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Tunnel Design Using Continuum and Discontinuum Approaches and the Effect of Joint Orientation on the Design
Massoud Palassi Department of Civil Engineering, University of Tehran, Iran and Pooyan Asadollahi Department of Civil, Architectural and Environmental Engineering,
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ABSTRACT
In this paper, using continuum and discontinuum approaches, numerical analyses are conducted for the design of Gavoshan tunnel and the results are compared with each other. Also, in order to investigate the effect of joint orientation on the tunnel design using different approaches, joints with various orientations are analyzed using continuum and discontinuum methods.
This research shows that using continuum equivalent media for simulating the behavior of jointed rock mass may result in unsafe predictions and the use of discontinuum solutions are recommended for the proper design of tunnels in the jointed rock.
Keywords: tunnel, jointed rock, numerical simulation, distinct element method
INTRODUCTION
Gavoshan tunnel is a water conveyance tunnel excavated through Hajikesh mountain located in the west of Iran. In this research, continuum and discontinuum numerical simulations are performed for this tunnel and the effect of joint orientation on the stresses and deformations is investigated by these approaches.
Numerical methods for the stress analysis can be classified into two main categories: (1) Domain or differential methods, consisting of the finite element, finite difference and distinct element methods. (2) Integral or boundary methods represented by the several versions of the boundary element methods. The appropriate numerical method for elastic and elasto-plastic analyses of closely jointed rock masses is Distinct Element Method (Brady 1992).
In this paper, closely jointed rock mass is modelled using the distinct element method as an appropriate numerical approach. UDEC (The Universal Distinct Element Code) is used for conducting the analyses for both continuum and discontinuum simulations.
ANALYZED TUNNEL AND ROCK MASS PROPERTIES
Gavoshan project, including Gavoshan dam and its 20.2km conveyance tunnel, is located in the west of Iran and will provide irrigation and drinking water for the north of Kermanshah province and the south of Kordestan province. Gavoshan tunnel will convey water gravitationally from Gavoshan dam to Kermanshah city and its surrounding areas.
In this paper, the mechanical properties of Gavoshan tunnel from km 10+500 to km 13+830 are used in the analyses. This part of the tunnel, with a diameter of 6m, is excavated by an open Tunnel Boring Machine (TBM). The tunnel has northwest-southeast direction and the maximum depth of the tunnel is about 500m. The rock mass is subjected to the hydrostatic in-situ stresses (k=1).
Geology of the Region
Moving from km 10+500 to km 13+830 of Gavoshan tunnel, one can observe diabase, gabbro, peridotite, and amphibolite. In this region, there are several major faults, including Bovaneh and Yakhte Khan faults, and several minor and random faults.
The rock mass in the region has four major discontinuities which their dips and strike angles as well as their length and spacing are summarized in Table 1.
Table 1. Joint physical characteristics
Intact Rock Properties
To evaluate the rock mass properties, two boreholes were drilled at km 10+803 and 13+190. The value of the RQD was evaluated from detailed study of the core logs. The average value for RQD index was 54%. Mechanical and physical properties of the intact rocks were evaluated from laboratory tests performed on several samples obtained from the boreholes. The specific gravity, porosity, water absorption, water content, uniaxial compressive strength, tensile strength, modulus of elasticity, and Poisson’s ratio were determined from these tests (Table 2). The uniaxial compressive strength of the specimens was measured using specimens with natural water content. The tensile strength of the intact rock was evaluated using the Brazilian test. The Poisson’s ratios mentioned in Table 2 were determined at 50% of failure stress.
Table 2. Mechanical and physical properties of intact rocks
1 UCS = Uniaxial compressive strength; 2 UTS= Uniaxial tensile strength;
3 E= Young modulus; 4 n= Possion's ratio
The intact rock Mohr-Coulomb strength parameters were determined using the relationships proposed by Hoek et al. (2002), in which a=0.5 and s=1. For the depth of Gavoshan tunnel, the equations suggest intact rock cohesion of 10.7MPa and friction angle of 44°.
Mechanical Properties of Equivalent Continuum Rock Media
Proper estimation of the strength and deformability characteristics of the rock mass is essential in the numerical analysis of the tunnels. Since the mechanical properties of the rock mass can be hardly evaluated directly, empirical relations based on the rock mass classification systems are used for this purpose. After determination of the rock mass Geological Strength Index (based on RMR system) and the intact rock constants, mi, the rock mass constants (mb, s, and a) can be evaluated (Hoek & Brown, 1997). As shown by Holland and Lorig (1997), this approach has proved to give an appropriate estimate of mechanical properties of the rock mass.
The Geological Strength Index (GSI), introduced by Hoek (1994) and Heok et al. (1995) can be used for estimating the rock mass strength for different geological conditions. Using 1989 version of RMR (Bieniawski 1989), GSI = RMR'89 – 5 where RMR'89 has the “groundwater rating” set to 15 and the “adjustment for joint orientation” set to zero. The GSI calculation for the rock mass is presented in Table 3.
