Eigenvalues and modes of distributed-mass symmetric multispan bridges with restrained ends for seismic response analysis

Closed-form solutions are developed for the elastic response of symmetric bridges, continuous over two, three or four spans, for unidirectional earthquake perpendicular to the deck, i.e., in the transverse or the vertical direction. The ends of the deck are restrained along the earthquake component, but free to rotate within the plane of bending. The mass and flexural rigidity of the deck are uniform along its length and modelled as continuous. The piers may support the deck through flexible bearings or are rigidly connected to it, in which case a lumped mass is considered at the connection. For each bridge geometry a nonlinear transcendental equation is derived for the modal circular frequency and solved numerically. Closed formulas are given for participation factors, participating masses, the curvature and end slope of the deck and the end reactions. The frequency and cumulative participating mass of the important modes are plotted as a function of the dimensionless total stiffness of the piers. Parametric analyses are presented for the mass of the piers, the location of the intermediate piers and their relative stiffness. They suggest that the eigenvalues are close to those of a simply supported beam with uniformized transverse restraint and mass, except for relatively high pier stiffness and for a single pier of medium stiffness. Even if this approximation is made, modal shapes and elastic forces or deformations of the deck may differ significantly from those of a simply supported beam with uniformized parameters and should be calculated with the closed-form expressions presented. The analytical first-mode frequency and the one from the single-mode method of the AASHTO code in general agree very well, but the first mode participating masses from the two methods may differ.