Abstract
Torsional problems commonly arise in frame structural members subjected to unsymmetrical loading. Saint-Venant proposed a semi inverse method to develop the exact theory of torsional bars of general cross sections. However, the solution to the problem using an analytical method for a complicated cross section is cumbersome. This paper presents the adoption of the Saint-Venant theory to develop a simple finite element program based on the displacement and stress function approaches using the standard linear and quadratic triangular elements. The displacement based approach is capable of evaluating torsional rigidity and shear stress distribution of homogeneous and nonhomogeneous; isotropic, orthotropic, and anisotropic materials; in singly and multiply-connected sections. On the other hand, applications of the stress function approach are limited to the case of singly-connected isotropic sections only, due to the complexity on the boundary conditions. The results show that both approaches converge to exact solutions with high degree of accuracy.
Highlights
Torsional problems commonly arise in three dimensional structural members subjected to unsymmetrical loading or internal twisting moment exerted to satisfy local equilibrium in members’ connections [1]
The twisting moment causes rotational deformation of cross section; which values are highly attributed to the resistance of cross section against twisting i.e. torsional rigidity
This research aims to develop a simple finite element method (FEM)-based program in MATLAB to deal with elastic torsional problems of noncircular cross sections, which formulation is based on the displacement and stress approaches
Summary
Torsional problems commonly arise in three dimensional structural members subjected to unsymmetrical loading or internal twisting moment exerted to satisfy local equilibrium in members’ connections [1]. The twisting moment causes rotational deformation of cross section; which values are highly attributed to the resistance of cross section against twisting i.e. torsional rigidity. Torsional rigidity of elastic circular sections can be evaluated using conventional mechanics and elasticity since the sections along member’s length remain plain and undistorted [2,3]. In a noncircular cross sectional member, the sections along its length warp due to various twisting. Applying the torsion theory for circular to non-circular sections results in violation of the equilibrium equations and boundary conditions. Saint-Venant proposed a semiinverse approach to solve torsional problems by assuming the unknown displacements to fulfill the equilibrium equations and boundary conditions
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