Transcription

1 Orgnal caon: Wang, H. N., Ul, Sfano, Jang, M. J. and H, P. (4) Analycal soluons for unnls of llpcal cross-scon n rhologcal rock accounng for squnal xcaaon. Rock Mchancs and Rock Engnrng. Prmann WRAP url: hp://wrap.warwck.ac.uk/747 Copyrgh and rus: Th Warwck Rsarch Arch Poral (WRAP) maks hs work by rsarchrs of h Unrsy of Warwck aalabl opn accss undr h followng condons. Copyrgh and all moral rghs o h rson of h papr prsnd hr blong o h nddual auhor(s) and/or ohr copyrgh ownrs. To h xn rasonabl and praccabl h maral mad aalabl n WRAP has bn chckd for lgbly bfor bng mad aalabl. Cops of full ms can b usd for prsonal rsarch or sudy, ducaonal, or no-for prof purposs whou pror prmsson or charg. Prodd ha h auhors, l and full bblographc dals ar crdd, a hyprlnk and/or URL s gn for h orgnal madaa pag and h conn s no changd n any way. Publshr s samn: "Th fnal publcaon s aalabl a Sprngr a hp://dx.do.org/.7/s A no on rsons: Th rson prsnd hr may dffr from h publshd rson or, rson of rcord, f you wsh o c hs m you ar adsd o consul h publshr s rson. Plas s h prmann WRAP url abo for dals on accssng h publshd rson and no ha accss may rqur a subscrpon. For mor nformaon, plas conac h WRAP Tam a: publcaons@warwck.ac.uk hp://wrap.warwck.ac.uk

2 Analycal soluons for unnls of llpcal cross-scon n rhologcal rock accounng for squnal xcaaon H. N. Wang *, S. Ul, M. J. Jang 3, P. H 3, School of Arospac Engnrng and Appld Mchancs, Tongj Unrsy, Shangha, 9, P.R. Chna School of Engnrng, Unrsy of Warwck, Conry, CV4 7AL, U 3 Dparmn of Gochncal Engnrng, Collg of Cl Engnrng, Tongj Unrsy, Shangha, 9, P.R. Chna * Corrspondng auhor H. N. Wang E-mal addrss: wanghn@ongj.du.cn (H.N. Wang)

3 Absrac: Tm dpndncy n unnl xcaaon s manly du o h rhologcal proprs of rock and squnal xcaaon. In hs papr, analycal soluons for dply burd unnls wh llpcal cross-scon xcaad n lnar scolasc mda ar drd accounng for h procss of squnal xcaaon. For hs purpos, an xnson of h prncpl of corrspondnc o sold mda wh m aryng boundars s formulad for h frs m. An nal ansoropc srss fld s assumd. To smula ralscally h procss of unnl xcaaon, soluons ar dlopd for a m dpndn xcaaon procss wh h major and mnor axs of h llpcal unnl growng from zro unl a fnal alu accordng o m dpndn funcons o b spcfd by h dsgnrs. Explc analycal xprssons n ngral form ar oband assumng h gnralzd ln scolasc modl for h rhology of h rock mass, wh Maxwll and ln modls sold as parcular cass. An xns paramrc analyss s hn prformd o nsga h ffcs of arous xcaaon mhods and xcaaon ras. Also h dsrbuon of dsplacmn and srss n spac a dffrn ms s llusrad. Sral dmnsonlss chars for as of us of praconrs ar prodd. y words: rhologcal rock; non-crcular unnl; analycal soluon; squnal xcaaon.

4 Ls of symbols A, A and jb A and jm A and a a a A ( =,, L, ) Coffcns n nrs conformal mappng kb A Coffcns corrlad o coordnas and maral paramrs of gnralzd ln modl n Appndx km A Coffcns corrlad o coordnas and maral paramrs of Maxwll modl n Appndx Funcon of half major axs wh rspc o m Inal alu of half major axs (a m =) Fnal alu of half major axs B ( =,, L,9 ) Coffcns n dsplacmn soluons j B ( =, ; j =, ) Trms dfnd n Eqs. (A.4) and (A.6) b Funcon of half mnor axs wh rspc o m b b Inal alu of half mnor axs (a m =) Fnal alu of half mnor axs B C ( =, ) Coffcns corrlad o maral paramrs of gnralzd ln modl n Appndx M C Coffcns corrlad o maral paramrs of Maxwll modl n Appndx c() Paramr n conformal mappng (dfnd n Eq. (6)) D ( =, ) Coffcns n srss soluons (n Eq. (37)) j F ( =, ; j =, ) Trms dfnd n Eqs. (A.3), (A.4), (A.5) and (A.6) f Inrs conformal mappng wh rspc o arabl z f Inrs conformal mappng wh rspc o arabl z G Tm-dpndn rlaxaon shar modulus for scolasc modl G G H G G S Shar modulus of lasc problm Shar lasc modulus of h Hookan lmn n h Gnralzd ln modl Shar lasc modulus of h ln lmn n Gnralzd ln modl Prmann shar modulus of h gnralzd scolasc modl: G = G G ( G + G ) S H H H Funcon dfnd n Eq. (48) I l Funcon dfnd n (A.7) Tm-dpndn rlaxaon bulk modulus n h rock scolasc modl Bulk modulus of lasc problm Numbr of ms n nrs conformal mappng m() Paramr n conformal mappng (dfnd n Eq. (6)) n j ( nχ, nτ ) Local coordnas n r Vcor ndcang h drcon normal o h boundary Normalzd xcaaon ra for gnralzd ln modl 3

5 M n r Normalzd xcaaon ra for Maxwll modl P ( =,,3 ) Prscrbd m-dpndn srsss a srss boundary P ( P ) x p y p x ( p y ) q * R S σ ( S ) s s, s, T T M u u Tracon (surfac forc) along h x (y) drcon on srss boundary Vrcal comprss srss a nfny Boundary racons (surfac forcs) appld on h unnl fac o calcula h xcaaon nducd dsplacmns and srsss Numbr of adopd pons n drmnaon of coffcns of nrs conformal mappng Radus of axsymmrc problm usd n normalzaon of dsplacmns Tm-dpndn srss (dsplacmn) boundars Varabl n h Laplac ransform Tnsors of h srss and sran daors of lasc cas Tnsors of h srss and sran daors of scolasc cas Rardaon m of ln componn of gnralzd ln scolasc modl Rlaxaon m of Maxwll scolasc modl Tm arabl (= s h bgnnng of xcaaon) End m of xcaaon Tm arabl ( = s h m h nal prssur appld) Bgnnng m of xcaaon Prscrbd dsplacmns a dsplacmn boundary ( A ) ( ) u ( u A x y ) Dsplacmn corrspondng o scolasc problm of cas A (n Carsan coordnas) ( C ) ( ) u ( u C x y ) Excaaon nducd dsplacmn for scolasc problm (n Carsan coordnas) ( C) ( C ) u χ ( u τ ) Excaaon nducd dsplacmn for scolasc problm (n local u ( u ) x s x y u ( u ) x s y u ( u ) y u ( σ ) coordnas) Dsplacmn along x (y) drcon for lasc problms Prscrbd dsplacmn along x and y drcon on dsplacmn boundary Dsplacmn along x (y) drcon for scolasc problms Dsplacmns (srsss) nsor for h scolasc problm * u ( σ * ) Dsplacmns (srsss) nsor oband by rplacng G wh sl [ G() ] and wh sl [ () ] n h gnral soluon for h assocad lasc problm r X X (x, y) z z A Cross-scon xcaaon ra Poson cor of a pon on h plan Poson cor of a pon on h boundary Carsan coordnas Complx arabl: z=x+y Gnrc pon on h boundary 4

