Higher Order Approximations for Bending of FG Beams Using B-Spline Collocation Technique

Abstract

In the present study, a functionally graded cantilever beam has been analyzed to observe its deformation behavior and stress variations. While the material properties (modulus of elasticity, modulus of rigidity, and density) have been varied along the height of the beam, Poisson’s ratio has been considered a constant. The governing equations have been derived using Hamilton’s Principle in the framework of higher-order beam theory. The derived equations are then simplified to a single equation, which is similar to that of isotropic beams. However, the work is extended to include a few higher-order terms and to study the effect of the incorporation of these terms on the resulting FG beam behavior. The development of a single governing equation for studying the statics and dynamics of an FG beam with the incorporation of higher-order terms is a unique part of the report. The solution of the governing equation is approached using approximate methods; in this work, the B-spline collocation technique is used to arrive at the results. A sixth-order B-spline basis function is used as an approximating polynomial, and the Greville abscissa has been used to generate collocation points. The obtained results have been verified against standard literature to find a satisfactory match. The results include comparative plots for normalized bending and transverse shear stresses, with and without the inclusion of higher-order terms. The results are then extended to study the effect of material index on the deformation and stress behavior of FG beams. The effect of aspect ratio on results is also studied for comparison of various beam theories.

Keywords

• functionally graded material
• hamilton principle
• b-spline collocation
• power law
• axial stress
• shear stress