System reliability analysis of tunnels reinforced by rockbolts

System reliability analysis of tunnelling problems is illustrated by using an iterative closed-form formulation for a circular tunnel reinforced by end-anchored rockbolts. Three performance functions, namely, the mobilized tensile force in the rockbolt, the maximum displacement and the plastic zone size of the tunnel, are considered. The bimodal bounds method and the multivariate normal cumulative distribution function (mvncdf) method are applied to the solution. Results show that the system probability of failure (Pf) estimated from the mvncdf method lies within the lower bound and upper bound set by the bimodal bounds method for the cases studied. The second-order reliability method (SORM) is used to refine the reliability indices and to improve the accuracy of the estimated system Pf. The influence of the correlation coefficient between the cohesion and the friction angle and the influence of the rockbolt installation position on the system Pf are discussed. A modified hybrid approach using the linear response surface method (RSM) to locate the design point and the artificial neural network (ANN) to approximate the actual limit state surface (LSS) is suggested to be used in the system reliability analysis for problems where closed-form solutions are not available. Traditional second-order RSM without cross terms may be inaccurate and may suffer from the problem of low convergence rate and the second-order RSM with cross terms may encounter the “false branch” problem. It is also shown in this paper that the “false branch” problem can be overcome by the importance sampling technique. Comparison with the second-order RSM shows that the proposed approach is efficient, accurate and robust for the system reliability analysis.