PSI - Issue 41

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

623

6

where max  is the critical normal stress and max  is the critical tangential stresses at the cohesive interfaces. An equivalent displacement jump 0 m  was defined (valid only for 0 n   ) as in the following:

2

1





0     0 0 m n s

m

,

(9)

   2 0 s  



2

0   m n

where m s n     is the ratio between the tangential and normal displacement jump. 2.2. Dynamic indicators for damage detection in reinforced concrete beams Based on the literature, in structural systems the variation of the natural vibration frequencies between the undamaged 0 j f and damaged d j f configuration can be taken into account as damage indicator and it is called the inverse eigenvalue sensitivity method (Balageas et al., 2006).:

0

d

f

f



j

j

.

(10)

0

f

j

An alternative strategy to investigate the damage effects on the mechanical behavior of structural systems is based on the comparison between the natural vibration modes before and after the occurrence of damage. In this regard, it is possible to use the so-called “Modal Assurance Criterion” (MAC) introduced by Allemang and Brown in 1982 (Allemang and Brown, 1982). This criterion was originally introduced to verify the degree of similarity (also called correlation) between the experimental and analytical modes. The generic component of the MAC matrix referring to the comparison between the modes in the undamaged configuration and those in the damaged one is defined as:       2 ,0 , ,0 ,0 , , T i j d ij T T i i j d j d MAC        (11)

,0 i  denotes the component vector of the

th i  natural vibration mode in the undamaged configuration th j  natural vibration mode in the damaged configuration. The

where

, j d  denotes the component vector of the

and

obtained values of ij MAC ranges from 0 to 1, specifically a value equal to unity corresponds to a perfect correlation between the two considered modes, while a value equal to zero indicates that there is no similarity between the modes. Since the damage in structural elements induces a variation in the natural vibration modes, through the MAC values it is possible to obtain a scalar measure of the damage as a function of the correlation between the natural vibration modes before and after the damage occurrence. Another effective damage indicator is the “Curvature Damage Factor” (CDF) (Abdel Wahab and De Roeck, 1999) which represents an average of the modal curvature difference at the generic coordinate th i  evaluated through the following expression:

1

N 

,0 ''    ij

''

CDF



,

(12)

, ij d

i

N

1

j



where the modal curvature is defined by means of the central difference method: