PSI - Issue 66

Umberto De Maio et al. / Procedia Structural Integrity 66 (2024) 459–470 Author name / Structural Integrity Procedia 00 (2025) 000–000

463 5

UD D i i UD i f f f 

(10)

An alternative strategy is based on the comparison between the natural vibration modes before and after the occurrence of damage. In this context, it is possible to detect the damage using two different strategies: the “Modal Assurance Criterion” (MAC) introduced by (Pastor et al., 2012) and the “Modal Curvature” (Yang et al., 2017). The first one is a mathematical measure used to quantify the degree of correlation between two mode shapes. The MAC value ranges from 0 to 1, where 1 MAC  means that the two mode shapes are perfectly correlated, while 0 MAC  means that the two mode shapes are completely uncorrelated. In this case, the correlation is between the mode shapes in the undamaged and damaged configuration. The generic element of the MAC matrix is defined as:      2 , , , , , , T iUD j D ij T T iUD iUD jD jD MAC  φ φ φ φ φ φ (11) where , i UD φ is the vector of the i -th natural vibration mode for the undamaged configuration, and , j D φ is the vector of the j -th natural vibration mode shape in the damaged configuration. The second strategy adopted in this work involves the use of the curvature of the mode shapes. The curvature of the j -th mode shape at the i -th coordinate can be evaluated by using the central difference approximation that leads to the following expression: 3. Results In this Section, the numerical results in terms of static and dynamic analysis are shown. The study is focused on the mechanical behavior of a reinforced concrete beam subjected to a four-point load that leads to flexural damage scenario. Several damage levels were considered, starting from the load associated with the first crack to the load causing collapse. In Section 3.1 the material and geometric properties of the beam are reported, while in Section 3.2 and 3.3 the outcomes of the static and dynamic analysis are shown, respectively, and validated through the comparison with experimental results from the literature (Hamad et al., 2015). 3.1. Material and geometric properties The present numerical simulations involve a four-point bending test analyzed by (Hamad et al., 2015). The mechanical properties of the materials are described in Table 1, while all the information about the geometrical setup can be seen in Fig. 1.   1 i h          1 i 2 2 ij j j MC  (12)

Table 1. Mechanical properties of the materials. Material Young’s Modulus [GPa]

Poisson’s ratio

Compressive strength [MPa]

Tension strength [MPa]

Yield strength [MPa]

Concrete Plane steel Ribbed steel

40.3 210 210

0.2 0.3 0.3

36.5

2.10

- -

- -

393.6

490