# Sustainability of Using Steel Fibers in Reinforced Concrete Deep Beams without Stirrups

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

_{f}= 0.5 or 1.0%. The study revealed that significant shear stress improvement was observed for high-strength concrete with a 0.5% steel fiber ratio. Although the shear strength of high-strength deep beams was also increased, the strength increase compared with high-strength beams with V

_{f}= 0.5% was minor. This indicates that an optimum V

_{f}of approximately 0.5% value is important to save materials cost. Also, the addition of steel fibers into brittle in nature, high-strength concrete increases its tensile strength, thereby considerably improving the shear strength of SFRC deep beams.

## 2. Shear Strength Models of SFRC Deep Beams

#### 2.1. Background

#### 2.2. Available Shear Strength Models

_{u}, MPa) of SFRC deep beams, where e = 1.0 for a/d > 2.8, e = 2.8(d/a) for a/d ≤ 2.8, τ = 4.15 MPa, F = d

_{f}V

_{f}(l

_{f}/d

_{f}), d

_{f}= 0.5 for round fibers, 0.75 for crimped or hooked fibers, and 1.0 for indented fibers. f

_{spfc}is the splitting tensile strength of SFRC, MPa, and can be estimated by f

_{spfc}= f

_{cu}/(20 − $\sqrt{F}$) + 0.7 + $\sqrt{F}$ (MPa), the parameter f

_{cu}(MPa) is cube concrete compressive strength (=1.25 f

_{cm}), where f

_{cm}is the measured cylinder concrete compressive strength (MPa) and ${v}_{b}=0.41\tau F$

_{u}, MPa) with a/d ≤ 2.5, where f′

_{c}is the concrete compressive strength (MPa), F and v

_{b}are similar to the definition of Narayanan and Darwish [10]:

_{v}, MPa) with a/d ≤ 2.5 using Equation (4), where f

_{f}is the flexural strength of SFRC, MPa.

## 3. Research Significance

## 4. Methodology

#### 4.1. Available Experimental Database

_{cm}, ρ, V

_{f}, and l

_{f}/d

_{f}by square root, cubic root, cubic root, and linear functions, respectively. On the other hand, the shear stresses are inversely related to the shear span ratio, as shown in Figure 5E. Figure 5 presents valuable information to help select the regression model components and save the amount of time needed to find the most fitting function for each variable.

#### 4.2. Comparison between RC and SFRC Databases

#### 4.3. Development of the Nonlinear Regression-Based Model and Features Selection

_{1}to X

_{12}), where X

_{1}to X

_{3}, X

_{4}to X

_{6}, X

_{7}, X

_{8}, and X

_{9}to X

_{10}were utilized in Equations (1)–(5), respectively. However, X

_{11}and X

_{12}are proposed in this study, where X

_{11}= $\sqrt{{{f}^{\prime}}_{c}}$ and X

_{12}= (V

_{f})

^{1/3}. These two parameters can be justified based on the information presented in Figure 5 which reveals the trendline between concrete compressive strength and shear stress is approximately a function of the square root of f′

_{c}. Similarly, the steel fibers volume ratio is related to the shear stress with a function of almost cubic root of V

_{f}. As f

_{spfc}= f

_{cu}/(20 − $\sqrt{F}$) + 0.7 + $\sqrt{F}$ (MPa), the value of X

_{2}= f

_{spfc}and X

_{8}= (f

_{spfc})

^{2/3}are correlated to the parameter F, hence, X

_{2}and X

_{8}can be dropped out. Based on Table 4, X

_{12}is selected as the first potential variable, since X

_{1}is significantly correlated (R ≥ 0.7) to X

_{10}and X

_{12}. Similarly, X

_{3}is the second potential variable, as it is correlated to X

_{4}and X

_{5}. Moreover, X

_{11}is the third potential variable, as X

_{6}is correlated to X

_{9}and X

_{11}. Therefore, the variables X

_{1}, X

_{4}, X

_{5}, X

_{6}, X

_{7}, and X

_{9}can be dropped out of the potential variables as they can be represented by X

_{3}, X

_{11}, and X

_{12}.

#### 4.4. Derivation and Evaluation of the Nonlinear Regression Model

_{s}, v

_{a}, v

_{c}, and v

_{f}are the shear stress contribution of the longitudinal steel reinforcement, shear span-to-depth ratio, concrete, and steel fibers, respectively. Based on the analysis conducted in the section of selecting the features, the three variables $\left(\rho \frac{d}{a}\right),\sqrt{{{f}^{\prime}}_{c}}$, and ${\left({V}_{f}\right)}^{1/3}$ are found to be representative of the other parameters. The vs. and v

_{a}components in Equation (6) can be correlated to the variable $\left(\rho \frac{d}{a}\right)$, while v

_{c}and v

_{f}can be related to $\sqrt{{{f}^{\prime}}_{c}}$, and ${\left({V}_{f}\right)}^{1/3}$, respectively. The general form of the proposed shear strength of SFRC deep beams is shown in Equation (7), where α

_{1}, α

_{2}, and α

_{3}are correlation constants determined by conducting the nonlinear regression analysis.

