# New Prediction Model for the Ultimate Axial Capacity of Concrete-Filled Steel Tubes: An Evolutionary Approach

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## Abstract

**:**

## 1. Introduction

## 2. Comparison of Genetic Programming vs. Genetic Engineering Programming

## 3. Experimental Database

## 4. Development of Model

## 5. Results and Discussion

_{GEP}is the ultimate axial moment capacity of the column calculated from Equation (6) and f

_{c}’ is the compressive strength of infilled concrete. D, t, Land f

_{y}are the diameter, thickness, length and yield strength of the steel tube, respectively.

^{2}value was increased from 0.97 to 0.99 while MAE and RMSE decreases 134 to 124 and 210 to 173, respectively. Moreover, that the error value for testing is lesser as compared with other training and validation set. This illustrates that the present GEP model can accurately predict the axial capacity of CFST members and can be used for the generalization purpose [72].

#### Model Performance, Validity and Comparative Study

^{2}accuracy of about 0.94 as compared to other models. This is due to simplified nature of GEP in prediction. Moreover, Glakoumelis et al. [80] predict the compressive nature of CFST by giving an empirical relation with a strong correlation value R

^{2}of about 0.895. Also, Goode et al. [79] and Lu et al. [78] give same empirical equation with some modification with R

^{2}value of 0.807 and 0.903, respectively as illustrated in Figure 7. This study show us that GEP-based empirical equations can be used in prediction of different variables.

## 6. Conclusions

^{2}proved the accuracy and reliability of GEP-based derived equations. In addition, this supervised machine learning algorithm can be used in many other domains. As they help us in making the forecast prediction by training and testing of data. This artificial intelligence-based algorithm then helps scientific community by taking measures and overcome the issues associated in mechanical work or in experimental work. Though, the comparison between the MAE, RMSE and R

^{2}of GEP model, AS5100.6, EC4, AISC, BS, DBJ and AIJ shows that GEP model performs best for all sets (learning, training and validation) of data. Even though the GEP-based model can calculate short CFST shear strength, it is restricted to long circular columns. The findings from this new research will give civil engineers and structural designers some useful information and can be used as a modern and powerful method to help decision-making in concrete construction fields.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Nomenclature

$\mathrm{CFST}=$ Concrete-filled steel tube |

$\mathrm{ANN}=$ Artificial neutron network |

$\mathrm{GP}=$ Genetic programming |

$\mathrm{GEP}=$ Genetic engineering programming |

$ETs=$ Expression trees |

$MAE=$ Mean absolute error |

$RMSE=$ Root mean square error |

${N}_{u}=$ Ultimate axial moment capacity |

${N}_{n}=$ Nominal axial moment capacity |

${N}_{e}=$ Euler’s bucking load |

${N}_{o}=$ Nominal axial compressive strength exclusive of length effects |

${A}_{s}=$ Steel section areas |

${A}_{c}=$ Concrete area |

${A}_{g}=$ Total composite cross-section area |

$D=$ Diameter of concrete core |

${E}_{c}=$ Concrete elastic modulus $=0.043{\mathsf{\omega}}_{c}^{1.5}\sqrt{{f}_{c}^{\prime}}\mathrm{MPa}$ |

${E}_{s}=$ Steel elastic modulus $=\mathrm{200,000}\text{}\mathrm{MPa}$ |

${f}_{c}^{\prime}=$ Concrete compressive strength |

${f}_{y}=$ Steel section minimum yield strength |

${I}_{c}=$ Concrete section moment of inertia |

${I}_{s}=$ Steel section moment of inertia |

$K=$ Length effectiveness factor |

$L=$ Length of laterally braced member |

${\left(\mathrm{EI}\right)}_{eff}=$ Composite section effective stiffness |

${N}_{e}=$ Elastic bucking load |

${\alpha}_{c}=$ Concrete contribution factor |

${f}_{cu}=$ 28-day characteristic strength of concrete cube |

${f}_{cc}=$ Triaxially contained concrete improved characteristic strength |

${f}_{scy}=$ Steel-tube nominal yield strength |

${f}_{ck}=$ Concrete characteristic strength |

${f}_{y}^{\prime}=$ Reduced nominal yield strength of the steel casing |

${l}_{e}=$ Effective length $=0.7l$ |

$l=$ Actual length |

${\eta}_{c}=$ Concrete confinement coefficient |

${\eta}_{a}=$ Steel tube confinement coefficient |

$\overline{\lambda}$ = Relative slenderness |

${\left(EI\right)}_{eff2}=$ Effective flexural stiffness |

${K}_{c}=$ Correction factor |

$\eta =$ Confinement factor $=0.27$ |

$\xi =$ Confinement factor |

${W}_{scm}=$ Section modulus of composite cross section |

${\mathsf{\gamma}}_{m}=$ Flexural strength index |

${f}_{min}$= Objective function |

## Appendix A

S. No | Diameter | Thickness | Yield Strength | Compressive Strength | Length | Length/Diameter | Axial Capacity |
---|---|---|---|---|---|---|---|