6 z z z Gnrc pon on h m-dpndn boundary a m Pon on m-dpndn dsplacmn boundary Pon on m-dpndn srss boundary z Complx arabl dfnd n Eq. (3) z j Boundary pons n z plan drmnd by Eq. (33) corrspondng o ζ j Grk symbols α δ δ Angl n local coordnas bwn n τ and x drcon Drac dla funcon Un nsor γ Funcon wh rspc o s oband by rplacng lasc modul G wh sl [ G() ] and wh sl [ () ] n κ Δ P x ( Δ Py ) Prscrbd srsss along h boundars n calculaon of xcaaon nducd dsplacmn and srsss Δ ( Δ ) Incrmnal srsss (srans) nducd by h unnl xcaaon s Δ u x, Δ u y Excaaon nducd dsplacmns of scolasc cas (x, y drcon) Δ u χ, Δ u Excaaon nducd dsplacmns of scolasc cas τ ( nχ, n drcon) τ Δ u Radal dsplacmn a h nnr boundary of axsymmrc lasc s * problm wh radus R and shar modulus G Δ u Radal dsplacmn a h nnr boundary of axsymmrc lasc s * problm wh radus R usd for normalzaon n Maxwll modl and shar modulus G Δ, Δ, Δ Excaaon nducd srsss of scolasc cas (x, y drcon) σ x σ χ σ y σ xy Δ, Δ, Δ Excaaon nducd srsss of scolasc cas ( nχ, n drcon) τ σ τ σ χτ ζ Complx arabl: ζ = ξ + η ζ Pons n ζ plan drmnd by Eq. (34) corrspondng o z j η η Imagnary par of ζ H Vscosy coffcn of h dashpo lmn n h gnralzd ln modl κ Maral coffcn dfnd by Eq. (4) λ Rao of horzonal and rcal srsss ξ ( ρθ, ) σ ( ε ) σ ( ε ) kk kk σ ( ε ) kk kk Ral par of ζ Polar coordnas Srss (sran) nsor for scolasc cas Man srss (sran) for lasc cas Man srss (sran) for scolasc cas S 5

7 σ x, σ x, xy σ σ y y Normal srss along x and y drcon for scolasc cas Normal srss along x and y drcon for lasc cas σ ( σ ) Shar srss for scolasc (lasc) cas x xy σ, σ, σ Inal normal and shar srsss a nfny y xy ( A ) ( A) ( A) σ x, σ y, σ Srsss corrspondng o scolasc problm of cas A (n xy Carsan coordnas) ( C ) ( C ) σ, σ, σ ( C ) x ( A ) χ y xy ( A) ( A) σ, σ, σ ( C ) χ τ χτ Excaaon nducd srsss (n Carsan coordnas) Srsss corrspondng o scolasc problm of cas A(n local coordnas) ( C) ( C ) σ, σ, σ Excaaon nducd srsss (n local coordnas) τ ϕ and ψ χτ Two complx ponals n analyss of lascy ϕ and ψ Two ponals oband by rplacng lasc modul G wh sl [ G() ] and wh sl [ () ] n ϕ and ψ ϕ ϕ ( A) ( B) and and ψ Two complx ponals for h lasc problm A ( A) ψ Two complx ponals for h lasc problm B ( B) ( C) ( C) ϕ and ψ Two complx ponals for calculang h xcaaon nducd dsplacmns and srsss n lasc cas ω Conformal mappng drmnd n Eq. (5) 6

8 Inroducon Analycal soluons ar naluabl o gahr undrsandng of h physcal gnraon of dformaons and srsss akng plac durng h xcaaon of unnls. Closd form soluons allow hghlghng h fundamnal rlaonshps xsng bwn h arabls and paramrs of h problm a hand, for nsanc bwn appld srsss and ground dsplacmns. Moror, alhough numrcal mhods such as fn lmn, fn dffrnc and o a lssr xn boundary lmn ar ncrasngly usd n unnl dsgn, full 3D analyss for xndd longudnal porons of a unnl sll rqur long runms, so ha h concpual phas of h dsgn procss rls on D analycal modls. In fac, analycal soluons allow prformng paramrc snsy analyss for a wd rang of alus of h dsgn paramrs of h problm so ha prlmnary smas of h dsgn paramrs o b usd n h succss phass of h dsgn procss can b oband. In addon, hy prod a bnchmark agans whch h orall corrcnss of sophscad numrcal analyss prformd n h fnal dsgn sag can b assssd. Mos yps of rocks ncludng hard rocks xhb m-dpndn bhaors [Malan ], whch nduc gradual dformaon or m n afr complon of h unnl xcaaon procss. Elasc and lasoplasc modls gnor h ffc of m dpndncy whch may conrbu n som cass up o 7% of h oal dformaon [Sulm al., 987]. In cas of squnal xcaaon, h obsrd m-dpndn conrgnc s also a funcon of h nracon bwn h prscrbd xcaaon sps and h naural rock rhology. Thrfor, propr smulaon of h whol squnc of xcaaon s of gra mporanc for h drmnaon of h opmal alus of h unnllng paramrs o ach opmal dsgn [Tonon, ; Sharfzadh al., ]. Squnal xcaaon s a chnqu bcomng ncrasngly popular for h xcaaon of unnls wh larg cross-scon n sral counrs (Tonon, ; Mura al., 3). For nsanc, km of unnls along h nw Tom and Mshn xprssways n Japan, ha bn bul a h so-calld cnr drf adancd mhod. 7

9 Ths squnal xcaaon chnqu has bn adopd by h Japans auhors as h sandard xcaaon mhod of mounan unnl (Mura, 3). In hs papr, h rock rhology s accound for by lnar scolascy. Th so calld gnralzd ln, Maxwll and ln rhologcal modls accordng o h classcal rmnology usd n rock mchancs (Jagr al., 3) wll b consdrd. Unlk h cas of lnar lasc marals wh consu quaons n h form of algbrac quaons, lnar scolasc marals ha hr consu rlaons xprssd by a s of opraor quaons. In gnral, s ry dffcul o oban analycal soluons for mos of h scolasc problms, spcally n cas of m-dpndn boundars alhough som closd-form soluons ha bn dlopd [Brady al., 985; Gnrk al., 964; Ladany al., 984]. Howr, n all hs works, only unnls wh crcular cross-scon ar consdrd, wh h xcaaon bng assumd o ak plac nsananously. In h lraur, h procss of squnal xcaaon s usually gnord snc prns h us of h prncpl of corrspondnc whch has bn radonally rsrcd o sold bods wh m naran gomrcal boundars [L, 955; Chrsnsn, 98; Gurn al., 96]. Howr, rcnly, analycal mhods ha bn nroducd o oban analycal soluons for crculars unnls xcaad n scolasc rock accounng for squnal xcaaon [Wang and N ; Wang and N ; Wang al. 3, Wang al. 4]. Bu for unnls of complx cross-sconal gomrs, (.g. llpc, rcangular, sm-crcular, nrd U-shapd, crcular wh a noch, c.), analycal soluons ar aalabl only n cas of lasc mdum [L al., ; Exadakylos al., ; Exadakylos al., 3], hnc dsrgardng h nflunc of h m-dpndn rhologcal bhaor of h rock and squnal xcaaon. In hs papr nsad, an analycal soluon s drd for squnally xcaad unnls of non-crcular (llpcal) cross-scon n lnarly scolasc rock subjc o a non-unform nal srss sa. Th srss fld consdrd s ansoropc so ha complx gologcal condons can b accound for. Th soluon s achd mployng complx arabl hory and h Laplac ransform. 8

10 Ellpcal and hors-sho scons wh h longr axs n h rcal drcon ar rahr common for ralway unnls (Snr, 996; Ambrg, 3; Anagnosou and Ehrbar, 3) and carns n rock,.g. h Eas Sd Accss Projc n Nw York (Won al., 3). Squnal xcaaon s mployd for hs yps of scons much mor ofn han for crcular scons snc Tunnl Borng Machnng s no aalabl for non-crcular scons. Also subway unnls ar ofn faurd by llpcal or hors-sho cross-scons (Hochmuh al., 987). Moror, sral road unnls rqur an llpcal or narly llpcal cross-scon wh h longr axs n h horzonal drcon o mnmz h xcaaon olum whls mng h gomrcal consrans rqurd for h consrucon of h road and rlad walk-ways (Mura al., 3). In Japan, llpcal scons ar spcfcally prscrbd for mounanous rgons (Mura, 3). Fnally, llpcal scons can also b h rsul of oalsaon of crcular scons n ansoropc rhologcal rock (Vu al., 3a; Vu al., 3b). A lmaon of h analycal soluons hr proposd s du o h absnc of lnng n h cross-scon consdrd. Th prsnc of lnng maks h problm mahmacally nracabl du o h consqun srucur ground nracon. Also n cas of non-crcular cross-scons h confnmn conrgnc mhod canno b appld du o h ansoropy of h dsplacmn fld. Howr, h analycal soluons hr nroducd can b mployd o prdc unnl conrgnc o assss whhr h prsnc of a lnng would b ncssary n h prlmnary dsgn phas. Also hy allow obanng a frs sma of h magnud of h xcaaon nducd dsplacmn fld. Moror, for dply burd unnls, lnng s ofn no ncssary. In h papr, analycal soluons ar prodd for a gnrc m dpndn xcaaon procss wh h major and mnor axs of h cross-scon ncrasng monooncally or m accordng o a funcon o b spcfd by h dsgnrs. Th analycal soluons ha bn drd n ngral form for h cas of a gnralzd ln scolasc rock. Th cas of Maxwll and ln modls can b oband as parcular cass of h soluon oband for h gnralzd ln modl. To calcula h dsplacmn and srss flds, numrcal ngraon 9

