A Three-Dimensional (3D) Semi-analytical Solution for the Ultimate End-Bearing Capacity of Rock-Socketed Shafts

This study proposes a new semi-analytical solution for the ultimate end-bearing capacity of rock-socketed shafts with the consideration of three-dimensional (3D) strength and 3D geometry. The rock mass is assumed to be rigid-plastic and governed by a 3D Hoek–Brown (HB) criterion, and the method of characteristics is utilized to derive the governing equations in a cylindrical coordinate system. A Runge–Kutta-based iterative approach is adopted to solve the derived governing equations as an initial value problem via MATLAB. Eight test shafts with measured ultimate end-bearing capacity were analyzed to validate the proposed solution. Then, the proposed solution was compared with an analytical solution based on 2D HB criterion and an empirical factor for considering 3D effect, and a numerical simulation based on 2D HB criterion and 3D geometry to investigate the effects of 3D strength and 3D geometry. The results indicate that ignoring 3D strength and 3D geometry could significantly underestimate the ultimate end-bearing capacity of rock-socketed shafts. Finally, extensive parametric analyses were conducted with the proposed solution to explore the effects of rock mass properties, shaft dimensions, and the disturbance factor. The results show that the ultimate end-bearing capacity factor $${N}_{\sigma }$$ (which is the ratio of the ultimate end-bearing capacity $${q}_{\mathrm{u}}$$ to the unconfined compressive strength of the intact rock $${\sigma }_{\mathrm{c}}$$ ) of rock-socketed shafts increases with rock mass constant $${m}_{\mathrm{i}},$$ geological strength index (GSI), the length of the shaft within soil layer $${H}_{\mathrm{s}}$$ and the length of the shaft within rock layer $${H}_{\mathrm{r}}$$ , while decreases with $${\sigma }_{\mathrm{c}}$$ . Also, the disturbance factor D significantly affects the ultimate end-bearing capacity factor $${N}_{\sigma }$$ , especially for shafts in rock masses with a low GSI.