Meshless formulation using NURBS basis functions for eigenfrequency changes of beam having multiple open-cracks

Abstract

This paper presents a meshless formulation using non-uniform rational B-spline (NURBS) basis functions, and its applications to evaluate natural frequencies of a beam having multiple open-cracks. Node-based NURBS basis functions are used to construct the approximation function. The characteristic differentiability of the NURBS basis functions allows it to represent a function having specific degrees of smoothness and/or discontinuity. The discontinuity can be incorporated simply by assigning multiple knots at those locations. Hence, it can yield exact solutions having interior discontinuous derivatives. These advantages of NURBS are well known, and have been used extensively in graphical approximation of geometrical surfaces. However, it is seldom used in other engineering applications. To model the multiple open-cracks in a beam, quartic NURBS basis functions are employed and quadruplicate knots are assigned at the crack locations. Hence, it is capable to model the abrupt changes of slope (the first derivative of displacement) across a crack. In the present applications, additional equivalent massless rotational springs are inserted at the crack locations to represent the local flexibility caused by the cracks. As such, the cracked beam can be treated in the usual manner as a continuous beam. By adopting the meshless Petrov–Galerkin formulation, a generalized stiffness matrix for the cracked beam can be derived. Compared to the conventional finite element method, the present method does not require a finite element mesh for the purposes of interpolation and numerical integration. The advantages and effectiveness of the present method is illustrated in solving the eigenfrequencies of a beam having multiple open-cracks of different depths.