Analytical solution for a deep tunnel excavated in a porous elasto-plastic material considering the effects of seepage forces

Transcription

1 Anlytil solution fo deep tunnel exvted in poous elsto-plsti mteil onsideing the effets of seepge foes. E. Bbos EngSolutions, In., Ft. Ludedle, Floid, USA ABSTACT: Seepge foes geneted duing tunnel exvtion ffet the gound esponse nd my use sevee instbility poblems, inluding totl gound ollpse nd flooding into the tunnel. This ppe pesents ne losed-fom nlytil solution, fo stesses nd displements ound ylindil tunnel exvted in poous te-being elsto-plsti mteil, hih inludes the effets of seepge foes. The nlysis is ied out in tems of effetive stesses nd poe te pessues. The solution shos tht seepge foes my hve mjo effet on the effetive stesses ound the tunnel, on the extent of the plsti zone, on the dil displements, nd on the tunnel suppot equiements. The onditions fo development of floing gound ondition t the tunnel fe hve been identified. The ppe pesents poedue fo the hydomehnil gound-suppot intetion nlysis, to detemine the exvtion indued inflo nd the pessues ting on the lining, thoughout the pogessive diving of the tunnel, inluding the effetive pessues exeted by the gound nd the pessues geneted by the te. 1 INTODUCTION Tunneling in te being gound ffets the hyduli equilibium of the suounding gound leding to seepge into the tunnel. The seepge foes geneted in the gound by the movement of te tods the tunnel n hve signifint effet on the behvio of the opening. Goundte inflo n use sevee instbility poblems, inluding totl gound ollpse nd flooding into the tunnel. Numeous se histoies of tunnel filues duing onstution, t setions ith high te pessues, hve been epoted (e.g. Mulnd, et l, 8, Singh et l, 6. Filue often onsists of sudden mud inush floing violently into the opening, dgging equipment, filities nd sometimes even okes, nd ompletely invding long stethes of tunnel. Even in less stiking ses, seepge foes n hve stong effet on the gound suppot equiements nd on the stutul equiements of the suppot system. Nevetheless, no dequte nlytil solutions tht onside the effet of seepge foes e uently vilble fo nlyzing the inelsti gound-suppot intetion to dimension tunnel suppot elements, nd fo popely ssessing ses of floing gound. A numbe of esehes hve investigted the effet of seepge foes duing exvtion on the stess nd displement fields ound tunnels. The nlysis tehniques used by these uthos vy fom dvned numeil methods to simplified elsti solutions. Alvez (1997 used the oupled hydo-mehnil disete-finite-element model developed by Bbos (199, Shin et l (5 used oupled finite element model ith nonline pemebility, Fenndez nd Moon (6 used oupled distint element model, Lee nd Nm (1, 4 used unoupled finite element nlyses, nd Li (1999 nd Bobet (3 developed elsti losed-fom solutions. In this ppe pope elsto-plsti solution is deived, fo deep ylindil tunnel exvted in Moh-Coulomb pefetly plsti mteil unde the te tble, onsideing seepge PAPE 493 1

2 foes. In ddition to estblishing the dius of the plsti zone, dil onvegene, the stess nd displement fields ound the tunnel, nd identifying the ondition fo development of floing gound ondition t the tunnel fe, poedue fo hydomehni gound-suppot intetion is poposed. The bsi ssumptions e stted fist nd then the solution is developed. BASIC ASSUMPTIONS The bsi onfigution onsideed in this study is shon in Figue 1. The nlysis ssumes deep ylindil tunnel of dius, exvted in ntully stessed te-being gound tht obeys the piniple of effetive stess. A suppoting lining is instlled fte exvtion. A ylindil oodinte system ith the oigin t the ente of the opening is used in the nlysis. The length of the tunnel is suh tht the poblem n be teted to-dimensionlly. Figue 1. ( Effetive stesses (b Poe te pessues ( Tnsient hyduli dius of influene. The hoizontl nd vetil in situ effetive stesses e ssumed to be equl nd to hve mgnitude P. The instlled suppot is ssumed to exet unifom dil effetive suppot pessue P i on the lls of the tunnel. The oiginl goundte pessue is P. The pessue inside the suppoted tunnel is equl to the tmosphei pessue. The te pessue t the lining-tunnel ll intefe is P i. Goundte inflo is ssumed to be dil. It is ssumed tht the effet of the te-pessue-dop vnishes beyond no-flo moving boundy loted t time-dependent distne (t.beyond suh instntneous distne, the tunnel hs no effet on the oiginl goundte ondition nd the poe te pessue beomes equl to the oiginl gound te pessue. The oiginl gound is ssumed to be poous line-elsti mteil hteized by Young s modulus E, Poisson s tio ν, Dy s oeffiient of pemebility K, nd speifi stoge S s. The filue hteistis of this mteil e defined by the Coulomb-Nvie eqution. σ 1 σ + σ 3. Nφ φ 1+ sinφ N (1 1 sinφ hee σ unonfined ompessive stength nd φ effetive fition ngle. The filed gound suounding the tunnel is ssumed to be pefetly plsti nd to stisfy the folloing filue iteion 1 σ + σ N 3 φ N φ 1 sinφ σ. 1+ sinφ ( It is ssumed tht the stength edues suddenly fom the pek defined in eqution (1 to the esidul defined in eqution (. PAPE 493

