Buckling analysis of axially functionally graded columns with exponentially varying cross-section

Abstract

Owing to efficiency, rising stability of structures, reduction in structural weight and cost, and the improvements in fabrication process, tapered beams are extensively adopted in civil and mechanical structures. Furthermore, the use of functionally graded (FG) materials has been increasing in many mechanical components due to their conspicuous characteristics such as high strength, thermal resistance and optimal distribution of weight. In the present paper, a numerical model combining the power series expansions and the energy method is adopted for stability and free vibration analyses of exponentially tapered columns made up of axially functionally graded materials. The main purpose of this paper is also calculating the critical buckling loads and natural frequencies concurrently for AFG members with exponentially-varying geometrical properties. For this, a mixed power series expansions and the principle of stationary total potential energy as a first endeavor is presented. In this study, the material properties of the non-prismatic beam including Young’s modulus of elasticity and density of material are assumed to be graded smoothly along the beam axis by a power-law distribution of volume fractions of metal and ceramic. Moreover, the cross-sectional area and moment of inertia vary exponentially over the member’s length. In this regard, the power series approximation is applied to solve the fourth order differential equation of motion, since in the presence of variable cross-section and axially non-homogeneous material, stiffness quantities are not constant. All geometrical and material properties and displacement component are developed based on power series of an identified degree. The natural frequencies of the AFG beam with variable cross-section are derived by imposing the boundary conditions and solving the eigenvalue problem. The explicit expression of vibrational shape function is then derived based on this rigorous numerical method. The vibrational mode shapes of an elastic member are similar to the buckling ones. Therefore, the obtained deflected shapes of the considered non-prismatic beams can be used as deformation shape of member for the linear buckling analysis. The critical buckling load of exponentially tapered beam made of AFGMs can be then estimated by adopting the principle of stationary total potential energy. According to the steps mentioned above, for measuring the accuracy and competency of the proposed numerical procedure, two numerical examples including axially non-homogeneous and homogeneous column with non-uniform section are represented. Numerical results of the critical buckling loads and natural frequencies for various boundary conditions, different gradient index and cross-section variation are represented. Due to lack of similar research for the stability and free vibration analyses of elastic AFG beams with exponential variation of the cross-sectional area and moment of inertia, outcomes of homogeneous members are compared with the results presented in other available numerical and analytical references and those related to tapered beams with material variation are then reported. The accuracy of the method is then remarked. This method has many positive points consisting of efficiency, accuracy and simplicity contrasted with more complex numerical methods. It has to be noticed that the present novel numerical technique can be applied to determine the critical buckling loads and natural frequencies of axially functionally graded (FG) prismatic beams as well as non-prismatic ones.