11 of h analycal xprssons n ngral form has bn carrd ou. Thn, a paramrc sudy nsgang h nflunc of arous xcaaon mhods, as wll as xcaaon ras, on h xcaaon nducd dsplacmns and srsss ar llusrad. Sral dmnsonlss chars of rsuls ar plod for h as of us of praconrs.. Formulaon of h problm Th prsn sudy focuss on h xcaaon of an llpcal unnl n a rhologcal rock mass. In h analyss, h followng assumpons wr mad: () Th rock mass s consdrd o conss of homognous, soropc, and lnarly scolasc maral undr sohrmal condons. () Th nal srss fld n h rock mass s dalzd as a rcal srss p and horzonal srss λ p, whr λ s h rao of horzonal and rcal srsss, as shown n Fgur. (3) Th unnl s dply burd, hnc no lnar araon of h srsss wh dph s consdrd. (4) Th xcaaon spd s low nough ha no dynamc srsss ar r nducd so ha srss changs occur n a quas-sac fashon a all ms. (5) Th cross-scon of h unnl s squnally xcaad, ha s, h half major and mnor axs of h llpcal unnl scon, a and b rspcly, ar m-dpndn. Th unnlng procss may b ddd no wo sags: h frs (.. xcaaon) sag, spans from m = o =, wh bng h nd m of h cross-scon xcaaon whls h scond sag runs from = onwards. In h frs sag, h sz of h major and mnor axs ars accordng o h m dpndn funcons, a() and b() rspcly, ha ar lkly o b dsconnuous or m du o chnologcal rqurmns snc squnal xcaaons nd o occur sp-lk. So, an mporan faur of h analycal soluons prodd n hs papr s ha hy ar applcabl o any yp of squnal xcaaons ncrasng hr spws or connuously or m. Th scond sag spans from = onwards, wh h alus of h major and mnor llpcal axs bng qual o a( = )= a and b( = )= b, rspcly. No

12 ha n cas h rao of h llps axs rmans consan, h scon grows homohcally, whras f h rao changs or m h shap of h scon ols oo (for nsanc from an nal crcular plo unnl o a fnal llpcal scon). Snc n mos of h cass h shap of h cross-scon changs or m, h gnral cas of m () = a () b () wll b consdrd. In h analyss, h ffc of h adancmn of h unnl along h longudnal drcon s no accound for. Th ffc of unnl adancmn can asly b consdrd mployng a fcous prssur as shown n (Wang al. 4; Pan and Dong 99), bu s hr omd for sak of smplcy n h draon of h soluon. So h cross-scon consdrd n h analyss s locad a a suffcn dsanc from h longudnal unnl fac ha srsss and srans ar unaffcd by hr-dmnsonal ffcs. Accordng o h aformnond assumpons, h problm can b formulad as plan sran n h plan of h unnl cross-scon. Ths plan wll b assumd o b of nfn sz wh an llpcal hol growng or m, subjc o a unform ansoropc srss fld, and mad of a scolasc mdum. Snc h hol s no crcular, polar coordnas ar no longr adanagous for h draon of h analycal soluon. Hnc, n hs papr Carsan coordnas (x, y) ar mployd for h draon of h soluon (s Fgur ) whch ar hn ransformd no polar coordnas ( ρθ, ) o show ha h (alrady known) soluon for a crcular cross-scon can b oband as a parcular cas. A sysm of local coordnas ( nχ, nτ ) s also mployd n h papr, wh n χ and n τ bng h normal and angnal drcons rspcly along h llpcal boundary (s Fgur ). In h followng analyss, sgn connon s dfnd as pos for nson and nga for comprsson. 3. Draon of h analycal soluon In ordr o fnd analycal soluons for boundary alu problms of lnar scolascy, h mos wdly usd mhods ar basd on h Laplac ransform of h dffrnal quaons and boundary condon quaons gornng h problm, whch n hs cas ar m-dpndn snc squnal xcaaon s accound for. In L [955] h classcal form of h corrspondnc

13 prncpl bwn lnar lasc and lnar scolasc soluons for boundary alu problms s dscrbd. Th prncpl sablshs a corrspondnc bwn a scolasc sold and an assocad fcous lasc sold of h sam gomry. Bu unl now, hs mhod has bn appld only o sold bods wh m naran boundars bcaus whn boundars ar funcons of m, h boundary condons canno b Laplac ransformd. In hs scon, w dscrb an xnson of h prncpl o m aryng srss boundars ha wll b mployd o ach h sough analycal soluon for h squnal xcaaon of unnls of llpcal cross-scons n scolasc rock. In h followng h rm gnral soluon s usd o ndca h mahmacal soluon o h s of dffrnal quaons rulng h problm whou any boundary condons mposd whras parcular soluon ndcas a soluon whch sasfs boh h s of dffrnal quaons rulng h problm and h boundary condons. 3. Solng procdur Assumng h Ensn s connon (.. rpad ndcs ndca summaon), h consu quaons of a gnral lnar scolasc sold can b xprssd n h form of conoluon ngrals, as shown blow: whr X s h poson cor and s ( X, ) = G( ) d ( X, ), σ ( X, ) = 3 ( ) d ε ( X, ). kk s and kk () ar h nsors of h srss and sran daors, 46 rspcly for h scolasc cas (hr h suprscrp sands for scolasc), dfnd as: wh σ and s = σ δσkk, 3 () = ε δεkk. 3 ε bng h nsors of srsss and srans rspcly. G() and () n Eq. (), rprsn h shar and bulk rlaxaon modulus, rspcly. Th asrsk ( ) n Eq. () ndcas h conoluon ngral, dfnd as: df ( τ ) f( ) df ( ) = f( ) f () + f( τ ). (3)

14 Th Laplac ransform of Eq. () ylds h followng: [ ] [ ] L s ( X, ) = sl G( ) L ( X, ), (4) L σkk ( X, ) = 3 sl ( ) L εkk ( X, ). L f() s a funcon of h arabl s dfnd n h Laplac ransform of h m whr [ ] funcon f(), dfnd as: [ f ] s L () = xp f() d, (5) Th Laplac ransform of h lnar lasc consu quaons s as follows (hr h suprscrp sands for lasc): 59 6 wh G and L s ( X, ) = GL ( X, ), (6) L σkk ( X, ) = 3 L εkk ( X, ). bng h lasc shar and bulk modulus, rspcly. No ha Eq. (4) s 6 oband from Eq. (6) by rplacng G wh s [ G() ] L and wh sl [ () ]. Thrfor, h 6 63 gnral soluon for a scolasc sohrmal problm, sasfyng h s of dffrnal quaons gornng sac qulbrum, knmac compably and h consu rlaonshp of h rock 64 n h m-dpndn doman, may b oband by rplacng G wh s [ G() ] L and wh [ () ] sl n h gnral soluon for h assocad lasc problm. Thn, prformng h Laplac nrs ransform, w oban: whr u * ( X,, s) and ( X ) ( σ X s ) * u ( X, ) = L u (,, s) L (7) (7a) * σ ( X, ) = L L (,, ), (7b) σ * ( X, s, ) ar h dsplacmns and srsss rspcly oband by 7 rplacng G wh s [ G() ] L and wh s [ () ] L n h gnral soluon for h 7 assocad lasc problm and L [ gs ( )] ndcas h nrs Laplac ransform, dfnd as: s L [ g( s)] = g( s)xp d π. (8) β In Eq.(7), h gnral scolasc soluon conans y unknown funcons of m, whch ha o b drmnd by mposon of h boundary condons. Dsplacmn boundary condons β + 3

