A NEW SOLUTION FOR SHALLOW AND DEEP TUNNELS BY CONSIDERING THE GRAVITATIONAL LOADS

Transcription

1 A NEW SOLUTION FOR SHALLOW AND DEEP TUNNELS BY CONSIDERING THE GRAVITATIONAL LOADS MOHAMMAD REZA ZAREIFARD and AHMAD FAHIMIFAR about th authos Mohammad Rza Zaifad Amikabi Univsity of Tchnology Than, Ian cosonding autho Ahmad Fahimifa Amikabi Univsity of Tchnology Than, Ian Abstact A nw, lasto-lastic, analytical-numical solution, considing th axial-symmty condition, fo a cicula tunnl xcavatd in a stain-softning and Hok Bown ock mass is oosd. To xamin th ffct of initial stss vaiations, and also th bounday conditions at th gound sufac, th fomulations a divd fo diffnt dictions aound th tunnl. Futhmo, th ffct of th wight of th lastic zon is takn into account in this gad. As th divd diffntial uations hav no xlicit analytical solutions fo th lastic zon, th finit-diffnc mthod (FDM) is usd in this study. On th oth hand, analytical xssions a divd fo th lastic zon. Sval illustativ xamls a givn to dmonstat th fomanc of th oosd solution, and to xamin th ffct of vaious bounday conditions. It is concludd that th classic solutions, basd on th hydostatic fa-fild stss, and nglcting th ffct of th bounday conditions at th gound sufac, giv alicabl sults fo a wid ang of actical oblms. Howv, ignoing th wight of th lastic zon in th analyss can lad to lag os in th calculations. Kywods gound-sons cuv, lasto-lastic analysis, bounday condition, axial symmty, gavitational loads 1 INTRODUCTION A numb of mthods a cuntly usd fo th dsign and analysis of tunnls. Among thm, th convgncconfinmnt mthod (th C.-C. mthod) has layd an imotant ol in oviding an insight into th intaction btwn th lining suot and th suounding gound mass. Th C.-C. mthod is basd on a conct that involvs an analysis of th gound-stuctu intaction by indndnt studis of th bhavio of th gound and of th tunnl suot. In this gad, th gound bhavio is sntd by a gound-sons cuv; which dscibs th gound convgnc in tms of th alid intnal ssu. Howv, to maintain simlicity, a numb of simlifying assumtions a mad in its divation. Ths assumtions mak th mthod alicabl only to d tunnls in hydostatic stss filds. In th ast, a numb of classic solutions fo dtmining th gound-sons cuv hav bn ublishd. Ths solutions may b catgoizd into two gous of analytical closd-fom solutions and analytical-numical unclosd-fom solutions. Although a numb of closd-fom solutions a availabl (such as that oosd by Bown t al. [1]; Shaan []; Caanza- Tos [3]; Pak and Kim [4]), ach solution suffs fom a lvl of aoximation in th sns that it incooats vaious simlifying assumtions. Fo xaml, ths solutions hav bn oosd fo th ock masss with siml bhavio modls, including th lastic-fctlastic o lastic-bittl-lastic bhavio modls. In fact, fo mo comlicatd bhavio modls obtaining an xact closd-fom solution is imossibl. On th oth hand, in th unclosd-fom solutions (Bown t al., [1]; Guan t al., [5]; L and Pituszczak, [6]; Fahimifa and Zaifad, [7]), considation of mo comlicatd and gnal matial-bhavio modls a ossibl. Howv, all th mntiond solutions (both th closdand unclosd-fom solutions) a basd on th classic assumtions mad in th C.-C. mthod. Diffnt ascts of th C.-C. mthod hav bn invstigatd by sachs, both analytically and numically. ACTA GEOTECHNICA SLOVENICA, 01/ 37.

