A limit equilibrium method for the assessment of the tunnel face stability taking into account seepage forces

Transcription

1 A limit equilibrium method for the aement of the tunnel face tability taking into account eepage force P. Perazzelli (1), T. Leone (1), G. Anagnotou (1) (1) ETH Zurich, Switzerland World Tunnel Congre 2013 Geneva Underground the way to the future! G. Anagnotou & H. Ehrbar (ed) 2013 Taylor & Franci Group, London ISBN ABSTRACT: The preent paper invetigate the problem of the tability of the tunnel face under eepage flow condition baed upon the o-called method of lice. Thi computational model improve the limit equilibrium method of Anagnotou & Kovári (1996) by treating the equilibrium in the wedge conitently with the overlying prim and doe not need an a priori aumption concerning the ditribution of the vertical tree. Auming a implified ditribution of the hydraulic head in the ground ahead of the face, a cloed-form olution i derived which can be ued for aeing face tability. 1 Introduction Seepage flow condition increae the rik of a collape of the tunnel face becaue they are aociated with the occurrence of pore preure gradient in the ground ahead of the face. The preure gradient act a a body force, o-called eepage force, which i directed toward the face and i, therefore, unfavorable with repect to it tability. The effect of eepage flow on the tability of the face can be invetigated by conidering the limit equilibrium of a mechanim (Anagnotou & Kovári 1996, Broere 2001), by method baed on platicity theorem (Lee et al. 2001, 2003, 2006) or by numerical tre analye (Ströhle and Vermeer 2009, Vermeer et al. 2002). The preent paper invetigate tunnel face tability under drained condition by conidering a wedge and prim mechanim (Fig. 1a, cf. Anagnotou and Kovári 1996) and analying the equilibrium of the wedge baed upon the method of lice (Anagnotou 2012). In analogy to the ilo theory, the method of lice aume proportionality between the horizontal tre y and the vertical tre z : ' y ', (1) where i the coefficient of lateral tre which i aumed to be contant. In order to calculate the ditribution of the vertical tree z inide the wedge, the equilibrium of an infiniteimally thin lice i conidered (Fig. 1b). The method of lice repreent an improvement of the model of Anagnotou & Kovári (1996) by eliminating the need for an a priori aumption of the ditribution of the vertical tre z in the wedge and offering the poibility of analyzing cae with non-uniform face upport, heterogeneou ground coniting of horizontal layer and non-uniform ditribution of the eepage force along the height of the face. In thi paper we conider a homogenou oil obeying the Mohr- Coulomb failure criterion, a uniform upport preure and an approximate ditribution of the hydraulic head, which i obtained by fitting the reult of three dimenional eepage flow analye (Fig. 2b). The computational prediction of the method of lice agree very well with publihed reult of experimental tet in dry oil when the coefficient of lateral tre i taken equal to 1.0 (Anagnotou 2012). For thi reaon, the calculation according to the method of lice will be carried-out for = 1.0 in the preent paper. Section 2 decribe the eepage-flow analyi and preent the implified ditribution of the hydraulic head, which will be introduced in the limit equilibrium analyi. Section 3 outline the limit equilibrium z 715

2 analyi of the conidered mechanim. Section 4 compare the computational reult with thoe obtained baed upon Anagnotou and Kovári (1996). (a) (b) Figure 1. (a) Failure mechanim. (b) Force acting upon an infiniteimal lice (a) (b) Figure 2. (a) Finite element meh for the computation of the hydraulic head field. (b) Contour line of the hydraulic head for an example with B = H = 10 m, T/H = 5, h = 100 m and h 0 > T+H (blue: head h cloe to the head h F of the face, red: head h cloe to the initial head h 0 ) 716

