# Evaluation of Shear Capacity of Steel Fiber Reinforced Concrete Beams without Stirrups Using Artificial Intelligence Models

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## Abstract

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## 1. Introduction

- First, the influence of using steel fibers on the SFRC’s basic mechanical properties and the reinforced SFRC beam’s shear bearing capacity are briefly reviewed. In addition, the database of SFRC beams containing no stirrups established by Lantsoght [28] is succinctly analyzed.
- A gray relational analysis is then performed to identify the parameter importance.
- AI models, including back-propagation artificial neural network (BPANN), random forest (RF) and multi-gene genetic programming (MGGP), are developed to simulate the shear strength of the reinforced SFRC beam without stirrups.
- A parametric study is finally carried out to validate and explain the AI models.

## 2. Literature Review, Experimental Database and Typical Prediction Models

#### 2.1. Literature Review

#### 2.2. Experimental Database for Shear Testing of SFRC Beams without Stirrups

_{c}is the SFRC’s compressive strength, and s

_{max}is the maximum dimension of coarse aggregate), the area ratio of longitudinal tensile reinforcement (ρ

_{s}) and the attributes and amounts of steel fibers (including fiber type, fiber tensile strength f

_{f}, fiber diameter d

_{f}, fiber aspect ratio l

_{f}/d

_{f}and fiber volume fraction V

_{f}). Each specimen’s ultimate shear strength is denoted as V

_{u}. Note that the parameter ρ

_{f}is also included in the table and is used to represent the bond condition between the steel fiber and concrete. The value of ρ

_{f}is 1.00 for hooked fibers, 0.75 for crimped fibers and 0.50 for straight fibers, as suggested in [30].

_{u}(defined as V

_{u}/bd) should be identified first. Previous research [40] on ordinary concrete beams has suggested that both concrete and longitudinal bars play a key role in resisting applied shear force. To describe these contributions, the following parameter selection scheme was adopted: (i) the parameters f

_{c}and s

_{max}were selected to quantify the contributions of concrete and aggregate interlock action, and (ii) the parameter ρ

_{s}was employed to reflect the shear contribution of the longitudinal bar. The reinforcement index F, defined as V

_{f}ρ

_{f}l

_{f}/d

_{f}[22], was used to characterize the internal confining effect of steel fibers. Furthermore, several other parameters, including d/b and a/d, were used in subsequent analyses, mainly to incorporate the effects of beam geometry and loading type. The above-mentioned parameters have been well documented for each collected specimen.

_{c}, s

_{max}, ρ

_{s}and F, together with the shear strength v

_{u}, for all 488 specimens. It is clear that the database contains both slender and deep beams, and most of the specimens were manufactured with normal-strength concrete. In addition, for SFRC beams, the volume fraction and geometric length of steel fibers were mostly distributed in the ranges of 0–1.5% and 25–60 mm, respectively. The most frequently used shapes of steel fibers were hooked (63% of all compiled beams), crimped (22%) and straight smooth (3%).

#### 2.3. Assessing Existing Prediction Models for Shear Capacity of SFRC Beams without Stirrups

## 3. Parameter Sensitivity Evaluation Using GRA

#### 3.1. Gray Relational Analysis Principle

_{u}) calculated for the database was designated as the reference matrix, A

_{0}(j), in which j = 1, 2, … n. The key tested parameters, including the sectional effective depth-to-width ratio (d/b), the shear span-to-effective depth ratio (a/d), the concrete’s compressive strength (f

_{c}), the maximum aggregate size (s

_{max}), the area ratio of longitudinal reinforcement (ρ

_{s}) and the fiber factor (F), were assigned as the comparative matrix, A

_{i}(j), in which, i = 1, 2, … m. The mathematical formula for constructing the relation between the reference matrix and the comparative matrix is as follows:

_{i}, which represents the gray relational coefficient, is determined by:

#### 3.2. Assessment of Parameter Sensitivity

_{c}> ρ

_{s}> d/b > F > s

_{max}> a/d.

