Designing Tunnel Support in Jointed Rock Masses Via the DEM

Abstract

A systematic approach of using the distinct element method (DEM) to provide useful insights for tunnel support in moderately jointed rock masses is illustrated. This is preceded by a systematic study of common failure patterns for unsupported openings in a rock mass intersected by three independent sets of joints. The results of our simulations show that a qualitative description of the failure patterns using specific descriptors is unattainable. Then, it is shown that DEM analyses can be employed in the preliminary design phase of tunnel supports to determine the main parameters of a support consisting of rock bolts or one lining or a combination of both. A comprehensive parametric analysis investigating the effect of bolt bonded length, bolt spacing, bolt length, bolt pretension, bolt stiffness and lining thickness on the tunnel convergence is illustrated. The highlight of the proposed approach of preliminary support design is the use of a rock bolt and lining interaction diagram to evaluate the relative effectiveness of rock bolts and lining thickness in the design of the tunnel support. The concept of interaction diagram can be used to assist the engineer in making preliminary design decisions given a target maximum allowable convergence. In addition, DEM simulations were validated against available elastic solutions. To the authors’ knowledge, this is the first verification of DEM calculations for supported openings against elastic solutions. The methodologies presented in this article are illustrated through 2-D plane strain analyses for the preliminary design stage. More rigorous analyses incorporating 3-D effects have not been attempted in this article because the longitudinal displacement profile is highly sensitive to the joint orientations with respect to the tunnel axis, and cannot be established accurately in 2-D. The methodologies and concepts discussed in this article, however, have the potential to be extended to 3-D analyses.

Appendices

Appendix 1: Solution for Bolt Support from Carranza-Torres (2009)

The equations of the analytical solution are provided in Carranza-Torres (2009), and are not reproduced here. The solution requires two dimensionless variables as input, namely α and β, defined as:

$$\alpha = \frac{{n_{\text{b}} A_{\text{b}} }}{{2\pi as_{\text{l}} }}$$

(14)

$$\beta = \frac{{\alpha E_{\text{b}} }}{2G}$$

(15)

where n _b is the number of rock bolts assuming that the bolt pattern covers the entire circumference, a is the opening radius, s _l is the bolt spacing in the longitudinal direction of the tunnel (tunnel axis), A _b is the cross-sectional area of the bolt, E _b is the Young’s modulus of the bolt, and G is the shear modulus of the rock mass.

Substituting α into β, we obtain:

$$\begin{aligned} \beta = & \left( {\frac{{n_{\text{b}} A_{\text{b}} }}{{2\pi as_{\text{l}} }}} \right)\frac{{E_{\text{b}} }}{2G} \\ = & \left( {\frac{{n_{\text{b}} }}{{2\pi as_{\text{l}} }}} \right)\frac{{\left( {{\raise0.7ex\hbox{${E_{\text{b}} A_{\text{b}} }$} \!\mathord{\left/ {\vphantom {{E_{\text{b}} A_{\text{b}} } {L_{\text{b}} }}}\right.\kern-0pt} \!\lower0.7ex\hbox{${L_{\text{b}} }$}}} \right)L_{\text{b}} }}{2G} \\ = & \left( {\frac{1}{{s_{\uptheta} s_{\text{l}} }}} \right)\frac{{K_{\text{a}} L_{\text{b}} }}{2G} \\ = & \left( {\frac{1}{{s_{\uptheta} s_{\text{l}} }}} \right)\frac{{K_{\text{a}} s_{\text{mean}} }}{2G} \\ \end{aligned}$$

(16)

where s _θ is the bolt spacing in the circumferential direction, L _b is the characteristic length of the bolt put into tension across the rock discontinuities (or the mean distance between reinforced rock joints along the bolt direction), s _mean is the mean rock discontinuity spacing, and K _a is the bolt stiffness acting across a rock joint which can be derived experimentally based on the procedures recommended by Stillborg (1994). Note that the assumption of E _b A _b = K _a s _mean is conceptually consistent with the equations that are used in Goodman (1989) to derive the deformability of a jointed rock mass from knowledge of the rock joint stiffness (Eqs. 6 and 7). In order to calculate the bolt forces from Eq. (15) in Carranza-Torres (2009), we replaced the group $\beta A_{\text{b}} /\alpha$ with $\frac{{K_{\text{a}} s_{\text{mean}} }}{2G}$.

Appendix 2: Solution for Axial Forces of Lining Under Isotropic Pressure

The expressions to calculate the axial lining forces are hereafter derived. For an elastic medium of shear modulus, G, subjected to isotropic pressure, p ₀, under plane-strain conditions the radial displacements for an opening of radius, a, can be calculated through the expression:

$$\frac{{\Delta u_{\text{ground}} }}{a} = \frac{{p_{0} - \Delta p}}{2G}$$

(17)

where $\Delta u_{\text{ground}}$ is the radial displacement and Δp is the internal support pressure. The hoop strain of a thin-walled cylinder subjected to an external pressure is:

$$\varepsilon_{\theta \theta } = \frac{a\Delta p}{{E_{\text{L}} t}}\left( {1 - \nu_{\text{L}}^{2} } \right)$$

(18)

where t is the thickness, E _L is the Young’s modulus and v _L is the Poisson’s ratio of the cylinder. The radial displacement of the lining, Δu _lining, can then be calculated as:

$$\frac{{\Delta u_{\text{lining}} }}{a} = \frac{a\Delta p}{{E_{\text{L}} t}}\left( {1 - \nu_{\text{L}}^{2} } \right)$$

(19)

In the algorithm to model lining support, the pressure exchanged between the lining and the ground is calculated from the penetration distance, Δu _node, of the lining nodes into the ground (see Fig. 29). The penetration distance can be expressed as:

$$\frac{{\Delta u_{\text{node}} }}{a} = \frac{\Delta p}{{K_{\text{L}} a}}$$

(20)

where K _L is the contact stiffness between the ground and lining. The displacements have to satisfy the following compatibility equation:

Fig. 29

Schematic showing the significance of lining-rock interface compliance

$$\frac{{\Delta u_{\text{ground}} }}{a} = \frac{{\Delta u_{\text{lining}} }}{a} + \frac{{\Delta u_{\text{node}} }}{a}$$

(21)

Combining the equations and ignoring $\nu_{\text{L}}^{2}$ to be consistent with the DEM models, we obtain:

$$\Delta p = \frac{{p_{0} E_{\text{L}} tK_{\text{L}} }}{{2K_{\text{L}} Ga + \frac{{2E_{\text{L}} tG}}{a} + E_{\text{L}} tK_{\text{L}} }}$$

(22)

Then, knowing that N = aΔp, we can deduce the expression for the axial force provided in Eq. (13).