Based on the three-dimensional generalized Zhang-Zhu (GZZ) strength criterion and a non-associated flow rule, this study employs the finite difference method to derive nonlinear semi-analytical solutions for the stress and displacement fields in elastoplastic surrounding rock. The analysis explicitly incorporates axial stress, demonstrating that irrespective of its initial magnitude, the post-yield stress state invariably features a consistent principal stress ordering: the axial stress becomes the intermediate principal stress, with the circumferential and radial stresses as the maximum and minimum principal stresses, respectively. Detailed computational results reveal a significant and non-monotonic influence of axial stress on excavation response. Both the plastic zone radius and tunnel wall displacement exhibit a distinct trend—first decreasing and then increasing—as axial stress rises, confirming the existence of an optimal axial stress value that minimizes rock deformation and failure extent. A comparative analysis of characteristic curves under different criteria shows marked differences. Unlike the conventional two-dimensional Hoek-Brown criterion, the three-dimensional GZZ criterion effectively accounts for the strengthening effect of the intermediate principal stress (σ2). Consequently, the GZZ criterion predicts a smaller plastic zone radius and reduced displacements, quantitatively demonstrating that the intermediate principal stress enhances rock strength and substantially inhibits both plastic zone development and surrounding rock deformation. This finding provides valuable theoretical insight for support design in deep underground engineering.