_{1}= 172.85, α

_{2}= 0.15, and α

_{3}= 1.73. The final version of the proposed shear strength (v

_{proposed}, MPa) is listed in Equation (8).

## 5. Results and Discussions

#### 5.1. Performance of the Proposed Model

_{1}, α

_{2}, and α

_{3}in Equations (7) and (8) were calibrated to attain the minimum value of Root Mean Square Error (RMSE), Equation (9), where N is the total number of specimens (172), v

_{test}and v

_{p}are the experimental shear stress and predicted shear strength, respectively. The RMSE is calculated to the predicted shear stress using Equations (1)–(3), and (5) and compared with the proposed equation. Table 5 reveals that the proposed model can predict the shear strength of SFRC deep beams with the least RMSE, which indicates the high accuracy of the forecasting model, while Equation (3) has the lowest RMSE with high scatteredness in predicting the shear stresses. Figure 7 shows the comparison of the experimental results of shear stresses with the predicted shear stresses using Equations (1)–(3), (5), and (8).

_{min}), maximum (v

_{max}), and average (v

_{avg}) of the experimental shear stresses in the database are 1.56, 13.95, and 4.95, respectively. The maximum predicted shear stress using Equations (1)–(3) are approximately 65% more than v

_{max}, while Equations (5) and (8) have maximum forecasted shear stresses of nearly 50% and 14% less than v

_{max}, respectively. The average of the predicted shear stresses using Equation (8) matches the average of tested specimens of 4.95 MPa. However, the average of shear stresses using Equations (1)–(3) are 5.59, 5.09, and 6.62 MPa which are higher than v

_{avg}, but Equation (5) has average shear stress (3.25 MPa) lower than v

_{avg}. The highest accuracy of the proposed equation (Equation (8)) can be explained by the fact it has the lowest RMSE value (Table 5). Figure 7 shows predictions using Equation (8) have the lowest scatteredness around the blue dashed line (this refers to the perfect prediction).

#### 5.2. Importance of Key Parameters

_{w}, a/d, ρ

_{w}, l

_{f}/d

_{f}, f

_{cm}, and V

_{f}. are indicated as crucial variables. The shear strength of SFRC deep beams without stirrups is most significantly affected by the parameters d and b

_{w}, with significance values of 90% and 78%, respectively. This is because the shear strength is greatly influenced by the size of the beam. Additionally, the compression strength of the concrete and the ratio of steel reinforcement have an important factor of approximately 75%. This is because these two factors are extremely important in resisting the applied shear forces through the dowel action and compression zone contribution on the beam. The shear span-to-depth ratio, steel fiber volume ratio, and steel fiber length-to-diameter ratio all have lower significance values than the shear span-to-depth ratio (64%, 48%, and 43%, respectively), but the designer should still consider their contributions, because doing so can make it more difficult to predict the shear strength. As a result, when developing the suggested regression-based model for this study, the contribution of these seven factors was considered.

#### 5.3. Effect of Selected Parameters on the Proposed Model

## 6. Summary and Conclusions

- Longitudinal steel reinforcement significantly boosts the shear strength of SFRC deep beams without stirrups. This can be justified, as the steel fibers improve deep beams’ capacity to carry loads by helping them bridge cracks.
- Even though the shear stresses are inversely related to the span-to-depth ratio, SFRC deep beams experience higher shear loads than RC deep beams because when the span-to-depth ratio of beams rises, the failure mode shifts from crushing of struts to diagonal shear failure.
- The results also show that the concrete compressive strength and depth of beams have only a small impact on the performance of SFRC and RC beams, which is understandable given that the strut-and-tie model, which primarily relies on the concrete compressive strength, provides the most accurate simulation of the shear strength of deep beams.
- A survey was conducted to explore the input parameters of the available shear strength models in literatures, and the investigation revealed the three variables $\left(\rho \frac{d}{a}\right),\sqrt{{{f}^{\prime}}_{c}}$, and ${\left({V}_{f}\right)}^{1/3}$ are significant to quantify the shear strength contribution of steel reinforcement, concrete, and steel fibers ratio. Therefore, these three variables were utilized to build the proposed shear strength model for SFRC deep beams without stirrups.
- The proposed model outperformed the other equations in the literature, since it was able to anticipate the shear strength of SFRC deep beams with the lowest RMSE (=1.58) and lowest scatteredness, demonstrating the model’s high accuracy.
- The shear stresses predictions using the proposed model revealed that the trendlines of a/d, $\rho $, and ${f}_{cm}$ versus the predicted shear stresses match the test results in terms of the average fitting line and the scatteredness of data. This demonstrates the suggested model’s superior performance when considering various crucial factors that affect the shear strength of SFRC deep beams without stirrups.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 5.**Effect of various variables on measured shear stress; (

