# Forecasting Strength of CFRP Confined Concrete Using Multi Expression Programming

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## Abstract

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## 1. Introduction

## 2. Research Methodology

#### 2.1. Multi Expression Programming

#### 2.2. Experimental Database

_{co}; E

_{FRP}}

_{cc}}

_{cc}is the confined compressive strength of CFRP. d; h; nt; f′

_{co}; and E

_{FRP}are the respective section diameter, the corresponding height of specimen, the CFRP layers thickness, unconfined concrete strength, and finally, the elastic modulus of fibers, appear to be potentially effective parameters in predicting the ultimate load values and thus be utilized as the input parameters to establish the model. Moreover,

**ε**and

_{co}**ε**are the corresponding strain values of unconfined concrete and CFRP confined concrete of respective specimens.

_{cc}#### 2.3. Modeling Parameters

- To exist a correlation between the observed and expected values |R| needs to be between 0.2 < |R| < 0.8.
- If |R| evaluated to be < 0.2, that depicts a weak correlation among the actual and predicted values.
- |R| has to be larger than 0.8 to maintain a strong correlation between expected and actual values.

**ρ**and OF, lie between 0 and infinity. However, for the reputation of a good model,

**ρ**and OF must be less than 0.2 [92]. The parameter OF has significant importance as it considers the effect of three main statistical parameters involved in training and testing datasets, i.e., RRMSE, R, and relative percentage.

## 3. Results and Discussion

#### 3.1. Mechanical Properties and Formulation

_{cc}for all data sets: train, validate, and test phase. Furthermore, the slope of the best fit line for all three data sets and the slope for an ideal fit scenario are displayed in the graph. The slope of the best fit line should pass through the origin and approach one for a perfect fit. Figure 3 shows a significant correlation between actual and projected results for all datasets in the created model. The corresponding slopes for train, validate, and test phases are evaluated as 0.9299, 0.9357, and 0.9517, respectively. The results are quite identical and closer to a good fit throughout all sets. This fitting indicates that the model has been efficiently developed and thus has a strong generalization ability, as it behaves well enough on unknown data when forecasting output. The generalization of the established model suggests that the problem of model over-fit has been minimized and reduced on a broad scale. It is also worth noting that the quantity of data points needed for forecasting is eminently reliant on the efficiency and applicability of produced models [103,104,105]. So far, 828 data points have been added into the assembled database for forecasting f′

_{cc}, which is another fascinating part of this work; as a result, high precision with few discrepancies has been obtained.

#### 3.2. Model Performance and Evaluation:

_{cc}model has a ratio of 116. While in the testing stage, the f′cc model has a ratio of 24.8. As previously stated, numerous statistical measures assess the efficiency of the developed models such as R, RMSE, RSE, MAE, ρ, RRMSE, and OF. The values of various statistical measures or Indices for training, validation, and testing sets are demonstrated in Table 6 for the generated f′

_{cc}model.

_{cc}as outstanding. Thus computed RRMSE indices for each set are less than 0.10, i.e., 0.0045, 0.0098, and 0.0097, respectively.

**ρ**value remains less than 0.20 for all three sets, the model will be inferred as reliable and proficient for forecasting output. In addition to these statistical measures, another indicator OF was incorporated in this study to counter the overfitting problems. Overfitting not only alters the results but is also responsible for forecasting inaccurate outputs. However, the OF computed for a developed model is 0.009. This value is exceptionally close to zero, indicating the validity and overall performance of the model and overrule the issues of overfitting by addressing it satisfactorily. The statistics of absolute errors along with respective data points are plotted in Figure 4.

_{cc}is 6.37 Mpa, with a maximum error recorded as 16.48 MPa. As quoted initially, the database employed in this study contains 828 points. Among such a huge database, it is worth noting that only 12% of instances have an error greater than 8%. However, the maximum error density obtained based on these data points is not considered high. It is pertinent to mention that approximately 88% of the predictions obtained have errors computed less than 8% for the f′

_{cc}model.

