# Shear Strength Estimation of Reinforced Concrete Deep Beams Using a Novel Hybrid Metaheuristic Optimized SVR Models

^{1}

^{2}

^{3}

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^{5}

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## Abstract

**:**

## 1. Introduction

## 2. Background of Variables Impacts the Shear Strength of RC Deep Beams

**V**depends on (1) Concrete quality ${f}_{c}^{\prime}$ (for the diagonal strut), (2) Main steel yield strength f

_{y}(for the main tie), and (3) Web reinforcement (horizontal and vertical). As some variables have a big role in forming the deep beam’s shear strength, the main variables considered in this study are the shear span to depth ratio, the main reinforcement, ratio and yield strength, concrete compressive strength, and web reinforcement characteristics.

_{c}was above a certain limit. The non-proportional increase in shear strength compared to the increase in concrete compressive strength can be attributed to two reasons. First, the limited contribution of the aggregate interlock mechanism in members with high strength concrete compared to the one with normal strength concrete, as the cracks cross the aggregate particles in high strength concrete and do not go around them as in normal strength concrete. Second, the formed tensile strains perpendicular to the main diagonal strut work on reducing the benefits of using high strengths. Oh and Shin [28] noticed a brittle failure of deep beams with concrete of 74 MPa without any warning, which is different from the failure of other beams with 23 MPa. They also observed a decrease in the rate of increase in the ultimate strength of beams with high-strength concrete.

## 3. Material and Data Collection

^{2}between Vu and the main variables is 0.39 for the ${f}_{c}^{\prime}$ variable. These results indicate that the relationship between the main variables and Vu cannot be estimated directly, and a complex relationship may be detected by using all main variables. Similarly, for the other variables, the relationship between Vu and variables varies. The best R

^{2}is estimated using a number of main bars, R

^{2}= 0.37. The variation in R

^{2}indicates the complex relationship between Vu and all variables. Table 2 and Figure S1 show the increase of the resistance with beam effective width and height, number of main bars, and concrete strength up to a certain limit.

## 4. Methods and Development Models

#### 4.1. Support Vector Regression

_{i}x) = (φ(x

_{i})φ(x)). In the current study, the radial basis kernel (RBF) is applied.

#### 4.2. Optimization Methods

#### 4.2.1. PSO

- First, it initializes the particle of the swarm, then defines the maximum number of iterations, and finally defines the cost function.
- After defining the cost function, it evaluates the swarm in order to identify the global and local best.
- Lastly, it calculates the velocity of each particle and then updates its position using the following equations:

#### 4.2.2. HHO

_{0}$\in \left(-1,1\right)$, and they indicate that energy falls for the prey with their escapes. Thus, the hawk can decide the solution based on the E computation and starting in phase 3 when $\left|E\right|\ge 1$, and exploiting the neighborhood when $\left|E\right|<1$. Once starting phase 3, hawks decide to apply a soft or hard besiege. $\left|E\right|\ge 0.5$ indicates the prey still has enough energy to escape, but maybe some misleading jumps occur in it to fail, so a soft besiege works. On the other hand, in the case of $\left|E\right|<0.5,$ the prey is too fatigued to escape, so hard besiege works. Here, the HHO is used to optimize the SVR parameters.

#### 4.2.3. AVOA

_{2}. This can be processed as follows:

_{3}. This can be defined as follows:

#### 4.3. Models’ Development and Accuracy Assessment

^{2}), variance account factor (VAF), variance inflation factor (VIF), mean absolute error (MAE), root mean square error (RMSE), performance index (PI), mean bias error (MBE), and percentage error (PE). The R

^{2}and VAF are used to measure the correlation between the measured and predicted values. VIF is used to evaluate the collinearity between the measured and predicted values; VIF > 10 indicates high collinearity. The models’ errors are evaluated using MAE, RMSE, and MBE, and the PE is used to estimate the accuracy of the proposed model error in predicting the shear strength of RC deep beams. The mathematical expression of these indices can be expressed as follows:

^{2}, and N is the number of the data sample.

