- Email: [email protected]
Upper bound limit analysis of collapse shape for circular tunnel subjected to pore pressure based on the Hoek–Brown failure criterion
Tunnelling and Underground Space Technology 26 (2011) 614–618
Contents lists available at ScienceDirect
Tunnelling and Underground Space Technology journal homepage: www.elsevier.com/locate/tust
Upper bound limit analysis of collapse shape for circular tunnel subjected to pore pressure based on the Hoek–Brown failure criterion F. Huang ⇑, X.L. Yang School of Civil Engineering, Central South University, Changsha, Hunan 410075, PR China
a r t i c l e
i n f o
Article history: Received 19 March 2010 Received in revised form 22 November 2010 Accepted 3 April 2011 Available online 6 May 2011 Keywords: Limit analysis Pore pressure Collapsing block Nonlinear failure criterion
a b s t r a c t Based on Hoek–Brown failure criterion and using the upper bound theorem of limit analysis, a numerical solution for the shape of collapsing block in circular tunnel subjected to pore pressure is derived. The effect of water pressure which is assumed to be a work rate of external force is included in the upper bound analysis. By employing variational calculation to minimize the objective function, the upper solution of collapsing block is obtained. In order to evaluate the validity of the method used in this paper, the result for pore pressure coefficient ru = 0, with no effect of pore pressure taken into account, is compared with previous work. The good agreement shows that the method of calculating the upper solution for the shape of collapsing block subjected to pore pressure is valid. Ó 2011 Elsevier Ltd. All rights reserved.
1. Introduction The stability of tunnel face for a tunnel excavated in urban area is a major problem in tunnel engineering, which has been studied by many scholars since limit analysis theory is introduced to estimate the lower and upper bound stability solutions by Davis et al. (1980). Compared with the traditional stability analysis techniques, slice method and limit equilibrium method, solutions derived from limit analysis are more rigorous and no assumption with respect to forces is required (Yang and Yin, 2006). Due to these advantages, limit analysis method is used widely to analyze the stability of the front of tunnels driven in shallow strata. Leca and Dormieux (1990) derived upper and lower bound solutions of critical retaining pressure applied to the tunnel face by constructing a kinematically admissible failure mechanism in the framework of the limit analysis theory. In order to obtain a precise upper bound solution, Leca and Dormieux (1990) developed three types of three-dimensional failure mechanisms which represented collapse failure and blow-out failure respectively, and the failure mechanisms are cited frequently by other authors (Li et al., 2009; Lee et al., 2004; Lee and Nam, 2001; Buhan et al., 1999). For the purpose of improving the upper and lower bounds to describe the stability of a range of heading sizes, Augarde et al. (2003) employed finite element limit analysis methods to derive rigorous bounds on load parameters. At present, the limit analysis theory in tunnel engineering is mainly applied to analyzing the stability of tunnel face excavated ⇑ Corresponding author. Tel.: +86 731 82656248. E-mail address: [email protected] (F. Huang). 0886-7798/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.tust.2011.04.002
in shallow strata by using linear Mohr–Coulomb criterion. However, the collapse of deep tunnel is a complex nonlinear evolution process, thus, the difference of mechanical characteristics between deep tunnel and shallow tunnel is significant. Consequently, the linear failure criterion is not suited to solve the problem of roof collapsing for a deep-buried tunnel. Since Hoek–Brown failure criterion is developed for estimating the inherent characteristics and nonlinear failure feature of tightly hard rock mass, this nonlinear failure criterion has been widely applied in a variety of geotechnical engineering (Yang et al., 2004; Jimenez et al., 2008). Based on the generalized Hoek–Brown failure criterion, Merifield et al. (2006) used limit theorems to evaluate the ultimate bearing capacity of a surface footing on a rock mass, and the actual collapsing load is bracketed precisely by computing both the upper and lower bound solutions for the bearing capacity. To investigate the possible collapse of a rectangular cavern roof, Fraldi and Guarracino (2009) derived the exact solution of detaching profile by means of the upper bound theorem of limit analysis and Hoek–Brown failure criterion. The effect of pore pressure should be included in the stability analysis for tunnels excavated in areas where the underground water table changes with season. When the limit analysis theory is used to analyze the stability of tunnels, with the effect of pore pressure taken into account, the main problem is how to calculate the rate of work done by pore pressure. The water pressure in the soil pores is regarded as an external force loaded on the soil skeleton by Michalowski (1995). In the light of this assumption, the work of water pressure can be expressed as a sum of pore pressure work on skeleton and the work of the water pressure on boundary. Due to its simplicity and validity, several authors (Viratjandr and
615
F. Huang, X.L. Yang / Tunnelling and Underground Space Technology 26 (2011) 614–618
Michalowski, 2006; Kim et al., 1997) used this assumption to analyze the stability of slope in the framework of limit analysis subjected to pore water pressure. Located in high ground stress environment, the deformation, looseness and collapse of rock mass over the roof of deep-buried tunnel are confined in a certain region. Furthermore, the collapsing block over the roof of the tunnel forms a ‘collapsing arch’ which bears the whole load originated from the weight of rock mass. As a result, the study of collapsing shape for deep-buried tunnel will contribute to understanding the failure property of surrounding rock mass and providing theoretical basis for developing a supporting structure of deep-buried tunnels. The purpose of this paper is to make use of the upper bound theorem of limit analysis to derive the formula for constructing an effective shape of the collapsing block, taking the effect of pore pressure into account in the medium subjected to Hoek–Brown failure criterion.
surrounding rock mass subjected to pore pressure
f(x)
L
σn
V collapsing block
g(x)
θ w
o
H
τ
x
circular tunnel cross section y Fig. 1. Collapsing pattern of rock mass for circular tunnel roof with pore pressure.