**A**) Effective depth, (

**B**) Concrete compressive strength, (

**C**) Longitudinal steel reinforcement ratio, (

**D**) Steel fibers volume ratio, (

**E**) Shear span-to-depth ratio, and (

**F**) Steel fibers length-to-diameter ratio.

**Figure 6.**Effect of various parameters on the measured shear stresses for SFRC and RC deep beams; (

**A**) Longitudinal steel reinforcement ratio, (

**B**) Shear span-to-depth ratio, (

**C**) Concrete compressive strength, and (

**D**) Effective depth.

**Figure 7.**Statistical information for tested and predicted shear stresses using Equations (1)–(3), (5), and (8).

Reference | No. of Specimens | d (mm) |
${{\mathit{f}}^{\prime}}_{c}$ (MPa) | ρ (%) | V_{f}(%) | a/d | l_{f}/d_{f} |
---|---|---|---|---|---|---|---|

[6] | 1 | 102 | 22.7 | 1.1 | 1.0 | 1.5 | 60 |

[7] | 2 | 197 | 29.1;29.9 | 1.34 | 0.5; 0.75 | 2.0 | 60 |

[13] | 5 | 221 | 34 | 1.1; 2.2 | 0.5; 1.0 | 1.5; 2.5 | 60 |

[10] | 8 | 130; 126 | 49–78.8 | 2.0, 5.72 | 0.25–2.0 | 2.0 | 100;133 |

[5] | 8 | 215 | 92–99.1 | 0.37–4.58 | 0.5–1.5 | 1.0;2.0 | 75 |

[14] | 9 | 345 | 37.8–68.2 | 3.55 | 0.25–1.0 | 0.7–0.93 | 100 |

[15] | 2 | 175 | 80 | 3.59 | 0.5; 1.0 | 2.0 | 100 |

[16] | 4 | 186 | 23–26 | 1.2 | 0.5; 1.0 | 2.0 | 50;100 |

[17] | 6 | 557 | 40.8–56.5 | 2.15 | 0.4–1.5 | 1.35 | 60;100 |

[11] | 3 | 212 | 30.8–68.6 | 1.5 | 0.5;0.75 | 2.0 | 62.5 |

[18] | 3 | 127 | 39.8 | 3.09 | 1.76 | 1.2;1.8 | 66.8 |

[19] | 1 | 300 | 109.5 | 3.08 | 0.75 | 1.75 | 75 |

[20] | 9 | 135 | 60–64.2 | 1.16 | 0.5;1.0 | 2.2 | 65;80 |

[21] | 2 | 210 | 44.6;57.2 | 1.5 | 0.5 | 2.0 | 62.5 |

[22] | 4 | 222 | 40.4–45.6 | 1.43 | 0.5;0.75 | 1.8;2.25 | 80 |

[23] | 2 | 219 | 40.9;43.2 | 1.9 | 1.0;2.0 | 2.0 | 60 |

[24] | 4 | 260 | 34.5–37.1 | 2.52 | 0.5–2.0 | 2.0 | 35 |

[25] | 3 | 175 | 82–83.8 | 1.0 | 0.5–1.5 | 1.5 | 80 |

[26] | 6 | 170 | 31.3;38.3 | 2.35 | 0.85;1.3 | 0.8–2.4 | 100 |

[27] | 1 | 219 | 80 | 1.91 | 1.0 | 2.0 | 55 |

[28] | 1 | 275 | 28.4 | 0.35 | 0.5 | 2.0 | 75 |

[8] | 12 | 165 | 25.3–89.4 | 1.3;2.9 | 0.5;1.0 | 1.45 | 60 |

[29] | 11 | 260–305 | 26.5–50 | 0.93–1.64 | 0.25–0.75 | 1.54–2.48 | 45–80 |

[30] | 4 | 127 | 20.7 | 2.0 | 1.0 | 2.0;2.4 | 62.5–100 |

[31] | 6 | 360–657 | 33.5 | 1.06–1.78 | 1.0 | 0.46–1.11 | 60 |

[32] | 9 | 275 | 24.9–31.2 | 0.63–2.47 | 1.0–3.0 | 1.5;2.0 | 60 |

[33] | 18 | 362 | 38.4–47.4 | 1.11–2.32 | 0.4–1.2 | 1.0–2.0 | 37.5 |

[34] | 9 | 170;178 | 52.5–54.6 | 1.23;1.35 | 0.5–1.5 | 1.97;2.35 | 80 |

[35] | 4 | 250 | 53.1;55.4 | 0.53 | 0.5;1.0 | 2.4 | 50 |

[36] | 8 | 270 | 40.4;43.2 | 1.21 | 0.26;0.51 | 0.5;1.5 | 50 |

[37] | 7 | 80–280 | 45.8–51.5 | 2.0–3.54 | 0.75;1.5 | 2.0 | 60;67.7 |

