An analytical algorithm for deriving the lateral bending’s natural frequencies and modal shapes of a three-tower double-cable suspension bridge

Abstract

With increasing span lengths of modern suspension bridges, their response increasingly depends on lateral bending vibration, particularly natural frequencies and modal shapes. These are currently assessed by the finite element method (FEM), which is less intuitive and physically meaningful than the analytical approach. This study attempts to fill this gap by developing an analytical algorithm for solving natural frequencies and modal shapes of lateral bending of a three-tower double-cable suspension bridge with unequal-length main spans. Vibrations of the upper and lower main cables and stiffening girder are interrelated using the transverse and vertical mechanical equilibrium equations of the upper and lower main cables. Then, using D'Alembert's principle, the differential equation of lateral bending of the stiffening girder continuum is derived. Further, the natural frequencies and modal shapes are solved by separating variables. The calculation results for an engineering example prove that the natural frequencies and modal shapes solved analytically for lateral bending are highly consistent with those solved by FEM. The lateral bending deformations of the first and the second orders are S-shaped. The deformations in the secondary main span and main span are in opposite phases, while those in the secondary main span are in the same phase with the right-side span ones. In the first-order lateral bending vibration mode, the deformation of the secondary main span is smaller than that of the main span, with the opposite trend in the second-order one. In the third-order lateral bending vibration mode, deformations in the secondary main span and the main span are in the same phase, while those in the right-side span are in the opposite phase to those in the secondary main span and the main span. The deformation of the secondary main span is smaller than that of the main span.