• The improved Pasternak foundation beam model incorporates generalized shear force to account for soil continuity beyond beam ends, eliminating anomalies in existing umbrella arch analytical models.
• Analytical solutions for finite-length umbrella arches under arbitrary loads were derived using the initial parameter method and validated against existing test data.
• Dimensionless characteristic parameters representing relative stiffness between arch and soil were proposed; soil parameters significantly affect arch deformation and internal forces.
• Tunnel excavation height and advance have contrasting influences on unsupported and buried umbrella arch sections, informing design optimization.