The wavelet finite element method in the dynamic analysis of a functionally graded beam resting on a viscoelastic foundation subjected to a moving load

Over recent years there has been a growing demand for materials that possess a wide variation of constitutive properties, which may not naturally occur within homogeneous materials. The evolution of composite materials has led to the development of a relatively new class, commonly referred to as functionally graded materials, which consist of two or more materials (often metals and ceramics) with properties varying continuously with respect to spatial coordinates. In this paper, the dynamic response of a functionally graded (FG) beam is analysed using the wavelet finite element method (WFEM). The scaling functions of the Daubechies wavelet and B-spline wavelet on the interval (BSWI) families are employed as interpolating functions for the construction of the wavelet-based FG beam elements; based on Euler Bernoulli beam theory. The FG beam, comprising of steel and alumina, is assumed to vary continuously in the transverse and axial directions according to the power law. The free vibrations behaviour of a FG beam with different material distributions is compared with other approaches from published data to validate and assess the performance of this formulation. A FG beam resting on a viscoelastic foundation is analysed when subjected to a moving point load. The dynamic responses are evaluated using the Newmark time integration method. The effects of the material distribution, velocity of the moving load and damping of the system are discussed based on the numerical examples presented. The results indicate that WFEMs achieve higher levels of accuracy with fewer elements implemented, in comparison to the classical finite element method, in the analysis of the FG beam. Furthermore, the BSWI wavelet-based approach performs better than the Daubechies-based WFEM.