3 2 Seepage flow analyi We determine numerically the three dimenional, teady tate hydraulic head field around the tunnel face auming Darcy law with a uniform ground permeability. The permeability coefficient doe not influence the hydraulic head field. A no-flow boundary condition and a contant piezometric head h F are precribed to the tunnel wall (imperviou lining) and to the tunnel face, repectively. At the far-field boundary, the piezometric head i taken equal to the water table elevation h 0. Thi condition applie alo to the water table (no draw-down, i.e. ufficient groundwater recharge from the urface). Figure 2a how the central part of the finite element meh adopted for the calculation, which were performed by the finite element program COMSOL. A quare tunnel cro-ection i conidered for implicity (analogue to the limit equilibrium model). The computational domain conit of one half of the ytem due to the vertical ymmetry plane. Figure 3 how the normalized ditribution of the hydraulic head ahead of the face and above the tunnel along two characteritic line (the tunnel axi and the vertical axi z) for a quare tunnel (H=B) with overburden T=5H auming that the water table i located above the oil urface (h 0 > T+H). The normalized ditribution of the hydraulic head depend in general on the normalized overburden T/H, but remain practically contant for T/H value higher than 10 (Zingg and Anagotou 2012). In the comparative calculation of the following ection, T/H i taken equal to 5. The numerically computed hydraulic head ahead of the face (i.e., inide the wedge) and above the tunnel (i.e., inide the prim) i given approximately by the equation and x -b h x, y, z hx h 1- H F e h (2) x z -b a 1- H H hx, y, z hx, z hf 1-e h, (3) repectively, where h i the difference between the far field hydraulic head h 0 and the hydraulic head at the face h F. The contant a and b are obtained by curve fitting the analytical olution to the numerical reult and depend in general on the normalized overburden T/H. For the comparative calculation in the following Section, the contant a and b are taken to 2.85 and 1.64, repectively, which apply to T/H = 5 (Fig. 3). z/h [-] 4 (h-h F )/(h 0 -h F ) [-] numerical reult approximate ditribution Tunnel (h-h F )/(h 0 -h F ) [-] x/h [-] Figure 3. Ditribution of the normalized hydraulic head (h-h F )/(h 0 -h F ) along two characteritic line (B = H, T/H = 5, h 0 > T+H) 717

4 3 Computational model 3.1 Outline The mechanim under conideration (Fig. 1a) fail, if the load exerted by the prim upon the wedge exceed the force which can be utained by the wedge at it upper boundary. At the limit equilibrium the prim load i equal to the bearing capacity of the wedge. The prim load i calculated baed on the ilo theory (Section 3.2), while the bearing capacity of the wedge i calculated by conidering the equilibrium of an infiniteimal lice (Section 3.3). Both, the load of the prim and the bearing capacity of wedge, depend on the inclination of the lip plane. The critical value of the angle (Fig. 1a), i.e. the value that maximize upport requirement, will be determined iteratively. 3.2 The prim loading Auming that the water level i above the oil urface and that the ground i homogeneou and obey the Mohr-Coulomb failure condition with coheion c and angle of internal friction, the effective vertical load of the prim read a follow: T R '- c - tan R V' ilo max0, 1-e whbhtan, (4) tan where the coefficient R i the ratio of the area to the circumference of a horizontal cro-ection of the prim, while the coefficient depend on the hydraulic head ditribution (Anagnotou and Kovári 1996). The coefficient can be determined either numerically on the bai of the numerically computed hydraulic head ditribution (Anagnotou and Kovári 1996) or analytically auming the approximate ditribution of Eq. (3): a tan a T H R 1e 1e a tan Hb tan H R b tan. (5) 3.3 The bearing capacity of the wedge In order to determine the bearing capacity of the wedge we conider the equilibrium of an infiniteimal lice (Fig. 1b). The following force act upon the lice: it ubmerged weight dg ; the upporting force V (z) reulting from the effective normal tre exerted by the underlying ground; the loading force V (z) + dv reulting from the effective normal tre exerted by the overlying ground; the effective normal force dn reulting from the effective normal tre at the inclined lip urface; the hear force dt on the inclined lip urface; the hear force dt on the two vertical lip urface; the reultant eepage force df x and df z (df y i equal to zero due to the ymmetry) and the upporting force ds reulting from the effective upport preure at the tunnel face. The equilibrium equation parallel and perpendicular to the liding direction are: and where and ' in ( ' ' )co dt dt ds df dv dg df (6) x z dn ' ( dv ' dg ' df )in ds '-df co, (7) df x dfz ztan B/2 ztan B/2 z w 0 B/2 w 0 B/2 h x, y, z z x (8) dx dy dz. (9) Expreion for the other force appearing in Eq. (6) and (7) can be found in Anagnotou (2012). The equilibrium condition lead to the following differential equation for the effective vertical force V (z): h x, y, z x dx dy dz 718