3 The volumeti esponse of the mteil in the plsti zone is ontolled by the diltion ngle α. If the diltion ngle is equl to the fition ngle (α φ then the plsti flo ule is sid to be ssoited nd the mteil undegoes volumeti expnsion hile it plstifies. If the diltion ngle is less thn the fition ngle the plsti flo ule is sid to be non-ssoited. If the diltion ngle is zeo, thee is no volumeti hnge hile the mteil plstifies. The diltion ngle entes the solution though the pmete N α tht ontols the tio beteen the iumfeentil (o tngentil nd the dil omponents of the plsti stin inement. ε ε plsti θ plsti 1 N α 1+ sinα N α (3 1 sinα 3 FUNDAMENTAL ELATIONS 3.1 Equilibium nd omptibility Figue. ( Equilibium of n infinitesiml element (b Wte ontinuity on infinitesiml element. Figue ( illusttes the stesses ting on typil gound element. The bsi diffeentil eqution of equilibium fo suh element is σ σ σ + θ P + hee σ nd σ θ epesent the effetive dil nd tngentil stesses espetively t distne, nd the thid tem is the seepge foe. The dil nd tngentil stins fo plne stin onditions n be stted in tems of the dil displement u, s follos u u ε ε θ (5 Stin omponents e elted by the stin omptibility eqution, hih is obtined by eliminting the dil displement in Eq. (5 ε θ ε ε θ 3. Continuity Figue (b illusttes the mount of te tht entes diffeentil element nd the mount of te tht leves the element, pe unit time. The diffeene beteen these to quntities epesents the hnge in the volume of te stoed in the element. Fo stedy stte ondition, the mount of te tht leves the element is equl to the mount of te tht entes it. In the tnsient stge, the flo te tht exits the element is lge thn the flo te tht entes, s some te stoed in voids nd joints of the gound is elesed. (4 (6 PAPE 493 3

4 Using Dy s l to ite the flo tes in nd out of the element nd iting the hnge in volume of te in tems of the volumeti speifi stoge of the gound S s, the ontinuity ondition yields the folloing diffusion eqution P 1 P + S s P K t The volumeti speifi stoge is the mount of te pe unit volume of stuted fomtion tht is elesed, pe unit deline in hyduli hed. Speifi stoge n be expessed in tems of ompessibility of te C, ompessibility of skeleton C s, nd poosity n, s S γ ( C + nc. s s (7 4 METHOD OF ANALYSIS The method of nlysis follos the ppoh used by Peohet (4 to deive tnsient solution fo tunnel dishge unde onstnt ddon. The nlysis is bsed on the ide tht the tnsient solution n be omputed s suessive stedy stte snpshots using time dependent dius of influene,. Theefoe, in deiving the solution fo stesses nd displements, the equilibium eqution is integted, by teting the dius of influene s knon onstnt distne beyond hih the te pessues beomes equl to the oiginl te pessue P. The tnsient ntue of the solution is epesented by the time dependeny of, hih is detemined fom the solution of the diffusion eqution. The poposed solution inludes oupling beteen effetive stesses nd poe te pessues ith gound displements. Hoeve, the solution neglets ny vition of the oeffiient of pemebility ith hnges in effetive stess. Moe peisely the nlysis ssumes tht the tio S s /K is onstnt in time nd spe. A simil ppoximtion is impliit in the onventionl solution fo onsolidtion of soft lys, hih is simil diffusion poess involving elese of poe te fom stuted gound. The mehnism fo poe te elese in the lte diffusion poess is hoeve diffeent, due to diffeenes in the eltive ompessibility of the skeleton nd te, nd to the mgnitude of the te pessues. While duing onsolidtion of lys, the mehnism fo te elese is ompession of the soil skeleton; the tnsient flo into deep tunnel exvted in te-being ok fomtion is podued minly by te expulsed fom the voids due to expnsion of poe te, s the poe te pessue is edued. 5 ELASTIC ANALYSIS Substituting the stess-stin eltions fo elsti behvio nd the pessue gdient fo dil flo into the equilibium eqution, nd integting the esulting diffeentil eqution, yields: Fo : ( P Pi σ P ( P Pi. ln + ( 1 ν (1 ln (8 (1 ν ln ( P Pi σ P + ( P Pi +. ln + ( 1 θ ν (1 + ln (8b (1 ν ln (1 + ν ( P Pi( 1+ ν 1 u ( P Pi +. ln + ( ν( 1 + (1 ν ln E E(1 νln Fo, Eq. (8 edues to: (1 + ν. u [( P Pi + ( P Pi ] E (8 (9d PAPE 493 4

