A NEW SOLUTION FOR SHALLOW AND DEEP TUNNELS BY CONSIDERING THE GRAVITATIONAL LOADS mohammad reza zareifard and ahmad fahimifar about the authors Mohammad Reza Zareifard Amirkabir University of Technology Tehran, Iran E-mail: zareefard@aut.ac.ir corresponding author Ahmad Fahimifar Amirkabir University of Technology Tehran, Iran E-mail: fahim@aut.ac.ir Abstract A new, elasto-plastic, analytical-numerical solution, considering the axial-symmetry condition, for a circular tunnel excavated in a strain-softening and Hoek-Brown rock mass is proposed. To examine the effect of initial stress variations, and also the boundary conditions at the ground surface, the formulations are derived for different directions around the tunnel. Furthermore, the effect of the weight of the plastic zone is taken into account in this regard. As the derived differential equations have no explicit analytical solutions for the plastic zone, the finite-difference method (FDM) is used in this study. On the other hand, analytical expressions are derived for the elastic zone. Several illustrative examples are given to demonstrate the performance of the proposed solution, and to examine the effect of various boundary conditions. It is concluded that the classic solutions, based on the hydrostatic far-field stress, and neglecting the effect of the boundary conditions at the ground surface, give applicable results for a wide range of practical problems. However, ignoring the weight of the plastic zone in the analyses can lead to large errors in the calculations. Keywords ground-response curve, elasto-plastic analysis, boundary condition, axial symmetry, gravitational loads 1 INTRODUCTION A number of methods are currently used for the design and analysis of tunnels. Among them, the convergence-confinement method (the C.-C. method) has played an important role in providing an insight into the interaction between the lining support and the surrounding ground mass. The C.-C. method is based on a concept that involves an analysis of the ground-structure interaction by independent studies of the behavior of the ground and of the tunnel support. In this regard, the ground behavior is represented by a ground-response curve; which describes the ground convergence in terms of the applied internal pressure. However, to maintain simplicity, a number of simplifying assumptions are made in its derivation. These assumptions make the method applicable only to deep tunnels in hydrostatic stress fields. In the past, a number of classic solutions for determining the ground-response curve have been published. These solutions may be categorized into two groups of analytical closed-form solutions and analytical-numerical unclosed-form solutions. Although a number of closed-form solutions are available (such as that proposed by Brown et al. [1]; Sharan [2]; Carranza-Torres [3]; Park and Kim [4]), each solution suffers from a level of approximation in the sense that it incorporates various simplifying assumptions. For example, these solutions have been proposed for the rock masses with simpler behavior models, including the elastic-perfect-plastic or elastic-brittle-plastic behavior models. In fact, for more complicated behavior models obtaining an exact closed-form solution is impossible. On the other hand, in the unclosed-form solutions (Brown et al., [1]; Guan et al., [5]; Lee and Pietruszczak, [6]; Fahimifar and Zareifard, [7]), consideration of more complicated and general material-behavior models are possible. However, all the mentioned solutions (both the closed-and unclosed-form solutions) are based on the classic assumptions made in the C.-C. method. Different aspects of the C.-C. method have been investigated by researchers, both analytically and numerically. ACTA GeOTeCHNICA SLOVENICA, 2012/2 25. M. R. ZflRËIFflRD & A. FAHIMIFAR: A NEW SOLUTION FOR SHALLOW AND DEEP TUNNELS BY CONSIDERING THE GRAVITATIONAL LOADS With the development of computer codes, numerical analyses have become common methods for the analysis of tunnels. Various cases of analyses, including two-dimensional, three-dimensional and time-dependent behaviors, can be performed using commercial codes. Nevertheless, utilizing numerical methods for investigating the C.-C. method, such as that presented by Carranza-Torres and Fairhurst [8] and that presented by Gonzalez-Nicieza et al. [9], do not easily reveal the actual effects that the simplifying assumptions have on the mechanical response of a tunnel. Therefore, analytical solutions of simpler cases, such as that proposed by Lu et al. [10], Detournay and Fairhurst [11], and Reed [12], can help to realize various aspects of the ground-response curve of tunnels much better. In the C.-C. method the effects of the boundary conditions at the ground surface are neglected and, also, the variation of the initial stresses (i.e., in-situ stresses) are not taken into account, classically. However, for shallow tunnels, the initial stresses cannot be assumed to be constant over the tunnel section, and thus the assumption of hydrostatic, far-field stresses may not be applicable. On the other hand, in the C.