## 3 Constitutive Tension Model  
The proposed model in this study is based on cycles of numerical analysis and experimental-numerical deflection error minimization with respect to material model parameters until convergence to the experimental response. A descending model was critical to represent the gradual loss of tension stiffness with the propagation of the beam cracks. This model represents the contribution of concrete in tension between two successive cracked sections. This approach is known as the smeared crack approach, which simulates the global response of the entire beam. Therefore, this response could be viewed as an averaged behavior for all beam sections. The tension model may be a function of the structural element and the loading patterns considered. A uniaxial plain concrete element would show a different behavior than a reinforced concrete flexural beam due to the steel-concrete bond in the latter member. The presented constitutive tension model for reinforced concrete flexural beams is a summation of various reinforced concrete features. Plain concrete softening and steel tension stiffening are considered to be the main phenomena incorporated to develop the constitutive model. Tension softening occurs due to the development of concrete flexural cracks. The fracture energy, transformed from the strain energy, advances existing major cracks and initiates minor micro cracks, which weaken the concrete at relatively high rates. Steel-Reinforced concrete tension stiffening is the second important characteristic that represents the local bond-slip between the concrete and the steel bars. This feature causes the effective or averaged tensile stresses in the concrete to gradually reduce in between two successive primary cracks. In addition to tension softening and tension stiffening, shrinkage and aggregate size also participates in altering the concrete tensile behavior. Shrinkage builds up residual stresses in the concrete while hardening. This results in lowering the tensile strength and tension stiffening effects. On the other hand, the aggregate size/shape effects are mainly observed when shear and tensile stresses are combined. Therefore, a numerical constitutive tension model based on calibrating the global experimental response is important to capture all these characteristics in one model. The proposed constitutive concrete tensile model relates the tensile strain (\(\varepsilon_{t}\)) with the tensile concrete stresses (\(f_{t}\)) as a function of the material parameters calibrated based on experiments for flexural reinforced concrete beams. The model is presented as a function of concrete cracking and steel yielding parameters, Equation (23). Stress and strain constants in the equation were extracted by actively minimizing the deflection error between the numerical analysis vs. a large pool of four points bending tests on flexural beams--see Tables 2 and 3. \[\frac{f_{t}}{0.5f_{r}}=1-\frac{1}{\ln\left(\frac{1.4\varepsilon_{f}}{ \varepsilon_{cr}}\right)}\ln\left(\frac{\varepsilon_{t}}{\varepsilon_{cr}}\right) \tag{23}\]  
This proposed descending function on the domain (\(\varepsilon_{cr}\leq\varepsilon_{t}\leq 1.4\varepsilon_{y}\)) shows a sudden drop to (\(0.5f_{r}\)) at the cracking strain (\(\varepsilon_{cr}\)) upon strain to fracture energy conversion and continues to descend till zero when the tensile strain reaches (\(1.4\varepsilon_{y}\))--see Figure 7. The sudden drop in stress at cracking is a function of the tension softening effect while the exhaustion of the tension stiffening effect is expected to be related steel yielding.