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Mathematical Problems in Engineering
Volume 2015, Article ID 681257, 13 pages
Research Article

Collapsed Shape of Shallow Unlined Tunnels Based on Functional Catastrophe Theory

1MOE Key Lab of Urban Underground Engineering, Beijing Jiaotong University, Beijing 100044, China
2School of Civil Engineering, Beijing Jiaotong University, Beijing 100044, China

Received 18 December 2014; Accepted 25 March 2015

Academic Editor: Yuri Vladimirovich Mikhlin

Copyright © 2015 Chengping Zhang and Kaihang Han. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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.