Coupled analytical solutions for circular tunnels considering rock creep effects and time-dependent anchoring forces in prestressed bolts

The instability of the surrounding rock in underground tunnels often induces major engineering disasters. The application of prestressed bolts is an effective reinforcement method for underground structures. In this work, we consider the interaction between the rock creep and the time-dependent anchoring forces in prestressed bolts. We derive theoretical solutions for rock creep displacement caused by excavation and for the anchoring forces of the prestressed bolts, and verify the solutions using a numerical simulation and an engineering example. First, based on the coordinated deformation between the prestressed bolts and the creeping rock mass, we establish a coupled model that takes into account the rock creep and the evolving anchoring forces. We then use the superposition principle to derive elastic solutions for rock displacement and anchoring force. Second, to reflect the effect of rock creep and time-dependent anchoring force, the Burgers model is used for the rock mass and the elastic model is used for prestressed bolts. According to the coordinated deformation between rock and bolts, we obtain the analytical solutions under the coupled actions in the Laplace space. The viscoelastic solutions for rock displacement and anchoring force considering the coupling effect are then solved by using the inverse Laplace transform. Finally, we compare the analytical solutions with numerical simulation results from FLAC3D and monitoring results from an engineering example to verify the accuracy of the analytical solutions. The theoretical model provides a reference for studying tunnel reinforcement, analyzing rock creep behavior and long-term stability of the reinforcement structure.