An analytical framework with closed-form solutions for hybrid curved-straight tunnel segments with deformation joints under arbitrary dynamic loads

The safety assessment of tunnels comprising both curved and straight segments connected by deformation joints remains a critical challenge in underground engineering, due to the complex interplay of geometric curvature, structural discontinuity, and arbitrary dynamic loading. This paper presents a novel analytical framework that delivers explicit closed-form solutions for the dynamic response of a hybrid curved-straight beam system on a Pasternak viscoelastic foundation under arbitrary dynamic loads. The model integrates curved segments, straight segments, and deformation joints into a unified piecewise beam formulation. A key contribution is the use of a Dirac-delta-based virtual-force representation to capture displacement and rotation jumps at deformation joints without homogenization, thus preserving physically meaningful risk indicators across transitions. Using modal superposition, we derive fully analytical expressions for displacement, velocity, and bending moment. The framework is validated through degenerative analysis, high-fidelity finite element simulations, and a real-world case study. Parametric studies reveal significant dynamic-response discontinuities along the tunnel, indicating that curved segments become more critical when the curvature radius is small relative to the characteristic deformation wavelength, with deformation joints acting as key amplification interfaces. The proposed solution offers not only computational efficiency but also a transparent, mechanics-based tool for performance-driven design and risk-informed monitoring of combined curved-straight tunnel systems, effectively bridging high-fidelity analysis with practical engineering decision-making.