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This paper presents a new sectional flexibility factor to simulate the reduction of the stiffness of a single-edge open cracked beam. The structural model for crack of the beam is considered as a rotational spring which is related to the ratio of crack depth to the beam height,

A crack in the structure will reduce the structural strength and result in severe damage under critical loading conditions. The major issue of the structural health monitoring is to detect crack depth and location in the present study. The model with linear elastic fracture mechanics and Euler-Bernoulli beam theory are being widely used in the recent research literatures. The cracked beam is modeled as two-segment beam with the crack simulated as a rotational spring. The crack of the beam is considered as a local flexibility which is a function of the crack depth.

Sih [

In order to obtain the natural frequencies of the crack beam, finite element method was used to compute the eigensolutions in the recent literatures. The order of the determinants increases as the degree of freedom increases in finite element method. In order to reduce the order of the determinants, Lin et al. [

A prismatic beam is considered with an open and nonpropagating crack of depth

Geometry of a beam with a single-edge crack.

The stress condition is considered as

Let

The strain energy has the form [

A simple beam with length

Schematic diagram of a simply supported cracked beam.

Model for the cracked beam with sectional flexibility.

The equation of motion for the cracked simple beam system can be expressed as (

Using (

In order to verify the procedure presented in this paper, results obtained by applying this method are compared with the available data for single-edge open cracked beam. A 300 mm simple supported beam of cross-section

A simplified stress intensity factor

In this research, small crack depth ratio which implies early stage of structure damage is considered. The comparison of data obtained by the proposed method and previous research is shown in Figure

Stress intensity factor variations.

The nondimensional cracked section flexibility

Three generalized loading conditions, bending, tension, and torsion, on a cracked beam were considered to evaluate sectional flexibility in the past literature. Beams are mainly affected by bending moment in most loading cases; therefore only bending effects are considered in evaluating the simplified cracked section flexibility. The results of the proposed method of non-dimensional cracked section flexibility are compared with those of previous research in Figure

Nondimensional cracked section flexibility variations.

The first five natural frequencies of the uncracked beam are calculated as

Natural frequency ratio with the crack depth at location 0.8.

Crack location | Crack depth | Natural frequency ratio | ||||

0.8 | 0.0 | 1.0000 | 1.0000 | 0.9999 | 1.0000 | 0.9999 |

0.8 | 0.1 | 0.9966 | 0.9912 | 0.9913 | 0.9967 | 0.9999 |

0.8 | 0.2 | 0.9918 | 0.9792 | 0.9803 | 0.9927 | 0.9999 |

0.8 | 0.3 | 0.9850 | 0.9628 | 0.9660 | 0.9876 | 0.9999 |

0.8 | 0.4 | 0.9747 | 0.9394 | 0.9474 | 0.9814 | 0.9999 |

0.8 | 0.5 | 0.9581 | 0.9052 | 0.9237 | 0.9737 | 0.9999 |

0.8 | 0.6 | 0.9292 | 0.8544 | 0.8943 | 0.9648 | 0.9999 |

0.8 | 0.7 | 0.8735 | 0.7800 | 0.8605 | 0.9551 | 0.9999 |

0.8 | 0.8 | 0.7525 | 0.6809 | 0.8271 | 0.9459 | 0.9999 |

0.8 | 0.9 | 0.4791 | 0.5825 | 0.8021 | 0.9392 | 0.9999 |

Variation of natural frequency ratio with the crack depth of a simply supported beam (

With a crack located at the center of the beam, the ratios of natural frequencies between cracked beam and uncracked beam are listed in Table

Natural frequency ratio with the crack depth at location 0.5.

Crack location | Crack depth | Natural frequency ratio | ||||

0.5 | 0.0 | 1.0000 | 1.0000 | 0.9999 | 1.0000 | 0.9999 |

0.5 | 0.1 | 0.9902 | 1.0000 | 0.9904 | 1.0000 | 0.9905 |

0.5 | 0.2 | 0.9770 | 1.0000 | 0.9778 | 1.0000 | 0.9786 |

0.5 | 0.3 | 0.9586 | 1.0000 | 0.9613 | 1.0000 | 0.9636 |

0.5 | 0.4 | 0.9321 | 1.0000 | 0.9391 | 1.0000 | 0.9448 |

0.5 | 0.5 | 0.8927 | 1.0000 | 0.9094 | 1.0000 | 0.9218 |

0.5 | 0.6 | 0.8314 | 1.0000 | 0.8699 | 1.0000 | 0.8948 |

0.5 | 0.7 | 0.7328 | 1.0000 | 0.8199 | 1.0000 | 0.8654 |

0.5 | 0.8 | 0.5723 | 1.0000 | 0.7634 | 1.0000 | 0.8377 |

0.5 | 0.9 | 0.3245 | 1.0000 | 0.7144 | 1.0000 | 0.8175 |

Variation of natural frequency ratio with the crack depth of a simply supported beam (

It is quite obvious that the natural frequencies decrease due to the existence of cracks. That is due to the cracked beam becoming more flexible due to the reduction of moment of inertia of the section property.

Figures

(a) Normalized mode shape of the first mode with a crack location

The simplified stress intensity factor and flexibility were derived utilizing the crack beam theorem of Nobile [

If the crack is right on the position of nodal point of certain modes, frequencies ratio shows no difference with

The simplified stress intensity factor and flexibility of this method can be further extended to construct frequencies ratio contours for beams with cracks. The natural frequencies obtained by applying this model can be used to verify the experimental measurements in a similar way to that in [