-C. method, the above solutions neglect the weight of the plastic zone developed around the tunnel. However, very few works concerning the effect of gravitational forces acting on the ground have been conducted [13-15]. In fact, gravitational loading differs for various directions around the tunnel periphery, and, for the same internal pressure, convergence of the crown is expected to be larger than that of the walls, because of the weight of the failed material on the top of the tunnel. In this paper, an unclosed-form analytical solution is presented for the stress and displacement fields around a circular tunnel excavated in an elasto-plastic strain-softening and Hoek-Brown rock material. In this method the effects of boundary conditions at the free surface of the ground, and also the effect of the weight of the plastic zone are taken into account. For this purpose, the formulations are derived for both the horizontal and the vertical directions that are passing through the tunnel center. In addition, the gravitational loadings are considered as radial body forces being applied to the rock mass. 2 ANALYTICAL SOLUTION OF TUNNELS CONSIDERING THE GRAVITATIONAL LOADS Fig. 1 shows the general case of a tunnel excavated in a homogeneous and isotropic rock mass under an in-situ stress field below a horizontal ground surface. In general cases the initial stresses in the Cartesian coordinate system are Oy = yy + ps and ox = oz = KOy , where x, y and z are the Cartesian coordinate axes (as shown in Fig. 1), y is the specific weight of the rock, K is the so-called lateral stress coefficient, and ps is the uniform vertical stress that is applied to the ground surface from infrastructures or embankments (surcharge load). In cylindrical coordinates (r,Q,z), the stress field around a tunnel (see Fig. 2) has to fulfill the equilibrium equations for each element of the rock mass, as in [16]: darr . 1 da, Br dr r d6 ■ + Fr = 0 da. rB 1 daee , ar6 + a6r dr d6 + Fq = 0 (1) (2) Ground *4i 11 n 1111111 Initial condition: ............ ay = W+Ps 0x=K(w+ps) Figure 1. Circular shallow tunnel in a semi-infinite medium under an initial stress field. y i y 38. ACTA GEOTECHNICA SLOVENICA, 2012/2 M. R. ZAREIFARD & A. FAHIMIFAR: A NEW SOLUTION FOR SI M. R. ZflRËIFflRD & A. FAHIMIFAR: A NEW SOLUTION FOR SHALLOW AND DEEP TUNNELS BY CONSIDERING THE GRAVITATIONAL LOADS where Fr = -ysin9, and Fg = -ycosQ are the gravitational body forces in the radial and circumferential directions, respectively, y is the unit weight of the ground and 9 is the angle measured clockwise from the horizontal direction. Obtaining an exact elasto-plastic analytical solution for this problem is extremely complicated and even unsolvable in more cases (as shown by Detournay and Fairhurst [11]), because, in this case, the principal stresses may rotate in each direction. Tunnel Figure 2. Body forces and stress components corresponding to an element of the rock mass. As mentioned in the C.-C. method for simplifying the problem, the ground-response curve is constructed based on the elasto-plastic solution for a circular opening subjected to hydrostatic far-field stresses and a uniform internal pressure (see Fig. 3).

Co ¡B Plastic Elastic Figure 3. Circular deep tunnel excavated in a hydrostatic stress field. 1 ds6r Thus, in equilibrium Eq. (1) the term r qq vanishes. In this case it is assumed that the initial stresses in the vicinity of the tunnel are constant a^o = °x0 = Yh0 + Ps and do not increase linearly. In this paper an analytical solution for deriving the ground-response curve of a tunnel under equal initial stresses (K = 1) is proposed by considering the effects of the initial stress variations due to the gravitational loads and taking the effects of the boundary conditions into account at the free surface of the ground. In this solution the analyses are performed for the horizontal and the vertical directions for axial symmetry conditions 1 da, (see Fig. 4). Thus, the term--— will vanishes. In this r dO regard, the governing equilibrium equation becomes more straightforward as: P. Atjnfinity <7„ = Yhfl + p5 Figure 4. Analysis of the shallow tunnels along the horizontal and the vertical directions by considering the axial symmetry condition. HALLOW AND DEEP TUNNELS BY CONSIDERING THE GRAVITATIONAL LOADS 38. ACTA GEOTECHNICA SLOVENICA, 2012/2 M. R. ZflRËIFflRD & A. FAHIMIFAR: A NEW SOLUTION FOR SHALLOW AND DEEP TUNNELS BY CONSIDERING THE GRAVITATIONAL LOADS dar dr - + Fr = 0 (3) where Fr is the applied radial body force, which depends on the gravitational loads through the considering direction. As mentioned, since the axial symmetry condition is assumed, only the radial component of the gravitational loads, i.e. , Fr = -ysin0, is taken into account, and the circumferential component is neglected. Thus, Fr = -y for the vertical direction and Fr = 0 for the horizontal direction are obtained (see Fig. 4). For the polar coordinates defined in Fig. 1, the initial equal stress field (K = 1) is given by: J0(r,e r0(r,ß