_{m}) to analyze a model’s external reliability. The R

_{m}value must be greater than 0.5 to meet the criterion. However, it is shown in Table 7 that the proposed models meet the external validation criteria, demonstrating that they are credible, resilient, and not just another simple correlation of input and output variables.

#### 3.3. Parametric Analysis

- (i)
- Material properties such as modulus of elasticity (E) of FRP, unconfined concrete (f′
_{co}) strength; - (ii)
- Geometric properties such as thickness (nt) of FRP composites, and cylinder diameter (d).

_{cc}for each input parameter are shown in Figure 6. These observations are aligned with the experimental study conducted in the past.

#### 3.3.1. Effect of Diameter (d)

_{cc}due to a change in concrete diameter (d) is shown in Figure 6a. When the diameter value ‘d’ increases from 50 mm, the f′

_{cc}for concrete increases at first, then continues to decline up to 400 mm, leaving all other parameters constant. When d increases from the initial to the final value, an overall decline of approximately 44% was recorded in f′

_{cc}value. In addition, it can be inferred from the graph that there is a gradual decrement observed in the value of f′

_{cc}from 118.528 Mpa to 66.977 Mpa up to a final value of d. Therefore, it is convenient to envision that the effect on f′

_{cc}is significant at smaller diameters.

#### 3.3.2. Effect of Thickness of FRP Layers (nt)

_{cc}was outstandingly 147 percent, when nt was increased from 0.15 mm to 1 mm, demonstrating that the impact of raising nt is more considerable at lower levels. Moreover, for values of nt beyond 1 mm to 5.9 mm, there is no significant increment observed in f′

_{cc}, i.e., 120.40 Mpa to 140.84 Mpa accounting for a 17% increase in f′

_{cc}. It can also be concluded that for higher values of f′

_{cc}the effect of raising the thickness of the wraps appears to be less significant. This trend is consistent with prior studies conducted on CFRP confined concrete, as the increase observed in strength between min and max thickness is 188%. However, confined concrete strength (f′

_{cc}) generally has a linear relation with the unconfined concrete strength f′

_{co}, thus increasing proportionally [107,108]. Apart from this, increasing the thickness of FRP wraps (nt) has the same effect as observed in the current study.

#### 3.3.3. Effect of Elastic Modulus of FRP (E_{f})

_{f}. The E

_{f}values vary from 10 GPa to 663 GPa, with each increment of 50 GPa. When the parameter E

_{f}value exceeds from 10 GPa up to 390 GPa, the f′

_{cc}increases by 143 percent. Similarly, when E

_{f}increases beyond 390 Gpa, a slight increase in f′

_{cc}can be observed up to 663 Gpa. However, it should be noted that with the increment in E

_{f}from its initial to the final value, an overall improvement in f′

_{cc}can be observed. Furthermore, it may be stated that an increase in E

_{f}has a considerable influence on f′

_{cc}.

#### 3.3.4. Effect of Concrete Compressive Strength (f′_{co})

_{co}ranging from 6 MPa to 190 MPa with a 25 MPa increase. Figure 6d shows the variable effect on f′

_{co.}When f′

_{co}is increased from minimum to maximum while all other parameters remain constant, the forecasted model’s concrete strength rises by 271 percent. In general, the confined concrete strength f′cc increases linearly by increasing the unconfined concrete strength (f′

_{co}).

_{cc}grows as f′

_{co}increases, as plotted in Figure 6d. This pattern is also consistent with current design trends. From the foregoing discussion, it can be stated that the constructed MEP model successfully captured the complicated behavior of confined concrete strength and can thus be used widely for future prediction.

## 4. Conclusions

**ρ**and OF indicators which confirms that the problem of overfitting has been managed effectively. The parameter R is exceptionally in the range between (0.9948–0.9953), which depicts the firm relationship among the forecasted and experimental findings for all the datasets incorporated in the study. Higher R and lower MSE, on the other hand, imply that a high degree of prediction was anticipated for all sets, demonstrating the universality of the suggested model.

