A study of the safety thickness of cave roof under rock-socketed pile in karst area based on the cusp catastrophe theory

According to the engineering characteristics of rock-socketed pile in karst areas, the existing beam-slab structure is optimized, the structural system formed by the rock-socketed pile and the cave in the karst areas is simplified as a flat-beam model constrained by the sliding bearing and horizontal stress at both ends of the roof. The energy potential function and bifurcation set equation are derived based on the cusp catastrophe theory, and the equilibrium surface of the cusp catastrophe model under the rock-socketed pile in karst regions is also established. Based on the current research on horizontal stress of rock mass, combined with the necessary condition and sufficient condition of catastrophe instability of the cave roof under rock-socketed pile, calculation formula of the minimum safety thickness of cave roof in karst areas is deduced. The method is verified with an engineering example and compared with the existing beam-slab theory. The results show that the method is reasonable and feasible, and can be used to provide a reference for the design of rock socketed pile in karst areas.