The vibration response of five beam theories under eccentric moving harmonic loads

Abstract

The shear rotary inertia caused by the shear deformation affects the natural frequencies and the vibration characteristics of bridges. To guarantee the safety of bridges, it is necessary to investigate the shear rotary inertia as a crucial content. Unlike previous studies, the motivation of this paper is to improve the Timoshenko beam model and further explore the influence of the shear rotary inertia on flexural–torsional natural frequencies and flexural–torsional vibration properties of bridges. Based on the Timoshenko beam theory, a modified dynamic equation of the Timoshenko beam is established. Its natural frequencies are obtained, and its analytical solutions under moving harmonic loads are derived using the integral transform method. In order to further understand the shear rotary inertia, the other four beams (namely, the Euler–Bernoulli beam, the shear beam, the Rayleigh beam, and the Timoshenko beam) are introduced to make comparisons with the modified Timoshenko beam. Subsequently, in the parameter analysis, the natural frequencies and vibration responses of five different beams with various lengths are analyzed numerically. It is discovered that the shear deformation and the shear rotary inertia have great effect on the high-mode vertical and bridge flexural–torsional frequencies, and it can improve the accuracy of calculating the natural frequencies of beams by about 6.00%. Furthermore, the shear deformation effect on vertical dynamic responses under harmonic moving loads intensifies with the beam spans increasing. The shear beam theory and the Timoshenko beam theory are appropriate for the vibration analyses of short beams, and the Rayleigh beam theory and the Timoshenko beam theory are suitable for the vibration analyses of long beams, while the modified Timoshenko beam theory, being more generally, is suitable for both beams.