

5 Excavation in jointd rock 751 xcavation. If th displacmnts and th strsss of th block systm ar ½D m 1 Š and ½r m 1 Š rspctivly aftr th (m-1)th stp of xcavation, thn th nxt xcavation stp will mak th displacmnts and th strsss of th systm bcom ½D m Š¼½D m 1 Šþ½dDŠ; ½r m Š¼½r m 1 Šþ½drŠ ð14þ whr [dd] and [dr] ar th incrmnts of displacmnts and incrmntal strsss du to xcavation, rspctivly. Dnoting th xcavatd zon and th surrounding rock mass aftr th mth stp of xcavation as X m and Xr m, rspctivly. Th lastic modulus of th xcavatd blocks and rtaind blocks ar rprsntd by ke and E, rspctivly. For convninc, lt v b an xcavation impact factor dfind as follows: v ¼ 1; in X r m ; v ¼ k; in X m : ð15þ 3m 6m In ordr to solv [dd] and [dr], th itration algorithm for th mth stp of xcavation using th block softning approach is givn by ½K n Šþ 2½Mn Š Dt 2 ½DD n Š¼½F n Šþ 2 Dt ½Mn Š½V0 n Š ð16þ ½K n Š¼ X Z v½bš T ½EŠ½BŠdxdy ð17þ X V n ½M n Š¼ X Z q½tš T ½TŠdxdy ð18þ X V n ½F n Š¼ X Z ð½tš T ½b Š ½BŠ T ½r m 1 ŠÞdxdy X r V n m X Z ½BŠ T ½dr n 1 Šdxdy ð19þ X V n Fig. 2 Gomtry of tunnl xcavation ½dD n Š¼½dD n 1 Šþ½DD n Š ½dr n Š¼½dr n 1 Šþv½EŠ½BŠ½DD n Š ½A n Š¼2½DD n Š=Dt 2 2½V n 0 Š=Dt ð20þ ð21þ ð22þ ½V0 nþ1 Š¼½V0 n Šþ½An ŠDt ð½v0 nþ1 Š¼½0Šfor static problmsþ ð23þ whr th suprscript n stands for tim stp; and [A] is th acclration vctor. Assuming that thr ar k blocks in th calculation domain. At th start of itration, [dd 0 ] = 0 and [dr 0 ] = 0. For static problms, th displacmnts of th blocks mt th convrgnc critrion givn by ðmax i¼1;k jjddn i jjþ=ðmax i¼1;k jjddn i jjþ 1 ð24þ in which 1 dnots th usr-spcifid rror tolrancs; valu of 1 = 10-4 is usd in this study. Th itration procds to th mth stp of xcavation with updatd strss and displacmnts vctors givn by Fig. 3 Block mshs for DDA analysis ½D m Š¼½D m 1 Šþ½dD n Š; ½r m Š¼½r m 1 Šþ½dr n Š: ð25þ If displacmnts of th blocks cannot mt th convrgnc critrion, failur of rock masss may tak plac. Aftr th failur of rock masss occurs, it is ncssary to adopt a dynamic analysis mod of DDA to simulat th motion bhavior. Slction of softning factor How to dtrmin th propr valus for th softning factor k is an important problm as th softning factor has a significant influnc on th accuracy of th solution. As an illustration, a rctangular tunnl xcavation is studid to invstigat th influnc of th softning factor on th disturbd displacmnts. Figur 2 shows a rctangular