5 Fo : σ P ( P σ (9 σ θ P + ( P σ (9b (1 + ν u ( P σ E (9 The set of equtions 8 sho the detimentl effet of seepge foes on the stesses ound the tunnel. Seepge foes edue the effetive dil stess, hih is the mino pinipl stess, nd inese the tngentil stess, hih is the mjo pinipl stess. Theefoe, seepge foes inese the devitoi stess σ -σ θ, ound the tunnel, hile eduing the effetive onfining stess σ 3. On the othe hnd, Eq. 8 shos tht even if the effetive suppot pessue is equl to the ffield effetive stess, thee is still gound displement podued by seepge foes. Aoding to Eq. (8d, the displement t the ll fe tuns out to be the sme s tht omputed fo the totl net pessue. Hoeve, the bove esult is tue only fo elsti onditions nd only t the ll fe. 6 ELASTO-PLASTIC ANALYSIS The elsti solution is vlid fo effetive suppot pessues gete thn itil pessue P i it fo hih the effetive stesses t the tunnel ll eh the filue iteion of the oiginl gound. Using Eqs. (8 fo nd substituting into Eq. 1 yields P σ ( P Pi Pi it + (1 N + 1 (1 ν ( N + 1 φ φ Fo effetive suppot pessues loe thn the itil, the effetive stesses indued in the gound folloing exvtion ill exeed the oiginl gound stength, nd plsti zone of dius p ill develop ound the tunnel. The gound outside the boundy defined by p is ssumed to emin elsti. 6.1 Stesses nd displements in the elsti zone The elsti solution deived bove is vlid in the elsti zone. If the plsti zone is ithin the dius of influene of the tunnel (e.g. p < the solution is set of equtions simil to Eqs. (8, in hih the tunnel dius, is epled by p, nd the effetive intenl pessue P i, is epled by the effetive dil stess t the elsti/plsti intefe, σ p. On the othe hnd, if the plsti zone extends beyond, the solution is set of equtions simil to Eqs. (9, in hih the dil distne is epled by p nd the effetive stess σ is epled by σ p. 6. Stesses in the plsti zone Substituting the filue iteion fo the mteil in the plsti zone, into the equilibium eqution, nd integting the esulting diffeentil eqution, yields the folloing stess distibution in the plsti zone. σ P i * σ + Nφ 1 Nφ 1 σ N φ * 1 (11 σ σ + σ. hee θ N φ * ( P Pi σ ln σ (11b PAPE 493 5