Jeo(r,i = gy + Ps = g(h0 -rsinq) + ps (4) where ag0 and ar0 are the initial circumferential and radial stresses, respectively, and ps is the surcharge load. It should be noted that when there is a very weak or a heavily weathered layer of rock or residual soils on the upper levels, their effect can be considered as surcharge loads applied to the underlying ground. In this study, the formulations are derived for both the horizontal (through the tunnel springline) and the vertical (through the tunnel crown) directions. In this manner, for both directions, because of the axial symmetry conditions, the geometry, boundary conditions and the applied loads are generalized to all directions (see Fig. 4). The problem for both the horizontal and vertical directions are shown in Figs. 5(a) and 5(b), respectively. As observed in Figs. 4 and 5(a), for the horizontal direction, the problem is similar to a circular tunnel in an infinite medium. In this case, at the tunnel radius (i.e., at r = r) the internal pressure p is applied, and at an infinite radius (i.e., at r = ¥ ), the pressure yh0 + ps is applied. In addition, the radial body forces through this direction are equal to zero. It is observed that this case is similar to the problem of a deep tunnel. On the other hand, as observed in Figs. 4 and 5(b), for the vertical direction, the problem of a thick-walled cylinder is the result. In this case, at the tunnel radius the internal pressure pj is applied, and at radius r = h0 , the pressure ps is applied. Furthermore, the radial body forces through this direction are Fr = -y . As shown in Fig. 5, two different zones may be formed around the tunnel (for both directions): the external elastic zone, and the internal plastic zone, which may be divided into the softening zone and the residual zone. The strain-displacement relations in the polar coordinate system for the axial symmetric problem are given by [17]: Sq = ^, Sr = £ (5) r dr a) Horizontal direction: circular hole in an infinte medium under initial uniform sress a0 = yh0 + ps b) Vertical direction: thick walled cylinder under radial gravity loading and a surcharge loading Figure 5. Geometry, applied loads and boundary conditions for the horizontal and vertical directions. 38. ACTA GEOTECHNICA SLOVENICA, 2012/2 M. R. ZflRËIFflRD & A. FAHIMIFAR: A NEW SOLUTION FOR SHALLOW AND DEEP TUNNELS BY CONSIDERING THE GRAVITATIONAL LOADS where ur is the radial component of the displacement and £g and er are the circumferential and radial strains, respectively. Furthermore, the stress state at a distance r is defined by the radial stress ar and the circumferential stress oq, which are the minor o3 and the major ffj principal stresses, respectively, as shown in Fig. 5. 3 BEHAVIOR MODEL The rock mass is assumed to exhibit the strain-softening behavior, in this study, which can be reduced to the perfect elasto-plastic or elasto-brittle-plastic cases. Generally, this behavior is characterized by a transitional failure criterion and a plastic potential. A softening parameter controls the gradual transition from an initial failure criterion (or a potential one) to a residual one. In the present work, the deviatoric plastic strain gp = ep — ep is employed as the softening parameter. Although there is no universal way of defining the strain-softening parameter, as pointed out by Alonso et al. [18], the above softening parameter is the most widely accepted. The plastic strain increments can be obtained from the plastic potential function, g(or, oq, yp) according to: where Ky is the dilation factor, and is given as: sp = l (6) and: da. dg ée = (7) da pr where A is a plastic multiplier, er = --— and ee = (t is a fictitious 'time' variable). Equations (6) and (7) are the constitutive equations in the plastic regime, and are usually termed the flow rule. If the plastic potential coincides with the failure criterion, then it is called an associated flow rule; otherwise it is called a non-associated flow rule. In this regard the incremental plasticity involves a consideration of a fictitious 'time' variable, even if it does not have any physical meaning. This variable controls the evolution of the plasticity and the plastic strain rates. In the formulation presented in this research, the plastic radius, Rp, will be assumed to be the time variable. This choice allows the acquisition of a simple formulation for the problem, in order to obtain a certain kind of solution, as illustrated by Alonso et al. [18]. Here, the Mohr-Coulomb criterion is selected as a plastic potential function for a non-associated flow rule: dep de p . p _ dee g = ae — KY ar (8) Ky = 1 + sin Y g 1-sin Y „ (9) fg , in Eq. (9), is termed the dilation angle and varies as a function of the softening parameter yp. The rock mass is assumed to obey the Hoek-Brown failure criterion, given by [19]: a — ar = (macar + sac ) (10) in which oq is the circumferential stress, or is the radial stress, oc is the uniaxial compressive strength of the intact rock material, and m and s are the Hoek-Brown constants that depend on the properties