6 100m 752 Q. Jiang t al. Tabl 1 Comparison btwn disturbd displacmnts from th proposd softning block approach and th rmoving block approach (unit: mm) k Block 1 Block 2 Block 3 Block 4 Displacmnt Error (%) Displacmnt Error (%) Displacmnt Error (%) Displacmnt Error (%) (41.78) (42.18) (49.26) (49.58) (41.78) (42.18) (49.26) (49.58) (41.78) (42.18) (49.26) (49.58) (41.78) (42.18) (49.26) (49.58) (41.78) (42.18) (49.26) (49.58) (41.78) (42.18) (49.26) (49.58) 0.38 Valus in parnthss ar obtaind using th rmoving block approach 40m Excavation zon 30m m Fig. 6 Initial gomtry for th xcavation of a slop Fig. 4 Fig. 5 Plot of displacmnt vctors Plot of principal strss vctors tunnl xcavation in a jointd rock mass. Th tunnl is m and th plan strain condition is assumd. It is furthr assumd that th rock is dividd into blocks by horizontal and vrtical joints. Th following paramtrs ar adoptd for all rock blocks: Young s modulus E = 420 MPa, Poisson s ratio m = 0.26, and dnsity q = 2500 kg/m 3. Th friction angl, cohsion, and tnsil strngth of th joints ar / = 16, c = 0 MPa, and T = 0 MPa, rspctivly. Th normal and shar stiffnss of th joints ar Kn = 4,200 MN/m and Ks = 1,680 MN/m, rspctivly. Th maximum allowabl displacmnt ratio within a tim stp is Sr = Th xtrnal boundaris of th modl ar fixd in thir rspctiv normal dirctions. Th initial strsss in th rock mass bfor xcavation ar r x = r y =-3 MPa, whil th slf-wight of th rock blocks is not considrd. To improv th dformation capability of th considrd rock blocks, it is allowabl to us an artificial joint for dividing rock blocks into s a st of uniformly dformabl sub-blocks (K 1996). Ths subblocks ar connctd with on anothr by vry strong springs, which ar usd to nsur continuity across any artificial joint. As a consqunc, a picwis constant

7 Excavation in jointd rock 753 (a) Softning block approach (a) Softning block approach (b) Rmoving block approach (b) Rmoving block approach Fig. 7 Plot of displacmnt vctors Fig. 8 Plot of principal strss vctors strain fild is introducd into ach parnt block, thrby nhancing its dformability. Th block mshs for DDA analysis ar shown in Fig. 3. Th proposd softning block approach is usd to valuat th dformation of jointd rock mass du to xcavation. In ordr to invstigat th influnc of softning factor k on th accuracy of solution, six valus of softning factor, 10-1,10-2,10-3,10-4,10-5, and 10-6, ar adoptd for th analysis. Tabl 1 summarizs th displacmnts of th cntr of rprsntativ blocks in th surrounding rock mass of th tunnl aftr xcavation for th diffrnt valus of softning factor. For comparison, th displacmnts of th cntr of rprsntativ blocks obtaind from th rmoving block approach ar also shown in Tabl 1. Assuming th rsults from th rmoving block approach ar th xact solution, thn th rlativ rror can b calculatd by E r = d s -d r /d r %, whr d s and d r ar th xcavation inducd displacmnts prdictd by th softning and rmoving block approachs, rspctivly. As can b sn from Tabl 1, thr is a gnral trnd that th displacmnts of th considrd blocks incras with dcrasing softning factor. Th rsulting rrors ar smallr than 3 % whn th valu of th softning factor is smallr than 10-3, which indicats that th accuracy of th solution obtaind from th proposd softning block approach can b accptabl for a softning factor smallr than It is suggstd that softning factors ranging from 10-5 to 10-4 b adoptd for th proposd softning block approach. From th practical point of viw, it can b undrstood that an mpty zon producd by th xcavation is littl diffrnt from a zon filld with a matrial having much smallr stiffnss valus compard to th surrounding rocks. For th purpos of illustration, a valu of k = 10-4 is usd in th subsqunt analyss. This cas is charactrizd by th symmtris of mshs, loads and fixd points. For th cas of k = 10-4, Figs. 4 and 5 show th distribution of displacmnts and principal strsss vctors rspctivly. Not that th rsults from DDA analysis also show a good symmtry, which indicats that th rsults obtaind from th softning block approach ar rasonabl.

