Abstract
Abstract In this paper, a new hybrid approach is presented based on the combination of the power series expansions and the Rayleigh-Ritz method for stability and free vibration analyses of axially functionally graded non-uniform beams resting on constant Winkler-Pasternak elastic foundation. In the proposed novel technique, the power series approximation is first adopted to solve the motion equation. Regarding this numerical methodology, the transverse displacement and all mechanical properties are expanded in terms of power series of a known degree. By solving the eigenvalue problem, one can acquire the fundamental natural frequencies. According to aforementioned method, the expression of vibrational mode shape is also determined. Based on the similarities existing between the vibrational and buckling deformation shapes, Rayleigh-Ritz method is finally employed to construct eigenvalue problem for obtaining the critical loads. In order to illustrate the correctness and convergence of the method, several numerical examples of axially non-homogeneous and homogeneous beams are conducted. The obtained outcomes are compared to the results of Finite Element Analysis in terms of ANSYS software and those of other available numerical and analytical solutions. The accuracy of the method is then remarked.
Highlights
Elastic flexural members whose cross-sectional profile changes partially or gradually along their length, known as non-prismatic beams, are widely spread in many engineering applications, due to their ability in improving both strength and stability, satisfaction aesthetic necessities and optimization weight of structures
In the absence of elastic foundation and non-homogeneous material, the validity of the formulation for analyzing the static buckling and free vibration is accomplished by comparing the results to those obtained by Ansys software and to other analytical and numerical solutions presented in the literature
The critical buckling load of non-prismatic beam resting on an elastic foundation can be estimated by adopting the principle of stationary total potential energy
Summary
Elastic flexural members whose cross-sectional profile changes partially or gradually along their length, known as non-prismatic beams, are widely spread in many engineering applications, due to their ability in improving both strength and stability, satisfaction aesthetic necessities and optimization weight of structures. Kim et al (2005) developed the dynamic stiffness matrix based on power series method to study the free vibration behavior of thin-walled beam with non-symmetric cross-section resting on two-parameter elastic foundation by considering shear deformation. Adomian decomposition method (ADM) was applied by Hassan and Nassar (2015) in order to determine the critical buckling loads in the static analysis and the natural frequencies for free vibration behavior of Timoshenko beam resting on a two parameter elastic foundation. In the present study, an improved and efficient hybrid numerical method to exactly evaluate the critical buckling loads and free transverse frequencies is proposed for any types of AFG members with linear, polynomial or exponential variation of mechanical properties and resting on uniform elastic foundation. Comments and conclusions are presented towards the end of the manuscript
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