# Machine Learning Prediction Model for Shear Capacity of FRP-RC Slender and Deep Beams

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## Abstract

**:**

_{c}), section depth (d), section width (b), modular ratio (n), reinforcement ratio (ρ

_{f}), shear span-to-depth ratio (a/d). The proposed model for slender beams resulted in an average tested-to-predicted ratio of 0.98 and a standard deviation of 0.21, while the deep beams model resulted in an average tested-to-predicted ratio of 1.03 and a standard deviation of 0.29. For deep beams, the model provided superior accuracy over all models. However, this can be attributed to the fact that the investigated models were not intended for deep beams. The deep beams model provides a simple method compared to the strut-and-tie method.

## 1. Introduction

## 2. Existing Shear Strength Models

#### 2.1. Shear Model by ACI 440-15

_{c}) is as follows:

#### 2.2. Shear Model by ACI 440-22

#### 2.3. Shear Model by Canadian Standard Code CAN/CSA S806 (2012)

#### 2.4. Shear Model by JSCE-97

_{c}is calculated following the expression presented in Equation (15). It is worth noting that Equation (21) accounts for the size effect and the axial stiffness of FRP reinforcement using the ${\beta}_{d}$ and ${\beta}_{p}$factors.

#### 2.5. Shear Model by Nehdi et al. (2007)

#### 2.6. Shear Model by Hoult et al. (2008)

_{fl}A

_{fl}(Equation (25)). The proposed model was verified by a database of 398 specimens of FRP-RC beams tested under shear force.

#### 2.7. Shear Model by Kara (2011)

#### 2.8. Shear Model by Mari et al. (2013)

#### 2.9. Shear Model by Bažant and Yu (2005)

#### 2.10. Shear Model by ACI 318-19

_{c}can be taken as (Equation (34)). The size effect factor, λ

_{s}, is given by (Equation (35)), which identifies that the reduction in shear strength with depth begins at 250 mm.

#### 2.11. Shear Model by Kaszubska et al. (2018)

#### 2.12. Shear Model by Ebid and Deifalla (2021)

## 3. Surveyed Database

_{c}), FRP type, reinforcement ratio (ρ), and modulus of elasticity of FRP bars (E

_{f}). The range of the variables in the database is presented in Figure 1, and a summary of the collected database is listed in Table 1 and Table 2.

## 4. Performance of Shear Strength Models

_{c test}/V

_{c predicted}) for each of the models discussed earlier with respect to the shear-span ratio and the concrete compressive strength. The conservatism and scatteredness for each method are presented using the average model error (V

_{c test}/V

_{c predicted}) and the standard deviation of each method. Results for slender beams (a/d < 2.5) are presented in Figure 2 and Figure 3, while Figure 4 shows the results of deep beams (a/d > 2.5).

_{c test}/V

_{c predicted}) of 1.77, 1.69, and 1.83, respectively. Including the size effect in the ACI 440-22 and a minimum shear capacity has notably improved the prediction by reducing model error (V

_{c test}/V

_{c predicted}) and the standard deviation from 1.83 to 1.68 and from 0.37 to 0.34, respectively, highlighting the importance of including this effect. The ACI 440-22 changes improve the predictions for members with low reinforcement ratios.

_{c test}/V

_{c predicted}) in Figure 4 shows that all selected models fail to present the behavior of deep beams. The steep negative slope trendlines in Figure 4 demonstrate a high underestimation of the shear capacity in the short shear-span-to-depth ratio. Ebid’s (2021) model resulted in an average normalized shear capacity (V

_{c test}/V

_{c predicted}) and standard deviation of 5.44 and 3.17, respectively, which are the highest among all models. In contrast, Nehdi’s (2007) model resulted in the lowest average V

_{c test}/V

_{c predicted}and standard deviation of 1.66 and 0.90, respectively.

