# Flexural Tensile Strength of Concrete with Synthetic Fibers

^{1}

^{2}

^{*}

## Abstract

**:**

^{3}of synthetic fibers with different geometry and form was presented. There was a 5.5–13.5% increase in the flexural tensile strength depending on the mixture type. Moreover, in the case of fiber-reinforced concretes, the ductility was enhanced and the samples were characterized by significant residual flexural tensile strengths. Additionally, from the workability tests it was concluded that after the incorporation of fibers, the consistency class decreased by one, two or three. Nevertheless, the compressive strengths of concrete with and without fibers were very similar to each other, and varied from 58.05 to 61.31 MPa. Moreover, it was concluded that results obtained from three-point bending tests significantly differed from empirical formulas for the calculation of the flexural tensile strength of fiber-reinforced concretes with dispersed steel fibers present in the literature. As a result, the new formula determined by the authors was proposed for concrete with polymer fibers with a nominal fiber content ≤1.0% and slenderness of up to 200. It must be mentioned that the formula gave a very good agreement with studies presented in different literature positions. In addition, an attempt was made to evaluate the strengths of tested mixes in accordance with the Model Code 2010. However, it occurred that the proposed fiber-reinforced concrete mixtures would not be able to replace traditional reinforcement in a form of steel bars. Furthermore, in uniaxial tensile tests, it was not possible to determine the σ–w graphs, and received results for maximum tensile strength did not show the clear influence of fibers incorporation on concrete. Then, the fracture energy enhancement (from about 16 to 22 times) and dependencies: crack mouth opening displacement–deflection; crack mouth opening displacement–crack tip opening displacement; and crack tip opening displacement–deflection were analyzed. Finally, the results from flexural tensile tests were compared with measurements of the surface displacement field obtained through the Digital Image Correlation technique. It was concluded that this technique can be successfully used to determine the crack mouth and crack tip opening displacements with very high accuracy.

## 1. Introduction

_{f}-fiber length; d

_{f}-fiber diameter), type and nominal volume content (V

_{f}) of fibers in the concrete. In addition, the bonding mechanism between the fiber and concrete also has an impact, i.e., whether the fibers are monofilament or fibrillated. Synthetic fibers can be divided into micro-and macrofibers. They are distinguished by their diameter: the microfibers have d

_{f}< 30 mm and the macrofibers have d

_{f}> 30 mm [20]. Their function is also different: microfibers prevent from microcracks occurring at the beginning of the concrete hardening process and increase tensile strength, and macrofibers often play an additional load-bearing function, protect against macrocracks and increase concrete ductility [4]. Furthermore, concrete is often formed with the addition of these two types of fibers to obtain a mixture that both replaces the traditional steel bar reinforcement and counteracts shrinkage [21].

^{3}, for which the limit of proportionality and residual strengths were described. Furthermore, the proposition of new formula, despite the one presented in the EN 14651 standard [28] for metallic fibers, to describe the relationship between deflection and crack mouth opening displacement for synthetic fibers was introduced. Other dependencies such as crack mouth opening displacement-crack tip opening displacement and crack tip opening displacement-deflection were analyzed to help other researchers in the case when only one parameter would be measured during the 3PBT, and the calculation of residual strengths would be required. Additionally, formulas presented in the literature for the calculation of flexural tensile strength for FRC are usually dedicated to reinforcement with metallic fibers [42,43,44,45]. This study will fill the knowledge gap when it comes to the estimation of bending strength for polymer FRC with a nominal fiber content ≤ 1.0% and slenderness of up to 200. Furthermore, an attempt was made to evaluate the tensile strengths in uniaxial tensile tests on samples significantly bigger than the one usually used in this kind of test [45,46]. It must be mentioned that the proprietary set-up was designed and used to execute this type of test. Concerning the classification of FRC according to Model Code 2010 [26], only a limited number of articles was found. Namely, in [46] Carlesso et al. classified the post-cracking strengths of FRC, but with a higher amount of polypropylene fibers from 5.0 to 10.0 kg/m

^{3}. On the other hand, for similar V

_{f}the classification is done in [47]. Other research [48] also mentioned the toughness class, but for steel FRC. Moreover, since little work has been done using the DIC technique for polymer FRC subjected to 3PBT, the current study will help to enhance the knowledge in this area. The conclusions from the comparison of results from clip gauges and the DIC system will be beneficial for other researchers. It must be also mentioned that Glinicki in [13] performed similar studies with 2.0–3.9 kg/m

^{3}of macro synthetic fibers for industrial floors. However, they covered just the compressive strength test and four-point bending test according to ASTM C 1018-94 standard [49], so were different than presented in the current research. An extensive study on the effect of polypropylene fibers on the properties of concrete is also presented in [50]. Nevertheless, it did not deal with the issue of residual flexural tensile strengths and fracture energy. In [46], 3PBT was performed according to [29], though the amount of fibers was higher than the one proposed in the current study and was equal to 5.0–10.0 kg/m

^{3}. Moreover, the work focused more on the fatigue behavior of polymer FRC. To sum up, according to the author’s knowledge and prepared Table 1, there is no research that simultaneously covers aspects of influence of synthetic fibers on workability; compressive, flexural, tensile strength and toughness, together with classification of post-cracking strengths according to Model Code 2010 [26] and complex study on empirical equations to explain the improvement in bending strength for no-metallic fibers.

## 2. Materials and Methods

#### 2.1. Materials

^{3}of PM fibers (PM_3); 2.0 kg/m

^{3}of PM fibers (PM_2); 3.0 kg/m

^{3}of PD fibers (PD_3); 2.0 kg/m

^{3}of PD fibers (PD_2); and 2.0 kg/m

^{3}of FF fibers (FF_2). The fibers added to the concrete mix are characterized in Table 2. All fibers had similar tensile strength (f

_{t}), chemical resistance, and their density was equal to 0.91 g/cm

^{3}, melting point 160–170 °C. On the other hand, they differed in dimensions and form, e.g., PM and FF fibers were longer and slender than PD fibers and were added to the mixture in the form of bundles, which disintegrate during mixing. In addition, Portland cement type I of the strength class 42.5 with high early strength (R)-CEM I 42.5R meeting the requirements of EN 197-1 [57] was used. Moreover, coarse pebble aggregate with a diameter of 2–8 mm was used, and as a fine-grained aggregate sand with a maximum diameter of 2 mm. In order to ensure proper workability, a superplasticizer based on a modified acrylic polymer–MAPEI Dynamon SX 08 was added to the mixture. Pure water from the water supply network was used. The water-cement ratio (w/c) was equal to 0.50. Each batch of mixes was prepared in a single excipient. All mixes were made on the same day, at a similar temperature and humidity. The composition of all concrete mixes is shown in Table 3. It is also worth noting that it was constant, and the only variable was nominal volume content V

_{f}.

_{f}was also visible, as in a mixture with 2.0 kg/m

^{3}of PM fibers, a greater drop of the cone was recorded than in the same mixture with 3.0 kg/m

^{3}of the same fibers. Furthermore, in the case of a mixture with hybrid fibers, the workability was the most severe.

