Stress distribution and deflection of symmetric tapered beams

This paper discusses stress distributions and deflections of tapered beams. Assuming the elementary axial stress distribution and applying Jourawski theory, the shear stress is calculated. In an analogous manner, the transverse normal stress is computed. In contrast to prismatic beams, the shear and the transverse normal stress do not vanish over the surfaces if the beam height varies, even if surface tractions are not present. Their values also depend on the tapering angle and the axial stress at the boundary. Then analytical expressions for the deflections are computed by applying Castigliano's second theorem and considering a fictitious (dummy) force. The complementary strain energy is computed from the derived stress relations as a function of the real load and the dummy forces and moments. Taking the partial derivative with respect to the dummy force, the analytical results for the axial and vertical deflections are calculated. The outcome of the derived tapered beam model is compared to elementary results from Bernoulli-Euler and Timoshenko. The target solutions are obtained by two-dimensional finite element calculations for a tapered cantilever and a clamped-hinged beam subjected to various loads. It is shown that both the shear stress and also the transverse normal stress are correctly predicted by the new method and the deflections computed by Castigliano's theorem are in a very good agreement with numerical solutions. Errors are significantly reduced compared to the errors of the Timoshenko solution.