2. Pore pressure in upper bound limit analysis According to the upper bound theorem, when the velocity boundary condition is satisfied, the load derived by equating the external rate of work to the rate of the energy dissipation in any kinematically admissible velocity field is no less than the actual collapsing load (Michalowski, 1995). To achieve the effects of pore pressure in the framework of the upper bound theorem of limit analysis for slope stability, Viratjandr and Michalowski (2006) assumed that the work of water pressure is equal to the sum of pore pressure work on skeleton expansion and the work of the water pressure on boundary. Therefore, the upper bound theorem can be expressed as follows when the effect of pore pressure is taken into account.
Z v
rij e_ ij dV P
Z s
T i vi ds þ
Z v
X i vi dV
Z v
ue_ ij dV
Z
ni vi u ds
ð1Þ
s
where rij and e_ ij are the stress tensor and strain rate in the kinematically admissible velocity field respectively, Ti is a surcharge load on boundary s, X is the body force, V is the volume of the mechanism, vi is the velocity along the velocity discontinuity surface, ni is the unit vector and u is the pore pressure. Furthermore, some assumptions are required when the upper bound theorem is used to analyze the stability of geotechnical engineering. Firstly, the material is perfectly plastic and obeys an associated flow rule; secondly, the blocks bounded by the velocity discontinuity line and boundary surface are regarded as rigid materials (Yang and Zou, 2006). According to the assumption, the strain rate e_ ij in the blocks are equal to zero, which means only the last term on the right-hand side of Eq. (1) contributes to the effect of pore pressure.
4. Limit analysis of collapsing shape with Hoek–Brown failure criterion The widely used Hoek–Brown failure criterion is always represented in terms of the major and minor principal stresses. However, the relationship between normal and shear stresses can also be expressed by Hoek–Brown failure criterion, which is given by (Hoek and Brown, 1997)
s ¼ Arci
rn þ rtm rci
B ð2Þ
where rn is the normal stress, s is the shear stress, A and B are material constants, and rci rtm are the uniaxial compressive strength and the tensile strength of the rock mass respectively. According to the Hoek–Brown failure criterion represented in terms of normal and shear stresses and associated flow rule, the normal and shear stresses and strain on the detaching surface are calculated. Consequently, the energy dissipation rate which is determined by the internal forces on the detaching surface can be written as (Fraldi and Guarracino, 2009)
D ¼ rn e_ n þ sn c_ n n ov 1 1 0 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ rci ½ABf ðxÞ1B ð1 B1 Þ rtm t 1 þ f 0 ðxÞ2
ð3Þ
where e_ n and c_ n are normal and shear plastic strain rate respectively, f0 (x) is the first derivative of f(x), t is the thickness of the detaching surface. Due to the collapsing block is symmetrical with respect to the y-axis, the work rate of collapsing block produced by weight can be expressed as
3. Failure mode of rock mass for circular tunnel
Pc ¼ Constructing a kinematically admissible failure mechanism is the key factor for upper bound theorem of limit analysis. According to the actual mechanical characteristics of rock mass over the roof of a deep-buried tunnel, an arched detaching curve f(x) is used to describe the velocity discontinuity surface, which can be seen from Fig. 1. (Fraldi and Guarracino, 2010). Due to the slide between the collapsing block and surrounding rock mass, the plastic flow occurs along the velocity discontinuity surface. Thus, the energy dissipation rate along the surface can be calculated based on Hoek–Brown failure criterion and associated flow rule. By equating the energy dissipation rate to the rate of external work, the virtual work equation satisfying the velocity boundary condition is obtained. In order to determine the effective shape of the collapsing block in a limit state, the variational calculation is employed to minimize the objective function.
Z
L
c½f ðxÞ gðxÞv dx
ð4Þ
0
in which c is the dry unit weight of the rock mass, L is the half width of the collapsing block, and g(x) is the equation describing the circular tunnel profile which is given by
gðxÞ ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 b x2 b L2
ð5Þ
where b is the radius of the circular tunnel. Based on the study of Michalowski (1995), pore pressure is represented as
u ¼ r u cz
ð6Þ
where ru is the pore pressure coefficient, z is the vertical distance between the roof of the tunnel and the top of the collapsing block. According to Eq. (1), the pore pressure inducing work rate along the detaching surface is
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F. Huang, X.L. Yang / Tunnelling and Underground Space Technology 26 (2011) 614–618
Wu ¼
Z
ni vi u ds
ð7Þ
s
Z
s
1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi v r u c½f ðxÞ gðxÞ ds 1 þ f 0 ðxÞ2
0
¼
Z
v ru c½f ðxÞ gðxÞ dx
ð8Þ
0
As a result, the difference of the total energy dissipation rate and the total work rate done by external forces can be written as
Z
f½f ðxÞ; f 0 ðxÞ; x ¼
c½f ðxÞ gðxÞv dx
L
r u c½f ðxÞ gðxÞv dx
¼
1
AB
w½f ðxÞ; f 0 ðxÞ; x dx
ð9Þ
0
where w[f(x), f (x), x] is a functional which can be expressed as
n
1
rci ½ABf 0 ðxÞ1B ð1 B1 Þ rtm
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 ð1 þ r u Þc f ðxÞ b x2 þ b L2 v