- (a)
- The values of the gray relational factor (λ) for concrete strength f
_{c}and maximum aggregate size s_{max}are 87.79% and 82.61%, respectively. The value for the fiber-related parameter F is 84.23%. These results suggest that, compared with steel fibers, concrete has a greater bearing on the shear strength. This is probably because concrete can resist external loads throughout the entire loading course [49]. Meanwhile, with the widening of the critical crack, the role of steel fiber will be weakened [30]. - (b)
- The shear contribution provided by the longitudinal reinforcement cannot be ignored. This can be stressed by assigning ρ
_{s}with secondary importance (λ = 86.69%). - (c)
- Additionally, the parameters d/b and a/d have notable effects on the shear capacity of SFRC beams containing no stirrups, where they are registered with λ values of 85.69% and 81.96%, respectively. This phenomenon is consistent with that observed in [30].

## 4. Shear Capacity Prediction Using Artificial Intelligence Models

#### 4.1. Back-Propagation Artificial Neural Network (BPANN)

_{ij}denotes the weight from the lower-layer neuron i to the upper-layer neuron j; x

_{i}represents neuron i’s output; n indicates the total number of i; and net

_{j}is the weighted sum of the upper-layer neuron j.

_{n}and Δw

_{n}

_{−1}denote the variations in the weight w at m and m−1 iterations, respectively; and η and α are, respectively, the learning rate and the momentum coefficient.

#### 4.2. Random Forest (RF)

^{−6}.

#### 4.3. Multi-Gene Genetic Programming (GP)

_{1}, x

_{2}and x

_{3}, while the functions mainly include arithmetic operations, such as addition, minus, multiplication and protected division. As can be seen from this subplot, the original expression is (x

_{1}+ x

_{2})(x

_{1}− 1) + sin(x

_{3}) + 3/x

_{2}. After exchanging gene segments (x

_{1}− 1) and 3/x

_{3}and mutating x

_{1}as x

_{3}, the final predictive expression becomes (x

_{2}+ x

_{3}) + 3/x

_{2}+(x

_{1}− 1)sin(x

_{3}).

#### 4.4. Prediction Results and Discussion

^{2}), the root mean squared error (RMSE) and the mean absolute percentage error (MAPE) were used to reflect the prediction errors. A lower RMSE or MAPE value indicates a better prediction accuracy, and a higher R

^{2}value corresponds to a closer fit [31]. These metrics can be calculated using the following equations:

_{avg}is the average value of the measured outputs; and N is the total number of data in the training or testing sets.

^{2}values were all higher than 0.95. The RF model showed slightly higher accuracy than the other models. It can also be observed from Figure 6b that the error frequency of the RF model conformed to a normal distribution.

## 5. Parametric Study

_{c}, s

_{max}, ρ

_{s}, F] = [2, 3.0, 50, 10, 2.5%, 50%]. Compared with the statistical indicators summarized in Table 1, it is not difficult to find that the parameter values adopted by this beam are quite close to the average values in the database. In addition, the maximum and minimum parameter values for each beam in Table 5 are also close to the upper and lower limits in the test database. Thus, we believe that the selected beams in Table 5 still fall within the applicability of the database.

#### 5.1. Influence of the SFRC’s Compressive Strength

_{u}increases with an increase in the concrete strength f

_{c}. It is evident from the predictions in Table 5, Group I, that all eight empirical formulas (Methods 1–8), together with the developed AI models (Methods 9–11), correctly predict this trend.

_{u}/(f

_{c})

^{1/3}and the compressive strength f

_{c}. As can be seen from this plot, once it is normalized with respect to the cubic root of f

_{c}(i.e., (f

_{c})

^{1/3}), the shear strength of the tested beams is no longer sensitive to the concrete strength. This implies that the prediction model to predict the shear strength of SFRC beams without stirrups should be based on the cubic root of the compressive strength of SFRC, which is consistent with the recommendation in [31]. Clearly, the predictions obtained using the AI models can well reflect this trend.

