PSI - Issue 54

J.V. Araújo dos Santos et al. / Procedia Structural Integrity 54 (2024) 575–584 Author name / Structural Integrity Procedia 00 (2019) 000 – 000

580

6

slot). Scenario 3 is a multiple damage scenario, where localized damage (square slot) and extensive damage (two rectangular slots) occur. Five damage cases for each damage scenario are studied and the thickness reduction of the slots for each one is listed Table 4.1. As the damage identification procedure proposed in this paper rely on the post processing of modal data, four finite element analysis were carried out (one for the plate in the undamaged state and three for the set of damage scenarios) in order to obtain modal displacements (mode shapes) of the plate for free-free condition. The finite element mesh consists of 300 dž 300 four node non-conforming finite elements, yielding a distance of 1 mm between each two consecutive nodes in each direction.

(a) (c) Figure 4.1: Geometry and location of slots: (a) damage scenario 1, (b) damage scenario 2, and (c) damage scenario 3 (b)

Table 4.1: Thickness of slots Case

Effective Thickness

Thickness Reduction (mm)

Percentual Thickness Reduction (%)

1 2 3 4 5

2.7 2.4 2.1 1.8 1.5

0.3 0.6 0.9 1.2 1.5

10 20 30 40 50

4.2. Mode Shapes Sensitivity to Damage The differences between the root-mean-square (RMS) of the mode shapes in the undamaged and damaged states is used to assess the changes in each mode shape induced by the stiffness reduction. These differences were computed for the first 12 modes, and it was found out that, in most situations, the 3rd and 9th modes present the largest and lowest differences, respectively. Table 4.2 shows the differences between the RMS for these two modes and all damage scenarios and cases considered in this study. It is clear that the changes in the 9th mode due to damage are lower than those of the 3rd mode, in particular for all damage cases of scenario 1, where the differences in the RMS of the 9th mode are four orders of magnitude lower than the difference in the RMS of the 3rd mode. This behavior agrees with the findings by Rucka (2011) and dos Santos et al. (2019), which show that the identification with modes where the damage is located in areas with zero or small curvatures is very difficult to accomplish. In order to support this conclusion and for brevity’s sake, it was decided to focus our attention on the post-processing of these two modes. Figure 4.2 shows the norm of modal strains of scenario 1 extreme cases of damage (case 1 and 5), i.e. the slots have reductions in thickness of 10% and 50%, respectively. We can see that the norms of strains of the 3rd mode present perturbations at the location of the slot in both smallest and largest damage cases (Figure 4.2(a)). However, the norms of strains of the 9th mode do not present any visible perturbations for either small or large damage (Figure 4.2(b)). Table 4.2: Percentual differences between RMS of mode shapes in undamaged and damaged states RMS Percentual Difference [%] Scenario 1 Scenario 2 Scenario 3 Case Mode 3 Mode 9 Mode 3 Mode 9 Mode 3 Mode 9 1 0.3475 0.0001 0.7210 0.0128 2.1355 2.5687 2 0.7545 0.0002 1.6535 0.0253 3.9287 3.9902 3 1.2139 0.0002 2.9587 0.0372 5.5221 4.3468 4 1.7017 0.0003 4.9349 0.0481 7.1486 5.6361 5 2.1750 0.0003 8.0543 0.0582 9.1297 6.9013