Machine Learning Approach for Prediction of Lateral Confinement Coefficient of CFRPWrapped RC Columns
Abstract
:1. Introduction
2. Dataset Interpretation
 Corner radii are most significantly due to reduction/curtailments of the stress attack and improved strain distribution during extreme load application. At this moment, RC columns jeopardized maximum load, causing damage to weak zones due to uneven stirrup distribution, proper reinforcement arrangement or mixing proportion.
 By reducing corner radii and wrapping with CFRP material, we can technically ensure that our RC columns have enhanced performance, with improved ductility and comprehensive strength.
 Specimens examined by Ref. [54] showed that the compressive strength ratio (f’cc/f’co) of relatively largescale square columns confined by CFRP increases almost linearly along with the increase of corner radius.
 Demonstrating that with CFRP, confinement is inconsequential to enlarge the compressive strength of RC columns with sharp corners (r = 0 mm) at the highest loading extents, although the ductility can be increased.
 Ref. [57] pointed out that the strength and strain augmentation effect of sporadically wrapped specimens can be perfected with evenlydistributed overlap regions. Thereupon, respective overlapping zones were staged on a different side and ducked the corner zones.
 Ref. [58] demonstrated the confinement potency model by considering lateral confinement level, corner radius ratio and size effect, proposed for FRPconfined square columns. Juxtaposed with other extant models, the contemplated one provides an enhanced examination of FRPconfined square columns.
3. Methodology
3.1. Genetic Programming (GP)
3.2. Minimax Probability Machine Regression (MPMR)
3.3. Deep Neural Network (DNN)
4. Results and Discussion
5. Conclusions
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
References
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References  Equations of K_{s} 

Ref. [13]  ${\mathrm{K}}_{\mathrm{s}}=\frac{{{\mathrm{f}}^{\prime}}_{\mathrm{cc}}}{{{\mathrm{f}}^{\prime}}_{\mathrm{co}}}=1+4.1\frac{{\mathrm{f}}_{\mathrm{l}}}{{{\mathrm{f}}^{\prime}}_{\mathrm{co}}}$ 
Ref. [14]  ${\mathrm{K}}_{\mathrm{s}}=1+\frac{{\mathsf{\rho}}_{\mathrm{s}}{\mathrm{f}}_{\mathrm{yh}}}{{{\mathrm{f}}^{\prime}}_{\mathrm{c}}}$ ${\mathsf{\rho}}_{\mathrm{s}}=\frac{{\mathrm{A}}_{\mathrm{sh}}{\mathrm{l}}_{\mathrm{s}}}{{{\mathrm{sb}}_{\mathrm{c}}}^{\prime}{\mathrm{d}}_{\mathrm{c}}}$ 
Ref. [15]  ${\mathrm{K}}_{\mathrm{s}}=1+\frac{{\mathrm{b}}_{\mathrm{c}}^{2}}{140{\mathrm{P}}_{\mathrm{occ}}}\left[\left(1\frac{{\mathrm{mc}}_{\mathrm{i}}^{2}}{5.5{\mathrm{b}}_{\mathrm{c}}^{2}}\right){\left(1\frac{\mathrm{s}}{{\mathrm{b}}_{\mathrm{c}}}\right)}^{2}\right]\sqrt{{\mathsf{\rho}}_{\mathrm{s}}{\mathrm{f}}_{\mathrm{yh}}}$
${\mathrm{P}}_{\mathrm{occ}}=0.85{{\mathrm{f}}^{\prime}}_{\mathrm{c}}\left({\mathrm{A}}_{\mathrm{ck}}{\mathrm{A}}_{\mathrm{st}}\right)$ ${\mathrm{K}}_{\mathrm{s}}=1+\frac{6.7}{{{\mathrm{f}}^{\prime}}_{\mathrm{c}}}{\left({\mathrm{f}}_{\mathrm{le}}\right)}^{0.17}{\mathrm{f}}_{\mathrm{le}}$ 
Ref. [16]  ${\mathrm{K}}_{\mathrm{s}}=1\frac{{\left(\mathrm{b}2{\mathrm{r}}_{\mathrm{c}}\right)}^{2}+{\left(\mathrm{h}2{\mathrm{r}}_{\mathrm{c}}\right)}^{2}}{3\mathrm{bh}\left(1{\mathsf{\rho}}_{\mathrm{s}}\right)}$ 
Ref. [17]  $\frac{{\mathrm{f}}_{\mathrm{cc}}}{{\mathrm{f}}_{\mathrm{ucon}}}=1+\mathrm{k}\frac{{\mathrm{f}}_{\mathrm{lc}}}{{\mathrm{f}}_{\mathrm{ucon}}}$ 
Ref. [11]  ${\mathrm{K}}_{\mathrm{s}}=\mathrm{exp}\left[\left(7.16\times {10}^{4}\right)\left(\mathrm{b}\right)0.0023\left(\mathrm{h}\right)+0.493\left(\mathrm{t}\right)0.0223\left({\mathrm{f}}_{\mathrm{co}}^{,}\right)+\left(2.42\times {10}^{6}\right)\left({\mathrm{E}}_{\mathrm{CFRP}}\right)+0.652\right]$ 
Ref. [18]  ${\mathrm{f}}_{\mathrm{lc}}=\frac{{\mathsf{\rho}}_{\mathrm{FRP}}{\mathrm{F}}_{\mathrm{FRP}}}{2}$ ${\mathsf{\rho}}_{\mathrm{frp}}=\frac{8{\mathrm{t}}_{\mathrm{frp}}{\mathrm{b}}_{\mathrm{frp}}}{\mathrm{D}\left({\mathrm{b}}_{\mathrm{frp}}+{\mathrm{S}}_{\mathrm{frp}}\right)}$ 
Ref. [19]  ${\mathrm{f}}_{\mathrm{l}}=\frac{2{\mathrm{E}}_{\mathrm{f}}{\mathrm{t}}_{\mathrm{f}}{\mathrm{ne}}_{\mathrm{fe}}}{\mathrm{D}}$ $\mathrm{D}=\sqrt{{\mathrm{b}}^{2}+{\mathrm{h}}^{2}}$ 
Current Study  Doran et al. (2015) [11]  

