On stress function in Saint-Venant beams

Excerpt

The elastostatic problem of a beam subject to shear and torsion was solved by Saint-Venant [7, 8] in terms of a unknown scalar field \(\varphi:\varOmega\mapsto \mathcal{R}\) describing the warping of the cross-section Ω to be evaluated by solving a Neumann boundary value problem. This problem depends linearly on the twist \({\alpha }\in \mathcal{R}\) and on the shearing d′∈V, being V the two-dimensional linear space of translations in the plane π _Ω of Ω. The twist is the average of the local-twist field over Ω and the shearing is the derivative of the bending curvature vector along the centroidal z-axis [21]. The Neumann boundary value problem can then be decomposed in three scalar problems, corresponding to three independent sets of kinematic parameters. It is known that these problems admit a unique solution, to within an additive constant, for any simply or multiply connected cross-section [22]. The knowledge of the cross-section warping is basic for the evaluation of the tangential stress field τ(d′,α):Ω↦V which can be expressed as sum of shear and twist contributions: τ(d′,α)=τ ^sh(d′)+τ ^tw(α), setting by linearity τ ^sh(d′):=τ(d′,0) and τ ^tw(α):=τ(o,α). The resultant vector of the twist field τ ^tw(α) vanishes, while the one of the shear field τ ^sh(d′) is equal to the shear force. The shear centre is the pole in the plane π _Ω characterized by vanishing of the shear stress twisting moment [1, 4, 5, 9‐11, 13, 21‐23, 25]. By integrating the differential equation of equilibrium, Prandtl [14] showed that, for any simply or multiply connected cross-section, the twist field τ ^tw(α) can be expressed in terms of the gradient of a stress function. Its evaluation is based on the solutions of 1+n Dirichlet boundary value problems, being n≥0 the number of holes in the cross-section, and of an algebraic system providing the integration constants.¹ In the general case in which d′≠o, existence of a stress function for the tangential stress field τ(d′,α) can be ensured only under special conditions. The present note deals with the question concerning the requirements to be imposed on the cross-section to assure existence of the stress function for a given d′≠o. A full answer is provided in Proposition 1 by generalizing the condition enunciated in [12] which was based on the special assumption of a shear force acting along a principal direction. The interest for existence of stress function is still well motivated being this issue often not explicitly discussed in literature. Also, treatments in terms of stress function are considered for multiply connected cross-sections not fulfilling the necessary requirements [24]. The intrinsic treatment adopted hereafter is in line with the ones in [16‐20]. …