5 dv ' z dfz dfx B V' M + P B BP dz B dz dz. (10) The coefficient, M, P and P were introduced by Anagnotou (2012). The effective vertical force V (z) can be determined by olving the differential equation (10) for the boundary condition V (0) = 0. For the approximate ditribution of hydraulic head (Eq. 2), the effective vertical force i equal to V '( z) C ( ) B ' C ( ) B c - C ( ) B '- C B h, (11) c h w where i the normalized z coordinate ( =z/h), the coefficient C (), C c () and C () can be found in Anagnotou (2012), while the additional coefficient H 1 1 H 1 H tan 1 B b C h( ) P e e H B H B btan btan B B take into account the eepage force. The bearing capacity of the wedge i identical to the effective vertical force at z = H (=1). From the equilibrium equation (7), we can obtain the following expreion for the ditribution of the effective normal tre n on the liding urface: dn ' z dv '( z) 1 Bdz dz B b tan ' n z ' H tan in ' w h 1e co (12). (13) One can readily verify, that n become negative (tenion), if the hydraulic head difference h i ufficiently high. Thi reult i expected becaue the eepage force are directed toward the face and their magnitude increae with h. Note that even if the ground exhibit a tenile trength, tenile tree higher than c/tan (in combination with the Mohr-Coulomb failure criterion) would lead to negative hear trength value, which doe not make ene. In thi cae, which may happen at high h value, the olution for the effective vertical force V (z) i clearly wrong. The minimum effective upport preure 1, which enure that the tenile effective tre doe not exceed c/tan at any point of the liding urface can be determined from Eq. (13): ' P h P ' H P c, (14) where 1 1 w 2 3 b tan 1 e C h(1) P, (15) 1 C (1) cot C (1) 1 cot P 2 C (1) B cot C (1) H, (16) c H c Cc(1) B cot P3. (17) cot C (1) ( Pinco ) cot C (1) tan The coefficient P, P c and M c can be found in Anagnotou (2012). For oft ground without tenile trength, the effective normal tre n mut be higher than 0, which i a more trict condition than the one mentioned above. The minimum effective upport preure, which i needed in order to atify thi condition, i given by the following equation: ' P h P ' H P c, (18) where 2 1 w 2 4 c H c Cc(1) P4 B. (19) cot C (1) 719

6 Eq. (11), which give the effective vertical force inide the wedge (and for z=h correpond to the bearing capacity), i only correct for effective upport preure higher than 1 (or 2 in the cae of a no-tenion material). 3.4 The effective upport preure In order for the face to remain table, the bearing capacity of the wedge ha to be higher than the vertical load exerted by the prim. To enure thi, the effective upport preure ha to be higher than a minimum value 3 which can be determined from the limit equilibrium condition V (H)=V ilo : where ' F h F ' H F c, (20) 3 1 w H tan F1 C h(1) C (1) B, (21) C (1) B R e F2 C (1) H Btan tan T/ R 1 tan (1 ), (22) tan T/ R 1 tan H(1 e ) F3 Cc (1). (23) C (1) Btan The neceary effective upport preure mut enure all criteria mentioned above, i.e.: ' max ( 1 or ), 2 3. (24) 4 Comparative calculation Figure 4 how the effective upport preure a a function of the hydraulic head difference h for the example of a tunnel croing coheionle oil with a friction angle of 15, 25 or 35. Figure 5 how the effect of coheion on the upport preure for the ame example and for a fixed value of the hydraulic head difference (h = 100 m). The olid line are calculated according to the method of lice. The dahed line are baed on the method of Anagnotou and Kovári (1996) for a uniform coefficient of lateral tre = 1 (a in the method of lice). The dotted line conider the conervative aumption of = 0.8 for the prim and w = 0.4 for the wedge, which wa propoed by Anagnotou and Kovári (1996). Conequently, the difference between dotted and dahed line i due to the different value of, while the reaon for the difference between the olid and the dahed line i that the two computational model (equilibrium of the entire wedge v. equilibrium of lice) lead to different horizontal tree at the lateral liding urface of the wedge. A the horizontal tree generate frictional reitance, their effect increae with the friction angle. The difference between the two model are, therefore, maller for low to moderate friction angle and bigger for the cae of =

