Experimental Volume Incidence Study and the Relationship of Polypropylene Macrofiber Slenderness to the Mechanical Strengths of Fiber-Reinforced Concretes

Abstract

An experimental study was conducted to examine the mechanical strengths of concretes with straight high-strength knurled polypropylene macrofibers. Incidences of concrete mechanical strengths were determined for three different fiber dosages and lengths. In addition, compressive, indirect-splitting-test tensile, and flexural strengths were determined through testing. The results showed no statistically significant correlation between the volume and length of fibers with the compressive strength of polypropylene fiber-reinforced concrete (PPFRC). However, there was a statistically significant correlation between the split tensile strength, the volume, and the length of the fibers when the volume was greater than 0.80%, and the length of the fibers was greater than 50 mm. Furthermore, the modulus of rupture increased when the volume of fibers was greater than 0.80% and the length of the fibers was 60 mm. Finally, equations were proposed to determine the tensile strength by split test and the modulus of rupture as a function of the mixture’s resistance without fibers, the fibers’ volume and length.

1. Introduction

Concrete is one of the most widely used materials in the construction industry [1,2,3,4]. Hydraulic cement concrete results from mixing and binding stone aggregates with a mixture of water and Portland cement. The elemental composition of concrete is aggregates plus a binder [5]. The main advantages of concrete are accessibility, versatility, and strength. Concrete has good compressive strength but low tensile and flexural strength [6]. Moreover, concrete is a quasi-brittle material as it does not feature good inelastic deformation capacities [1]. Its low tensile strength and fragility are especially evident in the appearance of cracks in slabs. These cracks engender tensile stresses that exceed the tensile strength of concrete [7].

In recent years, breakthroughs have been reported in diverse types of concrete, encouraged by the demands of the construction industry. Therefore, currently, the main types of concrete are high-strength [8], self-compacting [9,10,11,12], permeable [13,14,15,16], light [5,17,18] and fiber-reinforced concretes. The development of fiber-reinforced concrete (FRC) started at the end of the 1950s. In the 1960s, various fiber types were introduced: vegetal, mineral, synthetic, steel, and carbon, among others [19]. FRC is mainly designed for constructing elements prone to cracking, such as sprayed concrete in tunnels, beams, slabs, and structures, which withstand impact forces [20,21,22] or pavement construction. Fibers reduce slab thickness, control cracks and decay ratios due to cracking, increase joint spacing, and improve cracking properties [23,24,25]. Fibers mainly enhance the post-cracking behavior of concrete as they help control cracks and constrain deformations caused by tensile forces because they can withstand generated tensile forces. Consequently, fibers increase concrete toughness, i.e., the amount of energy required for concrete to fail [20].

Many fiber families exist according to classification criteria [26]. The main criteria are materials, geometry, shape, cross-section, and texture [23]. Each type of fiber behaves differently when hardened concrete is subjected to tensile stresses [27,28]. According to their materials, fibers are classified into synthetic and metallic [26]. Synthetic fibers are classified into polypropylene, carbon, glass, and basalt polymer fibers. Given their affordable cost, some of the most used synthetic fibers are polypropylene (PP) fibers [29]. PP fibers are not strong and feature a low elasticity modulus, but they are flexible, ductile, and have high toughness due to their large inelastic deformation capacity [29]. PP fibers have lower strength, elasticity modulus, and specific weight than synthetic and metallic fibers [30,31]. The tensile strengths of PP and steel fibers range from 150 to 700, and 400 to 1200 MPa, respectively. The elasticity modulus of PP and steel fibers range from 3.5 to 9.5 and 180 to 200 GPa, respectively [24]. The specific weights of PP and steel fibers are 0.91 and 7.86 g/cm³, respectively. Moreover, PP fibers exhibit a better capacity for tensile elongation (21%) than steel fibers (3.5%) [30]. PP fibers are also chemically neutral, making them resistant to alkali and acids and non-corrosive [30,32]. Further, PP fibers have high melting temperatures (165–170 °C) because they are thermostable polymers [30,33]. These fibers increase the durability, plastic deformation, deformation capacity, toughness, cracking control, shear strength, and ductility of concrete [21].

Based on their geometry, fibers are classified into macrofibers and microfibers [26]. Macrofiber lengths range from 19 to 60 mm, and their diameter usually exceeds 0.3 mm, whereas the length and diameter of microfibers are usually under 19 and 0.3 mm, respectively [34]. According to their structure, microfibers are classified into monofilament and fibrillated [35]. Longer fibers are more efficient at improving the mechanical properties of concrete than shorter fibers [36]. One of the most important properties of FRC is its slenderness ratio [37]. The larger the slenderness ratio, the higher the likelihood of fiber flocculation and a more heterogeneous mixture distribution [38]. According to their shape, fibers are classified into crimped, twisted, stapled, sinusoidal, and straight. Furthermore, fibers are classified by cross-section according to the following types: rectangular, round, trilobal, cross-shaped, star, and hexagonal [35]. According to their direction, fibers can be parallel to regular stress or angled [26]; fibers with a parallel orientation exhibit better efficiency than angled fibers [36]. According to texture, fibers can be wavy, stretched, knotty, or knurled [39,40]. Based on their strength, fibers can be regular ranging from 400 to 450 MPa [41]. According to their volume, fibers are classified into low (from 0.05% to 0.5%), medium (from 0.5% to 5%) and high (from 5% to 20%) matrix volumes [31]. Suppose an average matrix volume of 0.6 m³ is considered for a cubic meter of concrete—in that case, the average equivalence in volume per concrete cubic meter ranges from 0.03% to 0.3% for the low average, from 0.3% to 3% for the medium average, and from 3% to 12% for the high average. The type of PP fibers, geometric properties, and dosage affect the mechanical properties of PPFRC (polypropylene fiber-reinforced concrete) [23].

