# Symbolic Regression Model for Predicting Compression Strength of Prismatic Masonry Columns Confined by FRP

^{*}

## Abstract

**:**

## 1. Introduction

^{2}equals to 0.91 which is superior compared to the existing models.

## 2. Analytical FRP Confined Masonry Models

^{3}, in the estimation of ${f}_{mc}$. In particular, the parameter approximately estimates the effect of porosity and voids of both the constituent materials as well as texture of masonry on the axial compressive strength.

## 3. Experimental Database

#### 3.1. Collected Database from Literature

#### 3.2. Processed Database

## 4. Symbolic Regression Model

## 5. Results and Discussion

#### 5.1. Complex Symbolic Regression Modelling

#### 5.2. Simplified Model

^{2}is 0.91 against 0.83. In addition, the simplified model is much simpler to inspect and implement, particularly for hand calculations. Statistical indictors for the mathematical expression obtained by simplified symbolic expression are presented in Table 6 for the whole set of data.

## 6. Comparison of Proposed Symbolic Regression Model with Existing Formulas

^{2}value, ARR, MAE, MSE and RMSE for each model along with the proposed model are calculated for obtaining the approach’s reliability (Table 7).

## 7. Conclusions

- The complex symbolic regression-based black-box model can be considered good with MAE and MSE for training data are 1.51 MPa and 3.48 MPa.
- The average ratio for the proposed model is 1.03 which is very good as only 3% variation is there for prediction compared to the experimental date.
- For the number of predictions less than 10% error output, it is evident that the proposed model is showing very excellent results compared to the other models. The number of predictions less than 10% error is 53 for the proposed model which is well above the immediate lower value of Di Ludovico et al. [20] for tuff as 24.
- The proposed simplified analytical model can predict the compressive strength of FRP-confined masonry prisms subjected to axial loading for any type of masonry better than available analytical models in the literature.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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References | Formulations |
---|---|

Krevaikas and Triantafillou [21] | ${f}_{mc}={f}_{m}\xb7\left(0.6+1.65\left(\frac{{f}_{l,eff}}{{f}_{m}}\right)\right)if\frac{{f}_{l,eff}}{{f}_{md}}\ge 0.24$ |

Corradi et al. [19] | ${f}_{mc}={f}_{m}+{f}_{l,eff}\times 2.4\times {\left(\frac{{f}_{l,eff}}{{f}_{m}}\right)}^{-0.17}$ |

Di Ludovico et al. [20] for Clay | ${f}_{mc}={f}_{m}+{f}_{l,eff}\times 1.53\times {\left(\frac{{f}_{l,eff}}{{f}_{m}}\right)}^{-0.10}$ |

Di Ludovico et al. [20] for Tuff | ${f}_{mc}={f}_{m}+{f}_{l,eff}\times 1.09\times {\left(\frac{{f}_{l,eff}}{{f}_{m}}\right)}^{-0.24}$ |

Faella et al. [14] (Simplest) | ${f}_{mc}={f}_{m}\xb7\left(1+\left(\frac{{g}_{m}}{1000}\right)\xb7{\left(\frac{{f}_{l,eff}}{{f}_{m}}\right)}^{0.662}\right)$ |

Faella et al. [14] (More accurate) | ${f}_{mc}={f}_{m}\xb7\left(1+0.416{\left(\frac{{g}_{m}}{1000}\right)}^{2.064}\xb7{\left(\frac{{f}_{l,eff}}{{f}_{m}}\right)}^{0.507}\right)$ |

CNR-DT 200 R1 [35] | ${f}_{mc}={f}_{m}\xb7\left(1+\left(\frac{{g}_{m}}{1000}\right)\xb7{\left(\frac{{f}_{l,eff}}{{f}_{m}}\right)}^{0.5}\right)$ ${\epsilon}_{fd,rid}=\mathrm{min}\left\{{n}_{a}\xb7\frac{{\epsilon}_{fk}}{{\gamma}_{f}};0.004\right\}$ |

Rao and Pavan [36] | ${f}_{mc}={f}_{m}\xb7\left(1+1.53\times {\left(\frac{{f}_{l,eff}}{{f}_{m}}\right)}^{0.92}\right)$ |

Ramaglia et al. [22] for Clay | ${f}_{mc}={f}_{m}\xb7\left(-0.57+1.57\sqrt{1+10.3\times \left(\frac{{f}_{l,eff}}{{f}_{m}}\right)}-2\times \left(\frac{{f}_{l,eff}}{{f}_{m}}\right)\right)$ |

