Exact solution based finite element formulation of cracked beams for crack detection

In this study, a new finite element formulation is presented for straight beams with an edge crack, including the effects of shear deformation, and rotatory inertia. The main purpose of the study is to present a more accurate formulation to improve the beam models used in crack detection problems. Stiffness matrix, consistent load vector, and mass matrix of a beam element is obtained using the exact solution of the governing equations. The formulation for frame structures is also presented. Crack is modelled utilizing from the concepts of linear elastic fracture mechanics. Several numerical examples existing in the literature related to the vibrations of such structures are solved to validate the proposed model. Additionally, an experimental modal analysis is performed to see the superiority of the present method for high modes of vibration, which are generally not taken into account in crack detection problems. The inverse problem is also solved using a well–known optimization technique called genetic algorithms. Effects of shear deformation, rotatory inertia, and number of natural frequencies considered, on the accuracy of the estimation of crack parameters are investigated. It is found that considering more number of frequencies yields better estimation of crack parameters, but require a better modelling of the dynamics of the beam. Therefore, the present formulation is found to be an essential tool in crack detection problems.