1 | 120.9 | 3.73 | 312 | 30.22 | 2311 | 19.11 | 725 |

2 | 166 | 5 | 288.1 | 63.70 | 1040 | 6.27 | 1862 |

3 | 88.9 | 5.842 | 406 | 50.50 | 1117.6 | 12.57 | 715.56 |

4 | 114.3 | 3.1 | 348 | 62.78 | 670 | 5.86 | 898 |

5 | 95 | 3.68 | 392 | 31.44 | 860 | 9.05 | 686 |

6 | 166 | 5 | 288.1 | 36.55 | 1040 | 6.27 | 1495 |

7 | 168.2 | 4.52 | 302 | 52.80 | 813 | 4.83 | 2113 |

8 | 114.3 | 3.1 | 348 | 62.78 | 670 | 5.86 | 904 |

9 | 219 | 7 | 273 | 46.50 | 1200 | 5.48 | 3200 |

10 | 114 | 6.34 | 486 | 45.00 | 850 | 7.46 | 1608 |

11 | 100 | 2.5 | 433.2 | 54.78 | 600 | 6.00 | 750 |

12 | 108 | 4 | 338.88 | 35.71 | 5400 | 50.00 | 210.7 |

13 | 219 | 7 | 273 | 46.50 | 1420 | 6.48 | 3070 |

14 | 215.9 | 4.08 | 292 | 28.67 | 2220 | 10.28 | 1650 |

15 | 152.4 | 1.55 | 294 | 43.25 | 914 | 6.00 | 721.5 |

16 | 114 | 6 | 486 | 45.00 | 850 | 7.46 | 1334 |

17 | 114.3 | 3.1 | 340 | 73.10 | 3370 | 29.48 | 379 |

18 | 95 | 3.66 | 338 | 30.00 | 2032 | 21.39 | 463 |

19 | 216 | 4.04 | 293 | 36.89 | 2220 | 10.28 | 2289 |

20 | 114.3 | 3.19 | 414 | 35.44 | 838 | 7.33 | 734 |

21 | 95 | 12.4 | 277 | 26.22 | 1420 | 14.95 | 907 |

22 | 108 | 4 | 337.6 | 43.12 | 756 | 7.00 | 785 |

23 | 152.4 | 1.55 | 330 | 32.11 | 1499 | 9.84 | 734 |

24 | 166 | 5 | 288.1 | 65.17 | 1040 | 6.27 | 1852 |

25 | 166 | 5 | 289.1 | 34.68 | 2700 | 16.27 | 1117.2 |

26 | 110 | 1.9 | 350 | 14.44 | 2200 | 20.00 | 252 |

27 | 120.9 | 3.76 | 312 | 26.78 | 1049 | 8.68 | 721 |

28 | 216 | 4.11 | 291 | 36.89 | 2220 | 10.28 | 2239 |

29 | 108 | 4.5 | 348.1 | 46.87 | 4023 | 37.25 | 318 |

30 | 190.7 | 6 | 505 | 57.40 | 3450 | 18.09 | 2130 |

31 | 166 | 5 | 288.1 | 53.11 | 1040 | 6.27 | 1695 |

32 | 108 | 4 | 338.88 | 35.71 | 864 | 8.00 | 766.36 |

33 | 95 | 12.75 | 277 | 26.22 | 1420 | 14.95 | 938 |

34 | 114 | 5.94 | 486 | 45.00 | 1750 | 15.35 | 1138 |

35 | 216 | 4.11 | 304 | 29.11 | 2220 | 10.28 | 1834 |

36 | 152.4 | 3.17 | 415 | 26.56 | 2271 | 14.90 | 939 |

37 | 114 | 4.68 | 332 | 45.00 | 850 | 7.46 | 1049 |

38 | 108 | 4.5 | 259.7 | 25.48 | 1620 | 15.00 | 524 |

39 | 108 | 4 | 338.88 | 35.71 | 3240 | 30.00 | 478.24 |

40 | 110 | 1.9 | 350 | 14.44 | 2200 | 20.00 | 219 |

41 | 76.48 | 1.73 | 369 | 32.56 | 609.45 | 7.97 | 330.04 |

42 | 166 | 5 | 274.4 | 36.43 | 1100 | 6.63 | 1985 |

43 | 127.1 | 2.95 | 376 | 77.20 | 711 | 5.59 | 1305 |