@w d @w ¼0 @f ðxÞ dx @f 0 ðxÞ
ð10Þ
ð11Þ
By substituting Eq. (10) into Eq. (11), and integrating the result, the analytical expression of the detaching curve f(x) is obtained
1B 1 ð1 þ ru Þc B c0 B xþ H
rci
c
ð12Þ
in which c0 and H are integration constants. As the detaching curve f(x) is symmetrical with respect to the y axis, the integration constant c0 is equal to zero. Therefore, the final form of the detaching curve f(x) is 1
1B ð1 þ ru Þc B
rci
1
xB H
ð13Þ
Combining Eqs. (13) and (10), the functional w becomes
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 ð1 þ r u Þc H þ b x2 b L2 o B1 1 1 1 rtm B1 rciB AB ½ð1 þ r u ÞcB xB v
rci
1
LB H ¼ 0
ð16Þ
B1 1 1 1þB 1 r B AB ½ð1 þ ru ÞcB L B ¼ 0 B þ 1 ci
w½f ðxÞ; f 0 ðxÞ; x ¼
ð14Þ
The expression of f is obtained by calculating the integral of w along the interval [0, L]
ð17Þ
By substituting Eq. (16) into Eq. (17), a nonlinear equation with respect to parameter L is obtained
(
1
ð1 þ ru ÞcAB
According to the upper bound theorem, the effective shape of collapsing block in a limit state can be determined by minimizing the difference of the total energy dissipation rate and the rate of external work. Moreover, the expression, w, is a integral functional, which can be transformed into a differential equation by variational calculation. As a result, determining the extremum of w is in search of the boundary value of the corresponding differential equation. Based on variational principle, the necessary condition for searching the extremum of the functional w is given by
f ðxÞ ¼ AB
1B ð1 þ r u Þc B
L
0
w½f ðxÞ; f 0 ðxÞ; x ¼
ð15Þ
2 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3 2 2 ð1 þ ru Þcb 4 L L L 5 1 ½ð1 þ r u ÞcH rtm L þ arcsin b b b 2
L qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi D 1 þ f 0 ðxÞ2 ð1 þ r u Þc½f ðxÞ gðxÞ v dx
Z
B1 1 1 1þB 1 r B AB ½ð1 þ ru ÞcB L B B þ 1 ci
in which the parameter L and H are still undetermined. Nevertheless, another equation which includes L and H is derived by equating the external rate of work to the rate of the energy dissipation.
0
Z
0
w½f ðxÞ; f 0 ðxÞ; x dx
From Fig. 1 it can be found that f(x = L) = 0, thus, the Eq. (13) becomes
L
0
Z
¼
Z qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi D 1 þ f 0 ðxÞ2 v dx
L
0
1
L
¼ ½ð1 þ r u ÞcH rtm L 2 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3 2 2 ð1 þ r u Þcb 4 L L L 5 þ 1 arcsin b b b 2
L
f ðxÞ ¼ AB
Z 0
By substituting Eqs. (5) and (6) into Eq. (7), the explicit form of the work rate of the pore pressure is
Wu ¼
f¼
1B ð1 þ r u Þc B 2
2
rci
1
ð1 þ ru Þcb 4 L L þ arcsin b b 2
)
LB rtm L sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3 2 L 5 1 b
B1 1 1 1þB 1 r B AB ½ð1 þ ru ÞcB L B ¼ 0 B þ 1 ci
ð18Þ
Though the analytical solution of nonlinear equation cannot be found, the numerical solution of L is calculated easily with the help of numerical software. 5. Numerical result of collapsing shape for circular tunnel In order to investigate the influence of the different rock parameters and pore pressure coefficients on the shape of collapsing block for circular tunnels, the collapsing blocks subjected to pore pressure using Hoek–Brown failure criterion are drawn in Fig. 2. With reference to the work of Fraldi and Guarracino (2010), the rock mass parameter B varies from 3/4 to 1 and pore pressure coefficient ranges from 0 to 0.75, which is clearly illustrated in Fig. 2. It can be seen from the figure that, the width of the collapsing block decreases with the increasing values of B while its height tends to increase when the pore pressure coefficient is the same. In addition, when the parameter B is the same, the width and the height of the collapsing block both decrease with the increasing values of pore pressure coefficient. For the condition of ru = 0, the influence of pore pressure is not taken into account on the stability analysis and the shape of collapsing block is in complete accord with the result of Fraldi and Guarracino (2010), which means the method of calculating pore pressure used in this paper is valid and effective. To study the changing regularity of collapsing mechanism with the effect of pore pressure, the collapsing blocks of rock parameters corresponding to A = 2/3, rci = 2.5 MPa, rtm = rci/100, c = 25 KN/m3, B = 3/4 with tunnel radius varying from 3.75 to 7.5 m are illustrated in Fig. 3, while the pore pressure coefficient ru is equal to 0, 0.25 and 0.5 respectively. As is shown in Fig. 3, both the width and the height of the collapsing block increase with the increase
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F. Huang, X.L. Yang / Tunnelling and Underground Space Technology 26 (2011) 614–618
6
6 ru=0
ru=0 ru=0.25
5
ru=0.25
5
ru=0.5
ru=0.5 ru=0.75
4
h/m
h/m
B=3/4 3
2
1
1
-3
-2
-1
0 L/m
1
2
3
0 -4
4
6
B=5/6
3
2
0 -4
ru=0.75
4
-3
-2
-1
0 L/m
1
2
ru=0
ru=0.25
5
ru=0.25
5
ru=0.5
ru=0.5 ru=0.75
4
ru=0.75
4
B=1
h/m
B=4/5
h/m
4
6 ru=0
3
3
2
2
1
1
0 -4
3
-3
-2
-1
0 L/m
1
2
3
4
0 -4
-3
-2
-1
0 L/m
1
2
3
4
Fig. 2. Shape of collapsing blocks of circular tunnel roof for different rock parameters and pore pressure coefficients (A = 2/3, rci = 2.5 MPa, rtm = rci/100, c = 25 KN/m3, b = 3.75 m).
of tunnel radius whether the effect of pore pressure is taken into account or not. Similar to the preceding conclusion, the width and the height of the collapsing block decrease with the increasing values of pore pressure coefficient when tunnel radius is at a certain value.