#### 4.4. Sensitivity Analysis

_{i}, in the data array K is a vector of lengths j, i.e.,:

## 5. Results and Discussion

#### 5.1. All Variables Impact on Vu Estimation

^{2}= 0.98 and RMSE = 32.20 kN in the training stage. The comparison between all models using performance indices of all statistical indices in the training and testing stages shows that the AVOA-SVR model has a high index. However, the performance of HHO-SVR is better in terms of PI, RMSE, and PE in the testing stage. Moreover, Figure 4 illustrates the linear correlation between experimental and predicted Vu values for the proposed models. From Figure 4, it is shown that the performance of the AVOA-SVR model is more accurate than other proposed models in the training and testing stages. The distortion of data points around best fitting is small in modeling Vu with the AVOA-SVR model, and the VIF is higher than in other models, as presented in Table 5. This means that when we used all variables, the AVOA-SVR model can be used to estimate Vu with a model error approach of 6.95%.

#### 5.2. Selected Variables Impact on Vu Estimation

^{2}= 0.97, and low model error, PE = 2.25%, are observed. In the testing stage, a low distortion around best fitting is observed with the AVOA-SVR. In addition, the statistical correlation factors are high, R

^{2}= 0.97 and VAF = 94.46, as presented in Table 7 and Figure 5. The VIF values of AVOA-SVR in the training and testing stages are higher than other models. This means the accuracy of AVOA-SVR is acceptable with low distortion around the observed values. Although the PI and RMSE of the HHO-SVR models are lower than for the AVOA-SVR model, the distortion of HHO-SVR datasets is high, as presented in Table 7 and Figure 5. Meanwhile, the performance of the proposed models is shown to be low to estimate the high shear strength (as presented in Figure 4 and Figure 5). However, the performance of AVOA-SVR is seen as more robust. This indicates that AVOA-SVR can overcome the variation change in the data used. Figure 6 also shows a faster convergence rate of the AVOA-SVR model compared to the other two hybrid models. Therefore, the AVOA-SVR can be used to estimate the shear strength of RC deep beams with 3.4% model accuracy. The statistical comparison indices in Table 5 and Table 7 show that the performance of proposed models with the selected variables is better than that for using all variables in modeling the proposed techniques. This means that the selected variables are more influential in the shear strength of RC deep beams.

#### 5.3. Comparison with Previous Studies and Codes

#### 5.4. Sensitivity Analysis of Input Variables

## 6. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**(

**i**) Real case of using deep beams (

**left**) and deep beam terminology (

**right**); V: Shear strength, a: Shear span, d: Effective depth of beam. (

**ii**) Basic reinforcement details of simple RC deep beam. (

**iii**) Failure pattern in deep beam (

**left**), Different mechanism components (

**middle**), and Flow of forces in deep beam (

**right**); where, C = compression force in concrete, T = tensile force in the main steel, Hs = tensile force in horizontal web reinforcement, Vs = tensile force in vertical web reinforcement; 1: Main diagonal strut, 2: Splitting tension force, 3: Main tensile force.

**Figure 4.**Scatter plot of model’s performances in the training (

**upper row**) and testing (

**lower row**) stages.

**Figure 5.**Scatter plot of model’s performances in the training (

**upper row**) and testing (

**lower row**) stages.

**Figure 7.**Comparison of model’s performances, (

**a**) boxplot, (

**b**) Q-Q plot, (

**c**) scatter plot of relative shear strength with measured shear strength, and (

**d**) zoom in for upper plot with +20% limits for the best models.

Ref. | Equation | Explanation |
---|---|---|

ACI [10] | ${V}_{u}=0.17\sqrt{{f}_{c}^{\prime}}bd+\frac{Av{f}_{y}d(sin\mathsf{\theta}+\mathit{cos}\mathsf{\theta})}{s}$ | $\mathsf{\theta}$ is the angle between the stirrups and the beam longitudinal axis |

Russo [8] | ${V}_{u}=0.545\left(k{\rm X}{f}_{c}^{\prime}cos\alpha +0.25{\rho}_{h}{f}_{yh}cot\alpha +0.35\frac{a}{d}{\rho}_{v}{f}_{yv}\right)bd$ | $k=\sqrt{{\left(n\rho \right)}^{2}+2n\rho}-n\rho $ $\mathrm{tan}\alpha =\frac{a}{0.9d}$ ${\rm X}=0.74{\left(\frac{{f}_{c}^{\prime}}{105}\right)}^{3}-1.28{(\frac{{f}_{c}^{\prime}}{105})}^{2}+0.22\left(\frac{{f}_{c}^{\prime}}{105}\right)+0.87$ |

Liu [4] | ${V}_{u}={V}_{CLZ}+{V}_{ci}+{V}_{s}+{V}_{d}$ | ${V}_{CLZ}$ is the shear resisted at the critical loading zone, ${V}_{ci}$ represents the contribution of aggregate interlock, ${V}_{s}$ is the shear resisted by web reinforcement and ${V}_{d}$ is the dowel action in the main longitudinal bars. |

