## 4 Numerical Analysis
### Model Geometry, Material Models and Finite Element Mesh  
Models for numerical analysis were formed in accordance with experimentally tested models of continuous beams (Figure 8). Figure 8a shows the geometry of the concrete beam which is the  
Figure 6: Map of cracks of B\(-\)1, B\(-\)2 and B\(-\)3 beams. Figure 7: Comparative load vs. crack width diagrams obtained using LVDT 3 in Section I and LVDT 4 in Section III. same for all analyzed beams (B-1, B-2, B-3). Figure 8b shows the geometry of the reinforcement for Beams B-1 and B-2 (the geometry is the same, the difference is in the diameter of the reinforcement), and Figure 8c shows the geometry of the reinforcement of the Beam B-3. Bilinear kinematic hardening is adopted as a material reinforcement model (Figure 9a). The modulus of elasticity of steel is 200 GPa, and the Poisson's ratio is 0.30. The material behaves ideally plastic up to the yield point which is \(\mathrm{f_{y}}=500\) MPa, and after that, plasticization of material occurs, with the adopted force of rupture \(\mathrm{f_{u}}=560\) MPa at the strain of \(\mathrm{\varepsilon_{u}}=0.20\). Concrete is modeled with a different stress-strain relation under compression and tension. Based on the results of the experimental test, the relationship between stress and compression strain was determined (Figure 2), which was also used in numerical modeling. Figure 8: Model geometry in the software package Abaqus: (**a**) concrete beam; (**b**) reinforcement of Beams B–1 and B–2; (**c**) reinforcement of Beam B–3. Figure 9: Stress–strain diagram for: (**a**) steel reinforcement; (**b**) concrete in tension. Based on the results of experimental tests, the compressive strength of concrete was defined as 50.13 MPa, while the modulus of elasticity was calculated as the ratio of stress and strain at 40% of the value of concrete compressive strength. It has been adopted that concrete behaves linearly elastic up to a compressive stress corresponding to 40% of the value of compressive strength, which is in accordance with European regulations [37]. The relationship between stress and strain during concrete tension has a great influence on the results of the analysis due to the sensitivity of the model to the development of cracks in concrete. After the analysis of different models, and due to the lack of experimental testing of tensile strength of concrete, the stress-strain relationship was chosen, which was proposed by [PERSON], [PERSON], [PERSON], [PERSON] [40]. According to this model, the relationship of stress and strain (Figure 9b) is linear up to the tensile strength of concrete f\({}_{\text{t}}\) with the corresponding strain \(\epsilon_{\text{cr}}\), and with the increase of stress, the relation has a form:  
\[\sigma_{\text{t}}=\text{f}_{\text{t}}\Big{(}\frac{\epsilon_{\text{cr}}}{ \epsilon_{\text{t}}}\Big{)}^{\text{c}};\;\text{c}=0.1;\;\epsilon_{\text{t}}> \epsilon_{\text{cr}} \tag{1}\]  
The concrete damaged plasticity (CDP) material model was used to define the plastic behavior of concrete. The CDP model introduces concrete damage due to compression and tension in the analysis. It is based on models of fracture mechanics (Drucker-Prager hyperbolic plastic flow potential function) and is adequate for modeling quasi-brittle materials [41]. A more detailed analysis of the CDP material model can be found in the literature [42; 43; 44; 45]. For adequate application of the CDP model, it is necessary to define the following parameters:  
* Dilation angle \(\psi\) (\(10^{\circ}\leq\psi\leq 40^{\circ}\)), whereby the adopted value in the paper is \(\psi=35^{\circ}\). The analyses indicated that satisfactory results are obtained when the dilation angle is between \(30^{\circ}\) and \(40^{\circ}\), as well as the differences are minimal for the different values of dilation angle in the mentioned range [43; 45];
* eccentricity of the plastic potential flow \(\varepsilon\), whereby the adopted value in the paper is \(\varepsilon=0.1\) (default value).