# Experimental Testing of Reinforced Concrete Deep Beams Designed by Strut-And-Tie Method
## 4 Finite Element Modeling  
As part of this research, FEM models were developed in Abaqus 2016 [15] for the simulation of the reinforced concrete deep beams behavior, which were validated on the basis of the experimental test results. In future researches, one of the FEM models will be useable for analyzing the behavior of this type of members, but with different geometric characteristics and reinforcement layouts. In the models, the analyses were performed assuming a plane stress state. An analysis involving geometric and material nonlinearity was applied. In the numerical simulations of the behavior of the tested specimens, two FEM models were formed for each specimen individually. The first FEM model (FEM A) is shown in Figure 16. The static system of model FEM A corresponds to the static system of a simple beam applied in the program \"ST method\" for the design of walls, Figures 6-8. Figure 14: Steel frame for testing reinforced concrete members: (**a**) frame appearance and (**b**) geometric characteristics. Figure 13: The arrangements of comparators for measuring deflection and strain gauges for specimen W3. Figure 15: Out-of-plane stability of the specimen: (**a**) position of the spacers and (**b**) spacer. According to the results, crack arrangements, Figure 22, and mid-span deflection, Figure 24, determined on the numerical model FEM A and experimental results, it is shown that the differences were large. The main reason is the existence of an interaction between the steel frame and the tested specimens, Figure 14, which is explained in detail in Chapter 5. Regarding this, a more complex FEM model (FEM B) was formed, which is shown in Figure 17. In the models FEM A and FEM B, the same dimensions of the tested samples were applied, as well as the same reinforcement layouts. Both models used the same constitutive model to simulate the nonlinear behavior of concrete and reinforcement, as well as the same load application scheme. The model FEM B is expanded with a model of steel frame, contact with the floor base, and the spacers for reinforcement. The nonlinear behavior model for reinforced concrete, the so-called continuum mechanics-based model, was applied. In such model, concrete can be discretized with three-dimensional or two-dimensional solid finite elements, and the reinforcement is modeled with truss finite elements. The constitutive model for concrete is the concrete damaged plasticity (CDP) model, formed by a combination of plasticity theory and damage mechanics according to recommendations [33; 34; 35; 36]. In all nonlinear analyses, the relationship between stress and deformation in concrete at uniaxial pressure prescribed in EN1992-1-1 [21] was applied. Figure 17 shows the relationships between engineering stresses and strains at uniaxial pressure and tension for concrete and reinforcement. The values of the constituent parameters of the CDP model were adopted based on the recommendations given [37] and summarized as \(K_{c}\) = 0.7, \(\psi\) = 13\({}^{\circ}\), \(\sigma_{\text{Io}}\)/\(\sigma_{co}\) = 1.16, and \(\epsilon\) = 0.1. \(K_{c}\) defines the relationship between the second invariant of the stress in the meridional plane at tensile and pressure, \(\psi\) is dilatancy angle, \(\sigma_{\text{Io}}\)/\(\sigma_{\text{co}}\) is the ratio of the biaxial to the uniaxial compressive strength of concrete and \(\epsilon\) is the eccentricity of the plastic potential surface. Damage variables for compression and tension were determined according to the recommendations given in [37]. The modulus of elasticity of the concrete was 30 GPa and the density 2500 kg/m\({}^{3}\). CPS4 finite elements were used to model the concrete (C--continuum stress/displacement; PS--plane stress; 4-4-node) [15], for the adopted uniform mesh size of \(20\times 20\) mm. The constitutive behavior of steel for reinforcement is modeled by a bilinear relationship between stress and strain. The modulus of elasticity was 210 GPa and the density 7850 kg/m\({}^{3}\).