^{3}for flexural tensile strength tests in accordance with EN 14651 [28] for each type of concrete were concreted. Additionally, in order to test the tensile strength, one dog-bone specimen was concreted for each series with a height of 750 mm, a middle cross-section of 100 × 100 mm, and a lower/upper cross-section of 100 × 140 mm

^{2}. It must be noted that dog-bone specimens were specially reinforced by two layers of stirrups on both ends—on the 1/3 depth and 2/3 depth of the sample what is presented in Figure 2. This was done to force the crack occurrence in the middle of the sample. A total of 36 cubes, 18 beams and 6 dog-bone specimens were concreted (Table 6). All samples were stored under a foil and systematically watered to avoid drying and the appearance of shrinkage cracks. After 17 days, the cubes and beams were demolded and left in a room temperature at 20 °C ± 2 °C and ≥95% humidity according to [62] until the test day (36 days for PC; 37 days for PM_3, PM_2, PD_3; 38 days for PD_2, FF_2.). Additionally, on day 18, all beams in the middle of the span were cut-5 mm wide and 25 mm deep along the entire beam width with a diamond saw in accordance with EN 14651 [28].

#### 2.2. Methods

^{3}were tested in accordance with EN 206 [60]. The tests were performed on a Controls Model 50-C46CO2 machine (Controls, Liscate, Italy) where the stress was increased at a rate of 0.5 MPa/sec (Figure 3).

^{3}were tested. The span between the supports (l) was equal to 500 mm. The samples were loaded with a force (F) in the middle of the span with a constant increment of δ equal to 0.2 mm/min until reaching δ = 5 mm, i.e., until the end of the test. During the test, three curves were recorded, F-CMOD, F-δ and F-CTOD, though according to EN 14651 [28], the first one is used for the strength characteristics of FRC. Nevertheless, there is a formula (Equation (1)) to calculate the CMOD value using δ.

_{,}(Equation (2)) and residual flexural tensile strengths: f

_{R,1}, f

_{R,2}, f

_{R,3}and f

_{R,4}(Equation (3)), using the formulas included in the EN 14651 standard [28]. It must be mentioned that for F-CMOD plot obtained during testing of FRC, CMOD

_{1}= 0.5 mm, CMOD

_{2}= 1.5 mm, CMOD

_{3}= 2.5 mm and CMOD

_{4}= 3.5 mm.

- ${\mathrm{f}}_{\mathrm{ct},\mathrm{L}}^{\mathrm{f}}\u2014$ limit of proportionality [N/mm
^{2}]; - ${\mathrm{f}}_{\mathrm{R},\mathrm{j}}$—residual flexural tensile strength corresponding to CMOD = CMOD
_{j}or δ = δ_{j}(j = 1, 2, 3, 4) [N/mm^{2}]; - ${\mathrm{F}}_{\mathrm{L}}$—load corresponding to the limit of proportionality [N];
- ${\mathrm{F}}_{\mathrm{j}}$—load corresponding to CMOD = CMOD
_{j}or δ = δ_{j}(j = 1, 2, 3, 4) [N]; - $\mathrm{l}$—span length [mm]—500 mm;
- $\mathrm{b}$—width of the specimen [mm]—150 mm;
- ${\mathrm{h}}_{\mathrm{sp}}$—distance between the tip of the notch and the top of the specimen [mm]—150 mm–25 mm = 125 mm.

## 3. Results and Discussion

#### 3.1. Compressive Strength

^{3}were tested for compressive strength (f

_{c}) in accordance with EN 206 [60]. The tests were performed on a Controls Model 50-C46CO2 machine (Controls, Liscate, Italy) where the stress was increased at a rate of 0.5 MPa/sec. The mean f

_{cm}, standard deviation s

_{fc}, and the coefficient of variation V

_{fc}of the compressive strength for PC, PM_3, PM_2, PD_3, PD_2, and FF_2 are summarized in Table 7. It should be noted that samples no. 5 for PC and PD_2 and sample no. 6 for PM_3 were rejected during the analysis, as they significantly differed from the other results. This could be due to inaccurate mixing of the concrete components at the bottom of the mixer, as the samples were concreted as one of the last in the series.

_{cm}of the PM_2; PD_3; PD_2, and FF_2 by respectively 4.5; 2.4; 5.6, and 4.9% and did not affect the f

_{cm}of the PM_3 series when comparing with the PC (Figure 9). Moreover, it can be seen that the reduction of V

_{f}from 3 kg/m

^{3}to 2 kg/m

^{3}had a positive effect on f

_{cm}. On the other hand, the means of individual FRC mixes were from 58.05 to 61.31 MPa, standard deviations from 0.72 to 2.77 MPa, and the coefficients of variation from 1.19 to 4.66%, which showed that the variation between the series was not very large. The conclusion can be made that the type and the amount of fibers did not significantly affect the f

_{cm}, which is in agreement with many other studies, e.g., [65,66,67]. It is also worth noting that the clear deterioration of the workability of the concrete did not have a negative effect on the f

_{c}. Additionally, the 5% quantile of the concrete compressive strength (f

_{c;0.05}) was calculated, which means the probability of occurring the value of f

_{c}less than 0.05. For the distribution of t-non-centrifugal with a one-sided rejection region, the quantile can be determined from Equation (4). The results of the f

_{c;0.05}calculations are presented in Table 7.

- n—number of samples (n = 6 or 5) [-];
- s
_{fc}—standard deviation of compressive strength [MPa]; - t
_{n-1,α}—with a confidence level of α = 0.05, for six samples (n − 1 = 5) − t_{5;0.05}= 1.67, for five samples (n − 1 = 4) − t_{4;0.05}= 1.46 [-].

#### 3.2. Flexural Tensile Strength

_{L}) and to CMOD = 0.5 (F

_{1}); 1.5 (F

_{2}); 2.5 (F

_{3}), and 3.5 mm (F

_{4}) were determined for the tested concrete mixes (Table 8). Subsequently, the flexural tensile strength in the range of limit of proportionality (${\mathrm{f}}_{\mathrm{ct},\mathrm{L}}^{\mathrm{f}}$) and residual strengths: f

_{R,1}; f

_{R,2}; f

_{R,3}; f

_{R,4}were calculated and are shown in Figure 12.

_{f}. Mixes with slightly shorter fibers (l

_{f}= 48 mm) performed better, in particular where fewer fibers (2 kg/m

^{3}) have been used. However, the best results were obtained for a hybrid blend with V

_{f}= 2 kg/m

^{3}. This may be because of the fact that the combination of two types of fibers more effectively bridged the cracks that appear in the initial stage of cracking, and the lower fiber content allowed for more even distribution of all fibers in the concrete. Furthermore, when comparing the results with the ones obtained in the other literature positions, it occurred that the level of enhancement in the flexural tensile strength (f

_{fl}) is in agreement with many of them. For example in [68], where V

_{f}was equal to 0.33%, fibers were 60 mm long and with a diameter of 1.0 mm, f

_{fl}increased by 6.24%. Additionally, Meza et al. reported in [69] that the addition of 2.7 kg/m

^{3}of copolymer (l

_{f}/d

_{f}= 38 mm/2.0 × 0.5 mm) fibers resulted in 6.44% increase of f

_{fl}. Finally, Rucka et al. in [40] tested exactly the same type of fibers PM with the dosage equaled to 2 kg/m

^{3}what resulted in a 6.83% f

_{fl}increase. There also exist studies in which the improvement in strength was smaller, even though fibers with similar length and content were used, namely f

_{fl}, which increased from 2.96 to 4.41%, depending on the V

_{f}in [70]. In the other case, comparable enhancement was obtained for almost twice higher amount of fibers, like in [71] where after the addition of 0.50% of polypropylene fibers (l

_{f}/d

_{f}= 38 mm/0.91 mm) f

_{fl}increased from 3.25 to 3.70 MPa, so by 13.8%. One should also take into account the fact that in many studies the decrease of f

_{fl}was noted despite the addition of fibers, like in [72,73,74].