8 754 Q. Jiang t al m Bandd Or Schistos Or Hanging Wall Quartzit Foot Wall Quartzit (a) k = m Fig. 10 Initial gomtry for blasthol opn stoping opration (ITASCA Consulting Group Inc, 1996) solutions ar compard with th rsults from th rmoving block approach. Cas 2 is a larg blastd opn stoping opration problm and th solutions ar compard with th rsults from UDEC analyss Tst cass (b) k = (c) k =2.0 Fig. 9 Contours of horizontal displacmnt for diffrnt latral prssur cofficints Two typical tst cass ar prsntd to dmonstrat th capability and validity of th proposd softning block approach to modl xcavation problms. Cas 1 is th xcavation of a slop in a jointd rock mass and th Cas 1: Excavation of a slop Tst cas 1 involvs xcavation of a slop with a hight of 40 m and a slop angl of 50, as shown in Fig. 6. Th dimnsion of th calculation domain is m and th plan strain condition is assumd. It is furthr assumd that th rocks ar cut by two sts of joints, spacd at intrvals of 5 and 5 m, rspctivly, and at angls of 60 and 150, rspctivly, to th horizontal dirction. Th matrial constants for all rock blocks ar: Young s modulus E = 20 GPa, Poisson s ratio m = 0.35, and unit wight c = 27 KN/m 3. Th intrfac proprtis for th joints ar: friction angl / = 45, cohsion c = 0.6 MPa, tnsil strngth T = 0.2 MPa. Th lft and right boundaris of th slop ar rstraind in th horizontal dirction and th bottom boundary of th slop is fixd. Th initial strsss in th jointd rock mass bfor xcavation ar considrd as r yy ¼ ch; r xx ¼ kch; s xy ¼ 0 ð26þ whr h is th vrtical distanc from th ground surfac; and k is th latral prssur cofficint. Both th softning block and rmoving block approachs ar usd to valuat th dformation of th slop du to xcavation. For th cas of k = 1.0, Figs. 7 and 8 show th distributions of vctor displacmnts and principal strsss rspctivly. It can b obsrvd that th two

9 Excavation in jointd rock 755 Tabl 2 Summary of paramtrs for various typs of rock (ITASCA Consulting Group Inc, 1996) Hanging wall Bandd or Schistos or Footwall Young s modulus (GPa) Poisson s ratio Unconfind comprssiv strngth (MPa) Fig. 11 Block mshs for DDA analysis diffrnt mthods produc almost th sam rsults. In ordr to invstigat th influnc of th latral prssur cofficint k on th slop dformation, thr diffrnt latral prssur cofficints (k = 1.0, 1.5, 2.0) ar adoptd for th analyss. For diffrnt latral prssur cofficints, Fig. 9 givs th contours of horizontal displacmnts from th softning block approach. It is clarly sn that th latral prssur cofficint has a significant influnc on th dformation of th slop. Th horizontal displacmnts inducd by xcavation unloading gradually incras with th incrasing k valus, which would affct th stability of th slop to som xtnt. Cas 2: Opn stoping using vrtical rtrat A numrical xampl is prsntd to dmonstrat th capability of th proposd softning block approach to modl squntial xcavation stps. A larg blast hol opn stoping opration problm (ITASCA Consulting Group Fig. 12 Distributions of displacmnt vctors aftr th first xcavation stp Inc., 1996) was slctd. As shown in Fig. 10, a mining opration with a stply dipping or body with an avrag dip of 80 was analyzd btwn th 990 m lvl and th 1,190 m lvl of th min. Th avrag thicknss of th or body is 14 m. Thr is a low-angl discontinuous joint st with an avrag dip of 10 and an avrag spacing of 30 m. Four typs of rock (hanging wall quartzit, footwall