Variable | Equation (R^{2}) | Variable | Equation (R^{2}) | Variable | Equation (R^{2}) |
---|---|---|---|---|---|

a/d | y = 358.43x^{−0.803} (0.26) | b | y = 117.38 × 10^{0.0052x} (0.21) | Ag | y = 476.32x^{−0.164} (0.03) |

ρ | y = 65.95ln(x) + 345.74 (0.01) | d | y = 0.7899x + 35.304 (0.35) | Std | y = 165.06 × 10^{0.0731x} (0.05) |

${f}_{y}$ | y = 396ln(x) − 2024.7 (0.16) | h | y = 0.6985x + 32.425 (0.33) | Bd | y = 524.3x^{−0.262} (0.03) |

${f}_{c}^{\prime}$ | y = 495.94ln(x) − 1234.7 (0.39) | a | y = 259.88 × 10^{0.0003x} (0.015) | where: y represents the Vu x represents input variables R ^{2} is the coefficient of determination | |

ρ_{v} | y = 325.96 × 10^{−0.159x} (0.01) | Lp | y = 169.42 × 10^{0.0054x} (0.15) | ||

$s$ | y = 0.2678x + 344.21 (0.06) | Sp | y = 169.42 × 10^{0.0054x} (0.15) | ||

${f}_{yv}$ | y = 19.734x^{0.4588} (0.07) | V/P | y = 199.58x + 199.66 (0.010) | ||

${\rho}_{h}$ | y = −371.18x + 429.11 (0.10) | # bars | y = 483.29ln(x) − 199.43 (0.37) |

Variable | RA | M | SD | KU | SK | Variable | RA | M | SD | KU | SK |
---|---|---|---|---|---|---|---|---|---|---|---|

a/d | 1.93 | 1.28 | 0.46 | −0.03 | 0.38 | b (mm) | 200.00 | 188.18 | 66.50 | −0.94 | 0.28 |

$\rho $ (%) | 3.50 | 2.00 | 0.82 | 0.16 | 0.65 | d (mm) | 1374.00 | 443.74 | 212.19 | 11.52 | 3.14 |

${f}_{y}$ (MPa) | 502.00 | 459.71 | 147.09 | 0.50 | 1.26 | h (mm) | 1550.00 | 505.91 | 235.73 | 12.91 | 3.32 |

${f}_{c}^{\prime}$ (MPa) | 66.10 | 28.33 | 13.75 | 7.04 | 2.64 | a (mm) | 1600.00 | 543.97 | 242.31 | 2.87 | 1.09 |

${\rho}_{v}$ (%) | 1.25 | 0.29 | 0.32 | 0.69 | 1.10 | Lp (mm) | 210.00 | 113.11 | 45.63 | 3.05 | 2.04 |

$s$ (mm) | 330.00 | 155.33 | 80.63 | 0.54 | 1.07 | Sp (mm) | 210.00 | 113.11 | 45.63 | 3.05 | 2.04 |

${f}_{yv}$ (MPa) | 791.00 | 430.68 | 171.05 | 6.12 | 2.55 | V/P | 0.50 | 0.93 | 0.16 | 2.96 | −2.18 |

${\rho}_{h}$ (%) | 0.91 | 0.12 | 0.24 | 3.58 | 2.15 | #bars | 10.00 | 3.61 | 1.70 | 10.50 | 2.95 |

Vu (kN) | 1869.00 | 385.80 | 285.02 | 6.25 | 2.09 | Ag (mm) | 22.00 | 14.20 | 5.67 | 0.53 | 1.29 |