44 | 114.3 | 3.1 | 348 | 62.67 | 1020 | 8.92 | 888 |

45 | 110 | 1.9 | 350 | 40.50 | 2200 | 20.00 | 437 |

46 | 355.6 | 11.18 | 361 | 47.00 | 1880 | 5.29 | 11,460 |

47 | 88.9 | 5.85 | 400 | 49.75 | 508 | 5.71 | 992 |

48 | 127.3 | 1.63 | 334 | 77.20 | 711 | 5.59 | 1285 |

49 | 210 | 3 | 233.2 | 33.52 | 1040 | 4.95 | 1705 |

50 | 114.3 | 3.1 | 340 | 64.56 | 3720 | 32.55 | 293 |

51 | 355.6 | 4.72 | 281 | 27.00 | 1880 | 5.29 | 3517 |

52 | 114 | 3.41 | 291 | 43.75 | 2750 | 24.12 | 569 |

53 | 95 | 3.66 | 332 | 31.44 | 860 | 9.05 | 656 |

54 | 95 | 12.7 | 277 | 26.22 | 860 | 9.05 | 1034 |

55 | 108 | 4.5 | 358 | 106.00 | 1188 | 11.00 | 1194 |

56 | 121 | 3.73 | 333 | 27.11 | 1050 | 8.68 | 746 |

57 | 219 | 7 | 273 | 46.50 | 990 | 4.52 | 3278 |

58 | 168.4 | 4.52 | 302 | 52.80 | 813 | 4.83 | 2233 |

59 | 160 | 2.5 | 433.2 | 39.40 | 960 | 6.00 | 1426 |

60 | 121 | 3.71 | 313 | 30.67 | 2310 | 19.09 | 695 |

61 | 88.9 | 5.842 | 406 | 50.50 | 1422.4 | 16.00 | 712 |

62 | 88.9 | 5.72 | 400 | 48.25 | 1422 | 16.00 | 712 |

63 | 165.2 | 4.1 | 353 | 49.88 | 3965 | 24.00 | 1019 |

64 | 168.1 | 4.52 | 298 | 52.30 | 813 | 4.84 | 2233 |

65 | 92 | 3 | 260.7 | 26.07 | 1380 | 15.00 | 409 |

66 | 114 | 5.94 | 486 | 31.11 | 1280 | 11.23 | 1285 |

67 | 114 | 6.11 | 486 | 40.00 | 2750 | 24.12 | 941 |

68 | 168.8 | 5 | 302.4 | 40.50 | 2135 | 12.65 | 1130 |

69 | 108 | 4.5 | 348.1 | 31.91 | 4158 | 38.50 | 342 |

70 | 82.55 | 1.397 | 482.3 | 47.29 | 1422.4 | 17.23 | 294.59 |

71 | 114 | 3.23 | 290 | 36.67 | 1751 | 15.36 | 706 |

72 | 121 | 5.41 | 348 | 27.11 | 1050 | 8.68 | 1018 |

73 | 165.2 | 4.17 | 358.7 | 49.82 | 1321.6 | 8.00 | 1445 |

74 | 95 | 3.86 | 332 | 31.44 | 1420 | 14.95 | 567 |

75 | 95 | 12.6 | 279 | 26.22 | 860 | 9.05 | 1018 |

76 | 166 | 5 | 289.1 | 34.68 | 2700 | 16.27 | 1271.06 |

77 | 114.3 | 3.1 | 348 | 65.56 | 1335 | 11.68 | 794 |

78 | 108 | 4.5 | 358 | 106.00 | 1620 | 15.00 | 1018 |

79 | 114.3 | 3.1 | 348 | 67.22 | 2040 | 17.85 | 688 |

80 | 250 | 7 | 243 | 55.58 | 1480 | 5.92 | 4116 |

81 | 76.5 | 1.74 | 364 | 49.88 | 610 | 7.97 | 423 |

82 | 95 | 3.91 | 392 | 31.44 | 1420 | 14.95 | 606 |

83 | 108 | 4.5 | 358 | 106.00 | 756 | 7.00 | 1286 |

84 | 165 | 4.7 | 355 | 33.40 | 2475 | 15.00 | 1058 |

85 | 114 | 1.72 | 266 | 43.75 | 2750 | 24.12 | 353 |

86 | 95 | 12.6 | 275 | 25.89 | 1981 | 20.85 | 903 |

