*4.2. Beam Failure Models and Experimentally Determined Shear Capacity*

Various failure models were observed during the tests, depending on the type of shear reinforcement and the combinations (Table 3) that were used [44]. The failure of B beams (without shear reinforcement) and beams having only fibre reinforcement (BF beams) had a shear tension character (Figure 17). Once the first diagonal cracks appeared, one increased its width as the load was increased. A number of small cracks occurred at the main reinforcement level, which means that gradual loss of the main reinforcement adhesion to the fibre composite occurred. Consequently, the depletion of shear capacity of such elements resulted from the main reinforcement steel slide at the place of anchoring on the support.

**Figure 17.** Example of a shear tension failure in the shear area of BF series beams.

In the case of BFSa series beams (elements with 120 mm stirrup spacing), the failure was similar to that described above, but with the diagonal tension failure feature. Such a failure mode resulted from the high shear capacity of BSFa beams, and occurred in the case of beams with insufficient flexural reinforcement over their entire span. In this case, the flexural reinforcement steel yield point was observed not in the middle of the element span, but at the section of the simultaneous action of the bending moment and transversal force. This resulted in an increase in width of one of the diagonal cracks, leading to its elongation and penetration into the compressed cross-section area. This crack caused, in effect, the crashing of concrete in the cross-section above the crack. Therefore, for this beam series, the second test stage—which was aimed to define the shear capacity of the undamaged shear area—failed. In this particular case, the beams failed because of bending (Figure 18). In most cases of BFSa series beams (featuring greater stirrup spacing equal to 120 mm) and BSa series beams (with stirrup spacing of 120 mm, no steel fibres added), the stirrups through which the diagonal crack passed had fractured. In the destruction of BFSb series beams (stirrups spaced at 90 mm), which showed the highest shear capacity, failed due to yielding of the flexural reinforcement (flexural failure). Finally, due to high deformation, the compressed area of the element crashed. In effect, a secondary shear failure of the diagonal tension type occurred (Figure 19).

**Figure 18.** Typical secondary failure of a BFSa series beam.

**Figure 19.** Example of a secondary BFSb series beams failure.

Table 4 shows the mean values of forces *Vcr* (shearing force at which a diagonal crack occurs) and *Vult* (the ultimate shear force) obtained from tests for particular beam series.

Table 5 shows the effects of reinforcement of SFRWSC beams obtained in the tests. The highest values of the shear force (*Vcr*), for which the initial diagonal cracking has been observed, and the ultimate shear force (*Vult*) were observed for the beams reinforced with both stirrups and steel fibres. The reduction of stirrup spacing from 120 mm to 90 mm contributed to an increase in the shear capacity of elements by 23%, on average, for the beams without fibres, and 13% for the beams with fibres. The analysis of the results obtained for the beams reinforced only with stirrups and only with steel fibres led to the conclusion that slightly higher carrying capacities occurred in the case of the beams without stirrups. Compared to BFSa beams, the tested forces, *Vult*, were lower by 40% for BSa beams and by 26% for BF beams. Despite the fact that addition of the second type of

shear reinforcement provided a lesser reinforcement effect (see Table 5), the impact of steel fibres and stirrups on the shear capacity added up, which originated from the results for BFSa and BFSb series beams. Similar conclusions have also been drawn by other authors, who analysed the co-operation of stirrups with steel fibres in resistance to transversal forces [21–23,73].Our research also showed that fibre reinforcement had an advantageous impact on the occurrence of the first diagonal crack, in comparison to those beams that contained no steel fibres. The value of the transversal cracking force (*Vcr*) in the fibre composite elements was higher (by approximately 38%), compared to the beams without fibre reinforcement. In beams of BF, BSa, and BFSa series, the ratio of the cracking force (*Vcr*) to the ultimate shearing force (*Vult*) was constant and amounted to approximately 0.56. The insignificant increase in *Vult*, compared to *Vcr*, for B series beams originated from the fact that the shearing force was reduced by occurrence of the so-called "dowel action", engaging the aggregate and the compressed area.

