Vibration, Buckling and Dynamic Stability of Stepped Beams with Multiple Transverse Cracks

Abstract

The present study deals with the vibration, buckling and parametric instability behavior of both uniform and stepped beams with single and multiple transverse cracks. The variation of natural frequency, buckling load and dynamic stability regions with different parameters including relative crack depth, position of cracks, static and dynamic load factors are analyzed using finite element method (FEM) through Bolotin’s approach. Experiments are also conducted to determine the natural frequencies of vibration of aluminum and steel beams with open cracks and compared with numerical predictions. The computation of natural frequencies and plotting of mode shapes of the vibration of cracked beams are done using ANSYS. Besides this, the extent of cracks in a beam are predicted using vibration data by ANN technique. The loading on the beam is considered as axial with a simple harmonic fluctuation with respect to time. During the instability, 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 finite element method. Stiffness matrix of the cracked beam element is obtained from the flexibility matrix of the intact beam and the additional flexibility matrix due to crack. The frequencies of vibration and buckling loads of cracked cantilever beams reduce for both uniform and stepped beams with increase in crack depth and number of cracks. However the crack near the step reduces natural frequency and buckling load more than other locations. The onset of instability occurs earlier with introduction of more number of cracks due to reduction of stiffness. The instability region for the beam with crack located nearer to the fixed end occurs at lower excitation frequency than near free end of a cantilever beam. The vibration and instability results can be used as a technique for structural health monitoring or testing of structural integrity, performance and safety. It is observed that the extent of damage can be detected using ANN models with reasonable accuracy considering only frequencies as input. Out of the three models used, BRNN model is found to be the best model as compared to DENN and LMNN considering regression values, root mean square error values and over fitting ratios for simply supported beams.