Abstract
The free vibration of beams, subjected to a constant axial load and end moment and various boundary conditions, is examined. Based on the Euler-Bernoulli bending and St. Venant torsion beam theories, the differential equations governing coupled flexural-torsional vibrations and stability of a uniform, slender, isotropic, homogeneous, and linearly elastic beam, undergoing linear harmonic vibration, are first reviewed. The existing formulations are then briefly discussed and a conventional finite element method (FEM) is developed. Exploiting the MAT LAB-based code, the resulting linear Eigenvalue problem is then solved to determine the Eigensolutions (i.e., natural frequencies and modes) of illustrative examples, exhibiting geometric bending-torsion coupling. Various classical boundary conditions are considered and the FEM frequency results are validated against those obtained from a commercial software (ANSYS) and the data available in the literature. Tensile axial force is found to increase natural frequencies, indicating beam stiffening. However, when a force and an end moment are acting in combination, the moment reduces the stiffness of the beam and the stiffness of the beam is found to be more sensitive to the changes in the magnitude of the axial force compared to the moment. A buckling analysis of the beam is also carried out to determine the critical buckling end moment and axial compressive force.
Highlights
Hashemi and Richard [2] used the frequency dependent dynamic finite element (DFE) method, a hybrid method developed in [3] that combines the accuracy of analytical methods to the versatility of numerical methods, to conduct a vibration analysis of beams that are geometrically coupled in bending and torsion
Frequency and stability analyses of two illustrative examples are carried out and the finite element method (FEM) results are compared with those obtained from ANSYS software and the data available in the literature
The presented FEM results showed excellent agreement with those obtained from ANSYS and experimental data
Summary
Beams are important and versatile structural elements as many civil, mechanical, and aerospace structures are commonly modeled as preloaded beams or beam assemblies [1]. Hashemi and Richard [2] used the frequency dependent dynamic finite element (DFE) method, a hybrid method developed in [3] that combines the accuracy of analytical methods to the versatility of numerical methods, to conduct a vibration analysis of beams that are geometrically coupled in bending and torsion. Hashemi and Richard [22] formed a DFE solution for the free vibration analysis of axially loaded bending-torsion coupled beams. The FEM is adaptable to many complex systems, including those with material and geometric variations, for example, nonuniform geometry It seems that a FEM-based thorough investigation of the geometrically coupled flexural-torsional vibration of beams, simultaneously subjected to both axial force and end moment, and the coupling effects caused by end moments have not been reported in the open literature. The presented FEM formulation, can be extended to thin walled beams with open cross sections, where torsion-related warping effects cannot be neglected
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