15 may b xprssd as follows: ( ) u ( X, ) = u,wh X Su (), (9)(9b) and srss boundary condons as: whr j ( ) σ ( X, n ) = P,wh X S (), (9a) n j s a cor ndcang h drcon normal o h boundary, X s h poson of a σ 8 pon on h boundary, Sσ () and Su () ar h boundary surfacs whr srss and 8 dsplacmn condons rspcly ar appld, and P () and u () ar wo prscrbd funcons of m. Unlk problms wh m naran gomrcal boundars, X and n j n Eq. (9), ar funcons of m, hnc hy ar no consan wh rspc o h Laplac ransform, so ha hy canno b akn ou of h ransform opraor. Thrfor, h rlaonshp bwn h parcular soluon of h scolasc problm hr xamnd and h soluon of h assocad lasc on s unknown. Rplacng rwrn as: u and σ wh h xprssons n Eq. (7), Eq. (9) can b ( X ) u ( X, ) = L L u (,, s) = u, X S (), () (a) * X= X X= X ( σ X ) σ ( X, n ) = L L (, s, ) n = P, X S (). (b) * j X= X j X= X Th sysm of quaons () oghr wh Eq. (7) dfn h s of quaons o b sasfd by h parcular soluon ha w sk. To fnd h soluon, complx ponal hory wll b mployd (s h nx scon). 3. Problm formulaon Complx ponal hory has bn wdly usd o analyz mahmacal problms assocad wh undrground consrucons, spcally n h analyss of non-crcular opnngs. For a wo dmnsonal (D) lasc problm, dsplacmns and srsss can b xprssd n rms of wo analycal funcons of complx arabl,.. ϕ ( z) and ( z) 4 u σ ψ wh z = x + y and =, whch ar calld ponal funcons. So srsss and dsplacmns can b wrn as

16 3 4 (Muskhlshl 963): ϕ (,) z G( ux + uy) = κϕ( z, ) z ψ ( z, ), () z (,) ϕ z σx + σ y = 4R[ ], () z ϕ(,) z ψ(,) z σ y σx + σxy = z +. (3) z z wh x,y bng Carsan coordnas n h unnl cross-scon plan (s Fgur ) and G + n cas of plan srans 3 + G κ =, (4) 5 + 8G n cas of plan srsss 9 and gz (,) s h conjuga of h complx funcon g g(,) z ϕ and =. Th ponals ( ) ψ ( z) n Eqs. (-3) ar m dpndn snc h gomrc boundars of our problm ar m-dpndn. Accordng o h formulaon of h problm llusrad n h prous scon, h Laplac ransforms of h quaons rulng h scolasc problm ar prformd as follows: z ϕ (,,) zs L ( ux) + L ( uy) = L γ() s ϕ(,,) z s z ψ(,,) z s sl [ G( ) ] z (5) ( ) ( ) (,,) ϕ zs σx σ y L + L = 4L R[ ] (6) z ϕ(,,) zs ψ(,,) zs L ( σ ) ( ) ( y L σx + L σxy) = L z + (7) z z whr h funcon γ () s apparng n Eq. (5) s oband by rplacng wh s [ () ] G wh s [ G() ] L and L. Analogously, h analycal xprssons for ϕ (,,) zs and ψ (,,) zs ar oband by rplacng h lasc modul wh sl [ G() ] and s [ () ] L n ϕ (,) z and ψ ( z, ) rspcly. Thn, prformng h nrs Laplac ransform of Eqs. (5)-(7) and mposng h boundary condons, h quaons for h unknown funcons wll b sablshd, as shown n h followng. Snc n our problm only boundary condons on h srsss ar prsn, from hr onwards 5

17 w consdr only h srss boundary, S (). Th quaon mposng h boundary condon on h srsss s as follows: σ 3 whr T x and () (,,) z ϕ zs σ L L ϕ(,,) z s + z + ψ(,,) z s = ( Tx + Ty ) ds z, (8) za y z= zσ () T dno h racons acng on h (srss) boundary along h x and y drcons rspcly; zσ () s a gnrc pon on h (srss) boundary,.. zσ() Sσ() ; and z A s an arbrary pon on h boundary. Accordng o h hory of complx arabl rprsnaon (Muskhlshl 963), n cas of a smply conncd doman subjc o a consan body forc (n our cas no body forc s prsn), h wo analycal funcons ϕ and ψ ar maral paramr ndpndn so ha ϕ = ϕ and 9 3 ψ = ψ. Moror, also h analycal xprssons for h srsss ar ndpndn of h maral paramrs (s Eqs. (6) and (7)). Hnc w can smplfy Eq. (8) no: ϕ () (,) z zσ ϕ(,) z + z + ψ(,) z = ( Px + Py ) ds z, (9) za z= zσ () Thrfor, h boundary condons appld on h scolasc mdum ar h sam as h boundary condons appld on h assocad lasc mdum. Hnc, also h analycal soluon for h srss fld s h sam for boh h scolasc mdum and h assocad lasc on. 35 Concrnng dsplacmns nsad, hy can b oband by rplacng G wh s [ G() ] L and wh s [ () ] L n h Laplac ransformd xprssons oband for h lasc cas. 3.3 Calculaon of srsss and dsplacmns nducd by h xcaaon L us consdr a rock mass nally subjc o h followng gosac ansoropc srss sa: 39 σ = λ, x p σ y = p, σ =, snc a rfrnc nal m =. Th rock mass s subjc o xy growng dsplacmns or m du o s scosy. In Fgur (b), h nnr dashd ln ndcas h boundary Sσ ( ) of h unnl a a m, wh xcaaon. Pror o h bgnnng of h xcaaon (a m bng h sar m of = ), h racons p ( ( ) x z ) 6

18 and ( ( ) y ) p z (wh z ( ) dnong a gnrc pon on h m-dpndn boundary a m ) xchangd bwn h wo bods along A h bgnnng of h xcaaon, a m > S σ ( ) may b asly calculad mposng qulbrum., p ( ( ) x z ) and ( ( ) y ) p z go o zro along h boundary of h xcaad zon nducng dsplacmns n h rock. Th xcaaon nducd srss, sran and dsplacmn ncrmns wll b calculad snc h bgnnng of h xcaaon. To hs nd, h consu quaon (s Eq. ()) for h daorc srss nsor may b rwrn as follows (h draon for h soropc par of h srss nsor s analogous): 5 5 d d + s ( ) ( ) G( ) G( τ ) d G( ) d [ ( ) ( )] G( ) τ τ τ + = wh whls for = : () 5 53 d + s ( ) = ( ) G( ) + G( τ) () Th analycal xprssons of h shar rlaxaon modulus G for h consdrd scolasc 54 modls (s Fgur ) ar lsd n Tabl. Th rfrnc m can b chosn suffcnly larg 55 so ha. In cas of modls wh lmd scosy,.g. gnralzd ln and ln modls, hr calld Typ A modls, h frs wo rms n Eq. () urn ou o b qual o h frs wo rms n Eq. () (for h dmonsraon of hs qualy s Appndx A.), so ha: d + s ( ) = s ( ) + G( τ) + [ ( ) ( )] G( ) () + Insad, n cas of modls wh unlmd scosy,.g. Burgrs and Maxwll modls, hr calld Typ B modls, hs s no h cas so ha Eq. (3) no longr holds ru (s Appndx A.3). Now, 6 for Typ B modls, w dfn Δs ( ) s ( ) s ( ) and Δ ( ) ( ) ( ), as ncrmnal 6 63 srsss and srans rspcly nducd by h unnl xcaaon. Inroducng a nw rfrnc m, wh =, hn Eq. () may b rwrn as follows: No ha h rlaonshp bwn dδ + Δ s () = G( τ) + Δ ( ) () ()* () G = G dδ (3) Δ s and Δ s h sam as ha n Eq. (). For h fld of 7

