A state-switching PDE model for contact separation induced response amplification in girder bridges under bidirectional seismic excitation

In this study, we develop a state-switching partial differential equation (PDE) model to investigate contact–separation-induced response amplification in girder bridges subjected to bidirectional seismic excitation. The girder and pier were idealized as Euler–Bernoulli beams, while the bearing was modeled as a Kelvin–Voigt element in the horizontal direction and a compression-only unilateral constraint in the vertical direction. A complementarity condition was introduced to describe the vertical girder–bearing interaction, and the horizontal transient response was formulated through state-dependent beam-type PDEs under contact and separation states. Based on modal expansion and state-switching conditions, a semi-analytical solution procedure was established to reveal the coupling mechanism between vertical contact loss and horizontal response amplification. Our numerical results showed that vertical excitation had a limited influence when continuous contact is maintained. However, when the excitation period approached the first vertical natural period, contact loss could occur and cause boundary-condition switching, leading to a significant increase in horizontal displacement. The dynamic V/H model produced stronger amplification than the fixed V/H assumption in the short-period range, while increasing span length further intensified the response. Local demand–capacity indicators, including the pier-base flexural demand ratio and the shear-key impact demand ratio, also increased when vertical excitation was considered. An OpenSees finite element (FE) comparison confirmed that the proposed analytical model could reasonably reproduce the main horizontal response characteristics. The proposed framework provides an effective reduced-order approach for analyzing state-dependent bridge dynamics under bidirectional seismic excitation.