_{cc}/f′

_{co}) published and recommended in the literature by various scholars. Thus, the proposed strength model outperformed all existing models by minimum error projection. The proposed study evaluates the compressive strength of CFRP confined concrete very well by utilizing the developed model. Thus, the corresponding developed empirical relationships have the ability to forecast confined concrete behavior efficiently, which would be useful for analyzing and designing various composite concrete structural members.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 5.**Average absolute error (AEE) of strength enhancement ratio (f′

_{cc}/f′

_{co}) in predicted model.

**Figure 6.**Variations in presented strength model using Input parameters: (

**a**) d, (

**b**) nt, (

**c**) E

_{f}, (

**d**) f′

_{co}.

Researcher | Year | Developed Model |
---|---|---|

Richart et al. [58] | 1928 | $\frac{{f}_{cc}^{\prime}}{{f}_{co}^{\prime}}=1+4.1\frac{{\mathrm{f}}_{\mathrm{l}}}{{f}_{co}^{\prime}}$ |

Newman and Newman [57] | 1969 | $\frac{{f}_{cc}^{\prime}}{{f}_{co}^{\prime}}=1+3.7{(\frac{{\mathrm{f}}_{\mathrm{l}}}{{f}_{co}^{\prime}})}^{2}$ |

Fardis and Khalili [56] | 1982 | $\frac{{f}_{cc}^{\prime}}{{f}_{co}^{\prime}}=1+3.3{(\frac{{\mathrm{f}}_{\mathrm{l}}}{{f}_{co}^{\prime}})}^{0.86}$ |

Karbhari and Gao [59] | 1997 | $\frac{{f}_{cc}^{\prime}}{{f}_{co}^{\prime}}=1+2.1{(\frac{{\mathrm{f}}_{\mathrm{l}}}{{f}_{co}^{\prime}})}^{0.87}$ |

Samaan et al. [60] | 1998 | $\frac{{f}_{cc}^{\prime}}{{f}_{co}^{\prime}}=1+6.0{\frac{({\mathrm{f}}_{\mathrm{l}})}{{f}_{co}^{\prime}}}^{0.70}$ |

- | - | thus ${\mathrm{f}}_{\mathrm{o}}$ = 0.872${f}_{co}^{\prime}$ + 0.371${\mathrm{f}}_{\mathrm{l}}$ + 6.258 |

- | - | E_{2} = 245.61${f}_{co}^{\prime}$^{0.2} + 1.3456$\frac{{\mathrm{E}}_{\mathrm{f}}\mathrm{t}}{\mathrm{D}}$ |

Saafi et al. [61] | 1999 | $\frac{{f}_{cc}^{\prime}}{{f}_{co}^{\prime}}=1+2.2{(\frac{{\mathrm{f}}_{\mathrm{l}}}{{f}_{co}^{\prime}})}^{0.84}$ |

Lam and Teng [62] | 2003 | $\frac{{f}_{cc}^{\prime}}{{f}_{co}^{\prime}}=1+3.3\frac{{\mathrm{f}}_{\mathrm{l}}}{{f}_{co}^{\prime}}$ |

Mander et al. [63] | 2005 | $\frac{{f}_{cc}^{\prime}}{{f}_{co}^{\prime}}=\sqrt[2.254]{1+7.94\frac{{\mathrm{f}}_{\mathrm{l}}}{{f}_{co}^{\prime}}}-2\frac{{\mathrm{f}}_{\mathrm{l}}}{{f}_{co}^{\prime}}-1.254$ |

Bisby et al. [64] | 2005 | $\frac{{f}_{cc}^{\prime}}{{f}_{co}^{\prime}}=1+2.425\frac{{\mathrm{f}}_{\mathrm{l}}}{{f}_{co}^{\prime}}$ |

Matthys et al. [65] | 2006 | $\frac{{f}_{cc}^{\prime}}{{f}_{co}^{\prime}}=1+2.3{(\frac{{\mathrm{f}}_{\mathrm{l}}}{{f}_{co}^{\prime}})}^{0.85}$ |