2 With th dvlomnt of comut cods, numical analyss hav bcom common mthods fo th analysis of tunnls. Vaious cass of analyss, including twodimnsional, th-dimnsional and tim-dndnt bhavios, can b fomd using commcial cods. Nvthlss, utilizing numical mthods fo invstigating th C.-C. mthod, such as that sntd by Caanza-Tos and Faihust [8] and that sntd by Gonzalz-Niciza t al. [9], do not asily val th actual ffcts that th simlifying assumtions hav on th mchanical sons of a tunnl. Thfo, analytical solutions of siml cass, such as that oosd by Lu t al. [10], Dtounay and Faihust [11], and Rd [1], can hl to aliz vaious ascts of th goundsons cuv of tunnls much btt. In th C.-C. mthod th ffcts of th bounday conditions at th gound sufac a nglctd and, also, th vaiation of th initial stsss (i.., in-situ stsss) a not takn into account, classically. Howv, fo shallow tunnls, th initial stsss cannot b assumd to b constant ov th tunnl sction, and thus th assumtion of hydostatic, fa-fild stsss may not b alicabl. On th oth hand, in th C.-C. mthod, th abov solutions nglct th wight of th lastic zon dvlod aound th tunnl. Howv, vy fw woks concning th ffct of gavitational focs acting on th gound hav bn conductd [13-15]. In fact, gavitational loading diffs fo vaious dictions aound th tunnl ihy, and, fo th sam intnal ssu, convgnc of th cown is xctd to b lag than that of th walls, bcaus of th wight of th faild matial on th to of th tunnl. aound a cicula tunnl xcavatd in an lasto-lastic stain-softning and Hok Bown ock matial. In this mthod th ffcts of bounday conditions at th f sufac of th gound, and also th ffct of th wight of th lastic zon a takn into account. Fo this uos, th fomulations a divd fo both th hoizontal and th vtical dictions that a assing though th tunnl cnt. In addition, th gavitational loadings a considd as adial body focs bing alid to th ock mass. ANALYTICAL SOLUTION OF TUNNELS CONSIDERING THE GRAVITATIONAL LOADS Fig. 1 shows th gnal cas of a tunnl xcavatd in a homognous and isotoic ock mass und an in-situ stss fild blow a hoizontal gound sufac. In gnal cass th initial stsss in th Catsian coodinat systm a σ y = γy + s and σ x = σ z = Kσ y, wh x, y and z a th Catsian coodinat axs (as shown in Fig. 1), γ is th scific wight of th ock, K is th so-calld latal stss cofficint, and s is th unifom vtical stss that is alid to th gound sufac fom infastuctus o mbankmnts (suchag load). In cylindical coodinats (,θ,z), th stss fild aound a tunnl (s Fig. ) has to fulfill th uilibium uations fo ach lmnt of th ock mass, as in [16]: s 1 s s -s F = 0 (1) In this a, an unclosd-fom analytical solution is sntd fo th stss and dislacmnt filds s 1 s s + s F = 0 () Figu 1. Cicula shallow tunnl in a smi-infinit mdium und an initial stss fild. 38. ACTA GEOTECHNICA SLOVENICA, 01/ M. R. ZAREIFARD & A. FAHIMIFAR: A NEW SOLUTION FOR SH

3 wh F = γsinθ, and F θ = γcosθ a th gavitational body focs in th adial and cicumfntial dictions, sctivly, γ is th unit wight of th gound and θ is th angl masud clockwis fom th hoizontal diction. Obtaining an xact lasto-lastic analytical solution fo this oblm is xtmly comlicatd and vn unsolvabl in mo cass (as shown by Dtounay and Faihust [11]), bcaus, in this cas, th incial stsss may otat in ach diction. Figu 3. Cicula d tunnl xcavatd in a hydostatic stss fild. 1 s Thus, in uilibium E. (1) th tm vanishs. In this cas it is assumd that th initial stsss in th vicinity of th tunnl a constant σ y0 = σ x0 = γh 0 + s and do not incas linaly. Figu. Body focs and stss comonnts cosonding to an lmnt of th ock mass. As mntiond in th C.-C. mthod fo simlifying th oblm, th gound-sons cuv is constuctd basd on th lasto-lastic solution fo a cicula oning subctd to hydostatic fa-fild stsss and a unifom intnal ssu (s Fig. 3). In this a an analytical solution fo diving th gound-sons cuv of a tunnl und ual initial stsss (K = 1) is oosd by considing th ffcts of th initial stss vaiations du to th gavitational loads and taking th ffcts of th bounday conditions into account at th f sufac of th gound. In this solution th analyss a fomd fo th hoizontal and th vtical dictions fo axial symmty conditions (s Fig. 4). Thus, th tm 1 s will vanishs. In this gad, th govning uilibium uation bcoms mo staightfowad as: Figu 4. Analysis of th shallow tunnls along th hoizontal and th vtical dictions by considing th axial symmty condition. ALLOW AND DEEP TUNNELS BY CONSIDERING THE GRAVITATIONAL LOADS ACTA GEOTECHNICA SLOVENICA, 01/ 39.