19 nducd srsss, srans and dsplacmns, also h sam quaons of qulbrum and compably mus b sasfd. Howr, h corrspondng boundary condons dffr from h boundary condons shown n Fgur, and h srsss prscrbd along h boundars (s Eq. (8)) may b wrn as follows: ( () ) ( () ) ΔP +Δ P = p z p z, along h nnr m-dpndn boundary; and x y x y Δ P + Δ P =, along h our (nfn) boundary (4) x y Th boundary condons for calculang h nducd srsss and srans ar shown n Fgur (b). 73 No ha h racons p x and p y appld on h nnr boundary n Fgur 3 (b) and (c) ar of qual absolu alu, bu of oppos drcon. Th soluon procdur mployd for yp A modls canno b usd snc Eq. () no longr holds ru. In cas of Typ B modls, h rock bfor xcaaon undrgos connuous dsplacmns (s Eq. (A.9) n Appndx A.). So n ordr o calcula h xcaaon nducd dsplacmns h rock wll b assumd lasc bfor h xcaaon aks plac. As oulnd n Scon 3., h soluon for h dsplacmns can b oband from h soluon of h assocad lasc problm. Th lasc soluon for our problm wll b oband as h combnaon of wo fcous cass hr calld cas A and B accordng o h prncpl of suprposon. In cas of no racons on h nnr boundary (s Fgur (a)), w oban Soluon A-la (lasc soluons of cas A); whl h cas of a plan whou hol subjc o h dsplayd boundary srsss n Fgur (b) s rfrrd o as Soluon B-la (lasc soluons of cas B). Thrfor, h lasc nducd soluons,.. Soluon C-la, may b oband by subracng Soluon B-la from Soluon A-la. In h followng scon, h soluons wll b drd by mans of complx ponal hory. 3.4 Draon of h analycal soluon Th mhod of conformal mappng prods a ry powrful ool o sol problms nolng complx gomrs. L us consdr h complx plan z=x+y wh x and y 8

20 rprsnng h horzonal and rcal drcons rspcly n h plan of h unnl cross-scon (s Fg. ). Also l us dfn a funcon o map h (nfn) doman (n h z plan) of h rock surroundng h llpcal cross-scon no a fcous doman (n h ζ-plan wh ζ = η + ξ ) wh a un crcular hol. Snc h llpcal cross-scon ars or m, h mappng funcon s m-dpndn oo: 96 m () z = ωζ (, ) = c( ) ζ+ ζ (5) 97 whr: 98 a () + b () c () = and a () b () m () =. (6) a () + b () 99 If a () b () s consan durng h xcaaon sag, h xcaaon xpands homohcally and m rmans consan or m. Accordng o h boundary condons shown n Fgur a, wo complx ponals for h lasc problm A wh m-dpndn boundars may b drd as follows (Muskhlshl, 963): [ λ+ + λ ] ( ) ( ) ( ) ( ) ( ) A ( + λ) pc ( ) m () ϕ ζ, = ζ m pc 4 + ζ + ζ ( λ ) ( ) m () ( ) ( ) ( ) ( A) pc pc ψ ζ, = ζ ( λ) m ( ) ( λ) m( ) + ζ ζ [ λ λ ] + ( + ) m ( ) + m ( ) pc ( ) + ζ ζ m ( ) (7) (8) Accordng o lascy hory, h wo ponals usd o calcula h lasc dsplacmns of h nfn plan subjc o h ansoropc nal srss sa pror o xcaaon (Soluon B-la) ar as follows [Ensn and Schwarz, 979]: ( B) ( λ) pc ( ) m () ϕ ( z) = + ζ 4 + ζ, ψ ( z) B ( λ) pc ( ) m () = ζ + ζ ( ) Accordng o h suprposon prncpl of lascy, h ponals for calculang h xcaaon nducd dsplacmns ar as follows (Soluon C-la): (9) 9

21 [ λ+ + λ ] ( C ) ( ) ( A ) ( ) ( B ) ( ) m ( ) pc ( ) ϕ ζ, = ϕ ζ, ϕ ( ζ, ) = (3) ζ (, ) (, ) (, ) ( C) ( A) ( B) ψ ζ = ψ ζ ψ ζ = [ λ λ ] ( ) m ( ) m ( ) pc ( ) pc () ( ) = + λ ( + m ( ) ) + ( λ ) m ( ) + ζ ζ ζ m ( ) Afr subsung Eqs. (3) and (3) no Eqs. (), () and (3) rspcly, h lasc dsplacmns and srsss (Soluon C-la) on h plan ζ may b calculad. Accordng o h analyss n Scon 3., h soluon for h scolasc cas can b oband by applyng h prncpl of corrspondnc, and h Laplac nrs ransform of h arabls (srsss, srans, c.) calculad for h lasc cas wh h arabl z rad as a consan n h Laplac ransform. Howr, n Eqs. (3) and (3) h arabl ζ appars rahr han z, hnc accordng o Eq. (5), Eqs. (3) and (3) ar m dpndn and canno b Laplac ransformd. To rplac ζ wh z and, h nrs funcon of h conformal mappng f ( z ) b found. If ζ n Eqs. (3) and (3) s rplacd wh f ( ),, (3) ζ = nds o z, hn all h m-dpndn funcons n Eqs. (3) and (3) may b Laplac ransformd, and h scolasc soluon may b drd from Eqs. (5), (6) and (7). Thn, dfnng: z z = (3) c () and subsung n Eq. (5) h followng s oband: m () z = ζ + (33) ζ If h xcaaon procss s homohc,.. m s a consan, hn hr s no arabl n Eq. (33), and h nrs conformal mappng may b xprssd as [Zhang ]: wh h y undrmnd coffcns, (,,..., ) k= ( ) k ζ = f ( z, ) = Az + A z (34) k k AA k=. For numrcal rasons, h srs wll b runcad o a fn numbr, l, of rms o calcula h funcon approxmaly. Du o h fac ha h nrs conformal mappng s drd from h corrspondng drc conformal mappng,

22 333 hr s a on-o-on corrspondnc bwn all h alus of on funcon, wh h alus of h 334 ohr funcon. L us choos a numbr of pons ζ ( j=, L, q), wh q=6, lyng on h j 335 nnr boundary of h un crcl n h ζ plan o calcula h corrspondng pons z lyng j 336 on h nnr boundary n h z plan usng Eq. (33). Thn, q lnar quaons for A and A k 337 can b oband by subsung ζ j and z no Eq. (34): j ζ ζ M ζ = + ζ l k = Az + Ak z k = l k = Az + Ak z k = l k j Az j Akz j k = M l k q = Azq + Akz q k = Snc h numbr of ndpndn quaons s largr han h numbr of unknown coffcns ( A, A, A,, A l ), h sysm s ndrmna. To sol h sysm,.. o drmn h unknown coffcns, w mployd h mhod of mnmum las squars. Th non-zro coffcns oband for h llpcal shaps hr consdrd, ar lsd n Tabl for l=5. In Fgur 4, h (35) 343 curs on plan z and ζ drmnd by drc and nrs conformal mappng rspcly ar 344 plod for arous shaps of h llpcal unnl boundary. Th llpss on h z plan (plod n Fgurs 4 (a-), (b-) and (c-)), map no h crcls plod as dashd lns on h ζ plan (Fgurs 4a-, b- and c-), whch ar drmnd a Eq. (5). Th curs wh connuous ln on h ζ plan ha bn oband by nrs conformal mappng (s Eq. (34)), appld o h 348 llpss on h z plan. I can b obsrd ha curs drmnd by nrs conformal mappng, ar ry clos o crcular. Howr, w can obsr ha h nrs conformal mappng s lss accura for h nnr boundary whn m s largr han.4. Accordng o h drc and nrs conformal mappngs, a on-o-on corrspondnc for pons on h z and ζ plan s

