Analytical and numerical analysis of coupled linear vibrations of fgm beams and frames

Abstract

Functionally graded materials (FGM) are composite materials with continuously varying material properties. They have been used in aerospace, defense, aviation industries as structural elements such as rods, beams, frames, plates or shells due to their advantages. Researchers have been investigating mechanical behavior of FGM structures for static deflection, buckling, free or forced vibration cases. In this study, vibration analysis of FGM beam and frames are studied via analytical and numerical methods for different beam theories. In the first chapter, literature review is presented about FGM beam vibrations. Scope, purpose and the method of the thesis are stated. In the second chapter, analytical model is derived for Euler-Bernoulli and Timoshenko beam theories. To this aim, Hamilton Principle is utilized for obtaining the governing equations and boundary conditions of beams. In the third chapter, finite element method (FEM) is revisited and a brief theoretical background is provided. For analysis of Euler-Bernoulli beam theory; linear and quadratic formulations are derived. FEM modelling of Timoshenko beam theory requires methods to eliminate "shear locking" phenomenon which is a numerical error that overestimates shear strain potential energy. In order to eliminate that the shear locking error; interdependent interpolation element method and reduced order integration element method are utilized. Another numerical problem, arising for linear elements of reduced order integration, which is called "hourglassing", is eliminated by using quadratic elements in case of reduced order integration. In the fourth chapter, analytical and FEM results of beams are compared with each other and verified by comparison with literature. Once the models are verified, a parametric studies are conducted for Euler-Bernoulli beam theory and Timoshenko beam theory in order to observe non-dimensional natural frequency change with respect to variations of $L/h$ ratio and power index (k) for clamped-clamped, simply-supported, clamped-pinned and clamped free ends. Mass normalized mode shapes are also presented for numerical and analytical solutions with Modal Assurance Criterion (MAC) matrices. Verified FEM formulations are applied to frames which are more complex structures than beams the analytical investigation of which cumbersome. FEM results of frames are also compared with literature studies. For frame analysis three different scenarios are presented with varying power index (k). In fifth chapter, results of parametric study are discussed and commented on. Effects of L/h ratio, power index (k) and boundary conditions on non-dimensional natural frequencies are explained and highlighted. Final remarks are made and possible next steps are recommended.