Several researchers have studied the influence of the volume of polypropylene macrofibers on the mechanical strength of concrete. These investigations concluded that the macrofibers, in some cases improve, and in other cases deteriorate, the mechanical resistance. However, none of these investigations used statistical tests to analyze their data to support their conclusions. Additionally, no equations were found to predict the mechanical strengths of concrete with polypropylene fibers. This study used three different dosages of volume (0.40%, 0.80%, and 1.20%) and slenderness (46, 58, and 70) of fibers to determine their influence on the mechanical strengths of fiber-reinforced concrete. The mechanical strengths evaluated were the compressive strength, the tensile strength by the split test, and the flexural strength or modulus of rupture. Concrete with a compressive strength of 40 MPa to 45 MPa was designed. The Grubbs test was used to analyze and eliminate the resistance outliers of each sample. The two-sample t-test evaluated the strength difference between the standard concrete sample without fibers and the concrete samples with fibers. Pearson’s test was used to assess the relationship between variables. In addition, a comprehensive statistical analysis between variables was developed. Empirical equations were proposed to determine the mechanical resistance of the variables studied. For these purposes, three specimens were prepared for each type of concrete to ensure the reliability of the results.

2. Materials and Methods

2.1. Materials

This study used Type I Portland cement manufactured according to ASTM C150 standard [42]. According to the manufacturer’s data, the cement had a density of 3.12 g/mL and a specific surface area of 336 m²/kg. The amounts of the chemical compounds were: tricalcium silicate (C3S), 54.2%; dicalcium silicate (C2S), 11.9%; tricalcium aluminate (C3A), 10.1%; and, tetracalcium ferroaluminate (C4AF), 9.7%.

This study used 0–4.69 mm natural river fine and 4.69–25.4 mm coarse crushed aggregate. Both the aggregates met the granulometric standards required by ASTM C33 (American Society for Testing and Materials) [43]. The coarse aggregate was HUSO N° 56. The properties of the aggregates are shown in

Table 1. Physical properties of aggregates.

Property	Unit	Fine Aggregate	Coarse Aggregate
Top size		3/8″	1 1/2″
Nominal maximum size		8.00 mm	1″
Fineness modulus	-	2.95	7.41
Mesh #200 content	Percent	8.69	-
Specific weight of dry mass	g/cm3	2.66	2.69
Specific weight of mass SSD	g/cm3	2.69	2.72
Specific weight of bulk mass	g/cm3	2.73	2.78
Absorption	Percent	1.06	1.15
Loose unit weight	kg/m3	1716.46	1465.82
Compacted unit weight	kg/m3	1969.75	1610.22

, and the granulometric curves in Figure 1.

PP fibers with 540 MPa tensile strength were used. The fiber diameter was 0.86 mm, and their lengths were 40, 50, and 60 mm. Further, slenderness ratios were calculated as 46, 58, and 70.

Table 2. Fiber properties.

Feature	Property
Base material	Polypropylene (PP)
Texture	Knurled
Shape	Straight
Length	40, 50, and 60 mm
Equivalent diameter	0.86 mm
Slenderness ratio	46, 58 and 70
Relative density	0.92
Tensile strength	540 MPa
Number of fibers per kilo	>38,500
Weight per fiber	0.032 g

denotes the properties of fibers, and Figure 2 depicts their length and shape.

Superplasticizer additive was used, manufactured according to ASTM C494 standard [44]. According to the manufacturer’s data, the additive had a density of 1.20 g/mL and appeared as a dark brown liquid.

2.2. Mixture Design

The mixture design was conducted using the ACI (American Concrete Institute) method to meet a specified compressive strength of 40 MPa [45]. The water/cement ratio was 0.45. The exact amounts of water, cement, coarse aggregate, fine aggregate, and superplasticizer additive (SP) were used for all cases. Only the fiber dosage was modified in volumes of 0.4%, 0.8%, and 1.20%. The design slump was 100 mm, commonly used in the construction field.

Table 3. Design of concrete mixtures using the ACI method (kg/m³).

Type	Water	Cement	Sand	Stone	Super-Plasticizer	PP Fibers 40 mm	PP Fibers 50 mm	PP Fibers 60 mm
Pattern	226.61	502.83	721.12	891.20	7.10	-	-	-
D:0.4–40	226.61	502.83	721.12	891.20	7.10	3.60	-	-
D:0.8–40	226.61	502.83	721.12	891.20	7.10	7.20	-	-
D:1.2–40	226.61	502.83	721.12	891.20	7.10	10.80	-	-
D:0.4–50	226.61	502.83	721.12	891.20	7.10	-	3.60	-
D:0.8–50	226.61	502.83	721.12	891.20	7.10	-	7.20	-
D:1.2–50	226.61	502.83	721.12	891.20	7.10	-	10.80	-
D:0.4–60	226.61	502.83	721.12	891.20	7.10	-	-	3.60
D:0.8–60	226.61	502.83	721.12	891.20	7.10	-	-	7.20
D:1.2–60	226.61	502.83	721.12	891.20	7.10	-	-	10.80

denotes the mixture proportions for each type of concrete.

2.3. Specimens

The specimens tested under compressive and indirect-tensile-splitting tests were prepared according to ASTM C39 [46] and ASTM C496 [47] standards. For flexural tests, fiberless specimens were prepared according to ASTM C78 [48], and the specimens containing fibers were prepared according to ASTM C1609 [49]. The specimens were prepared according to ASTM C192 [50]. The pouring sequence was as follows: first, stone, sand, and cement were poured, followed by water and the additive. Finally, the fibers were progressively added until uniformly distributed in the mixture. The sequence of placing materials is shown in Figure 3. The sizes of the cylindrical specimens for compressive and splitting tests were 150 mm in diameter and 300 mm in height. The sizes of the beams were parallelepipeds with a base of 150 mm, a height of 150 mm, and a length of 550 mm. The specimens were wet-cured after the first day of pouring, in a curing pond for 27 days until testing. In all cases, the specimens were fractured on the 28th day.

2.4. Experimental Method

The compressive testing of concrete specimens was performed according to ASTM C39 [46]. A cylindrical specimen was used, and the test was performed with the base leaning against the bottom loading plate in the compression equipment. An unbonded capping system comprising two neoprene pads was placed at the bottom and top of each specimen to evenly distribute the loads. After that, the loading piston moved downwards until it contacted the specimen. The load application rate was 15 MPa/min until reaching specimen failure. According to ASTM C39, compressive strength is the maximum load applied to the specimen between its cross-sectional area. Therefore, it can be calculated using Equation (1), where $f_{cm}$ is the compressive strength in MPa; $P_{m\mathrm{á}x}$ is the maximum load in N; and $D$ is the average diameter in mm.

$$f_{cm}=\frac{4 P_{m\mathrm{á}x}}{\pi D^2}$$

(1)

The indirect tensile strength test was performed using the diametral compression of cylindrical specimens or the splitting test according to ASTM C496 [47]. Here, a cylindrical specimen was used. This specimen was tested on its side with its longitudinal axis parallel to the support points. Two neoprene pads were also placed in the load application zone to distribute the loads. The load piston moved downwards until it contacted the specimen. The load application rate was 1.2 MPa/min until the specimen failed. According to ASTM C496, the splitting-test tensile strength is defined as the load applied on the specimen between the failure area comprised of the load application lines. Therefore, it can be calculated using Equation (2), where $f_{sp}$ is the splitting-test tensile strength in MPa; $P_{m\mathrm{á}x}$ is the maximum load in N; and $D$ is the average diameter in mm.

$$f_{sp}=\frac{2 P_{m\mathrm{á}x}}{\pi lD}$$

(2)

The flexural test is an indirect tensile test that measures the ability of a concrete beam to withstand loads in flexural strength or modulus of rupture [51]. The four-point beam flexural test was performed according to ASTM C78 [48] for regular concrete and ASTM C1609 [49] for fiber concrete. A parallelepiped specimen simply supported at its ends with a spacing of 450 mm was used. Loads were applied to three beam sections at a spacing of 150 mm. The load was applied at a 1.2 MPa/min rate until the specimen failed. The flexural tensile strength of beams is defined as the flexural moment in the pure bending zone, i.e., in the central third within the section modulus of the beam. Therefore, it can be calculated using Equation (3), where $f_r$ is the modulus of rupture in MPa; $P_{m\mathrm{á}x}$ is the maximum load in N displayed on the machine; $L$ is the length of the beam in mm; $b$ is the base of the cross-section; and $h$ is the beam depth.

$$f_r=\frac{P_{m\mathrm{á}x}L}{b{h}^2}$$

(3)

3. Results

Experimental results for the mechanical properties of high-strength FRC with different fiber volumes, $V_f$, and slenderness ratios, $l/d$, are shown in

Table 4. Results of the mechanical strengths of concrete.

Type	Slump (mm)	${\mathit{V}}_{\mathit{f}}$ (%)	l (mm)	${\mathit{f}}_{\mathit{c}\mathit{m}}$ (MPa)	CV%	${\mathit{f}}_{\mathit{s}\mathit{p}}$ (MPa)	CV%	${\mathit{f}}_{\mathit{r}}$ (MPa)	CV%
Pattern	99.00	-	-	40.27	1.38	2.55	10.36	5.79	21.78
M1_0.4_40	100.00	0.40	40.00	39.64	1.83	2.82	10.83	6.20	8.19
M2_0.8_40	68.00	0.80	40.00	45.15	1.33	2.75	8.50	5.81	1.75
M3_1.2_40	56.00	1.20	40.00	42.74	1.88	3.08	8.44	6.39	3.48
M4_0.4_50	100.00	0.40	50.00	43.97	0.77	3.10	1.96	5.23	3.29
M5_0.8_50	78.00	0.80	50.00	43.02	1.43	3.15	2.40	6.40	0.60
M6_1.2_50	35.00	1.20	50.00	43.89	1.53	3.42	6.69	6.14	4.44
M7_0.4_60	80.00	0.40	60.00	41.49	1.47	2.50	15.93	6.01	4.04
M8_0.8_60	68.00	0.80	60.00	42.63	0.47	3.15	7.97	6.28	4.66
M9_1.2_60	37.00	1.20	60.00	40.93	0.75	3.45	6.34	6.49	6.29

. The compressive strength, $f_{cm}$, splitting-test tensile strength, $f_{sp}$, and modulus of rupture, $f_r$, were the result of the average rupture of three specimens. The average compressive strength was 42.36 MPa with a standard deviation of 1.76 MPa. The average splitting-test tensile strength was 3.00 MPa with a standard deviation of 0.33 MPa. The average modulus of rupture was 6.07 MPa with a standard deviation of 0.38 MPa. For a significant correlation between variables, each sampling of the different types of concrete must be different from the sampling of the standard concrete or at least the majority of the correlated series, that is, two of three types. In addition, there must be a high level of correlation. This was defined with Pearson’s correlation parameter, $r$. If the absolute value of the Pearson coefficient is between 0 and 5%, the correlation is non-existent, between 5 and 20% it is very weak, between 20 and 40% it is weak, between 40 and 60% it is medium, between 60 and 80% it is considerable, between 80 and 95% it is strong, between 95 and 99.9%, it is very strong, and at 100% it is perfect. Conversely, the coefficient of determination $r^2$ was also evaluated. To obtain a good model, the $r^2$ ratio should be at least 0.70 [52]. Finally, the correlation must be statistically significant. This means that the p-value must be greater than the level of significance established as 0.100 for this investigation.

3.1. Compressive Strength

Figure 4 denotes the relationship between the compressive strength of PPFRC and fiber volume dosage. The lines shown in the figure represent the compressive strength ($f_{cm}$) trend based on linear regression for the fiber lengths of 40, 50, and 60 mm. Pearson’s correlation coefficients, $r$, were 0.59, 0.99, and 0.76, respectively, showing medium, very strong, and considerable positive correlations. Here, the trend was more determined when the length of the fibers was 50 mm.

Table 5. The t-test between results of compressive strength $f_{cm}$ with a significance level of 0.10.

Item	Sample 1	Sample 2	${\mathit{f}}_{\mathit{c}\mathit{m}}$		${\mathit{f}}_{\mathit{s}\mathit{p}}$		${\mathit{f}}_{\mathit{r}}$
			p	Null Hypothesis	p	Null Hypothesis	p	Null Hypothesis
1	Pattern	M1_0.4_40	0.167	no rejection	0.329	no rejection	0.062	rejected
2	Pattern	M2_0.8_40	0.415	no rejection	0.389	no rejection	0.010	rejected
3	Pattern	M3_1.2_40	0.328	no rejection	0.089	rejected	0.011	rejected
4	Pattern	M4_0.4_50	0.545	no rejection	0.071	rejected	0.271	no rejection
5	Pattern	M5_0.8_50	0.254	no rejection	0.064	rejected	0.025	rejected
6	Pattern	M6_1.2_50	0.143	no rejection	0.023	rejected	0.022	rejected
7	Pattern	M7_0.4_60	0.869	no rejection	0.923	no rejection	0.024	rejected
8	Pattern	M8_0.8_60	0.328	no rejection	0.064	rejected	0.020	rejected
9	Pattern	M9_1.2_60	0.636	no rejection	0.020	rejected	0.027	rejected

shows the results of the t-test performed between each type of fiber concrete and the standard concrete. The results showed that in no case could the null hypothesis be rejected, which stated that there was no statistically significant difference between the samples.

Table 6. Pearson’s hypothesis test of the correlation between $f_{cm}$ and $V_f$.

MV	IV	DV	${\mathit{r}}^2$	$\mathit{r}$	p	Null Hypothesis
l = 40 mm	$V_f$	$f_{cm}$	0.344	0.587	0.413	no rejection
l = 50 mm	$V_f$	$f_{cm}$	0.975	0.987	0.012	rejected
l = 60 mm	$V_f$	$f_{cm}$	0.581	0.762	0.238	no rejection

MV—Moderating variable. IV—Independent variable. DV—Dependent variable.

denotes the results of Pearson’s statistical hypothesis test to reject or not reject the null hypothesis about the correlation between compressive strength and fiber volume. The p-value for the concrete samples where 40 mm and 60 mm fibers were used was 0.413 and 0.238, respectively. Considering a significance level of 0.100, if the p-value is greater than the value of the significance level, the possibility of the null hypothesis cannot be rejected. Therefore, the results were not statistically significant in these two types of concrete.

On the other hand, the concrete sample with 50 mm fibers had a p-value of 0.012. Therefore, the null hypothesis was rejected, and the results were statistically significant. This analysis showed that if there was a positive correlation between variables, the greater the volume of fibers, the greater the resistance to compression. However, the hypothesis that the samples were identical could not be rejected statistically. This was because the results between samples were similar. Figure 5 denotes the results of this study when compared with other studies [32,53,54,55,56]. Several authors have concluded that the addition of plastic macrofibers in low volumes does not have significant effects on compressive strength because they reported increases of less than 10% [55,57,58,59,60,61]. Meddah and Bencheikh [32] discovered that adding fibers in volumes greater than 2% decreases compressive strength. Some authors, such as Noushini et al. [62], found a 1–7% reduction in compressive strength when using a fiber volume of 0.5%.

Figure 6 denotes the relationship between the compressive strength of PPFRC and fiber length. As only one fiber diameter (0.86 mm) was used in this study, fiber lengths and slenderness ratios were directly related. The lines shown in the figure represent the compressive strength ($f_{cm}$) trend obtained from a linear regression for fiber dosages at 0.40%, 0.80%, and 1.20%; Pearson’s correlation coefficients, $r$, were 0.12, 0.81, and 0.63, respectively, showing very weak, strong, and considerable positive correlations. The determination ratios, $r^2$, were 0.01, 0.65 and 0.40, respectively. Therefore, no good fit existed between the lines and strength behavior. The three trend lines indicated that the compressive strength increased with the fiber length. The strength increase ratios were 0.69%, 2.75%, and 2.64%, yielding an average increase of 2.03%. This implied that the difference in increase using the 40–50 mm fibers was 0.20 MPa, a negligible value, as only the average standard deviation of the compressive test on specimens was 0.81 MPa. Therefore, the compressive strength remained constant with the variation in the fiber length for the same dosage. Table 5 and

Table 7. Pearson’s hypothesis test of the correlation between $f_{cm}$ and l.

MV	IV	DV	$r^2$	$r$	p	Null Hypothesis
$V_f$ = 0.4%	l	$f_{cm}$	0.015	0.124	0.876	no rejection
$V_f$ = 0.8%	l	$f_{cm}$	0.654	0.809	0.191	no rejection
$V_f$ = 1.2%	l	$f_{cm}$	0.398	0.631	0.369	no rejection

MV—Moderating variable. IV—Independent variable. DV—Dependent variable.

show the hypothesis t and Pearson tests, confirming the previous analysis. The “t” hypothesis test showed that the null hypothesis that the samples were not statistically different, could not be rejected. The Pearson hypothesis test showed that the null hypothesis with no correlation between the variables, could not be rejected. The summary of this analysis can be seen in

Table 8. Summary of statistical tests on compressive strength results.

Variables		Q1	Q2	Q3	Variables		Q1	Q2	Q3
IV1	IV2				IV1	IV2
l = 40 mm	$V_f$ = 0.00%	-	medium	no	$V_f$ = 0.40%	No fiber	-	very weak	no
	$V_f$ = 0.40%	no				l = 40 mm	no
	$V_f$ = 0.80%	no				l = 50 mm	no
	$V_f$ = 1.20%	no				l = 60 mm	no
l = 50 mm	$V_f$ = 0.00%	-	very strong	yes	$V_f$ = 0.80%	No fiber	-	strong	no
	$V_f$ = 0.40%	no				l = 40 mm	no
	$V_f$ = 0.80%	no				l = 50 mm	no
	$V_f$ = 1.20%	no				l = 60 mm	no
l = 60 mm	$V_f$ = 0.00%	-	considerable	no	$V_f$ = 1.20%	No fiber	-	considerable	no
	$V_f$ = 0.40%	no				l = 40 mm	no
	$V_f$ = 0.80%	no				l = 50 mm	no
	$V_f$ = 1.20%	no				l = 60 mm	no

IV1—Independent variable 1. IV2—Independent variable 2. Q1—Does it differ from the mix without fibers? Q2—What type of correlation is it? Q3—Is the correlation statistically significant?

