PSI - Issue 54

578 J.V. Araújo dos Santos et al. / Procedia Structural Integrity 54 (2024) 575–584 Author name / Structural Integrity Procedia 00 (2019) 000 – 000 modal displacements ( , ) and other quantities. To compute derivatives, however, it is necessary to use Kuo et al. (1991) approach, namely because we are interested in evaluating cross-derivatives, such as 2 ( , )/ . Indeed, in this case Equation (2.12) leads to erroneous results since the values of the second-order derivatives would be zero. The weighting factors of a (2 + 1) × (2 +1) points data set for Ratzlaff and Johnson (1989) and Kuo et al. (1991) approaches are given as a function of the one-dimensional weighting factors by Nikitas and Pappa-Louisi (2000) ℎ ( +1, ),( +1, ) = ℎ +1, + ℎ +1, − 1 (2.14) and ℎ ( +1, ),( +1, ) =ℎ +1, ℎ +1, (2.15) respectively, where and are the indices of coordinates and . Finally, we can define the two-dimensional discrete convolutions to compute smoothing values in a point with coordinates ( 0 , 0 ) as [cf. Equation (2.8)] ( 0 , 0 )= ∑ ∑ ℎ (1, ),(1, ) ( 0 − , 0 − ) =− =− (2.16) Similarly, we can write the two-dimensional discrete convolutions [cf. Equation (2.11)] ( ) ( 0 , 0 )= ∑ ∑ ℎ ( +1, ),(1, ) ( 0 − , 0 − ) =− =− (2.17) ( ) ( 0 , 0 )= ∑ ∑ ℎ (1, ),( +1, ) ( 0 − , 0 − ) =− =− (2.18) ( + ) ( 0 , 0 )= ∑ ∑ ℎ ( +1, ),( +1, ) ( 0 − , 0 − ) =− =− (2.19) for the computation of approximations to th order derivatives in respect to , th order derivatives in respect to and ( + ) th order cross-derivatives, respectively. 3. Damage Identification Procedure According to the Kirchhoff assumptions for the analysis of a plate, the bending (flexural) strains ( , ), ( , ), and ( , ) of the -th mode are given by the following equation ( ) ( , ) = { ( ) ( , ) ( ) ( , ) ( ) ( , )} = ℎ 2 { − 2 ( ) ( , ) 2 − 2 ( ) ( , ) 2 −2 2 ( ) ( , ) } (3.1) where ( ) ( , ) is the transverse displacement of the midplane and ℎ is the plate thickness. The three components 4