PSI - Issue 66

Umberto De Maio et al. / Procedia Structural Integrity 66 (2024) 459–470 Author name / Structural Integrity Procedia 00 (2025) 000–000

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of structures, taking into account both the elastic-plastic and fracture components (Capitán and Garijo, 2024; Omidi and Lotfi, 2013). In this study, a numerical model based on finite element method is developed to simulate the behavior of reinforced concrete beams subjected to monotonic loads. The implemented model uses the Coupled Damage Plasticity Model to accurately describe the damage evolution and degradation of the dynamic properties of the structure. The goal is to study in detail how damage phenomena affect the natural frequencies of vibration and modal shapes of the beam, thus providing an effective structural monitoring strategy to ensure the safety and durability of existing infrastructure. 2. Adopted numerical modelling The degradation of the modal characteristics in reinforced concrete (RC) beam structures, depending on damage phenomena caused by monotonic and cyclic loads affecting the structure, has been analyzed by using a finite element (FE) model implemented in the commercial software COMSOL Multiphysics and consists of two different mathematical sub-models. In particular, as explained in Section 2.1, the Coupled Damage Plasticity model (CDPM) (Grassl et al., 2011), used to simulate the structural behavior of a cracked RC beam, is employed in combination with the Embedded Truss Model (ETM), already employed by some of the authors in (De Maio et al., 2023) to capture the interaction between the concrete and the reinforcement steel bars. In Section 2.2 the dynamic damage indicators employed to detect the occurrence of damage phenomena are briefly described. 2.1. Coupled Damage Plasticity Model The Coupled Damage Plasticity Model is an advanced approach able to describe the mechanical behavior of degraded materials. In this model, plasticity and damage are coupled to accurately represent both permanent deformation due to plastic stresses and reduction in material stiffness due to micro-cracking phenomena or other forms of degradation. It is used in advanced numerical simulations to predict material behavior under cyclic loading or under monotonic loading to failure. The general constitutive equations for the coupled damage-plasticity model are based on the concept of the damaged stress tensor d σ and the undamaged stress tensor un σ .       1 1 : d t un c un un p d d         σ σ σ σ C ε ε (1) un  σ represent the positive and negative components of the effective stress tensor un σ , respectively. The two scalar damage variables, t d and c d , range from 0 (no damage) to 1 (fully damaged), and describe the extent of damage affecting the material. The plasticity model is independent of damage, and it is described through the classical constitutive relations of solid mechanics, in which a yield function   , p un p F  σ defines the start condition of plasticization:   1 2 , , p un h h F q q σ (2) where   1 h p q  and   2 h p q  are the dimensionless variables controlling the evolution of the size and shape of the yield surface and plastic potential p Q . The plastic flow rule is not-associated, and it can be written as:   , p un p pl p un Q      σ ε σ   (3) In this context, un  σ and