On the Exact Solution of Non-Uniform Torsion for Beams with Axial Symmetric Cross-Section
In the traditional theory of non-uniform torsion the axial displacement field is expressed as the product of the unit twist angle and the warping function. The first one, variable along the beam axis, is obtained by a global congruence condition; the second one, instead, defined over the cross-section, is determined by solving a Neumann problem associated to the Laplace equation, as well as for the uniform torsion problem. So, as in the classical theory the warping function doesn’t punctually satisfy the first indefinite equilibrium equation, the principal aim of this work is to develop a new theory for non-uniform torsion of beams with axial symmetric cross-section, fully restrained on both ends and loaded by a constant torque, that permits to punctually satisfy the previous equation, by means of a trigonometric expansion of the axial displacement and unit twist angle functions. Furthermore, as the classical theory is generally applied with good results to the global and local analysis of ship structures, two beams having the first one an open profile, the second one a closed section, have been analyzed, in order to compare the two theories. Keywords—Non-uniform torsion, Axial symmetric cross-section, Fourier series, Helmholtz equation, FE method.
Nonuniform elastic torsion and flexure of members with asymmetric cross-section
Abstract In a previous paper, the authors considered the linear response of a solid prismatic elastic member to a torque which varied along the length of the member. The cross-section considered was only restricted by the requirement of two axes of symmetry. That restriction is removed in the present work, but its removal implies that the response to torque will generally now involve bending, so the treatment has been extended to allow arbitrary loading in torsion/flexure. It transpired that a satisfactory analysis is incompatible with the concept of a “shear centre”, and the existence of a shear centre, as part of a rigorous theory or as an approximation, is clarified in the paper.
The General theory of Cylindrical and Conical Tubes Under Torsion and Bending Loads
Summary Parts 1 to 5 of this paper (February, September and November 1947 issues of the JOURNAL) investigated the stresses and deformations of closed tubes in which the thicknesses were governed by the t s * and t * laws. In the present part, the analysis is extended to multi-cell tubes with openings, open tubes with or without St. Venant torsional stiffness, and to tubes formed by joining elements of different cross-sections. To illustrate the theory a numerical example of the stressing of a four-boom wing consisting of seven joined elements is fully worked out. Finally, an appendix gives practical methods of dealing with tubes which do not conform to the t s * and t * laws, and of finding approximate solutions for four-boom tubes with direct stress carrying covers.