Free vibration analysis of a cracked beam by finite element method

Abstract

In this paper, the natural frequencies and mode shapes of a cracked beam are obtained using the finite element method. An ‘overall additional flexibility matrix’, instead of the ‘local additional flexibility matrix’, is added to the flexibility matrix of the corresponding intact beam element to obtain the total flexibility matrix, and therefore the stiffness matrix. Compared with analytical results, the new stiffness matrix obtained using the overall additional flexibility matrix can give more accurate natural frequencies than those resulted from using the local additional flexibility matrix. All the elements in the overall additional flexibility matrix are computed by 128-point (1D) or (128×128)-point (2D) Gauss quadrature, and then further best fitted using the least-squares method. The explicit form best-fitted formulas agree very well with the numerical integration results, and are very convenient for use and valuable for further reference. In addition, the authors constructed a shape function that can perfectly satisfy the local flexibility conditions at the crack locations, which can give more accurate vibration modes.

Introduction

The cracked beam problem has attracted the attention of many researchers in recent years. Various kinds of analytical, semi-analytical and numerical methods have been employed to solve the problem of a cracked beam [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12]. A common method is to use the finite element method (FEM). The key problem in using FEM is how to appropriately obtain the stiffness matrix for the cracked beam element. The most convenient method is to obtain the total flexibility matrix of the element first and then take inverse of it. The total flexibility matrix of the cracked beam element includes two parts. The first part is the original flexibility matrix of the intact beam. The second part is the additional flexibility matrix due to the existence of the crack, which leads to energy release and additional deformation of the structure. Papadopoulos and Dimarogonas [1] elegantly presented the ‘local flexibility matrix’ of a beam due to the existence of the crack by the integration of stress intensity factors. Their obtained flexibility matrix is indeed a ‘local’ one, as we can see in their paper that K_I2=K_I3=0, where P₂ and P₃ are the shearing forces. The local flexibility matrix is especially appropriate for the analysis of a cracked beam if one employs an analytical method by solving the differential equations piecewisely [2]. It is also appropriate to use a semi-analytical method by using the modified Fourier series [3], [4], [5], mechanical impedance method [6], Rayleigh–Ritz method [7], or transfer matrix method [8]. When FEM is used, to obtain the stiffness matrix it is necessary to take into account the effect of the distance between the right hand side end node of the element and the crack location, i.e., L_c (see Fig. 1, Fig. 2). The reason is that the shearing force P₂ also contributes to the opening type (i.e., K_I2) of the crack through the bending moment P₂L_c. This problem has been previously ignored in [9], [10], in which the local flexibility matrix is directly added to the flexibility matrix of the corresponding un-cracked element to obtain the total flexibility matrix. This is not very accurate because the former matrix is to describe the local behaviour in the vicinity of the crack region, while the later matrix describes the overall behaviour of the beam element. In this paper, the authors derived new FEM formulas to overcome the existing shortcomings by adding an ‘overall additional flexibility matrix’, which describes the overall behaviour of flexibility due to the presence of the crack, onto the flexibility matrix of the corresponding intact beam. By comparing the FEM results obtained in this paper with available existing analytical methods, the new stiffness matrix can indeed give more accurate results than those obtained from using the local flexibility matrix. Moreover, all the elements of the overall additional flexibility matrix are computed by 128-point (1D) or 128×128 (2D) Gauss quadrature and then further best fitted using the least-squares method. The best-fitted formulas agree very well with the numerical integration results. They are convenient for use and valuable for further reference.

Once the stiffness matrix of a cracked beam element is successfully obtained by ‘going detour’ (i.e., obtaining the total flexibility matrix first and then taking inverse of it), standard FEM procedure can be followed, which will lead to a generalized eigenvalue problem and thus the natural frequencies can be obtained. However, it is worth noting that it is not appropriate to compute the vibration modes for the elements having cracks by still using the standard Hermitian interpolation as in the common FEM method. This problem is seemly ignored by many researchers when using the ‘detour method’. In this paper, the relationship between the displacements at the crack locations and those at the two end nodes of a cracked beam element is derived (see Eq. (65)). The presented shape function in this paper can perfectly satisfy the local flexibility conditions as well as continuity conditions at the crack locations, which can give more accurate vibration modes.