Nonlinear frequency behavior of cracked functionally graded porous beams resting on elastic foundation using Reddy shear deformation theory

Abstract

This paper presents the nonlinear frequency behavior of cracked functionally graded porous beams subjected to various boundary conditions using Reddy shear deformation theory and Green’s tensor together with the Von Karman geometric nonlinearity. The material properties of the beam vary exponentially in thickness direction. A generalized differential quadrature method (GDQM) in conjunction with a direct numerical iteration method is developed to solve the system of equations derived by means of Hamilton’s principle. Demonstrating the convergence of this method, the verification is performed by using extracted results from a previous study based on the Euler and Timoshenko beam theory. The results for extensive studies are provided to understand the influences of the different gradient indexes, vibration amplitude ratios, porosity indexes, shear and elastic foundation parameters, and boundary conditions on the nonlinear frequency behavior. The location of crack plays the significant role on the nonlinear frequency ratios. The frequency ratios hit the minimum when the crack is located in mid-span of the beam. In this regard, the effect of shear stiffness of foundation is much more than that of Winkler one in the increasing mid-span value of nonlinear frequency ratio. It is also shown that the crack location results in decline of frequency ratios in the mid-span of beam specially for the cracks located closer to surface, the deeper ones lead to get nonlinear frequency ratios rise much more due to the fact that the deeper the crack is, the weaker the crack section becomes. The results of this paper and the effects of these parameters can be used in the optimal design of functionally graded beams and in crack prediction, detection, and monitoring techniques.

Nonlinear frequency analysis of porous Bi directional functionally graded beams utilizing reddy shear deformation theory

“…(40) [ 33 , 35 , [46] , [47] ] using this methodology: where ; ; ; are the first, second, third, and fourth-order weighting coefficients of the DQM, respectively Eq. (41) [ 33 , [46] , [47] ]. …”

“…Malekzadeh explained how to use the harmonic balancing approach, and this results in Eq. (55) [ 33 , [35] , [47] ]: …”

Robust boundary control approaches to the stabilization of the Euler–Bernoulli beam under external disturbances

“…There are great differences between different beam systems due to their materials, end support methods, and so on. For example, single material (Liu and Guo (2018); Guo et al (2018)) and composite beams (Berczyński and Wróblewski (2005); Ghasemi et al (2016); Bouazza et al (2019a,b); Bouazza and Zenkour (2020); Derbale et al (2021); Forghani et al (2022)), hinged-hinged (H-H) and clamped-clamped (C-C) beams (Taeprasartsit (2015)), as well as strain-driven and stress-driven beams (Zhang and Qing (2021)) are widely investigated. The Euler–Bernoulli beam is a kind of flexible structure system which has attracted a lot of attentions in terms of the controller design and the stability analysis over the past few decades.…”