PSI - Issue 41

Andrea Pranno et al. / Procedia Structural Integrity 41 (2022) 618–630 Author name / Structural Integrity Procedia 00 (2019) 000–000

625

8

(point B). We observed a slight variation between the numerical and experimental results from 0 to 4 mm of deflection, where the numerical model highlighted a slightly stiffer structural response due to the common toughening effect induced by the DIM which is discussed in depth in the following scientific papers (De Maio et al., 2020c, 2019b, 2019a).

Exp. envelope Loading path Unloading paths

L10

10 15 20 25 30 35 40

L9

L8

L7

L6

L5

Load [kN]

L4

L3

L2

0 5

L2'-L6'

L8' L9' L10' L7'

0

2

4

6

8

10

12

14

Deflection [mm]

Fig. 3. Load versus mid-span deflection curves compared with the envelope of the experimental results by (Hamad et al., 2015) during the loading and unloading phase.

In addition, ten damage levels, corresponding to a percentage of the maximum load level which is equal to 42.1 kN, were identified and listed in Tab.3 based on the data reported in (Hamad et al., 2015). For each level of damage, an unloading process was performed and the obtained load-deflection curve is reported in Fig. 3 by mean of a gray dashed line with the exception of the first level of damage L1 corresponding to the undamaged configuration.

Tab. 3: Percentages of maximum load (“Max. load %”) associated with the investigated damage levels (“damage level”).

Damage level Max. load %

L1

L2

L3

L4

L5

L6

L7

L8

L9

L10

0 0

15.90

22.73

27.87

35.10

47.27

59.17

71.13

83.43

95.07

Load [kN]

6.69

9.58

11.6

14.7

19.9

24.9

29.9

35.1

40

Subsequently, a modal analysis was carried out by superimposing a linearized eigenvalue problem over the solution of the quasi-static analysis whose results were reported in Fig.3. The modal analysis, giving the natural vibration frequencies of the damaged beam, was superimposed on the static solution at the unloading phase (Points L1'-L10') to consider the effects of partially closed cracks caused by contact phenomena. As can be seen in Fig. 4, the obtained natural vibration frequencies, expected to be influenced by the presence of diffuse damage, were normalized with respect to the frequencies related to the undamaged configuration. The normalized frequencies were plotted as a function of the levels of damage and compared with the experimental envelope reported in (Hamad et al., 2015). For the first and fourth natural vibration modes, the numerical and experimental results are in perfect agreement. However, a slight difference was detected in high-order natural vibration modes (6 th and 7 th modes), because they are typically most affected by measurement and dispersion errors. The highest deviation is less than 5% and it has been detected at