Analytical solutions for bolted tunnel structures with voids behind the lining

Understanding the mechanical behaviour of tunnels reinforced by end-anchored bolts and lining with voids behind the lining is crucial in determining the appropriate supporting measures. In this study, we constructed a geometric model for a tunnel with a single or double void. We used the substructure method and point matching method to obtain the elastic analytical solutions for the model. We examined the stress, internal force, and deformation characteristics of the tunnel and support structure under the influence of single and double voids. To validate the analytical model, we established a numerical model using the finite difference method. Additionally, we compared and analyzed the mechanical behaviour of tunnels reinforced by bolts and lining versus lining-only, and determined the contribution factor of the bolts.