# Gene Expression Programming for Estimating Shear Strength of RC Squat Wall

^{1}

^{2}

^{3}

^{4}

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## Abstract

**:**

## 1. Introduction

## 2. Research Significance

## 3. Effect of Barbells and Flanges on the Ultimate Shear Strength

## 4. Factors Affecting Shear Strength

#### 4.1. Aspect Ratio

#### 4.2. Horizontal and Vertical Web Reinforcement

#### 4.3. Axial Load

## 5. Previously Proposed Models

## 6. Experimental Database

_{w}), web length (l

_{w}), web thickness (t

_{w}), flange elements length (l

_{f}), flange thickness (t

_{f}), aspect ratio (h

_{w}/l

_{w}), compressive strength of concrete (${{f}_{c}}^{\prime}$), yield strength of the horizontal steel (f

_{yh}), and yield strength of vertical steel (f

_{yv}).

_{w}/l

_{w}) of all the selected walls is equal to or less than 2.0. The figure shows that concrete compressive strength (${{f}_{c}}^{\prime}$) varies from 10 to 137 MPa, while the horizontal bars yield strength (f

_{yh}) and the vertical bars yield strength (f

_{yv}) ranges from 224 to 1420 MPa. Similarly, the horizontal bars reinforcement ratio (ρ

_{h}) and the vertical bars reinforcement ratio (ρ

_{v}) vary between 0.01% and 2.98%, while the flange element’s longitudinal reinforcement ratios (ρ

_{f}) vary between 0.074% and 9.57%. Also considered in the database is the axial load (P) which varies from 0 to 2770 kN.

## 7. Gene Expression Programming Algorithm

## 8. Proposed GEP Model for Estimating Joint Shear

_{h}and ρ

_{v}represent the horizontal and vertical reinforcement ratios, respectively.

## 9. Statistical Performance Criteria

## 10. Results and Discussion

#### 10.1. Parametric Study

- This section describes the parametric study for identifying the influence of input variables on the shear strength capacity of squat walls. Therefore, a variable of interest is perturbed whilst other variables are locked at the mean values, considering the other variables are uncorrelated.
- If the wall height (i.e., h
_{w}) is incrementally increased, the shear strength of the squat flanged RC wall is expected to decrease, Figure 6a. This is because the lateral stiffness of the wall reduces with the increase in wall height. - The web dimensions, including the length and thickness, are one of the significant parameters controlling the shear capacity of RC walls. As shown in Figure 6b,c, the trend for shear strength is upward which is associated with an increase in shear area.
- Similar to web dimensions, Figure 6c shows that the thickness and the length of flanges vary sharply with respect to the shear strength of the wall. So, it can be argued that an increase in flange dimensions significantly improves the shear capacity of the squat flanged RC wall.
- It can be seen from Figure 6d,e that the shear strength rises with the increase in flange element thickness (t
_{f}) and flange element length (l_{f}). - It can be observed in Figure 6g,i,j that the shear strength increases sharply with the increase in reinforcements.
- Figure 6f,l shows the compressive strength of concrete, and the axial load has a marginal influence on the shear capacity of the walls.

#### 10.2. Performance of GEP Models

^{2}= 0.95, which outclass all the models as shown in Figure 7. Moreover, the performance factor and the average absolute error of the proposed model are statistically better than the previously proposed model. Among the previously proposed model, the least predictive ability is obtained from the model proposed by Eurocode and Ma et al. [15]. The main reason for its low performance is that the Eurocodes [10] neglect the influence of flanges. Similarly, Figure 8 shows statistical performance checks of various models. A clear satistical excellence of the proposed model is observed having a performance factor of 0.9, coefficient of determination of 0.25, and average absolute error of 1.1.

