Complementation of analytical solutions for the quantitative roof stability analysis of rock cavities and tunnels

This study revisited a well-founded analytical solution describing the collapse profiles of tunnel roofs and estimated traditional stability measures to be used for quantitative safety assessment and practical engineering design. The original analytical approach was proposed by Fraldi and Guarracino based on a variational approach in the context of limit analysis, and subsequently extended to account for pore-pressure, seismic excitation, shallow buried depth, and axis-symmetric analysis, etc. In this study, the optimal bounds for the stability number, shear strength-based factor of safety, and support pressure were derived from the available original and extended analytical expressions of the failed block in both rectangular and circular tunnels. The computational results were in good agreement with those obtained with semi-analytical and numerical approaches but were obtained with much less computational effort. Regarding the tunnel at a shallow depth, a revised profile function was suggested by employing a different boundary condition that allowed the geometric restriction in the previous study to be eliminated; as a result, improved solutions could be obtained. All computation outcomes, including those of the extensions, were validated with those based on the piecewise linear failure mechanisms. The obtained failure mechanisms demonstrated an identical shape of the profile function in terms of curvature, and differences in safety measures were attributed to the dimension of the failed blocks. The proposed techniques in this study enrich the existing theoretical method while retaining its extreme simplicity.