ملخص
Abstract
Finite element method is used to analyze the flexural vibrations of beams using Galerkin approach for both Bernoulli and Timoshenko theories.The methodology is started by integral formulation of the partial differential equations for the physical problem, and then the element stiffness, mass matrices and load vector have been obtained. Free vibrations analysis is used for the predictions of natural frequencies and mode shapes for, first flexural vibrations by classical theory, second flexural vibrations by including the effect of shear deformation and rotary inertia in the analysis. Finally, the axial and torsional vibrations, in addition to flexural vibrations, in both transverse and lateral directions together have been taken into consideration, thus the finite element equations and the element stiffness, mass, and load vector matrices have been developed for axial, torsional, and flexural vibration together.The natural frequencies are tabulated for all classical types of classical boundary conditions. The mode shapes corresponding to the natural frequencies were presented graphically. The exact solutions and results obtained by other standard methods are used for comparison with the current numerical results. The forced vibrations were introduced for the analysis of dynamic behavior of beams under several types of loading. The finite element code with Taylor series method and mode superposition for the Bernoulli and Timoshenko beams has been carried out. Numerical results are presented in the form of comparisons between the Bernoulli and Timoshenko and demonstrate how the effect of shear deformation reduced the displacement response with increasing the excitation frequency, excitation amplitude, and length of the beam. Comparisons are made with the exact and existing classical methods and checked with those obtained IVusing standard methods (ANSYS package) and are found in good agreement. It should be noted that all models of beams have always convergence to the correct solution.