Dynamic Analysis of Cracked Viscoelastic Beam with Circular Cross Section using an Operator Based Finite Element Approach

Abstract

The occurance of crackin any structure introduces change in its physical properties which further tends to alter its behavior during dynamic condition. Under such condition, a structure tends to lose stiffness and decreases energy. The as pects that affect the structural dynamics most is depth of crack and number of cracks. So it is necessary to analyze the dynamic behavior of structure in presence of a crack. Generally, most of the metal are considerd as viscoelastic material and most of the time, damping effect of material assist to isolate vibrations. Consequently, the present study focuses on to explore the dynamic analysis of viscoelastic cantilever beam of circular cross-section with single open crack. The ultimate compliance matrix of the cracked element is produced by formulating the local flexibility matrix of the cracked part and further adding it to the compliance matrix of the intact part. The elastics train energy release rate theory is employed to form the elements of the compliance matrix of the cracked part knowing the fact that the flexibility of a structure increases in presence of crack and its further propagation, which results to addition of deformation in loading direction as stated in Castigliano’s theorem. The flexibility coefficients of the compliance matrix are formulated assuming that the loading conditions contribute only to the opening mode of crack. The complexity informulation referring to perform area integral occurs due to the beam cross-section i.e. circular. Operator based method is applied to build up the equation for the general linear viscoelastic system. Both, Euler Bernoulli and Timoshenko beam theories are applied to obtain the higher order equation of motion. The continuum is discretized using finite element method. Further, the eigen analysis of the system is performed by converting the higher order equation to state space form. The cracked element stiffness matrix is obtained from the addition of the cracked compliance matrix and total compliance matrix of the system depending on the crack position and number of elements the system is discretized into. The results displayed basically compares the outcome between two beam theories and it is observed that due to shear effect, the Timoshenko circular beam be likely to shows a lower response in contrast to the viscoelastic Euler-Bernoulli beam. Further, the natural frequency of vibration diminishes due to crack occurrence when matched up to an intact viscoelastic beam.