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

E Chuan Yang,Xiang Zhao,Ying Hui Li

doi:10.1155/2015/197049

E Chuan Yang, Xiang Zhao + Show 1 more

Open Access

https://doi.org/10.1155/2015/197049

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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.

Highlights

Graded materials (FGMs) are composite materials made from at least two components that follow a certain material gradient distribution
Free vibration of Functionally graded materials (FGMs) beams with open edge cracks is investigated in this paper based on Euler-Bernoulli beam theory and the continuous beam model
The numerical discussion shows that FGM beams with more stubby shapes and lower Young’s modulus ω1c /ω1 ω2c /ω2

Summary

Introduction

Graded materials (FGMs) are composite materials made from at least two components that follow a certain material gradient distribution. Xiang et al [13, 14] investigated the natural frequencies and identification problems of cracked shafts; a lumped flexibility model and the wavelet theories are employed to construct the rotating Rayleigh-Euler and Rayleigh-Timoshenko beam elements. They researched the plane problem in infinite plates and drew the conclusion that the effect of Poisson’s ratio on the SIFs is very limited [22] They investigated the stress distribution near the crack-tip of the FGM plates containing open edge cracks and tabulated the dimensionless SIFs of the cracks of different crack depths and different material gradients [25]. Based on Euler-Bernoulli beam theory and the rotational spring model, Aydin [30] used a third-order determinant to analyze the natural frequencies of damaged FGM beams containing arbitrary numbers of cracks.

Theoretical Models

Validation of the Proposed Method

Numerical Results and Discussions

Conclusions