Viscoelastic analytical solution for the surrounding rock-support system in ultra-deep shafts under non-hydrostatic in-situ stress and groundwater pressure

The surrounding rock-support system of an ultra-deep shaft exhibits pronounced time-dependent mechanical behavior under conditions of high in situ stress and groundwater pressure. To investigate the stability of the surrounding rock under the combined effects of non-hydrostatic in situ stress and groundwater pressure, a mechanical model of the surrounding rock-support system of an ultra-deep shaft was established by considering the creep behavior and installation timing of the support structure, and the corresponding viscoelastic analytical solution was derived. The validity of the proposed solution was verified through numerical calculations, and the effects of the key parameters on the mechanical response of the system were further analyzed. The results indicate that the radial displacements of the shaft wall and surrounding rock initially increase rapidly with time and subsequently approach stable values. The analytical and numerical results are in good agreement, with the absolute relative errors at all sampled comparison points remaining below 5%. Non-hydrostatic in situ stress induces pronounced angular nonuniformity in the system response, and the radial and circumferential stresses of the surrounding rock reach their maximum values in the 𝜃 = 0° and 𝜃 = 90° directions, respectively. Delayed support installation increases the early free convergence of the surrounding rock and correspondingly delays the development of the interface load and internal forces in the support structure. Among the parameters considered, the horizontal principal stress ratio exerts the most significant influence on the response magnitudes and angular differences. When this ratio reaches 2.0, the differences in the interface contact stress and circumferential stress in the support structure between the 𝜃 = 0° and 𝜃 = 90° directions reach approximately 7.79 MPa and 73.78 MPa, respectively. The support viscosity coefficient primarily governs the rate of response development and the early stress peaks, while having only a limited influence on the long-term stable state. For similar deep hard rock conditions, it is recommended that support closure be completed within 0.5-2 d after excavation, with approximately 1 d adopted as a preliminary control target. Intensified monitoring and differentiated support measures should also be implemented along both principal stress directions. The proposed analytical method has a concise formulation and high computational efficiency and can provide a theoretical basis for surrounding rock stability analysis and support parameter optimization in ultra-deep shafts.