Abstract: Tunnel convergence patterns, which serve as the key boundary condition for analyzing strata deformation induced by tunnel excavation, are strongly influenced by complex geological conditions and construction processes. However, existing analytical theories for strata deformation are based on idealized convergence assumptions, such as uniform or elliptical convergence, which limit their applicability in complex engineering conditions. To overcome this limitation, a unified equation for tunnel convergence patterns is first established based on Fourier series and coordinate transformations, enabling an accurate description of the tunnel contour trajectory. Subsequently, the deformation of tunnel contour under arbitrary convergence patterns is derived, and the analytical relationship between the parameters of the unified convergence-mode equation and the tunnel contour deformation is obtained. Further, based on complex variable theory, a conformal mapping of tunnel contour deformation under arbitrary convergence patterns is achieved. Finally, a complex variable solution for strata deformation under arbitrary convergence patterns is developed. The results from model degeneration analysis and engineering applications demonstrate that the proposed method generalizes traditional complex variable solutions and more accurately characterizes strata deformation induced by tunnel excavation. Actual engineering conditions are complex and variable, and under complex convergence patterns, the deformation characteristics of the surrounding strata during tunnel excavation differ significantly from traditional predictions. The proposed method enables refined analysis of strata deformation induced by tunnel excavation and supports inverse analysis of complex tunnel convergence patterns.

Key words: tunnel excavation, strata deformation, complex variable function, tunnel convergence pattern, ground settlement