#### 5.2. Influence of the Shear Span-to-Effective Depth Ratio

#### 5.3. Effect of the Area Ratio of Longitudinal Reinforcement

_{s}has a positive influence on the beams’ shear capacity.

#### 5.4. Effect of the Maximum Aggregate Size

_{max}on the shear capacity of SFRC beams without stirrups. However, their predicted results show an increasing trend as the aggregate size increases, which is opposite to experimental observations.

_{max}on the shear strength, as presented in Figure 8d.

#### 5.5. Effect of the Fiber Factor

#### 5.6. Effect of the Sectional Effective Depth-to-Width Ratio

## 6. Conclusions

- (1)
- The empirical strength models evaluated in this paper cannot predict with desirable accuracy the shear bearing capacity of SFRC beams without stirrups. There are a number of reasons for this, including the inadequate account of the role of steel fibers (such as the effective fiber distributed area along the critical diagonal shear crack, the total amount of fibers in this area, the fiber type and orientation, and the individual fiber pull-out load–slip relationship) and the limited database that the models were derived from.
- (2)
- The GRA results indicate that the shear strength of the beams depends crucially on the following parameters: the material properties of concrete, the amount of longitudinal reinforcement, the attributes of steel fibers, and the geometrical and loading characteristics of SFRC beams in shear. The λ values of these parameters are all greater than 80%, indicating that all of these parameters should be considered for a more rational prediction of beam shear strength. Unfortunately, none of the empirical models evaluated take these parameters into full account.
- (3)
- The three AI models—BPANN, RF and MGGP—are effective in predicting the shear capacity of SFRC beams without stirrups. Their predictive performance is excellent, with all R
^{2}values higher than 0.95. By contrast, RF slightly surpasses BPANN and MGGP (mainly because RF is an ensemble learning method, which combines the results of multiple weak learners), while MGGP provides an unambiguous design expression. The AI models fit the experimental data in both the training and testing sets, showing good generalization capacity within the range of the data collected. - (4)
- The AI models were used to perform a parametric study to strengthen support for experimental trends. The results show that these models reveal the potential effects of all of the important factors affecting the shear capacity, and these effects can be reasonably explained. Therefore, the developed AI models can be used as fast, accurate and simulation-free tools for designing SFRC beams without stirrups.

## Supplementary Materials

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 3.**Illustration of BPANN [31].

**Figure 4.**Illustration of RDT and RF [31].

**Figure 5.**Illustration and flow chart of MGGP [31].

Parameter | d/b | a/d | f_{c} (MPa) | s_{max} (mm) | ρ_{s} (%) | F (%) | v_{u} (MPa) |
---|---|---|---|---|---|---|---|

