Modal analysis of damaged structures by the modified finite element method

Abstract

ABSTRACT. The classical 3D beam element has been modified and developed as a new finite element for vibration analysis of frame structures with flexible connections and cracked mem hers. The mass and stiffness matrices of the modified elements are established basing on a new form of shape functions, which are obtained in investigating a beam with flexible supports and crack modeled through equivalent springs. These shape functions remain the cubic polynomial form and contain flexible connection (or crack) parameters. They do not change standard procedure of the finite element method (FEM). Therefore, the presented method is easy for engineers in application and allows to analyze eigen-parameters of structures as functions of the connection (or crack) parameters. The proposed approach has been applied to calculate natural frequencies and mode shape of typical frame structures in presented examples.

The Dynamic Stiffness Matrix Method in Forced Vibration Analysis of Multiple-Cracked Beam

“…(2) Calculating the load vector F by using formulas (12). (3) Solving equation (15) resulted in the state vector +Z>(0),. (4) Determining the state vectors Z > (j), j"1, 2, 2 , n and afterward, the complex amplitude (or consequently, the spectrum) of displacement, slope, shear force and bending moment along the beam are calculated by using equation (16).…”

“…To overcome this the authors of reference [13] have conjugated the FEM with the idea of a compliance matrix to develop a speci"c technique for vibration analysis of cracked beam. This development made the FEM suitable for analysis of frame structures with the local #exibility due to a crack in its members [14,15]. The results of such developed FEM application to a cracked beam are still approximate in comparison with the analytical methods.…”