.

3.2. Splitting-Test Tensile Strength

Figure 7 denotes the relationship between the splitting-test tensile strength of PPFRC and the dosage of fiber volume. The lines shown in the figure represent the tensile strength ($f_{sp}$) trend obtained from a linear regression for the fiber lengths of 40, 50, and 60 mm; Pearson’s correlation coefficients, $r$, were 0.90, 0.94, and 0.93, respectively, showing strong, strong, and strong positive correlations. The determination ratios, $r^2$, were 0.80, 0.88 and 0.86, respectively. Therefore, an excellent fit existed between the lines and strength behavior. As can be observed, this trend was more determined when the fiber length increased. However, the three lines denoted good regression and evidence that the tensile strength increased as the fiber dosage increased. The strength increase ratio for each variation of 0.4% fiber was 0.15, 0.27, and 0.30 MPa, for the fiber lengths of 40, 50, and 60, respectively. Considering that the standard deviation for the splitting-test tensile strength was 0.22 MPa, meaningful strength increases were obtained with the 50 and 60 mm fiber lengths, with values close to each other. Conversely, the concrete prepared with 40 mm fibers evidenced no significant strength increase. Table 5 and

Table 9. Pearson’s hypothesis test of the correlation between $f_{sp}$ and $V_f$.

MV	IV	DV	${\mathit{r}}^2$	$\mathit{r}$	p	Null Hypothesis
l = 40 mm	$V_f$	$f_{sp}$	0.808	0.899	0.101	no rejection
l = 50 mm	$V_f$	$f_{sp}$	0.883	0.940	0.060	rejected
l = 60 mm	$V_f$	$f_{sp}$	0.867	0.931	0.069	rejected

MV—Moderating variable. IV—Independent variable. DV—Dependent variable.

show the hypothesis “t” and Pearson tests, confirming the previous analysis. The hypothesis test “t” showed that in the mixtures with fibers of 50 and 60 mm in length, the null hypothesis was rejected in most cases, which states that the results were not statistically different from the results of the pattern mix without fibers. On the other hand, in the mixture with 40 mm fibers, the null hypothesis could not be rejected in most cases. Concerning the Pearson test, the null hypothesis was rejected in mixtures with 50 and 60 mm fibers, stating there was no correlation between the results. On the other hand, this hypothesis could not be rejected in mixtures with 40 mm fiber length.

Figure 8 denotes the results of the present study compared with other studies [54,56,63]. The observed trend was consistent with the results from previous studies. Several authors have concluded that adding plastic macrofibers in moderate volumes has a sensible influence on splitting-test tensile strength, from 20% to 50% [52]. A linear trend was observed in all the cases in the regression curves.

Figure 9 denotes the relation between the splitting-test tensile strength of PPFRC and fiber length. The lines shown in the figure represents the splitting-test tensile strength ($f_{sp}$) trend obtained after a linear regression for the fiber dosages of 0.40%, 0.80%, and 1.20%; Pearson’s correlation coefficients, $r$, were 0.29, 0.91, and 0.98, respectively, showing weak, strong, and very strong positive correlations. The determination ratios, $r^2$, were 0.09, 0.83 and 0.97, respectively. Therefore, a good fit existed between the lines and strength behavior. Here, it can be observed that this trend was more determined as the fiber length increased. The three trend lines showed that the tensile strength increased as the fiber length or slenderness ratio increased. For each 10 mm fiber length variation, both the strength and slenderness increased at a rate of $l/d$= 10.63, which, in the ranges studied, were 0.07, 0.10, and 0.16 MPa, for 0.40%, 0.80%, and 1.20% fiber dosages, respectively. Considering that the standard deviation of the splitting-test tensile strength was 0.22 MPa, no significant strength increase was observed between the concretes prepared with fiber dosages of 0.40% and 0.80%. However, a slight increase in strength existed in the concretes with a fiber dosage of 1.20%, specifically if concretes with 40 and 60 mm were compared (when the slenderness increased by 23.25). The latter was stronger. Table 5 and

Table 10. Pearson’s hypothesis test of the correlation between $f_{sp}$ and $l$.

VM	VI	VD	${\mathit{r}}^2$	$\mathit{r}$	p	Null Hypothesis
$V_f$ = 0.4%	l	$f_{sp}$	0.087	0.295	0.706	no rejection
$V_f$ = 0.8%	l	$f_{sp}$	0.825	0.908	0.092	rejected
$V_f$ = 1.2%	l	$f_{sp}$	0.966	0.983	0.017	rejected

MV—Moderating variable. IV—Independent variable. DV—Dependent variable.

show the hypothesis “t” and Pearson tests, respectively, that confirmed the previous analysis. The hypothesis t-test showed that in the mixtures with fiber volume of 0.8%, in most cases, and 1.2%, in all cases, the null hypothesis was rejected, which stated that the results were not statistically different from the results of the standard mix without fibers. On the other hand, when mixing with volume, the null hypothesis could not be rejected in most cases. Concerning the Pearson test, the null hypothesis was rejected in the mixtures with fibers of 0.8% and 1.2%, stating no correlation between the results. On the other hand, this hypothesis could not be rejected in mixtures with 0.4% by volume of fibers. The summary of this analysis can be seen in

Table 11. Summary of statistical tests on tensile strength results by split test.