6 Eqution (11b shos tht the effet of the seepge foes n be visulized s tht of eduing the ompessive stength of the gound in the plsti zone. 6.3 dius of plsti zone Consideing equilibium of effetive dil stesses t the plsti/elsti boundy yields p N ( P ln * P i P σ σ p + + N φ N φ φ ( N φ + 1 ( 1 ν ln (1 * σ Pi + N 1 The dius of the plsti zone is detemined by solving itetively Eq. (1. Eqution (1 pesumes tht the plsti zone is ithin the dius of influene of the tunnel (e.g. p <. If the plsti zone extends beyond de dius of influene, dditionl omputtions e equied to detemine the tul extent of the plsti zone. Fist Eq. (8 is used to ompute σ, the effetive dil stess t. Next simplified vesion of Eq. (1 is used to ompute the dius of the plsti zone, in hih the effetive suppot pessue P i is epled by σ nd the te pessue gdient tem is ignoed. Suh eqution n be solved dietly ithout ny itetive poedue. 6.4 Floing gound ondition Eqution (1 ill not onvege (i.e. ontinuity of effetive dil stesses t the plsti/elsti boundy nnot be omplished if the te pessue gdient, effetive suppot pessue, nd stength of the gound in the plsti zone e suh tht P P ( i P i ln ( Nφ 1 + σ Fo the bove ondition, hih mkes the denominto of Eq. (1 negtive, the gound nnot esist the seepge foes nd ollpses into the tunnel. A floing gound ondition ill develop t the tunnel fe, ith the gound invding the tunnel. Eqution (13 inludes thee tems, hih mens thee e thee ys to hndle floing gound onditions. ( edue the net te pessue ith dinge (o ompessed i, inese effetive suppot pessue by dopting stiffe suppot, nd ( inese gound stength ith gouting. φ (13 Figue 3. Effetive stesses, te pessues nd displements ound tunnel. PAPE 493 6

7 6.5 Displements in the plsti zone Substituting the ssumed flo ule into the stin omptibility eqution, nd integting the esulting diffeentil eqution yields the folloing distibution of displement ithin the plsti zone N α p (1 + ν p u σ p E [( P + ( P Pp ] (14 hee σ p nd P p e the effetive dil stess nd poe te pessues t the plsti/elsti intefe. The bove solution llos omputing the stess nd displement fields ound the tunnel. Fig. 3 shos de distibution of poe te pessues, effetive stesses nd displements fo n pplition exmple desibed in setion 9. The effetive nd te pessues t the ll fe e Pi.84 MP nd P i 1.9 MP espetively, hile the in situ effetive stess nd poe pessues e P 4.9 MP nd P 3. MP espetively. 7 TANSIENT ANALYSIS 7.1 Tnsient dius of influene The dius of influene, hs been teted so f s onstnt distne beyond hih the poe te pessue beomes equl to the oiginl te pessue. Hoeve, in elity, suh dius is tnsient in ntue. Its initil vlue is the dius of the opening hen the exvtion is mde, nd it expnds dilly s time poeeds, until mximum vlue is ehed, oesponding to the development of stedy stte ondition. Job nd Lohmn (195 deived the fist tnsient solution of the diffusion eqution, fo tunnel subjeted to sudden, onstnt hyduli hed. The solution is omplited one, hih involves fist nd seond kind zeo-ode Bessel funtions. eently, Peohet (5 developed muh simple nlytil solution, hih yields essentilly the sme esults. Fom the fomul fo the tnsient dishge in suh solution, nd ounting fo the pesene of line, the folloing eqution fo the tnsient dius of influene is obtined.5 πk ( P Pi 1 + t S s P hee t is the time elpsed sine flo stted t the setion nlyzed, nd mx is the mximum dius of influene oesponding to the finl stedy stte ondition. mx (15 7. Pogessive exvtion Combining the bove elsto-plsti solution nd the onept of tnsient dius of influene, it is possible to evlute the vition in time of the tunnel suppot equiements, extent of the plsti zone, distibutions of poe te pessues, effetive stesses, nd displements, duing tunnel exvtion, in tems of the tunnel dvne te. The nlysis follos the ppoh used by Peohet (5 to detemine tnsient dishge into tunnel duing pogessive dilling. Conside the pogessive diving of tunnel into pemeble zone t n vege dvne te v, s illustted in Figue 4. At time t, the tunnel fe is loted t distne vt, nd the time t hih position x <vt s ehed is x/v. Thus, the time elpsed sine tht position s ehed nd duing hih inflo oued t tht setion is t-x/v, hih is the time tht goes into eqution 15. Theefoe, the tnsient dius of influene is non-unifom funtion of spe ove the distne vt. Thus fo long tunnel, it is possible tht by the time the tunnel fe is ne the end of the te-being zone (xl, hile the tnsient expnsion of the dius of influene is beginning t suh lotion, ne the entne to the te-being zone (x the dius of influene my be ppohing the stedy stte mximum. PAPE 493 7

