Analysis of behaviour of beams subjected to an eccentric compressive force under effect of a moving harmonic load

In this study, linear and non-linear vibrations of a simply-supported beam subjected to an eccentric compressive load and a moving harmonic load are investigated numerically in the framework of the Euler-Bernoulli beam theory (EBBT), Timoshenko beam theory (TBT) and Reddy-Bickford beam theory (RBBT). In the non-linear solution of the problem, only the geometrically non-linear effects are taken into account. Kelvin-Voigt model is utilized for the material of the beam. The unknown functions which represent the displacements of the beam and the rotations of the cross-section are separated into time-dependent and space-dependent coordinates. In the linear and the non-linear solutions, polynomials are chosen for the space-dependent coordinates of the displacements of the beam and the rotations of the cross-section. Lagrangian functional of the problem is obtained by satisfying the constraint conditions of supports with Lagrange multipliers. Equations of motion are derived by using Lagrange?s equations. By solving the time-dependent system of differential equations, the displacements, velocities and accelerations of the beam are calculated for any time. In the non-linear analysis, Newmark- method is used in conjunction with the Picard (direct iteration) and Newton-Raphson method for solving the equations of motion. The validity of the obtained results in this study is demonstrated by comparing them with the previously published results which are the special cases of the investigated problem. Convergence studies are performed. Free vibration analyses are made for the linear case. In this study, the effects of the shear deformation, the velocity and the excitation frequency of the moving harmonic load, the magnitude and the eccentricity of the eccentric compressive load and the damping of the material of beam are investigated in detail. In addition to these, in the non-linear solution of the problem, the effect of the large transverse displacements is examined in detail.