Total | 172 |

Statistical Measure | Features | |||||
---|---|---|---|---|---|---|

d (mm) | b_{w} (mm) | a/d | f_{cm} (MPa) | ρ (%) | V_{u} (kN) | |

Mean | 425 | 183 | 1.35 | 33.5 | 2.02 | 283.4 |

Median | 356 | 178 | 1.36 | 27.9 | 1.82 | 220.8 |

Mode | 305 | 178 | 1 | 31.4 | 1.13 | 192.7 |

Standard deviation | 234 | 64 | 0.4 | 16.7 | 1.1 | 206 |

Minimum | 132 | 51 | 0.5 | 11.3 | 0.26 | 20.7 |

Maximum | 1559 | 460 | 2 | 87 | 5.04 | 1240 |

Count | 281 | 281 | 281 | 281 | 281 | 281 |

Reference | Variable | Parameter | Note |
---|---|---|---|

[10] | X_{1} | $F$ | X_{2} is dependent on X_{1}, therefore, X_{2} will be dropped |

X_{2} | ${f}_{spfc}$ | ||

X_{3} | $\rho \frac{d}{a}$ | ||

[5] | X_{4} | ${\left(\rho \frac{d}{a}\right)}^{0.333}$ | |

X_{5} | $a/d$ | ||

X_{6} | $\sqrt[3]{{{f}^{\prime}}_{c}}$ | ||

[6] | X_{7} | ${\left(\rho \right)}^{1/3}$ | |

[11] | X_{8} | ${f}_{spfc}{}^{2/3}$ | X_{8} is dependent on X_{1}, therefore, X_{8} will be dropped |

[12] | X_{9} | ${{f}^{\prime}}_{c}$ | |

X_{10} | $\rho {V}_{f}$ | ||

Proposed in this study | X_{11} | $\sqrt{{{f}^{\prime}}_{c}}$ | |

X_{12} | ${\left({V}_{f}\right)}^{1/3}$ |

Variable | X_{1} | X_{3} | X_{4} | X_{5} | X_{6} | X_{7} | X_{9} | X_{10} | X_{11} | X_{12} |
---|---|---|---|---|---|---|---|---|---|---|

X_{1} | 1 | |||||||||

X_{3} | 0.23 | 1 | ||||||||

X_{4} | 0.20 | 0.96 | 1 | |||||||

X_{5} | −0.04 | −0.75 | −0.78 | 1 | ||||||

X_{6} | 0.10 | 0.24 | 0.23 | −0.05 | 1 | |||||

X_{7} | 0.29 | 0.69 | 0.78 | −0.26 | 0.32 | 1 | ||||

X_{9} | 0.12 | 0.23 | 0.23 | −0.05 | 0.99 | 0.33 | 1 | |||

X_{10} | 0.70 | 0.45 | 0.50 | −0.10 | 0.17 | 0.73 | 0.18 | 1 | ||

X_{11} | 0.10 | 0.23 | 0.23 | −0.05 | 1.00 | 0.33 | 0.99 | 0.17 | 1 | |

X_{12} | 0.79 | 0.03 | 0.06 | 0.04 | −0.05 | 0.20 | −0.04 | 0.69 | −0.05 | 1 |

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## Share and Cite

**MDPI and ACS Style**

Almasabha, G.; Murad, Y.; Alghossoon, A.; Saleh, E.; Tarawneh, A.
Sustainability of Using Steel Fibers in Reinforced Concrete Deep Beams without Stirrups. *Sustainability* **2023**, *15*, 4721.
https://doi.org/10.3390/su15064721

**AMA Style**

Almasabha G, Murad Y, Alghossoon A, Saleh E, Tarawneh A.
Sustainability of Using Steel Fibers in Reinforced Concrete Deep Beams without Stirrups. *Sustainability*. 2023; 15(6):4721.
https://doi.org/10.3390/su15064721

**Chicago/Turabian Style**

Almasabha, Ghassan, Yasmin Murad, Abdullah Alghossoon, Eman Saleh, and Ahmad Tarawneh.
2023. "Sustainability of Using Steel Fibers in Reinforced Concrete Deep Beams without Stirrups" *Sustainability* 15, no. 6: 4721.
https://doi.org/10.3390/su15064721