Simple approach for solution of the quasi-plane-strain problem in a circular tunnel in a strain-softening rock mass considering the out-of-plane stress effect

The out-of-plane stress is sometimes the major or intermediate principal stress in a circular tunnel opening. The influences of the out-of-plane stress and axial strain are often neglected in the stability analyses of tunnel excavation, which can induce significant errors in the determination of surrounding rock deformations. In this paper, the use of a simple approach is proposed to solve the quasi-plane-strain problem of circular tunneling considering the effect of the out-of-plane stress, which is deformation-dependent and influenced by the in situ stress. As the intermediate principal stress is deformation-dependent, to obtain the numerical solution of the intermediate principal stress, the quasi-plane-strain problem is defined based on assumptions that the initial axial total strain is a nonzero constant (ε0) and that the axial plastic strain is nonzero. With the numerical solution for the plastic strain, obtained using the plastic potential functions based on the three-dimensional failure criteria, the formula for the intermediate principal stress can be derived using Hooke’s law. The proposed approach can be utilized to obtain the numerical solution for the intermediate principal stress, which is deformation-dependent, and the numerical results can be simplified as the solution presented by Pan and Brown. The proposed approach can also be used to obtain the solution for the strain softening of the surrounding rock. To verify its validity and accuracy, the results obtained using the proposed approach are compared with the solution of Pan and Brown. In addition, parametric studies are performed to address the influences of the out-of-plane stress on the stress and displacement in the circular tunnel.