**Table 4.** The mean values of forces *Vcr* and *Vult* for particular beam series.


\*—flexural failure.

**Table 5.** Increase coefficients for particular beam series.


Consequently, our research work demonstrated that the addition of steel fibres to SFRWSC had an enormous impact on the resistance to transversal forces. It increased the shear capacity by approximately 90% in beams with no stirrups and that by approximately 65% in beams with stirrups. The fibre functioning character in the shear area was more beneficial than stirrup action, due to more ductile character of the material. An example of dependence of the side surface strain at element height (*εy*) on the transversal force (*V*) is shown in Figure 20.

We also ascertained that the impact of steel fibres and stirrups added up, in terms of resistance to transversal forces, in order to increase the element shear capacity. Furthermore, it was found that the concentration of stirrups had no significant impact on the cracking force (*Vcr*), for the beams with fibres and without. Similar results have also been obtained by Lim and Oh [74]. We also found that, as the shear capacity of BF and BSb series beams was comparable, stirrups #4.5 spaced at 90 mm intervals reinforced the beams to the same degree as the content of steel fibres amounting to 94 kg/m3 (1.2%).

**Figure 20.** An example of dependence of the side surface strain at element height (*εy*) on the transversal force (*V*) for B, BSa, and BF series beams at the first stage of the test.

#### *4.3. Diagonal Cracks*

The dependence of the shear force (*V*) on the width of the diagonal crack opening (*w*) in beams made of SFRWSC is shown in Figure 21. For elements reinforced with fibres (BF series) a higher cracking force than that in beams reinforced with stirrups (BS series) was noted, which may have a beneficial impact, in terms of reduction of the shear span requiring shear reinforcement. Curves describing the dependence of the transversal force on the width of the diagonal crack opening for beams with fibre reinforcement (BF series) and reinforced only with stirrups (BS series) featured a similar angle of inclination to the horizontal axis. Such a course of the curves, as shown in Figure 16, indicates the similar functions of steel fibres and stirrups after element cracking. At the same time, in the case of BFS series beams (reinforced with stirrups and steel fibres), the effects of the fibres and stirrups add up, as indicated by the slow increase in the crack opening width (*w*) with increasing transversal force value (*V*). It should be noted that the maximum width of the diagonal crack opening in BFS-type elements was significantly lower than that observed in BF and BS series beams. This resulted in a higher number of cracks in BFS-type elements, where the shear reinforcement was provided by stirrups and steel fibres. Ultimately, it can be stated that fibres have strong impact, not only on the shear capacity of SFRWSC elements, but also on their diagonal cracking. Comparing the test results obtained for BF and BS series beams, it can be stated that the same values of crack widths occurred at higher transversal force values (by approximately 40%) for the beams containing fibres. It should be noted that the *V-w* curves for BF and BS elements featured a similar angle of inclination to the horizontal axis, which may indicate a similar mechanism of resistance to transversal force after cracking by stirrups and steel fibres.

The highest number of cracks for one shear area was observed in beams reinforced with stirrups and fibres: either 3 or 4. For beams with only fibres, the number of cracks was similar to that for beams with only stirrups: 2 or 3, on average. The presence of greater number of cracks simultaneously resulted in smaller crack widths. For beams with stirrups and fibres, the opening width of diagonal cracks did not exceed 1 mm, while that for beams with only fibres was, on average, 1.25 mm.

In summary, in terms of diagonal cracking, the elements containing steel fibres and stirrups behaved best, as an increase in force caused a considerably lower increment of crack opening width, where as a constant value was observed in the case of beams reinforced with fibres (BF) or stirrups (BS).

**Figure 21.** Shear force (*V*) versus diagonal cracks opening width (*w*) for selected B, BF, BS, and BFS series beams.