11 10
ru=0
9
ru=0.25
8
ru=0.5
h/m
7
6. Conclusions
6 5 4 3 2 1 0
-8
-6
-4
-2
0 L/m
2
4
6
8
Fig. 3. Shape of collapsing blocks for tunnel radius varying from 3.75 m to 7.5 m subjected to pore pressure.
Based on Hoek–Brown failure criterion, a numerical solution for the shape of collapsing block in circular tunnel subjected to pore pressure is obtained in the framework of the upper bound theorem of limit analysis. The effect of water pressure which is regarded as a work rate of external force is included in the upper bound theorem. In order to derive the upper solution of collapsing block, variational calculation is employed to minimize the objective function. For the condition that the effect of water pressure is not taken into account, the solution in this paper is in complete accord with the result of Fraldi and Guarracino (2010), which means that the method of calculating the shape of collapsing block subjected to pore pressure is valid. This paper has extended the work of Fraldi and Guarracino (2010) who did not consider the effect of pore water on the
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F. Huang, X.L. Yang / Tunnelling and Underground Space Technology 26 (2011) 614–618
prediction of collapsing mechanism. According to the discussion above, some conclusions can be drawn: (1) The different rock parameters have significant influence on the collapsing shape of rock mass in circular tunnels. It is found that the width of the collapsing block decreases with the increasing values of rock mass parameter B. However, its height tends to increase with the increase of B when the pore pressure coefficient is the same. (2) The work rate of pore pressure which is incorporated into the upper bound theorem of limit analysis is measured by pore pressure coefficient ru. According to the changing regularity of the collapsing shapes for different pore pressure coefficients, both the width and height of the collapsing block decrease with the increasing values of pore pressure coefficient. (3) By studying the effect of different tunnel radius on the collapsing shape of circular tunnel, it is found that both the width and the height of the collapsing block increase with the increase of tunnel radius whether the effect of pore pressure is taken into account or not.
Acknowledgments This paper were supported by the Hunan Provincial Postgraduate Innovation Project (No. CX2009B043) and Doctoral Dissertation Innovation Project of Central South University (No. 2010bsxt07). The financial supports are greatly appreciated. References Augarde, C.E., Lyamin, A.V., Sloan, S.W., 2003. Stability of an undrained plane strain heading revisited. Comput. Geotech. 30 (5), 419–430.
Buhan, P., Cuvilier, A., Dormieux, L., Maghous, S., 1999. Face stability of shallow circular tunnels driven under the water table: a numerical analysis. Int. J. Numer. Anal. Methods. Geomech. 23 (1), 79–95. Davis, E.H., Dunn, M.J., Mair, R.J., Seneviratne, H.N., 1980. The stability of shallow tunnels and underground openings in cohesive material. Geotechnique 30 (4), 397–416. Fraldi, M., Guarracino, F., 2009. Limit analysis of collapse mechanisms in cavities and tunnels according to the Hoek–Brown failure criterion. Int. J. Rock. Mech. Min. Sci. 46 (4), 665–673. Fraldi, M., Guarracino, F., 2010. Analytical solutions for collapse mechanisms in tunnels with arbitrary cross sections. Int. J. Solids Struct. 47 (2), 216–223. Hoek, E., Brown, E.T., 1997. Practical estimate the rock mass strength. Int. J. Rock. Mech. Min. Sci. 34 (8), 1165–1186. Jimenez, R., Serrano, A., Olalla, C., 2008. Linearization of the Hoek and Brown rock failure criterion for tunnelling in elasto–plastic rock masses. Int. J. Rock. Mech. Min. Sci. 45 (7), 1153–1163. Kim, J., Salgado, R., Yu, H.S., 1997. Limit analysis of soil slopes subjected to porewater pressure. J. Geotech. Geoenviron. Eng. ASCE 125 (1), 49–58. Leca, E., Dormieux, L., 1990. Upper and lower bound solutions for the face stability of shallow circular tunnels in frictional material. Geotechnique 40 (4), 581–606. Lee, I.M., Nam, S.W., 2001. The study of seepage forces acting on the tunnel lining and tunnel face in shallow tunnels. Tunnel. Underground Space Technol. 16 (1), 31–40. Lee, I.M., Lee, J.S., Nam, S.W., 2004. Effect of seepage force on tunnel face stability reinforced with multi-step pipe grouting. Tunnel. Underground Space Technol. 19 (6), 551–565. Li, Y., Emeriault, F., Kastner, R., Zhang, Z.X., 2009. Stability analysis of large slurry shield-driven tunnel in soft clay. Tunnel. Underground Space Technol. 24 (4), 472–481. Merifield, R.S., Lyamin, A.V., Sloan, S.W., 2006. Limit analysis solutions for the bearing capacity of rock masses using the generalised Hoek–Brown criterion. Int. J. Rock. Mech. Min. Sci. 43 (6), 920–937. Michalowski, R.L., 1995. Slope stability analysis: a kinematical approach. Geotechnique 45 (2), 283–293. Viratjandr, C., Michalowski, R.L., 2006. Limit analysis of submerged slopes subjected to water drawdown. Can. Geotech. J. 43 (8), 802–814. Yang, X.L., Yin, J.H., 2006. Estimation of seismic passive earth pressures with nonlinear failure criterion. Eng. Struct. 28 (3), 342–348. Yang, X.L., Zou, J.F., 2006. Stability factors for rock slopes subjected to pore water pressure based on the Hoek–Brown failure criterion. Int. J. Rock. Mech. Min. Sci. 43 (7), 1146–1152. Yang, X.L., Li, L., Yin, J.H., 2004. Stability analysis of rock slopes with a modified Hoek–Brown failure criterion. Int. J. Numer. Anal. Methods Geomech. 28 (2), 181–190.