## 11. Conclusions

- An increase in the wall web and flange dimensions, and the reinforcement ratio increase the shear capacity of the wall. Whereas, an increase in the wall height reduces the wall shear capacity. Parameters such as concrete compressive strength and axial load moderately influence the wall capacity.
- The developed model successfully incorporates all the aforementioned key parameters and, given the experimental data, these parameters show a high contribution to the development of the model.
- From statistical investigation, a high correlation (R
^{2}) of 0.96 is observed. Likewise, the performance factor of 1.1 and average absolute error of 23%, all substantiate the higher accuracy of the proposed. - An important benefit of the proposed model is that a single model can effectively approximate the shear capacity of a barbell, flanged or rectangular wall.
- Finally, it is asserted that the proposed model offers a better prediction than the existing models of shear strength of flanged or barbed squat walls, leading toward the development of design guidelines for squat walls.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Ranges of dataset parameters. (

**a**) Height of wall (h

_{w}), (

**b**) Web length (l

_{w}), (

**c**)Web thickness (t

_{w}), (

**d**) Flange thickness (t

_{f}), (

**e**)Flange elements length (l

_{f}), (

**f**) Compressive strength of concrete (f

_{c}′), (

**g**) Longitudinal reinforcement ratios of the flanged element (ρ

_{f}), (

**h**) Yield strength of longitudinal reinforcement (f

_{yf}), (

**i**) Reinforcement ratio of wall in the horizontal direction (ρ

_{h}), (

**j**) Reinforcement ratios of wall in vertical direction (ρ

_{v}), (

**k**) Yield strength of the horizontal bars (f

_{yh}), (

**l**) Yield strength of vertical bars (f

_{yv}), (

**m**) Axial load in wall (P).

**Figure 4.**Comparison of estimated and experimental shear strength of flanged and barbell squat wall (

**a**) Training Data (

**b**) Validation Data (

**c**) All Data.

**Figure 5.**The predictive performance of the proposed shear strength model based on the key parameters. (

**a**) Height of wall hw, (

**b**) Web length (l

_{w}), (

**c**)Web thickness (t

_{w}), (

**d**) Flange thickness (t

_{f}), (

**e**) Flange elements length (l

_{f}), (

**f**) Compressive strength of concrete (f′

_{c}), (

**g**) Yield strength of flenged reinforcement (f

_{yf}), (

**h**) Reinforcement ratios of flenge (ρ

_{f}), (

**i**) Reinforcement ratio of wall in the horizontal direction (ρ

_{h}), (

**j**) Reinforcement ratios of wall in vertical direction (ρ

_{v}), (

**k**) Yield strength of the horizontal bars (f

_{yh}), (

**l**) Yield strength of vertical bars (f

_{yv}), (

**m**) Axial load in wall (P).

**Figure 6.**Sensitivity of each influencing parameter on the shear strength of RC flanged and barbell squat wall. (

**a**) Height of wall (h

_{w}), (

**b**) Web length (

_{lw}), (

**c**) Web thickness (t

_{w}), (

**d**) Flange thickness (t

_{f}), (

**e**) Flange elements length (l

_{f}), (

**f**) Compressive strength of concrete (f

_{c}), (

**g**) Reinforcement ratios of flenge (ρ

_{f}), (

**h**) Yield strength of flenged reinforcement (f

_{yf}), (

**i**) Reinforcement ratio of wall in the horizontal direction (ρ

_{h}), (

**j**) Reinforcement ratios of wall in vertical direction (ρ

_{v}), (

**k**) Yield strength of vertical bars (f

_{yv}), (

**l**) Axial load in wall (P).

No. | Reference | Proposed Formula | Comments |
---|---|---|---|

1 | ACI 318-14 [8] | ${V}_{u}={A}_{w}\left({\alpha}_{c}\sqrt{{f}_{c}^{\prime}}+{\rho}_{h}{f}_{yh}\right)\le 0.83{A}_{w}\surd {f}_{c}^{\prime}$ | Can be used for rectangular and flanged RC walls |

2 | ASCE 43-05 [9] | ${V}_{u}=\left(8.3\sqrt{{f}_{c}^{\prime}}-3.4\sqrt{{f}_{c}\left(\frac{{H}_{w}}{{L}_{w}}-0.5\right)}+\frac{N}{4{L}_{w}{b}_{w}}+{\rho}_{se}{f}_{yw}\right)d{b}_{w}\le \phantom{\rule{0ex}{0ex}}20\sqrt{{f}_{c}^{\prime}}d{b}_{w},{\rho}_{ze}=A{\rho}_{v}+B{\rho}_{h}$ | Can be used for rectangular and flanged RC walls |

