Free Vibration Analysis for Cracked FGM Beams by Means of a Continuous Beam Model

Abstract

Based on Euler-Bernoulli beam theory and a continuous stiffness beam model, the free vibration of rectangular-section beams made of functionally graded materials (FGMs) containing open edge cracks is studied. Assuming the material gradients follow exponential distribution along beam thickness direction, the conversion relation between the vibration governing equations of a FGM beam and that of an isotropic homogenous beam is deduced. A continuous function is used to characterize the bending stiffness of an edge cracked FGM beam. Thus, the cracked FGM beam is treated as an intact beam with continuously varying bending stiffness along its longitudinal direction. The characteristic equations of beams with different boundary conditions are obtained by transfer matrix method. To verify the validity of the proposed method, natural frequencies for intact and cracked FGM beams are calculated and compared with those obtained by three-dimensional finite element method (3D FEM) and available data in the literature. After that, further discussions are carried out to analyze the influences of crack depth, crack location, material property, and slenderness ratio on the natural frequencies of the cracked FGM beams.

Vibration analysis of cracked beam based on Reddy beam theory by finite element method

“…The crack depth and location affect the properties of a structure including the natural frequency, the vibration response amplitude, and the mode shape. These changes in the structure dynamical properties can be easily measured, and therefore, a number of papers have focused on the analysis of cracked beams (Ahmed and Uddin, 2019; Aria et al, 2019; Banerjee and Guo, 2009; Ebrahimi et al, 2005; Farghaly, 1994; Kim et al, 2018; Kisa et al, 1998; Labib et al, 2014; Loya et al, 2009; Mazaheri et al, 2018; Torabi and Nafar Dastgerdi, 2012; Yamuna and Sambasivarao, 2014; Yang et al, 2015; Yendhe et al, 2016) and this is also the aim of this paper.…”

“…Kim et al (2018) examined the free vibration behavior of a cracked Timoshenko beam model based on ultra-spherical polynomials for general boundary conditions modeled using elastic springs at both ends. Yang et al (2015) investigated the free vibration of rectangular cracked Bernoulli-Euler beam made of functionally graded materials (FGMs) using the transfer matrix method (TMM). Much research work considers the free vibration analysis of cracked beams using Bernoulli-Euler (Aria et al, 2019; Banerjee and Guo, 2009; Labib et al, 2014; Loya et al, 2009; Mazaheri et al, 2018) and Timoshenko (Farghaly, 1994; Kisa et al, 1998; Torabi and Nafar Dastgerdi, 2012) theories, leading to the conclusion that the presence of a crack reduces the natural frequency of vibration.…”

Free and forced vibration analysis of multiple cracked FGM multi span continuous beams using dynamic stiffness method

“…The basic concepts of fracture mechanics for FGM structures have been given in (Erdogan & Wu, 1997;Jin & Batra, 1996) that are essentials to develop the models of crack in FGM structures. Namely, the continuous stiffness model (Christides & Barr, 1984; E. C. Yang, Zhao, & Li, 2015) proposes to treat a cracked FGM beam as the beam with continuously varying bending stiffness along its span. Therefore, the governing equation of vibration of the cracked FGM beam are similar to those of nonuniform beam that increases the difficulty to study beams made of FGM.…”