Table 3. Evaluation of GSI for the Gavoshan tunnel
Table 4. Mechanical properties of the rock mass
Equivalent Mohr-Coulomb strength parameters (c', f') are determined using the relationships proposed by Hoek et al. (2002). The rock mass deformation modulus (Em) is calculated by Serafim and Pereira (1983) relationships modified by Hoek et al. (2002). The results are summarized in Table 4. Since the tunnel has been excavated by a TBM, the value of disturbance factor (D) used in the calculation of the rock mass strength and deformation properties, is taken as zero.
Joints Shear Strength
Mechanical properties of rock joints, which are necessary for simulating joint shear behavior, are obtained from laboratory tests. Joints cohesion, friction angle, normal and shear stiffness are given in Table 5.
Table 5. Joint mechanical properties
Poisson's Ratio of the Equivalent Continuum Rock Media
A single Poisson’s ratio, n, is defined only for isotropic elastic materials. Therefore, it is necessary to define a “Poisson effect” that can be used for anisotropic materials. The Poisson's effect is defined as the ratio of horizontal-to-vertical stress when a load is applied in the vertical direction and no strain is allowed in the horizontal direction. Plane strain conditions are assumed. As an example, the Poisson's effect for an isotropic elastic material is obtained from:
| (1) |
Consider the Poisson effect produced by the vertical jointing pattern shown in Figure 1. The intact rock elastic modulus and Poisson's ratio are presented by E and n. The joint normal and shear stiffness for the joint sets are shown by Kn and Ks.
Figure 1. Model for Poisson's effect in rock with two joint sets
The joints and intact rock act in series. In other words, the stresses acting on the joints and on the rock are identical. The total strain of the jointed rock mass is the sum of the strain due to the jointing and the strain due to the compressibility of the rock. The elastic properties of the rock mass as a whole can be derived as:
 |
(2) |
If then the Poisson's effect is determined from the following equation:
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(3) |
Where and for the rock mass shown in Figure 1, and are calculated as follow:
 |
(4) |
 |
(5) |
After calculation of the Poisson's effect from Equation 3, Poisson's ratio of the equivalent continuum rock mass can be determined using Equation 1.
Considering the tunnel direction, only two joint sets, J1 and J3, are taken into account in simulating the 2D model perpendicular to the Gavoshan tunnel axis. The Poisson's ratio for the rock mass of Gavoshan tunnel is estimated as 0.32.
NUMERICAL SIMULATIONS
Constitutive Model of Intact Rock and Joints
To represent the behaviour of intact material and continuum media, deformable blocks are used with constitutive model of Mohr-Coulomb plasticity. Also, Joint area contact-Coulomb slip constitutive model is employed for joint modelling, which is considered appropriate for the joints and faults analysis in the general rock mechanics.
The Constant Normal Stiffness (CNS) technique is more appropriate than the Constant Normal Load (CNL) technique for the stability analysis of an excavation and for simulation of the behaviour of bolted joints. In the analysis, the contribution of the surrounding rock mass stiffness is considered to be constant, while the normal stress continues to vary during deformation (Indraratna and Haque, 2000). The CNL technique is used in the analyses performed in this research.
Model Boundaries
The model boundaries should be far enough from the region of study so that the model response is not influenced by these artificial boundaries. In general, for the analysis of a single underground excavation, boundaries should be located roughly five excavation diameters from the excavation periphery. The boundary conditions in a numerical model consist of the values of field variables (e.g., stress, displacement) that are introduced at the boundary of model.
For the 6m diameter Gavoshan tunnel, a 36x36m square has been used for the 2D modelling of the rock mass in the simulations. Compressive stresses are applied to boundaries as shown in Figure 2.
Figure 2. Boundary conditions of the model
Mechanical Properties of Shotecrete
The mechanical properties of shotcrete which is used as the support system in the simulations, are presented in Table 6. Since the shotcrete is continuous in the direction perpendicular to the analysis plane, the value specified for the elastic modulus is divided by (1-n²) to account for the plane-strain conditions.
Table 6. Mechanical properties of shotcrete
Continuum Numerical Simulation
Employing the values obtained for the mechanical properties of the rock mass, the excavation is simulated in the equivalent continuum media. For the numerical analyses, UDEC program, based on the Distinct Element Method, is used. The Distinct Element Method requires at least a joint which produces two distinct blocks. Therefore, two joints are defined as shown in Figure 3. In addition, in order to make the mesh generated around the tunnel periphery finer, a circular joint with the radius of 7m is assigned. The joints are allocated high shear strength; therefore they do not affect the analysis results.
The model should be at an initial force-equilibrium state before the excavation can be simulated. The boundary and initial conditions are assigned so that the model is at equilibrium initially. However, it is necessary to calculate the initial equilibrium state under the given boundary and initial conditions. Therefore, after modeling the rock mass with the boundary conditions, the model is solved to obtain the initial equilibrium. Before modeling the excavation, the displacements are reset, so that the change in the displacements due to the excavation can be recognized. After fixing the bottom boundary in the vertical direction, the excavation is performed. Then, the model is solved for two cases:
Figure 3. Simulated 2D model for continuum analyses
1. Unsupported tunnel and 2. Supported tunnel. In the latter case, it is assumed that the plane which is modeled for simulating the support system is located 2m from the face of excavation. Therefore, the shotcrete is activated after some convergence and relief of some part of the stress.
