Effects of the Variational Consistency and Variational Inconsistency on Free Flexural Vibration Frequencies of Simply-supported Rectangular Isotropic Shear Deformable Beams

Abstract

In order to derive governing differential equations of a displacement-based shear deformation beam theory, one can either utilise the variationally consistent approach (VCA) or the variationally inconsistent approach (VIA). In the VCA, variational principles are utilised to derive beam governing equations. Whereas in the VIA, beam gross / integrated two-dimensional equilibrium equations are utilised to derive beam governing equations. Shimpi et al. [2] have proposed the single variable shear deformation beam theory (SVSDBT) which is based on the VIA. Whereas, Shimpi et al. [1] have proposed the two variable shear deformation beam theory (TVSDBT) which is based on the VCA. In the SVSDBT and TVSDBT, the beam axial displacement and beam transverse displacement consist of bending and shearing components. In both theories, bending components do not take part in the cross-sectional shearing force, and shearing components do not take part in the cross-sectional bending moment. Both theories utilise linear strain-displacement relations. Displacement functions of both theories give rise to the beam transverse shear strain (and hence to the beam transverse shear stress) which varies quadratically through the beam thickness and maintains transverse shear stress-free beam surface conditions. For the case of dynamics, the SVSDBT has only one governing equation, and the TVSDBT has two governing equations which are inertially coupled. The aim of this paper is to present the comparison study on free flexural vibration frequencies of simply-supported rectangular isotropic shear deformable beams obtained by utilising the SVSDBT and TVSDBT. These frequencies are obtained for various values of the beam thickness-to-length ratio and beam vibration mode parameter. This comparison shows that for both theories, the VCA and VIA have a minimal effect on resulting beam free flexural vibration frequencies.