Variables		Q1	Q2	Q3	Variables		Q1	Q2	Q3
IV1	IV2				IV1	IV2
l = 40 mm	$V_f$ = 0.00%	-	strong	no	$V_f$ = 0.40%	No fiber	-	weak	no
	$V_f$ = 0.40%	no				l = 40 mm	no
	$V_f$ = 0.80%	no				l = 50 mm	yes
	$V_f$ = 1.20%	yes				l = 60 mm	no
l = 50 mm	$V_f$ = 0.00%	-	strong	yes	$V_f$ = 0.80%	No fiber	-	strong	yes
	$V_f$ = 0.40%	yes				l = 40 mm	no
	$V_f$ = 0.80%	yes				l = 50 mm	yes
	$V_f$ = 1.20%	yes				l = 60 mm	yes
l = 60 mm	$V_f$ = 0.00%	-	strong	yes	$V_f$ = 1.20%	No fiber	-	very strong	yes
	$V_f$ = 0.40%	no				l = 40 mm	yes
	$V_f$ = 0.80%	yes				l = 50 mm	yes
	$V_f$ = 1.20%	yes				l = 60 mm	yes

IV1—Independent variable 1. IV2—Independent variable 2. Q1—Does it differ from the mix without fibers? Q2—What type of correlation is it? Q3—Is the correlation statistically significant?

.

3.3. Flexural Strength or Modulus of Rupture

Figure 10 illustrates the relationship between the PPFRC modulus of rupture and fiber volume. The lines shown represent the modulus of rupture ($f_r$) trend obtained from a linear regression at the fiber lengths of 40, 50, and 60 mm. Pearson’s correlation coefficients, $r$, were 0.62, 0.74, and 0.87, respectively, showing, considerable, and strong positive correlations. The determination ratios, $r^2$, were 0.62, 0.74 and 0.87, respectively. Therefore, good concordance existed between the lines and behavior of the strength. The three trend lines showed that the modulus of rupture increased as the fiber length increased. The strength increase ratios were 0.36, 0.44, and 0.46 MPa, for each 0.40% increase in fiber volume, respectively. These were considerable values considering that the standard deviation of the beam flexural test was 0.23 MPa. Table 5 and

Table 12. Pearson’s hypothesis test of the correlation between $f_r$ and $V_f$.

MV	IV	DV	${\mathit{r}}^2$	$\mathit{r}$	p	Null Hypothesis
l = 40 mm	$V_f$	$f_r$	0.622	0.789	0.211	no rejection
l = 50 mm	$V_f$	$f_r$	0.740	0.860	0.140	no rejection
l = 60 mm	$V_f$	$f_r$	0.869	0.932	0.067	rejected

MV—Moderating variable. IV—Independent variable. DV—Dependent variable.

show the hypothesis “t” and Pearson tests. The “t” hypothesis test showed that the null hypothesis was rejected in most cases, which stated that the results were not statistically different from the standard mixture without fibers. Concerning the Pearson test, in mixtures with 60 mm fibers, the null hypothesis was rejected, stating that there was no correlation between the results. On the other hand, this hypothesis could not be rejected in mixtures with 40 and 50 mm fiber length fibers. A strength increase was verified when the dosage increased to at least 60 mm fiber lengths. The strength increase for the 40 and 50 mm fiber length was negligible for the evaluated cases in this study.

Figure 11 denotes the results of the present study compared with other studies [53,54,55]. The observed trend was consistent with the results from previous studies. Yin et al. [51] reported that synthetic macrofibers have no apparent effects on flexural strength, which is dominated by the properties of the concrete matrix. Furthermore, low PP fiber volumes have no effects on flexural strength [61]. Other authors argue that the improvement in splitting-test tensile strength is similar to the improvement in the modulus of rupture [60]. The main advantage of using macrofibers is an improvement in the behavior of concrete post-cracking, i.e., on ductility and toughness. This is a paramount property regarding improving concrete behavior in the presence of cracking, as concrete is characterized as a quasi-fragile material. Cracks occur when the exerted stress overcomes the concrete flexural strength. When the concrete fails, loads are transmitted to fibers, which prevent cracks from spreading and delay collapse. This mechanism is called the bridging effect [64,65].

Figure 12 denotes the relationship between the modulus of rupture of PPFRC and fiber length. The lines shown represent the modulus of rupture ($f_r$) trend, which arose from a linear regression at fiber dosages of 0.40%, 0.80%, and 1.20%; Pearson’s correlation coefficients, $r$, were 0.59, 0.96, and 0.95, respectively, showing medium, very strong, and strong positive correlations. The determination ratios, $r^2$, were 0.35, 0.92 and 0.90, respectively. Therefore, a good fit existed between the lines and strength behavior. Therefore, a good match existed between these lines and the strength behavior. The three trend lines indicated different behaviors against the flexural tensile strength. The strength increase ratios were 0.13, 0.22, and 0.24 MPa for each 10 mm increase in the fiber length, or increases at $l/d$ = 10.63, respectively. These were negligible values considering that the standard deviation of the beam flexural test was 0.23 MPa. No significant strength increase was observed between the concretes prepared with the fiber dosage of 0.40%. However, a slight increase in strength existed in concretes with 0.80 and 1.20% fiber dosages. Table 5 and

Table 13. Pearson’s hypothesis test of the correlation between $f_r$ and $l$.

MV	IV	DV	${\mathit{r}}^2$	$\mathit{r}$	p	Null Hypothesis
$V_f$ = 0.4%	l	$f_r$	0.347	0.589	0.411	no rejection
$V_f$ = 0.8%	l	$f_r$	0.924	0.961	0.039	rejected
$V_f$ = 1.2%	l	$f_r$	0.896	0.947	0.053	rejected

MV—Moderating variable. IV—Independent variable. DV—Dependent variable.

show the hypothesis “t” and Pearson tests, confirming the previous analysis. The “t” hypothesis test showed that the null hypothesis was rejected in most cases, which stated that the results were not statistically different from the standard mixture without fibers. Concerning the Pearson test, the null hypothesis was rejected in mixtures with fiber lengths of 0.8% and 1.2%, stating no correlation between the results. On the other hand, this hypothesis could not be rejected in the mixture with 0.4% by volume of fibers. The summary of this analysis can be seen in

Table 14. Summary of statistical tests on flexural strength results.