_{R,j}reached higher values when the fiber content was higher, so V

_{f}= 0.33% (Figure 11 and Figure 12). Moreover, after cracking, the PM_3 beam behaved best, wherein higher V

_{f}and longer fibers with l

_{f}= 54 mm were added, so those that will more efficiently bridge the macrocracks appearing in the second phase of cracking. It is also noteworthy that among the mixtures with 2 kg/m

^{3}the highest f

_{R,2}; f

_{R,3}; and f

_{R,4}are achieved for FF_2 series, which may again indicate the positive effect of combining two types of fibers. The graphs of F-CMOD for FRC (Figure 10) also show some vertical jumps, which were the result of fibers’ failure in the crack cross-section. Failure mechanisms of concrete with fibers are presented and described in Figure 14. Namely, it can be said that in an uncracked concrete the fibers are inactive, however when micro-and macro-cracks occur (6 and 3, respectively), fibers start to bridge them and transfer tensile stresses [75]. Additionally, fibers can prevent further propagation of the crack’s tip (5). Later on, it can happen that fibers will debond from the matrix (4); pull-out (2), or rupture (1), and it will lead to absorption and dissipation of some energy, as a result, stabilization of crack growth in the concrete [75]. It must be noted that in the current study, the rupture of fibers was observed. Often, after breaking the fiber, there was a slight strengthening of the concrete, as the force that it was able to transfer increased with increasing CMOD. It was visible during the analysis of f

_{R,j}for: PM_3, where f

_{R,4}> f

_{R,3}; PM_2, where f

_{R,4}> f

_{R,3}; PD_2, where f

_{R,3}> f

_{R,2}and in Figure 11 and Figure 12. It should also be mentioned that the results for individual samples within the series indicated a significant dispersion of the F-CMOD curves, which was a result of a small fracture area, and thus a large statistical variability of the amount of fibers crossing this surface [76]. Therefore, when testing according to EN 14651 [28], a larger number of samples should be studied. Nevertheless, thanks to this method, which is intended for testing concretes with metallic fibers, a smaller scatter of results is obtained for samples with synthetic fibers than for steel ones [10,76,77]. This can be a result of a more homogeneous distribution of non-metallic fibers than metallic fibers in the concrete mix. It is also worth paying attention to the fact that after the end of the test the FRC beams did not split in half, but retained their integrity. Moreover, all the beams were damaged by a quasi-vertical crack beginning in the sample notch.

#### 3.2.1. Empirical Equations for Flexural Tensile Strength

_{fl}of FRC can be found. Legeron and Paultre [42] suggested that f

_{fl}may be calculated by Equation (5) using the characteristic compressive strength f

_{ck}of steel FRC and the coefficient λ being in the range of 0.35–0.65, but usually, the value of 0.5 is assumed. This formula raises doubts, however, as it does not take into account the material, geometry and volume content of the fibers, which are used in the concrete mix, and only the f

_{ck}itself. Another suggestion is Equation (6) presented by Glinicki in [43,44], also for FRC with steel fibers, which takes into account both the nominal fiber content (V

_{f}) and the fiber slenderness (l

_{f}/d

_{f}). On the other hand, Swamy and Mangat [45] presented Equation (7), which not only makes f

_{fl}dependent on the properties of the metallic fiber, but also on the flexural tensile strength of concrete without fibers (f

_{flc}).

_{fl}for concrete with metallic fibers. Therefore, it was checked whether their use in the case of the tested concretes with synthetic fibers would be justified, and the results of the calculations are presented in Table 9, where ${\mathrm{f}}_{\mathrm{ct},\mathrm{L}}^{\mathrm{f}}$ = f

_{fl}.

_{ck}gave results more than twice higher than those obtained in the current study. Better correlation is achieved when Equation (6) was used because the obtained results were lower by 17–41%, except for PM_3 for which they were 14% higher than those obtained in 3PBT. On the other hand, the best agreement was obtained when Equation (7) was applied since the values of f

_{fl}were 1.03–1.31 times greater than f

_{fl}from the performed tests. Given the fact that steel fibers have a greater influence on the mechanical properties of concrete than synthetic fibers [9,10,69] Equation (7) was adjusted in such a way that the concrete matrix would have a greater and the added fibers smaller influence on the calculated f

_{fl}. Therefore, the authors proposed Equation (8). Table 10 compares the results obtained for the conducted research and the results included in selected literature positions with the results obtained for the empirical formula for flexural tensile strength f

_{fl}proposed by the authors. It can be concluded that Equation (8) correlated very well with the results obtained during the PM_3; PM_2; PD_3; PD_2; and FF_2 tests, as the obtained values were respectively 2.4%; 0.2% greater and 0.3%; 7.1%; 7.3% lower than ${\mathrm{f}}_{\mathrm{ct},\mathrm{L}}^{\mathrm{f}}$ = f

_{fl}. As was mentioned before, the validity of using Equation (8) was checked, when synthetic fibers of a different length, diameter, volume content and concrete mixes of various compositions included in other literature references were used. The analysis showed that the fiber content in the concrete has a significant influence on the degree of correlation. Namely, for V

_{f}≤ 0.5%, there was usually quite a good agreement with the conducted research (f

_{fl}acc. to Equation (8)/f

_{fl}= 0.967–1.084). Moreover, when 0.5% ≤ V

_{f}≤ 1.0%, then f

_{fl}was from 7.2% lower to 11. 3% greater than the result obtained from the performed tests. In case of higher V

_{f}the correlation was rather poor. Therefore, it can be concluded that Equation (8) can be successfully used to estimate the value of f

_{fl}when V

_{f}≤ 1.0%. It is noteworthy that also the fiber geometry had a significant influence on the level of correlation with the proposed formula. Fibers with l

_{f}/d

_{f}greater than 200, i.e., equal to 480 and 600, gave values respectively 1.17–2.55 and 1.23–1.42 times greater than the ones from the tests depending on the V

_{f}. To sum up, Equation (8) can be profitably used when the slenderness of the polymer fibers is less than 200 and V

_{f}≤ 1.0% (Figure 15).

#### 3.2.2. Fiber-Reinforced Concrete Classification

_{R,1k}, and f

_{R,3k}, respectively. These values are calculated from Equation (9) and Equation (10). It is worth mentioning that f

_{R,1k}corresponds to the serviceability limit state (SLS), and f

_{R,3k}corresponds to the ultimate limit state (ULS). The rules for classifying Parameters 1 and 2 and those regarding the possibility of total or partial replacement of traditional reinforcement in a form of steel bars by fibers are described in Table 11.