of the rock mass and the extent to which it was broken before being subjected to the failure stresses oq and or . For the plastic zone, the above equation is given as: *g„r"c < "g"c )2 (11) ae — ar = (mgarac + s ar2 I2 where mg and Sg are the Hoek-Brown constants for the plastic zone and vary as a function of the softening parameter yp. In contrast to the solution presented by Brown et al. [1], the solution proposed in this work considers the elastic strains induced in the plastic zone. The relationships between the elastic strains eer , and e6e , and the stresses or and oq are given by Hooke's law [17]: er =- ee =- i +n sg 1 + n [( 1 — n )(sr — ao ) + v (a0 — ao )] (12) [(1 — v ](ae — ao ) + v (ar — ao )] (13) where o0 is the initial stress, and calculating from Eq. (4), Eg and v are the elasticity modulus and the Poisson's ratio of the rock mass, respectively. However, for the plastic zone, the elasticity modulus Eg varies as a function of the softening parameter yp. It should be noted that in the plastic zone, the failure and dilation parameters, appearing in Eqs. (9) and (11), and also the rock mass elastic modulus appearing in Eqs. (12) and (13), can be described by a bilinear function based on the deviatoric plastic strain yp: yp wi —(wi — wr 0 < 1P < g (14) g ^ g where w represents one of the parameters mg , sg , fg and Eg , and yp is the critical deviatoric plastic strain from g w = 38. ACTA GEOTECHNICA SLOVENICA, 2012/2 M. R. ZflRËIFflRD & A. FAHIMIFAR: A NEW SOLUTION FOR SHALLOW AND DEEP TUNNELS BY CONSIDERING THE GRAVITATIONAL LOADS which the residual behavior starts, and should be identified by experiments. The subscripts 'i' and 'r denote the initial and residual values, respectively. 4 ANALYSIS OF THE PLASTIC ZONE For both cases (the horizontal and the vertical directions) a plastic zone of radius Rp will be formed around the tunnel. The governing equations on the plastic zone are similar, but not identical, for both cases. It is important to highlight that the strains and stresses in the plastic zone depend on two factors: on a physical variable r, which is the distance to the centre of the excavation; and on a fictitious 'time' variable t = Rp , which is a measure of the plasticity evolution. In this regard, the dimensionless variable p is considered, that maps the physical plane (r, t) into a plane of coordinate p according to the following transformation (see Fig. 6): rr P = or p = (15) t Rp Based on the above transformation, the solutions for the strain and stress fields do not depend on the plastic radius. In this regard, the equilibrium Eq. (3) can be expressed r with respect to the normalized radius p = —— as: Rp dsr dp -FR = 0 y (16) where Fr is the radial body force. Fr is equal to -y for the vertical direction, and it is equal to zero for the horizontal direction (see Fig. 5). A combination of the failure criterion, i.e., Eq. (11), and equilibrium equation, i.e., Eq. (16), gives: 1 I . T~ dar t + 'r'P = mg s ac + sg ac )2 (17) It is assumed that in the plastic zone the total strains consist of the elastic and plastic parts: £r = £r + £p , Eg = Eg + Eg (18) where Er and Eg are total radial and circumferential strains, respectively, and the superscripts e and p denote the elastic and plastic parts of the strains, respectively. Thus, the total strain rates Eg and Er can be written in terms of the elastic (,Eer j and plastic (,Ep j components as: er = er + ep » ée = èe + ée (19) where the dot denotes the derivative of strain with dE respect to the fictitious time variable t = Rp (E = ), F Stand Eer and Eeg are obtained using Hooke's law, i.e., Eqs. (12) and (13). For the Mohr-Coulomb type of plastic potential function (8), elimination of the plastic multiplier A from the flow rule, i.e., equations (6) and (7), gives the relation between the plastic parts of the radial and circumferential strain rates as follows: Ep + ky Ep = 0 (20) where the coefficient of dilation Ky is obtained from Eq. (9). Based on the given transformation (Eq. (15)), the partial derivatives of the field functions with respect to the variables r and t = Rp are evaluated with the operators: ÊH=ÊH dr Rp dp ao = _ p_do dt Rp dp (21) (22) Eliminating ur from Eqs. (5) by applying Eqs. (15) and (21) develops the simple compatibility equation: er = p£0 + £e where e' is defined as: (23) Figure 6. Normalized plastic zone with a finite number of annular elements. e' = — (24) dp Since a multi-linear behavior model and the incremental theory of plasticity have been used, the governing equa- 38. ACTA GEOTECHNICA SLOVENICA, 2012/2 M. R. ZflRËIFflRD & A. FAHIMIFAR: A NEW SOLUTION FOR SHALLOW AND DEEP TUNNELS BY CONSIDERING THE GRAVITATIONAL LOADS tions on the stresses and strains in the plastic zone have no analytical solutions, and must be solved numerically, as presented in Appendix A. Defining the stresses and strains on the outer boundary of the plastic zone, where p = pi = 1, successive values of the stresses and strains are calculated from the formulations presented in Appendix A by successive increments of pj (see Fig. 6) until the value of the radial stress for a specific pn (i.e. or(n)) reaches pj . Thus, for an analysis of the plastic zone it is necessary to calculate the boundary stresses at the external radius of the plastic zone (plastic radius) by considering the interactions between the elastic and plastic zones. 