8 Figue 4. Flo egime, dius of influene, nd te inflo pe unit length of tunnel, fo tunnel exvted t onstnt veloity. As esult, the dius of influene t the tunnel fe, t the point hee the suppot is being instlled, nd elsehee, depend on the tunneling te. If the tunnel is diven fst, the dius of influene long the tunnel ill be smll, esulting in lge hyduli gdients nd seepge foes, hih ould use stbility poblems t the fe. Fo sloe dvne te, lge hyduli influene zone develops, eduing hyduli gdients nd seepge foes. Theefoe, in poo pemeble gound slo te of dvne is pefeed. Hoeve, on less pevious soft gound, ith shot-tem stength (undined gete thn the long-tem stength (dined, pid dvne impoves stbility. Anlysis fo the lte se is not onsideed in this study. In both ses tunnel suppot, should be instlled s fst s possible to povide ely mehnil suppot. On the othe hnd, s shon in Fig. 4, the lge initil heding inflo o flush flo (Heue, 1995, Fenndez et l 6, nd the ontinuous deese ith time of the inflos pe unit length tht hve been mesued t tunnel lls s the stedy stte ondition is ppohed, e ll onsistent ith the onept of tnsient dius of influene. As the dius of influene expnds the hyduli gdient deeses nd the inflo edues s ell. 7.3 Stedy stte dius of influene Poe pessues, seepge foes, goundte inflo, nd the isk fo instbilities e in genel getest immeditely fte exvtion. The long-tem ondition ould not be itil unless thee is signifint deteiotion of gound popeties ove time (e.g. due to etheing, slking, eep, et. Still, the long-tem ondition my be signifint fo the tunnel opetion in tems of inflo. Fo long times, s the dius of influene expnds y fom the tunnel ll, the flo devites fom iul fshion s the stedy stte ondition is ppohed, nd djust to the site goundte boundy onditions. Some possible stedy stte equilibium onfigutions e shon in Figue 5. The finl stedy stte onfigution of the te tble depends on the vilble ehge. In se ( ehge is fom bove nd thee is n unlimited mount of te vilble fo ehge. The tunnel does not modify the hoizontl te tble. This is se of onfined flo hee the gound emins stuted. In se (b the te tble is belo the gound sufe nd the gound eeives limited ehge te dietly fom the sufe, minly fom peipittion. The ehge hoeve is insuffiient to mintin hoizontl te tble nd signifint ddon develops. This is se of unonfined flo hee dinge of poe te mkes pt of the gound ptilly stuted. In se ( the ehge is lso limited, but gete thn tht in se (b, esulting in smll ddon. In se (d ehge is fom n quife undelying the gound ound the tunnel. ehge is unlimited nd the type of flo is onfined. Bsed on the flo net, one ould estimte stedy stte dius of influene mx s distne beyond hih the poe pessue is not ffeted signifintly by the pesene of the tunnel. Typilly mx n be estimted s fto (1 to of the oiginl goundte pessue hed P o /γ. Altentively, n equivlent mx ould be omputed s the dius of influene fo symmetil flo egime tht yields the sme inflo s tht of the tul flo net. PAPE 493 8

9 Fo se ( in Fig. 5, the equivlent stedy stte dius of influene n be omputed in tems of the depth to the tunnel D, using the tul nlytil solution fo flo te (H, 196, s D mx (16 Figue 5. Stedy stte flo egime. 7.4 Unonfined flo In ses of unonfined flo, suh s in ses (b nd ( in Figue 5, thee e tully to diffeent tnsient poesses: expnsion of the dius of influene nd ddon of the te tble. Although both tnsient poesses involve elese of stoed te, the mehnism of te elese is diffeent. Duing expnsion of the dius of influene, te is elesed fom stuted gound tht emins stuted. The mehnism fo te elese is expulsion of poe te due minly to expnsion of poe te, s poe te pessue is edued, nd to less extent, to ompession of the gound skeleton. Duing the ddon of the te tble, te fom stoge is elesed by the mehnism of tully dining the poes of the gound, hih mkes pt of the gound ptilly stuted (the depession one. Futhemoe, the te of the to tnsient poesses is diffeent. The te of expnsion of the dius of influene is ontolled by the speifi stoge S s, mong othe ftos. On the othe hnd, the te of ddon is ontolled by the stotivity S (stoge oeffiient. The stotivity is the peentge (by volume of te tht the gound ould yield. Fo ose-gined gound, the stoge oeffiient is equl to the poosity. Fo fine-gined gound the stoge oeffiient is smlle thn the poosity beuse moleul nd subsufe tension foes in the poe spes keep some of the te in the voids. The finl idth X mx, of the depession one, fo the stedy stte ondition, n be detemined bsed on the ehge (e.g. mm/dy nd the tunnel inflo s: X mx q/. A solution fo the tnsient dius of the depession one of n unonfined ell fom hih te is pumped s deived by Chi (1994 s PAPE 493 9