Maximum | 4.90 | 6.00 | 215.0 | 22.0 | 5.72 | 285.75 | 13.96 |

Mean | 1.81 | 2.92 | 49.0 | 10.7 | 2.46 | 53.95 | 3.64 |

Minimum | 0.42 | 0.46 | 9.8 | 0.4 | 0.37 | 7.50 | 0.60 |

Standard deviation | 0.77 | 0.98 | 25.2 | 5.1 | 1.01 | 35.96 | 2.14 |

Standard error | 0.03 | 0.04 | 1.1 | 0.2 | 0.05 | 1.63 | 0.10 |

Median | 1.57 | 3.00 | 40.3 | 10.0 | 2.54 | 48.75 | 3.01 |

Mode | 1.26 | 2.00 | 33.2 | 10.0 | 3.09 | 60.00 | 2.61 |

Kurtosis | 3.99 | 0.37 | 7.8 | −0.1 | 0.98 | 7.14 | 4.72 |

Skewness | 1.94 | 0.00 | 2.2 | 0.0 | 0.77 | 2.02 | 2.06 |

Reference | Equation |
---|---|

CECS38-2004 [41] | ${v}_{\mathrm{u}}=\frac{1.75}{1+a/d}{f}_{\mathrm{t}}(1+{\beta}_{\mathrm{v}}{V}_{\mathrm{f}}\frac{{l}_{\mathrm{f}}}{{d}_{\mathrm{f}}})$ $\frac{a}{d}=\mathrm{min}[\mathrm{max}(\frac{a}{d},1.5),3.0]$ ${\beta}_{\mathrm{v}}=\{\begin{array}{c}0.70\mathrm{shear}-\mathrm{flat}\mathrm{fibers}\hfill \\ 0.50\mathrm{shear}-\mathrm{cut}\mathrm{profied}\mathrm{fibers}\hfill \\ 0.60\mathrm{cut}-\mathrm{off}\mathrm{profied}\mathrm{fibers}\hfill \\ 0.90\mathrm{mill}-\mathrm{cut}\mathrm{profied}\mathrm{fibers}\hfill \end{array}$ ${f}_{\mathrm{t}}=\{\begin{array}{c}0.3{({f}_{\mathrm{c}}-8)}^{\frac{2}{3}}{f}_{\mathrm{c}}\le 58\mathrm{MPa}\hfill \\ 2.12\mathrm{ln}(1+0.1{f}_{\mathrm{c}}){f}_{\mathrm{c}}58\mathrm{MPa}\hfill \end{array}$ |

DAfStB-2012 [42] | ${v}_{\mathrm{u}}={v}_{\mathrm{c}}+{v}_{\mathrm{f}}$ ${v}_{\mathrm{c}}=0.12k{[{\rho}_{\mathrm{s}}({f}_{\mathrm{c}}-8)]}^{\frac{1}{3}}$ $k=1+\sqrt{\frac{200}{d}}$ $(d:\mathrm{in}\mathrm{mm})$ ${v}_{\mathrm{f}}=0.68{f}_{\mathrm{ctR},\mathrm{u}}^{\mathrm{f}}\frac{h}{d}$ ${f}_{\mathrm{ctR},\mathrm{u}}^{\mathrm{f}}=0.185{k}_{\mathrm{G}}^{\mathrm{f}}{f}_{\mathrm{cfIk},\mathrm{l}2}^{\mathrm{f}}$ ${k}_{\mathrm{G}}^{\mathrm{f}}=\mathrm{min}(1.0+0.5{A}_{\mathrm{ct}}^{\mathrm{f}},1.7)$ ${A}_{\mathrm{ct}}^{\mathrm{f}}=b/1000\times \mathrm{min}(d,1500)/1000$ $(b:\mathrm{in}\mathrm{mm})$ ${f}_{\mathrm{cfIk},\mathrm{l}2}^{\mathrm{f}}=0.63\sqrt{{f}_{\mathrm{c}}/0.85}+2.88\times {10}^{-3}F\sqrt{{f}_{\mathrm{c}}/0.85}+5.20\times {10}^{-4}F$ |

Fib-2010 [43] | ${v}_{\mathrm{u}}=0.12k{[{\rho}_{\mathrm{s}}(1+7.5\frac{{f}_{\mathrm{ctR},\mathrm{u}}^{\mathrm{f}}}{{f}_{\mathrm{t}}})({f}_{\mathrm{c}}-8)]}^{\frac{1}{3}}$ ${f}_{\mathrm{t}}=\{\begin{array}{c}0.3{({f}_{\mathrm{c}}-8)}^{\frac{2}{3}}{f}_{\mathrm{c}}\le 58\mathrm{MPa}\\ 2.12\mathrm{ln}(1+0.1{f}_{\mathrm{c}}){f}_{\mathrm{c}}58\mathrm{MPa}\hfill \end{array}$ |

Greenough and Nehdi [44] | ${v}_{\mathrm{u}}=0.35(1+\sqrt{\frac{400}{d}}){({f}_{\mathrm{c}})}^{0.18}{[{\rho}_{\mathrm{s}}\frac{d}{a}(1+0.01F)]}^{0.4}+0.01531F$ $(d:\mathrm{in}\mathrm{mm})$ |