5 ANALYSIS OF THE ELASTIC ZONE It should be noted that the tunnel excavation induces additional stresses on the rock mass initially subjected to equal field stresses; thus, the final stresses in the rock mass will be equal to the sum of the initial stresses and the induced stresses. As the excavation is taking place, the stresses Sag and Sar are induced in the rock mass. By reducing the initial portion of the stresses from equilibrium Eq. (3), the governing equilibrium equation in terms of the induced stresses Sar and Sag is given by: dSar (Sag — Sar ) dr r = 0 (25) directions) (see Fig 5), and thus the corresponding analytical formulations will be different. For the case of the horizontal direction: S = S =1 + nS Rp 0er(r) = Seg(r) = 0ar(Rp R 2 Sar(r )=—Sag(r ) = SSr(Rp For the case of vertical direction: Sar(r )=—Sar(Rp , — R2p Rp Sag(r)=—Sar(Rp )h[—PRp 1—hL 1 r 2„ 'l+h2' T 2 r (28) (29) (30) (31) S 1 + nS Seg(r)=— 6arR Rp X 1 + n c oer(r) =--oa p )h2 —R2p R2 Eo r(Rp )ho2— R2p 1 — 2n + r 1 — 2n — r (32) (33) Based on the above equations, the same expressions for the induced stresses and strains are obtained, for both cases (the horizontal and the vertical directions), for a deep tunnel, namely where h0 >> Rp . On the other hand, it is observed that the gravitational loads will not affect the displacements in the elastic zone directly. In the above equations, dar(Rp^ is the induced radial stress at the plastic radius, and is obtained from: In the elastic zone, Hooke's law for plane-strain conditions can be used between the induced stresses and strains [17]: Sar = - ( 1 + n )(, 1 — 2n ) Sag = ( 1 + n )(, 1 — 2n ) [(1 — n ) er + n£g] (26) [[1 — n ) eg + ner (27) da /„ \= a /„ \ —a r(R r(Rp r 2(Rp (34) The final stresses (ffr(r), ty(r)), at any radius in the elastic zone, are obtained from the sum of the initial portions (fffO(r), ffdo(r)) and the induced portions (Sar(r), Sag(r)). ar(r ) = ar 2(r ) + Sar(r) ag(r ) = ago(r ) + Sag(r ) (35) (36) A combination of the equilibrium equation (25) with the above equations (Eqs. (26) and (27)); and then applying the strain-displacement Eqs. (5), gives the following equation for the unknown radial displacement ur : 1 dur r dr d 2i dr2 - = 0 (27) This differential equation has an analytical solution for the elastic zone by applying the boundary conditions at the internal and external radii. The boundary conditions are different for both cases (the horizontal and vertical The final radial and circumferential stresses at the plastic radius (og(Rp) and ffr(Rp)) must satisfy the strength criterion; therefore, substituting these stresses into the Hoek-Brown strength criterion (i.e., Eq. (10)), and solving the equation obtained, gives the final boundary radial stress at the plastic radius 0(r). For the vertical direction: ar(R ) = 1 (ac + 2b( 1 + a)) m^2 + 4m{ßac ( 1 + a) + 4as;a;2 (37) 2 + a + — a R 38. ACTA GEOTECHNICA SLOVENICA, 2012/2 M. R. ZflRËIFflRD & A. FAHIMIFAR: A NEW SOLUTION FOR SHALLOW AND DEEP TUNNELS BY CONSIDERING THE GRAVITATIONAL LOADS h2 — R2 h0 Rp h0 1 + "T RP ß = ( 1 + a) s0 And for the horizontal direction: (38) (39) 7 ILLUSTRATIVE EXAMPLES The solution described in this paper has been programmed in the FORTRAN language for use with a computer. This program was used to analyze several typical tunnels, and the results were then interpreted. Sr(Rp )= P X — I A2 + 4As0(q=0„ )+ Sisp o(e=oc (40) X= 4 (41) Where mi and si are the failure parameters for the original rock mass. It should be noted that the plastic zone around the circular opening is only formed when the internal support pressure pi is lower than a critical value of ariRp=r,) (where Rp = r) As mentioned, for deep tunnels, h0 >> r0 , the governing equations in the elastic zone for both the horizontal and vertical directions are identical. However, in this case, the weight of the plastic zone may be significant; and thus the gravitational loads must be taken into account. 6 COMPUTATION PROCEDURE As illustrated in Appendix A, in the plastic zone, the finite-difference calculations are carried out in terms of the normalized radius p . First, the bound- Rp ary stresses and strains at the plastic radius (pi=1 or r1=Rp) are calculated from the equations presented in Section 5. Then, the successive values of the stresses and strains in the plastic zone are computed from the equations presented in Appendix A. The computations of the stresses or(j) and Ogj and the strains er(j) and £g(j) are carried out until the equilibrium conditions at the tunnel radius are satisfied. Thus, when the value of radial stress, for a specific pn, satisfies equation or(j)=pi, the computations will be stopped. The new value of the plastic radius will then be obtained, by dividing the tunnel radius r, by this final value of pn. When