Ramaglia et al. [22] for Tuff | ${f}_{mc}={f}_{m}\xb7\left(-15.25+16.25\sqrt{1+0.46\times \left(\frac{{f}_{l,eff}}{{f}_{m}}\right)}-2\times \left(\frac{{f}_{l,eff}}{{f}_{m}}\right)\right)$ |

Napoli and Realfonzo [37] (Simplest) | ${f}_{mc}={f}_{m}\xb7\left(1+1.10\times {\left(\frac{{f}_{l,eff}}{{f}_{m}}\right)}^{0.4}\right)$ |

Napoli and Realfonzo [37] (More accurate) | ${f}_{mc}={f}_{m}\xb7\left(1+{\left(\frac{{g}_{m}}{1000}\right)}^{0.15}\xb7{\left(\frac{{f}_{l,eff}}{{f}_{m}}\right)}^{0.5}\right)$ |

Parameters | Min | Max | Median | Average | Std. | 25th Percentile | 75th Percentile | Common Value | Num. Diff. Values |
---|---|---|---|---|---|---|---|---|---|

${\gamma}_{m}\left(\mathrm{kg}/{\mathrm{m}}^{3}\right)$ | 1250 | 2000 | 1750 | 1656.50 | 172.43 | 1565 | 1750 | 1750 | 10 |

$B\left(\mathrm{mm}\right)$ | 115 | 550 | 250 | 274.52 | 87.68 | 240 | 290 | 250 | 38 |

$H\left(\mathrm{mm}\right)$ | 115 | 560 | 250 | 264.79 | 91.31 | 240 | 288 | 250 | 36 |

$H/B$ | 1 | 2 | 1 | 1.06 | 0.21 | 1 | 1.01 | 1 | 7 |

$L\left(\mathrm{mm}\right)$ | 300 | 1760 | 500 | 559.07 | 251.32 | 485 | 511 | 500 | 29 |

$B/L$ | 0.28 | 0.83 | 0.49 | 0.49 | 0.15 | 0.34 | 0.5 | 0.5 | 24 |

${r}_{c}\left(\mathrm{mm}\right)$ | 0 | 85 | 20 | 20.77 | 13.03 | 10 | 25 | 20 | 10 |

${f}_{f,u}\left(\mathrm{MPa}\right)$ | 1371 | 4830 | 2560 | 2717.68 | 1019.51 | 1605 | 3500 | 1600 | 16 |

${E}_{f}\left(\mathrm{GPa}\right)$ | 65 | 673 | 143 | 163.08 | 106.96 | 70 | 230 | 230 | 13 |

${\epsilon}_{f,u}\left(\%\right)$ | 0.29 | 3.2 | 1.99 | 1.96 | 0.60 | 1.5 | 2.5 | 1.5 | 14 |

${T}_{f}\left(\mathrm{mm}\right)$ | 0.117 | 0.96 | 0.379 | 0.41 | 0.20 | 0.24 | 0.48 | 0.48 | 22 |

${f}_{m}\left(\mathrm{MPa}\right)$ | 2 | 14.33 | 7.04 | 7.79 | 3.69 | 5.36 | 11.91 | 7.85 | 33 |

${K}_{h}$ | 0.32 | 51 | 0.51 | 0.95 | 4.67 | 0.46 | 0.57 | 0.49 | 23 |

${K}_{v}$ | 0.23 | 1 | 1 | 0.91 | 0.17 | 0.89 | 1 | 1 | 13 |

${f}_{l;eff}/{f}_{m}$ | 0.04 | 1.57 | 0.45 | 0.54 | 0.38 | 0.23 | 0.76 | 0.23 | 55 |

${f}_{mc}\left(\mathrm{MPa}\right)$ | 2.79 | 44.87 | 12.03 | 13.87 | 8.15 | 8.5 | 18.42 | 5.1 | 107 |

Parameters | Variable Impacts |
---|---|

${f}_{m}\left(\mathrm{MPa}\right)$ | 0.915 |

${f}_{l;eff}/{f}_{m}$ | 0.272 |

$B/L$ | 0.103 |

${T}_{f}\left(\mathrm{mm}\right)$ | 0.035 |

$H/B$ | 0.026 |

${\epsilon}_{f,u}\left(\%\right)$ | 0.010 |

${K}_{h}$ | 0.007 |

Indictors | Training Data | Test Data |
---|---|---|

R^{2} | 0.92 | 0.80 |

ARE | 15.76% | 29.14% |

MAE | 1.51 | 4.27 |

MSE | 3.48 | 30.31 |

RMSE | 1.87 | 5.51 |

Parameters | Variable Impacts |
---|---|

${f}_{m}\left(\mathrm{MPa}\right)$ | 0.906 |

${T}_{f}\left(\mathrm{mm}\right)$ | 0.179 |

${f}_{l;eff}/{f}_{m}$ | 0.17 |

$H\left(\mathrm{mm}\right)$ | 0.145 |

Indictors | Value over Dataset |
---|---|

R^{2} | 0.91 |

ARE | 16.82% |

MAE | 1.86 |

MSE | 5.93 |

RMSE | 2.44 |

References | R^{2} | Average Relative Error (ARR) | Mean Absolute Error (MAE) | Mean Squared Error (MSE) | Root Mean Squared Error (RMSE) |
---|---|---|---|---|---|