## 5. Gene Expression Programming (GEP)

## 6. Proposed Model

_{c}), section depth (d), section width (b), modular ratio (n), and reinforcement ratio (ρ

_{f}). In addition to the previous parameters, the deep beam model includes a shear span-to-depth ratio (a/d). The proposed model for slender beams resulted in an average tested-to-predicted ratio of 0.98 and a standard deviation of 0.21, while the deep beams model resulted in an average tested-to-predicted ratio of 1.03 and a higher standard deviation of 0.29 due to its complex behavior and the limited number of experimental data. To assess the accuracy of the proposed models and the consistency of the prediction with respect to other variables, the ratios of the test-to-predicted values were plotted for different variables, as shown in Figure 8 and Figure 9. The horizontal trendlines in Figure 8 and Figure 9 indicate consistent predictions over the entire range of the variables.

## 7. Conclusions

_{c test}/V

_{c predicted}) and the Standard deviation. The following conclusions can be drawn:

- For slender beams (a/d < 2.5), the ACI 318-19, Bažant and Yu, Kara, and Mari et al. models demonstrate less scatteredness compared to other models. Interestingly, the ACI 318 model resulted in high accuracy with an average V
_{c test}/V_{c predicted}= 0.92 and a standard deviation of 0.24, although it was developed for steel-RC beams. In contrast, the Ebid and Deifalla, and ACI 440 (2015) models provided the highest scatteredness and standard deviation. - Including the size effect in the ACI 440-22 (public draft) has notably improved the prediction by reducing model error (V
_{c test}/V_{c predicted}) and the standard deviation from 1.82 to 1.66 and from 0.37 to 0.34, respectively, highlighting the importance of including this effect. - Gene expression programming (GEP) has been utilized along with the compiled database to develop shear models for slender and deep beams. The proposed model yielded a superior accuracy over other models with an average V
_{c test}/V_{c predicted}= 0.98 and a standard deviation of 0.21 for slender beams. For deep beams, the proposed model resulted in an average V_{c test}/V_{c predicted}= 1.03 and a standard deviation of 0.29. The proposed models are a function of concrete compressive strength, reinforcement ratio, effective depth, modular ratio, and shear span-to-depth ratio. In addition to the accuracy of the GEP models, the proposed models are much simpler than some of the design models, such as the JSCE and CSA S806 models.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 2.**Experimental-to-prediction shear capacity of slender FRP-RC beams with respect to shear-span to depth ratio a/d.

**Figure 3.**Experimental-to-prediction shear capacity of slender FRP-RC beams with respect to the concrete compressive strength f′

_{c}.

**Figure 4.**Experimental-to-prediction shear capacity of deep FRP-RC beams with respect to shear-span to depth ratio a/d.

**Figure 8.**Prediction accuracy (V

_{c test}/V

_{c predicted}) of the proposed model for slender beams with respect to (

**a**) effective depth; (

**b**) concrete compressive strength; (

**c**) reinforcement ratio; (

**d**) modular ratio.

**Figure 9.**Prediction accuracy (V

_{c test}/V

_{c predicted}) of the proposed model for deep beams with respect to: (

**a**) effective depth; (

**b**) concrete compressive strength; (

**c**) reinforcement ratio; (

**d**) effective depth-to-shear-span ratio.

**Figure 10.**Bar chart comparison between the existing and proposed model for slender beams based on (

**a**) Average (V

_{c test}/V

_{c predicted}) (

**b**) Standard deviation.

**Figure 11.**Bar chart comparison between the existing and proposed model for deep beams based on (

**a**) Average (V

_{c test}/V

_{c predicted}) (

**b**) Standard deviation.

Reference | N | b | a/d | f′_{c} | ${\mathit{\rho}}_{\mathit{f}}\%$ | ${\mathit{E}}_{\mathit{f}\mathit{u}}$ |
---|---|---|---|---|---|---|

Razaqpur, A. G. et al. (2004) [31] | 6 | 200 | (2.67–4.5) | (40.5–49.5) | (0.25–0.88) | 145 |

El-sayed et al. (2005) [32] | 8 | 1000 | ((6–6.4)) | 40 | (0.39–2.44) | (40–114) |

