An Engineering Theory of thick Curved Beams loaded in-plane and out-of-plane: 3D Stress Analysis

Abstract

The complete theory of curved beams in terms of kinematic and stress, considering thickness effect and out-of-plane loads (torsion included) is provided in this paper. The engineering beam approach is followed; therefore, the key-point is the invariance of the cross-section shape during the undeformed-deformed configuration change. The normal circumferential stress and the tangential stresses due to torsion derive from the latter assumption; the other two normal stress (radial and axial) and tangential stresses due to pure shear are obtained through a generalization of Jourawsky's approach for curved beams. The obtained stress state is triaxial and takes into account five of the six components of the stress tensor. The kinematic equations obtained are valid for any compact (non-thin-walled) and non-hollowed cross-section. The complete stress solution, instead, is limited to the further assumption of double-symmetric cross-sections. For the case of rectangular sections an analytical solution in closed-form is available. For other sections, the procedure requires the numerical evaluation of some geometric integrals. The reliability of the proposed solution is tested in the planar loading case comparing it with the analytical solution in 2D Elasticity; for the out-of-plane loading cases the comparisons are carried out with FEM analyses (3D solid elements). The proposed model agrees very well for both comparisons. An interesting advantage of this solution is the capability to carry out post-processing stress analyses on curved beam structures, using the results obtained from beam finite elements (one-dimensional) and avoiding the recourse to higher element types (3D solid elements).