Orthotropic Column Buckling: The Euler and Engesser/Haringx/Timoshenko Formulas Versus an Elasticity Solution

Abstract

The well known column buckling formulas of Euler and these of Engesser / Haringx / Timoshenko (which correct for transverse shear) were derived for isotropic materials, and are routinely used in composite structural applications. The accuracy of these formulas, when orthotropic composite material and moderate thickness are involved, is investigated in the present study by comparing the critical loads from these formulas with the predictions of a three-dimensional orthotropic elasticity solution. The column is assumed to be in the form of a hollow circular cylinder and the Euler or Timoshenko loads are based on the axial modulus. As an example, the cases of an orthotropic material with stiffness constants typical of glass/epoxy or graphite/epoxy and the reinforcing direction along the periphery or along the cylinder axis are considered. First, it is found that the elasticity approach predicts in all cases a lower than the Euler value critical load. Moreover, the degree of non-conservatism of the Euler formula is strongly dependent on the reinforcing direction; the axially reinforced columns show the highest deviation from the elasticity value. Second, the Engesser or first Timoshenko shear correction formula is conservative in all cases examined, i.e. it predicts a lower critical load than the elasticity solution. The Haringx or second Timoshenko shear correction formula is in most cases (but not always) conservative. However, in all cases considered, the second estimate is always closer to the elasticity solution than the first one. For the isotropic case both Timoshenko formulas are conservative estimates.