23 sablshd. For a gnral non-homohc xcaaon procss, h paramr m s a funcon of m, so ha an analycal xprsson for h nrs conformal mappng canno b oband. Howr, dscr alus of h nrs conformal mappng or m may b calculad accordng o h prscrbd m() and c(). Subsung Eqs. (3), (3), (34) no Eqs. (5) and (6), h xcaaon nducd dsplacmns and srsss n lnarly scolasc rock (Soluon C-s) can b drd as follows: ( C ) ( C ) ( ux ) + ( uy ) = p [ B (,) z s + B (,) z s + B3 (,) z s + B4 (,) z s ] L L (36) wh σ σ ( C) x ( C) y [ λ+ + λ m] γ () s ( ) ( ) c ( ) B (,) z s = L, sl [ G() ] f( z) { } { } = p R D ( z, ) m p R D ( z, ), (37) { } ( C) σ xy = p Im D( z, ). (38) z λ+ ( + λ) m( ) B (,) z s = L, sl [ G() ] f ( z) m( ) ( + m ( ) )( + λ) + m( )( λ) c( ) ( ) B3 (,) z s = L, sl [ G() ] f z [ λ ( + λ) m ( )] + m ( ) c( ) B4 (,) z s = L sl [ G() ] f( z) f ( z) m( ) D [ λ λ ] 3 z ( ) m( ) f ( z) λ (,) z = 3, D (,) z = + ( + ) m ( ) + + ( λ) m( ) c ()[ f ( z) m ()] [ f ( z) m ()]. + m ( ) [ λ ( + λ) m( ) ][3 f ( z) m( )] + 3 f ( z) m( ) λ ( + λ) m ( ), [ f ( z) m( )] Bcaus h srsss of h scolasc and lasc cass ar h sam, h srsss of cas A (soluon A) ar h oal srsss n h rock, and can b calculad by h wo ponals of Soluon A, as: ( A) x σ m( λ ) λ = p R + p D( z, ) m p R { D( z, ) }. (39) σ ( A) y { } ( A) σ xy = p Im D( z, ) (4)

24 If α s h angl bwn h horzonal axs x and h normal drcon (s Fgur ), h angnal and normal dsplacmns and srsss around h boundary of h xcaaon may b calculad as follows: ( ( C ) ( ) ( C ) ) α ( ) ( ) u ( C ) ( C χ + uτ = ux + uy ) L L L L (4) ( A) χ ( A) τ σ σ ( C) χ ( C) τ α { } { } = p R D ( z, ) m p R D ( z, ), (4) { α } ( C ) χτ σ = p Im D ( z, ). (43) σ m( λ ) λ α = p R + p D( z, ) m p R { D( z, ) }. (44) σ { α } ( A ) χτ σ = p Im D ( z, ) (45) Th xprssons for srsss hr prodd ar suabl for all lnar scolasc modls, snc h srss sa dpnds only on h shap and sz of h opnng; conrsly dsplacmns dpnd on h scolasc modl consdrd. Th analycal soluon for h dsplacmns s prodd n h nx scon. 3.5 Soluon for h dsplacmns Rock masss whch ha srong mchancal proprs or ar subjc o low srsss xhb lmd scosy. For hs yp of bhaor, h gnralzd ln scolasc modl (s Fgur a) s commonly mployd [Da 4]. On h ohr hand, wak, sof or hghly jond rock masss and/or rock masss subjc o hgh srsss ar pron o xcaaon nducd connuous scous flows. In hs cas, h Maxwll modl (s Fgur b) s suabl o smula hr rhology, du o h fac ha hs modl s abl o accoun for scondary crp. In hs scon, h analycal soluon for h gnralzd ln modl s dlopd. Th consu paramrs of hs modl ar as follows: ) h lasc shar modul G H, du o h Hookan lmn n h modl; ) G, du o h sprng lmn of h ln componn; ) h scosy coffcn η, du o h dashpo lmn of h ln componn (s Fgur c). Th soluon for h Maxwll modl may b oband as a parcular cas of h gnralzd ln modl, for G =. No ha 3

25 395 h soluon for h ln modl (s Fgur c) may also b oband as anohr parcular cas 396 of h Gnralzd ln modl for GH Assumng ha h rock s ncomprssbl,.. ( ), h wo rlaxaon modul apparng n h consu quaons (s Eq. ()) ar as follows: ( ) G GH+ G G η = + H G H G G + G G + G H H, ( ) = (46) Th nducd dsplacmns, Soluon C-s, may b drd by subsung Eq. (46) no Eq. (4): wh ( C ) ( C ) α p χ τ [ (,) (,) (,) (,)] u + u = B5 z + B6 z + B7 z + B8 z (47) 4 ( )[ + + ] H (, τ) cτ λ ( λ) m( τ) λ+ ( + λ) m( τ) B5 (,) z =, B 6(,) z = z H(,) d f[ z( τ )] τ τ, f [ z( τ)] m( τ) ( ) ( ) H (, τ) c τ + m ( τ) ( + λ) + m ( τ)( λ) B7 (,) z =, f [ z( τ )] ( )[ ] H (, τ) cτ λ ( + λ) m( τ) + m( τ) B (,) z =, and f [ z( τ)] f [ z( τ)] m( τ) 8 { } ( τ ) η H (, τ) δ( τ) G η H G = +. (48) Whn m= and λ=, h problm rducs o a crcular unnl subjc o a hydrosac sa of srss, and h dgnra soluon n Eq. (47) concds wh h soluon prodd n (Wang and N ), hnc h problm bcoms axsymmrc. 4. Comparson wh FEM rsuls Two yps of FEM analyss wr run mployng h FEM cod ANSYS (rson., mployng h modul of srucur mchancs). Th frs FEM analyss wans o rplca h scolasc problm of soluon A whras h scond on h problm of soluon C. All FEM analyss wr carrd ou wh a small dsplacmn formulaon o b conssn whn h draon of h analycal soluon. Analycal soluon A-s for gnralzd ln scolasc modl can b drd by subsung Eqs. (7), (8), (34) and (46) no Eqs. (5)-(7). Th xprssons for dsplacmns 4

26 47 ar as follows: ( A ) ( A ) p ux + uy = [ B5(,) z + B6(,) z + B7(,) z + B8(,) z + B9(,) z ] (49) 4 whr B9 (,) z = ( λ) z H(,) τ. Dsplacmns and srsss of soluon C-s and srsss of soluon A-s can b found n Eqs. (47), (37), (38), (39) and (4), rspcly. Frs, w shall compar dsplacmns and srsss of soluon A-s oband by h analycal soluon wh h FEM analyss along 3 drcons (horzonal, rcal, 45 o or h horzonal). Scond, h xcaaon nducd srsss and dsplacmns from h analycal soluon C-s and FEM along Ln ( 45 o or h horzonal) wll b compard o alda h corrcnss of h analycal soluon hr achd. In h FEM analyss of cas A-s, nal srsss ar appld on a planar doman hang an llpcal hol wh h major axs bng a long and mnor axs b long (Par Ⅰ n Fgur 5). Thn, h rock s squnally xcaad a dffrn ms (s Par Ⅱ o Ⅶ n Fgur 5), as lsd n Tabl 3. In h scond smulaon nsad, nal srsss ar frs appld on h fn rcangular doman whou hol, hn an xcaaon sarng afr 5 days s smulad. Par Ⅰ o Ⅶ ar xcaad a =5 h day, 5 h day,, 56 h day, rspcly. In h nd, h xcaaon nducd srsss and dsplacmns can b oband by subracng h nal alus bfor xcaaon from h ons calculad n h xcaaon sag. In FEM analyss, lmns ar dld a h m of xcaaon by sng h sffnss of h dld lmns o zro (by mulplyng h sffnss marx by -6 ). A rcal srss, p = MPa, and a horzonal srss, λ p wh λ =.5, wr appld a h boundars of h doman of analyss. Th rock was smulad as a gnralzd ln mdum, wh h followng consu paramrs adopd: GH = MPa, G = MPa and η = MPa day. Th xcaaon squnc hr consdrd s spcfd by h alus of h major and mnor axs of h llpcal scon lsd n Tabl 3 wh an nal alu of a =3.m for h major axs and b =.m for h mnor axs. No ha h rao m()=cons,.. h 5

