## 4 Numerical Modeling and Methodology
### Finite-Element Modeling Strategy
's [15] work, thereby providing compelling justification for the new model's implementation in assessing the feasibility of ACPH applications in structurally deficient precast frameworks encountered in real-world scenarios. In this study, to thoroughly investigate the effect of ACPH, an ABAQUS model was initially developed for four PRC frames with different characteristics. For the purpose of calibrating these finite element models, the ABAQUS model of the frame with ACPH, as detailed in the experimental study by [PERSON] et al. [15], was established. Subsequently, the cyclic load applied in the referenced experiment was imposed on this model. The Concrete Damaged Plasticity (CDP) material model offers a well-established framework for simulating the nonlinear post-cracking behavior of RC members under cyclic or dynamic loads. The CDP model transitions from an elastic state to one defined by tension cracking, compression softening, and stiffness degradation under cyclic loading. This approach significantly differs from a simple elastic-perfectly plastic model, as it incorporates tensile damage, strain-softening effects, and the gradual reduction in concrete's load capacity post-microracking. It also considers compressive crushing and the associated decline in concrete stiffness. Figure 3 illustrates the monotonic compressive stress-strain relationship for C35 concrete, which constitutes the envelope curve in the CDP model used in ABAQUS [33]. While the curve does not directly depict cyclic behavior, the CDP model employs internal damage factors to replicate stiffness deterioration, a decrease in unloading slope, and inelastic residual stresses under reversed cyclic loading. The effects are regulated by  
factors including the compression damage variable, plastic strain thresholds, and viscosity regularization, as originally defined by [PERSON] and [PERSON] [34], enabling the model to shift from elastic to softening regimes while ensuring numerical stability. The depiction of cyclic compressive behavior in concrete is essential for appropriately modeling stress redistribution at the beam-column contact in ACPH-equipped frames. As seismic stresses change direction, the CDP model accurately reflects the gradual reduction in stiffness and strength in the cover and core areas, allowing for the identification of localized compressive failure when the concrete's capacity is surpassed. This enables the simulation to ascertain if plastic rotation is effectively diverted into the ACPH steel plates, instead of manifesting inside the concrete itself. The calibrated compressive response used in this work, sourced from the literature and design standards, was adequate for replicating the desired damage localization behavior under cyclic settings. Concrete exhibits a relatively low tensile strength when compared to its compressive strength; however, this characteristic plays a crucial role in influencing post-cracking stiffness and the pattern of crack propagation. Utilizing a standard uniaxial tensile stress-strain relationship without any softening consideration may lead to abrupt brittle failure when tensile stresses surpass the cracking threshold. Conversely, empirical data indicate that, after the onset of cracking, concrete can sustain some tensile stress across the cracks due to reinforcement interaction, which is referred to as \"tension stiffening\". A widely recognized approach for defining post-cracking softening behavior in materials is the modified tension stiffening law, as elaborated by [PERSON] and [PERSON] [35], among others. By integrating tension stiffening into analytical frameworks, the potential for sudden crack initiation is diminished, resulting in a more even distribution of damage within structural components subjected to flexural or tensile stress.