Issue 29

P. Casini et alii, Frattura ed Integrità Strutturale, 29 (2014) 313-324; DOI: 10.3221/IGF-ESIS.29.27 315 and s = a / h is the non dimensional severity of the crack. The simple supported beam, Fig. 1a, has been forced under two different loading condition: almost symmetric 0.45 F l l        and asymmetric 0.06 F l l        . Also for the spanned beam two different locations for the load are considered: 2 3 F l l  and 2 5 F l l  , where 1 2 l l l   . The beams are modeled using standard one dimensional Euler-beam elements, with three degrees of freedom for each node, as well as the damaged element; the mesh consists of 60 finite elements. a) b) Figure 1 : Systems under investigation. a) Simple supported beam, l =0.6 m, l F =0.26 m (quasi symmetric) or l F =0.04 m (asymmetric). b) Spanned beam, l 1 =0.4 m, l 2 =0.2 m, l F =0.3 m or l F =0.5 m. To characterize the dynamic properties of the structure, distinction should be made between the first natural frequencies, 1 1 2 u u T    and 1 1 2 d d T    , of the two constituent sub-models (open crack, superscript d , and closed crack, superscript u ) and the first natural frequency 1  of the system: this frequency can be obtained as the average of the natural periods of the two sub-models 1 u T and 1 d T : 1 1 1 1 1 1 4 u d u d u d i T T           . This relation strictly holds for a single-degree-of-freedom oscillator with a bilinear stiffness and is therefore called first bilinear frequency of the system [3]. As demonstrated in the literature [3, 20, 21] the same equation approximates with good accuracy the natural frequencies of a structures with a breathing crack. In conclusion the resonance of the damaged systems occurs at the bilinear frequency i  given by: u d i i i u d i i        (1) where u i  is the i th natural frequency of the undamaged system and d i  is the i th natural frequency of the linear damaged system when crack is always open. Model of the breathing crack In several applications, the equilibrium configuration of the system coincides with the switching condition between closed and open behaviour for the crack. Under this condition, the cracked zone of the beam can be modelled with a finite element that has a bilinear element matrix with the discontinuity passing through the origin: this leads to an overall bilinear model where the nonlinear characteristics are independent of the vibration amplitude [3, 4, 21]. The standard element stiffness matrix K for a plane beam element with three degrees of freedom per node, collected in the vector U , once shear deformability is neglected, is described by: 2 2 2 / 0 0 0  0 /  6 / 0 12 / 12 / 6 / 4    0 6 /    0 ,       / 0    0 12 / 6 /    4 i u u i i u j u j j u A I A I v l l l l l EI K U u A I l v symm l l                                                 (2)