PSI - Issue 47

Umberto De Maio et al. / Procedia Structural Integrity 47 (2023) 469–477 Author name / Structural Integrity Procedia 00 (2019) 000–000

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13 unloading paths at different load levels (L1-L12) evaluated as a fraction of the failure load, have been performed. It can be seen that before rebar yielding, which occurs at the L10 load level, small residual plastic deformations are present due to the stiffening effect provided by the FRP plate; after this stage, however, large residual plastic deformations are recorded until the damage reaches the L12 level, at which debonding of the reinforcement system occurs and the loading curve drops to that of the control beam. After the static analysis, a linearized eigenvalue problem has been solved at the endpoint of each unloading path in order to take into account the effects of the complex nonlinear phenomena, i.e. concrete plasticity and crack partial closure behaviour, on the natural vibration frequencies. The results of the modal analysis in terms of degradation of the natural frequencies as the damage level increases, normalized with respect to the value obtained for the undamaged configuration, are reported in Fig. 4. It is important to note that only those mode shapes that exhibit purely flexural behaviour have been chosen, going to avoid those with coupled bending-axial behaviour and rigid motions.

Fig. 4. Variation of the normalized natural vibration frequencies for the 1 st ,4 th , 6 th , and 7 th mode shapes as the damage level increases.

In order to evaluate the effects of diffuse damage on the dynamic characteristics of FRP-strengthened RC beams, “Modal Assurance Criterion” MAC (Allemang and Brown, 1982), has been used. Such an indicator allows to compare the vibration modes obtained in the damaged and undamaged configurations by the following expression:      2 ,0 , ,0 ,0 , , T i j d ij T T i i j d j d MAC        , (6) , j d  is the component vector in the damaged configuration of the j -th vibration mode. The results, for all damage levels, are reported in Table 2. The indicator has been evaluated for the four investigated mode shapes based on the displacement obtained from the points placed at the upper side of the beam. It is important to remind that, a MAC value of 1 represents a perfect match between the damaged and undamaged vibration modes, while a value near to 0 indicates a complete mismatch between the modes. As can be seen, perfect correspondence occurs only for the first load level; in fact, in this case, all contributions along the main diagonal are equal to 1. For subsequent damage levels, small differences can be seen, with the values deviating only slightly from the unit value. wherein ,0 i  is the component vector in the undamaged configuration of the i -th vibration mode, while