Std (mm) | 12.70 | 8.67 | 2.42 | −0.38 | −0.11 | ||||||

Bd (mm) | 6.20 | 7.66 | 2.08 | −0.74 | −0.65 |

Metaheuristic Algorithm | Parameters | Value |
---|---|---|

AVOA | Population | 5 |

Iteration | 15 | |

P_{1} | 0.9 | |

P_{2} | 0.3 | |

P_{3} | 0.6 | |

Alpha | 0.8 | |

Beta | 0.2 | |

Gamma | 2.5 | |

Range of C | [10^{3}, 10^{−3}] | |

Range of ε | [10^{3}, 10^{−3}] | |

Range of γ | [10^{3}, 10^{−3}] | |

PSO | Population | 5 |

Iteration | 15 | |

C_{1} | 1 | |

C_{2} | 2 | |

Range of C | [10^{3}, 10^{−3}] | |

Range of ε | [10^{3}, 10^{−3}] | |

Range of γ | [10^{3}, 10^{−3}] | |

HHO | Population | 5 |

Iteration | 15 | |

N | 3 | |

Range of C | [10^{3}, 10^{−3}] | |

Range of ε | [10^{3}, 10^{−3}] | |

Range of γ | [10^{3}, 10^{−3}] |

Training | R^{2} | VAF | VIF | PI | RMSE | MAE | MBE | PE |

AVOA-SVR | 0.984 | 97.330 | 64.510 | −30.241 | 32.198 | 24.377 | −0.047 | 1.723 |

PSO-SVR | 0.813 | 78.261 | 5.358 | −89.973 | 91.568 | 31.605 | 26.960 | 4.899 |

HHO-SVR | 0.818 | 66.278 | 5.500 | −62.003 | 63.483 | 105.885 | −6.032 | 3.397 |

Testing | R^{2} | VAF | VIF | PI | RMSE | MAE | MBE | PE |

AVOA-SVR | 0.756 | 67.921 | 4.102 | −76.076 | 77.505 | 101.702 | −13.001 | 6.949 |

PSO-SVR | 0.630 | 52.981 | 2.706 | −75.687 | 76.837 | 106.357 | 17.850 | 6.889 |

HHO-SVR | 0.715 | 45.786 | 3.514 | −46.690 | 47.856 | 162.579 | −56.320 | 4.290 |

Model | M | Maximum | Minimum | SD | COV |
---|---|---|---|---|---|

Liu [4] | 1.10 | 1.54 | 0.65 | 0.15 | 0.13 |

Russo [8] | 1.00 | 1.63 | 0.48 | 0.19 | 0.19 |

ACI [10] | 0.59 | 2.06 | 0.09 | 0.41 | 0.69 |

AVOA-SVR | 0.95 | 1.87 | 0.34 | 0.16 | 0.17 |

Training | R^{2} | VAF | VIF | PI | RMSE | MAE | MBE | PE |

AVOA-SVR | 0.974 | 96.726 | 39.202 | −40.095 | 42.036 | 26.728 | −0.360 | 2.249 |

PSO-SVR | 0.834 | 81.625 | 6.042 | −90.753 | 92.402 | 32.755 | 18.958 | 4.944 |

HHO-SVR | 0.816 | 71.805 | 5.427 | −72.442 | 73.975 | 92.860 | −7.926 | 3.958 |

Testing | R^{2} | VAF | VIF | PI | RMSE | MAE | MBE | PE |

AVOA-SVR | 0.970 | 94.460 | 33.512 | −35.876 | 37.790 | 43.168 | −7.149 | 3.388 |

PSO-SVR | 0.950 | 91.774 | 20.118 | −45.091 | 46.958 | 44.085 | 0.475 | 4.210 |

HHO-SVR | 0.948 | 79.860 | 19.147 | −33.841 | 35.586 | 106.952 | −50.633 | 3.190 |

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**MDPI and ACS Style**

Kaloop, M.R.; Roy, B.; Chaurasia, K.; Kim, S.-M.; Jang, H.-M.; Hu, J.-W.; Abdelwahed, B.S.
Shear Strength Estimation of Reinforced Concrete Deep Beams Using a Novel Hybrid Metaheuristic Optimized SVR Models. *Sustainability* **2022**, *14*, 5238.
https://doi.org/10.3390/su14095238

**AMA Style**

Kaloop MR, Roy B, Chaurasia K, Kim S-M, Jang H-M, Hu J-W, Abdelwahed BS.
Shear Strength Estimation of Reinforced Concrete Deep Beams Using a Novel Hybrid Metaheuristic Optimized SVR Models. *Sustainability*. 2022; 14(9):5238.
https://doi.org/10.3390/su14095238

**Chicago/Turabian Style**

Kaloop, Mosbeh R., Bishwajit Roy, Kuldeep Chaurasia, Sean-Mi Kim, Hee-Myung Jang, Jong-Wan Hu, and Basem S. Abdelwahed.
2022. "Shear Strength Estimation of Reinforced Concrete Deep Beams Using a Novel Hybrid Metaheuristic Optimized SVR Models" *Sustainability* 14, no. 9: 5238.
https://doi.org/10.3390/su14095238