Narrow Results

This paper presents a rigorous study on the static and dynamic behavior of beams affected by cracks. The theory of distributions developed by Laurent Schwartz is adopted as it is particularly suitable for the treatment of discontinuities in functions for the deflection and derivatives of the beam. Thus, this paper presents a contribution towards the understanding and application of the theory of distributions to the static and dynamic behavior of structural elements affected by cracks. A simple, computationally efficient and accurate algorithm is developed for the problems of concern. Numerical results are presented for beams with two cracks. The algorithms developed for beams with discontinuities are obtained in a rigorous framework for static and vibration problems.

In this paper, the effect of an open edge crack on the instability of rotating non-uniform beams subjected to uniform distributed tangential compressive load is studied. The local stiffness due to the presence of crack is considered in the global stiffness matrix of the structure using the finite element method. The cracked beam element is modeled as two equal sub-beam elements connected by a massless rotational spring. Based on the fracture mechanics, the strain energy release rate and the stress intensity factors are employed to investigate the stiffness of the rotational spring. Then, the modified shape functions are developed to reflect the crack stiffness in the finite element analysis. To validate the accuracy of the finite element model and results obtained, comparisons have been made between the results obtained and those available in the literature. The effects of several parameters, including the linear and nonlinear thickness variations, angular velocity, crack location and size, on the instability of cracked rotating non-uniform cantilevers are also examined. The results show that the location of crack significantly influences the critical magnitude of the follower force that destabilizes the cantilevers. In addition, geometric non-uniformity reduces the stability of the cracked cantilevers. For the same amount of cantilever mass, different patterns of mass distribution result in different stability diagrams.

In this paper, based on distributed transfer function method (DTFM), the closed-form analytical solutions for vibrations of Euler–Bernoulli beam and frame structures with arbitrary number of cracks are studied. First, generalized DTFM is employed to characterize the dynamical model for a single cracked beam and its analytical solutions for eigenvalue problem and frequency response are obtained. Then, a new DTFM cracked element that encapsulates one crack of arbitrary location inside the beam is proposed. Using the DTFM cracked element and global dynamic stiffness matrix assembly technique, damaged frame structures of arbitrary form can be modeled for vibration analysis. Previous analytical methods only addressed low-frequency vibration of simple cracked beam structures, the proposed method can yield analytical solutions in the medium- and high-frequency regions, which is critical for the small crack detection in complex frames. Lastly, three numerical examples are given to illustrate the correctness and effectiveness of the DTFM in analyzing natural frequencies, modal shapes and frequency responses for cracked structures. By comparing with the Finite Element Method (FEM) and benchmarks from literatures, we proved that the DTFM has better performances in terms of accuracy and efficiency.

The vibration analysis of beams with cracks is an important problem in the structural dynamics community. In this study, a general model for the vibration analysis of a cracked beam with general boundary conditions was developed and investigated, emphasizing its vibration and power flow characteristics. The beam crack was introduced via torsional and translational coupling springs, which separated the beam structure into two segments, and the corresponding vibration characteristics were investigated via an energy-based formulation in conjunction with the Lagrangian procedure. A boundary-smoothed Fourier series was employed to construct the beam displacement field to avoid boundary differential discontinuities. Various crack statuses, including their depths or positions can be easily considered by adjusting the stiffness coefficient of the artificial springs. Several examples were presented to validate the effectiveness and accuracy of the proposed model. The modal characteristics and forced response of a cracked beam were predicted and analyzed, respectively, with a detailed depiction of the power flow around the crack. The results indicate that the presence of a crack has an important effect on the modal characteristics of an elastically restrained beam, as well as on the power flow distribution across the beam structure. This study can provide an effective tool for the dynamic analysis and power flow mechanism of beam structures with various cracks and complex boundary conditions.

In this study, the dynamic responses of a cracked beam layed on a visco-elastic foundation subjected to moving loads are calculated. An Euler–Bernoulli beam model is used to describe the beam behaviors. In addition, the beam has several open cracks one-sided with different depths. By using the Fourier transform, the dynamic responses of the beam are determined analytically in the frequency domain with the help of Green’s function. By coupling with the periodic supported beam model, an analytical model of the railway sleeper is developed. This model allows the fast calculation of the dynamic responses of a damaged sleeper. A dynamic computational model using the FEM method was also developed and compared to the analytical model. The results from the two methods are relatively comparable for three cases of beams without cracks, beams with one crack, and beams with two cracks.