Torsion Analysis of Functionally Graded Iso/Orthotropic Sections Using Differential Quadrature and pb2-Rayleigh Ritz Methods

This study investigates the determination of torsional rigidity and maximum shear stresses in arbitrarily shaped isotropic and orthotropic composite sections with functional grading of material. Numerical examples validate the approach, and a comprehensive parametric study is conducted. The paper demonstrates the practical application of the differential quadrature method and pb-2 Rayleigh–Ritz methods for evaluating shear stress and torsional rigidity in various section shapes, including quadrilaterals and triangles with straight and curved edges, as well as elliptical, circular, and rounded polygonal shapes. Boundary conditions are specified as Dirichlet. The analysis employs four-nodded or eight-nodded iso-parametric rectangular elements, and three-nodded or six-nodded triangular elements to map irregular physical domains into square or triangular computational domains. This approach reduces the governing equation for torsion, relying on the Prandtl stress function, to a second-order differential equation (Poisson's equation), significantly decreasing computational effort. The primary unknowns within the computational domain and along its boundary are the values of Prandtl's stress function (for Poisson's equation) or the warping displacement function (for Laplace's equation). This streamlined method effectively analyzes the torsional behavior of functionally graded isotropic and orthotropic composite sections with complex shapes.