PSI - Issue 68

Gomez-Mancilla Julio C. et al. / Procedia Structural Integrity 68 (2025) 318–324 Gomez-Mancilla J / Structural Integrity Procedia 00 (2025) 000–000

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3. Criterion, BSFI j Modal Indexes Formulae for Onset and Damage Detection Onset damage detection criterion. First, use the criterion to detect and identify the initiation of damage. The criterion is based on frequency tests on the structure and expressions below included in the BFSI, considering that orthotropic cross-section beams cause bending frequencies splitting in pristine integral structures. 0 subindex implies measured data at initial pristine integral conditions. The computed recorded Δω 0 m allows notched, imperfect manufactured yet integral beams to be differentiated from those already damaged; the damage criteria imply a non zero result, 0m = 0 ղ m − 0ξm ≠ 0, and any BSFI (k) m > 0 for sub-index m= 1,2,3..., # m_mx (5,6) Table 1 . Damaged model, literature comparisons for the first three flex modes ω ξ1 , ω ξ2 , ω ξ3 , test data from Dirr et al. (1988) vs. Matlab calculations, three crack depth cases: a c / D = 0 (integral), 0.26 and 0.39 are compared; the damage adds two ω ղ . a c /D=0, int ω ξ =ω ղ Matlab a c /D=0.26, ω ξ, test Dirr a c /D=0.26, ω ξ , Matlab a c /D=0.26, , Matlab a c /D=0.39, ω ξ, test, Dirr a c /D=0.39, ω ξ , Matlab a c /D=0.39, , Matlab The six BSFI (k) ( k =1-6) comprise ten expressions. Yet, since in circular beams both integral bending mode frequencies repeat and have the same value, i.e., all j-modes = 0, then they reduce to six different indexes. Moreover, to work only with positive BSFI (k) values and remove orthotropic influence, Eqs. (7b-10b) are the BSFI ( a c /D, m ) indexes which remove any notched and imperfect cross-sections and are computed at the p-assumed crack locations Lc/L . Such first four 9 9 9 9 9 9 (/) 1 ( k =1-4) allow for Onset Detection and Damage detection in non circular non-isotropic beams. The six BSFI m indexes are as follows, 9 9 9 9 9 9 (2) 1 =; 34 #$% 5 4 #& % 6 5 34 $% 5 4 & % 6 4 & % <100 , (2) 1 = > 999999 (2) 1 −; 4 #& % <> (7a, b) 9 9 9 9 9 9 9 (9) 1 =; 34 #$% 5 4 #& % 6 5 34 $% 5 4 & % 6 4 $% <100 , (9) 1 = ? 999999 (9) 1 −@ 4 #$% A? (8a, b) (:) 9999999999999 = ; 34 #$% 5 4 #& % 6 5 34 $% 5 4 & % 6 4 #& % <100 , (:) 1 = > 999999 (:) 1 −; 4 #& % <> (9a, b) (;) 1 =; 34 #$ * 54 #& * 6 5 34 $ * 5 4 & * 6 4 #$ * <100, (;) < = ? 999999 (;) < −@ % 4 #$% A? (10a, b) (=) 1 =>@ 4 #$ * 5 4 $ * 4 $ * A−; 4 #& * 5 4 & * 4 & * <>100, (>) 1 = > 34 #$ * 5 4 $ * 6 5 34 #& * 5 4 & * 6 4 #$ * 5 4 #& * >100 (11,12) where m = 1 to m_mx, is the maximum number of considered flexural modes, which is limited by the quality of the testing equipment. Onset Damage Criterion : Structural damage starts whenever any SFBI (k) m > 0, i.e., only a flat plane indicates an integral structure. ω o1 =225.8 ω ξ1 =224.9 ω ξ1 =224.7 =225.76 ω ξ1 = 223.3 ω ξ1 =223.3 = 225.66 ω o2 =616.4 ω ξ2 =614.7 ω ξ2 =614.7 % = 616.34 ω ξ2 = 613.4 ω ξ2 =612.67 % = 616.18 ω o3 =1191.7 ω ξ3 =1,190.4 ω ξ3 =1,190.92 = 1,191.7 ω ξ3 = 1,188.7 ω ξ3 =1,189.9 % = 1191.6