Table 7. Summarized results of the continuum numerical simulations
The results are summarized in Table 7. Figure 4 shows the plastic zone around the unsupported and supported tunnel. The distribution of the axial force and bending moment in the shotcrete are shown in Figure 5.
Figure 4. Plastic zone around the excavation
(continuum numerical simulation)
Figure 5. Bending moment and axial force developed in shotcrete
(continuum numerical simulation)
Discontinuum Numerical Simulation
In this section, a discontinuum model is used for the tunnel analyses. Considering the tunnel direction, two joint sets deviating +64 and -49 from the horizontal, with an average spacing of 0.6m and 0.8m respectively are taken into account. Joint mechanical properties are presented in Table 5.
The analysis process is the same as the continuum numerical simulation and similarly, the analyses are performed for unsupported tunnel and the tunnel supported by 30cm thick shotcrete. The results are summarized in Table 8. The failure zone, where the loose blocks are subjected to falling into the tunnel, is shown in Figure 6. The joints which experience shear displacement in unsupported and supported tunnel are presented in Figure 7. The distribution of the axial force and moment in the shotcrete are shown in Figure 8.
Table 8. Summarized results of the discontinuum numerical simulations
Figure 6. Failure zone, with falling blocks in unsupported tunnel
(discontinuum numerical simulation)
Figure 7. Shear zone around the excavation
(discontinuum numerical simulation)
Figure 8. Moment and axial force in shotcrete
(discontinuum numerical simulation)
EFFECT OF THE JOINT ORIENTATION ON THE ANALYSES
As mentioned previously, in obtaining the mechanical parameters of equivalent continuum media for jointed rock mass, the joint orientation affects the results only in calculating the Poisson's ratio. Therefore, if the joint orientations change, the mechanical properties of equivalent continuum media do not change except the Poisson's ratio. But in discontinuum approaches, the joints can be simulated by its geometry and mechanical properties. So the results of the numerical analyses based on discontinuum approaches can show the effect of changing joint orientation.
In order to consider the effect of joint orientation on the analyses performed by different approaches, four combinations of two joint sets are considered. Each combination is analyzed using numerical continuum and discontinuum approaches. The joint orientation and spacing for each joint combination as well as their appropriate Poisson's ratio for equivalent continuum media is presented in Table 9. All other parameters, including the mechanical properties of intact rock and joints, tunnel depth and diameter, are the same as those mentioned for Gavoshan water conveyance tunnel.
Table 9. Geometry of joints and equivalent continuum media Poisson's ratio
The results of numerical continuum approach analyses performed for 4 different joint set combinations are presented in Figure 9. This figure shows that the change of joint orientation does not affect displacements and radius of plastic zone, while it can change shotcrete safety factors.
The results of numerical discontinuum analyses performed for 4 different joint set combinations are presented in Figure 10 and 11. The figures show that the change of the joint orientation has a significant effect on the displacements. In addition, the analyses indicate that there are some points with compressional or shear failure in some parts of the shotcrete layer in all joint set combinations.
As shown in the figures, in the numerical continuum approach changing the joint orientation does not affect the results except the safety factor of shotcrete. But, the analyses based on numerical discontinuum approach show that the joint orientation has a significant effect on the results. Also, the analyses indicate that in continuum numerical simulations the shotcrete safety factors are between 1 and 2 and in the discontinuum numerical analyses the shotcrete has some compressional and shear failure points.
Figure 9. The results of numerical continuum approach analyses for different joint set combination
Figure 10. The results of numerical discontinuum approach analyses for different joint set combination
CONCLUSIONS
In this paper, the Gavoshan tunnel was analyzed using continuum and discontinuum numerical approaches. The results obtained from these approaches were compared with each other.
In order to investigate the effect of joint orientation on the analyses performed by different approaches, four combinations of two joint sets were considered. Each combination was analyzed using numerical continuum and discontinuum approaches.
In obtaining the mechanical parameters of equivalent continuum media for jointed rock mass, the joint orientation affects the results only in calculating Poisson's ratio. Therefore, if the joint orientations change, the mechanical properties of equivalent continuum media do not change except the Poisson's ratio. But in discontinuum approaches, the joints can be simulated by its geometry and mechanical properties. So the results of the numerical analyses based on discontinuum approaches can show the effect of changing joint orientation.
In addition, in the continuum numerical simulations, the change of joint orientation does not affect displacements and radius of plastic zone, while it can change shotcrete safety factors. But, simulating the rock mass using the discontinuum numerical approach has shown that the change of joint orientation has a significant effect on the displacements. Finally, the investigations indicate that in the continuum numerical simulations the shotcrete safety factors are between 1 and 2 and in the discontinuum numerical analyses the shotcrete has some compression and shear failure points.
This research shows that using continuum equivalent media for simulating the behavior of jointed rock masses may lead to unsafe predictions and the use of discontinuum solutions are recommended for proper design of tunnel support system in jointed rock.
REFERENCES
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