4 ds s -s - + F = 0 (3) d wh F is th alid adial body foc, which dnds on th gavitational loads though th considing diction. As mntiond, sinc th axial symmty condition is assumd, only th adial comonnt of th gavitational loads, i.., F = γsinθ, is takn into account, and th cicumfntial comonnt is nglctd. Thus, F = γ fo th vtical diction and F = 0 fo th hoizontal diction a obtaind (s Fig. 4). Fo th ola coodinats dfind in Fig. 1, th initial ual stss fild (K = 1) is givn by: s0 (, ) = s0 (, ) = s0 (, ) = gy+ s = g( h0 - sin) + s (4) wh σ θ0 and σ 0 a th initial cicumfntial and adial stsss, sctivly, and s is th suchag load. It should b notd that whn th is a vy wak o a havily wathd lay of ock o sidual soils on th u lvls, thi ffct can b considd as suchag loads alid to th undlying gound. In this study, th fomulations a divd fo both th hoizontal (though th tunnl singlin) and th vtical (though th tunnl cown) dictions. In this mann, fo both dictions, bcaus of th axial symmty conditions, th gomty, bounday conditions and th alid loads a gnalizd to all dictions (s Fig. 4). Th oblm fo both th hoizontal and vtical dictions a shown in Figs. 5(a) and 5(b), sctivly. As obsvd in Figs. 4 and 5(a), fo th hoizontal diction, th oblm is simila to a cicula tunnl in an infinit mdium. In this cas, at th tunnl adius (i.., at = i ) th intnal ssu i is alid, and at an infinit adius (i.., at = ), th ssu γh 0 + s is alid. In addition, th adial body focs though this diction a ual to zo. It is obsvd that this cas is simila to th oblm of a d tunnl. On th oth hand, as obsvd in Figs. 4 and 5(b), fo th vtical diction, th oblm of a thick-walld cylind is th sult. In this cas, at th tunnl adius th intnal ssu i is alid, and at adius = h 0, th ssu s is alid. Futhmo, th adial body focs though this diction a F = γ. As shown in Fig. 5, two diffnt zons may b fomd aound th tunnl (fo both dictions): th xtnal lastic zon, and th intnal lastic zon, which may b dividd into th softning zon and th sidual zon. Th stain dislacmnt lations in th ola coodinat systm fo th axial symmtic oblm a givn by [17]: u du =, = (5) d σ 0 at = σ 0 at = a) Hoizontal diction: cicula hol in an infint mdium und initial unifom sss σ 0 = yh 0 + s b) Vtical diction: thick walld cylind und adial gavity loading and a suchag loading Figu 5. Gomty, alid loads and bounday conditions fo th hoizontal and vtical dictions. 40. ACTA GEOTECHNICA SLOVENICA, 01/

5 wh u is th adial comonnt of th dislacmnt and and a th cicumfntial and adial stains, sctivly. Futhmo, th stss stat at a distanc is dfind by th adial stss σ and th cicumfntial stss σ θ, which a th mino σ 3 and th mao σ 1 incial stsss, sctivly, as shown in Fig BEHAVIOR MODEL Th ock mass is assumd to xhibit th stain-softning bhavio, in this study, which can b ducd to th fct lasto-lastic o lasto-bittl-lastic cass. Gnally, this bhavio is chaactizd by a tansitional failu cition and a lastic otntial. A softning aamt contols th gadual tansition fom an initial failu cition (o a otntial on) to a sidual on. In th snt wok, th dviatoic lastic stain g = - is mloyd as th softning aamt. Although th is no univsal way of dfining th stain-softning aamt, as ointd out by Alonso t al. [18], th abov softning aamt is th most widly acctd. Th lastic stain incmnts can b obtaind fom th lastic otntial function, g(σ, σ θ, γ ) accoding to: g = l (6) s and: g = l (7) s whl is a lastic multili, = and = t t (τ is a fictitious tim vaiabl). Euations (6) and (7) a th constitutiv uations in th lastic gim, and a usually tmd th flow ul. If th lastic otntial coincids with th failu cition, thn it is calld an associatd flow ul; othwis it is calld a non-associatd flow ul. In this gad th incmntal lasticity involvs a considation of a fictitious tim vaiabl, vn if it dos not hav any hysical maning. This vaiabl contols th volution of th lasticity and th lastic stain ats. In th fomulation sntd in this sach, th lastic adius, R, will b assumd to b th tim vaiabl. This choic allows th acuisition of a siml fomulation fo th oblm, in od to obtain a ctain kind of solution, as illustatd by Alonso t al. [18]. H, th Moh Coulomb cition is slctd as a lastic otntial function fo a non-associatd flow ul: g = s - K s (8) Y wh K ψ is th dilation facto, and is givn as: 1+ sinyg K Y = (9) 1 - sin Y g ψ g, in E. (9), is tmd th dilation angl and vais as a function of th softning aamt γ. Th ock mass is assumd to oby th Hok Bown failu cition, givn by [19]: ( m s ) s - s = s s + s (10) c c in which σ θ is th cicumfntial stss, σ is th adial stss, σ c is th uniaxial comssiv stngth of th intact ock matial, and m and s a th Hok-Bown constants that dnd on th otis of th ock mass and th xtnt to which it was bokn bfo bing subctd to th failu stsss σ θ and σ. Fo th lastic zon, th abov uation is givn as: ( ) 1 m s s - s = s s + s (11) g c g c wh m g and s g a th Hok-Bown constants fo th lastic zon and vay as a function of th softning aamt γ. In contast to th solution sntd by Bown t al. [1], th solution oosd in this wok consids th lastic stains inducd in th lastic zon. Th lationshis btwn th lastic stains, and, and th stsss σ and σ θ a givn by Hook's law [17]: 1+ n = é ( 1-n)( s - s0) + n( s -s0) ù ë û (1) E g 1+ n = é ( 1-n)( s - s0) + n( s -s0) ù ë û (13) E g wh σ 0 is th initial stss, and calculating fom E. (4), E g and ν a th lasticity modulus and th Poisson s atio of th ock mass, sctivly. Howv, fo th lastic zon, th lasticity modulus E g vais as a function of th softning aamt γ. It should b notd that in th lastic zon, th failu and dilation aamts, aaing in Es. (9) and (11), and also th ock mass lastic modulus aaing in Es. (1) and (13), can b dscibd by a bilina function basd on th dviatoic lastic stain γ : ìï g wi -( wi - w) 0 < g < g * w = ï í g ïïï * ïî w g ³ g * (14) wh ω snts on of th aamts m g, s g, ψ g and E g, and γ * is th citical dviatoic lastic stain fom ACTA GEOTECHNICA SLOVENICA, 01/ 41.