Shehata et al. [66] | 2007 | $\frac{{f}_{cc}^{\prime}}{{\mathrm{f}}_{\mathrm{co}}}=1+2.4\frac{{\mathrm{f}}_{\mathrm{l}}}{{f}_{co}^{\prime}}$ |

Al-Salloum and Siddiqui [67] | 2009 | $\frac{{f}_{cc}^{\prime}}{{f}_{co}^{\prime}}=1+2.312\frac{{\mathrm{f}}_{\mathrm{l}}}{{f}_{co}^{\prime}}$ |

Teng et al. [68] | 2009 | $\frac{{f}_{cc}^{\prime}}{{f}_{co}^{\prime}}=1+3.5\left({\mathsf{\rho}}_{\mathrm{k}}-0.01\right){\mathsf{\rho}}_{\mathsf{\epsilon}}$ |

Realfonso and Napoli [69] | 2011 | $\frac{{f}_{cc}^{\prime}}{{f}_{co}^{\prime}}=1+3.57\frac{{\mathrm{f}}_{\mathrm{l}}}{{f}_{co}^{\prime}}$ |

Parameters | d | h | nt | E | f′_{co} | f′_{cc} | ε_{co} | ε_{cc} |
---|---|---|---|---|---|---|---|---|

- | (mm) | (mm) | (mm) | (Gpa) | (Mpa) | (Mpa) | (%) | (%) |

Mean | 154.62 | 307.88 | 0.82 | 182.52 | 40.56 | 74.58 | 0.26 | 1.53 |

Median | 152 | 304 | 0.38 | 230 | 36.3 | 66.78 | 0.24 | 1.35 |

Mode | 150 | 300 | 0.33 | 230 | 24.5 | 63 | 0.24 | 0.95 |

Sample Variance | 1927.85 | 7552.62 | 0.992 | 12,592.78 | 469.98 | 1125.324 | 0.0155 | 0.716 |

Skewness | 2.71 | 2.85 | 2.355 | 0.4467 | 2.603 | 2.05988 | 7.428 | 0.957 |

Standard Error | 1.53 | 3.02 | 0.03 | 3.899 | 0.75 | 1.17 | 0.004 | 0.031 |

Kurtosis | 12.59 | 13.48 | 5.784 | 0.3353 | 11.696 | 8.54763 | 60.888 | 0.658 |

Standard Deviation | 43.907 | 86.91 | 0.996 | 112.218 | 21.68 | 33.546 | 0.1246 | 0.846 |

Minimum | 51 | 102 | 0.09 | 10 | 6.2 | 17.8 | 0.1676 | 0.083 |

Maximum | 406 | 812 | 5.9 | 663 | 188.2 | 302.2 | 1.53 | 4.62 |

Range | 355 | 710 | 5.81 | 653 | 182 | 284.4 | 1.3624 | 4.537 |

- | d | h | t | E | f′_{co} |
---|---|---|---|---|---|

d | 1 | 0.99 | 0.02 | 0.07 | −0.09 |

h | 0.99 | 1 | 0.02 | 0.07 | −0.09 |

t | 0.02 | 0.02 | 1 | −0.49 | 0.19 |

E | 0.07 | 0.07 | −0.49 | 1 | −0.10 |

f′_{co} | −0.09 | −0.09 | 0.19 | −0.10 | 1 |

Parameters | Settings |
---|---|

Size of subpopulations | 150 |

Number of subpopulation | 100 |

Mathematical operators | +, −, ×, ÷, Cosθ, Sinθ, tanθ |

Crossover probability | 0.92 |

Mutation probability | 0.01 |

Variables | 0.5 |

Operators | 0.5 |

Number of generations | 10,000 |

S. No. | Equation | Condition | Suggested by |
---|---|---|---|

1 | ${R}_{m}={R}^{2}\times \left(1-\sqrt{\left|{R}^{2}-{R}_{o}^{2}\right|}\right)$ | ${R}_{m}>0.5$ | (Roy and Roy, 2008) [101] |

