*3.1. Failure Modes*

Two failure modes featured by the fiber pullout and the mortar spalling were observed in this study.

For series IA, the steel fibers of IA0 and IA1 were pullout with straightened hook-end; the steel fibers of IA2, IA3 and IA4 were pullout with straightened hook-end accompanied by the mortar spalling surrounded the fibers. As presented in Figure 3, the volume of mortar spalling significantly increases with the inclination angles. Although a similar peeling off area of the mortar presented on specimens with steel fibers inclined at angle 45◦ and 60◦, a larger peeling off depth happened on the specimens with steel fibers at greater inclination angle. This phenomenon is also reported in the reference [23]. With the increase in inclination angle of steel fiber, the peeling off force perpendicular to transversal section increased during the pull-out of steel fibers, which would be much more increased with the process of steel fibers straightened. This results in the tensile stress that could be over the tensile strength of mortar. If that occurs, the mortar near surface of transversal section will peeling off.

**Figure 3.** Typical failure mode of Series IA.

Figure 4 presents the typical failure mode of Series HIA. The spalling areas of mortar near the straight and inclined fibers are highlighted as red and blue circles, respectively. All the steel fibers of HIA0, HIA1 and HIA2 were pullout with straightened hook-end. The aligned fibers of HIA3 and HIA4 were pullout with the straightened hook-end, while the inclined steel fibers were pullout with straightened hook-end accompanied by the mortar spalling surrounded the fibers. This also indicted the effect of inclination angle on the bond performance of steel fibers in mortar. With the increase in inclination angle, the straightening degree of the hook-end decreases, while the spalling volume of mortar significantly increases.

Figure 5 presents the typical failure mode of Series NA. All steel fibers of NA0, NA1, NA2 and NA3 were pullout with straightened hook-end, while the mortars of NA4 peeled off accompanied with slightly straightened hook-end of steel fibers. The tensile strength of the mortar near the fiber end is not enough to resist the stress transmitted by the fiber to the matrix. This indicates a rational spacing among steel fibers is necessary to ensure

296

a sufficient surrounding mortar which can provides anchorage shear resistance of the interface between steel fiber and mortar.

**Figure 4.** Typical failure mode of Series HIA.

NA0 NA1 NA2 NA3 NA4

**Figure 5.** Typical failure mode of Series NA.

Based on the above description, in condition of the mortar with a compressive strength around 74.5 MPa, the steel fibers will be pullout with the straightened hook-end. If the inclination angle of steel fiber is over than 30◦, the mortar will be peeled off from the transversal section. When the fiber spacing is 3.5 mm, the mortar surrounded the steel fibers can be scraped out during the pulling out of steel fibers. This indicates the bond performance will be affected by the orientation and distribution of steel fiber in concrete matrix.

## *3.2. The Characteristic Pullout Load-Slip Curve*

Figure 6 presents the characteristic pullout load-slip (*PL-S*) curves of Series IA, HIA and NA. As explained in previous studies [14,16], the bond performance at debonding, peak and residual cases can be expressed by the corresponding point at the characteristic *PL-S* curve with a change of slope in ascending portion, at peak and in descending portion. They respectively represent the debonding load *P*<sup>d</sup> and slip *s*d, the peak load *P*<sup>p</sup> and slip *s*p, and the residual load *P*r and slip *s*r. Values of them are listed in Table 3.

**Figure 6.** The characteristic PL-S curves. (**a**) Series IA, (**b**) Series HIA and (**c**) Series NA.


**Table 3.** Test values at key points of the characteristic *PL-S* curves.

In Series IA, the curve of IA1 with the inclination angle of 15◦ almost coincides with that of IA0. With the inclination angle increased from 15◦ to 60◦, the slope of the ascending portion decreases, the *P*<sup>p</sup> decreased about 45.2%, while the peak-slip *s*<sup>p</sup> increased about 26.2%. The regularity is also reported by the reference [25]. The *P*<sup>d</sup> increased 13.4% with the inclination angle increased to 15◦, then obviously decreased 78.8% with the angle continuously increased to 60◦. The *P*r increased 29.5% with the inclination angle increased to 30◦, then decreased 20.3% with the angle continuously increased to 60◦. Both slips *s*<sup>d</sup> and *s*r presented the decrease trends with the increased inclination angle. The fluctuation of the descending portion of the curve gradually increases, and the area under curve gradually decreases. The variation of peak-slip is consistent with the reference [36]. Similar regularities are observed in the characteristic curves of Series HIA, the change degree of which almost reduced by half, due to a pair of steel fibers aligned to the pullout direction of series HIA.

In Series NA, the complete *PL-S* curves of NA0, NA1, NA2 and NA3 are obtained with failure mode of fibers pullout. The *P*d, *P*<sup>p</sup> and *P*<sup>r</sup> of NA0, NA1 and NA2 are positively correlated with the number of fibers of 1, 2 and 9, successively. The similar slips *s*d, *s*<sup>p</sup> and *s*<sup>r</sup> of NA0, NA1, NA2 were observed, while, the *P*<sup>d</sup> and *P*<sup>p</sup> of NA3 are significantly higher than the load values of NA0 multiplying the fiber number. NA4 only got the ascending portion of the curves, due to the failure mode of mortar peeled off.

