###### Abstract  
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 \(R_{p}\), the residual radius \(R_{d}\), 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 \(K_{\phi}\) 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. t 202212021202212021202120212021202120212212212212212212212221  
of intermediate principal stress on the deformation and failure resulting from tunnel excavation. However, both theory and practical observations indicate that most of the rock mass remains in a three-dimensional (3D) stress state after the tunnel excavation and unloading. Many studies, both theoretical and experimental [14; 15; 16], have shown that the strength of geotechnical materials is closely related to intermediate principal stress. Therefore, ignoring the effect of intermediate principal stress by rock mass strength can lead to deviations in engineering design. There is currently only a gradual application of strength criteria considering the influence of intermediate principal stresses within the study of surrounding rock of tunnels and related rock engineering. These applications mainly include the Drucker\(-\)Prager (DP) series strength criterion and the Unified Strength Theory (UST). Based on UST, [PERSON] et al. [17] proposed that intermediate principal stress could be represented by a parameter \(b\) and the other two principal stresses in the plastic zone. However, this assumption lacks a strong theoretical basis. A study by [PERSON] [18] that considered the effect of intermediate principal stress applied the unified strength theory of [PERSON] et al. [17] to the calculation of a circular tunnel. [PERSON] [19] applied the Levy-Mises constitutive relationship to deduced the expression of intermediate principal stress in the plastic zone as the average of the other two principal stresses. However, the Levy-Mises constitutive relationship for metallic materials is not suitable for geotechnical materials. [PERSON] et al. [20] applied the UST to study the effect of intermediate principal stress on the range of the fracture zone and deformation of the surrounding rock of a deeply buried tunnel. However, they did not provide a method for calculating intermediate principal stress. A finite difference method, which is used to calculate the strain-softening behavior of rock masses considering the effect of intermediate principal stress, has been proposed to calculate intermediate principal stress based on the unified failure criterion and non-flow associated flue [21].