#### **5. Computational Analysis**

The shear capacity of beams made of SFRWSC was computed through the application of two methods—those of RILEM TC-162-TDF [39] and Model Code 2010—using the SMCFT [17] method for the second approximation levels, as well as the former method. The objective of this computation was to prove the applicability of European standards for shear design of fibre concrete cross-sections; that is, to verify whether Model Code and RILEM can be used in the shear design of SFRWSC cross-sections.

Average values of features of SFRWSC with and without steel fibres, as well as average values of features of reinforcing steel, were used in the computations (Tables 2 and 3). Values of the axial tensile strength of SFRWSC were determined using the relationship of Amin and Foster [75], which allows for the transformation of residual strengths to axial tensile strengths. The angle of inclination of compression struts (*θ*) for the SMCFT [17] method was assumed as the minimum; whereas, for the two other methods (RILEM and Model Code 2010—former method), it was *θ* = 30◦. Safety coefficients in the shear capacity computations were as follows: *γ<sup>f</sup> = γ<sup>c</sup> =* 1.0.

To determine the shear capacity of BSa and BSb series beams through application of the SMCFT method, the second level of approximation was used for elements containing steel fibres, with the method for computation of the coefficient *kv* taking into account the share of the maximum aggregate grain [17]. To determine the total shear capacity (*VRd*), notation for the third level of approximation was used, taking into account the influence of the cross-sectional shear capacity in the element without transversal reinforcement(*VRd,c*), as well as the shear capacity due to transversal reinforcement (*VRd,s*).

To assess the usability of the RILEMTC-162-TDF and Model Code 2010 methods to determine the shear capacity of elements made of SFRWSC, the criterion of Baghi and Barosso [76] was used. The comparative quantity in this criterion is the ratio of the value of the experimentally fixed shear capacities (*Vexp*) to computational values (*Vcal*). The shear capacity assessment criteria are shown in Table 6 and Figures 22–27.


**Table 6.** The criteria for assessment of experimentally fixed (*Vexp*) and computational (*Vcal*) shear capacity values [76].

**Figure 22.** Values of experimentally fixed (*Vexp*) versus computational (*Vcal*) shear capacity determined according to Model Code using the SMCFT method (*θ* = min *θ*).

**Figure 23.** Values of experimentally fixed (*Vexp*) versus computational (*Vcal*) shear capacity determined according to the former method Model Code 2010 (*θ* = 30◦).

**Figure 24.** Values of experimentally fixed (*Vexp*) versus computational (*Vcal*) shear capacity determined according to the RILEM TC-162-TDF method (*θ* = 30◦).

**Figure 25.** Values of experimentally fixed (*Vexp*) versus computational (*Vcal*) shear capacity determined according to the former Model Code 2010 SMCFT method (measured angle *θ*).

The experimentally determined shear capacity values (*Vexp*) are set, in Figure 22, together with the analytically determined (*Vcal*) values based on the FIB Model Code 2010 using the SMCFT method. After analysis of the graph, it can be stated that the theoretical shear capacity values for B and BFSb series beams are lower than those which were determined experimentally. The computational shear capacity values for B series beams at 75%, as well as those for BFSb beams at 100%, fit into the "conservative" classification interval (Table 6). The best compatibility with the *Vexp* = *Vcal* straight line occurred for BSa and BSb series elements, for which the percentages of results complying with the "appropriate safety" criterion were 88% and 100%, respectively. The computational shear capacities for BF and BFSa series beams were classified as 50% within the 0.85–1.15 interval and 50% within the 1.15–2.0 interval.

**Figure 26.** Values of experimentally fixed (*Vexp*) versus computational (*Vcal*) shear capacity determined according to Model Code 2010 (measured angle *θ*).

**Figure 27.** Values of experimentally fixed (*Vexp*) versus computational (*Vcal*) shear capacity determined according to RILEM TC-162-TDF (measured angle *θ*).

