# Statistical Modeling of Compressive Strength of Hybrid Fiber-Reinforced Concrete—HFRC

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

^{3}. Basalt coarse aggregate with a fineness modulus of 6.82 and specific mass of 2.74 g/cm

^{3}was also used. The mix ratio of concrete was 1:2.90:3.60 (cement, sand, and gravel, by mass) with a water/cement ratio equal to 0.66. The mixture showed a slump of 150 mm without the addition of fibers.

## 3. Results

#### 3.1. Compressive Strength of Hybrid Fiber Mixtures

_{c}

^{HFRC}) are shown in Figure 7, Figure 8 and Figure 9. In each figure, axial compressive strength results of a control test sample (0 fiber reinforcement) are shown alongside those for 0.60%, 0.80% and 1.15% total hybrid fiber replacement, respectively. In Figure 7, mixture B′ with 3/8 polypropylene, 1/8 carbon, and 1/2 steel fibers for a total percentage of 0.60% in volume of concrete was the one that presented the greatest increase in compression strength. Due to the low workability of fiber-reinforced concrete, there is great difficulty in molding the specimens. This difficulty is greater in some mixtures; this effect is potentiated by fiber hybridization. Due to this fact, the specimens may have greater variation, and this was reflected in the results, as can be seen in the variation of resistance between the specimens.

#### 3.2. Statistical Modeling of Compressive Strengths of Hybrid Fiber Mixtures

^{HFRC}) and several independent variables (fibers: steel, carbon, and polypropylene). The independent variables were used with known values (proportions of different fibers) to predict the values of the selected dependent variable (compressive strength of the resulting composite). Each independent variable was weighted by the regression analysis procedure to ensure the best prediction from the set of independent variables presented. It is important to note that the coefficients calculated for the compressive strength model indicate the relative contribution of each of the independent variables (fraction of steel—S, carbon—C, and polypropylene—P) to the overall value of compressive strength. The analysis was facilitated by the regression considered and contributed to the interpretation of the influence of each variable (S, C, and P) in the definition of compressive strength; the process was interpretive. This modeling was performed considering that the values of S, P, and C were adjusted to each other through a linear combination. Thus, it was possible to obtain the best equation to represent the compressive strength. Furthermore, the modeling reproduced the effect of binary and tertiary combinations between the fibers. On the basis of the compressive strength results, a multiple linear regression model was applied to describe the influence of each type of fiber and its respective percentages. This behavior model is shown in Equation (1).

_{c}

^{HFRC}) in MPa, and S, C, and P are the portions of steel, carbon, and polypropylene fibers added to the mixture, respectively, as a percentage of the volume.

^{2}) was 91.08%. In addition, the standard error of the estimate yielded a standard deviation of 0.132102 of the residuals and an absolute mean error of 0.10199. Residual autocorrelation was evaluated and, in the process of processing the data and obtaining a model capable of representing the behavior of the HFRC about axial compression, no serious autocorrelation was found in the residues.

_{c}

^{HFRC}value is expressed as a function of a single fiber acting as a reinforcement element (isolated fiber).

_{c}

^{HFRC}values for the composites reinforced with them had higher levels.

## 4. Conclusions

^{HRFC}results were obtained with combinations containing proportions in the order of 50% to 75% of metallic fibers complemented with 25% to 37.5% of polypropylene fiber.

## Author Contributions

## Funding

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**(

**a**) Crack coalescence and fiber action stages; (

**b**) respective stress–strain diagram. Adapted from [15].

**Figure 14.**Comparison of compressive strength between the determined behavior equation and the tests performed—total fiber content: (

**a**) 0.60%; (

**b**) 0.80%; (

**c**) 1.15%.

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**MDPI and ACS Style**

Quinino, U.C.d.M.; Christ, R.; Tutikian, B.F.; Silva, L.C.P.d.
Statistical Modeling of Compressive Strength of Hybrid Fiber-Reinforced Concrete—HFRC. *Fibers* **2022**, *10*, 64.
https://doi.org/10.3390/fib10080064

**AMA Style**

Quinino UCdM, Christ R, Tutikian BF, Silva LCPd.
Statistical Modeling of Compressive Strength of Hybrid Fiber-Reinforced Concrete—HFRC. *Fibers*. 2022; 10(8):64.
https://doi.org/10.3390/fib10080064

**Chicago/Turabian Style**

Quinino, Uziel Cavalcanti de Medeiros, Roberto Christ, Bernardo Fonseca Tutikian, and Luis Carlos Pinto da Silva.
2022. "Statistical Modeling of Compressive Strength of Hybrid Fiber-Reinforced Concrete—HFRC" *Fibers* 10, no. 8: 64.
https://doi.org/10.3390/fib10080064