A Dynamic Stiffness Element for Vibration of Cracked Composite Timoshenko Beams

摘要

A Dynamic Stiffness Matrix (DSM) is developed to analyse the free vibrationrncharacteristics of cracked, thick, laminated, unidirectional, unbalanced compositernbeams. Based on the closed form solution of the governing differential equations, anrnexisting ‘exact’ DSM formulation for bending-torsion vibration of an intactrncomposite beam is first briefly discussed. Stress intensity factors, corrected forrngeometry and material anisotropy, are used to develop the local flexibility of arnthrough-thickness cracked uniform beam. The system is modelled using tworninterconnected intact beams and the crack is modelled by implementing its localrnflexibility. The intact elements’ DSMs exhibiting both mass and stiffness propertiesrnare then assembled and the boundary conditions are applied to form the nonlinearrneigenproblem of the overall system. The natural frequencies and modes are extractedrnusing the well-known Wittrick–William (W-W) root counting algorithm. The modelrnis first validated for both intact and defective thin (Euler-Bernoulli), flat, uniform,rncantilever, laminated composite beam of solid rectangular cross-section, exhibitingrnmaterial couplings. Numerical tests are then conducted for a defective uniformrnTimoshenko (thick) laminated composite beam, with a crack located at 50% thernlength, crack ratio a/b=0.5, and the shear correction factor k=5/6. Numerical resultsrnon natural frequencies and modes are presented and discussed.