Imam et al. [45] | ${v}_{\mathrm{u}}=\frac{0.70}{\sqrt{1+\frac{d}{25{s}_{\mathrm{max}}}}}{(0.01{\rho}_{\mathrm{s}})}^{1/3}\{{f}_{\mathrm{c}}{}^{0.44}[1+{(0.01F)}^{1/3}]+870\sqrt{\frac{0.01{\rho}_{\mathrm{s}}}{{(\frac{a}{d})}^{5}}}\}$ $(d:\mathrm{in}\mathrm{mm})$ |

Kuntia et al. [46] | ${v}_{\mathrm{u}}=(0.167+0.25F/100)\sqrt{{f}_{\mathrm{c}}}$ |

Sharma [47] | ${v}_{\mathrm{u}}=0.533{(\frac{d}{a})}^{\frac{1}{4}}\sqrt{{f}_{\mathrm{c}}}$ |

Yakoub [48] | ${v}_{\mathrm{u}}=\{\begin{array}{c}0.83\xi {(0.01{\rho}_{\mathrm{s}})}^{1/3}(\sqrt{{f}_{\mathrm{c}}}+249.28\sqrt{\frac{0.01{\rho}_{\mathrm{s}}}{{(\frac{a}{d})}^{5}}}+0.162{R}_{\mathrm{f}}{V}_{\mathrm{f}}\sqrt{{f}_{\mathrm{c}}}\frac{{l}_{\mathrm{f}}}{{d}_{\mathrm{f}}})a/d2.5\\ 0.83\xi {(0.01{\rho}_{\mathrm{s}})}^{1/3}(\sqrt{{f}_{\mathrm{c}}}+249.28\sqrt{\frac{0.01{\rho}_{\mathrm{s}}}{{(\frac{a}{d})}^{5}}}+0.405{R}_{\mathrm{f}}{V}_{\mathrm{f}}\sqrt{{f}_{\mathrm{c}}}\frac{{l}_{\mathrm{f}}}{{d}_{\mathrm{f}}}\frac{d}{a})a/d\le 2.5\end{array}$ $\xi =\frac{1}{\sqrt{1+\frac{d}{25{s}_{\mathrm{max}}}}}$ $(d:\mathrm{in}\mathrm{mm})$ ${R}_{\mathrm{f}}=\{\begin{array}{c}0.83\mathrm{crimped}\mathrm{fibers}\hfill \\ 1.00\mathrm{hooked}\mathrm{fibers}\hfill \\ 0.91\mathrm{rounded}\mathrm{fibers}\hfill \end{array}$ |

_{s}and F in above equations are expressed as %.

Parameter Definition | Setting |
---|---|

Population size | 1000 |

Number of generations | 1000 |

Max number of genes | 10 |

Max genes’ tree depth | 6 |

Function set | plus, minus, times, divide, sqrt, square, cube, sin |

Tournament size | 20 |

Elitism | 5% of population |

Probability of crossover event | 0.85 |

Probability of mutation event | 0.10 |

Probability of reproduction event | 0.05 |

Term | Value |
---|---|

Bias | 53.4 |

Gene 1 | $-1.141{[{x}_{6}+{x}_{2}\sqrt{{x}_{1}}]}^{1/4}$ |

Gene 2 | $-35.32{x}_{1}^{1/4}$ |

Gene 3 | $-({x}_{1}+2\sqrt{{x}_{1}})\sqrt{\mathrm{sin}({x}_{1}^{2})+{x}_{2}^{2}+\sqrt{{x}_{2}}}+0.003772(3{x}_{1}+{x}_{3}+{x}_{1}{x}_{2})$ |

Gene 4 | $-0.008006{x}_{4}{}^{1/4}(2{x}_{2}+{x}_{6}-\mathrm{sin}({x}_{1}{x}_{4})+2{x}_{1}{x}_{2})$ |

