Abstract
Abstract In the present paper, the free and forced vibration of multiple cracked multi span continuous beams made of functionally graded material (FGM) is investigated by the dynamic stiffness method. First, there are constructed dynamic stiffness matrix and nodal load vector of multiple cracked FGM beam element using Timoshenko beam theory and massless double spring model of crack. Then, effect of cracks parameters on free and forced vibration of the FGM continuous beams is examined. The theoretical developments are validated by numerical examples. The obtained results provide an efficient method to analyze free and forced vibration of multiple cracked FGM framed structures and assessment of the behavior of damaged structures.
Highlights
Graded materials (FGMs) have proved to be more benefit in comparison with the laminate composites and found widespread application in the high-tech industries such as aerospace, automobile, electronics, optics, chemistry, biomedical engineering etc
For vibration analysis of functionally graded material (FGM) structures, a number of methods have been proposed such as analytical (Nguyen, Vo, Nguyen, & Lee, 2016; Sina, Navazi, & Haddadpour, 2009), Rayleigh-Ritz method (Pradhan & Chakraverty, 2013), Galerkin (Sheng & Wang, 2018), combined Fourier series – Galerkin method (Zhu & Sankar, 2004), differential quadrature method (DQM) (Sundaramoorthy Rajasekaran, 2013a; Sınır, Çevik, & Sınır, 2018; Tang, Lv, & Yang, 2019), perturbation method (Sınır et al, 2018), asymptotic development method (Cao, Gao, Yao, & Zhang, 2018), discrete singular convolution and Taylor series expansion method (Wang & Yuan, 2017)
While the aforementioned methods are limited to apply for analysis of simple FGM beam structures, the Finite Element Method (FEM) developed in (Alshorbagy, Eltaher, & Mahmoud, 2011; Eltaher, Abdelrahman, Al-Nabawy, Khater, & Mansour, 2014; Mashat, Carrera, Zenkour, Al Khateeb, & Filippi, 2014; Naccache & El Fatmi, 2018) could be used for more complex structures such as that are composed from different beam elements
Summary
Graded materials (FGMs) have proved to be more benefit in comparison with the laminate composites and found widespread application in the high-tech industries such as aerospace, automobile, electronics, optics, chemistry, biomedical engineering etc,. The natural frequencies and mode shapes of multiple cracked FGM Timoshenko beam were investigated in (Lien, Duc, & Khiem, 2017a), (Lien, Đuc, & Khiem, 2017b) using the rotational spring model of cracks and actual position of neutral plane (Eltaher, Alshorbagy, & Mahmoud, 2013). The most important novelty of the present study is to use general solution for vibration shape of an FGM beam element with arbitrary number of cracks for constructing the dynamic stiffness matrix and load vector. This allows one to investigate crack continuous FGM beam with minimal number of elements equal to the number of spans in the beam. Dynamic stiffness matrix and the nodal load vector of a multiple cracked FGM Timoshenko beam element
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