Tunnel reinforcement with rockbolts

Closed-form solutions are presented for a tunnel supported with Discretely Mechanically or Frictionally Coupled (DMFC) and with Continuously Mechanically Coupled (CMC) or Continuously Frictionally Coupled (CFC) rockbolts. The formulation is based on the following assumptions: (1) circular cross section; (2) deep tunnel; (3) Ko = 1, i.e. axisymmetric problem; (4) homogeneous and isotropic ground; (5) elasto-plastic ground, with brittle failure governed by the Coulomb criterion and non-associated flow rule; (6) elastic reinforcement and (7) construction effects approximated with the β-method (reduction of internally applied stresses). The analytical solutions are compared with results obtained with a Finite Element Method. The comparisons show that the analytical solutions provide reasonable results for DMFC rockbolts and for CMC/CFC rockbolts with low to moderate spacing. For the same rock properties, tunnel geometry and construction, and reinforcement characteristics, CMC/CFC rockbolts result in slightly smaller convergence but somewhat larger rockbolt stresses than DMFC rockbolts. The behavior of DMFC rockbolts and rock response depends on “average” response of the rock between the two end points, while for CMC/CFC rockbolts the maximum stress in the rockbolt occurs at the location of maximum radial strain in the rock. In both cases the solution strongly depends on the relative stiffness between the rockbolt and the deformed rock. Practically speaking, placement of the reinforcement while the rock undergoes elastic deformations may not result in a substantial reduction of tunnel convergence. Instead, it appears that best results are obtained by placing the rockbolts while the rock undergoes plastic deformations. It has also been observed that when the distribution of rockbolts around the tunnel perimeter is linked to the far-field stresses, smaller convergence and reduced reinforcement stresses are possible.