the value of the plastic radius Rp is not initially determined, the computations must be performed iteratively. Thus, the value of plastic radius Rp , obtained in each step, is used for computations of the subsequent step. EXAMPLE 1 In this example the same tunnel as in Brown et al. [1] and Sharan [2] was analyzed, and the results were then compared. In Brown et al.'s, and Sharan's closed-form solutions, an elastic brittle rock mass behavior model has been used. Brown et al. neglected the elastic strain distribution in the plastic zone, while Sharan utilized an approximate formula for the elastic strains. In the proposed method, the analyses were performed for two values of yp , i.e., yp = 0 (corresponding to a brittle behavior) and yp = 0.01 (corresponding to a strain-softening behavior). A typical deep tunnel, h0 >> r0 , with the following typical properties is considered: mi = 1.7, mr = 1, Si = 0.0039, sr = 0, oc = 30 MPa, v = 0.25, E0 = Er = 5500 Mpa, y = 0.028 MN/m3, a0 = 30 MPa, r = 5 m, pi = 5 MPa, fi = 30°, fr = 20° where E0 and Er are the elastic modulus for the original and residual rock masses, respectively, and fi and fr are the dilation angles for the original and residual rock masses, respectively. 35 30 25 (O CL 5 ¥ 20 ca a> S 15 10 5 0 -proposed method(wall, Y =0) a proposed method(wall,Y =.01 ) ......Sharan (2003) --Brown and Bray (1983) 50 100 radial displacement (mm) 150 Figure 7. Ground-response curves for the tunnel of the example 1. 38. ACTA GEOTECHNICA SLOVENICA, 2012/2 M. R. ZflRËIFflRD & A. FAHIMIFAR: A NEW SOLUTION FOR SHALLOW AND DEEP TUNNELS BY CONSIDERING THE GRAVITATIONAL LOADS Figure 8. Distribution of stresses around the tunnel of example 1. As this case is a deep tunnel, only the formulations for the horizontal direction are utilized. Figs. 8 and 9 show the ground-response curves and the stress distributions obtained from the three theoretical methods. It is observed that Sharan's approximation overestimates the displacements, while Brown et al.'s approximation underestimates the displacements, as illustrated by Lee and Pietruszczak [6]. Furthermore, in the proposed method, the strain-softening behavior can also be considered, and as shown in Figs. 8 and 9, for Yp = 0.01, the ground-response curves coincide, while the plastic radii are different. EXAMPLE 2 In this example, the effect of gravitational loads being applied in the plastic zone is examined. In addition, the proposed method is compared with a numerical method. Figure 9. Distribution of stresses around the tunnel of example 2. 38. ACTA GEOTECHNICA SLOVENICA, 2012/2 M. R. ZflRËIFflRD & A. FAHIMIFAR: A NEW SOLUTION FOR SHALLOW AND DEEP TUNNELS BY CONSIDERING THE GRAVITATIONAL LOADS The following data set is used: yP*= 0.004, mt = 0.3, mr = 0.1, si = 0.0001, sr = 0, ac = 30 MPa, f = fr = 0, v = 0.25, E0 = 10000 MPa, Er = 4000 MPa, y = 0.028 MN/m3, ps = 0 MPa, a0 = 10 MPa, ri = 3 m, pi = 0.5 MPa This tunnel is also a deep tunnel; thus, for the analysis of the elastic zone the formulations proposed for the horizontal direction are utilized. However, because of the effect of gravitational loads in the plastic zone the results for the horizontal and vertical directions can be different. Figure 10. The ground-response curves for the tunnel of example 2. "El tr 10 9 8 7 6 5 4 3 2 1 0 -wall (pi=0) ----crown (pi-0) 0 wall :pi i MPa) .......crown (pi=1 MPa) V___—" ..... ~ ............... 100 200 tunnel depth (m) 300 Figure 11. Variation of the plastic radius versus h0. In Figs. 10 and 11 the ground-response curves and the stress distribution through the horizontal and vertical directions are depicted. In these figures, the results obtained from the FLAC 2D program [20] are also plotted, which show a very proper agreement with the proposed solution. Fig. 9 shows that due to the weight of the plastic zone, the plastic radii are not the same for the different directions, and the plastic radius increases from the floor to the crown. In Fig. 10, the ground-response curves obtained from Hoek and Brown's [19] simplified method are also plotted. As is clear from this figure, Hoek and Brown's method overestimates the tunnel convergence for the crown, and underestimates it for the floor. EXAMPLE 3 In this example, the effect of tunnel depth and horizontal and vertical directions on the results are examined. For this purpose, the following data set is used: yp*= 0.0, mt = 0.7, mr = 0.3, si = 0.001, sr = 0, ac = 30 MPa, f = fr = 0, v = 0.25, E0 = 1500 MPa, Er = 1500 MPa, y = 0.028 MN/m3, ps = 5.0 MPa, rj = 5 m, pi = 0.0 MPa and 1.0 MPa Fig. 11, shows the variations of the plastic radii for different values of the tunnel depth, for two cases of pi= 0.0 MPa and 1.0 MPa. In the case ofp, = 0.0 MPa, the plastic deformations in the plastic zone are large enough, and so the effect of gravitational loads is significant. Consequently, the plastic radii, through the horizontal and vertical directions, are not the same. On the other hand, in the case of pi = 1.0 MPa , the deformations in the plastic zone are small; thus, the plastic radii, through