1  No of Dataset  293  100 
2  Models 
 Fuzzy Logic 
3  No of Inputs  6  5 
4  Type of CFRP RC Columns  Rectangular and Square  Rectangular 
5  Corner Radii  Considered  Not Considered 
b (mm)  H (mm)  r (mm)  t_{w} (mm)  E_{frp} (mm)  f_{co} (mm)  K_{s} (mm)  

Min  20  108  5  0.056  10,500  10.83  0.94 
Mean  167.15  277.07  25.16  0.55  187,852.90  30.54  1.69 
Std  57.66  149.73  12.74  0.50  87,680.16  11.61  0.69 
Max  457  1200  60  3  640,000  55.36  4.79 
skewness  1.46  2.26  0.41  2.36  0.24  0.28  1.74 
Kurtosis  7.05  11.71  2.72  10.16  6.56  2.56  6.38 
GP  MPMR  DNN 




Statistical Parameters  Description  Ideal Condition 

Coefficient of Determination, ${R}^{2}=\frac{{\left({{\displaystyle \sum}}_{i=1}^{n}\left({U}_{i}{U}_{m}\right)\left({L}_{i}{L}_{m}\right)\right)}^{2}}{{{\displaystyle \sum}}_{i=1}^{n}{\left({U}_{i}{U}_{m}\right)}^{2}{{\displaystyle \sum}}_{i=1}^{n}{\left({L}_{i}{L}_{m}\right)}^{2}}$  Coefficient of determination calculates the constancy of collaboration between the actual and the predicted values.  The ideal value must be near to unity. 
Mean Absolute Error $MAE=\frac{1}{n}{{\displaystyle \sum}}_{i=1}^{n}\left{L}_{i}{U}_{i}\right$  MAE enumerates the accuracy error of the predicted and actual data.  MAE value should be 0. When the value of R overtures to 0. 
Root Mean Square Error $RMSE={\left(\frac{1}{n}{{\displaystyle \sum}}_{i=1}^{n}{\left[{L}_{i}{U}_{i}\right]}^{2}\right)}^{0.5}$  Analyze the measured value to the estimated value and calculate the square root of the mean residual error.  RMSE has to be 0. When the value of R overtures to 1, the RMSE value will be near to 0, and vice versa. 
Index of Agreement, $IA=1\frac{{{\displaystyle \sum}}_{i=1}^{n}{\left({L}_{i}{U}_{i}\right)}^{2}}{{{\displaystyle \sum}}_{i=1}^{n}{\left(\left{L}_{i}{U}_{m}\right+\left{U}_{i}{U}_{m}\right\right)}^{2}}$  Index was employed to analyze the precision of the measurable models in this investigation.  The IA value should be 1 to enumerate the performance model. 
Fractional Variance $FV=2\left({\sigma}_{u}{\sigma}_{l}\right)/\left({\sigma}_{u}+{\sigma}_{l}\right)$  FV emphases computed variance of actual and predicted data.  FV ideal value must be 0. 
Factor of Two (FA2) $0.5\u2a7dFA2=\frac{1}{n}{{\displaystyle \sum}}_{i=1}^{n}\left(\frac{{U}_{i}}{{L}_{i}}\right)\u2a7d2$  Indicates the range of the output results data between 0.5–2 as benchmark model accuracy.  Based on the model performance, the range output result data should lie between 0.5–2. 
Coefficient of Variation (%) $CV=\frac{RMSE}{{U}_{m}}\ast 100$  It symbolizes the ratio of the RMSE variance to the actual data variance. It is exhibited in percentage.  The ideal value of CV should be 0. RMSE is also 0. 