87 | 200 | 3 | 303.5 | 55.80 | 2002 | 10.01 | 1882 |

88 | 169 | 7.5 | 360 | 80.80 | 1768 | 10.46 | 2870 |

89 | 152.4 | 3.17 | 415 | 26.56 | 2271 | 14.90 | 881 |

90 | 121 | 3.86 | 332 | 30.67 | 2310 | 19.09 | 755 |

91 | 165.2 | 4.17 | 358.7 | 49.82 | 1982.4 | 12.00 | 1305 |

92 | 95 | 12.5 | 279 | 26.22 | 1420 | 14.95 | 947 |

93 | 108 | 4.5 | 348.1 | 31.91 | 3510 | 32.50 | 400 |

94 | 88.9 | 5.82 | 400 | 48.75 | 1727 | 19.43 | 614 |

95 | 88.9 | 5.842 | 406 | 50.50 | 812.8 | 9.14 | 918.925 |

96 | 166 | 5 | 288.1 | 52.90 | 1040 | 6.27 | 1764 |

97 | 121 | 5.44 | 327 | 30.67 | 2310 | 19.09 | 865 |

98 | 169.3 | 2.62 | 338.1 | 41.38 | 1830 | 10.81 | 689 |

99 | 121.01 | 3.66 | 300 | 27.11 | 1050 | 8.68 | 695 |

100 | 108 | 4 | 338.88 | 35.71 | 2160 | 20.00 | 672.28 |

101 | 166 | 5 | 284.2 | 51.24 | 870 | 5.24 | 1862 |

102 | 108 | 5 | 379.8 | 40.91 | 548 | 5.07 | 1084 |

103 | 114 | 3.35 | 291 | 45.00 | 850 | 7.46 | 785 |

104 | 108 | 4 | 338.88 | 35.71 | 1620 | 15.00 | 646.8 |

105 | 76.5 | 1.73 | 364 | 32.11 | 610 | 7.97 | 330 |

106 | 355.6 | 7.98 | 361 | 29.78 | 2083 | 5.86 | 7433 |

107 | 114 | 1.79 | 266 | 45.00 | 850 | 7.46 | 515 |

108 | 267.4 | 7 | 461 | 57.40 | 4800 | 17.95 | 3900 |

109 | 95 | 12.6 | 294 | 26.22 | 1980 | 20.84 | 917 |

110 | 114.3 | 3.1 | 348 | 62.67 | 1020 | 8.92 | 849 |

111 | 108 | 4.5 | 358 | 106.00 | 1188 | 11.00 | 1232 |

112 | 76 | 2 | 275 | 50.60 | 1556 | 20.47 | 330 |

113 | 216 | 6.3 | 411 | 36.89 | 2220 | 10.28 | 2932 |

114 | 114 | 5.73 | 486 | 40.00 | 2750 | 24.12 | 824 |

115 | 110 | 1.9 | 350 | 33.40 | 2200 | 20.00 | 374 |

116 | 219 | 4 | 325 | 61.44 | 1000 | 4.57 | 1980 |

117 | 267.4 | 7 | 461 | 57.40 | 1600 | 5.98 | 5190 |

118 | 88.9 | 5.81 | 400 | 47.62 | 1118 | 12.58 | 716 |

119 | 121 | 3.76 | 313 | 30.67 | 1050 | 8.68 | 837 |

120 | 108 | 4 | 338.88 | 35.71 | 2160 | 20.00 | 676.2 |

121 | 127 | 2.413 | 336 | 32.56 | 914 | 7.20 | 658.3 |

122 | 120.83 | 4.09 | 451.3 | 36.18 | 1050.04 | 8.69 | 1091.91 |

123 | 200 | 3 | 303.5 | 55.80 | 2001 | 10.01 | 1806 |

124 | 108 | 4 | 338.88 | 35.71 | 1080 | 10.00 | 783.02 |

125 | 82.55 | 1.397 | 482.3 | 47.29 | 1727.2 | 20.92 | 224.725 |

126 | 121.01 | 3.71 | 300 | 27.11 | 2310 | 19.09 | 641 |

127 | 140 | 2.5 | 433.2 | 47.43 | 840 | 6.00 | 1124 |

128 | 152.4 | 1.57 | 330 | 26.67 | 1499 | 9.84 | 681 |