Variables		Q1	Q2	Q3	Variables		Q1	Q2	Q3
IV1	IV2				IV1	IV2
l = 40 mm	$V_f$ = 0.00%	-	considerable	no	$V_f$ = 0.40%	No fiber	-	medium	no
	$V_f$ = 0.40%	yes				l = 40 mm	yes
	$V_f$ = 0.80%	yes				l = 50 mm	no
	$V_f$ = 1.20%	yes				l = 60 mm	yes
l = 50 mm	$V_f$ = 0.00%	-	considerable	no	$V_f$ = 0.80%	No fiber	-	very strong	yes
	$V_f$ = 0.40%	no				l = 40 mm	yes
	$V_f$ = 0.80%	yes				l = 50 mm	yes
	$V_f$ = 1.20%	yes				l = 60 mm	yes
l = 60 mm	$V_f$ = 0.00%	-	strong	yes	$V_f$ = 1.20%	No fiber	-	strong	yes
	$V_f$ = 0.40%	yes				l = 40 mm	yes
	$V_f$ = 0.80%	yes				l = 50 mm	yes
	$V_f$ = 1.20%	yes				l = 60 mm	yes

IV1—Independent variable 1. IV2—Independent variable 2. Q1—Does it differ from the mix without fibers? Q2—What type of correlation is it? Q3—Is the correlation statistically significant?

. Noushini et al. [62] stated that to improve the modulus of rupture, the suitable value of the slenderness ratio is 100. Herein, the maximum slenderness ratio is 70.

4. Statistical Evaluation of Results

4.1. Correlation between the Volume and Slenderness Ratio and Mechanical Strength

Multiple regression was used to obtain the following relations between the fiber variables (slenderness ratio, $l/d$, and fiber dosage, $V_f$) and normalized mechanical strengths (normalized splitting-test tensile strength, $n_{sp}$, and normalized flexural strength, $n_r$). These relationships are observed in empirical Equation (4), nonlinear, and (6), linear. No expression of the compressive strength is proposed as a function of the fiber variables because, as seen in the previous section, there was no statistically significant correlation.

$$n_{sp}=1.006-0.040\times V_f(\%)-0.000426\times \frac{l}{d}+0.00535\times V_f(\%)\times \frac{l}{d}\hspace{1em}{R}^2=0.787$$

(4)

$$f_{sp-calc}=n_{sp}\times f_{sp-pattern}$$

(5)

$$n_r=1.022+0.193\times V_f(\%)+0.000407\times \frac{l}{d}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}{R}^2=0.746$$

(6)

$$f_{r-calc}=n_r\times f_{r-pattern}$$

(7)

The relationships between the measured strength parameters and results from Equations (5) and (7) and the 95% reliability intervals are shown in Figure 13 and Figure 14. The correlation between the measured and calculated values was 0.787 and 0.746 for $f_{sp}$ and $f_r$, respectively. The correlation ratios for the slenderness ratio and fiber dosage exhibited a significant positive ratio for $f_{cm}$ and $f_r$ and a strong relation for $f_{sp}$. Moreover, the typical error between the measured and calculated values was 16.97% and 19.52%, for $f_{sp}$ and $f_r$, respectively. Figure 15 denotes the regression graphs resulting from applying Equations (1)–(3) and the standardized prediction profile graphs. Here, increases of up to 30% and 10% were predicted for $f_{cm}$, $f_{sp}$ and $f_r$, respectively, for the types of concrete and ranges of the parameters used in the present study.

4.2. Correlation between Tensile and Compressive Strengths

Figure 16 denotes the relationship between the compressive strength ($f_{cm}$) and splitting-test tensile strength ($f_{sp}$) of PPFRC. It can be observed that the splitting-test tensile strength increased with the compressive strength. After 28 days of measurements, the splitting-test tensile strength ranged from 2.75 to 3.45 MPa. Previous studies [23,54,56,59,60,63,66,67,68] have demonstrated that the split tensile strength of PPFRC after 28 days was in the range of 1.56 to 5.60 MPa. The tensile-strength values measured through the splitting test were within the typical strength range. The splitting-test tensile strength of PPFRC at 28 days was within 6.09–8.43% of the compressive strength. Generally, the splitting-test tensile strength of regular concrete is within 8.00% to 14.00% [52,69,70,71]. However, previous studies have indicated that splitting-test tensile strengths range from 6.78% to 13.30% of compressive strength for PPFRC. Although the 6.09% value was out of range, this value matched the type of concrete with the highest compressive strength. All the other values were within the typical values for PPFRC. Furthermore, the split tensile strength/compressive strength ratio decreased as the compressive strength increased. This trend was reported by Neville [72] in regular-weight concretes, Caldarone [70], and Dewar [73] for high-strength concretes, and [71] for lightweight concretes. According to these studies, the splitting-test strength at low compressive strengths is approximately 10%, whereas, for high strengths, this value is reduced by up to 5%. This trend was also observed in the present study results, as shown in Figure 17.

The regression equations between the splitting-test tensile strength and modulus of rupture were generated from the general expression of the form shown in Equation (8) [70]. Other authors and standards widely use this general expression [74].

$$f_t=a f_{cm}{ }^b$$

(8)

where $f_t$ is the tensile strength, and $f_{cm}$ is the compressive strength. Finally, a and b are constants.