Issa et al. (2016) [33] | 7 | 300 | (5.65–7) | 35.9 | (0.68–3.4) | (48–53) |

Tureyen and Frosch (2002) [18] | 6 | 457 | 3.4 | 34.5 | (0.96–1.92) | (37.5–47.1) |

El Refai and Abed (2016) [34] | 5 | 152 | 3.3 | 49 | (0.31–1.52) | 50 |

Tomlinson and Fam (2015) [35] | 3 | 150 | 4 | (56.5–60) | (0.39–0.81) | 70 |

Khaja and Sherwood (2013) [36] | 11 | 400 | (3–8) | (47.8–51.4) | (0.57–4.1) | (47.5–51.9) |

Abdul-Salam (2014) [37] | 18 | 100 | (5.92–6.18) | (47.9–86.2) | (0.45–1.42) | (69.5–144) |

Razaqpur et al. (2011) [38] | 6 | 300 | (3.5–6.5) | 52.3 | (0.28–0.35) | 114 |

Chang and Seo (2012) [39] | 6 | 1200 | 5.8 | 30 | (0.73–1.22) | (44–50) |

Kaszubska et al. (2018) [29] | 7 | 150 | (2.9–3) | (28.8–31.7) | (0.99–1.85) | (50.2–50.9) |

Ashour and Kara (2014) [40] | 6 | 200 | (2.7–5.9) | (21.6–28) | (0.12–0.51) | (32–38) |

Ashour (2006) [41] | 6 | 150 | (2.7–3.7) | (34–59) | (0.45–1.15) | 32 |

Tariq and Newhook (2003) [42] | 12 | (130–160) | (2.7–3.3) | (34.7–43.2) | (0.72–1.54) | (42–120) |

Kim and Jang (2014) [43] | 24 | 150 | (3–4.5) | 30 | (0.31–0.71) | (48.2–146.2) |

El-sayed A et al. (2009) [44] | 2 | 600 | 6.68 | 68 | (0.53–0.77) | 48 |

Zhao et al. (1995) [45] | 3 | 150 | 3 | 34 | (1.51–3.02) | 105 |

Alkhrdaji et al. (2001) [46] | 3 | 178 | (2.61–2.69) | 24 | (0.77–2.3) | 40 |

Deitz et al. (1999) [47] | 3 | 305 | (4.5–5.8) | (27–30) | 0.73 | 40 |

Duranovic et al. (1997) [48] | 2 | 150 | 3.65 | (33–38) | (1.31–1.36) | 45 |

Matta et al. (2013) [49] | 9 | (114–229) | 3.1 | (32.1–59.7) | (0.13–0.28) | (43.2–48.2) |

Joseph R et al. (2001) [50] | 18 | (178–279) | 4.06 | 36.3 | (0.86–1.75) | 40.336 |

Alam (2010) [51] | 2 | 250 | 3.5 | (34.5–39.8) | (0.42–0.84) | (48–120) |

Olivito and Zuccarello (2010) [52] | 20 | 150 | 5.56 | (30–40) | (0.786–1.3) | 115 |

Bentz et al. (2010) [53] | 4 | 450 | (3.48–3.05) | 35 | (0.48–1.91) | 37 |

Gross et al. (2003) [54] | 12 | (152–203) | 4.06 | 79.6 | (1.25–2.1) | 40.3 |

Ali et al. (2014) [55] | 4 | 130 | 3 | (31–33.5) | (0.6–0.91) | 51.5 |

Guadagnini et al. (2006) [3] | 1 | 150 | 3.3 | 40 | 1.1 | 45 |

Nakamura and Higai (1995) [56] | 2 | 300 | 4 | (23–28) | (1.34–1.79) | 29 |

Swamy (1997) [57] | 1 | 155 | 3.15 | 39 | 1.55 | 34 |

Liu (2011) [58] | 20 | (635–1854) | 6.04 | (65–87) | (0.54–0.94) | 43.3 |

Farahmand (1996) [59] | 6 | 200 | (3,4) | (31–35) | (0.51 1.08) | 41.3 |

Gross et al. (2004) [60] | 4 | (89–159) | (6.35–6.45) | (60–81) | (0.33–0.76) | 139 |

