# Investigation into the Flexural Toughness and Methods of Evaluating Ductile Concrete
## 4 Methods of Evaluating Bending Toughness of High-Ductility Concrete  
### Methods of Evaluating Bending Toughness  
The American ASTMC-1018 standard [39] evaluation method is based on the entire process of stress, as shown in Figure 9a, using the mid-span deflection \(\delta\), toughness index \(I\), and residual strength index R of the test specimen when the first crack appears to evaluate the toughness of fiber-reinforced concrete; this is suitable for concrete with a large fiber content and a stable load-deflection curve after cracking. However, all calculations depend on the existence of a large subjective initial crack deflection \(\delta\) and the initial defects of the test piece, which have a great impact on the production of an accurate solution as well as calculation errors. The Japanese Civil Society Standard [40] (JSCE-SF4) defines a toughness factor to characterize the toughness of fiber-reinforced concrete materials, which is the average bending strength when the deflection \(\delta\) of the beam span reaches \(1/150\), as shown in Figure 9b. The method of evaluating the flexural toughness of fiber-reinforced concrete  
Figure 8: Effect of the water-binder ratio of HDC on load-deflection curves: (a) 56 d; (b) 28 d.  
\begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline
**Specimen** & \(f_{\text{max}}\) (MPa) & \(\delta\) (mm) & \(\delta_{\alpha}\) (mm) & \(f_{\text{ass}}\) (MPa) & \(f_{1}\) (MPa) & \(R_{1}\) & \(f_{\text{ass}}\) (MPa) & \(R_{\text{ass}}\) (MPa) & \(R_{\text{ass}}\) \\ \hline HCS6-0-29 & 4.62 & 0.05 & 0.05 & 1.25 & 2.27 & 1.82 & — & — & — & — \\ HCS6-1-29 & 8.88 & 0.64 & 0.09 & 1.61 & 6.15 & 3.82 & 6.40 & 3.98 & 5.45 & 3.39 \\ HCS6-1-5-29 & 11.49 & 0.95 & 0.06 & 2.00 & 8.60 & 4.29 & 8.81 & 4.39 & 7.90 & 3.94 \\ HCS6-2-29 & 15.32 & 1.18 & 0.08 & 2.67 & 11.56 & 4.33 & 11.80 & 4.42 & 9.76 & 3.65 \\ HCS6-2-26 & 15.04 & 1.28 & 0.08 & 2.45 & 11.94 & 4.87 & 12.03 & 4.91 & 11.04 & 4.51 \\ HCS6-2-32 & 13.32 & 1.29 & 0.08 & 2.05 & 10.63 & 5.18 & 10.86 & 5.29 & 9.09 & 4.43 \\ HCS8-1-29 & 7.33 & 0.55 & 0.06 & 1.27 & 5.27 & 4.16 & 5.58 & 4.40 & 5.08 & 4.01 \\ HCS8-1-5-29 & 11.67 & 1.09 & 0.10 & 2.31 & 8.25 & 3.57 & 8.46 & 3.66 & 7.08 & 3.07 \\ HCS2-2-29 & 15.14 & 1.85 & 0.12 & 2.88 & 11.72 & 4.07 & 11.87 & 4.13 & 11.35 & 3.95 \\ HCS2-24 & 15.55 & 0.87 & 0.08 & 2.23 & 11.14 & 4.99 & 11.51 & 5.16 & — & — \\ HCS2-2-26 & 15.80 & 1.17 & 0.08 & 3.06 & 11.61 & 3.79 & 12.15 & 3.97 & 11.52 & 3.76 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Test results and flexural toughness indexes of specimens.  
Figure 7: Effect of fiber content on load-deflection curves: (a) 56 d; (b) 28 d.  
in the Chinese CECS 13:2009 standard [41] was improved on this basis. These methods can avoid errors when determining the first crack and evaluating the toughness of concrete with different fiber contents.  
It can be seen from Figure 10a that HDC had good deformation capacity. It might still be in the rising section of the load-deflection curve at the specified mid-span deflection. At this time, the material still had a large load-bearing and deformation capacity as well as traditional bending. The toughness evaluation method could not effectively evaluate the bending toughness of HDC. In addition, as shown in Figure 10b, at the same deflection limit, the different specimens might have been in different stress stages, and a bending toughness index calculated using the same deflection limit was not comparable. Therefore, there is an urgent need for a bending toughness evaluation method that can fully reflect the properties of HDC materials, including the aspects of strength and energy, and that can reasonably and accurately evaluate strengthening and toughening effects to control structural strength and ductility from a materials perspective.