Contents lists available at ScienceDirect
Tunnelling and Underground Space Technology journal homepage: www.elsevier.com/locate/tust
Upper bound limit analysis of collapse shape for circular tunnel subjected to pore pressure based on the Hoek–Brown failure criterion F. Huang ⇑, X.L. Yang School of Civil Engineering, Central South University, Changsha, Hunan 410075, PR China
a r t i c l e
i n f o
Article history: Received 19 March 2010 Received in revised form 22 November 2010 Accepted 3 April 2011 Available online 6 May 2011 Keywords: Limit analysis Pore pressure Collapsing block Nonlinear failure criterion
a b s t r a c t Based on Hoek–Brown failure criterion and using the upper bound theorem of limit analysis, a numerical solution for the shape of collapsing block in circular tunnel subjected to pore pressure is derived. The effect of water pressure which is assumed to be a work rate of external force is included in the upper bound analysis. By employing variational calculation to minimize the objective function, the upper solution of collapsing block is obtained. In order to evaluate the validity of the method used in this paper, the result for pore pressure coefficient ru = 0, with no effect of pore pressure taken into account, is compared with previous work. The good agreement shows that the method of calculating the upper solution for the shape of collapsing block subjected to pore pressure is valid. Ó 2011 Elsevier Ltd. All rights reserved.
1. Introduction The stability of tunnel face for a tunnel excavated in urban area is a major problem in tunnel engineering, which has been studied by many scholars since limit analysis theory is introduced to estimate the lower and upper bound stability solutions by Davis et al. (1980). Compared with the traditional stability analysis techniques, slice method and limit equilibrium method, solutions derived from limit analysis are more rigorous and no assumption with respect to forces is required (Yang and Yin, 2006). Due to these advantages, limit analysis method is used widely to analyze the stability of the front of tunnels driven in shallow strata. Leca and Dormieux (1990) derived upper and lower bound solutions of critical retaining pressure applied to the tunnel face by constructing a kinematically admissible failure mechanism in the framework of the limit analysis theory. In order to obtain a precise upper bound solution, Leca and Dormieux (1990) developed three types of three-dimensional failure mechanisms which represented collapse failure and blow-out failure respectively, and the failure mechanisms are cited frequently by other authors (Li et al., 2009; Lee et al., 2004; Lee and Nam, 2001; Buhan et al., 1999). For the purpose of improving the upper and lower bounds to describe the stability of a range of heading sizes, Augarde et al. (2003) employed finite element limit analysis methods to derive rigorous bounds on load parameters. At present, the limit analysis theory in tunnel engineering is mainly applied to analyzing the stability of tunnel face excavated ⇑ Corresponding author. Tel.: +86 731 82656248. E-mail address: [email protected] (F. Huang). 0886-7798/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.tust.2011.04.002
in shallow strata by using linear Mohr–Coulomb criterion. However, the collapse of deep tunnel is a complex nonlinear evolution process, thus, the difference of mechanical characteristics between deep tunnel and shallow tunnel is significant. Consequently, the linear failure criterion is not suited to solve the problem of roof collapsing for a deep-buried tunnel. Since Hoek–Brown failure criterion is developed for estimating the inherent characteristics and nonlinear failure feature of tightly hard rock mass, this nonlinear failure criterion has been widely applied in a variety of geotechnical engineering (Yang et al., 2004; Jimenez et al., 2008). Based on the generalized Hoek–Brown failure criterion, Merifield et al. (2006) used limit theorems to evaluate the ultimate bearing capacity of a surface footing on a rock mass, and the actual collapsing load is bracketed precisely by computing both the upper and lower bound solutions for the bearing capacity. To investigate the possible collapse of a rectangular cavern roof, Fraldi and Guarracino (2009) derived the exact solution of detaching profile by means of the upper bound theorem of limit analysis and Hoek–Brown failure criterion. The effect of pore pressure should be included in the stability analysis for tunnels excavated in areas where the underground water table changes with season. When the limit analysis theory is used to analyze the stability of tunnels, with the effect of pore pressure taken into account, the main problem is how to calculate the rate of work done by pore pressure. The water pressure in the soil pores is regarded as an external force loaded on the soil skeleton by Michalowski (1995). In the light of this assumption, the work of water pressure can be expressed as a sum of pore pressure work on skeleton and the work of the water pressure on boundary. Due to its simplicity and validity, several authors (Viratjandr and
615
F. Huang, X.L. Yang / Tunnelling and Underground Space Technology 26 (2011) 614–618
Michalowski, 2006; Kim et al., 1997) used this assumption to analyze the stability of slope in the framework of limit analysis subjected to pore water pressure. Located in high ground stress environment, the deformation, looseness and collapse of rock mass over the roof of deep-buried tunnel are confined in a certain region. Furthermore, the collapsing block over the roof of the tunnel forms a ‘collapsing arch’ which bears the whole load originated from the weight of rock mass. As a result, the study of collapsing shape for deep-buried tunnel will contribute to understanding the failure property of surrounding rock mass and providing theoretical basis for developing a supporting structure of deep-buried tunnels. The purpose of this paper is to make use of the upper bound theorem of limit analysis to derive the formula for constructing an effective shape of the collapsing block, taking the effect of pore pressure into account in the medium subjected to Hoek–Brown failure criterion.
surrounding rock mass subjected to pore pressure
f(x)
L
σn
V collapsing block
g(x)
θ w
o
H
τ
x
circular tunnel cross section y Fig. 1. Collapsing pattern of rock mass for circular tunnel roof with pore pressure.
