Despite the extensive investigation on the stress and displacement distributions in tunnels, few have considered the influences of the damaged zone around a tunnel on the strength and stiffness parameters of the surrounding rock, including the gradual variation in the damaged factor D and dimensionless damaged radius $$\rho^{{\text{d}}}$$ ρ d , under the effect of excavation disturbance. In this paper, a numerical solution is presented for the stresses and displacement of a circular tunnel excavated in a Hoek–Brown rock mass considering the progressive destruction of the damaged zone. First, the results obtained using the proposed algorithm are compared with those obtained using other numerical solutions, demonstrating a high degree of accuracy through some examples. Second, the influences of the damaged factor $$D$$ D and dimensionless damaged radius $$\rho^{d}$$ ρ d on the stresses, radial displacement, plastic radii, and ground response curve are investigated. The results show that, as the damaged factor D increases, the radial displacement and plastic radius increase, whereas the tangential stress decreases. Both the plastic radius and displacement decrease with decreasing $$\rho^{{\text{d}}}$$ ρ d . This shows that the damaged factor D has a significant effect on tunnel convergence.

Past studies on deep-lying tunnels under the assumption of plane strain have generally neglected the influence of intermediate principal stress even though this affects the surrounding rocks in the plastic zone. This study proposes a finite difference method to compute the stress strain plastic region and displacement of a tunnel based on the Drucker–Prager (D–P) yield criterion and non-associated flow rule and considering the influences of intermediate principal stress and the strain-softening behavior of surrounding rock. The computed results were compared with those of other well-known solutions and the accuracy and validity of the method were confirmed through some examples. Parameter analysis was conducted to investigate the effects of intermediate principal stress on stress-strain, the plastic region, the ground response curve, and the dilatability of surrounding rock. The results showed that the plastic radius , the residual radius , and radial displacement of surrounding rock first decreased and then increased with increasing intermediate principal stress coefficient b from 0 to 1, with the minimums occurring at b = 0.75. On the contrary, the peak and rate of variation of the dilatancy coefficient first increased and then decreased with increasing b and the dilatancy coefficient gradually transitioned from nonlinear to linear variation. Meanwhile, the inhibition of the plastic radius and radial displacement gradually weakened with increasing support pressure, whereas the dilatancy coefficient of the tunnel opening gradually increased.

The effects of aspect ratio on the stresses and deformations of radial passenger tires under inflation pressure are analyzed with the nonlinear finite element method. The stiffness properties of the cord-rubber composites used in this paper were measured by a Shimadzu Universal Testing Machine DCS-500. The analysis shows that when the aspect ratio decreases from 80 to 70, while the properties of the materials and the inflation pressure of the tires remain the same, the stresses in the carcass decrease by 5%, and those in the belt increase by 10–15%. Illustrations are given for inflated shapes, cord loads, and lower sidewall displacements.