10 X 4K ( P P Sγ X mx t X mx 1 e i t.5 Thus, oding to Eq. (17 the idth of the ddon one expnds exponentilly ith time, t deesing te. Although Eq. (17 is pplible to ell o vetil shft insted of hoizontl tunnel, it suggests tht the te of ddon is muh sloe thn the te of expnsion of the dius of influene. Theefoe, it ppes tht fo unonfined flo ses, the expnsion of the dius of influene pedomintes, in the ely stges of flo fte tunnel exvtion, hees ddon domintes in the lte stges. (17 8 HYDOMECHANICAL GOUND-SUPPOT INTEACTION A gound-intetion nlysis is used to detemine the pessues ting on the lining, inluding the hyduli poe te pessue P i nd the mehnil effetive pessue P i. The nlysis is ied out fo time t, t hih the tunnel suppot is instlled. The suppot system onsideed inludes onete lining nd (o okbolts. Figue 6. ( Hyduli gound-suppot intetion (b mehni gound-suppot intetion The onete o shotete lining is ssumed to be poous nd elsti-pefetly plsti. The popeties of the lining e thikness t, Young s modulus E, Poisson s tio ν, ompessive stength f, nd oeffiient of pemebility K. The mximum suppot pessue P mx is omputed bsed on the speified ompessive stength. The ungouted mehnilly o hemilly nhoed bolts e hteized by stiffness K b nd mximum suppot pessue P bmx. These popeties e omputed bsed on the sping, length, individul pity, nd othe hteistis of the individul elements (Hoek & Bon, Hyduli intetion The flo te seeping though the gound is PAPE 493 1

11 q gound πk ( P Pi (18 γ ln Whees the flo te though the lining is q lining πk Pi γ ln t Figue (6 illusttes the vition ith P i of both, the flo te though the lining nd the seepge though the gound. Both e line eltions. Fom ontinuity, both flo te quntities must be equl. Thus, s shon in Figue 6, the inflo into the tunnel nd the tul te pessue ting on the lining e epesented by the intesetion of the bove to lines. If eltively impevious lining is seleted, the esulting te pessue ting on the lining is high, lose to oiginl te pessue P nd the tunnel inflo is miniml On the onty, if pemeble lining is seleted, the te pessue on the lining is miniml but the tunnel inflo is lge. 8. Hydomehnil intetion 8..1 Gound esponse uve Figue 6 illusttes the gound esponse uve s omputed ith the solution developed in the peeding setion, by omputing the ll displement fo the poe te pessue P i detemined bove, nd fo vible effetive suppot pessue P i. The effetive suppot povided by the fe effet is shon in Fig. 6 s P f. 8.. Suppot hteisti uve The suppot hteisti uve is obtined by subjeting the lining to the hyduli pessue P i nd to vible mehnil pessue P i. The hteisti uve is positioned t n initil displement u io epesenting the displement tht hs oued in the gound pio to the instlltion of the suppot. Suh displement n be estimted fom longitudinl defomtion pofile, s usul in the Convegene Confinement Method. The solution fo poous line subjeted to hyduli lod P i nd mehnil lod P i is obtined using the sme ppoh folloed to obtin the elsti solution in setion 5. The hteisti uve is: P P u ( i i u + + K K hee 1 (1 + ν (1 ν + ( t K E ( t [ ] [ ] i nd The tngentil stess t the inne ll of the lining is 1 (1 + ν (1 ν + ( t K E (1 ν ( t (1 ν ( 1 ln + ln ν t t (19 (1 P i Pi 1 ν σ θ + + ( ( t t 1 (1 ν ( 1 ln t The mximum suppot pessue geneted by the onete lining is the one tht mkes the tngentil stess equl to the ompessive stength of the mteil (i.e.. Although the nlysis ssumes both suppot systems e instlled t time t, in ombining thei hteisti uves, the tul sequene must be onsideed. As Fig. 7 illusttes, the sequene detemines hethe o not okbolts help to suppot the extenl te pessue ting on the onete lining. If shoteting is done fist folloed by ok-bolting, okbolts help to suppot σ θ f PAPE