- ${\mathrm{f}}_{\mathrm{ct},\mathrm{Lk}}^{\mathrm{f}}$—characteristic limit of proportionality [N/mm
^{2}], - ${\mathrm{f}}_{\mathrm{ct},\mathrm{Lm}}^{\mathrm{f}}$—mean limit of proportionality [N/mm
^{2}], - f
_{R,jk}—characteristic residual flexural tensile strength corresponding to CMOD = CMOD_{j}or δ = δ_{j}(j = 1,2,3,4) [N/mm^{2}], - f
_{R,jm}—mean residual flexural tensile strength corresponding to CMOD = CMOD_{j}or δ = δ_{j}(j = 1,2,3,4) [N/mm^{2}], - ${\mathrm{S}}_{\mathrm{ct},\mathrm{L}}^{\mathrm{f}}$—standard deviation of ${\mathrm{f}}_{\mathrm{ct},\mathrm{L}}^{\mathrm{f}}$ from ${\mathrm{f}}_{\mathrm{ct},\mathrm{Lm}}^{\mathrm{f}}$ [N/mm
^{2}] calculated from Equation (11); - s
_{R,j}—standard deviation of f_{R,j}from f_{R,jm}[N/mm^{2}] calculated from Equation (12)$${\mathrm{s}}_{\mathrm{ct},\mathrm{L}}^{\mathrm{f}}=\sqrt{\frac{\sum {\left({\mathrm{f}}_{\mathrm{ct},\mathrm{Lm}}^{\mathrm{f}}-{\mathrm{f}}_{\mathrm{ct},\mathrm{L}}^{\mathrm{f}}\right)}^{2}}{\left(\mathrm{n}-1\right)}}$$$${\mathrm{s}}_{\mathrm{R},\mathrm{j}}=\sqrt{\frac{\sum {\left({\mathrm{f}}_{\mathrm{R},\mathrm{jm}}-{\mathrm{f}}_{\mathrm{R},\mathrm{j}}\right)}^{2}}{\left(\mathrm{n}-1\right)}}$$

_{R,1k}were smaller than 1.0. Additionally, only for the mixtures PM_3 and PD_2 the ratio f

_{R,3k}/f

_{R,1k}was greater than 0.5, so it is possible to assign them to letters c and a of Parameter 2, respectively. It is also not possible to completely or partially replace the traditional reinforcement with the fibers, as none of the concrete met the two above-mentioned requirements simultaneously: f

_{R,1k}/${\mathrm{f}}_{\mathrm{ct},\mathrm{Lk}}^{\mathrm{f}}$ > 0.4 and f

_{R,3k}/f

_{R,1k}> 0.5. It is also worth mentioning that the characteristic values of the residual strengths were significantly reduced as a result of the large standard deviation (${\mathrm{S}}_{\mathrm{ct},\mathrm{L}}^{\mathrm{f}}$ and s

_{R,j}) between samples in a given series. In future tests, it is advisable to increase the number of samples within each mixture. Furthermore, concrete mixes with different amounts of cement, coarse and fine aggregate, water and superplasticizer should be tested to verify made conclusions.

#### 3.3. Fracture Energy and Toughness

_{F-CMOD}). In the article, this area is counted up to CMOD = 3.5 mm. Figure 16 shows a graph of the mean fracture energies (G

_{F-CMOD=3.5}) calculated for all three specimens within each mixture. It can be concluded that specimens with more fibers performed best. The calculated toughness index T

_{i}, which is equal to the ratio of G

_{F-CMOD=3.5}for FRC to G

_{F-CMOD=3.5}for PC samples, indicates that the fracture energy for PM_3 and PD_3 increased 22 and 19.6 times, respectively (Table 13). On the other hand, among the mixtures with V

_{f}= 2kg/m

^{3}, the best results were achieved for samples FF_2, where a hybrid blend was used. Compared to other studies in the literature such as [76,77], for similar or higher macrofibers contents in the mix, the ratio of G

_{F-CMOD}of plain concrete to G

_{F-CMOD}of FRC was much lower. There are also studies [80] where microfibers with V

_{f}= 0.50% were used, and the G

_{F-CMOD}value was two times lower than the fracture energy presented in this article, despite the higher fiber content. However, it is related to the fact that shorter fibers have a greater influence when concrete is in the initial stage of flexural tensile test, so up to the point of crack formation. It is because they more effectively bridge the smaller cracks and their amount in the concrete mixture is higher. Moreover, in [71] for lightweight concretes with the addition of slightly shorter polypropylene fibers with l

_{f}/d

_{f}= 38 mm/0.91 mm but with higher V

_{f}= 0.5; 0.7; 0.9; 1.1; 1.3% similar values of G

_{F-CMOD}= 13.88; 16.60; 18; 13; 23.26; 18.27 Nm, respectively, were obtained. All this indicates the logicality of the obtained results for the PC, PM_3, PM_2, PD_3, PD_2 and FF_2.

_{GF-CMOD=3.5}) and the coefficient of variation (V

_{GF-CMOD=3.5}) of the fracture energy. All the mixtures were characterized by quite high s

_{GF-CMOD=3.5}and V

_{GF-CMOD=3.5}, which is also visible in Figure 10. In order to investigate the cause of such a large discrepancy between the samples, each beam was broken and photographed in the crack cross-section. Figure 17 presents three samples of the mixture PM_3 (PM_3_1, PM_3_2, PM_3_3) and PD_3 (PD_3_1, PD_3_2, PD_3_3) with the largest s

_{GF-CMOD=3.5}. In the case of the PM_3 mix, it was noticeable that some of the twisted fibers did not disintegrate, which influenced the distribution of the fibers in the crack cross-section and make it uneven. In order to avoid this situation in the future, the fibers should be added at the initial stage along with aggregates so that they have the possibility of breaking the bundles of fibers. A similar situation occurred in the case of PM_2. On the other hand, in the PD_3 mixture, there was no tendency to create bundles of fibers and the PD_2 mixture was characterized by the lowest V

_{GF-CMOD=3.5}. Thus, the reason for such large differences in G

_{F-CMOD=3.5}in PD_3 may be connected with a significantly different number of fibers in the crack cross-section. It is noteworthy that the F-CMOD plot for the sample PD_3_3 clearly differed from F-CMOD plots for PD_3_1 and PD_3_2 and took much larger force values for the same values of CMOD. Additionally, by comparing Figure 17d–f it can be seen that the latter shows a greater amount of fibers in the failure cross-section. In order to verify this conclusion more precisely, it would be necessary to count the fibers in the crack cross-section for all PD_3 samples and compare them.

#### 3.4. Tensile Strength

_{t}obtained during the UTT are presented in Figure 18. It can be concluded that the addition of fibers did not have a clear influence on f

_{t}. Despite PM_3 and FF_2, the samples had lower strength by around 11.5%. On the other hand, the small effect of V

_{f}is visible, since mixes with 3 kg/m

^{3}behaved better than the ones with 2 kg/m

^{3}of the same fibers. Additionally, the hybrid mixture had f

_{t}2.6% higher than the mix with no fibers. As was seen before, FF_2 performed usually better than other concrete mixtures with 2 kg/m

^{3}of fibers. In future research, a higher number of tested samples should be used to limit the influence of different amounts of fibers in the cracked cross-section within one mixture.

#### 3.5. CMOD-δ, CMOD-CTOD and CTOD-δ

^{2}= 0.999). Figure 20 also shows the line representing Equation (1), which describes the relationship between CMOD and δ contained in the EN 14651 standard [28]. As can be seen, it differs from the lines determined for FRC mixtures and correlates with them worse than Equation (13) proposed by the authors. Then, in Figure 21, CMOD-CTOD plots are visible, which practically overlap each other. As in the case of CMOD-δ, the formula of the line describing the mean CMOD-CTOD (R

^{2}= 0.998) was determined (Equation (14)). For the CTOD-δ relationship (Figure 22) the procedure algorithm was the same as for CMOD-CTOD and the formula proposed by the authors is marked as Equation (15). Equations (13)–(15) can be used to calculate δ, CMOD and/or CTOD when only one of the values is measured during the three-point bending test. Additionally, it is worth noting that all the discussed relationships are linear and directly proportional.