## *3.3. Bond Strengths and Strength Ratio*

The nominal debonding strength *τ*d, bond strength *τ*max and residual bond strength *τ*res are calculated using Formulas (1)–(3), and the test values of them are present in Figure 6. To reflect the debonding resistance of steel fiber from mortar and the loss rate of bond strength in the descending portion, the nominal fiber utilization efficiency *u*sf, the nominal strength ratios *u*de and *u*res are calculated using Formulas (4)–(6), and the test values of them are listed in the Table 4.

$$\tau\_{\rm d} = \frac{P\_{\rm d}}{n\pi d\_{\rm f} (l\_{\rm f,cm} - s\_{\rm d})} \tag{1}$$

$$\tau\_{\text{max}} = \frac{P\_{\text{P}}}{n\pi d\_{\text{f}}(l\_{\text{f,em}} - s\_{\text{P}})} \tag{2}$$

$$\tau\_{\rm res} = \frac{P\_{\rm r}}{n \pi d\_{\rm f} (l\_{\rm f,em} - s\_{\rm r})} \tag{3}$$

$$
\mu\_{\rm sf} = \frac{4P\_{\rm P}}{n\pi d\_{\rm f} f\_{\rm sf}} \tag{4}
$$

$$
\mu\_{\rm de} = \frac{\tau\_{\rm d}}{\tau\_{\rm max}} \tag{5}
$$

$$
\mu\_{\rm res} = \frac{\tau\_{\rm res}}{\tau\_{\rm max}} \tag{6}
$$

**Table 4.** The nominal strength ratios.


Figure 7a shows the variations of *τ*d, *τ*max and *τ*res of Series IA with the inclination angle. Slight inclination of steel fibers benefits to the resistance of debonding and the residual bond. This leads that *τ*<sup>d</sup> and *τ*res reach the maximum at the angle of 15◦ and the angle of 30◦ with an increment of 14% and 17%, respectively, compared to that of IA0. However, the *τ*max trends to decrease with the increase in inclination angle from 0◦ to 15◦, and decreases linearly with the inclination angle increased from 15◦ to 60◦. The *τ*max of IA4 is about 44% lower than that of IA0. This is similar to the pullout test result of single hook-end steel fiber [31].

**Figure 7.** The bond strengths. (**a**) Series IA, (**b**) Series HIA and (**c**) Series NA.

With the increase in the inclination angle, the risk of mortar cracking increases and the fiber utilization rate decreases. The nominal fiber utilization efficiency *u*sf decreases by 45.2% with the angle increased to 60◦. The nominal strength ratio *u*de increases by 15% with the inclination angle increased from 0◦ to 15◦, and then decreases by 64% with the angle increased from 15◦ to 60◦. This illustrates that the debonding resistance is sensitive to the inclination angle. The nominal strength ratio *u*res increases by 38% with the inclination angle increased from 0◦ to 30◦, and then changes a little with the continuous increase in

the angle. This indicates that the loss rate of bond strength can be reduced with a larger inclination angle of steel fiber, due to the better bond retention by the compressive action of peeling off force perpendicularly on the steel fibers.

Figure 7b shows the variations of *τ*d, *τ*max and *τ*res of Series HIA with the hybrid inclination angle. Compared with Series IA, half of steel fibers aligned to the pullout direction. This lightened the effect of inclination angle on the bond of steel fibers. With the hybrid action of aligned and inclined steel fibers, the *τ*<sup>d</sup> of HIA2 and HIA3 are 24% and 21% higher than that of HIA0. Although the *τ*max still trends to decrease with the increase in the inclination angle, the decrement becomes slower. The decrement is 25% with the inclination angle increased from 15◦ to 60◦, which is 56.8% that of the IA4. Therefore, the reduction of *τ*max comes from the decreased bond strength of the inclined steel fiber in Series HIA, no hybrid effect exists among inclined and aligned steel fibers. A slight increase in the *τ*res appears with the increase in inclination angle. This is due to the better bond retention of inclined steel fiber during the pullout.

Except for the large inclination angle of 60◦, the smaller angle benefits to the debonding resistance and the retention of residual bond. The strength ratio *u*de increases 50.7% with the inclination angle from 0◦ to 30◦, and then decreases 35.7% with the inclination angle continuously increased from 30◦ to 60◦. The strength ratio *u*res increases 31.4% with the inclination angle from 0◦ to 60◦. The nominal fiber utilization efficiency *u*sf decreased by 27.3% with the angle increased to 60◦. It is almost half decrease rate compared with the regularity of *u*sf for series IA. It also illustrates that no hybrid effect exists among inclined and aligned steel fibers.