3 | EN 1998-1 [10] | ${V}_{Rs}={b}_{w}z{\rho}_{h}{f}_{yh}cot\theta ,{V}_{R,max}=\frac{{\alpha}_{cw}{b}_{w}z{f}_{c}^{\prime}v}{\left(cot\theta +tan\theta \right)}$ | Can be used for rectangular and flanged RC walls |

4 | Gulec and Whittaker [5] | ${V}_{u}=\frac{1.5\sqrt{{f}_{c}^{\prime}}{A}_{w}+0.25{F}_{v}+0.20{F}_{be}+0.40N}{\sqrt{\frac{{H}_{w}}{{L}_{w}}}}\le 10\sqrt{{f}_{c}^{\prime}}{A}_{w},For1.0\le \frac{{A}_{w,tot}}{{A}_{w}}\ge 1.25,$ ${V}_{u}=\frac{0.04{f}_{c}^{\prime}{A}_{w,tol}+0.40{F}_{v}+0.15{F}_{be}+0.35N}{\sqrt{\frac{{H}_{w}}{{L}_{w}}}}\le 15\sqrt{{f}_{c}^{\prime}}{A}_{w,tol},For\frac{{A}_{w,tol}}{{A}_{w}}\ge 1.25$ | Can be used for rectangular and flanged RC walls |

5 | Ma et al. [15] | ${V}_{u}=\frac{\left(0.32{f}_{yf}{\rho}_{f}{t}_{f}{l}_{f}+0.18{f}_{yv}{\rho}_{v}{t}_{w}{z}_{w}+\frac{p}{2}\right){d}_{w}}{{h}_{w}}+0.54{f}_{yh}{\rho}_{h}{t}_{w}{h}_{w}\le \phantom{\rule{0ex}{0ex}}1.44{A}_{t}\sqrt{{f}_{c}^{\prime}},{d}_{w}={l}_{w}-{t}_{f}-0.5\left(\frac{0.32{f}_{yf}{\rho}_{f}{t}_{f}{l}_{f}+P}{0.59{f}_{c}^{\prime}{t}_{w}}-\frac{{t}_{f}{l}_{f}}{{t}_{w}}\right)$ | Can be used for flanged RC walls |

6 | Wood et al. [3] | $0.5{A}_{cv}\sqrt{{f}_{c}^{\prime}}\le {V}_{u}=\frac{{A}_{vf}{f}_{y}}{4}\le 0.83{A}_{cv}\sqrt{{f}_{c}^{\prime}}$ | Can be used for flanged RC walls |

7 | Barda et al. [2] | ${V}_{u}=\left(0.67\sqrt{{f}_{c}^{\prime}}-\frac{0.21\sqrt{{f}_{c}^{\prime}}{h}_{w}}{{l}_{w}}+\frac{P}{4{l}_{w}{t}_{w}}+{\rho}_{v}{f}_{y}\right){t}_{w}d\left(d=0.6{l}_{w}\right)$ | Can be used for flanged RC walls |

8 | Adorno-Bonilla [7] | ${V}_{u}=(0.54+0.19{f}_{c}^{\prime}-\frac{0.17{f}_{c}^{\prime}{h}_{w}}{{l}_{w}}+\frac{0.45P}{{A}_{g}}+0.39{\rho}_{se}{f}_{yse}+\phantom{\rule{0ex}{0ex}}0.31{\rho}_{be}{f}_{ybe}){A}_{cv},{f}_{yse}=A{f}_{yv}+B{f}_{yh},{\rho}_{se}=A{\rho}_{v}+B{\rho}_{h},{\rho}_{be}=\phantom{\rule{0ex}{0ex}}\frac{{A}_{sbe}}{{A}_{cv}},if\frac{{h}_{w}}{{l}_{w}}\le 0.5A=1andB=0,if0.5\frac{{h}_{w}}{{l}_{w}}1.5A=-\frac{{h}_{w}}{{l}_{w}}+\phantom{\rule{0ex}{0ex}}1.5andB=\frac{{h}_{w}}{{l}_{w}}-0.5,if\frac{{h}_{w}}{{l}_{w}}\ge 1.5A=0andB=1$ | Can be used for flanged RC walls |

