A Modified Unlu–Gercek Longitudinal Deformation Profile for Circular Inclined Roadway: Incorporating Spatial and Geomechanical Parameters

Determining the optimal timing for support installation is of critical importance to the stability of surrounding rock in inclined mine roadways. Existing expressions for the longitudinal deformation profile (LDP) commonly overlook the coupled effects of spatial parameters and geomechanical parameters in inclined roadways. In this study, the LDP expression for a circular inclined roadway under three-dimensional in situ stress conditions is firstly derived analytically. Subsequently, the coupled influence of spatial and geomechanical parameters on the LDP is investigated using FLAC3D numerical simulations. Finally, building upon the work of Unlu and Gercek, a modified LDP formula tailored to circular inclined roadways is proposed. The results indicate the following: (1) The parameters exerting a significant influence on the LDP shape can be ranked in descending order of importance as cohesion, internal friction angle, β, and α. Parameter α affects the LDP curve within the range of X/R = −6 to 6. As α increases from 5° to 35°, the displacement release coefficient at the tunnel face (u0*) drops sharply from 15.830% to 5.107%. In contrast, β, cohesion, and internal friction angle influence the LDP within the range of X/R = −6 to 9. Specifically, as β varies from 0° to 90°, u0* decreases from 15.830% to 8.920%; as cohesion increases from 2 MPa to 3.5 MPa, u0* declines from 18.80% to 14.38%; and as the internal friction angle rises from 20° to 35°, u0* drops from 18.40% to 12.08%. (2) Poisson’s ratio, the axial lateral pressure coefficient, the horizontal lateral pressure coefficient, and the elastic modulus exhibit relatively minor effects. Poisson’s ratio primarily influences the displacement release coefficient in the vicinity of the excavation face, within the range of X/R = −1 to 3. As Poisson’s ratio increases from 0.2 to 0.35, u0* decreases from 15.83% to 13.54%. Although its effect at the tunnel face is modest, it should not be disregarded. (3) Validation against existing LDP formulas and field data demonstrates that the proposed modified formula more effectively characterizes the spatiotemporal evolution of displacement release in the surrounding rock of circular inclined roadways, thereby providing a more reliable reference for determining the optimal support installation timing and optimizing the support design of inclined shafts.