#### 3.6. Digital Image Correlation

_{x}strain maps are presented for selected CMOD, corresponding with ${\mathrm{f}}_{\mathrm{ct},\mathrm{L}}^{\mathrm{f}}$ and f

_{R,1}–f

_{R,4}. It is easily visible that the crack propagation developed within increasing F and CMOD. The conclusion mentioned before that the beams were destroyed by a quasi-vertical crack beginning in the sample notch is also confirmed by the DIC technique. When it comes to strain maps, only for small CMOD up to 0.5 mm the horizontal strains could have been noticed in the images. When the crack width was increasing, two parts of the sample were detaching from each other, making the camera not able to recognize the speckle pattern.

## 4. Conclusions

^{3}of synthetic fibers of different geometry and form. The following conclusions were drawn from the conducted research:

- As a result of adding synthetic fibers to concrete, the workability of the mixes deteriorated significantly. The consistency class decreased by three for FF_2, two for PM_3, PD_3 and PD_2, and one for PM_2, compared to plain concrete. Nevertheless, the type and fiber content did not significantly affect the compressive strength, which ranged from 58.06 to 61.31 MPa, depending on the mixture, while for concrete without fibers it was equal to 58.05 MPa.
- The fibers contributed to an increase in the flexural tensile strength of approximately 5.5% for PM_3, PM_2, and PD_3; 11.5% for PD_2 and 13.5% for FF_2. The mixtures with slightly shorter fibers (l
_{f}= 48 mm) performed better, and in particular the one where a smaller amount of fibers (2 kg/m^{3}) has been used. However, the best results were obtained with a hybrid blend with V_{f}= 2 kg/m^{3}. This may be due to the fact that the combination of two types of fibers more effectively bridged the microcracks, and the lower fiber content allowed fibers for their more even distribution in the concrete. - The damage of beams without fibers was brittle and sudden, unlike for the fiber-reinforced concrete samples, which were still able to transfer a significant load with increasing crack mouth opening displacement, and possessed some ductility. The residual flexural tensile strengths of concrete with synthetic fibers usually reached higher values, when the fiber content was higher. Additionally, the mixture with the incorporation of the larger amount of fibers which were longer (l
_{f}= 54 mm) performed best. It was because longer fibers more efficiently bridged the macrocracks. It is also worth noting that among the mixtures with 2 kg/m^{3}, the highest values for almost all residual strengths were achieved for the hybrid blend, which may indicate the positive effect of combining two different types of fibers. - As a result of the addition of fibers to concrete, the fracture energy, calculated up to the crack mouth opening displacement of 3.5 mm, increased from about 16 to 22 times depending on the type of the mixture.
- Significant dispersion of the force-crack mouth opening displacement curves was the result of a small fracture area (high statistical variability of the number of fibers crossing this surface) resulting from the used test method-according to EN 14651. Additionally, in the case of some mixtures, it was noticeable that some of the twisted fibers did not disintegrate, which resulted in the uneven distribution of fibers. The reason for large deviations in the results of individual samples may be connected with a different number of fibers in the cross-section of the crack.
- The formula proposed by the authors for calculating the flexural tensile strength for concrete with polymer fibers with a nominal fiber content of ≤ 1.0% and slenderness up to 200 correlates very well with the results of the presented studies, as well as with those contained in various literature positions. Regarding the formula proposed in the EN 14651 standard to determine the relationship between the deflection and the crack mouth opening displacement, it was concluded that it differs from the relationship recorded during the presented tests of beams with synthetic fibers. As a result, a new formula was proposed by the authors, together with equations describing crack mouth opening displacement–crack tip opening displacement and crack tip opening displacement–deflection dependencies.
- According to Model Code 2010, the use of 2 and 3 kg/m
^{3}of tested fibers in the concrete mix would not partially or fully replace traditional reinforcement in a form of steel bars. Furthermore, it was not possible to obtain σ-w curves from uniaxial tensile tests, and received results for maximum tensile strength did not show the clear influence of fibers’ incorporation on concrete. - The DIC technique can be successfully used to determine the crack mouth and crack tip opening displacements with very high accuracy.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 2.**Casting of dog-bone specimens for uniaxial tensile test: (

**a**) first layer of stirrups on the 1/3 depth of the sample; (

**b**) second layer of stirrups on the 2/3 depth of the sample.

**Figure 3.**Set-up for compressive strength test: 1—sample; 2—fixed head; 3—movable head; 4—frame; 5—motor; 6—digital display meter.

**Figure 4.**Set-up for three-point bending test: 1—sample; 2—loading roller; 3—supporting roller; 4—rigid frame to install LVDT; 5—LVDT to measure δ; 6—clip gauge to measure CTOD; 7—clip gauge to measure CMOD.

**Figure 5.**Picture of the sample during the three-point bending test: 1—sample; 2—loading roller; 3—supporting roller; 4—rigid frame to install LVDT; 5—LVDT to measure δ; 6—clip gauge to measure CTOD; 7—clip gauge to measure CMOD.

**Figure 6.**Set-up for uniaxial tensile test: 1-sample; 2—manual load application; 3—grip; 4—bar assembling to prevent falling down of the sample after failure; 5—LVDT to measure w; 6—rigid frame.

**Figure 7.**Picture of the sample: (

**a**) before the uniaxial tensile test; (

**b**) after the uniaxial tensile test: 1—sample; 3—grip; 4—bar assembling to prevent falling down of the sample after failure; 5—LVDT to measure w; 6—rigid frame.

**Figure 8.**Set-up for monitoring the samples using the DIC system: (

**a**) computer for storage the data with the lamp to lightened the studied surface; (

**b**) camera.

**Figure 10.**Graphs of mean F-CMOD curves for (

**a**) PC; (

**b**) PM_3; (

**c**) PM_2; (

**d**) PD_3; (

**e**) PD_2; (

**f**) FF_2 with graphs of the individual samples within the series.

**Figure 12.**Comparison of the mean values of the flexural tensile strength in the range of limit of proportionality (${\mathrm{f}}_{\mathrm{ct},\mathrm{L}}^{\mathrm{f}}$) and residual flexural tensile strengths (f

_{R,j}) for the tested concrete mixtures.

**Figure 14.**Failure mechanisms of FRC: 1—fiber rapture; 2—fiber pull-out; 3—fiber bridging the macrocrack; 4—matrix/fiber debonding; 5—fiber counteracting crack propagation; 6—fiber bridging the microcracks.

**Figure 15.**Graphical representation of the correlation of presented research and selected literature positions with the empirical formula for the flexural tensile strength f

_{fl}proposed by the authors.

**Figure 17.**FRC beams broken in the crack cross-section: (

**a**) PM_3_1; (

**b**) PM_3_2; (

**c**) PM_3_3; (

**d**) PD_3_1; (

**e**) PD_3_2; (

**f**) PD_3_3.

**Figure 20.**CMOD-δ diagram for the tested concrete mixes with the formula described in EN 14651 [28] and Equation (13) proposed by the authors.