2. Pore pressure in upper bound limit analysis According to the upper bound theorem, when the velocity boundary condition is satisfied, the load derived by equating the external rate of work to the rate of the energy dissipation in any kinematically admissible velocity field is no less than the actual collapsing load (Michalowski, 1995). To achieve the effects of pore pressure in the framework of the upper bound theorem of limit analysis for slope stability, Viratjandr and Michalowski (2006) assumed that the work of water pressure is equal to the sum of pore pressure work on skeleton expansion and the work of the water pressure on boundary. Therefore, the upper bound theorem can be expressed as follows when the effect of pore pressure is taken into account.
Z v
rij e_ ij dV P
Z s
T i vi ds þ
Z v
X i vi dV
Z v
ue_ ij dV
Z
ni vi u ds
ð1Þ
s
where rij and e_ ij are the stress tensor and strain rate in the kinematically admissible velocity field respectively, Ti is a surcharge load on boundary s, X is the body force, V is the volume of the mechanism, vi is the velocity along the velocity discontinuity surface, ni is the unit vector and u is the pore pressure. Furthermore, some assumptions are required when the upper bound theorem is used to analyze the stability of geotechnical engineering. Firstly, the material is perfectly plastic and obeys an associated flow rule; secondly, the blocks bounded by the velocity discontinuity line and boundary surface are regarded as rigid materials (Yang and Zou, 2006). According to the assumption, the strain rate e_ ij in the blocks are equal to zero, which means only the last term on the right-hand side of Eq. (1) contributes to the effect of pore pressure.
4. Limit analysis of collapsing shape with Hoek–Brown failure criterion The widely used Hoek–Brown failure criterion is always represented in terms of the major and minor principal stresses. However, the relationship between normal and shear stresses can also be expressed by Hoek–Brown failure criterion, which is given by (Hoek and Brown, 1997)
s ¼ Arci
rn þ rtm rci
B ð2Þ
where rn is the normal stress, s is the shear stress, A and B are material constants, and rci rtm are the uniaxial compressive strength and the tensile strength of the rock mass respectively. According to the Hoek–Brown failure criterion represented in terms of normal and shear stresses and associated flow rule, the normal and shear stresses and strain on the detaching surface are calculated. Consequently, the energy dissipation rate which is determined by the internal forces on the detaching surface can be written as (Fraldi and Guarracino, 2009)
D ¼ rn e_ n þ sn c_ n n ov 1 1 0 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ rci ½ABf ðxÞ1B ð1 B1 Þ rtm t 1 þ f 0 ðxÞ2
ð3Þ
where e_ n and c_ n are normal and shear plastic strain rate respectively, f0 (x) is the first derivative of f(x), t is the thickness of the detaching surface. Due to the collapsing block is symmetrical with respect to the y-axis, the work rate of collapsing block produced by weight can be expressed as
3. Failure mode of rock mass for circular tunnel
Pc ¼ Constructing a kinematically admissible failure mechanism is the key factor for upper bound theorem of limit analysis. According to the actual mechanical characteristics of rock mass over the roof of a deep-buried tunnel, an arched detaching curve f(x) is used to describe the velocity discontinuity surface, which can be seen from Fig. 1. (Fraldi and Guarracino, 2010). Due to the slide between the collapsing block and surrounding rock mass, the plastic flow occurs along the velocity discontinuity surface. Thus, the energy dissipation rate along the surface can be calculated based on Hoek–Brown failure criterion and associated flow rule. By equating the energy dissipation rate to the rate of external work, the virtual work equation satisfying the velocity boundary condition is obtained. In order to determine the effective shape of the collapsing block in a limit state, the variational calculation is employed to minimize the objective function.
Z
L
c½f ðxÞ gðxÞv dx
ð4Þ
0
in which c is the dry unit weight of the rock mass, L is the half width of the collapsing block, and g(x) is the equation describing the circular tunnel profile which is given by
gðxÞ ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 b x2 b L2
ð5Þ
where b is the radius of the circular tunnel. Based on the study of Michalowski (1995), pore pressure is represented as
u ¼ r u cz
ð6Þ
where ru is the pore pressure coefficient, z is the vertical distance between the roof of the tunnel and the top of the collapsing block. According to Eq. (1), the pore pressure inducing work rate along the detaching surface is
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Wu ¼
Z
ni vi u ds
ð7Þ
s
Z
s
1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi v r u c½f ðxÞ gðxÞ ds 1 þ f 0 ðxÞ2
0
¼
Z
v ru c½f ðxÞ gðxÞ dx
ð8Þ
0
As a result, the difference of the total energy dissipation rate and the total work rate done by external forces can be written as
Z
f½f ðxÞ; f 0 ðxÞ; x ¼
c½f ðxÞ gðxÞv dx
L
r u c½f ðxÞ gðxÞv dx
¼
1
AB
w½f ðxÞ; f 0 ðxÞ; x dx
ð9Þ
0
where w[f(x), f (x), x] is a functional which can be expressed as
n
1
rci ½ABf 0 ðxÞ1B ð1 B1 Þ rtm
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 ð1 þ r u Þc f ðxÞ b x2 þ b L2 v
