A 3D analytical solution for deep shafts with elliptical cross-sections under depth-dependent in-situ stress

The shaft is a critical component in mine infrastructure, serving as the main conduit for access and material transport, and plays a pivotal role in maintaining operational stability. However, non-uniform pressure distributions pose a significant threat to shaft integrity, especially within the complex geomechanical context of deep multi-field coupling. To address this challenge, a depth-dependent 3D analytical model is proposed to evaluate the elastic response of deep shafts under non-uniform stress, with a specific focus on elliptical cross-section. The model establishes a 3D complex potential function framework for a depth-dependent stress field and derives the elastic stress–strain equation for elliptical shafts within depth-dependent in-situ stress field. Analytical solutions are obtained for stress and strain at typical burial depths of 300, 800, 1500, and 1800 m. A comparative analysis is conducted to investigate the influence of shaft geometry on its stability under non-uniform pressure. The elliptical axis ratio m and burial depth h significantly affect the stress and strain distributions around the shaft. At the unsupported shaft boundary, the radial stress is zero because of the traction-free boundary condition, whereas the hoop and vertical stresses vary with the boundary position, axis ratio, lateral pressure coefficients, and burial depth. The maximum hoop stress and the maximum radial, hoop, and vertical strains increase with increasing m, lateral pressure coefficient λ 1 , and burial depth h . Under the investigated stress conditions and model assumptions, elliptical shafts with smaller values of m exhibit lower stress and strain concentrations than the circular shaft, together with reduced compressive response and substantially lower tensile stress and strain. Furthermore, the analytical solutions show strong agreement with numerical simulation results. These findings provide valuable theoretical insights for the analysis, design and support of deep elliptical shafts.