6 which th sidual bhavio stats, and should b idntifid by ximnts. Th subscits i and dnot th initial and sidual valus, sctivly. 4 ANALYSIS OF THE PLASTIC ZONE Fo both cass (th hoizontal and th vtical dictions) a lastic zon of adius R will b fomd aound th tunnl. Th govning uations on th lastic zon a simila, but not idntical, fo both cass. It is imotant to highlight that th stains and stsss in th lastic zon dnd on two factos: on a hysical vaiabl, which is th distanc to th cnt of th xcavation; and on a fictitious tim vaiabl τ = R, which is a masu of th lasticity volution. In this gad, th dimnsionlss vaiabl ρ is considd, that mas th hysical lan (, τ) into a lan of coodinat ρ accoding to th following tansfomation (s Fig. 6): = o = (15) t R Basd on th abov tansfomation, th solutions fo th stain and stss filds do not dnd on th lastic adius. In this gad, th uilibium E. (3) can b xssd with sct to th nomalizd adius = as: R ds s -s - + FR = 0 (16) d wh F is th adial body foc. F is ual to γ fo th vtical diction, and it is ual to zo fo th hoizontal diction (s Fig. 5). A combination of th failu cition, i.., E. (11), and uilibium uation, i.., E. (16), givs: ( ) 1 m gss c + sgsc ds + FR = (17) d It is assumd that in th lastic zon th total stains consist of th lastic and lastic ats: = +, = + (18) wh and a total adial and cicumfntial stains, sctivly, and th suscits and dnot th lastic and lastic ats of th stains, sctivly. Thus, th total stain ats and can b wittn in tms of th lastic (, ) and lastic (, ) comonnts as: = +, = + (19) wh th dot dnots th divativ of stain with sct to th fictitious tim vaiabl τ = R ( = ), t and and a obtaind using Hook s law, i.., Es. (1) and (13). Fo th Moh Coulomb ty of lastic otntial function (8), limination of th lastic multili l fom th flow ul, i.., uations (6) and (7), givs th lation btwn th lastic ats of th adial and cicumfntial stain ats as follows: KY + = 0 (0) wh th cofficint of dilation K ψ is obtaind fom E. (9). Basd on th givn tansfomation (E. (15)), th atial divativs of th fild functions with sct to th vaiabls and τ = R a valuatd with th oatos: () 1 () = R () () =- t R (1) () Eliminating u fom Es. (5) by alying Es. (15) and (1) dvlos th siml comatibility uation: Figu 6. Nomalizd lastic zon with a finit numb of annula lmnts. = + (3) wh is dfind as: d = (4) d Sinc a multi-lina bhavio modl and th incmntal thoy of lasticity hav bn usd, th govning ua- 4. ACTA GEOTECHNICA SLOVENICA, 01/

7 tions on th stsss and stains in th lastic zon hav no analytical solutions, and must b solvd numically, as sntd in Andix A. Dfining th stsss and stains on th out bounday of th lastic zon, wh ρ = ρ 1 = 1, succssiv valus of th stsss and stains a calculatd fom th fomulations sntd in Andix A by succssiv incmnts of ρ (s Fig. 6) until th valu of th adial stss fo a scific ρ n (i.. σ (n) ) achs i. Thus, fo an analysis of th lastic zon it is ncssay to calculat th bounday stsss at th xtnal adius of th lastic zon (lastic adius) by considing th intactions btwn th lastic and lastic zons. 5 ANALYSIS OF THE ELASTIC ZONE It should b notd that th tunnl xcavation inducs additional stsss on th ock mass initially subctd to ual fild stsss; thus, th final stsss in th ock mass will b ual to th sum of th initial stsss and th inducd stsss. As th xcavation is taking lac, th stsss δσ θ and δσ a inducd in th ock mass. By ducing th initial otion of th stsss fom uilibium E. (3), th govning uilibium uation in tms of th inducd stsss δσ and δσ θ is givn by: dds ( ds -ds) - = 0 (5) d In th lastic zon, Hook s law fo lan-stain conditions can b usd btwn th inducd stsss and stains [17]: E0 ds = ( )( ) ( 1- n 1 1 ) + n + n - n [ ] E0 ds = ( )( ) ( 1- n 1 1 ) + n + n - n [ ] (6) (7) A combination of th uilibium uation (5) with th abov uations (Es. (6) and (7)); and thn alying th stain-dislacmnt Es. (5), givs th following