Gene 5 | $6.794\sqrt{4{x}_{1}{x}_{2}+{x}_{2}\sqrt{{x}_{1}}}$ |

Gene 6 | $28.78{({x}_{2}^{3}+{x}_{2})}^{1/4}$ |

Gene 7 | $-2.88{x}_{1}^{1/4}{x}_{2}{x}_{5}^{1/8}$ |

Gene 8 | $0.6659\sqrt{{x}_{5}({x}_{3}+\mathrm{sin}{x}_{4})-{x}_{5}^{3/2}}$ |

Gene 9 | $-59.93\sqrt{{x}_{2}}$ |

Gene 10 | $0.6062\sqrt{{x}_{6}+\sqrt{{x}_{2}}}$ |

_{1}, x

_{2}, x

_{3}, x

_{4}, x

_{5}and x

_{6}are, respectively, d/b, a/d, f

_{c}, s

_{max}, ρ

_{s}and F.

Group | Influence Parameters | v_{u} Predicted by Different Methods | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

No. | [d/b, a/d, f_{c}, s_{max}, ρ_{s}, F] | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 |

I | [2, 3.0, 30, 10, 2.5%, 50%] | 1.55 | 1.44 | 1.22 | 2.18 | 2.14 | 1.60 | 2.22 | 1.27 | 2.29 | 2.49 | 2.74 |

[2, 3.0, 50, 10, 2.5%, 50%] | 2.38 | 1.81 | 1.45 | 2.31 | 2.39 | 2.06 | 2.86 | 1.53 | 2.74 | 3.15 | 3.37 | |

[2, 3.0, 70, 10, 2.5%, 50%] | 2.89 | 2.10 | 1.64 | 2.41 | 2.60 | 2.44 | 3.39 | 1.74 | 2.92 | 3.43 | 3.64 | |

[2, 3.0, 100, 10, 2.5%, 50%] | 3.34 | 2.45 | 1.89 | 2.52 | 2.85 | 2.92 | 4.05 | 2.01 | 2.93 | 3.67 | 3.68 | |

II | [2, 0.5, 50, 10, 2.5%, 50%] | 3.81 | 1.81 | 1.45 | 3.93 | 100.1 | 2.06 | 4.48 | 35.05 | 12.80 | 11.49 | 14.59 |

[2, 1.0, 50, 10, 2.5%, 50%] | 3.81 | 1.81 | 1.45 | 3.17 | 18.73 | 2.06 | 3.77 | 7.21 | 8.40 | 7.26 | 9.11 | |

[2, 3.0, 50, 10, 2.5%, 50%] | 2.38 | 1.81 | 1.45 | 2.31 | 2.39 | 2.06 | 2.86 | 1.53 | 2.74 | 3.15 | 3.37 | |

[2, 6.0, 50, 10, 2.5%, 50%] | 2.38 | 1.81 | 1.45 | 1.94 | 1.47 | 2.06 | 2.41 | 1.22 | 2.55 | 2.62 | 3.22 | |

III | [2, 3.0, 50, 10, 0.5%, 50%] | 2.38 | 1.41 | 0.85 | 1.58 | 1.04 | 2.06 | 2.86 | 0.77 | 1.77 | 1.65 | 1.67 |

[2, 3.0, 50, 10, 1.5%, 50%] | 2.38 | 1.66 | 1.22 | 2.03 | 1.81 | 2.06 | 2.86 | 1.22 | 2.20 | 2.72 | 2.61 | |

[2, 3.0, 50, 10, 2.5%, 50%] | 2.38 | 1.81 | 1.45 | 2.31 | 2.39 | 2.06 | 2.86 | 1.53 | 2.74 | 3.15 | 3.37 | |

[2, 3.0, 50, 10, 5.0%, 50%] | 2.38 | 2.07 | 1.83 | 2.81 | 3.60 | 2.06 | 2.86 | 2.13 | 3.66 | 4.29 | 4.44 | |