12 the fluid pessue. On the onty, if the tunnel ll is bolted befoe shoteting, the onete line lone hs to suppot the fluid pessue. Figue 7. Combined suppot hteisti uve fo to sequenes of suppot instlltion. ( shoteting-bolting, (b bolting-shoteting 9 EXAMPLE A 6 m dimete tunnel is exvted in jointed sndstone t depth of 3 m belo the sufe hee the totl in situ stess is P o 8.1 MP nd the poe te pessue is P o 3. MP (pessue hed m bove gound sufe, Cse ( Fig. 5. The folloing mteil dt is given: Oiginl ok mss: E 1.5 GP, ν.33, σ 1 MP, φ 3º, K.1 mm/se, S s.5 m -1. ok mss in plsti zone: σ.5 MP, φ 5º, α º. Shotete lining: t mm, E 16 GP, ν.5, f 5 MP, K 3x1-5 mm/se. okbolts: 5 mm okbolt, sping ptten: 1. m x 1. m. The suppot is instlled t distne of m fom the dvning tunnel fe, bout 4 hous fte tunnel fe dvne. Bsed on the bove distne nd longitudinl defomtion pofile, the initil gound displement pio to the instlltion of the suppot is estimted to be.477 the mximum displement y fom fe (displement t hih n unstble floing gound ondition develops. The equivlent stedy stte dius of influene is mx x 3 6 m. This exmple s solved fo: (1 the oiginl pmetes, ( line ith numeous shinkge ks nd gps epesented by inesing its pemebility n ode of mgnitude (K 3x1-4 mm/se, (3 no poe te pessue o onventionl solution in tems of totl stesses ithout onsideing poe te pessues nd seepge foes. The esults of the nlysis e pesented in Fig. 8 nd summized in Tbles 1 nd. Ptil esults fo se intemedite beteen 1 nd (K 6x1-5 mm/se ee pesented in Fig. 3. Comping to se 3 (no poe pessues/ onventionl nlysis, se 1 demonsttes the effet of poe pessues nd se demonsttes the effet of seepge foes. In se 1, the line is eltively impemeble s omped to the gound, thus seepge is limited. As Fig. 8 shos, the extenl te pessue in the line is eltively high in this se. Hene the poe pessues in the gound ne the tunnel emin high. On the othe hnd, in se, the ked lining llos moe inflo nd the elese of the extenl pessue. As Fig 8 shos, the extenl te pessue in the lining is eltively lo. Thus, thee is moe signifint hyduli gdient in the gound in this se, leding to lge seepge foes. Fig. 8b shos the distibution of poe te pessue in the gound. The dius of influene is lge in the se of the ked line beuse the net hyduli hed is lge leding to fste te of expnsion of. Tble shos the expnsion ith time of. Thus, even in pemeble gound, it tkes fe months to develop stedy stte ondition. The gound etion uves e shon in Fig 8. The te pessue fo eh se is shon s dotted line. The point of equilibium is ehed fo muh lge displements in se 3, due to poe pessues nd seepge foes, next in se, due to poe pessues nd lest in se 3, hih inludes neithe poe pessues no seepge foes. Effetive stesses nd the extent of the plsti zone e shon in Fig. 8d. The onventionl nlysis gives the most optimisti esults. The dius of the plsti zone ineses fom.1 to.64 beuse of the te pessues, nd to.86 due to seepge foes. PAPE 493 1

13 Finlly, the dil displements e shon on Figue 8e. Poe pessues inese the dil displement t the tunnel ll fom 114 mm to 164 mm, nd seepge foes to 43 mm. Figue 8. Hydomehnil intetion, nd effetive stesses, poe pessues, nd displement ound tunnel fo (1 oiginl pmetes ( ked line (3 onventionl nlysis ith no seepge foes Tble 1. Summy of esults. No P P P i P i U p/ / MP MP MP MP mm PAPE

