• 标题：Upper bound analysis of collapse failure of deep tunnel under karst cave considering seismic force
• 关键词：Upper bound theorem, Karst tunnel, Nonlinear failure criterion, Karst cave,Variational principle
• 作者：Cheng Lyu, Li Yu, Mingnian Wang, Pengxi Xia, Yuan Sun

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When tunnels are excavated in karst areas, the existence of karst caves in the surrounding rock usually results in the collapse failure of tunnels. Tunnel collapse has always been a challenging issue because it seriously threatens the safety of tunnel builders. To investigate the collapse of a deep tunnel under a karst cave, we constructed the collapse mechanisms of the rectangular and circular tunnels under a karst cave, and obtained the expressions of the collapse failure in tunnel considering seismic force and seepage force by means of the variational principle and limit analysis theory. Compared with previous results and numerical solutions, it is found that the results in this paper are consistent with those results and numerical solutions, which indicates that the proposed method in this work is rational. According to the analytical solutions, the shape of the collapsing blocks of the deep rectangular and circular tunnels under a karst cave are drawn, and the effects of different parameters, especially the vertical seismic coefficient kv, on the collapsing blocks are analyzed. The findings in this paper indicate that the range of collapsing blocks decreases with increased kv. The two formulas for calculating the critical height between the tunnel and the karst cave are given by derivation.