10 756 Q. Jiang t al. Fig. 13 Distributions of displacmnt vctors aftr th scond xcavation stp quartzit, bandd or and a wakr schistos or) ar dfind in th modl. Th paramtrs for all th rocks considrd ar summarizd in Tabl 2, basd on th rsults from laboratory tsts. Th proprtis of joint ar assumd as follows: friction angl / = 27, cohsion c = zro, tnsil strngth T = zro. Th pr-mining stat of strss was stimatd to b hydrostatic. At th 1,190 m lvl, th Fig. 14 Distributions of displacmnt vctors aftr th third xcavation stp strss is 33 MPa. Both th DDA and DEM (using cod UDEC) wr mployd for th dformation analysis. In th DDA analysis, th softning block approach with a softning factor of k = 10-4 was usd to modl th squntial mining procss. Th strss concntration in th crown pillar is of intrst aftr th mining of th stop is compltd. Th block mshs shown in Fig. 11 wr usd for DDA

11 Excavation in jointd rock 757 Fig. 15 Distributions of displacmnt vctors aftr th fourth xcavation stp Fig. 16 Distributions of displacmnt vctors aftr th final xcavation stp analysis. For simplicity, th boundary condition was simulatd with two fixd blocks rprsnting an infinit lastic mdium in th far fild. Th mining progrssd in fiv stags. Firstly, th uppr or blocks abov th 1,090 m lvl ar mind for a stop hight of 45 m. Thn th lowr stop is mind in four stags of 17, 15, 15 and 18 m, and a 10 m crown pillar is lft. For comparison, th rsults from both DDA and UDEC analyss ar plottd in Figs. 12, 13, 14, 15, 16, 17 and 18. Figurs 12, 13, 14, 15 and 16 show th distributions of displacmnt vctors aftr ach xcavation stag is compltd. Figur 17 shows th distributions of principal

12 758 Q. Jiang t al. Fig. 17 stp Distributions of principal strsss aftr th first xcavation Fig. 18 Distributions of principal strsss aftr th final xcavation strss vctors aftr th mining of th uppr lvl is compltd whil Fig. 18 shows th final strss concntration at th nd of th mining squnc. At this stag, most of th strsss ar transfrrd to th abutmnts. It can b sn from Figs. 12, 13, 14, 15, 16, 17 and 18 that th distributions of strss and displacmnt from th proposd softning block approach ar almost idntical to thos from th UDEC analysis. Compard with UDEC analysis, th softning block approach is an implicit analysis mthod. Th xcavatd blocks nd not b rmovd from th calculation modl and th pr-procssing work for ach xcavation stp is simplifid substantially.

13 Excavation in jointd rock 759 Conclusions Th instability of rock masss causd by xcavation is vry important in rock nginring. As th ovrall stability of th rock mass dpnds on th xcavation squnc and th corrsponding strss history, th simulation of th xcavation should trac construction procsss. This papr proposs a softning block approach for th simulation of squntial xcavations in jointd rock masss. Th concpt of th softning block is prsntd, followd by th algorithm and implmntation in th original DDA program. Two typical tst cass ar invstigatd to dmonstrat th proposd approach. Th rsults indicat that th softning block approach liminats th nd to rmov th xcavatd blocks from th calculation domain aftr xcavation. In this way, th numbr of blocks and thir storag positions in th global stiffnss matrix rmain th sam. As th ratio of th xcavatd zon to th whol calculation domain is vry small, th computational costs of th global stiffnss matrix for both th rmoving and softning block approachs ar of th sam ordr of magnitud. Howvr, th tim consuming and tdious pr-procssing work associatd with ach xcavation stp is significantly simplifid and th stp-wis xcavation can b simulatd mor fficintly by th softning block approach. For th nginring problms involving many xcavation stps in th calculation modl, this advantag bcoms particularly attractiv. Th distributions of displacmnt and strss from th proposd softning block approach ar idntical to thos from th rmoving block approach and UDEC analyss for th dmonstration cass. Th accuracy of solution obtaind from th softning block approach is accptabl for a softning factor smallr than It is suggstd that th softning factors ranging from 10-5 to 10-4 b adoptd for th proposd softning block approach. 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