PSI - Issue 5

J. V. Araújo dos Santos et al. / Procedia Structural Integrity 5 (2017) 1198–1204 J.V. Araújo dos Santos et al. / Structural Integrity Procedia 00 (2017) 000 – 000

1201

4

s

s

s

x h

x h

(

(     )

)





s

dx x d x ( ) [ ( )]  



(8)

h

2

where h is the distance between two consecutive nodes of the finite element discretization, i.e. the step is equal to the length of a finite element.

2.3. Damage simulation and indicator

The modal response of a free-free aluminum beam 400 mm long, 40 mm wide, and 3 mm thick, subjected to simulated damage is presented in this work. As depicted in Figure 1, the damaged area starts at coordinate x 1 = 0.1175 m and ends at coordinate x 2 = 0.1225 m. Therefore, the damage is centered on coordinate ( x 1 + x 2 )/2 = 0.12 m from the left end of the beam. Several damage cases are considered in the present work, being each one simulated by a reduction in the bending stiffness EI of finite elements located at the damaged area, as reported in Table 1.

Fig. 1. Localization of the damaged area in the free-free aluminum beam.

Table 1. Description of the four damage cases. Damage case

Reduction in EI [%]

1 2 3 4

10 20 30 40

The first modal displacement and rotation fields of the beam in the undamaged and damaged states are the results of a finite element analysis. The beam was discretized in 4000 equally spaced finite elements, based on the Euler Bernoulli beam theory. The damage indicator used with the data coming from shearography is the Modified Curvature Difference (MCD), proposed by Araújo dos Santos et al. (2011, 2014):

s

dx d x s 

[ ~ ( )] [ ( )] 

s

dx MCD x d x ( ) 



(9)

~ ( ) x s 

The quantity ( ) x s  denotes the rotation in the undamaged state, being both computed through Equation (8). The word modified is used here to indicate the changes of the well-known damage indicator, proposed by Pandey et al. (1991), which is based on the curvatures as second order derivatives of the modal displacements, and not as first order derivatives of the modal rotations. Since we obtain directly the reference modal curvature, according to Equation (7), the equivalent damage indicator for the reference data is the Direct Curvature Difference (DCD): denotes the modal rotations in the damage state, whereas