Proposed formula | 0.91 | 0.16 | 1.86 | 5.94 | 2.44 |

Krevaikas and Triantafillou [21] | 0.69 | 0.31 | 4.22 | 31.30 | 5.60 |

Corradi et al. [19] | 0.75 | 0.40 | 4.09 | 29.35 | 5.42 |

Di Ludovico et al. [20] for Clay | 0.79 | 0.23 | 2.75 | 13.46 | 3.67 |

Di Ludovico et al. [20] for Tuff | 0.82 | 0.20 | 2.71 | 15.86 | 3.98 |

Faella et al. [14] (Simplest) | 0.81 | 0.25 | 2.77 | 13.81 | 3.72 |

Faella et al. [14] (accurate) | 0.82 | 0.20 | 2.47 | 12.30 | 3.51 |

CNR-DT 200 R1 [35] | 0.82 | 0.28 | 3.08 | 16.42 | 4.05 |

Rao and Pavan [36] | 0.79 | 0.23 | 2.77 | 13.64 | 3.69 |

Ramaglia et al. [22] for Clay | 0.79 | 0.32 | 3.42 | 19.29 | 4.39 |

Ramaglia et al. [22] for Tuff | 0.80 | 0.22 | 2.73 | 13.83 | 3.72 |

Napoli and Realfonzo [37] (Simplest) | 0.81 | 0.18 | 2.29 | 12.67 | 3.56 |

Napoli and Realfonzo [37] (accurate) | 0.82 | 0.18 | 2.33 | 13.19 | 3.63 |

References | Average (Prediction/Experimental) | COV (%) | Number of Predictions Less than 10% Error | Highest Prediction | Lowest Predictions |
---|---|---|---|---|---|

Proposed formula | 1.03 | 22.4 | 53 | 1.74 | 0.46 |

Krevaikas and Triantafillou [21] | 0.82 | 37.2 | 16 | 3.12 | 0.23 |

Corradi et al. [19] | 1.33 | 32.3 | 10 | 4.56 | 0.4 |

Di Ludovico et al. [20] for Clay | 1.05 | 28.2 | 14 | 3.36 | 0.32 |

Di Ludovico et al. [20] for Tuff | 0.95 | 24.5 | 24 | 2.69 | 0.3 |

Faella et al. [14] (Simplest) | 1.16 | 26.2 | 12 | 3.55 | 0.34 |

Faella et al. [14] (accurate) | 1.00 | 24.6 | 16 | 2.95 | 0.29 |

CNR-DT 200 R1 [35] | 1.23 | 24.6 | 9 | 3.46 | 0.37 |

Rao and Pavan [36] | 1.04 | 28.4 | 16 | 3.37 | 0.32 |

Ramaglia et al. [22] for Clay | 1.26 | 27.1 | 10 | 3.33 | 0.42 |

Ramaglia et al. [22] for Tuff | 1.01 | 27.1 | 17 | 3.1 | 0.31 |

Napoli and Realfonzo [37] (Simplest) | 1.05 | 23.3 | 12 | 2.59 | 0.35 |

Napoli and Realfonzo [37] (accurate) | 1.01 | 23.4 | 16 | 2.6 | 0.33 |

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**MDPI and ACS Style**

Alotaibi, K.S.; Islam, A.B.M.S.
Symbolic Regression Model for Predicting Compression Strength of Prismatic Masonry Columns Confined by FRP. *Buildings* **2023**, *13*, 509.
https://doi.org/10.3390/buildings13020509

**AMA Style**

Alotaibi KS, Islam ABMS.
Symbolic Regression Model for Predicting Compression Strength of Prismatic Masonry Columns Confined by FRP. *Buildings*. 2023; 13(2):509.
https://doi.org/10.3390/buildings13020509

**Chicago/Turabian Style**

Alotaibi, Khalid Saqer, and A. B. M. Saiful Islam.
2023. "Symbolic Regression Model for Predicting Compression Strength of Prismatic Masonry Columns Confined by FRP" *Buildings* 13, no. 2: 509.
https://doi.org/10.3390/buildings13020509