Figure 23 shows values of shear capacity *Vexp* to *Vcal*, computed according to the former Model Code method. The results obtained show that the shear capacities for B series beams were much lower than the experimental shear capacity values. It appears, from our analysis, that 88% of the shear capacity results should be treated—in accordance with the adopted criterion—as "conservative". The values of shear capacity calculated for BSa and BSb series beams were close to those determined by application of the SMCFT method. In this case, the shear capacity values were qualified as being "appropriate safety". The highest discrepancies were observed for the shear capacity values of BF series beams calculated by application of the SMCFT method. A total of 75% of beam shear capacity values were classified within the 1.15–2.0 interval (Table 6). Having applied the former Model Code method, it is clear that the impact of steel fibres on the shear capacity of SFRWSC beams is lower.

Figure 24 shows the values of shear capacity *Vexp* to *Vcal*, computed according to the RILEM TC-162-TDF method. Having analysed the shear capacity computation results, we have found that, when using this method, the highest discrepancies between the computational and experimental shear capacity values were obtained, where the highest

discrepancy was observed for BF series beams. In this case, 100% of the shear capacity values were classified as "conservative". The impact of steel fibres on shear capacity fell into the1.15–2.0 interval (Table 6) for BFSa and BFSb series beams, making them also less effective. The computational shear capacity values, determined by application of the former Model Code method with adoption of the same shear capacity computational algorithm as in EC-2 [77], for elements containing no shear reinforcement and all beams with stirrups, overlapped. It should also be pointed out that Figures 25–27 show shear capacity values for BFSb series beams, despite their bending failure; however, assuming that shear force acting together with the destructive momentum is the minimum experimentally fixed shear capacity (this has been indicated by stirrups functioning in BFSb beams, as described below), it can be stated that the highest *Vexp* to *Vcal* differences occurred with the RILEM method (difference min. 33%), compared to the former Model Code method (min. 20%) and the SMCFT method (min. 5%). Large differences in shearing capacity values obtained by application of the RILEMTC-162-TDF method have also been confirmed by other authors, such as Matthys [78], Parmentier [41], and Arslan [79,80].

Figures 25–27 show comparisons of the experimentally fixed shear capacity values for SFRWSC beams with the values computed using the RILEM TC-162-TDF method [39], Model Code 2010 [17] using the SMCFT method, and the former method. The adoption of real (obtained from tests) angles *θ* caused a significant computational discrepancy in the shear capacity values. The SMCFT method (Figure 25) requires particular attention; in this method, the angle *θ* is used not only for computation of the shear capacity of beams with stirrups, but also in the elements containing only fibres. In this case, the majority of shear capacity values computed for BF series beams were higher than the shearing capacity values found experimentally, causing an opposite situation than if the minimum value of angle *θ* was adopted. Three of the determined shear capacity values qualified as "dangerous" (Table 6). A similar situation occurred in the case of BSa series beams, when using the RILEM and former Model Code methods. In the remaining cases, particularly for extremely high values of angle *θ*, the computational values of shear capacity were higher (even by 50%).

For better interpretation purposes, and for assessment of the computed shear capacity values determined using the minimum and measured angle *θ*, the Integral Absolute Error (*IAE*) index was used. Having analysed *IAE* indices for *θ = min* and *θ* = 30◦, the best compatibility was observed for BSa and BSb series beams, particularly in the case of the SMCFT method. In this case, the *IAE* index value did not exceed 10%. For BF series beams, the SMCFT method also showed the best compatibility (equal to 15%). The worst compatibility (exceeding 30%)was obtained with the RILEMTC-162-TDF method. *IAE* indices for BFSa and BFSb series beams had the lowest values with the Model Code 2010 method (12% and 16%, respectively); whereas, with the RILEM method, they exceeded 20%. The RILEMTC-162-TDF method, when analysed by other authors [78–80], also indicated significant discrepancies, with respect to computed values.