uation fo th unknown adial dislacmnt u : 1 u du d u = 0 (7) d d This diffntial uation has an analytical solution fo th lastic zon by alying th bounday conditions at th intnal and xtnal adii. Th bounday conditions a diffnt fo both cass (th hoizontal and vtical dictions) (s Fig 5), and thus th cosonding analytical fomulations will b diffnt. Fo th cas of th hoizontal diction: 1+ n R d () =- d() = ds R ( ) (8) E R ds () =- ds() = ds R ( ) (9) Fo th cas of vtical diction: R æ h ö ds () =-ds 1 R ( ) - h - R çè ø n R æ h 1 0 R 0 0 ç (30) R æ h ö ds() =- ds 1 R ( ) h0 R + (31) - ç è ø 1+ n R æ h 0 ö d() =- ds 1 n R ( ) - + E h R (3) - çè ø 1+ ö d () =- ds - n - E ( ) h - R ç è ø (33) Basd on th abov uations, th sam xssions fo th inducd stsss and stains a obtaind, fo both cass (th hoizontal and th vtical dictions), fo a d tunnl, namly wh h 0 >> R. On th oth hand, it is obsvd that th gavitational loads will not affct th dislacmnts in th lastic zon dictly. In th abov uations, ds R ( ) is th inducd adial stss at th lastic adius, and is obtaind fom: ds = s - s R ( ) R ( ) 0( R) (34) Th final stsss (σ (), σ θ() ), at any adius in th lastic zon, a obtaind fom th sum of th initial otions (σ 0(), σ θ0() ) and th inducd otions (δσ (), δσ θ() ). s () = s0() + ds () (35) s() = s0() + ds() (36) Th final adial and cicumfntial stsss at th lastic adius (σ θ(r ) and σ (R )) must satisfy th stngth cition; thfo, substituting ths stsss into th Hok Bown stngth cition (i.., E. (10)), and solving th uation obtaind, givs th final bounday adial stss at th lastic adius σ (R ). Fo th vtical diction: 1 s = ( ) ( ) ( misc + b 1+ a )- R æ æ 1 ö ö - misc + 4mibsc( 1 + a) + 4asisc + a + çè ç è a ø ø 1 (37) ACTA GEOTECHNICA SLOVENICA, 01/ 43.

8 R æ h ö 1 0 a = + h0 - R ç R çè ø (38) b= ( 1+ a) s 0 ( R, = 90 ) (39) And fo th hoizontal diction: 1æ ö s = 4 R ( ) ç l- l + ls + s 0( 0 ) is c s = + çè ø 0( = 0 ) (40) ( misc) l = (41) 4 Wh m i and s i a th failu aamts fo th oiginal ock mass. It should b notd that th lastic zon aound th cicula oning is only fomd whn th intnal suot ssu i is low than a citical valu of σ (R = i ) (wh R = i ). As mntiond, fo d tunnls, h 0 >> 0, th govning uations in th lastic zon fo both th hoizontal and vtical dictions a idntical. Howv, in this cas, th wight of th lastic zon may b significant; and thus th gavitational loads must b takn into account. 6 COMPUTATION PROCEDURE As illustatd in Andix A, in th lastic zon, th finit-diffnc calculations a caid out in tms of th nomalizd adius =. Fist, th bounday stsss and stains at th lastic adius (ρ 1 =1 o R 1 =R ) a calculatd fom th uations sntd in Sction 5. Thn, th succssiv valus of th stsss and stains in th lastic zon a comutd fom th uations sntd in Andix A. Th comutations of th stsss σ () and σ θ() and th stains ( ) and ( ) a caid out until th uilibium conditions at th tunnl adius a satisfid. Thus, whn th valu of adial stss, fo a scific ρ n, satisfis uation σ () = i, th comutations will b stod. Th nw valu of th lastic adius will thn b obtaind, by dividing th tunnl adius i by this final valu of ρ n. 7 ILLUSTRATIVE EXAMPLES Th solution dscibd in this a has bn ogammd in th FORTRAN languag fo us with a comut. This ogam was usd to analyz sval tyical tunnls, and th sults w thn inttd. EXAMPLE 1 In this xaml th sam tunnl as in Bown t al. [1] and Shaan [] was analyzd, and th sults w thn comad. In Bown t al. s, and Shaan s closd-fom solutions, an lastic bittl ock mass bhavio modl has bn usd. Bown t al. nglctd th lastic stain distibution in th lastic zon, whil Shaan utilizd an aoximat fomula fo th lastic stains. In th oosd mthod, th analyss w fomd fo two valus of γ *, i.., γ * = 0 (cosonding to a bittl bhavio) and γ * = 0.01 (cosonding to a stainsoftning bhavio). A tyical d tunnl, h 0 >> 0, with th following tyical otis is considd: m i = 1.7, m = 1, s i = , s = 0, σ c = 30 MPa, ν = 0.5, E 0 = E = 5500 Ma, γ = 0.08 MN/m 3, σ 0 = 30 MPa, i = 5 m, i = 5 MPa, ψ i = 30, ψ = 0 wh E 0 and E a th lastic modulus fo th oiginal and sidual ock masss, sctivly, and ψ i and ψ a th dilation angls fo th oiginal and sidual ock masss, sctivly. γ γ Whn th valu of th lastic adius R is not initially dtmind, th comutations must b fomd itativly. Thus, th valu of lastic adius R, obtaind in ach st, is usd fo comutations of th subsunt st. Figu 7. Gound-sons cuvs fo th tunnl of th xaml ACTA GEOTECHNICA SLOVENICA, 01/

