Abstract
Free vibration analysis of beams with single delamination undergoing bending-torsion coupling is made, using traditional finite element technique. The Galerkin weighted residual method is applied to convert the coupled differential equations of motion into to a discrete problem, where, in addition to the conventional mass and stiffness matrices, a delamination stiffness matrix, representing the extra stiffening effects at the delamination tips, is introduced. The linear eigenvalue problem resulting from the discretization along the length of the beam is solved to determine the frequencies and modes of free vibration. Both “free mode” and “constrained mode” delamination models are considered in formulation, and it is shown that the continuity (both kinematic and force) conditions at the beam span-wise locations corresponding to the extremities of the delaminated region, in particular, play a great role in “free mode” model formulation. Current trends in the literature are examined, and insight into different types of modeling techniques and constraint types are introduced. In addition, the data previously available in the literature and those obtained from a finite element-based commercial software are utilized to validate the presented modeling scheme and to verify the correctness of natural frequencies of the systems analyzed here. The paper ends with general discussions and conclusions on the presented theories and modeling approaches.
Highlights
Materials or components built up in layers are used in various manufacturing sectors ranging from shipbuilding to aerospace. e ever-increasing application of such layered structural elements is primarily due to their many attractive features such as high specific stiffness, high specific strength, good buckling resistance, and formability into complex shapes, to name a few
A systematic finite element method (FEM)-based formulation and numerical solutions for the free vibration modeling and analysis of delaminated beams under axial compressive load and end moment were presented. e delaminated beam con gurations are modeled and analyzed as the assemblage of four Euler–Bernoulli beams attached at the defect edges
Both the “free mode” and “constrained mode” models were created to carry out buckling and vibration analyses, and the results were validated against the analytical data and other numerical results available in the open literature
Summary
Materials or components built up in layers are used in various manufacturing sectors ranging from shipbuilding to aerospace. e ever-increasing application of such layered structural elements is primarily due to their many attractive features such as high specific stiffness, high specific strength, good buckling resistance, and formability into complex shapes, to name a few. The characteristic equation of the defective system is obtained by dividing the multidelaminated beam-columns into segments and by imposing recurrence relation from the continuity conditions on each subbeam-column To verify their results, experimental results are obtained for isotropic single delaminated beamcolumns and confirmed that the sizes, location, and number of delaminations have significant effect on the system’s vibration and stability. By implementing the Galerkin weighted residual method, the governing di erential equations of motion are discretized, leading to element matrices As it will be explained later in this paper, in addition to the mass and sti ness matrices commonly resulting from the conventional FEM formulation, the presented procedure leads to three extra sti ness matrices: a geometric sti ness matrix caused by the axial load, a delamination sti ness matrix, resulting from the relationships between the two delamination region’s extremities, and a coupling geometric sti ness matrix caused by the applied end moment. While the FEM model presented in this paper assumes isotropic materials, further research is underway to extend it to sandwich and bre-reinforced laminated composite beams, characterized by an extensional response coupled with exural/torsional and coupled bending-torsion vibration
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