- | where ${R}_{o}^{2}=1-\frac{{\sum}_{i=1}^{n}{\left({m}_{i}-{e}_{i}^{o}\right)}^{2}}{{\sum}_{i=1}^{n}{\left({m}_{i}-{\underset{\_}{\overline{m}}}_{i}^{o}\right)}^{2}},{e}_{i}^{o}=k\times {m}_{i}$ | ${R}_{o}^{2}\cong 1$ | - |

- | $R{\prime}_{o}^{2}=1-\frac{{\sum}_{i=1}^{n}{\left({e}_{i}-{m}_{i}^{o}\right)}^{2}}{{\sum}_{i=1}^{n}{\left({e}_{i}-{\underset{\_}{\overline{e}}}_{i}^{o}\right)}^{2}},{m}_{i}^{o}=k\prime \times {e}_{i}$ | $R{\prime}_{o}^{2}\cong 1$ | - |

2 | $k=\frac{{\sum}_{i=1}^{n}\left({e}_{i}\times {m}_{i}\right)}{{\sum}_{i=1}^{n}{e}_{i}^{2}}$ | $0.85<k<1.15$ | (Golbraikh and Tropsha, 2002) [102] |

3 | $k\prime =\frac{{\sum}_{i=1}^{n}\left({e}_{i}\times {m}_{i}\right)}{{\sum}_{i=1}^{n}{m}_{i}^{2}}$ | $0.85<k\prime <1.15$ | [102] |

- | RMSE | RSE | MAE | RRMSE | R | ρ | OF |
---|---|---|---|---|---|---|---|

Training | 7.768321 | 0.010346 | 6.471356 | 0.005 | 0.9948 | 0.002291 | 0.009156 |

Validation | 7.17975 | 0.009859 | 5.944429 | 0.009 | 0.9950 | 0.004578 | - |

Testing | 7.719133 | 0.009733 | 6.33431 | 0.010 | 0.9953 | 0.004921 | - |

Database | 7.6756 | 0.0102 | 6.3719 | 0.004 | 0.9949 | 0.00189 | - |

Sr. No. | Parameters | Sets | Database | ||
---|---|---|---|---|---|

Training | Validation | Testing | |||

1 | k | 0.991410 | 0.993896 | 1.011315 | 0.994896 |

2 | k′ | 0.998249 | 0.981181 | 0.979612 | 0.994932 |

3 | R_{m} | 0.889943 | 0.892999 | 0.896258 | 0.891061 |

4 | ${R}_{\xb0}^{2}$ | 0.999809 | 0.999783 | 0.999783 | 0.999802 |

5 | ${R}_{\xb0}^{\prime 2}$ | 0.999823 | 0.999796 | 0.999794 | 0.999816 |

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## Share and Cite

**MDPI and ACS Style**

Ilyas, I.; Zafar, A.; Javed, M.F.; Farooq, F.; Aslam, F.; Musarat, M.A.; Vatin, N.I.
Forecasting Strength of CFRP Confined Concrete Using Multi Expression Programming. *Materials* **2021**, *14*, 7134.
https://doi.org/10.3390/ma14237134

**AMA Style**

Ilyas I, Zafar A, Javed MF, Farooq F, Aslam F, Musarat MA, Vatin NI.
Forecasting Strength of CFRP Confined Concrete Using Multi Expression Programming. *Materials*. 2021; 14(23):7134.
https://doi.org/10.3390/ma14237134

**Chicago/Turabian Style**

Ilyas, Israr, Adeel Zafar, Muhammad Faisal Javed, Furqan Farooq, Fahid Aslam, Muhammad Ali Musarat, and Nikolai Ivanovich Vatin.
2021. "Forecasting Strength of CFRP Confined Concrete Using Multi Expression Programming" *Materials* 14, no. 23: 7134.
https://doi.org/10.3390/ma14237134