## 3 Results and Discussion
### Ductility Analysis  
The ductility of a structural system is a measure of its plasticity-to-elasticity deformations and can be used to describe a system's toughness. This structural parameter is generally measured by the deflection ductility index (\(\mu\)) [1,8,36,48]. The amount of the apparent mid-span deflection is, therefore, required for the evaluation of \(\mu\). In the current investigation, two deflection ductility indices (i.e., \(\mu_{u}\) and \(\mu_{m}\)) were calculated to quantify the ductility properties of UHPC beams. Here, \(\mu_{u}\) and \(\mu_{m}\) were obtained by dividing the peak load and ultimate deflections by those corresponding to the bar-reinforcement yielding (\(\Delta_{y}\)), as given by Equations (4) and (5), respectively:  
\[\mu_{u} =\Delta_{u}/\Delta_{y} \tag{4}\] \[\mu_{m} =\Delta_{m}/\Delta_{y} \tag{5}\]  
The results of the ductility indices are presented in Table 7. In this table, the B-type beam was taken as the control beam for comparing the ductility properties of the bar-reinforced UHPC beams, as the unreinforced beam clearly had inferior ductility. In Table 7, the percentage differences between the ductility parameters and the control ones are given in parentheses. From both calculated indices, it can be concluded that the ductility of the UHPC beam decreased as its effective depth decreased. Based on \(\mu_{m}\), the ductility of the beam with a 75 mm effective depth (F-type) was surprisingly comparable to that of \(\mu_{m}\) with a 125 mm effective depth (B-type). Thus, \(\mu_{u}\) was found to be more practical for evaluating the ductility of UHPC beams. According to this property, the best tensile reinforcement effective depth after 125 mm was 105 mm. Based on this particular investigation, a 125 mm effective depth appears to provide the highest capability for fibers to control cracks.  
\begin{table}
\begin{tabular}{c c c c c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Beam**} & \multicolumn{2}{c}{**Elastic Phase**} & \multicolumn{2}{c}{**Vield Phase**} & \multicolumn{3}{c}{**Ultimate Phase**} & \multicolumn{3}{c}{**Ductility Indices**} \\  & \(P_{\sigma}\) & \(\Delta_{m}\) & \(P_{y}\) & \(\Delta_{y}\) & \(P_{u}\) & **Mean** & \(\Delta_{m}\) & \(\Delta_{m}\) & \(M_{\sigma}\) & \(\mu_{u}\) & **Mean** & \(\mu_{u}\) & **Mean** & \(\mu_{u}\) \\  & **(R-N)** & **(mm)** & **(R-N)** & **(mm)** & **(R-N)** & **(mm)** & **(mm)** & **(mm)** & **(R-N)** & **(mm)** & **(mm)** & **(mm)** & **(mm)** & **(mm)** \\ \hline A1 & 103.6 & 0.46 & - & - & 195.2 & 2.37 & 8.62 & 21.96 & 7.4 & - & - & - & - \\ A2 & 85.0 & 0.51 & - & - & 150.4 & 172.8 & 1.93 & 8.62 & 16.92 & 8.0 & 7.7 & - & - & - \\ \hline B1 & 93.8 & 0.50 & 207.3 & 1.60 & 265.2 & 269.7 & 2.92 & 13.17 & 298.44 & 10.4 & 10.8 & 1.83 & 8.26 & 9.38 \\ B2 & 104.4 & 0.52 & 179.6 & 1.20 & 274.1 & 269.7 & 2.70 & 13.92 & 30.83 & 11.2 & 2.25 & 2.04 & 11.61 & 9.93 \\ \hline C1 & 130.2 & 1.06 & 245.0 & 2.40 & 253.9 & & 2.76 & 17.66 & 26.28 & 16.4 & 16.3 & 1.15 & 1.22 & 7.36 & 7.55 \\ C2 & 186.3 & 1.42 & 265.1 & 2.35 & 282.9 & & 3.02 & 18.17 & 29.28 & 16.2 & (+50.9\%) & 1.29 & (\(-40.3\%\)) & 7.75 & (\(-24.0\%\)) \\ \hline D1 & 139.5 & 1.06 & 255.8 & 2.51 & 280.8 & & 3.77 & 17.89 & 24.00 & 12.4 & 12.6 & 1.50 & 1.58 & 7.14 & 8.27 \\ D2 & 128.8 & 0.90 & 255.5 & 2.26 & 318.7 & & 3.76 & 21.26 & 27.25 & 12.8 & (+16.6\%) & 1.66 & (\(-22.5\%\)) & 9.40 & (\(-16.7\%\)) \\ \hline E1 & 169.0 & 1.19 & 321.1 & 31.18 & 3551.1 & & 4.68 & 22.99 & 27.17 & 15.6 & 15.8 & 1.47 & 1.34 & 7.24 & 8.38 \\ E2 & 170.2 & 0.85 & 309.8 & 2.44 & 323.5 & & 2.93 & 21.47 & 16.0 & (+46.3\%) & 1.20 & (\(-34.5\%\)) & 9.52 & (\(-15.6\%\)) \\ \hline F1 & 170.4 & 1.21 & 301.4 & 2.93 & 341.2 & & 3.83 & 24.79 & 23.03 & 16.8 & 18.4 & 1.31 & 1.24 & 8.47 & 9.07 \\ F2 & 158.4 & 1.03 & 308.3 & 2.97 & 329.1 & & 3.49 & 28.65 & 22.21 & 20.0 & (+70.4\%) & 1.18 & (\(-39.1\%\)) & 9.66 & (\(-8.7\%\)) \\ \hline \hline \end{tabular}
\end{table}
Table 7: Summary of the four-point testing of UHPC beams.