The IAEs for the measured angle *θ* criterion were higher, which can also be confirmed through observation of Figures 25–27. A conclusion can be drawn here: the adoption of a proper angle of inclination for the concrete and SFRWSC compression struts constitutes a key aspect in the determination of shear capacity. A large spread of measured angles *θ* resulted in higher *IAE* index values. To illustrate the impact of angle *θ* on the computational shear capacity values, values of *Vexp*/*Vcal* were analysed, comparing them with the compression strut inclination angles observed during testing (Figure 28). In this analysis, only the SMCFT method was considered, in which the angle *θ* (apart from the stirrup shear capacity) has an impact on the shear capacity of the fibres alone. From Figure 25, it is apparent that, with an increase in the compression strut inclination angle, the shear capacity values decreased. The highest values of *Vexp*/*Vcal* occurred for angles *θ* exceeding 35◦, where the computational shear capacity value was lower than the experimentally fixed value by anywhere from 25% to over 50%. For the lowest angles *θ*, the situation was opposite, and the theoretical shear capacity values were exorbitant. However, in

this case, the maximum difference was approximately 30%. A conclusion can be drawn here: assuming the tested SFRWSC, computation using high angle *θ* values results in considerable underestimation of the shear capacity *VRd,s*, even in the event of an actual occurrence of such an angle. This has happened to appear in all beams with a high degree of shear reinforcement (BSb and BFSb series beams). The best compatibility (*Vexp*/*Vcal* = 1) was obtained when the angle fell within the interval of 20◦ to 32◦ for BF, BSb, and BFSa series beams. At this point, it should be noted that, in the case of beams containing only fibres, the best compatibility was achieved when the real angle was equal to 25◦. This is a very important conclusion because, according to the analytical method proposed by the Model Code, the minimum angle of inclination of compression struts calculated for those beams was approximately 33◦. The tests and computations performed further indicate a necessity to correct the procedure of computation for the minimum angle inclination value for compression struts, according to the SMCFT method, for the tested SFRWSC. This was indicated through analysis of the real and minimum angles *θ*, as well as the impact of the real angle *θ* on the experimentally fixed and computed shear capacity values for SFRWSC (Figure 28).

**Figure 28.** *Vexp/Vcal* values versus values of the measured angle of compression struts inclination (exp. *θ*) obtained in the tests.

#### **6. Conclusions**

Experimental testing and computation of the shear capacity of reinforced concrete elements, made using a novel fine aggregate composite with and without steel fibres, allowed us to formulate the following conclusions:


resistance to transversal forces, both in the aspect of increasing the shear capacity, as well the element deformability.


$$f\_{\text{Ftu}} = 2.18 \text{ MPa} > 0.08 \sqrt{f\_{\varepsilon}} = 0.6 \text{ MPa} \tag{1}$$


The conclusions presented in the article should not be generalized. Experimental tests and calculations were carried out for an innovative SFRWSC with a grain size of up to 2 mm, a specific cross-section of beams, the amount of longitudinal and transverse reinforcement and a static scheme. The obtained results of the research and the conducted literature studies in the field of the subject of the presented work allowed for the formulation of further research directions. The use of numerical modeling to predict the behavior of beams with different static patterns, for different types of element cross-sections, values of classical and fiber reinforcement, variable *a*/*d* shear slenderness, or various actions and their combinations is one of them. Thus, the analyzes will provide a large number of results that cannot be registered in an experiment with a measuring apparatus. Moreover, numerical analyzes will allow to identify the necessary areas for further experimental research and to introduce changes in the experimental program in order to improve the applied model. Due to the highly non-linear nature of the analyzed shear process in the fiber-reinforced concrete elements and the analysis until failure, it will be advantageous to use a quasi-static calculation strategy.

**Author Contributions:** Conceptualization, W.G.; data curation, M.L.; methodology, W.G. and M.L.; formal analysis, W.G. and M.L.; software, M.L.; validation, W.G.; investigation, M.L.; writing original draft preparation, W.G. and M.L.; writing—review & editing, W.G.; visualization, M.L.; supervision, W.G.; resources, M.L.; project administration, W.G. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding.

**Institutional Review Board Statement:** This study was not carried out on humans or animals.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** The data presented in this study are available on request from the corresponding author.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **Nomenclature**


#### **References**

