Keywords: torsion, finite element, wide flange sections, shear deformation.

Wide flange steel sections subjected to twisting moments are generally prone to large warping stresses and excessive angles of twist. It is therefore common practice in the steel construction industry to avoid twisting moments in steel assemblies consisting of wide flange sections whenever possible. In a number of practical applications however torsion cannot be avoided and the designers are compelled to count on the torsional resistance of wide flange sections in their design. Also, the design of wide flange sections under twisting moments is frequently governed by serviceability limit states arising from the member attaining excessive angles of twist before its material undergoes plastic deformations. Therefore, an accurate determination of the response of wide flange members under torsional moments in the elastic regime is of practical importance.

In this context, this paper aims at formulating the equilibrium conditions, boundary conditions, and developing a finite element formulation which accurately captures the elastic response of wide flange members of long and short spans. The classical Vlasov theory (reference [1]), commonly adopted in similar classes of problems, is based on two fundamental kinematic assumptions. These are: a) In-plane deformations of the section are negligible, and b) Shear strains along the section mid-surface are negligible.

Under these two assumptions, the torsional equilibrium equation describing the behaviour of thin-walled beams has been well established. Closed form solutions of the resulting equilibrium equations under various boundary conditions are provided in a number of texts. The second Vlasov assumption restricts the applicability of the theory for long and moderate span members where shear deformation effects are expected to have negligible effects. In contrast, the formulation proposed in the paper incorporates the effect of shear deformations by conceiving each flange of the section to act as a Timoshenko beam. By combining the first kinematic assumption of the Vlasov thin-walled beam theory with the kinematic assumption of the Timoshenko deep beam (which relaxes the angle between the beam centreline and the angle of rotation of the normal plane) for the flanges, the equilibrium equations and associated boundary conditions are formulated. The resulting differential equations of equilibrium involve two field variables (the angle of twist of the section and the rotation angle of the normal to the flange centreline) and are found to be coupled. The coupled equilibrium conditions are then explicitly integrated and the resulting integration constants are subsequently related to the generalized nodal displacements. A family of hyperbolic interpolation functions involving two parameters is derived. The interpolation functions are subsequently employed in conjunction with the principle of virtual work to formulate an element stiffness matrix which is exact within the limitations of the kinematic assumptions made. The resulting finite element features two end nodes with two degrees of freedom per node. The derived stiffness matrix and associated energy equivalent load vectors are shown to yield nodal displacement and force values in exact agreement with those based on the closed form solution of the governing equations of equilibrium, for any set of externally applied member twisting moments while maintaining the number of finite elements used to a minimum. For beams with large and moderate spans, the newly developed finite element predicts angles of twist, normal stresses and shearing stresses in close agreement with those based on the classical Vlasov solution. For short span beams however, the formulation reveals that the classical Vlasov solution under-predict the angle of twist.

1

V.Z. Vlasov, "Thin Walled Elastic Beams", Second Edition, Israel Program for Scientific Translations, Jerusalem, Israel (1961).

purchase the full-text of this paper (price £20)

go to the previous paper
go to the next paper
return to the table of contents
return to the book description
purchase this book (price £140 +P&P)