Parametric Instability of Beams with Transverse Cracks Subjected to Harmonic In-Plane Loading

Cracks in structural members lead to local changes in their stiffness and consequently their static, dynamic and stability behavior is altered. The influence of cracks on dynamic characteristics like free vibration, buckling and parametric resonance characteristics of a cracked beam with a transverse crack using finite element method (FEM) is investigated in the present work. Modal testing of beams with transverse open crack is conducted using FFT analyzer to verify the frequencies of vibration of beams. The crack is assumed to be open type and the analysis is linear based on small deformation theory neglecting damping. The loading on the beam is considered as axial with a simple harmonic fluctuation with respect to time. A two-noded Timoshenko beam element with provision of crack is used in this study. The equation of motion represents a system of second-order differential equations with periodic coefficients of the Mathieu–Hill type. The development of the regions of instability arises from Floquet's theory and the periodic solution is obtained by Bolotin's approach using FEM. It is observed that the frequencies of vibration and buckling load of the beam are influenced significantly by location and depth of cracks. It is observed that, for a given location of crack, the onset of instability occurs earlier with increase in depth of crack. As the location of crack moves from the fixed end to the free end the excitation frequency increases. The instability occurs later and the width of the instability regions reduces. When the damage is near to the free end, the instability region almost coincides with the instability region for the undamaged beam. This means that the damage near the fixed end is more severe on the dynamic instability behavior than that of the crack located at other positions. The vibration and instability results can be used as a technique for structural health monitoring or testing of structural integrity, performance and safety.