27 llpcal scon ols homohcally. Th FEM msh narby h hol s plod n Fgur 5. Th pons and lns slcd for comparson bwn h FEM analyss and h analycal soluon ar plod n Fgur 6: hr pons on h nnr boundary (pons,, 3 n h Fgur) and hr lns, on horzonal (ln ), on rcal (ln 3) and on nclnd a 49.8 or h horzonal (ln ), wr chosn. In Fgur 7, dsplacmns and srsss for pons, and 3 ar plod rsus m. In Fgurs 8 and 9 dsplacmns and srsss rspcly a four dffrn ms (= s, 3 rd, 6 h and h days) ar plod for lns, and 3 rsus h dsanc o h cnr of h llps. I mrgs ha h prdcons from h analycal soluon ar n xclln agrmn wh h rsuls from h FEM analyss. In Fgur 7 can b nod ha dsplacmns and srsss undrgo a spws ncras followng nsananous xcaaon ns ( s, nd, 3 rd,... 6 h days). In Fgur h xcaaon nducd dsplacmns and srsss along Ln oband from h analycal soluon and FEM analyss, ar plod. A good agrmn n rms of boh srsss and dsplacmns can b obsrd. Unlk h cas of soluon A, almos all h nducd dsplacmns ar dcrasng funcons of h dsanc o h cnr of h llps. 5. Paramrc nsgaon In ordr o sudy h nflunc of squnal xcaaon ra and mhods, as wll as h m-dpndn dsrbuon of dsplacmns and srsss, a paramrc nsgaon s llusrad n hs scon. Wh h sam noaon as n Scon 4, a and b ar alus of h half major and mnor axs a m =, rspcly, and a, b ar h alus of h axs whn, wh bng h nd m of xcaaon. Assumng an axsymmrc lasc problm,.. crcular unnl n nfn plan, subjcng o hydrosac nal srss p, wh unnl radus = ( + ) and shar modulus GS = GHG ( GH + G) whch s h prmann modulus of * R a b gnralzd ln modl (s Fgur a), h xcaaon nducd radal dsplacmn a h nnr boundary of h unnl can b calculad as follows: 6

28 u = p R G (5) s * ( S) In h followng analyss, h nducd dsplacmns of scolasc cass for llpcal unnl xcaaon wll b normalzd by h dsplacmn lsd n Eq. (5), and srsss ar normalzd by p. Thrfor, pos dmnsonlss normal srss s comprsson n h followng fgurs. Now, l us dfn h dmnsonal paramr T = η G, whch xprsss h rardaon m of h ln componn of h gnralzd ln modl. I s connn o normalz h m 47 as / T for h gnralzd ln modl. For Maxwll modl, G s qual o zro (s Fgur 473 ), hnc T can no b usd n normalzaon; nsad, h rlaxaon m TM = η G wll b H 474 mployd o normalz h m as / T. M Influnc of h xcaaon ra Concrnng squnal xcaaon, h alus of half major and mnor axs grow from zro o h fnal alus. In hs cas, a lnar ncras of h unnl axs or m s assumd whn s a+ r lss han,.. a () =, whr r s h (consan) spd of cross-scon a > xcaaon. I s connn o xprss h half major axs n dmnsonlss form as: 48 a a () + nr < = a T a (5) 48 whr n s h dmnsonlss xcaaon spd, dfnd as follows: r 48 n T = (5) r r a 483 In h paramrc analyss a a = 4 was assumd oghr wh h followng dmnsonlss 484 xcaaon spds: () nr, corrspondng o h cas of nsananous xcaaon (mplyng T = ); () n =.5 (mplyng T =.5 ); (3) n =.75 (mplyng T =. ); and (4) r n =.5 (mplyng T =.5). Concrnng h xcaaon mhod, a homohc xcaaon r wh h consan rao a () b () =. ( m = 3) s assumd n h analyss, wh h rao of r 7

29 horzonal and rcal srsss λ =/3. In ordr o cor h wd rang of rsponss for rock yps of dffrn scous characrscs, h m-dpndn dsplacmns and srsss wr analyzd for wo yps of rocks of dffrn sffnss raos: G =.5 and.. In Fgurs and h m-dpndn G H radal and angnal dsplacmns for h rock a h fnal unnl fac (.. h fac a h nd of xcaaon of cross-scon) wh angl θ = o, 45 o and 9 o ar plod for h yps of rock and xcaaon ras consdrd. Th symbol rprsns h nd m of xcaaon,. Th fgurs show ha h normal dsplacmn ncrass wh m and rachs a consan alu afr a cran prod of m; howr, h angnal dsplacmn frs dcrass wh m and hn ncrass rapdly owards h nd of h xcaaon, hn nually rachs a consan alu. Comparng Fgur wh Fgur, h fnal dsplacmns ar rachd lar for rocks wh smallr sffnss raos (Fgur ). I can also b nod ha h bggr h sffnss rao s, h largr h afr xcaaon dsplacmns ar. For boh yps of rock, h rsuls show ha a lowr xcaaon ra mpls a longr xcaaon m, whch n urn lads o a largr alu of normal dsplacmn a h unnl fac wh θ = 45 o and 9 o whn = ; howr, h angnal 53 dsplacmn a θ = 45 o and h normal dsplacmn a θ = o show no sgnfcan dffrnc among h arous xcaaon ras a m =. I can also b obsrd ha hghr xcaaon ras mply largr normal dsplacmn a any m, and h maxmum absolu alu of h angnal dsplacmn durng h xcaaon sag wll b largr. Th Maxwll modl s suabl o smula h rhology of wak, sof or hghly jond rock, wh connuous lnar scous rspons whn consan srsss ar appld. Whn G =, h Maxwll modl s oband (Fgur b). In hs cas, accordng o Eq. (5), G S =, andu s. Hnc, n ordr o normalz h dsplacmns, a dffrn normalzaon mus b mployd. To 5 ach hs, w chos o rplac G wh h nal lasc modulus S G of Maxwll modl n H 8

30 5 Eq. (5) o calcula h radal dsplacmn a h unnl fac for h axsymmrc lasc 53 * problm,.. us = p R ( GH). In hs cas, w adop n T = as h dmnsonlss M r M r a 54 xcaaon ra wh TM = η G, and w consdr h followng four xcaaon ras n our H 55 analyss: () M M M n ; () n =.5 ; (3) n =.75 ; and (4) n =.5. In Fgur 3, h M r r r r o normalzd dsplacmns a h fnal unnl fac a pon θ = 45 ar plod agans h normalzd m /T M. Snc h srsss of h rock ar consan afr xcaaon (s Eqs. (44) and (45)), n Fgur 3, h dsplacmns afr xcaaon grow lnarly or m. I also mrgs ha h nflunc of h xcaaon ra for Maxwll modl s smlar o ha for h gnralzd ln modl. Obsrng Eqs. (44) and (45), s shown ha h srsss dpnds only on h sz and shap of h opnng, hnc gn a prscrbd squnal xcaaon h srss fld s dncal for all h scolasc modls. In Fgur 4, h prncpal srsss of h rock a h unnl fac a pons θ = o, θ = 45 o and θ = 9 o, ar prsnd for arous xcaaon ras. As can b xpcd, h araons of srss wh m ar mor gradual for lowr xcaaon ras. In all h cass, h maxmum dffrnc bwn h wo prncpal srsss occurs afr xcaaon. 5. Influnc of h xcaaon mhods In hs scon, h fnal alus of h major and mnor axs and rao of horzonal and rcal srsss λ ar h sam as n h prous scon wh h nd m of xcaaon bng T =.. Th m-dpndn unnl nnr boundars, whch smula h ral across-scon xcaaon procss as cnr drf adancd mhod [aush and Hrosh 3] (.g. mhod C shown n Fgur 5), drllng and blasng mhod [Tonon ] (.g mhods A, B and B shown n Fgur 5), ar shown n Fgurs 5 (a), (b) and (c). Th funcons a () and b () ar plod n Fgurs 6 wh hr analycal xprssons prodd n Tabl 4. In ral projc applcaon, a () and b () may b drmnd by accounng for h acual xcaaon procss, 9

