Keywords: free vibration, cracked beam, composites, Timoshenko beam, dynamic stiffness matrix, compliance, stress intensity factor.

The application of the dynamic stiffness matrix (DSM) method for the dynamic analysis of flexible homogeneous and composite beams is well established. The DSM is based on the closed form solution of the governing equations of the system. In this study, the focus is on the modelling and free vibration analysis of cracked composite beam structures. The beam element is composed of composite material with a unidirectional plies, which results in a bending-torsion coupling [1]. Rigidities for bending, torsion and coupled bending-torsion are calculated from classical laminate theory. The element DSM is frequency dependent, containing properties of both stiffness and mass. The analysis of Euler-Bernoulli cracked beams have been previously developed by the authors [2] and Wang et al. [3]. To the best of authors' knowledge, no one has included the shear and rotary inertia effects in the analysis.

The explicit analytical expressions for the DSM of a bending-torsion thick (Timoshenko) beam, including the shear and rotary inertia effects, are exploited. If a transverse through thickness crack is present in the beam, the system can be split at the crack location into two elements. Following the same local flexibility concept as presented by Wang et al. [3], and assuming the crack can be modelled as a number of springs, a new cracked Timoshenko composite beam element is developed. The stress intensity factors (SIF) available for an Euler-Bernoulli composite beam are adjusted to include shear deformation by multiplying the stress component of the SIF by the shear correction factor, similar to Krawczuk et al. [4]. A compliance matrix can be formed and then inverted to generate the stiffness matrix of the equivalent linear and rotational spring elements representing the crack. The expressions for the DSM can be directly used to compute the natural frequencies of materially coupled bending-torsion vibrations of cracked composite beams. Intact beam and spring element matrices are then assembled in the usual way and the boundary conditions are applied to form the overall dynamic stiffness matrix of the defective structure. This practice results in a nonlinear eigenvalue problem. The determination of the natural frequencies then follows from the Wittrick-Williams (W-W) algorithm [5]. The cracked Timoshenko DSM is tested for beams with various thicknesses. A comparison is made between the natural frequencies for both intact and defective thin and Timoshenko composite beams. For a uniform beam only one element is required to model each intact portion of the defective beam.

1

J.R. Banerjee, "Free Vibration of Axially Loaded Composite Timoshenko Beams Using the Dynamic Stiffness Matrix Method", Computer & Structures, 69, 197-208, 1998. doi:10.1016/S0045-7949(98)00114-X

2

S.R. Borneman, S.M. Hashemi, H. Alighanbari, "Vibration Analysis of Doubly Coupled Cracked Composite Beams: An Exact Dynamic Stiffness Matrix", International Review of Aerospace Engineering, 1(3), 298-309, 2008.

3

K. Wang et al., "Modeling and Analysis of Cracked Composite Cantilever Beam Vibrating in Coupled Bending and Torsion", J. of Sound and Vibration, 284, 23-49, 2005. doi:10.1016/j.jsv.2004.06.027

4

M. Krawczuk, M. Palacz, W. Ostachowicz, "The dynamic analysis of a cracked Timoshenko beam by the spectral element method", Journal of Sound and Vibration, 264, 11391153, 2003. doi:10.1016/S0022-460X(02)01387-1

5

W.H. Wittrick, F.W. Williams, "A General Algorithm for Computing the Natural Frequencies of Elastic Structures", Quart. Jour. Mech. and Applied Math, Vol. XXIV, Pt. 3, 263-284, 1971. doi:10.1093/qjmam/24.3.263

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