9 γ γ γ γ Figu 8. Distibution of stsss aound th tunnl of xaml 1. As this cas is a d tunnl, only th fomulations fo th hoizontal diction a utilizd. Figs. 8 and 9 show th gound-sons cuvs and th stss distibutions obtaind fom th th thotical mthods. It is obsvd that Shaan s aoximation ovstimats th dislacmnts, whil Bown t al. s aoximation undstimats th dislacmnts, as illustatd by L and Pituszczak [6]. Futhmo, in th oosd mthod, th stain-softning bhavio can also b considd, and as shown in Figs. 8 and 9, fo γ * = 0.01, th gound-sons cuvs coincid, whil th lastic adii a diffnt. EXAMPLE In this xaml, th ffct of gavitational loads bing alid in th lastic zon is xamind. In addition, th oosd mthod is comad with a numical mthod. Figu 9. Distibution of stsss aound th tunnl of xaml. ACTA GEOTECHNICA SLOVENICA, 01/ 45.

10 Th following data st is usd: γ * = 0.004, m i = 0.3, m = 0.1, s i = , s = 0, σ c = 30 MPa, ψ i = ψ = 0, ν = 0.5, E 0 = MPa, E = 4000 MPa, γ = 0.08 MN/m 3, s = 0 MPa, σ 0 = 10 MPa, i = 3 m, i = 0.5 MPa This tunnl is also a d tunnl; thus, fo th analysis of th lastic zon th fomulations oosd fo th hoizontal diction a utilizd. Howv, bcaus of th ffct of gavitational loads in th lastic zon th sults fo th hoizontal and vtical dictions can b diffnt. In Figs. 10 and 11 th gound-sons cuvs and th stss distibution though th hoizontal and vtical dictions a dictd. In ths figus, th sults obtaind fom th FLAC D ogam [0] a also lottd, which show a vy o agmnt with th oosd solution. Fig. 9 shows that du to th wight of th lastic zon, th lastic adii a not th sam fo th diffnt dictions, and th lastic adius incass fom th floo to th cown. In Fig. 10, th gound-sons cuvs obtaind fom Hok and Bown s [19] simlifid mthod a also lottd. As is cla fom this figu, Hok and Bown s mthod ovstimats th tunnl convgnc fo th cown, and undstimats it fo th floo. EXAMPLE 3 In this xaml, th ffct of tunnl dth and hoizontal and vtical dictions on th sults a xamind. Fo this uos, th following data st is usd: γ * = 0.0, m i = 0.7, m = 0.3, s i = 0.001, s = 0, σ c = 30 MPa, ψ i = ψ = 0, ν = 0.5, E 0 = 1500 MPa, E = 1500 MPa, γ = 0.08 MN/m 3, s = 5.0 MPa, i = 5 m, i = 0.0 MPa and 1.0 MPa Figu 10. Th gound-sons cuvs fo th tunnl of xaml. Figu 11. Vaiation of th lastic adius vsus h 0. Fig. 11, shows th vaiations of th lastic adii fo diffnt valus of th tunnl dth, fo two cass of i = 0.0 MPa and 1.0 MPa. In th cas of i = 0.0 MPa, th lastic dfomations in th lastic zon a lag nough, and so th ffct of gavitational loads is significant. Consuntly, th lastic adii, though th hoizontal and vtical dictions, a not th sam. On th oth hand, in th cas of i = 1.0 MPa, th dfomations in th lastic zon a small; thus, th lastic adii, though th hoizontal and vtical dictions, a aoximatly th sam. It is concludd that th lastic adius fo th tunnl cown is lag than th tunnl wall, and th diffnc incass with th incasing th dth of th tunnl. Th valus of th adial dislacmnts at th distanc = h 0 along th hoizontal and vtical dictions may b utilizd as th low-bound and th u-bound valus, sctivly, fo aoximating th sufac sttlmnt. In Fig. 1 th adial dislacmnts at th distanc = h 0 fo diffnt valus of h 0 fo two cass of i = 0.0 MPa and 1.0 MPa w lottd. It is cla that at small valus of h 0, bcaus of th shot distancs btwn th gound sufac and th lastic zon, xcssiv sufac 46. ACTA GEOTECHNICA SLOVENICA, 01/

