Innovative Approaches to Torsion Analysis in Functionally Graded Iso/Orthotropic Sections: A Differential Quadrature and pb2-Rayleigh Ritz Exploration

Abstract This study focuses on the determination of torsional rigidity and maximum shear stresses in arbitrarily shaped isotropic and orthotropic composite sections with functional grading of material. A series of numerical examples are solved to validate this approach, with a comprehensive parametric study also conducted. The paper presents the practical application of the DQM and pb-2 Rayleigh-Ritz methods for evaluating shear stress and torsional rigidity in various section shapes, including quadrilaterals and triangles with both straight and curved edges, as well as elliptical, circular, and rounded polygonal shapes. The boundary conditions are specified as Dirichlet. To facilitate the analysis, four-nodded (for sections with straight edges) or eight-nodded (for sections with curved edges) iso-parametric rectangular elements, and three-nodded (for sections with straight edges) or six-nodded (for sections with curved edges) for triangular elements are used to map the irregular physical domain into a square or triangular computational domain, as appropriate. A noteworthy advantage of this approach is that the governing equation for torsion, which relies on the Prandtl stress function approach, reduces to a second-order differential equation (Poisson's equation). The transformation to this second-order equation is efficiently accomplished, resulting in reduced computational effort. Both within the computational domain and along its boundary, the primary unknowns are the values of Prandtl's stress functional (for Poisson's equation) or the warping displacement function (for Laplace's equation). This approach streamlines the analysis, making it a powerful tool for understanding the torsional behaviour of functionally graded isotropic and orthotropic and composite sections with complex shapes.