@w d @w ¼0 @f ðxÞ dx @f 0 ðxÞ
ð10Þ
ð11Þ
By substituting Eq. (10) into Eq. (11), and integrating the result, the analytical expression of the detaching curve f(x) is obtained
1B 1 ð1 þ ru Þc B c0 B xþ H
rci
c
ð12Þ
in which c0 and H are integration constants. As the detaching curve f(x) is symmetrical with respect to the y axis, the integration constant c0 is equal to zero. Therefore, the final form of the detaching curve f(x) is 1
1B ð1 þ ru Þc B
rci
1
xB H
ð13Þ
Combining Eqs. (13) and (10), the functional w becomes
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 ð1 þ r u Þc H þ b x2 b L2 o B1 1 1 1 rtm B1 rciB AB ½ð1 þ r u ÞcB xB v
rci
1
LB H ¼ 0
ð16Þ
B1 1 1 1þB 1 r B AB ½ð1 þ ru ÞcB L B ¼ 0 B þ 1 ci
w½f ðxÞ; f 0 ðxÞ; x ¼
ð14Þ
The expression of f is obtained by calculating the integral of w along the interval [0, L]
ð17Þ
By substituting Eq. (16) into Eq. (17), a nonlinear equation with respect to parameter L is obtained
(
1
ð1 þ ru ÞcAB
According to the upper bound theorem, the effective shape of collapsing block in a limit state can be determined by minimizing the difference of the total energy dissipation rate and the rate of external work. Moreover, the expression, w, is a integral functional, which can be transformed into a differential equation by variational calculation. As a result, determining the extremum of w is in search of the boundary value of the corresponding differential equation. Based on variational principle, the necessary condition for searching the extremum of the functional w is given by
f ðxÞ ¼ AB
1B ð1 þ r u Þc B
L
0
w½f ðxÞ; f 0 ðxÞ; x ¼
ð15Þ
2 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3 2 2 ð1 þ ru Þcb 4 L L L 5 1 ½ð1 þ r u ÞcH rtm L þ arcsin b b b 2
L qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi D 1 þ f 0 ðxÞ2 ð1 þ r u Þc½f ðxÞ gðxÞ v dx
Z
B1 1 1 1þB 1 r B AB ½ð1 þ ru ÞcB L B B þ 1 ci
in which the parameter L and H are still undetermined. Nevertheless, another equation which includes L and H is derived by equating the external rate of work to the rate of the energy dissipation.
0
Z
0
w½f ðxÞ; f 0 ðxÞ; x dx
From Fig. 1 it can be found that f(x = L) = 0, thus, the Eq. (13) becomes
L
0
Z
¼
Z qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi D 1 þ f 0 ðxÞ2 v dx
L
0
1
L
¼ ½ð1 þ r u ÞcH rtm L 2 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3 2 2 ð1 þ r u Þcb 4 L L L 5 þ 1 arcsin b b b 2
L
f ðxÞ ¼ AB
Z 0
By substituting Eqs. (5) and (6) into Eq. (7), the explicit form of the work rate of the pore pressure is
Wu ¼
f¼
1B ð1 þ r u Þc B 2
2
rci
1
ð1 þ ru Þcb 4 L L þ arcsin b b 2
)
LB rtm L sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3 2 L 5 1 b
B1 1 1 1þB 1 r B AB ½ð1 þ ru ÞcB L B ¼ 0 B þ 1 ci
ð18Þ
Though the analytical solution of nonlinear equation cannot be found, the numerical solution of L is calculated easily with the help of numerical software. 5. Numerical result of collapsing shape for circular tunnel In order to investigate the influence of the different rock parameters and pore pressure coefficients on the shape of collapsing block for circular tunnels, the collapsing blocks subjected to pore pressure using Hoek–Brown failure criterion are drawn in Fig. 2. With reference to the work of Fraldi and Guarracino (2010), the rock mass parameter B varies from 3/4 to 1 and pore pressure coefficient ranges from 0 to 0.75, which is clearly illustrated in Fig. 2. It can be seen from the figure that, the width of the collapsing block decreases with the increasing values of B while its height tends to increase when the pore pressure coefficient is the same. In addition, when the parameter B is the same, the width and the height of the collapsing block both decrease with the increasing values of pore pressure coefficient. For the condition of ru = 0, the influence of pore pressure is not taken into account on the stability analysis and the shape of collapsing block is in complete accord with the result of Fraldi and Guarracino (2010), which means the method of calculating pore pressure used in this paper is valid and effective. To study the changing regularity of collapsing mechanism with the effect of pore pressure, the collapsing blocks of rock parameters corresponding to A = 2/3, rci = 2.5 MPa, rtm = rci/100, c = 25 KN/m3, B = 3/4 with tunnel radius varying from 3.75 to 7.5 m are illustrated in Fig. 3, while the pore pressure coefficient ru is equal to 0, 0.25 and 0.5 respectively. As is shown in Fig. 3, both the width and the height of the collapsing block increase with the increase
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6
6 ru=0
ru=0 ru=0.25
5
ru=0.25
5
ru=0.5
ru=0.5 ru=0.75
4
h/m
h/m
B=3/4 3
2
1
1
-3
-2
-1
0 L/m
1
2
3
0 -4
4
6
B=5/6
3
2
0 -4
ru=0.75
4
-3
-2
-1
0 L/m
1
2
ru=0
ru=0.25
5
ru=0.25
5
ru=0.5
ru=0.5 ru=0.75
4
ru=0.75
4
B=1
h/m
B=4/5
h/m
4
6 ru=0
3
3
2
2
1
1
0 -4
3
-3
-2
-1
0 L/m
1
2
3
4
0 -4
-3
-2
-1
0 L/m
1
2
3
4
Fig. 2. Shape of collapsing blocks of circular tunnel roof for different rock parameters and pore pressure coefficients (A = 2/3, rci = 2.5 MPa, rtm = rci/100, c = 25 KN/m3, b = 3.75 m).
of tunnel radius whether the effect of pore pressure is taken into account or not. Similar to the preceding conclusion, the width and the height of the collapsing block decrease with the increasing values of pore pressure coefficient when tunnel radius is at a certain value.
