Free vibration analysis of bidirectional functionally graded double-tapered beam using the p-version of the finite element method

In this paper, we conducted a frequency analysis of bidirectional functionally graded double-tapered beam (BDFG beam) using the p-version of the finite element method, the beam was modeled by the Euler Bernoulli beam theory, the global matrices of the equation of motion are determined by applying the Lagrange equation on the kinetic and deformation energies. The material properties are assumed to vary along the thickness and width directions according the power-law distribution. The numerical results are obtained by a developed Matlab code and compared with Ansys Workbench and Solidworks for a pure metallic tapered cantilever beam and with previous study for a unidirectional functionally graded tapered cantilever beam (UDFG beam), case studies were carried out to analyze the influence of tapering angles, the power law indeces, and boundary conditions on the natural frequencies of the beam; these studies demonstrate the advantage of the proposed bidirectional functionally graded tapered beam over the metallic and unidirectional functionally (UDFG) beams.