An analytical investigation for compensation solutions in underground excavation at depth

The excavation compensation method (ECM) is used in deep tunneling to manage large deformations by applying high-prestressed active support immediately after excavation. Under asymmetric in-situ stresses, tunnel convergence becomes non-uniform, causing NPR cables to carry different axial forces even when the same pre-pretension is applied. To capture this circumferentially varying rock-support response, an equivalent compensation pressure along the tunnel perimeter is introduced and assumed to follow a first-harmonic (cosine-type) angular distribution. Based on this representation, analytical solutions are derived within a plane-strain, linear-elasticity framework for the equilibrium convergence displacement and the corresponding bolt axial forces, with the surrounding rock idealized as homogeneous and isotropic. The results show that the proposed first-harmonic distribution provides a rational estimate of cable force variation and enables practical design calculations for NPR-cable pretension and spacing. The analytical predictions are validated against field monitoring data from Changning Tunnel, demonstrating applicability for pretension design and post-excavation deformation assessment. These findings provide analytical guidance for designing NPR-cable pretension and allowable deformation in the ECM-based support design.