Durbin–Watson (DW) statistics, $DW=\frac{{{\displaystyle \sum}}_{i=2}^{n}{\left({j}_{i}{j}_{i1}\right)}^{2}}{{{\displaystyle \sum}}_{i=1}^{n}{j}_{i}^{2}}$ where,$\left({j}_{i}={U}_{i}{L}_{i}\right)$  It measures the predictive accuracy. To validate the predictive capability of the prediction models,  The ideal value of DW must be close to 2. 
Normalized Mean Bias Error (NMBE), $NMBE=\frac{\frac{1}{n}{{\displaystyle \sum}}_{i=1}^{n}\left({L}_{i}{U}_{i}\right)}{\frac{1}{n}{{\displaystyle \sum}}_{i=1}^{n}{U}_{i}}\ast 100$  NMBE estimates the aptitude of the model to anticipate a value, which is staged away from the mean value. It is expressed in percentage.  A positive NMBE reveals overprediction, and a negative value depicts underprediction 
Doran et al. (2015) [11] (Fuzzy Logic) Overall  Training GP  Testing GP  Training MPMR  Testing MPMR  Training DNN  Testing DNN  

Number of Dataset  100  220  73  220  73  220  73 
R^{2}  0.919  0.89  0.89  0.885  0.712  0.806  0.712 
MAE  0.133  0.041  0.054  0.041  0.064  0.036  0.070 
RMSE  0.174  0.056  0.073  0.057  0.097  0.076  0.117 
IA  0.976  0.970  0.960  0.969  0.937  0.947  0.883 
FV  0.111  0.116  0.389  0.123  0.067  0.071  0.499 
FA2  0.993  0.836  1.135  1.237  1.283  1.184  1.454 
CV(%)  10.74  30.777  31.933  31.185  42.794  41.421  51.411 
DW statistic  1.513  1.453  1.004  1.491  0.978  0.842  0.877 
NMBE (%)    0.001  −9.596  0.130  −3.974  −2.666  −20.767 
Training (GP)  Testing (GP)  Training (MPMR)  Testing (MPMR)  Training (DNN)  Testing (DNN)  

R^{2}  3  3  2  1  1  1 
MAE  1  3  2  2  3  1 
RMSE  3  3  2  2  1  1 
IA  1  3  3  2  2  1 
FV  2  2  1  3  3  1 
FA2  1  1  3  3  2  1 
CV (%)  3  3  2  2  1  1 
DW statistic  2  2  3  1  1  1 
NMBE (%)  3  1  2  2  1  1 
Total Points  19  21  20  18  15  9 
Overall Points  40  38  24 
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Xue, X.; Makota, C.; Khalaf, O.I.; Jayabalan, J.; Samui, P.; Abdulsahib, G.M. Machine Learning Approach for Prediction of Lateral Confinement Coefficient of CFRPWrapped RC Columns. Symmetry 2023, 15, 545. https://doi.org/10.3390/sym15020545
Xue X, Makota C, Khalaf OI, Jayabalan J, Samui P, Abdulsahib GM. Machine Learning Approach for Prediction of Lateral Confinement Coefficient of CFRPWrapped RC Columns. Symmetry. 2023; 15(2):545. https://doi.org/10.3390/sym15020545
Chicago/Turabian StyleXue, Xingsi, Celestine Makota, Osamah Ibrahim Khalaf, Jagan Jayabalan, Pijush Samui, and Ghaida Muttashar Abdulsahib. 2023. "Machine Learning Approach for Prediction of Lateral Confinement Coefficient of CFRPWrapped RC Columns" Symmetry 15, no. 2: 545. https://doi.org/10.3390/sym15020545