Shear Coefficients for Orthotropic Beams

Because of their high stiffness-to-weight ratios, fiber reinforced composites are being studied for use in weight-critical compression members. Davis [1] pointed out that unidirectional or nearly unidirectional composites typically have a low transverse shear stiffness and the calculation of the critical column load for com-pression members should account for transverse shear effects. He presents experi-mental data for unidirectional Boron/Epoxy tubes which fall significantly below the Euler curve at higher stress levels. The column equations with shear effects included, as presented in Reference [ 2] , include in their derivation the expression for shear strain NQIA G. Cowper [ 3] presents a derivation for 1 /n or K as he defines it for isotropic materials and presents equation results for many commonly used cross sections. The factor K was shown to be dependent on the column material’s Poisson’s Ratio (v). The presence of a material property in K suggested that a new derivation should be undertaken which would account for non-isotropic materials such as composites. Most com-posite columns are fabricated from an in-plane macro-scopically orthotropic class of laminates- the [0/±6/90] family. Therefore, the present derivation which is for homogeneous orthotropic materials is applicable for most composite columns and should be employed in the calculation of critical column loads. The beam under consideration has its axis in the z-direction, is straight and uniform, and the cross section and applied loads are symmetric about the x-z plane. Also it will be assumed that body force and surface traction do not have com-ponents in the z-direction. Following the approach of Reference [ 3] , the transverse deflection, W, of the beam is taken as the mean deflection of the cross section. Thus, where A is the area of the cross section, and Ux is the x-component of displacement of a point of the beam.