Warping shear stresses in nonuniform torsion by BEM

Abstract

In this paper a boundary element method is developed for the nonuniform torsion of simply or multiply connected prismatic bars of arbitrary cross section. The bar is subjected to an arbitrarily distributed twisting moment, while its edges are restrained by the most general linear torsional boundary conditions. Since warping is prevented, beside the Saint-Venant torsional shear stresses, the warping normal and shear stresses are also computed. Three boundary value problems with respect to the variable along the beam angle of twist and to the primary and secondary warping functions are formulated and solved employing a BEM approach. Both the warping and the torsion constants using only boundary discretization together with the torsional shear stresses and the warping normal and shear stresses are computed. Numerical results are presented to illustrate the method and demonstrate its efficiency and accuracy. The magnitude of the warping shear stresses due to restrained warping is investigated by numerical examples with great practical interest.

Bars under Torsional Loading: A Generalized Beam Theory Approach

“…In all the aforementioned references the analysis of the nonuniform torsional problem is not complete, since the secondary shear stresses due to warping are not evaluated. Finally, Sapountzakis and Mokos in [21,22] developed a boundary element solution for the general linear elastic nonuniform torsion problem of homogeneous or composite prismatic bars of arbitrary cross section subjected to an arbitrarily distributed or concentrated twisting moment and supported by the most general linear torsional boundary conditions. In these latter research efforts the evaluation of the secondary warping function is accomplished leading to the computation of the secondary shear stresses due to warping, while the developed procedure is a pure BEM [23], since it requires only boundary discretization.…”

“…From the examined example problems presented in [18][19][20][21][22] it is concluded that (i) the magnitude of the evaluated normal stresses, due to restrained warping compared with those due to bending, necessitates the consideration of these additional normal stresses near the restrained edges;…”

“…The extensive use of the aforementioned structural elements necessitates a reliable and accurate analysis of bars of arbitrary cross section subjected to torsional loading taking into account the geometrical nonlinearity. Though several researchers have dealt either with the linear nonuniform torsional behaviour of beams [17,21,22,40] or with the nonlinear uniform torsional behaviour of doubly symmetric beams [89,90], to the author's knowledge very little work has been done on the corresponding nonlinear nonuniform torsional problem of arbitrary cross section beams. Ghobarah and Tso [91], Attard [92], and Attard and Somervaille [93] have presented a set of displacement relationships for a straight prismatic thin-walled open beam applicable to situations where displacements are finite, the cross section does not distort, strains are small and flexural displacements are small to moderate while cross sectional twist can be large.…”

“…These last formulations do not stand on the assumption of a thinwalled structure and therefore the cross section's torsional rigidity is evaluated exactly without using the so-called SaintVenant's torsional constant. The torsional rigidity of the cross section is evaluated directly employing the primary warping function of the cross section [21] depending on its shape. Three boundary value problems with respect to the variable along the beam axis angle of twist, to the primary and to the secondary warping functions are formulated.…”

“…where the transverse displacement components V, are valid for large rotations [99]; , are the angles of rotation due to bending of the cross section with respect to its centroid; ( ) denotes the rate of change of the angle of twist regarded as the torsional curvature; V ( ), ( ) are the transverse displacement components of the shear center ; , are the primary and secondary warping functions with respect to the shear center , respectively [21]; ( ) is an "average" axial displacement of the cross section of the bar, that will be later explained.…”