Development of Elements for Analysis of Functionally Graded Beams Using Applied Element Method

Functionally graded material (FGM) is a substitute for laminated composite materials. It avoids the disadvantages of laminated composites. Unlike laminated composite materials, in FGM, there is a continuous change in properties across the depth. It is widely used in the areas of aerospace, biomechanics, automobile, etc. Hence, static and dynamic analysis of FGM is of interest today. Researchers have developed analytical methods for both static and dynamic analyses of FGM. But, numerical analysis is essential to avoid assumptions in the analysis. Usually, the most popular numerical method, the finite element method (FEM), is used for the analysis of FGM structures. To implement the variation in properties in finite element software, FGM is thoroughly meshed along the direction of variation in property. The applied element method (AEM) developed in 1997 has some advantages when compared to FEM. Faster processing, lesser memory requirement and simplicity of analysis are some of them. Usually, most of the research on AEM is on concrete and steel structures. From the formulation, it can be seen that AEM can be used most effectively for FGM. So, in this paper, this aspect is highlighted by conducting static analysis and solving eigen value problem in dynamic analysis of beams made of FGM. The approach in conventional AEM to determine stiffness matrix is simplified so that it is suitable for the analysis of functionally graded (FG) beams. Two-dimensional elements are developed for FG beams with various material property distribution laws. The stiffness matrix and mass matrix are derived, and the method to find strain and stress at various points in FG beams is discussed. The deflection and stress distribution obtained by AEM is compared with those determined by analytical means. Finite element analysis is also carried out and compared. The natural frequency of beams with different length-to-depth ratio and support conditions are found and compared with the results in the literature. From the study, it is seen that AEM can be effectively used for the analysis of structures made of FGM.