NURBS-based modeling of bidirectional functionally graded Timoshenko beams for free vibration problem

An extension of the isogeometric analysis method is used for the description of the material property and investigate the free vibration characteristics of bidirectional functionally graded (BDFG) Timoshenko beam. One-dimensional Non-uniform Rational B-Spline (NURBS) basis functions are used to construct the beam geometry as well as approximate the solution, whereas the gradations of material property are represented by two-dimensional basis functions. Different control nets are used for interpolation of material property variations throughout the domain as well as the geometry and the analysis. Four specific types of symmetrical and asymmetrical material property distributions are studied and the volume fractions of constituents are defined by power and exponential laws. In particular, the combination of both degree elevation and knot insertion, namely k-refinement, is implemented due to its ability to control the continuity. The symmetric material distribution is defined to be C-0 continuous at the material interface and thereby yields a more accurate description for symmetric BDFG beam types. Several numerical examples are presented to demonstrate the performance and efficiency of the proposed approach. Comparison of the obtained results with those of the other existing studies with the common type of distributions confirms the accuracy of the proposed model. (C) 2016 Published by Elsevier Ltd.