11 Figu 1. Vaiations of adial dislacmnt at = h 0 vsus h 0. sttlmnts occu. On th oth hand, at lag valus of h 0, th sufac sttlmnt may incas with dth du to an incas in magnitud of th in-situ stsss. Futhmo, in Fig. 1, it is cla that th dislacmnts though th tunnl cown a lag than th tunnl wall, and th diffnc bcoms gat fo th wid lastic zons. 8 CONCLUSIONS In od to xamin th C.-C. mthod, commonly usd fo th analysis of tunnls, and to dmonstat th ffct of th classic assumtions on th chaactistics of th goundsons cuv, an analytical solution was oosd. Th solution is lativly siml, asy to us, and can adily indicat its snsitivity though a ang of ossibl gound aamts, bounday conditions, and alid loads. In this solution, th fomulations w divd though hoizontal and vtical dictions, by taking th gavitational loads into account. It was shown that, fo actical cass vn fo shallow tunnls, th convgnc confinmnt mthod is asonably alicabl. Howv, th ffct of gavitational loads in th analyss can b noticabl, and ignoing th lastic load can lad to lag os in th calculations. REFERENCES [1] Bown, E.T., Bay, J.W., Ladanyi, B., Hok, E. (1983). Gound sons cuvs fo ock tunnls. Jounal of gotchnical Enginin, Vol. 109, No. 1, [] Shaan, S.K. (003). Elastic bittl lastic analysis of cicula onings in Hok Bown mdia. Int. J. Rock Mch. Min. Sci., Vol. 40, No.6, [3] Caanza-Tos, C., Faihust, C. (1999). Th lasto-lastic sons of undgound xcavations in ock masss that satisfy th Hok Bown failu cition. Int. J. Rock Mch. Min. Sci., Vol. 36, No. 6, [4] Pak, K.-H., Kim, Y.-J. (006). Analytical solution fo a cicula oning in, an lasto-bittl-lastic ock. Int. J. Rock Mch. Min. Sci., Vol. 43, [5] Guan, Z., Jiang, Y., Tanabasi, Y., (007). Gound action analyss in convntional tunnling xcavation. Tunnlling and Undgound Sac Tchnology, Vol., No., [6] L, Y.-K., Pituszczak, S. (008). A nw numical ocdu fo lasto-lastic analysis of a cicula oning xcavatd in a stain-softning ock mass. Tunnlling and Undgound Sac Tchnology, Vol. 3, No. 5, [7] Fahimifa, A., Zaifad, M.R. (009). A thotical solution fo analysis of tunnls blow goundwat considing th hydaulic mchanical couling. Tunnlling and Undgound Sac Tchnology, Vol. 4, No. 6, [8] Caanza-Tos, C., Faihust, C. (000). Alication th convgnc-confinmnt mthod of tunnl dsign to ock masss that satisfy th Hok Bown failu cition. Tunnlling and Undgound Sac Tchnology, Vol. 16, No., [9] Gonzálz-Niciza C, Álvaz-Vigil A.E., Mnéndz-Díaz A., Gonzálz-Palacio C. (008). Influnc of th dth and sha of a tunnl in th alication th convgnc-confinmnt mthod. Tunnlling and Undgound Sac Tchnology, Vol. 3, No. 1, ACTA GEOTECHNICA SLOVENICA, 01/ 47.

12 [10] Lu A-Z., Xu G-s., Sun, F., Sun, W. (010). Elastolastic analysis of a cicula tunnl including th ffct of th axial in situ stss. Int. J. Rock Mch. And Mining Sci., Vol. 47, No. 1, [11] Dtounay, E. and Faihust, C. (1987). Twodimnsional lastolastic analysis of along, cylindical cavity und non-hydostatic loading. Int. J. Rock Mch. Min. Sci. & Gomch. Abst., Vol. 4, No. 4, [1] Rd MB. (1988). Th influnc of out-of-lan stss on a lan stain oblm in ock mchanics. Int. J. Num Anal Mth Gomch., Vol. 1, No., [13] Hok E., Bown E.T. (1980). Undgound xcavations in ock. Th Institut of Mining and Mtallugy, London [14] Caanza-Tos, C. and Faihust, C. (1997). On th stability of tunnls und gavity loading, with ost-ak softning of th gound. Int. J. Rock Mch. Min. Sci., Vol. 34, No. 3-4, [15] Fahimifa, A., Zaifad, M.R., (010). Th lastolastic sons of cicula tunnl considing gavity loads fo two cass of lan stain and lan stss conditions. Gotchnical Chalngs in Mgacits, Moscow. V.. [16] Kolymbas D. (005). Tunnlling and Tunnl Mchanics. Sing-Vlag, Blin Hidlbg () [17] Timoshnko S.P., and Goodi J.N. (198). Thoy of Elasticity. McGaw-Hill, Nw Yok [18] Alonso, E., Alano, L.R., Vaas, F., Fdz-Manin, G., Caanza-Tos, C. (003). Gound