**Figure 21.**CMOD-CTOD diagram for the tested concrete mixes together with Equation (14) proposed by the authors.

**Figure 22.**CTOD-δ diagram for the tested concrete mixes together with Equation (15) proposed by the authors.

**Figure 23.**Comparison of CMOD and CTOD results from clip gauges and the DIC system for selected samples: (

**a**) PM_3_2; (

**b**) PD_3_2; (

**c**) FF_2_3.

Ref. | Workability | Compressive Strength | Flexural Tensile Strength acc. to EN 14651 [28] | Fracture Energy | Uniaxial Tensile Strength | Model Code Classification [26] | DIC System |
---|---|---|---|---|---|---|---|

[13] | x | x | x ^{1} | ||||

[22] | x | x ^{1} | x | ||||

[23] | x | x | x ^{1} | x | |||

[38] | x | x | |||||

[40] | x | x | x | x | |||

[41] | x | x | x | x | |||

[46] | x | x | x | x | |||

[47] | x | x | x | x | |||

[48] ^{2} | x | x | x | x | |||

[50] | x | x | x ^{3} | ||||

[51] | x | x ^{1} | |||||

[52] | x | x ^{1} | |||||

[53] | x | x | x ^{1} | x | |||

[54] | x | ||||||

[55] | x | x | |||||

[56] ^{2} | x | ||||||

This study | x | x | x | x | x | x | x |

^{1}Acc. to different standard.

^{2}Research on steel FRC.

^{3}No information about the standard.

PM | PD | FF | |
---|---|---|---|

l_{f} | 54 mm | 48 mm | 52 mm/52 mm |

d_{f} | 0.45 mm | 0.60 mm | 0.45 mm/- |

l_{f}/d_{f} | 120 | 80 | 115/- |

f_{t} | 550–650 MPa | 550–600 MPa | 620–758 MPa |

Material | copolymer | copolymer | 95% copolymer/5% polypropylene |

Form | twisted, monofilament | monofilament | twisted, monofilament/fibrillated |

Composition | PC | PM_3 | PM_2 | PD_3 | PD_2 | FF_2 |
---|---|---|---|---|---|---|

CEM 42.5R | 400 | |||||

Pebble aggregate 2–8 mm | 1052 | |||||

Sand 0–2 mm | 673 | |||||

Water | 200 | |||||

Superplasticizer | 1.43 | |||||

w/c | 0.50 | |||||

Fibers | 0.0 | 3.0 | 2.0 | 3.0 | 2.0 | 2.0 |

V_{f} | 0.00% | 0.33% | 0.22% | 0.33% | 0.22% | 0.22% |

Procedure | Mixing Time |
---|---|

50% of pebble aggregate + 50% of sand | 1 min |

50% of CEM I 42.5R | 1 min |

100% of water + 100% of superplasticizer | 2 min |

50% of pebble aggregate + 50% of sand + 50% of CEM 42.5R | 1 min |

100% of fibers | 2 min ^{1} + * + 1 min + * + 1 min |

^{1}Only for mixes with fibers. * Technical break to remove material on the walls and blades of the mixer.

**Table 5.**Slump classification according to EN 12350-2 [58].

- | PC | PM_3 | PM_2 | PD_3 | PD_2 | FF_2 |
---|---|---|---|---|---|---|

h [mm] | 230 | 130 | 160 | 140 | 140 | 65 |

Consistency class | S5 | S3 | S4 | S3 | S3 | S2 |

Type of the Test | PC | PM_3 | PM_2 | PD_3 | PD_2 | FF_2 |
---|---|---|---|---|---|---|

Compressive strength test | 6 | 6 | 6 | 6 | 6 | 6 |

Flexural tensile test | 3 | 3 | 3 | 3 | 3 | 3 |

Uniaxial tensile test | 1 | 1 | 1 | 1 | 1 | 1 |

No. | Property | PC | PM_3 | PM_2 | ||||||
---|---|---|---|---|---|---|---|---|---|---|

1 | f_{c} [MPa] | 57.49 | f_{cm} [MPa] | 58.05 | 56.61 | f_{cm} [MPa] | 58.06 | 61.50 | f_{cm} [MPa] | 60.64 |

2 | 58.40 | s_{fc} [MPa] | 1.33 | 58.65 | s_{fc} [MPa] | 1.10 | 64.57 | s_{fc} [MPa] | 2.20 | |

3 | 59.89 | V_{fc} [%] | 2.29 | 57.23 | V_{fc} [%] | 1.89 | 58.91 | V_{fc} [%] | 3.64 | |

4 | 58.21 | f_{c;0.05} [MPa] | 57.26 | 59.27 | f_{c;0.05} [MPa] | 57.34 | 60.03 | f_{c;0.05} [MPa] | 59.14 | |

5 | 58.52 | 58.48 | ||||||||

6 | 56.26 | 60.34 | ||||||||

No. | Property | PD_3 | PD_2 | FF_2 | ||||||

1 | f_{c} [MPa] | 56.71 | f_{cm} [MPa] | 59.44 | 59.79 | f_{cm} [MPa] | 61.31 | 60.73 | f_{cm} [MPa] | 60.87 |

2 | 59.59 | s_{fc} [MPa] | 2.77 | 64.10 | s_{fc} [MPa] | 1.81 | 60.71 | s_{fc} [MPa] | 0.72 | |

3 | 60.45 | V_{fc} [%] | 4.66 | 62.02 | V_{fc} [%] | 2.95% | 61.93 | V_{fc} [%] | 1.19 | |

4 | 61.80 | f_{c;0.05} [MPa] | 57.55 | 60.84 | f_{c;0.05} [MPa] | 60.23 | 61.43 | f_{c;0.05} [MPa] | 60.38 | |

5 | 55.57 | 60.59 | ||||||||

6 | 62.50 | 59.81 | 59.84 |

**Table 8.**Force values corresponding to the limit of proportionality (F

_{L}) and CMOD = 0.5 (F

_{1}); 1.5 (F

_{2}); 2.5 (F

_{3}) and 3.5 mm (F

_{4}) for tested concrete mixtures.

Ref. | PC | PM_3 | PM_2 | PD_3 | PD_2 | FF_2 |
---|---|---|---|---|---|---|

F_{L} [kN] | 10.17 | 10.74 | 10.70 | 10.75 | 11.34 | 11.55 |

F_{1} [kN] | - | 4.74 | 3.34 | 3.62 | 6.15 | 3.78 |

F_{2} [kN] | - | 3.33 | 2.30 | 2.91 | 2.15 | 2.61 |

F_{3} [kN] | - | 3.29 | 2.09 | 2.90 | 2.27 | 2.53 |

F_{4} [kN] | - | 3.53 | 2.12 | 2.88 | 2.25 | 2.43 |

**Table 9.**The flexural tensile strength of FRC calculated using empirical formulas and resulting from the tests carried out.

Mixture | f_{fl} acc. to Equation (5) [MPa] | f_{fl} acc. to Equation (6) [MPa] | f_{fl} acc. to Equation (7) [MPa] | ${\mathbf{f}}_{\mathbf{ct},\mathbf{L}}^{\mathbf{f}}={\mathbf{f}}_{\mathbf{fl}}\mathbf{from}\mathbf{Studies}\left[\mathbf{MPa}\right]$ |
---|---|---|---|---|

PM_3 | 7.43 (2.16) | 3.92 (1.14) | 4.50 (1.31) | 3.44 |

PM_2 | 7.59 (2.22) | 2.86 (0.83) | 4.05 (1.18) | 3.42 |

PD_3 | 7.45 (2.17) | 2.86 (0.83) | 4.05 (1.18) | 3.44 |

PD_2 | 7.68 (2.12) | 2.15 (0.59) | 3.75 (1.03) | 3.63 |

FF_2 | 7.70 (2.08) | 2.78 (0.75) | 4.02 (1.09) | 3.69 |

_{fl}from the given formula to that resulting from the performed studies is presented in the brackets.

