## 4 Results
### Simplified Non-Linear Sectional and Beam Analysis  
All previous comparisons showed a good agreement with the experimental data in terms of the global response of the beams. In this section, a detailed comparison is presented against two beams tested by [PERSON] (2007) [18] and [PERSON] (1997) [19]. [PERSON] reported the global and sectional responses in his work. The sectional response was obtained by locating strain gauges at the tensile steel level and the top surface of the beam, which allowed for a full strain profile development. Unfortunately, [PERSON] (2007) [18] reported the failure of the strain gauges in the post-yielding zone. The beams were simply supported and were tested in four-point bending. The supports were placed 3 inches from the edge of the beam, providing a clear span of 4.72 m (15.5 ft). Tables 4 and 5 show the geometrical and material properties of the beam. Similarily, [PERSON] (1997) [19] reported the experimental load-deflection and the experimental moment-curvature curves for a beam they tested. Tables 6 and 7 show the geometrical and material properties of the analyzed beam. The present analysis, of these two extra beams, was performed as described in this paper to obtain the load-mid span deflection graphs and the load-tensile steel strain curve as well as the moment-curvature and the moment-maximum compressive strain curves. This comparison was necessary to illustrate that the presented constitutive model matches not only the global response of the beam but also it matches the sectional local response. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**h (mm)** & **b (mm)** & **d (mm)** & **cc (mm)** & **d\({}^{\prime}\) (mm)** & **L/2 (mm)** & **La (mm)** \\ \hline
210.8 & 200.7 & 160 & 25.40 & 43.18 & 1350.01 & 1249.68 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: The geometrical parameters for [PERSON] (2007) beam. Figure 17: [PERSON] and [PERSON] (1991) [17] moment vs. maximum compressive strain analytical graphs. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**h (mm)** & **b (mm)** & **d (mm)** & **cc (mm)** & **d\({}^{\prime}\) (mm)** & **L/2 (mm)** & **La (mm)** \\ \hline
304.8 & 152.4 & 261.87 & 25.4 & 39.62 & 2362.2 & 1752.6 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: The material parameters for [PERSON] (2007) beam. A good agreement between the experimental and the present analytical load-deflection curve is shown in Figure 18. This agreement confirms the accuracy of the proposed model against the global behavior of the beam. However, it is important to validate the constitutive tensile model against the local sectional response as well. Figures 19 and 20 show load vs. maximum compressive strain and load vs. tensile steel strain. Excellent agreements between the present analytical and the experimental results were observed for the two comparisons. Furthermore, the present analytical sectional solution is compared against the trilinear sectional response developed by [PERSON] et al. (2013) [20] in Figures 21 and 22. These comparisons confirm the trilinear behavior of the moment-curvature and moment-maximum compressive strain graphs. Figure 21: [PERSON] (2007) [18] moment vs. curvature analytical graphs. Figure 22: [PERSON] (2007) [18] moment vs. maximum compressive strain analytical graphs. [PERSON] (1997) [19] beam was also analyzed by the present analytical procedure. Figure 23 shows the global analytical and experimental response. An excellent match was obtained using the proposed constitutive tension model for the analytical vs. experimental load-deflection data.