An Approximate Solution of Functionally Graded Timoshenko Beam Using B-Spline Collocation Method

Abstract

Collocation methods are popular in providing numerical approximations to complicated governing equations owing to their simplicity in implementation. However, point collocation methods have limitations regarding accuracy and have been modified upon with the application of B-spline approximations. The present study reports the stress and deformation behavior of shear deformable functionally graded cantilever beam using B-spline collocation technique. The material grading is along the beam height and varies according to power law. Poisson’s ratio is assumed to be a constant. The equations are derived using virtual work principle in the framework of Timoshenko beams to obtain a unified formulation for such beams. A sixth order basis function is used for approximation and collocation points are generated using Greville abscissa. Deformation and stresses; bending (axial) stresses and transverse (shear) stresses, and position of neutral axis are studied for a wide range of power law index values. The results are reported along the beam cross-section and beam length.