IV | [2, 3.0, 50, 2.5, 2.5%, 50%] | 2.38 | 1.81 | 1.45 | 2.31 | 1.42 | 2.06 | 2.86 | 0.91 | 3.26 | 3.38 | 3.72 |

[2, 3.0, 50, 5.0, 2.5%, 50%] | 2.38 | 1.81 | 1.45 | 2.31 | 1.88 | 2.06 | 2.86 | 1.20 | 3.00 | 3.18 | 3.47 | |

[2, 3.0, 50, 10, 2.5%, 50%] | 2.38 | 1.81 | 1.45 | 2.31 | 2.39 | 2.06 | 2.86 | 1.53 | 2.74 | 2.85 | 3.37 | |

[2, 3.0, 50, 20, 2.5%, 50%] | 2.38 | 1.81 | 1.45 | 2.31 | 2.88 | 2.06 | 2.86 | 1.84 | 2.47 | 2.63 | 3.30 | |

V | [2, 3.0, 50, 10, 2.5%, 0%] | 1.59 | 1.65 | 1.38 | 1.32 | 1.83 | 1.18 | 2.86 | 1.44 | 1.69 | 2.66 | 1.98 |

[2, 3.0, 50, 10, 2.5%, 50%] | 2.38 | 1.81 | 1.45 | 2.31 | 2.39 | 2.06 | 2.86 | 1.53 | 2.74 | 3.15 | 3.37 | |

[2, 3.0, 50, 10, 2.5%, 100%] | 3.17 | 1.98 | 1.51 | 3.27 | 2.54 | 2.95 | 2.86 | 1.62 | 3.41 | 3.71 | 3.86 | |

[2, 3.0, 50, 10, 2.5%, 200%] | 4.76 | 2.30 | 1.62 | 5.10 | 2.72 | 4.72 | 2.86 | 1.79 | 3.98 | 3.94 | 4.27 | |

VI | [0.5, 3.0, 50, 10, 2.5%, 50%] | 2.38 | 2.39 | 2.04 | 3.09 | 3.26 | 2.06 | 2.86 | 2.09 | 3.51 | 3.69 | 4.74 |

[1.0, 3.0, 50, 10, 2.5%, 50%] | 2.38 | 2.03 | 1.69 | 2.63 | 2.88 | 2.06 | 2.86 | 1.84 | 3.21 | 3.29 | 3.59 | |

[2.0, 3.0, 50, 10, 2.5%, 50%] | 2.38 | 1.81 | 1.45 | 2.31 | 2.39 | 2.06 | 2.86 | 1.53 | 2.74 | 2.85 | 3.37 | |

[5.0, 3.0, 50, 10, 2.5%, 50%] | 2.38 | 1.67 | 1.25 | 2.03 | 1.73 | 2.06 | 2.86 | 1.10 | 2.08 | 2.40 | 2.93 |

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## Share and Cite

**MDPI and ACS Style**

Yu, Y.; Zhao, X.-Y.; Xu, J.-J.; Wang, S.-C.; Xie, T.-Y.
Evaluation of Shear Capacity of Steel Fiber Reinforced Concrete Beams without Stirrups Using Artificial Intelligence Models. *Materials* **2022**, *15*, 2407.
https://doi.org/10.3390/ma15072407

**AMA Style**

Yu Y, Zhao X-Y, Xu J-J, Wang S-C, Xie T-Y.
Evaluation of Shear Capacity of Steel Fiber Reinforced Concrete Beams without Stirrups Using Artificial Intelligence Models. *Materials*. 2022; 15(7):2407.
https://doi.org/10.3390/ma15072407

**Chicago/Turabian Style**

Yu, Yong, Xin-Yu Zhao, Jin-Jun Xu, Shao-Chun Wang, and Tian-Yu Xie.
2022. "Evaluation of Shear Capacity of Steel Fiber Reinforced Concrete Beams without Stirrups Using Artificial Intelligence Models" *Materials* 15, no. 7: 2407.
https://doi.org/10.3390/ma15072407