A 2D warping theory for shear deformable elastic beams of axisymmetric cross section in flexure

Shear deformable elastic beam models in flexure are considered in the hypothesis of cross section symmetric with respect to the flexure axis, whereby the external transverse forces act as distributed body forces. A 2D warping theory is presented where the shear-warping process is driven by a shear-warping parameter, say 𝜏, 0 ≤ 𝜏 ≤ 1, in such a way that equilibrium between zero external (𝑡 (𝑛)𝑖 = 0) and internal (𝜎(𝑛)𝑖 = 0) tractions upon the lateral surface of the beam is complied with no matters the value of 𝜏 ∈ (0, 1). On letting 𝜏 vary within (0, 1), a continuous beam family, say  ∶= {𝜏 ∶ 0 ≤ 𝜏 ≤ 1}, is generated, which spans from the Euler–Bernoulli (EB) beam model 0 at 𝜏 = 0 (no shear strain, nor warping effects), to a series of shear deformable beam models 𝜏 up to 𝜏 = 1. For intermediate values of 𝜏, 0 < 𝜏 < 1, every beam model is featured by transversally nonuniform shear stresses accompanied by suitable 2D warping effects, while the boundary equilibrium condition with zero applied tractions is satisfied. The warping theory herein presented is an original generalization of a previous 1D type warping theory devised for rectangular cross sections. Applications are reported in which a circular cross section is considered and a beam problem in static conditions is worked out.