Modified finite elements method to investigate vibrations of the main cables in suspended bridges

•Vibrations of suspended bridges under moving vehicles.•Analytical and finite elements methods.•Coupled lateral and torsional vibrations of the deck.•The deck is considered as Euler-Bernoulli beam.•The suspended cable theory with sag ratio less than 10% is used to study main cables.•Using the finite elements method for vibration of the suspended cables.•The Newmark method is used to solve in the time domain.•The matrix equations of the cables are obtained.•The method of solving nonlinear matrix equations is discussed. In this paper, a modified finite elements method is presented to study vibrations of a suspended bridge under moving vehicles. In this study, coupled lateral and torsional vibrations of the deck and lateral vibrations of the main cables are considered. The deck is considered as an Euler-Bernoulli beam. The suspended cable theory with the sag ratio less than 10% is used to study the vibrations of the main cables. In the finite elements method used in this study, the vibrational equations of the curved elements of the main cables are calculated in the Cartesian coordinates and change of tension is considered in the equations. Finally, the Newmark method is used to solve the vibrational equations in the time domain. To verify the finite elements method, it is compared with an analytical solution which is applied to a numerical example of the problem.