According to Kosmatka [69], the splitting-test tensile strength is estimated as 0.4 to 0.7 times the square root of the compressive strength. Therefore, the proposed Equation (9) based on the results of this experimental study denotes a nonlinear exponential relationship between the splitting-test and compressive strengths, as shown below:

$$f_{sp}=0.46 \sqrt{f_{cm}}$$

(9)

where $f_{sp}$ is the splitting-test tensile strength, and $f_{cm}$ is the uniaxial compressive strength, both in MPa. Figure 18 denotes the comparison of the experimental values of the splitting-test tensile strength calculated using several equations given by Olukun [75] for regular-weight concretes in Equation (10), Choi and Yuan [52] for concretes with PP macrofibers with compressive strengths between 30 and 40 MPa in Equation (11), Aslani and Nejadi (Aslani & Nejadi, 2013) for self-compacting concretes with PP macrofibers between 45 and 50 MPa in Equation (12), Yew et al. [76] for high-strength lightweight concrete with PP fibers between 40 and 46 MPa in Equation (13) and Yap et al. [77] for palm-shell reinforced concrete with PP fibers in Equation (14).

$$f_{sp}=0.20 f_{cm}{}^{0.7}$$

(10)

$$f_{sp}=0.55 f_{cm}{}^{0.5}$$

(11)

$$f_{sp}=0.067 f_{cm}{}^{1.0889}$$

(12)

$$f_{sp}=0.55 f_{cm}{}^{0.5}$$

(13)

$$f_{sp}=0.52 f_{cm}{}^{0.5}$$

(14)

From Equations (9)–(14), Equation (9) predicted the splitting-test tensile strength from the compressive strength, values that were close to the results of this study, with an average of 2.11% error in overestimation. Equation (10) also predicted the results with a 6.27% underestimation error. Equations (11)–(14) predicted the results with 22.09%, 34.29%, 22.09% and 15.43% overestimation errors, respectively.

Figure 19 denotes the relationship between the compressive strength, $f_{cm},$ and modulus of rupture, $f_r,$ of PPFRC. As observed, the modulus of rupture remained constant as the compressive strength increased. After 28 days of measurements, the modulus of rupture ranged from 5.23 to 6.49 MPa. Previous studies [52,53,55,56,66,68,78,79,80,81] have demonstrated that the modulus of rupture after 28 days of PPFRC was within the 2.25 to 8.80 MPa range. The measured modulus of rupture was within the typical strength ranges. The modulus of rupture of PPFRC after 28 days ranged from 11.89% to 15.84% of the compressive strength. The same previous studies showed that the modulus of rupture strengths ranged from 6.67% to 16.53% of the PPFRC compressive strength. Therefore, the present values fell within the typical values for PPFRC. Finally, Figure 20 denotes that the modulus of rupture/compressive strength ratio decreased as the compressive strength increased.

According to Kosmatka, the tensile strength by the modulus of rupture is estimated as 0.7 to 0.8 times the square root of the compressive strength [69]. The proposed Equation (15), which was based on the results of this experimental study, indicates a nonlinear exponential relationship between the modulus of rupture and compressive strength, as shown below:

$$f_r=0.93 \sqrt{f_{cm}}$$

(15)

where $f_r$ is the modulus of rupture, and $f_{cm}$ is the uniaxial compressive strength, both in MPa. An extremely low value of determination ratio was observed because, in the results from this study, no correlation existed between the compressive strength and modulus of rupture. Figure 21 denotes the comparison of the experimental values of the modulus of rupture calculated using several equations given by Yap et al. [77] for palm-shell reinforced concretes with PP fibers in Equation (16), by Aslani and Nejadi [82] for self-compacting concretes with PP macro fibers between 45 and 50 MPa in Equation (17), and by Yew et al. [76] for high-strength lightweight concretes with PP fibers between 40 and 46 MPa in Equation (18).

$$f_r=0.385 {f_{cm}}^{0.6667}$$

(16)

$$f_r=0.670{f_{cm}}^{0.5818}$$

(17)

$$f_r=0.53{f_{cm}}^{0.6667}$$

(18)

From Equations (15)–(18), Equation (15) predicted the modulus of rupture from the compressive strength, values close to the results of this study, with an average of 0.39% error in overestimation. Equation (17) also predicted the outcomes with a 1.81% error in underestimation. Finally, Equations (16) and (18) predicted the results with underestimation and overestimation errors of 22.50% and 6.69%, respectively.

5. Conclusions

Herein, an experimental plan was executed to assess the effects of fiber dosage and length on the mechanical strengths of concretes. In addition, compressive, indirect-splitting-test tensile, and flexural (or modulus of rupture) strengths were evaluated. The conclusions of this study are as follows:

• There is no statistically significant correlation between the compressive strength of the PPFRC with fibers’ volume or length. Furthermore, the results show that the strength values of each type of concrete with fibers do not differ statistically from the strength value of concrete without fibers;
• Reinforced concrete with PP fiber lengths of 50 and 60 mm exhibits increases of 34% and 35%, respectively, in splitting-test tensile strength when the dosage is at 1.20%, which is equivalent to an increase of 0.27 and 0.30 MPa for each 0.40% increase in fiber dosage. Concrete manufactured with 40 mm fibers has a less than 22% strength increase. Fiber-reinforced concrete with volumes of 0.80 and 1.20% exhibits increases of 0.104 MPa and 0.157 MPa for every 10 mm increase in fiber length. Concrete reinforced with 0.40% does not present a statistically significant increase;
• Reinforced concrete with fiber lengths of 60 mm shows an increase of 3.80, 8.46, and 12.09% when the dosage is 0.4, 0.8, and 1.20%, respectively, in the modulus of rupture. The 40 and 50 mm mixtures do not show statistically significant increases. On the other hand, mixtures with dosages of 0.40 and 0.80% exhibit increases of 4.41 and 4.60% for every 10 mm increase in fiber length. The mixtures with a fiber dosage of 0.40% do not show statistically significant increases;
• As a result of this study, Equations (4)–(7) are presented, which allow prediction of the tensile strength by split test, and the modulus of rupture as a function of the strength of concrete without fibers, dosage of fibers, and length of fibers.

The results of this study were based on the experimental results with variables and specimens defined herein. Therefore, further experimental studies that include more amplitude in the variables used, such as compressive strengths, fiber volumes, and slenderness ratios, are recommended to provide a broader range of analyzed cases with their respective conclusions.