9 | Kassem 2015 [6] | ${V}_{u}=0.47{f}_{c}^{\prime}\left[\psi {k}_{s}\mathrm{sin}\left(2\alpha \right)+\frac{0.15{\omega}_{h}{H}_{w}}{{d}_{w}}+1.76{\omega}_{v}cot\alpha \right]{A}_{t}\le \phantom{\rule{0ex}{0ex}}1.25\sqrt{{f}_{c}^{\prime}}{A}_{t},\psi =0.95-\frac{{f}_{c}^{\prime}}{250},{k}_{s}=\frac{{a}_{s}}{{d}_{w}},{a}_{s}=\left(0.25+\frac{0.85N}{{A}_{w}{f}_{c}^{\prime}}\right){L}_{w},{\omega}_{h}=\phantom{\rule{0ex}{0ex}}\frac{{\rho}_{h}{f}_{yh}}{{f}_{c}^{\prime}},{\omega}_{v}=\frac{{\rho}_{v}{f}_{yv}}{{f}_{c}^{\prime}},\alpha ={\mathrm{tan}}^{-1}\left(\frac{{H}_{w}}{{d}_{w}}\right),{d}_{w}=d-\frac{{a}_{s}}{3}$ |

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

General Setting | |

Chromosomes | 130 |

Head sizes | 16 |

Genes | 3 |

Linking function | Multiplication |

Function set | Abs, Exp, Ln, +, −, /, ×, sqrt |

Numerical Setting | |

Data type | Floating numbers |

Constants per gene | 10 |

Lower/Upper bound of constants | −20/20 |

Genetic Operators | |

Mutation rate | 0.0014 |

Inversion rate | 0.1 |

One-point recombination rate | 0.0027 |

Two-point recombination rate | 0.0027 |

Gene recombination rate | 0.0027 |

Gene transposition rate | 0.0027 |

Author | $\mathit{P}\mathit{F}=\frac{{\mathit{v}}_{\mathit{j}\mathit{h}}^{\mathit{E}\mathit{x}\mathit{p}}}{{\mathit{v}}_{\mathit{j}\mathit{h}}^{\mathit{E}\mathit{s}\mathit{t}}}$ Mean | AAE (%) | ${\mathit{R}}^{2}$ |
---|---|---|---|

ACI 318M-14 [8] | 0.78 | 33.00 | 0.89 |

ASCE 43-05 [9] | 1.18 | 37.00 | 0.87 |

EN 1998-1 [10] | 1.25 | 45.00 | 0.78 |

Gulec &Whittaker [5] | 0.92 | 25.00 | 0.89 |

Ma, et al. [15] | 0.83 | 33.00 | 0.78 |

Wood, et al. [3] | 0.65 | 39.00 | 0.85 |

Barda, et al. [2] | 1.74 | 33.00 | 0.84 |

Adorno-Bonilla [7] | 1.37 | 48.00 | 0.84 |

Kaseem [6] | 1.57 | 61.00 | 0.88 |

Proposed Model | 1.10 | 23.00 | 0.96 |

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**MDPI and ACS Style**

Tariq, M.; Khan, A.; Ullah, A.; Zamin, B.; Kashyzadeh, K.R.; Ahmad, M.
Gene Expression Programming for Estimating Shear Strength of RC Squat Wall. *Buildings* **2022**, *12*, 918.
https://doi.org/10.3390/buildings12070918

**AMA Style**

Tariq M, Khan A, Ullah A, Zamin B, Kashyzadeh KR, Ahmad M.
Gene Expression Programming for Estimating Shear Strength of RC Squat Wall. *Buildings*. 2022; 12(7):918.
https://doi.org/10.3390/buildings12070918

**Chicago/Turabian Style**

Tariq, Moiz, Azam Khan, Asad Ullah, Bakht Zamin, Kazem Reza Kashyzadeh, and Mahmood Ahmad.
2022. "Gene Expression Programming for Estimating Shear Strength of RC Squat Wall" *Buildings* 12, no. 7: 918.
https://doi.org/10.3390/buildings12070918