Figure 7c shows the variations of *τ*d, *τ*max and *τ*res of Series NA with the fiber number. The influence of fiber number on bond strength essentially relates to the influence of fiber spacing. There are slight decreases of the *τ*d, *τ*max, *τ*res, *u*sf, *u*de and *u*res of NA1 compared with those of NA0. This may be due to the eccentric pullout on the two steel fibers during the loading process. In addition, the reduction rate of the bond strengths of multiply fibers compared with single fiber is smaller than that with reported in the references [32,33]. This may attribute to the different cementitious matrix and pull-out test method. When the fiber number increased to 9 for NA2, the *τ*d, *τ*max, *u*sf and *u*de are basically equal to those of NA0, while the *τ*res and *u*res slightly decrease. When the fiber number reached to 16 for NA3, the *τ*d, *τ*max, *u*sf and *u*de increase by 20.2%, 8.1%, 8.1% and 11.7% compared with those of NA0, respectively. This indicates that a group effect of parallel fibers with fiber spacing no less than 5 mm benefits to the bond performances. Therefore, an interaction exists among steel fibers in concrete matrix if the steel fibers are uniformly distributed in parallel with a volume fraction over 0.78% (corresponding to *L*sf of 5 mm).

The values of *τ*d, *τ*max and *τ*res of IA0 and HIA0 are also presented in the Figure 6c. Ignoring the effect of embedded length, comparing the values of *τ*d, *τ*max and *τ*res of IA0, HIA0, NA0, NA1 and NA2, all the bond strengths increase with the flexural strength of mortar. The result is consistent with the previous study [14].

In addition, to get the real bond performance of steel fiber without influenced by the eccentric loading, a reasonable number of fibers should be used in the pullout test [32,33,37,38]. Comprehensively considering the range and test accuracy of the test system in this study, four steel fibers symmetrically arranged to section centroid is a better of chose.

Considered the pullout process of inclined steel fibers, the bond of steel fiber to mortar not only comes from the shear stress on interface, but also from the perpendicular pressure on interface, as shown in Figure 8. The former is the same as that of aligned steel fiber, which comes from the chemical adhesion and the mechanical fraction on interface between fiber and mortar and the anchorage of hook-end [14,16]. The latter comes from the component force of pullout load, which directly relies on the pullout load and the peeling off resistance of mortar. If the component force of pullout load is over the peeling off resistance of mortar, the mortar will be peeled off. This leads to broken off mortar from the surface of transversal section to the inner along steel fiber, and results in a shortening of the bond length of steel fiber. The final presentation of the nominal bond strengths by Formulas (1) to (3) will be

decreased. With the increase in inclination angle, the component force of pullout load increases to rise the possibility of the peeling off of mortar, as presented in Figures 3 and 4.

**Figure 8.** Actions on bond interface of inclined steel fiber.

#### *3.4. Bond Works*

The debonding work *W*d, the slipping work *W*<sup>p</sup> and the pullout work *W*<sup>r</sup> are used for evaluating the energy dispersion during the bond-slip process. They are the areas under the characteristic *PL-S* curve with the slip from origin to the slips at debonding, peak and residual points, respectively. Formulas are listed as follow:

$$\mathcal{W}\_{\rm d} = \int\_0^{S\_{\rm d}} Pds\tag{7}$$

$$\mathcal{W}\_{\mathbb{P}} = \int\_0^{\mathcal{S}\_{\mathbb{P}}} P ds \tag{8}$$

$$\mathcal{W}\_{\mathbf{r}} = \int\_0^{\mathcal{S}\_{\mathbf{r}}} P ds \tag{9}$$

Table 5 presented the test values of *W*d, *W*<sup>p</sup> and *W*r. In Series IA, the *W*<sup>d</sup> increases by 30% with the inclination angle from 0 to 15◦, and then decreases by 92% with the inclination angle continuously increased from 15◦ to 60◦. The *W*<sup>p</sup> has no change with the inclination angle from 0 to 15◦, and then decreases with the increase in inclination angle. The *W*p of IA4 is 34% lower than that of IA0. The *W*r has a linear decrease with the increase in inclination angle. The *W*r of IA4 is 64% lower than that of IA0. Therefore, when the inclination angle of steel fibers is larger than 15◦, the bond energy will be sustainably decreased.

In Series HIA, the *W*<sup>d</sup> obviously increases with the inclination angle of steel fibers, while the *W*<sup>p</sup> and *W*<sup>r</sup> trend to decrease with the increase in the inclination angle. This is consistent to the influence of the hybrid action of inclined to aligned steel fibers on the bond strength.

The influence of fiber spacing on the bond energy can be reflected by the bond works per single fiber. Therefore, the bond works *W*d, *W*<sup>p</sup> and *W*<sup>r</sup> of Series NA divided the fiber number *n* are listed in Table 5. With the decrease in fiber spacing from 22.2 mm to 5 mm, the *W*d/*n*, *W*p/*n* and *W*r/*n* increase 263%, 22% and 47%, respectively. This means an improving effect of reasonable fiber number on the bond energy. In addition, the lower

bond works of NA1 indicate that the eccentric loading of pullout test for two steel fibers should be avoided to get real bond performance.


**Table 5.** Bond works.
