Dynamic stability analysis of Euler-Bernoulli and Timoshenko beams composed of bi-directional functionally graded materials

Abstract

In this paper, dynamic stability analysis of beams composed of bi-directional functionally graded materials (BDFGMs) rested on visco-Pasternak foundation under periodic axial force is investigated. Material properties of BDFGMs beam vary continuously in both the thickness and longitudinal directions based on the two types of analytical functions (e.g. exponential and power law distributions). Hamilton's principle is employed to derive the equations of motion of BDFGMs beam according to the Euler-Bernoulli and Timoshenko beam theories. Then, the generalized differential quadrature (GDQ) method in conjunction with the Bolotin method is used to solve the differential equations of motion under different boundary conditions. It is observed that a good agreement between the present work and the literature results. Various parametric investigations are performed to study the effects of the gradient index, exponential and power law distributions, static load factor, length-to-thickness ratio and viscoelastic foundation coefficients on the dynamic stability regions of BDFGMs beam. The results show that the influence of gradient index of material properties along the thickness direction is greater than gradient index along the longitudinal direction on the dynamic stability of BDFGMs beam for both exponential and power law distributions. Also, the system becomes more stable and stiffer when BDFGMs beam is resting on visco-Pasternak foundation. Moreover, by increasing static load factor, the dynamic instability region moves to the smaller parametric resonance. The results of presented paper can be used to the optimal design and assessment of the structural failure and thermal rehabilitation of turbo-motor and turbo-compressor blades.