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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Moreover, the proposed method takes into account both the concrete plasticity and the contact effect between the crack faces during the unloading stage. Both static and dynamic behaviors of FRP-plated RC beams have been studied, evaluating the mode shapes by solving small amplitude-free oscillations problems at the final point of each unloading path. Finally, the variation of the natural vibration frequencies as damage level increase has been investigated, and the most common dynamic damage indicators have been calculated to assess the location and magnitude of the damage within FRP-reinforced concrete elements. 2. Static and dynamic behavior of FRP-strengthened reinforced concrete beam under cyclic loading conditions The present work aims to investigate the degradation of vibration characteristics in FRP-strengthened RC beams subjected to cyclic loading conditions. An advanced numerical model has been developed and implemented in the commercial Finite Element software COMSOL Multiphysics ® 6.0. The mechanical behavior of the tested structures, including cracking and debonding phenomena, are simulated by employing a cohesive/volumetric finite element approach able to simulate the crack onset and propagation in the concrete phase and the debonding failure mechanisms of the FRP system (De Maio et al., 2022b). The mechanical interaction between concrete and steel rebars has been simulated through an embedded truss model (ETM). In the following sections, a detailed description of the cohesive laws adopted to opportunely take into account the cycling loading conditions, the concrete plasticity, and the contact phenomena will be provided. For further details about the DIM and ETM models applied to reinforced concrete structures see (De Maio et al., 2019b) and reference therein. 2.1. Cohesive modeling for FRP-strengthened RC beams subjected to cycling loading conditions In order to analyze the damage phenomena in existing structures subjected to cyclic loading conditions, an accurate numerical formulation that incorporates concrete plasticity, is necessary. As previously introduced by some of the authors in (De Maio et al., 2023a; Pranno et al., 2022b), such effect is considered by adopting the plastic contributions p s  and p n  , in the normal and tangential direction, for the components of the displacement jump   u , as follows:     max max max max n p n n p n n n s s p s s p s s                                    , (1) Such plastic contributions have been set as a fraction of the maximum values of the tangential and normal displacement jump, i.e. max p s s s     and max p n n n     . In particular, s  is set equal to 0.5 according to the approach reported by (Foster and Marti, 2003), meaning that a fixed plastic deformation is introduced for the tangential component. Instead, n  is a linear function rely upon the ratio between the actual tensile stress and the critical tensile strength of the material   / c ft ft and it has been defined by a calibration procedure to match the experimental results given in (Reinhardt, 1984). Fig. 1 shows the complete traction-separation laws for Mode I and Mode II fracture processes both characterized by a plastic stiffness thus defined:   0 max max 1 with , i i p i p i i D K K i n s        , (2)

where D is the scalar damage variable governed by an exponential function, and 0 stiffness, in its normal ( i n  ) and the tangential ( i s  ) components.

i K denoting the initial elastic