Optimal reinforcement design of a friction pendulum bearing–isolated continuous girder bridge under non-stationary seismic actions

Friction pendulum bearings (FPBs) have been widely adopted in bridge engineering, yet pier reinforcement design for large FPB-isolated bridges still suffers from correlation neglect and high computational cost. This study addresses substructure reinforcement design of multi-span continuous girder bridges equipped with FPBs under non-stationary seismic actions. A hybrid framework is developed by integrating an explicit time-domain scheme with a fully probabilistic reinforcement formulation. Structural hysteresis is described using the Bouc–Wen model, and seismic input is represented as non-stationary stochastic excitation; the explicit time-domain algorithm is employed to compute the substructure response coefficient matrix. Response envelope curves are constructed from the covariance of response quantities, and the required longitudinal reinforcement ratio is determined iteratively through the relationship between the response envelopes and the section bending moment–axial force (M–N) interaction envelope. Compared with code-prescribed and conventional probabilistic procedures, the proposed approach accounts for moment-axial force correlation, reduces repeated dynamic analyses, and improves efficiency while maintaining design-level accuracy. Application to an FPB-isolated continuous girder bridge and verification using code-specified Monte Carlo simulation demonstrate the effectiveness of the method for estimating required seismic reinforcement demand in bridge substructures.