the horizontal and vertical directions, are approximately the same. It is concluded that the plastic radius for the tunnel crown is larger than the tunnel wall, and the difference increases with the increasing the depth of the tunnel. The values of the radial displacements at the distance r = h0 along the horizontal and vertical directions may be utilized as the lower-bound and the upper-bound values, respectively, for approximating the surface settlement. In Fig. 12 the radial displacements at the distance r = h0 for different values of h0 for two cases of pi = 0.0 MPa and 1.0 MPa were plotted. It is clear that at small values of h0, because of the short distances between the ground surface and the plastic zone, excessive surface 38. ACTA GEOTECHNICA SLOVENICA, 2012/2 M. R. ZflRËIFflRD & A. FAHIMIFAR: A NEW SOLUTION FOR SHALLOW AND DEEP TUNNELS BY CONSIDERING THE GRAVITATIONAL LOADS Figure 12. Variations of radial displacement at r = h0 versus h0 . settlements occur. On the other hand, at large values of [2] h0, the surface settlement may increase with depth due to an increase in magnitude of the in-situ stresses. [3] Furthermore, in Fig. 12, it is clear that the displacements through the tunnel crown are larger than the tunnel wall, and the difference becomes greater for the wider plastic zones. [4] 8 CONCLUSIONS In order to examine the C.-C. method, commonly used for [5] the analysis of tunnels, and to demonstrate the effect of the classic assumptions on the characteristics of the ground-response curve, an analytical solution was proposed. [6] The solution is relatively simple, easy to use, and can readily indicate its sensitivity through a range of possible ground parameters, boundary conditions, and applied loads. [7] In this solution, the formulations were derived through horizontal and vertical directions, by taking the gravitational loads into account. It was shown that, for practical cases even for shallow tunnels, the convergence-confinement method is reasonably applicable. However, the [8] effect of gravitational loads in the analyses can be noticeable, and ignoring the plastic load can lead to large errors in the calculations. _ [9] REFERENCES [1] Brown, E.T., Bray, J.W., Ladanyi, B., Hoek, E. (1983). Ground response curves for rock tunnels. Journal of geotechnical Engineerin, Vol. 109, No. 1, pp. 15-39. Sharan, S.K. (2003). Elastic-brittle-plastic analysis of circular openings in Hoek-Brown media. Int. J. Rock Mech. Min. Sci., Vol. 40, No.6, pp. 817-824. Carranza-Torres, C., Fairhurst, C. (1999). The elasto-plastic response of underground excavations in rock masses that satisfy the Hoek-Brown failure criterion. Int. J. Rock Mech. Min. Sci., Vol. 36, No. 6, pp. 777-809. Park, K.-H., Kim, Y.-J. (2006). Analytical solution for a circular opening in, an elasto-brittle-plastic rock. Int. J. Rock Mech. Min. Sci., Vol. 43, pp. 616-622. Guan, Z., Jiang, Y., Tanabasi, Y., (2007). Ground reaction analyses in conventional tunneling excavation. Tunnelling and Underground Space Technology, Vol. 22, No. 2, pp. 230-237. Lee, Y.-K., Pietruszczak, S. (2008). A new numerical procedure for elasto-plastic analysis of a circular opening excavated in a strain-softening rock mass. Tunnelling and Underground Space Technology, Vol. 23, No. 5, pp. 588-599. Fahimifar, A., Zareifard, M.R. (2009). A theoretical solution for analysis of tunnels below groundwater considering the hydraulic- mechanical coupling. Tunnelling and Underground Space Technology, Vol. 24, No. 6, pp. 634-646. Carranza-Torres, C., Fairhurst, C. (2000). Application the convergence-confinement method of tunnel design to rock masses that satisfy the Hoek-Brown failure criterion. Tunnelling and Underground Space Technology, Vol. 16, No. 2, pp. 187-213. González-Nicieza C, Álvarez-Vigil A.E., Menéndez-Díaz A., González-Palacio C. (2008). Influence of the depth and shape of a tunnel in the application the convergence-confinement method. Tunnelling and Underground Space Technology, Vol. 23, No. 1, pp. 25-37. 38. ACTA GEOTECHNICA SLOVENICA, 2012/2 M. R. ZflRËIFflRD & A. FAHIMIFAR: A NEW SOLUTION FOR SHALLOW AND DEEP TUNNELS BY CONSIDERING THE GRAVITATIONAL LOADS a2 = [10] Lu A-Z., Xu G-s., Sun, F., Sun, W. (2010). Elasto- where: plastic analysis of a circular tunnel including the effect of the axial in situ stress. Int. J. Rock Mech. And Mining Sci., Vol. 47, No. 1, pp. 50-59. [11] Detournay, E. and Fairhurst, C. (1987). Two-dimensional elastoplastic analysis of along, cylindrical cavity under non-hydrostatic loading. Int. J. Rock Mech. Min. Sci. & Geomech. Abstr., Vol. 24, No. 4, pp. 197-211. [12] Reed MB. (1988) . The influence of out-of-plane stress on a plane strain problem in rock mechanics. Int. J. Numer Anal Meth Geomech., Vol. 12, No., pp 173-81. [13] Hoek E., Brown E.T. (1980). Underground excavations in rock. The Institute of Mining and Metallurgy, London [14] Carranza-Torres, C. and Fairhurst, C. (1997). On the stability of tunnels under gravity loading, with post-peak softening of the ground. Int. J. Rock Mech. Min. Sci., Vol. 34, No. 3-4, pp. 