31 as prscrbd by h dsgnrs. Squnal xcaaon mhods A and C ar spws xcaaons, n whch pars o 5 (or o 4) ar xcaad nsananously n succsson. In Fgur 5 s shown ha h shap of h opnng n mhod A frs changs from llps o crcl, and hn o llps, by squnal xcaaon along h major axs drcon. Obously, h xcaaon s nonhomohc wh m-dpndn rao a () b () durng xcaaon. Fgur 6 shows ha h adopd xcaaon ra s fasr n h bgnnng and slowr oward h nd of h xcaaon n hs mhod. In mhod C, h nal shap of h opnng s crcular, hn gradually changs o llpcal wh h ncras of h rao a () b () wh m. Th xcaaon ra s slowr n h bgnnng and bcoms fasr oward h nd of xcaaon, whch s oppos of ha n mhod A. Excaaon mhods B and B nsad, ar connuous homohc xcaaons ( a () b () =.). mhod B consss of a lnar xcaaon a unform spd, whras mhod B consss of an xcaaon funcon a () n quadrac form, wh a fasr xcaaon ra oward h nd. For h wo yps of rock (.. G G H =.5 and G G H =.), h m dpndn normal 55 and angnal dsplacmns a h fnal unnl fac wh angls o, 45 o and 9 o ar plod n Fgurs 7 and 8 for h four xcaaon mhods. I mrgs ha h nducd dsplacmns ar sns o h xcaaon mhod adopd. In parcular can b obsrd ha h mhods wh fasr spds n h arly sags lad o largr normal dsplacmn a gnrc m (xcp for h cass of θ = o ), as wll as a h nd of xcaaon m. For all h xcaaon mhods, h o normal dsplacmns a θ = 45 and 9 o ncras or m and rach a consan pos alu afr a cran prod of m; whras h normal dsplacmns a θ = o ar approxmaly zro n h arly sags of xcaaon, and ncras rapdly oward h nd of xcaaon for mhods o B, B and C. Th angnal dsplacmns a θ = 45 n Fgur 7(d) ar nga and frs dcras n h arly sags and hn ncras o pos alus. In ordr o analys h maxmum dffrnc of dsplacmns among arous xcaaon 3

32 mhods, h normalzd dsplacmns and dffrnc rados bwn mhods A and C (h dffrnc bwn hs wo mhods s h maxmum accordng o Fgurs 7 and 8) a m = ar lsd n Tabl 5. Th dffrnc raos of normal dsplacmn for h rock wh G G H =.5 rang from 6% o 33%, and rach up o 6% for angnal dsplacmn. Th raos rang from 7% o 3% for normal dsplacmn and % for angnal dsplacmn for h yp of rock wh G G H =., whch ar lss han h ons n cass whr G G H =.5. Fgur 9 prsns h normalzd prncpal srsss calculad a h fnal unnl fac. I may b obsrd ha h srsss show no dffrnc for all of h xcaaon mhods whn, bcaus h fnal shap and sz of h unnl ar h sam. Howr, durng h xcaaon sag h srss fld s clarly affcd by h xcaaon mhod adopd. Ths srss analyss accounng for squnal xcaaon s aluabl o chck for ponal falur mchansms snc prods h srss sa a any m for any pon n h rock. 5.3 Dsrbuon of dsplacmns and srsss for dffrn xcaaon mhods In hs scon, h dsrbuons of dsplacmn and srss for h rock wh G G H =.5 ar analyzd, adopng squnal xcaaon mhods A and C wh h sam nd m of xcaaon. Four pons n m ar consdrd n h followng analyss: m () : () T =., h bgnnng of xcaaon; m : () () T =.5, durng h xcaaon sag; m : (3) (3) T =., h nd of xcaaon; and m : (4) (4) T =.5, h m afr xcaaon a whch no furhr dsplacmns praccally occur. Fgur prsns h conour plos of h normal dsplacmn a ms and (3) for (4) mhods A and C, rspcly; and Fgurs prsns h conour plos of h angnal dsplacmns. Th Fgur shows ha, h dsrbuon rgulars of normal dsplacmn a sam m afr xcaaon,.g h dsrbuons n Fgurs (a) and (c) or n (b) and (d), ar ry smlar, whar mhod s adopd. I can also b nod ha h alus of dsplacmn a sam poson corrspondng o dffrn mhods ha sgnfcan dffrnc around h unnl 3

33 crown whn, whras a (3), h dffrnc s ry small. Fgur shows ha h maxmum (4) nga angnal dsplacmn occurs nsd of h ground, and h maxmum pos on occurs a h unnl fac wh θ approxmaly qual o -3 dgr. Furhrmor, h dsrbuons of angnal dsplacmn wh dffrn xcaaon mhods ar ry smlar xcp for h alus,.g. h ons n Fgurs (a) and (c) or n (b) and (d). In Fgurs (a) and (b), h conours of h major and mnor prncpal srsss rspcly ar plod a m. (3) Fgurs 3(a) and (b) prsn h dsrbuon of normal and angnal dsplacmns a h fnal unnl fac as a funcon of h angl θ for xcaaon mhod A and C a arous ms, whch du o symmry of h problm, s llusrad n h rang θ = o o 9 o only. I mrgs ha h normal dsplacmn s a monooncally ncrasng funcon of h angl, and h cur shaps ar smlar for arous xcaaon mhods. Howr, a ms and (), h alus of (3) normal dsplacmn for h wo xcaaon mhods ar sgnfcanly dffrn. Unlk h 598 normal dsplacmn, h angnal dsplacmn ncrass wh θ for < θ < θ, hn max 599 dcrass for θmax < θ < π. Furhrmor, h angl corrspondng o h maxmum 6 dsplacmn, θ, dcrass or m. A h m max (n h xcaaon sag), h sgn of h () 6 angnal dsplacmn s oppos n h wo xcaaon mhods, xhbng approxmaly h 6 sam θ. Consdrng h dffrnc of nducd angnal dsplacmns bwn h wo max xcaaon mhods, a h nd of xcaaon s smallr, whras bcoms largr afrwards wh a rapd dcras of dsplacmn n mhod C. In addon, h angl corrspondng o h 65 maxmal alu, θ, s largr n mhod A han ha n mhod C. Th dffrnc bwn h max dsplacmns of h wo mhods s smalls whn T =.5( (4) ). In Fgurs 4(a) and (b), h prncpal srsss a h fnal unnl fac as a funcon of h angl θ ar plod for mhod A and C. Bcaus h srsss dpnds only on h sz and shap of h opnng, h srsss a m, whch ar h sam as h ons a m (4), ar no (3) 3

34 ncludd n Fgur 4. I can b nod ha a h nd m of xcaaon, T =., h dsrbuon of srsss s h sam whar xcaaon mhod s adopd, wh largs comprss major prncpal srss a θ = o. Conrsly, h dsrbuon of srsss durng xcaaon s sgnfcanly dffrn for h wo xcaaon mhods. 6. Conclusons Analycal xprssons for h rock srss and dsplacmn of dply burd llpcal unnls xcaad n scolasc mda wr drd accounng for squnal xcaaon procsss. An nal ansoropc srss fld was assumd so ha complx gologcal condons can b accound for, wh h rock mass modld as lnarly scolasc. Soluons wr drd for a squnal xcaaon procss, wh h major and mnor axs of h unnl growng monooncally, accordng o a m-dpndn funcon o b spcfd by h dsgnrs. Frs, an xnson of h prncpl of corrspondnc o sol scolasc problms nolng m-dpndn srss boundars was lad ou mployng h Laplac ransform chnqu and complx ponal hory. From h problm formulaon mrgs ha h srss fld dpnds only on h shap and sz of h opnng, whras dsplacmns ar a funcon of h rock rhologcal proprs. Th mhodology dscrbd n hs papr may n prncpl b appld o oban analycal soluons for any ohr arbrary cross-sconal shaps of unnls xcaad n scolasc rock. Th soluon for squnally xcaad unnls of llpcal cross-scon was drd by nroducng an nrs conformal mappng whch allows lmnang h arabl from h conformal mappng n h wo complx ponals. Th analycal ngral xprssons of h soluon oband for h gnralzd ln scolasc modl nclud h Maxwll and ln modls as parcular cass. To alda h mhodology, FEM analyss wr run. A good agrmn bwn analycal soluon and FEM analyss was shown. Fnally, a paramrc analyss for arous xcaaon ras and xcaaon mhods was 33