Stochastic Perturbation-Based Finite Element for Free Vibration of Functionally Graded Beams with an Uncertain Elastic M
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STOCHASTIC PERTURBATION-BASED FINITE ELEMENT FOR FREE VIBRATION OF FUNCTIONALLY GRADED BEAMS WITH AN UNCERTAIN ELASTIC MODULUS
N. V. Thuan1* and T. D. Hien2,3
Keywords: finite-element methods, random field, composites, free vibration, beams. The stochastic eigenvalue problem for free vibrations of functionally graded beams with a random elastic modulus is investigated. The effective material properties and beam cross section are assumed to vary continuously in different directions according to the exponential law. The governing equations for the natural frequency of the functionally graded beams are derived from Hamilton’s principle. In the stochastic finite-element method, the random process was discretized by averaging random variables in each element to increase the accuracy of the natural frequency found. A solution of the stochastic eigenvalue problem formulated was obtained using the perturbation technique in conjunction with the finite-element method. The spectral representation was used to generate a random process to employ the Monte Carlo simulation. A good agreement was obtained between the results of the first-order perturbation technique and the Monte Carlo simulation.
1. Introduction Uncertainties in the material parameters of structures are natural, and their existence is inevitable. For a functionally graded material (FGM), the level of this uncertainty can be very high compared with that for isotropic homogeneous materials.
Department of Transportation Engineering, Nha Trang University, Nha Trang, Vietnam Department of Civil Engineering, University of Transport and Communications, Hanoi, Vietnam 3 Research and Application Center for Technology in Civil Engineering (RACE), University of Transport and Communications, Hanoi, Vietnam *Corresponding author; tel.: +84-0393007896; e-mail: [email protected] 1 2
Russian translation published in Mekhanika Kompozitnykh Materialov, Vol. 56, No. 4, pp. 715-730, July-August, 2020. Original article submitted May 16, 2019; revision submitted February 12, 2020. 0191-5665/20/5604-0485 © 2020 Springer Science+Business Media, LLC
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Functionally graded materials (FGMs) are a new class of advanced inhomogeneous composites. The FGM concept originated in Japan in 1984, during a space-plane project [1]. The composition of this material was designed to vary continuously in it. This continuity reduces the influence of interfaces and exclude high interfacial stress. Therefore, FGMs have received wide-spread recognition in various engineering applications, including the mechanical and aerospace engineering, medicine, defense industry, and energy production. Such systems make use of each type of structural member in the most efficient manner in order to maximize its benefits [2]. Various studies on the behavior of structures made of FGMs have been performed. Some researchers have investigated the statics, dynamics, and buckling of FG beams [3-6]. For FG plates, reports on bending [7, 8], dynamics [9-15], stability [8, 16, 17] problems have been published. The traditi
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