75.e1-75.e18. [15] Fahimifar, A., Zareifard, M.R., (2010). The elastoplastic response of circular tunnel considering gravity loads for two cases of plane strain and plane stress conditions. Geotechnical Chalenges in Megacites, Moscow. V. 2. [16] Kolymbas D. (2005). Tunnelling and Tunnel Mechanics. Springer-Verlag, Berlin Heidelberg (2) [17] Timoshenko S.P., and Goodier J.N. (1982). Theory of Elasticity. McGraw-Hill, New York [18] Alonso, E., Alejano, L.R., Varas, F., Fdez-Manin, G., Carranza-Torres, C. (2003). Ground response curves for rock masses exhibiting strain-softening behaviour. Int. J. Numer. Anal. Meth. Geomech., Vol. 27, No. 13, pp. 1153- 1185. [19] Hoek E., Brown E.T. (1980). Underground excavations in rock. The Institute of Mining and Metallurgy, London [20] Itasca. (2000). User manual for FLAC, Version 4.0. as: Itasca Consulting Group Inc.: Minnesota m mg = sg = g ( ( j) 2 sg( j)+ sg ( J) 2 A1 = FrRp (pJ - Pj-1 ) + 2mgsc (A2) (A3) PJ - PJ-1 P j + P j-1 Pj - Pj-1 P j + P j-1 —2 2 mg f Pj - Pj-1 P j + P j-1 (A4) (A5) + 2mg acFrRp (p j - p j-1 ) + 4 Sg a, (A6) A3 = 4mg sc "j rj P j-1 P j + P j-1 The failure criterion, i.e., Eq. (11), can then be used to calculate the corresponding values of or(j) as follows: se( j)= sr( j)+(mg ( j)sc sr( j)+ sg ( jf, (A7) ar(j) is a function of the plastic radius Rp for vertical direction, as observed in Eq. (A1); thus, the analyses are carried out alternately in a sequence of successive approximations, to achieve the appropriate convergence of the plastic radius Rp . Strain analysis The total strain rates e'g and e'r can be written in terms of the elastic (e'ge , e'rej and plastic (p, e'/^j components e' = e'e + e'rp e'e = e'e + e' (A8) APPENDIX A. STRESS AND STRAIN ANALYSES FOR THE PLASTIC ZONE Stress analysis The finite-difference method (FDM) is used to solve Eq. (17) for Oj and oqq) by selecting an annular element of the outer radius pj-1 and the inner radius pj shown in Fig. 6: 1 sr( j)= sr( j-1) + \ {12 + Vr( j-1)}2 (A1) Considering the Mohr-Coulomb type of plastic potential function, i.e., Eq. (8), the elimination of the plastic multiplier l from the flow rule, i.e., Eqs. (6) and (7), and using Eq. (15) gives the relation between the plastic parts of radial and circumferential strain rates: e'rp + Ky egp = 0 (A9) The finite-difference method (FDM) is used to solve the governing differential equation obtained from Eqs (23) and (A9) for e'0(p ) and e'r(p.) by using the annular elements shown in Fig. 6 as: 38. ACTA GEOTECHNICA SLOVENICA, 2012/2 M. R. ZAREIFARD & A. FAHIMIFAR: A NEW SOLUTION FOR SHALLOW AND DEEP TUNNELS BY CONSIDERING THE GRAVITATIONAL LOADS sq(Pi ) 2W(j) - ee(j-1)(3 + Ky(j) )Ir(j-i) + £e(j-)j(1 + KY(j) ) - 2W(j)P(j) -(3+K 1 + K Y j))P(j-1I (A10) Y(j) r(jI where: £r(j) - £q(Pi ) - £r( j)- £r( j-1) P( j)- P( j-1) 4( j)- 4( j-1) r( j)- r( j-1) KY(j) - KY( j) + Ky( j-1) W(j) - £i(j)+ Ky(j)e0(j) W( j)- W( j)+W( j-1) (A11) (A12) (A13) (A14) (A15) thus: eq( j)- ek j) (r( j)- r( j-1))+eo (j-1) (A16) In addition, the corresponding values of the strains £\j), £kj) , eH i) , £0( j) , er(j) and £r(j) are obtained using Eqs. (12), (13), (18), (A8), and (A9). Using Eqs. (18) and (23), the following boundary conditions are obtained for the plastic radius, where, p = p = 1: er(j-1)- er(j-1) e6(j-1) - ee(j-1) and ep. - 0 r(j-1) eP - eq(j-1) eq(j-1) er(j-1)- eq(j-1) (A17) (A18) (A19) nomenclature: E0 : Eg : ho : Fr : Fd : K : Ky : mhs(: mr,sr: mg,Sg: Deformability modulus of elastic rock mass Deformability modulus of plastic rock mass Depth of the tunnel Radial body force Circumferential body force Lateral stress coefficient Dilation factor Material constants for original rock mass. Material constants for residual rock mass Material constants for broken rock mass Surcharge load 1 r : Radial distance from the center of the tunnel rj : Tunnel radius Rp : Radius of plastic zone ur : Radial displacement (x,y,z): Cartesian coordinates Sag: induced circumferential stress Sar : induced radial stress Major Principal strain of rock mass Minor principal strain of rock mass Circumferential strain Radial strain Elastic circumferential strain Elastic radial strain Plastic circumferential strain Plastic radial strain Derivative of strain e with respect to r Derivative of strain e with respect to p Unit weight of rock mass Deviatoric plastic strain Critical deviatoric plastic strain Softening parameter. Plastic multiplier Poisson's ratio of rock mass Angle measured clockwise from the horizontal direction Normalized radius Major principal stress Minor principal stress Uniaxial compressive strength of intact rock Circumferential stress Radial stress Initial circumferential stress Initial radial stress Initial stress Fictitious time variable Dilation angle for original rock mass Dilation angle for residual rock mass Dilation angle for plastic rock mass Subscript e: Refers to elastic part Subscript p: Refers to plastic part eq ere eqP eP r e : e' : Y : Yp : Yp* n : l : Vr : 6 : P : al : ^3 : ac : 06 : ar : 060 0r0 0o : T : fi : fr : fg : 2 2 ACTA GeOTeCHNICA SLOVENICA, 2012/2 49 .