Book Description

The goal of this effort is to develop shear-deformable finite elements which can be used to find the natural frequencies of composite beams. The first objective of the study is to derive the mass and stiffness matrices for the elements of interest and incorporate them into computer programs which can be used to estimate the natural frequencies of composite beams. Composite beams of interest include sandwich beams and those of fiber-reinforced laminated construction. Elements based on the beam theories of Bernoulli-Euler, Timoshenko, Levinson-Bickford, as well as a general third-order beam theory are considered. The elements ignore transverse normal strain, coupling between longitudinal and lateral motion caused by Poisson effects, and damping, and are limited to linear, elastic materials. However, both isotropic and orthotropic layers in symmetric and nonsymmetric configurations can be accomodated. In addition, the elements can impose a kinematic constraint on the entire beam or on individual layers within the beam. This study refers to elements which employ the latter approach as "stacked elements". The second objective is to evaluate the performance of the elements to determine when higher-order elements, including stacked elements, are needed to account for the effect of shear deformation on the natural frequencies of composite beams. Efforts associated with this objective indicate all elements developed are accurate within the limits of their respective theories. All elements possess good monotonic convergence properties and do not lock in the thin-beam limit. In addition, the evaluation reveals that the Bernoulli-Euler beam element is generally limited to cases involving the lower natural frequencies of long, slender beams made out of homogeneous materials having a low degree of orthotropy. (The degree of orthotropy is given by the ratio of Young's modulus in the longitudinal direction to the transverse shear modulus in the plane of the beam.) The Timoshenko beam element can be used effectively for homogeneous and composite beams possessing fairly high degrees of orthotropy if the analyst is able to choose an appropriate value for the shear correction factor associated with Timoshenko's theory. The Levinson-Bickford theory does not require a correction factor, and the element based on this theory can be used with confidence as long as the degree of orthotropy is not too high. As the degree of orthotropy increases, the analyst must rely on the third-order element to attain an adequate level of accuracy. Finally, it is found that stacked elements must be used in the analysis of sandwich beams when the shear modulus of the facings is much larger than the shear modulus of the core. In addition to this condition, the facings must be thick enough to prevent the deformation of the core from dominating the strain energy of the beam.