Effect of shear on deflection and buckling of beams and plates

Abstract

The paper deals with the influence of shear stresses on bending and buckling of beams and plates of composite materials employing the hypothesis of ‘straight line’. It is assumed that a set of points lying on a line perpendicular to the neutral surface (the mid-plane) before deformation remains on the line and this line is non-perpendicular to the neutral surface after a load's application. This non-orthogonality is caused by the presence of shear stresses. Thus, this approach does not require the section to be normal to the neutral surface as in the classic Kirchhoff hypothesis. It is shown that the latter is inconsistent with the existence of shear stresses. The corresponding differential equations are derived from the non-linear differential equations of the theory of elasticity in Cartesian coordinates. This allows well-founded simplification of the equations for different partial cases and also correct determination of the sections' rotation angles in the equations for buckling. Several solutions for bending and buckling of beams and plates are given while considering the effect of shear stresses. Formulae are derived for the Euler force of compressed beams and simply supported orthotropic plates, which allows calculation of the error resulting from the application of the Kirchhoff hypothesis. For example, it is shown that for h/b < 0.1 (h = plate thickness, b = dimension of smallest side) the error is smaller than 5% and it does not depend on the plate's aspect ratio. Also, a solution is found for bending strength of beams and orthotropic plates subjected to the arbitrarily distributed lateral pressure considering the effect of shear stresses. Formulae for the deflection in the plate's centre are derived as well as for calculating the error when the classic Kirchhoff hypothesis is applied.