## 3 Finite-Element Modeling (FE) and Validating
The concrete damage plasticity (CDP) formulation has two primary failure mechanisms: compressive crushing and tensile cracking, which represent the nonlinear behavior of concrete under various loading conditions. The CDP model effectively represents these failure mechanisms, and the plasticity component describes the inelastic strains resulting from irreversible deformation, utilizing a yield criterion to identify the shift from elastic to plastic behavior. Yield surfaces, such as the Mohr-Coulomb or Drucker-Prager criteria, are commonly applied to concrete to accurately define this transition. Figure 8 presents the force-displacement curve of the numerical model, compared with the experimental test results. It is clearly visible that the two curves align very well, confirming that the numerical model has been validated and is suitable for further investigations. The push-out tests consistently concluded with the failure of the studs, causing the load-displacement curve to suddenly drop after reaching the peak point. In the finite-element modeling, specifically within the Static Riks analysis, we aim to identify this point, which marks the end of the plastic behavior of the steel-concrete composite structure. The plastic failure observed in the numerical model occurred at the same displacement recorded during the push-out test. Figure 6: Compression properties of concrete. Figure 7: Tension properties of concrete. In this research, the Python-based script for Abaqus has been developed, which integrates all essential model configurations, including geometry, material models, contact definitions, and meshing parameters. This comprehensive integration ensures consistency and reliability throughout the modeling process. Following validation, the objective is for this Python-based parametrically constructed model to serve as a foundation for extensive studies similar to those referenced in the introduction. The validated model, due to its highly refined mesh comprising 868.012 variables, demands substantial computational resources. To enhance computational efficiency, the model was refined by focusing on a quarter of the model space, applying appropriate boundary conditions and symmetry settings, as illustrated in Figure 9. This refinement reduced the number of variables to 215.409, significantly decreasing computational demands while maintaining model accuracy. Figure 10 presents the force-displacement results of the previously introduced numerical model and experimental test, along with the corresponding results from the quarter model. This clearly demonstrates that the model accurately simulates real-world conditions. In addition to the force-displacement curve obtained from the numerical modeling,  
Figure 8: FE model and experimental results comparison. Figure 9: Finite-element model of the quarter model and applied boundary conditions. ABAQUS allows for the visualization of the states (such as stresses, deformations, displacements, and damages) that develop during each iteration of the Newton-Raphson process, step by step, across the entire structure at any given point. Thus, beyond the force-displacement validation between the experimental and numerical models, we placed significant emphasis on verifying the numerical model against real-world observations, such as stud deformations and concrete damages. It was found that these observations were fully consistent, further supporting the validation. The quarter model will be further utilized to investigate the effect of parameters, such as stud head diameter, stud length, shank length, shank diameter, and stud positioning on both the elastic and plastic behavior of the connection. Detailed Analysis of the Effects of Geometric Variations on the Behavior of Steel-Concrete Composite Structures  
This chapter presents the modeling results of steel-concrete joint connections, utilizing the finite-element model discussed in the preceding sections.