Collapsed Shape of Shallow Unlined Tunnels Based on Functional Catastrophe Theory

This paper investigates the collapse mechanisms and possible collapsing block shapes of shallow unlined tunnels under conditions of plane strain. The analysis is performed following the framework from a branch of catastrophe theory, functional catastrophe theory. First, the basic principles of functional catastrophe theory are introduced. Then, an analytical solution for the shape curve of the collapsing block of a shallow unlined tunnel is derived using functional catastrophe theory based on the nonlinear Hoek-Brown failure criterion. The effects of the rock mass parameters of the proposed method on the shape and weight of the collapsing block are examined. Moreover, a critical cover depth expression to classify deep and shallow tunnels is proposed. The analytical results are consistent with those obtained by numerical simulation using the particle flow code, demonstrating the validity of the proposed analytical method. The obtained formulas can be used to predict the height and width of the collapsing block of a shallow unlined tunnel and to provide a direct estimate of the overburden on the tunnel lining. The obtained formulas can be easily used by tunnel engineers and researchers due to their simplicity.