**Table 10.**Comparison of the results of the presented research and the results from selected literature positions with the results obtained for the empirical formula proposed by the authors for the flexural tensile strength f

_{fl}.

Ref. | V_{f} [%] | l_{f} [mm] | d_{f} [mm] | l_{f}/d_{f} | f_{fl} [MPa] | f_{fl} acc. to Equation (8) [MPa] | f_{fl} acc. to Equation (8)/f_{fl} |
---|---|---|---|---|---|---|---|

Presented research | 0.00 | - | - | - | 3.26 | 3.26 | 1.000 |

0.33 | 54 | 0.45 | 120 | 3.44 | 3.52 | 1.024 | |

0.22 | 54 | 0.45 | 120 | 3.42 | 3.43 | 1.002 | |

0.33 | 48 | 0.60 | 80 | 3.44 | 3.43 | 0.997 | |

0.22 | 48 | 0.60 | 80 | 3.63 | 3.37 | 0.929 | |

0.22 | 52 | 0.45 | 115 | 3.69 | 3.43 | 0.927 | |

[69] | 0.00 | - | - | - | 4.36 | 4.36 | 1.000 |

0.30 | 50 | 0.660 | 76 | 4.64 | 4.51 | 0.971 | |

0.33 | 50 | 0.660 | 76 | 4.54 | 4.52 | 0.997 | |

[10] | 0.00 | - | - | - | 7.70 | 7.70 | 1.000 |

0.50 | 12 | 0.025 | 480 | 8.00 | 9.34 | 1.168 | |

1.00 | 12 | 0.025 | 480 | 7.00 | 10.98 | 1.569 | |

1.50 | 12 | 0.025 | 480 | 6.50 | 12.62 | 1.942 | |

2.00 | 12 | 0.025 | 480 | 5.60 | 14.27 | 2.548 | |

[13] | 0.00 | - | - | - | 3.95 | 3.95 | 1.000 |

0.22 | 50 | 1.000 | 50 | 3.77 | 4.02 | 1.066 | |

0.33 | 50 | 1.000 | 50 | 3.94 | 4.05 | 1.029 | |

0.43 | 50 | 1.000 | 50 | 3.88 | 4.08 | 1.052 | |

[77] | 0.00 | - | - | - | 4.20 | 4.20 | 1.000 |

0.51 | 40 | 0.706 * | 57 | 4.40 | 4.38 | 0.996 | |

0.59 | 40 | 0.706 * | 57 | 4.45 | 4.41 | 0.991 | |

[78] | 0.00 | - | - | - | 5.54 | 5.54 | 1.000 |

0.50 | 15 | 0.100 | 150 | 5.74 | 6.04 | 1.052 | |

[79] | 0.00 | - | - | - | 4.78 | 4.78 | 1.000 |

0.30 | 15 | 0.100 | 150 | 4.69 | 5.08 | 1.084 | |

[80] | 0.00 | - | - | - | 5.21 | 5.21 | 1.000 |

0.50 | 20 | 0.100 | 200 | 5.61 | 5.88 | 1.049 | |

[68] | 0.00 | - | - | - | 4.58 | 4.58 | 1.000 |

0.33 | 60 | 1.000 | 60 | 4.87 | 4.71 | 0.967 | |

0.67 | 60 | 1.000 | 60 | 5.23 | 4.83 | 0.925 | |

[70] | 0.00 | - | - | - | 2.20 | 2.20 | 1.000 |

0.20 | 55 | 0.850 | 65 | 2.27 | 2.29 | 1.009 | |

0.40 | 55 | 0.850 | 65 | 2.30 | 2.37 | 1.033 | |

0.60 | 55 | 0.850 | 65 | 2.25 | 2.46 | 1.092 | |

[74] | 0.00 | - | - | - | 3.09 | 3.09 | 1.000 |

0.70 | 60 | 0.580 | 103 | 3.44 | 3.58 | 1.039 | |

1.00 | 60 | 0.580 | 103 | 3.40 | 3.78 | 1.113 | |

0.00 | - | - | - | 3.79 | 3.79 | 1.000 | |

0.70 | 60 | 0.580 | 103 | 4.11 | 4.27 | 1.039 | |

1.00 | 60 | 0.580 | 103 | 4.32 | 4.48 | 1.036 | |

[81] | 0.00 | - | - | - | 4.73 | 4.73 | 1.000 |

0.32 | 40 | 0.354 * | 113 | 4.69 | 4.97 | 1.059 | |

0.48 | 40 | 0.354 * | 113 | 4.82 | 5.09 | 1.055 | |

[65] | 0.00 | - | - | - | 4.33 | 4.33 | 1.000 |

0.80 | 48 | 0.917 | 52 | 5.09 | 4.59 | 0.902 | |

0.60 | 48 | 0.917 | 52 | 4.49 | 4.52 | 1.008 | |

0.40 | 48 | 0.917 | 52 | 4.36 | 4.46 | 1.022 | |

[82] | 0.00 | - | - | - | 2.62 | 2.62 | 1.000 |

0.15 | 12 | 0.020 | 600 | 2.64 | 3.25 | 1.230 | |

0.30 | 12 | 0.020 | 600 | 2.84 | 3.87 | 1.363 | |

0.50 | 12 | 0.020 | 600 | 3.32 | 4.71 | 1.418 | |

[47] | 0.00 | - | - | - | 5.00 | 5.00 | 1.000 |

0.40 | 48 | 0.900 | 53 | 5.00 | 5.13 | 1.026 | |

0.60 | 48 | 0.900 | 53 | 5.60 | 5.19 | 0.928 | |

0.80 | 48 | 0.900 | 53 | 5.00 | 5.26 | 1.052 | |

1.00 | 48 | 0.900 | 53 | 5.10 | 5.32 | 1.044 | |

1.50 | 48 | 0.900 | 53 | 5.70 | 5.49 | 0.962 | |

[83] | 0.00 | - | - | - | 5.60 | 5.60 | 1.000 |

0.22 | 19 | 0.095 | 200 | 5.91 | 5.90 | 0.998 |

**Table 11.**Principles of FRC classification according to Model Code 2010 [26].

Parameter 1: | Parameter 2: | Total or Partial Replacement of Traditional Reinforcement Is Possible When: | |
---|---|---|---|

