*3.5. Energy Ratios*

The debonding energy ratio *R*d, the slipping energy ratio *R*dp and the pullout energy ratio *R*pr of hook-end steel fiber are used to represent the energy dissipation ability in different stages of the characteristic *PL-S* curve from the origin to the debonding point, from the debonding point to the peak point and from the peak point to the residual point [14]. Formulas can be written as follow,

$$R\_{\rm d} = \frac{W\_{\rm d}}{W\_{\rm r}} \tag{10}$$

$$R\_{\rm dp} = \frac{\mathcal{W}\_{\rm P} - \mathcal{W}\_{\rm d}}{\mathcal{W}\_{\rm r}} \tag{11}$$

$$R\_{\rm pr} = \frac{\mathcal{W}\_{\rm r} - \mathcal{W}\_{\rm P}}{\mathcal{W}\_{\rm r}} \tag{12}$$

As shown in Figure 9a, the *R*<sup>d</sup> increases by 65% with the inclination angle from 0 to 15◦, and then decreases by 84% with the inclination angle continuously increased to 60◦. *R*dp increases 110% and *R*pr decreases 36% with the inclination angle increased from 0 to 60◦. This indicates that the energy dissipation ability increases before peak-slip and decreases afterward. To be applied for the concrete structures, the inclined steel fiber is favorable to the crack control at the normal serviceability, while less toughness and lower energy dispersion ability at the ultimate bearing capacity.

As presented in Figure 9b, the *R*<sup>d</sup> of Series HIA increases with the inclination angle of steel fiber. The *R*<sup>d</sup> of HIA4 is 11.9 times that of HIA0. This indicates the hybrid effect of inclined and aligned steel fibers is favorable on the energy dissipation before cracking of concrete matrix. The *R*dp and the *R*pr of Series HIA present the similar changes to those of Series IA with the increase in inclination angle of steel fiber; however, the changes become slowly due to half number of steel fibers were inclined. The *R*dp increases by 40% with the inclination angle from 0 to 45◦, and then decreases by 16% afterward. The *R*pr decreases by 23.2% with the inclination angle from 0 to 60◦. This once again indicates that the energy dissipation ability increases before peak-slip and decreases afterward with the inclination angle of steel fibers.

As shown in Figure 9c, with the fiber spacing decreased from 22.5 mm to 5 mm, the *R*<sup>d</sup> increases by 146%, the *R*dp decreases by 37% and the *R*pr increases by 13%. This indicated that the energy dissipation ability increases at the debonding process and the residual bond stage, which is consistent to the strengthening of tensile strength and toughness of steel fiber reinforced concrete. Meanwhile, it shows that steel fibers in concrete matrix should keep a reasonable spacing to develop their reinforcing contribution. Therefore, the content of steel fiber in concrete should be optimized. At the same time, the variation of energy ratios between NA0 and NA1 can be concluded to the validity of the pullout test with two steel fibers.

**Figure 9.** Influence of various factors on the bond energy ratios. (**a**) Series IA, (**b**) Series HIA and (**c**) Series NA.