11 10
ru=0
9
ru=0.25
8
ru=0.5
h/m
7
6. Conclusions
6 5 4 3 2 1 0
-8
-6
-4
-2
0 L/m
2
4
6
8
Fig. 3. Shape of collapsing blocks for tunnel radius varying from 3.75 m to 7.5 m subjected to pore pressure.
Based on Hoek–Brown failure criterion, a numerical solution for the shape of collapsing block in circular tunnel subjected to pore pressure is obtained in the framework of the upper bound theorem of limit analysis. The effect of water pressure which is regarded as a work rate of external force is included in the upper bound theorem. In order to derive the upper solution of collapsing block, variational calculation is employed to minimize the objective function. For the condition that the effect of water pressure is not taken into account, the solution in this paper is in complete accord with the result of Fraldi and Guarracino (2010), which means that the method of calculating the shape of collapsing block subjected to pore pressure is valid. This paper has extended the work of Fraldi and Guarracino (2010) who did not consider the effect of pore water on the
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prediction of collapsing mechanism. According to the discussion above, some conclusions can be drawn: (1) The different rock parameters have significant influence on the collapsing shape of rock mass in circular tunnels. It is found that the width of the collapsing block decreases with the increasing values of rock mass parameter B. However, its height tends to increase with the increase of B when the pore pressure coefficient is the same. (2) The work rate of pore pressure which is incorporated into the upper bound theorem of limit analysis is measured by pore pressure coefficient ru. According to the changing regularity of the collapsing shapes for different pore pressure coefficients, both the width and height of the collapsing block decrease with the increasing values of pore pressure coefficient. (3) By studying the effect of different tunnel radius on the collapsing shape of circular tunnel, it is found that both the width and the height of the collapsing block increase with the increase of tunnel radius whether the effect of pore pressure is taken into account or not.
Acknowledgments This paper were supported by the Hunan Provincial Postgraduate Innovation Project (No. CX2009B043) and Doctoral Dissertation Innovation Project of Central South University (No. 2010bsxt07). The financial supports are greatly appreciated. References Augarde, C.E., Lyamin, A.V., Sloan, S.W., 2003. Stability of an undrained plane strain heading revisited. Comput. Geotech. 30 (5), 419–430.
Buhan, P., Cuvilier, A., Dormieux, L., Maghous, S., 1999. Face stability of shallow circular tunnels driven under the water table: a numerical analysis. Int. J. Numer. Anal. Methods. Geomech. 23 (1), 79–95. Davis, E.H., Dunn, M.J., Mair, R.J., Seneviratne, H.N., 1980. The stability of shallow tunnels and underground openings in cohesive material. Geotechnique 30 (4), 397–416. Fraldi, M., Guarracino, F., 2009. Limit analysis of collapse mechanisms in cavities and tunnels according to the Hoek–Brown failure criterion. Int. J. Rock. Mech. Min. Sci. 46 (4), 665–673. Fraldi, M., Guarracino, F., 2010. Analytical solutions for collapse mechanisms in tunnels with arbitrary cross sections. Int. J. Solids Struct. 47 (2), 216–223. Hoek, E., Brown, E.T., 1997. Practical estimate the rock mass strength. Int. J. Rock. Mech. Min. Sci. 34 (8), 1165–1186. Jimenez, R., Serrano, A., Olalla, C., 2008. Linearization of the Hoek and Brown rock failure criterion for tunnelling in elasto–plastic rock masses. Int. J. Rock. Mech. Min. Sci. 45 (7), 1153–1163. Kim, J., Salgado, R., Yu, H.S., 1997. Limit analysis of soil slopes subjected to porewater pressure. J. Geotech. Geoenviron. Eng. ASCE 125 (1), 49–58. Leca, E., Dormieux, L., 1990. Upper and lower bound solutions for the face stability of shallow circular tunnels in frictional material. Geotechnique 40 (4), 581–606. Lee, I.M., Nam, S.W., 2001. The study of seepage forces acting on the tunnel lining and tunnel face in shallow tunnels. Tunnel. Underground Space Technol. 16 (1), 31–40. Lee, I.M., Lee, J.S., Nam, S.W., 2004. Effect of seepage force on tunnel face stability reinforced with multi-step pipe grouting. Tunnel. Underground Space Technol. 19 (6), 551–565. Li, Y., Emeriault, F., Kastner, R., Zhang, Z.X., 2009. Stability analysis of large slurry shield-driven tunnel in soft clay. Tunnel. Underground Space Technol. 24 (4), 472–481. Merifield, R.S., Lyamin, A.V., Sloan, S.W., 2006. Limit analysis solutions for the bearing capacity of rock masses using the generalised Hoek–Brown criterion. Int. J. Rock. Mech. Min. Sci. 43 (6), 920–937. Michalowski, R.L., 1995. Slope stability analysis: a kinematical approach. Geotechnique 45 (2), 283–293. Viratjandr, C., Michalowski, R.L., 2006. Limit analysis of submerged slopes subjected to water drawdown. Can. Geotech. J. 43 (8), 802–814. Yang, X.L., Yin, J.H., 2006. Estimation of seismic passive earth pressures with nonlinear failure criterion. Eng. Struct. 28 (3), 342–348. Yang, X.L., Zou, J.F., 2006. Stability factors for rock slopes subjected to pore water pressure based on the Hoek–Brown failure criterion. Int. J. Rock. Mech. Min. Sci. 43 (7), 1146–1152. Yang, X.L., Li, L., Yin, J.H., 2004. Stability analysis of rock slopes with a modified Hoek–Brown failure criterion. Int. J. Numer. Anal. Methods Geomech. 28 (2), 181–190.