Two consecutive numbers in series 1.0; 1.5; 2.0; 2.5; 3.0; 4.0; 5.0; 6.0; 7.0; 8.0; … [MPa] define the strength range f_{R,1k}. | a | for 0.5 ≤ f_{R,3k}/f_{R,1k} < 0.7 | f_{R,1k}/${\mathrm{f}}_{\mathrm{ct},\mathrm{Lk}}^{\mathrm{f}}$ > 0.4 and |

b | for 0.7 ≤ f_{R,3k}/f_{R,1k} < 0.9 | f_{R,3k}/f_{R,1k} > 0.5 | |

c | for 0.9 ≤ f_{R,3k}/f_{R,1k} < 1.1 | ||

d | for 1.1 ≤ f_{R,3k}/f_{R,1k} < 1.3 | ||

e | for 1.3 ≤ f_{R,3k}/f_{R,1k} |

**Table 12.**FRC classification according to Model Code 2010 [26].

PM_3 | |||||||||
---|---|---|---|---|---|---|---|---|---|

${\mathrm{f}}_{\mathrm{ct},\mathrm{Lm}}^{\mathrm{f}}$ [MPa] | 3.44 | ${\mathrm{S}}_{\mathrm{ct},\mathrm{L}}^{\mathrm{f}}$ [MPa] | 0.37 | k | 1.89 | ${\mathrm{f}}_{\mathrm{ct},\mathrm{Lk}}^{\mathrm{f}}$ [MPa] | 2.74 | f_{R,3k}/f_{R,1k} | 0.95 |

f_{R,1m} [MPa] | 1.52 | s_{R,1} [MPa] | 0.61 | 1.89 | f_{R,1k} [MPa] | 0.36 | f_{R,1k}/${\mathrm{f}}_{\mathrm{ct},\mathrm{Lk}}^{\mathrm{f}}$ | 0.13 | |

f_{R,3m} [MPa] | 1.05 | s_{R,3} [MPa] | 0.38 | 1.89 | f_{R,3k} [MPa] | 0.34 | |||

PM_2 | |||||||||

${\mathrm{f}}_{\mathrm{ct},\mathrm{Lm}}^{\mathrm{f}}$ [MPa] | 3.42 | ${\mathrm{S}}_{\mathrm{ct},\mathrm{L}}^{\mathrm{f}}$ [MPa] | 0.42 | k | 1.89 | ${\mathrm{f}}_{\mathrm{ct},\mathrm{Lk}}^{\mathrm{f}}$ [MPa] | 2.64 | f_{R,3k}/f_{R,1k} | 0.43 |

f_{R,1m} [MPa] | 1.07 | s_{R,1} [MPa] | 0.20 | 1.89 | f_{R,1k} [MPa] | 0.68 | f_{R,1k}/${\mathrm{f}}_{\mathrm{ct},\mathrm{Lk}}^{\mathrm{f}}$ | 0.26 | |

f_{R,3m} [MPa] | 0.67 | s_{R,3} [MPa] | 0.20 | 1.89 | f_{R,3k} [MPa] | 0.30 | |||

PD_3 | |||||||||

${\mathrm{f}}_{\mathrm{ct},\mathrm{Lm}}^{\mathrm{f}}$ [MPa] | 3.44 | ${\mathrm{S}}_{\mathrm{ct},\mathrm{L}}^{\mathrm{f}}$ [MPa] | 0.31 | k | 1.89 | ${\mathrm{f}}_{\mathrm{ct},\mathrm{Lk}}^{\mathrm{f}}$ [MPa] | 2.86 | f_{R,3k}/f_{R,1k} | 0.13 |

f_{R,1m} [MPa] | 1.16 | s_{R,1} [MPa] | 0.25 | 1.89 | f_{R,1k} [MPa] | 0.69 | f_{R,1k}/${\mathrm{f}}_{\mathrm{ct},\mathrm{Lk}}^{\mathrm{f}}$ | 0.24 | |

f_{R,3m} [MPa] | 0.93 | s_{R,3} [MPa] | 0.44 | 1.89 | f_{R,3k} [MPa] | 0.09 | |||

PD_2 | |||||||||

${\mathrm{f}}_{\mathrm{ct},\mathrm{Lm}}^{\mathrm{f}}$ [MPa] | 3.63 | ${\mathrm{S}}_{\mathrm{ct},\mathrm{L}}^{\mathrm{f}}$ [MPa] | 0.10 | k | 1.89 | ${\mathrm{f}}_{\mathrm{ct},\mathrm{Lk}}^{\mathrm{f}}$ [MPa] | 3.43 | f_{R,3k}/f_{R,1k} | 0.66 |

f_{R,1m} [MPa] | 1.97 | s_{R,1} [MPa] | 0.71 | 1.89 | f_{R,1k} [MPa] | 0.63 | f_{R,1k}/${\mathrm{f}}_{\mathrm{ct},\mathrm{Lk}}^{\mathrm{f}}$ | 0.18 | |

f_{R,3m} [MPa] | 0.73 | s_{R,3} [MPa] | 0.16 | 1.89 | f_{R,3k} [MPa] | 0.42 | |||

FF_2 | |||||||||

${\mathrm{f}}_{\mathrm{ct},\mathrm{Lm}}^{\mathrm{f}}$ [MPa] | 3.69 | ${\mathrm{S}}_{\mathrm{ct},\mathrm{L}}^{\mathrm{f}}$ [MPa] | 0.11 | k | 1.89 | ${\mathrm{f}}_{\mathrm{ct},\mathrm{Lk}}^{\mathrm{f}}$ [MPa] | 3.48 | f_{R,3k}/f_{R,1k} | 0.39 |

f_{R,1m} [MPa] | 1.21 | s_{R,1} [MPa] | 0.04 | 1.89 | f_{R,1k} [MPa] | 1.14 | f_{R,1k}/${\mathrm{f}}_{\mathrm{ct},\mathrm{Lk}}^{\mathrm{f}}$ | 0.33 | |

f_{R,3m} [MPa] | 0.81 | s_{R,3} [MPa] | 0.19 | 1.89 | f_{R,3k} [MPa] | 0.45 |

**Table 13.**Standard deviation, coefficient of variation of fracture energy and toughness index for individual concrete mixes.

Parameter | PC | PM_3 | PM_2 | PD_3 | PD_2 | FF_2 |
---|---|---|---|---|---|---|

s_{GF-CMOD=3.5} [Nm] | 0.18 | 4.27 | 2.89 | 4.54 | 1.38 | 1.73 |

V_{GF-CMOD=3.5} [%] | 27.56% | 30.58% | 28.10% | 36.50% | 11.75% | 14.56% |

T_{i} [-] | 1.0 | 21.97 | 16.19 | 19.59 | 18.56 | 18.71 |

CMOD [mm] | 0.05→${\mathbf{f}}_{\mathbf{ct},\mathbf{L}}^{\mathbf{f}}$ | 0.5→f_{R,1} | 1.5→f_{R,2} | 2.5→f_{R,3} | 3.5→f_{R,4} | Contour Legend |
---|---|---|---|---|---|---|

PM_3_1 | ||||||

PM_3_2 | ||||||

PM_3_3 | ||||||

PM_2_1 | ||||||

PM_2_2 | ||||||

PM_2_3 | ||||||

PD_3_1 | ||||||

PD_3_2 | ||||||

PD_3_3 | ||||||

PD_2_1 | ||||||

PD_2_2 | ||||||

PD_2_3 | ||||||

FF_2_1 | ||||||

FF_2_2 | ||||||

FF_2_3 |

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## Share and Cite

**MDPI and ACS Style**

Blazy, J.; Drobiec, Ł.; Wolka, P.
Flexural Tensile Strength of Concrete with Synthetic Fibers. *Materials* **2021**, *14*, 4428.
https://doi.org/10.3390/ma14164428

**AMA Style**

Blazy J, Drobiec Ł, Wolka P.
Flexural Tensile Strength of Concrete with Synthetic Fibers. *Materials*. 2021; 14(16):4428.
https://doi.org/10.3390/ma14164428

**Chicago/Turabian Style**

Blazy, Julia, Łukasz Drobiec, and Paweł Wolka.
2021. "Flexural Tensile Strength of Concrete with Synthetic Fibers" *Materials* 14, no. 16: 4428.
https://doi.org/10.3390/ma14164428