14 Tble. Expnsion of dius of Influene ith Time. Time hs 4 hs 1 hs 1 dy 1 eek 1 month 3 months 6 months Cse 1, (m: Cse, (m: CONCLUSIONS Seepge foes geneted duing tunnel exvtion my use seious instbility poblems, hih nnot be ssessed ith uent gound intetion nlyses. An elsto-plsti solution hs been deived tht inludes the effet of poe te pessues nd seepge foes, on the stess nd displement field ound the opening, nd on the extent of the plsti zone. The onditions fo development of floing gound t the tunnel fe hve been identified. The tnsient ntue of the hyduli dius of influened hs been desibed nd poedue to pedit its tnsient expnsion hs been poposed. The effets of the pogessive exvtion hve been disussed. Poedues fo hydomehni gound intetion nlysis hve been developed. An pplition exmple demonstted the signifint effet of poe te pessues nd seepge foes on the behvio of tunnels. The solution is pplible to tunneling in ek oks, highly ftued oks, ushed oks (e.g. fult zones, nd genel soil-like mteils, subjeted to high te pessues. EFEENCES Alvez, T.A A Study of the Coupled Hydomehnil Behvio of Jointed ok Msses Aound Pessue Tunnels, Ph.D. Thesis, Univesity of Illinois t Ubn-Chmpign, Ubn, IL. Bbos,.E Disete Element Models fo Gnul Mteils nd ok Msses, Ph.D. Thesis, Univesity of Illinois t Ubn-Chmpign, Ubn, IL. Bbos,.E. 9. EngSolutions TUNNEL, Use s Mnul. EngSolutions, In., Ft. Ludedle, FL. Bon, E.T., By, J.W., Ldnyi, B., Hoek, E Gound esponse uves fo ok tunnels. ASCE J. Geoteh. Eng. Div. 19 (1, 15-3 Bobet, A. 3. Effet of poe te pessue on tunnel suppot duing stti nd seismi loding. Tunneling nd Undegound Spe Tehnology 18, Chu, S.T Tnsient dius of Influene Model. ASCE J. Iig. nd Dinge Eng. 1 (5, Fenndez, G. nd J.S. Moon 6. Evlution nd ssessment of inflo tes in tunnels exvted in jointed ok mss. VI South Amein ok Mehnis Congess, Ctgen, Colombi. H, M.E Goundte nd seepge. MG-Hill Book Co, In.., Ne Yok, N.Y. Hendon A.J., nd A.K. Aiye 197. Stesses nd Stins Aound Cylindil Tunnel in n Elsto- Plsti Mteil ith Diltny. Omh Distit, Cops of Enginees, Tehnil epot No. 1, Omh, NE.. Heue,.E Estimting ok tunnel te inflo. pid Exvtion nd Tunneling Confeene, Chp. 3, pp Job CE, Lohmn S.W Nonstedy flo to ell of onstnt ddon in n extensive quife. Tns AGU 33(4: Li, X Stess nd displement fields ound deep iul tunnel ith ptil seling. Computes nd Geotehnis 4, Lee, I. nd Nm, S. 1. The study of seepge foes ting on the tunnel lining nd tunnel fe in shllo tunnels. Tunneling nd Undegound Spe Tehnology 16, Lee, I. nd Nm, S. 4. Effet of tunnel dvne te on seepge foes ting on the undete tunnel fes. Tunneling nd Undegound Spe Tehnology 19, Mulnd, A., C. Mulnd nd. Gutieez 8. Expeienes in tunnel exvtion by the onventionl method nd by TBMs in the Andes Mountin nge. Cse histoies. I South Amein Symposium on ok Exvtions, Bogot, Colombi. Peohet, P. 5. A simple solution to tunnel o ell dishge unde onstnt ddon. Hydology Jounl 13, Peohet, P. 5. Confined flo into tunnel duing pogessive dilling: n nlytil solution. Gound Wte 43, No. 6, Shin, J.H., Potts D.M., nd Zdvkovi L. 5. The effet of poe-te pessue on NATM tunnel linings in deomposed gnite soil. Cndin Geotehnil Jounl 4, Singh, B. nd.k. Goel (6 Tunneling in Wek oks, Volume 5 Geo-Engineeing Book Seies, Elsevie. PAPE