Free vibration analysis of bi-directional functionally graded single/multi-cracked beams

Abstract

Cracks are the major cause for failure and dynamic fracture in different mechanical systems. For this reason, in this study, a comprehensive study on dynamic behavior of bi-directional functionally graded (BDFG) single/multi cracked beams is presented. Material variation is presented using different material gradation laws such as power-law, sigmoidal and exponential varying functions in both thickness and longitudinal directions to present a comprehensive model of BDFG beams. Euler–Bernoulli beam theory and a novel finite element approach is presented to model the beam with single or multiple cracks and accordingly, stiffness and mass matrices for BDFG cracked beams are derived. Further, a general analysis on BDFG beams with different types of cracks is presented and influence of different parameters such as position of the cracks, depth of the cracks, material variation type and number of cracks on natural frequencies and mode shapes of such structures is discussed. It is shown that incorporating BDFG material in structures with proper varying functions for both axial and thickness directions could lead to an optimized beam with better performance while being cracked in different positions. Also it is concluded that cracks in BDFG beams could cause noticeable variation in dynamic behavior due to change in local stiffness depending on the position and the depth of the cracks.