PSI - Issue 37

Elizabeth K. Ervin et al. / Procedia Structural Integrity 37 (2022) 6–16 Ervin and Zeng / Structural Integrity Procedia 00 (2021) 000 – 000

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where κij is the curvature, ϕ ij is the modal displacement at the jth measured node of the ith mode, h1 is the distance between (j − 1)th, and jth nodes and h2 is the distance between jth and (j + 1)th nodes. Another commonly used DSF is modal strain energy, which is the energy stored in structural components when deformed in a mode shape pattern. Ndambi et al. (2002) proposed a method to calculate strain energy of a sub-region within a beam, and their method is extended to application for a structure subdivided into n-1 divisions. The strain energy of the jth sub-region corresponding to the mode shape ϕ i is given by ( ) 2 2 2 0 j l i ij j U EI dx   =    ȋͶȌ where E and I are the Young’s modulus and moment of inertia of the jth sub -region, respectively, and lj is the length of the jth sub-region. The strain energy of each sub-region at the ith mode is then summed to yield ( ) 2 2 2 0 1 1 2 j n l i i j i U EI dx x    =   =     ȋͷȌ where Ui denotes the summation of strain energy of each structural member for the ith mode. Note that EI can be identity or relative values and not the actual current values. To derive more DIs than those from literature, difference, division and percentage are employed on DSFs. The three methods are defined as absolute difference, fractional absolute value, and relative absolute value percentage, and they are denoted as Diff, Div and Perc , respectively. These notations will be used throughout this work to clearly identify employed mathematics. Statistical concepts have also been used to generate DIs. Assuming that DI values follow a Gaussian distribution implies that the feature on damage also follows that same distribution; this is an unlikely assumption for most structural damage. However, indices have been normalized to a standard Z score to create secondary DIs in some papers [Sun et al. (2001), Cornwell et al. (1999)]. The absolute value of Z score represents the distance between raw score and population mean of the standard deviation. The threshold of damage versus noise is subjective but literature generally suggest values outside [-2, 2] indicate locations of damage with a 95% confidence interval. As this interval is an arbitrary designation, the standard probability density function P of a Gaussian distribution can also be directly used as a DI. An abnormal DI value at a measured node is represented by the absolute value of standard score which is correlated to damage possibility. 2.2. 3D resultant indices The current study utilizes 17 uni-axial DI algorithms; this includes six found in literature and eleven that are proposed. This work denotes an index calculated using only data in one of the x , y or z directions as a directional DI. The physical meaning is a representation of the relative weakness or strength of structural members in one direction, and thus directional DIs are directly applicable to structures with a single dominant direction. However, using only directional DIs may introduce uncertainties in decision-making because the presence of damage may show as coupled mode shape changes. In short, DIs generated using modal properties in three directions are necessary to provide an inspector with an overall damage indication. This also means that tri-axial data capture is recommended for a full picture of damage. Two combination methods calculate combined DIs generated from modal properties in three directions. The first method uses root sum square method to calculate a resultant DI ( DI j R ) using a directional DI in each x , y and z direction. The alternative second method ( DI j S ) computes the magnitude of mode shapes before entering them into any single DI. The first combination method takes the resultant of directional DIs, and the second method inserts the resultant mode shape into each DI algorithm; this is an open issue regarding the order of operations, so both approaches are input into GA for natural selection. Table 1 presents all combined DIs produced from data in three directions. Those DIs from literature are noted, 1 2 x       