sons cuvs fo ock masss xhibiting stain-softning bhaviou. Int. J. Num. Anal. Mth. Gomch., Vol. 7, No. 13, [19] Hok E., Bown E.T. (1980). Undgound xcavations in ock. Th Institut of Mining and Mtallugy, London [0] Itasca. (000). Us manual fo FLAC, Vsion 4.0. Itasca Consulting Gou Inc.: Minnsota APPENDIX A. STRESS AND STRAIN ANALYSES FOR THE PLASTIC ZONE Stss analysis Th finit-diffnc mthod (FDM) is usd to solv E. (17) fo σ () and σ θ() by slcting an annula lmnt of th out adius ρ -1 and th inn adius ρ shown in Fig. 6: { } 1 s() = s( -1) + l1- l + l3s( -1) (A1) wh: mg() + mg() mg = (A) sg() + sg() sg = (A3) æ ö - -1 l1= FR ( - -1) + mgs c ç çè + -1 ø æ 1 1 l - ö é æ mgs - ö - - = c + ç -1 è + ê ø çè + -1 ø êë ù + mgscfr( - -1 ) + 4sgsc ú úú û æ ö - -1 l3 = 4mgs c (A6) ç çè + -1 ø (A4) Th failu cition, i.., E. (11), can thn b usd to calculat th cosonding valus of σ () as follows: ( ) 1 s () = s() + mg() scs() + sg() sc (A7) σ () is a function of th lastic adius R fo vtical diction, as obsvd in E. (A1); thus, th analyss a caid out altnatly in a sunc of succssiv aoximations, to achiv th aoiat convgnc of th lastic adius R. Stain analysis Th total stain ats and can b wittn in tms of th lastic(, ) and lastic(, ) comonnts as: = + = + (A8) Considing th Moh Coulomb ty of lastic otntial function, i.., E. (8), th limination of th lastic multilil fom th flow ul, i.., Es. (6) and (7), and using E. (15) givs th lation btwn th lastic ats of adial and cicumfntial stain ats: KY + = 0 (A9) Th finit-diffnc mthod (FDM) is usd to solv th govning diffntial uation obtaind fom Es (3) and (A9) fo ( i ) and ( i ) by using th annula lmnts shown in Fig. 6 as: (A5) 48. ACTA GEOTECHNICA SLOVENICA, 01/

13 ( W() - ( -1) ( 3+ KY() )) ( -1 ) + ( -1 ) 1+ KY () -W ( i ) = 1+ KY() KY ( ) ( ) ( () ) () ( ) () () (A10) wh: thus: () - ( -1) () = () -( -1) () -( -1) ( i ) = () -( -1) (A11) (A1) KY() + KY( -1) KY() = (A13) W () = () + KY() () (A14) W () +W( -1) W () = (A15) ( ) ( ) () = () () - ( -1) + -1 (A16) In addition, th cosonding valus of th stains (), (), (), (), () and () a obtaind using Es. (1), (13), (18), (A8), and (A9). Using Es. (18) and (3), th following bounday conditions a obtaind fo th lastic adius, wh, ρ = ρ = 1: ( = 1) = ( = 1) and = 0 ( = 1) (A17) ( = 1) = ( = 1) and = 0 ( = 1) (A18) ( = 1) = ( = 1) - ( = 1) (A19) NOMENCLATURE: E 0 : Dfomability modulus of lastic ock mass E g : Dfomability modulus of lastic ock mass h 0 : Dth of th tunnl F : Radial body foc F θ : Cicumfntial body foc K : Latal stss cofficint K ψ : Dilation facto m i,s i : Matial constants fo oiginal ock mass. m,s : Matial constants fo sidual ock mass m g,s g : Matial constants fo bokn ock mass s : Suchag load : Radial distanc fom th cnt of th tunnl i : Tunnl adius R : Radius of lastic zon u : Radial dislacmnt (x,y,z): Catsian coodinats δσ θ : inducd cicumfntial stss δσ : inducd adial stss 1 : Mao Pincial stain of ock mass 3 : Mino incial stain of ock mass : Cicumfntial stain : Radial stain : Elastic cicumfntial stain : Elastic adial stain : Plastic cicumfntial stain : Plastic adial stain : Divativ of stain with sct to τ : Divativ of stain with sct to ρ γ : Unit wight of ock mass γ : Dviatoic lastic stain γ * : Citical dviatoic lastic stain η : Softning aamt. l : Plastic multili ν : Poisson s atio of ock mass θ : Angl masud clockwis fom th hoizontal diction ρ : Nomalizd adius σ 1 : Mao incial stss σ 3 : Mino incial stss σ c : Uniaxial comssiv stngth of intact ock σ θ : Cicumfntial stss σ : Radial stss σ θ0 : Initial cicumfntial stss σ 0 : Initial adial stss σ 0 : Initial stss τ : Fictitious tim vaiabl ψ i : Dilation angl fo oiginal ock mass ψ : Dilation angl fo sidual ock mass ψ g : Dilation angl fo lastic ock mass Subscit : Rfs to lastic at Subscit : Rfs to lastic at ACTA GEOTECHNICA SLOVENICA, 01/ 49.