Thermoelastic Stresses in Beams

The behavior of elastic, free beams of arbitrary constant cross section under an arbitrary temperature distribution is investigated. An analytical method of successive approximations for the solution of linear partial differential equations is applied to the solution for thermoelastic stresses in such beams. The stresses are obtained in terms of series, the nth terms of which are of the form βnℒn(∂nT/∂nx) where β is the ratio of the maximum linear dimension of the cross section to the length of the beam, T is the temperature, x is the ratio of the axial co-ordinate to the length of the beam, and ℒn is a homogeneous linear integral differential operator in y and z only. When Poisson’s ratio is zero, the first term of the series for the normal axial stress is identical with the solution based on a strength-of-materials analysis, and the first term of the series for the axial displacement leaves plane sections plane. The leading terms of the series provide good approximations when the temperature can be expanded in a rapidly convergent power series and in long bars (β ≪ 1). The case of beams of constant circular cross section under temperatures varying only with the axial co-ordinate is examined and a numerical example is presented.