PSI - Issue 78

Dario De Domenico et al. / Procedia Structural Integrity 78 (2026) 65–72

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is associated with negligible reduction (less than 3%) of the natural frequency as noted in the comparative table. This result was confirmed for both the flexural (mode 1, 4 and 5) and lateral (i.e., out-of-plane, mode 2) modal shapes. Table 1: Comparison of natural frequencies of PC beam at SLS conditions for various damage scenarios. Beam R03_2 (damaged at L/4) Mode 1 Mode 2 (lateral) Mode 4 Mode 5 Undamaged 7.12 8.07 18.77 43.21 Damaged_2cut 7.04 7.92 18.35 42.90 Damaged_4cut 7.02 ( -1.4% ) 7.84 ( -2.8% ) 18.27 ( -2.7% ) 42.48 ( -1.7% ) Considering that the natural frequency is not a key performance indicator of the damage status of the PC beam, other modal parameters are investigated. According to literature papers, the modal curvature could be a good indicator of the damage condition of a structure. The modal curvatures are determined from the mode shapes based on a second derivative operation. This derivation can be accomplished by means of a central difference approximation (Wahab and De Roeck, 1999) with ′′ ≈( +1 −2 + −1 )/ 2 , where represents the displacement mode shape ordinate at the ℎ measured location and Δ is the distance between two successive measured locations. As an alternative, the use of third-order polynomials (Dessi and Camerlengo, 2015) or spline interpolations (Quaranta et al., 2014) has also been proposed in the literature. In this paper, third-order piecewise-polynomial interpolation functions (i.e., cubic spline interpolants) are fitted to the discrete set of experimental data: such functions are selected within the class of C 2 functions, that is, continuous functions with continuity ensured also in the first and second derivatives. These functions can be generally expressed by a general analytical format ( ) = + ( − )+ ( − ) 2 + ( − ) 3 , used in its natural (or variational) form, that is, by setting homogeneous boundary conditions at the two extremes. The spline coefficients , , and are easily determined by solving a system of linear equations and imposing compatibility conditions between the analytical spline and the discrete representation of the data. The modal curvature is then obtained by second derivative of the resulting function, i.e., ′′ ( ) ≈ ′′ ( ) . Once the modal curvature is determined for the undamaged and damaged specimen, the following damage indicator is computed and plotted along the beam length: = | ′′ − ′ ′ | (5)

Fig. 5: Results in terms of displacement modal shapes (top) and corresponding damage indicator in terms of discrepancy in modal curvature (bottom) for the PC beam R03_2 under different damage scenarios.