## 3 Results
### Ductility  
The ductility of a material is defined as its susceptibility to deformation without affecting its load-bearing capacity. The following analysis is carried out on the basis of [24]. To evaluate the influence of reinforcement on the bending ductility, four different indices \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline  & **Beam A1** & **Beam A2** & **Beam A3** & **Beam A4** & **Beam A5** & **Mean Value** &
\begin{tabular}{c} **Standard** \\ **Deviation** \\ \end{tabular} \\ \hline Maximum load & 30.35 & 37.83 & 33.29 & 20.94 & 32.49 & 30.98 & 6.24 \\ Deflection at failure & 53.41 & 67.94 & 88.68 & 40.75 & 45.95 & 59.35 & 19.33 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Results of beam tests—Series A. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline  & **Beam B1** & **Beam B2** & **Beam B3** & **Beam B4** & **Beam B5** & **Mean Value** &
\begin{tabular}{c} **Standard** \\ **Deviation** \\ \end{tabular} \\ \hline Maximum load & 29.69 & 31.24 & 38.07 & 33.97 & 34.96 & 33.59 & 3.27 \\ Deflection at failure & 53.61 & 86.50 & 84.78 & 76.2 & 54.31 & 71.08 & 16.11 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Results of beam tests—Series B.  
Figure 8: (a) Average deflection and (b) maximum loading force values. Figure 7: Force–deflection relationship for strengthened beams (B-series). were used. Three of them can be classified as deflection or deformation indices, whereas one is based on energy absorption. Energy absorption values were evaluated using the load versus deflection curves. The ductility expressed by the index \(D\), which is an analysis of the deflection of reinforced beams relative to unreinforced beams, is represented as a ratio, as follows:  
\[D=\frac{u_{Fmax,uv}}{u_{Fmax,uv}}, \tag{1}\]  
where \(u_{Fmax,uv}\) is the deflection at maximum load for the reinforced beam, and \(u_{Fmax,uv}\) is the deflection at maximum load for the unreinforced beam. The deflection ductility index \(\mu_{d}\) and \(\mu_{\Delta}\) are expressed as a ratio of the respective deflection values, as follows:  
\[\mu_{d}=\frac{u_{u}}{u_{e}}, \tag{2}\]  
\[\mu_{\Delta}=\frac{u_{u}}{u_{y}}, \tag{3}\]  
where \(u_{u}\) is the value of maximum deflection while keeping the load-bearing capacity, \(u_{e}\) is the value of deflection at the proportional limit, and \(u_{y}\) is the value of the deflection at the moment of yielding. The ductility index \(\mu_{E}\) was calculated based on the values of total energy and elastic energy:  
\[\mu_{E}=\frac{1}{2}\cdot\left(\frac{W_{tot}}{W_{e}}+1\right) \tag{4}\]  
where \(W_{tot}\) is the total energy value, and \(W_{e}\) is the elastic energy value. The load-deflection relationship curve was approximated using a fifth-degree polynomial. The elastic limit is defined as the point at which the trend line of the relevant graph section intersects with the polynomial function. In order to ascertain the yield stress, two further functions were defined. The first function is linear and parallel to the x-axis, passing through the point of maximum load. The second function is also linear, inclined at an angle determined from the slope of the linear elastic part of the graph in relation to the first function. The yield stress was identified as the point of intersection between the calculated straight line and the trend line. The energy values from the origin to the characteristic points were calculated using labeled integrals. A detailed description of the evaluation method can be found in [24]. The results for the various methods of calculating the ductility index are shown in Table 4. Statistical analyses of the deflection values, estimated surface areas, and ductility indices are presented in Tables 5 and 6.