Abstract A general mathematical approach is developed for the free vibration behaviour analysis of multi-girder and multi-cell box bridges with a single or multi span, including the effects of the transverse deformations of the bridge cross-section. The governing equations of motion and the corresponding boundary and continuity conditions are derived via the variational principle of virtual work following Hamilton's principle. The model is general and valid for any boundary and continuity conditions, and is applicable for multi-girder bridges with longitudinal and cross beams and for multi-cell box bridges. The warping and the distortion of the bridge cross-section effects are included in the proposed model. Closed-form solutions of the governing equations are derived and the Newton–Raphson method is used to determine the eigenfrequencies. Numerical examples are presented to validate the proposed model, and are also used to examine the accuracy of other approximate models used in the analysis of bridges. The results of the proposed model are validated through comparison with three-dimensional finite element models. The results reveal that the transverse deformations decrease the magnitudes of the eigenfrequencies of the torsional mode shapes, as well as the high flexural modes.