Maruyama and Zhao (1994) [61] | 4 | 150 | 3 | (28–35) | (0.55–2.2) | 94 |

Caporale and Luciano (2009) [62] | 4 | 150 | 4.12 | (24–31) | (0.92–1.54) | 45.8 |

Kilpatrick and Easden (2005) [63] | 12 | 420 | (3.61–6.41) | (61–93) | (0.61–2.61) | (40–42) |

Kilpatrick and Dawborn (2006) [64] | 9 | 420 | (6–6.16) | (48–92) | (0.68–1.16) | 42 |

Zeidan et al. (2011) [65] | 1 | 150 | 5 | 49 | 0.105 | 148 |

Reference | N | B | a/d | f′_{c} | ${\mathit{\rho}}_{\mathit{f}}\mathit{\%}$ | ${\mathit{E}}_{\mathit{f}\mathit{u}}$ |
---|---|---|---|---|---|---|

Matthias F and Lubell. (2013) [66] | 8 | (300–310) | (1.1–2.06) | (39.9–68.5) | (1.47–2.13) | (64.1–72) |

Abed et al. (2012) [67] | 9 | 200 | (1–1.52) | (43–65) | (0.92–1.84) | 51 |

Thomas and S. Ramadass (2016) [68] | 8 | (100–170) | (0.5–1.75) | (40.6–59.5) | (1.16–1.75) | 40 |

Zeidan et al. (2011) [65] | 3 | 150 | 2.5 | (24–46) | (0.105–0.21) | 148 |

Omeman et al. (2008) [69] | 8 | (150–350) | (1.36–1.86) | (35–60) | (1.13–2.26) | 134 |

Kim and Jang (2014) [43] | 29 | (150–200) | (1.5–2.5) | 30 | (0.31–0.71) | (48.2–146.2) |

Alam (2010) [51] | 24 | (250–300) | (1.5–2.5) | (34.3–88.3) | (0.18–1.51) | (48–120) |

Ali et al. (2014) [55] | 4 | 130 | 2.3 | (13–33.5) | (0.6–0.91) | 51.5 |

Razaqpur, A. G. et al. (2004) [31] | 1 | 200 | 1.82 | 40.5 | 0.5 | 145 |

El Refai and Abed (2016) [34] | 3 | 152 | 2.5 | 49 | (0.31–0.69) | 50 |

Khaja and Sherwood (2013) [36] | 3 | 400 | 2 | (50.8–51.6) | (0.57–2.28) | 47.5 |

Parameter | Selected Value |
---|---|

Dependent variable (shear stress) | 1 |

Independent variables | 5,6 |

Genes | 3 |

Function set | $-,+,\times ,\xf7,\sqrt{},\sqrt[3]{},power,$ |

Head size | 6,8 |

Linking function between ETs | Multiplication |

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**MDPI and ACS Style**

Tarawneh, A.; Alghossoon, A.; Saleh, E.; Almasabha, G.; Murad, Y.; Abu-Rayyan, M.; Aldiabat, A.
Machine Learning Prediction Model for Shear Capacity of FRP-RC Slender and Deep Beams. *Sustainability* **2022**, *14*, 15609.
https://doi.org/10.3390/su142315609

**AMA Style**

Tarawneh A, Alghossoon A, Saleh E, Almasabha G, Murad Y, Abu-Rayyan M, Aldiabat A.
Machine Learning Prediction Model for Shear Capacity of FRP-RC Slender and Deep Beams. *Sustainability*. 2022; 14(23):15609.
https://doi.org/10.3390/su142315609

**Chicago/Turabian Style**

Tarawneh, Ahmad, Abdullah Alghossoon, Eman Saleh, Ghassan Almasabha, Yasmin Murad, Mahmoud Abu-Rayyan, and Ahmad Aldiabat.
2022. "Machine Learning Prediction Model for Shear Capacity of FRP-RC Slender and Deep Beams" *Sustainability* 14, no. 23: 15609.
https://doi.org/10.3390/su142315609