7 Method of Slice (=1) Anagnotou and Kovári 1996 ( = 1) Anagnotou and Kovári 1996 ( = 0.8, w = 0.4) ' [kpa] h [m] Figure 4. Effective upport preure a a function of the hydraulic head difference h for a coheionle oil (B = H = 10 m, T = 50 m, h 0 > T + H, = 12 kn/m 3 ) ' [kpa] c [kpa] Figure 5. Effective upport preure a a function of the coheion c for a hydraulic head difference of h = 100 m (B = H = 10 m, T = 50 m, h 0 > T + H, = 12 kn/m 3 ) 721

8 5 Cloing remark According to the reult of comparative calculation, the method of lice lead to lower effective upport preure than the method of Anagnotou and Kovári (1996), but the difference in the model prediction are mall (< 20%) for low to moderate friction angle. Thee concluion are true for tunnel croing oft ground at hallow or moderate depth. In the cae of an open tunnel face at great depth below the ground water table, very high hydraulic head gradient develop in the ground ahead of the face. In thi cae equilibrium i only poible if the face i upported or if the ground exhibit a ufficiently high tenile trength. If the coheion of the ground i high (which may be true for hard oil or weak rock) and it tenile trength i negligible, the neceary face upport preure may exceed by far the preure, which would be needed jut in order that the wedge i able to utain the load of the overlying prim. In thi cae, tenile failure rather than liding become the critical mode for the determination of the upport preure, which mean that the tandard limit equilibrium model may underetimate the neceary upport preure (cf. Anagnotou and Kovári 1994). 6 Reference Anagnotou, G., Kovari, K Die Stabilität der Ortbrut bei Erddruckchilden. Mitteilungen der Schweiz. Geellchaft für Boden- und Felmechanik, 129, Anagnotou, G., Kovári, K Face tability condition with Earth Preure Balanced hield. Tunnelling and Underground Space Technology, 11 (2), Anagnotou G The contribution of horizontal arching to tunnel face tability. Geotechnik, Vol. 35, No. 1, Broere, W Tunnel Face Stability & New CPT Application. PhD thei, Delft Univerity of Technology, Delft. COMSOL. Suburface Flow Module (Comol Multiphyic 4.2). COMSOL Lee, I.-M., Nam, S.-W The tudy of eepage force acting on the tunnel lining and tunnel face in hallow tunnel. Tunnelling and Underground Space Technology, No.16, pp Lee, I.-M., Nam, S.-W., Ahn, J.-H Effect of eepage force on tunnel face tability. Canadian Geotechnical Journal, No. 40, pp Lee, I.-M., Nam, S.-W Seepage force conideration in tunnelling. International Sympoium on Underground Excavation and Tunnelling, Bangkok, Thailand. Ströhle, P., Vermeer, P. A Die Stabilität der Ortbrut bei trömendem Grundwaer. Kolloquium Bauen in Boden und Fel, TAE, Otfildern 2010, pp (in German). Vermeer, P. A., Rue, N., Marcher, Th Tunnel Heading Stability in Drained Ground. Felbau, 20, No. 6, Zingg, S., Anagnotou, G The effect of advance drainage on face tability in homogeneou ground. ITA- AITES World Tunnel Congre, WTC 2012, Bangkok 722