129 | 120.65 | 4.09 | 451.3 | 41.72 | 1050.04 | 8.70 | 1155.7 |

130 | 215.9 | 6.02 | 350 | 36.44 | 2220 | 10.28 | 2869 |

131 | 121 | 5.49 | 348 | 27.11 | 2310 | 19.09 | 816 |

132 | 200 | 2 | 237.2 | 30.28 | 980 | 4.90 | 1411 |

133 | 82.55 | 1.397 | 482.3 | 47.29 | 812.8 | 9.85 | 400.5 |

134 | 108 | 4 | 338.88 | 35.71 | 2700 | 25.00 | 648.76 |

135 | 114.3 | 3.1 | 340 | 67.22 | 2700 | 23.62 | 516 |

136 | 108 | 4.5 | 348.1 | 31.91 | 3510 | 32.50 | 390 |

137 | 110 | 1.9 | 350 | 40.50 | 2200 | 20.00 | 368 |

138 | 114.3 | 3.19 | 414 | 35.44 | 838 | 7.33 | 756 |

139 | 114.3 | 3.1 | 340 | 67.22 | 2700 | 23.62 | 536 |

140 | 95 | 12.7 | 277 | 26.22 | 860 | 9.05 | 1008 |

141 | 108 | 4.5 | 348.1 | 46.87 | 3510 | 32.50 | 440 |

142 | 165 | 4.7 | 355 | 14.44 | 2477 | 15.01 | 800 |

143 | 140 | 5 | 378.3 | 37.53 | 840 | 6.00 | 1379 |

144 | 108 | 4.5 | 259.7 | 25.48 | 1994 | 18.46 | 495 |

145 | 152.4 | 1.55 | 330 | 32.11 | 1499 | 9.84 | 725 |

146 | 110 | 1.9 | 350 | 33.40 | 2200 | 20.00 | 368 |

147 | 219 | 4 | 325 | 56.60 | 1000 | 4.57 | 1931 |

148 | 114 | 4.44 | 332 | 45.00 | 850 | 7.46 | 902 |

149 | 108 | 4.5 | 348.1 | 46.87 | 4158 | 38.50 | 298 |

150 | 108 | 4 | 347.7 | 40.47 | 1620 | 15.00 | 672 |

151 | 152.7 | 3.15 | 421 | 26.89 | 1676.4 | 10.98 | 880.11 |

152 | 108 | 4 | 338.88 | 35.71 | 4320 | 40.00 | 294 |

153 | 108 | 4 | 338.88 | 35.71 | 1620 | 15.00 | 707.56 |

154 | 108 | 4.2 | 259.7 | 25.87 | 648 | 6.00 | 722 |

155 | 92 | 3 | 260.7 | 26.07 | 920 | 10.00 | 431 |

156 | 108 | 4 | 338.88 | 35.71 | 864 | 8.00 | 869.26 |

157 | 219 | 7 | 273 | 46.50 | 990 | 4.52 | 3278 |

158 | 108 | 4.5 | 344 | 40.91 | 548 | 5.07 | 917 |

159 | 107 | 4 | 379.8 | 38.32 | 542 | 5.07 | 889 |

160 | 108 | 4.5 | 259.7 | 25.48 | 648 | 6.00 | 665 |

161 | 219 | 7 | 273 | 46.50 | 990 | 4.52 | 3278 |

162 | 190.7 | 6 | 505 | 65.44 | 2300 | 12.06 | 2610 |

163 | 114 | 3.31 | 291 | 30.00 | 2320 | 20.35 | 535 |

164 | 95 | 12.6 | 275 | 25.89 | 861 | 9.06 | 1019 |

165 | 114.3 | 3.1 | 348 | 62.67 | 1020 | 8.92 | 845 |

166 | 140 | 5 | 378.3 | 42.63 | 840 | 6.00 | 1501 |

167 | 88.9 | 5.842 | 406 | 50.50 | 508 | 5.71 | 890 |

168 | 95 | 3.51 | 340 | 31.44 | 1980 | 20.84 | 488 |

169 | 108 | 4 | 338.88 | 35.71 | 4320 | 40.00 | 345.94 |

170 | 127.3 | 1.63 | 376 | 77.20 | 711 | 5.59 | 1285 |

171 | 95 | 3.76 | 332 | 31.44 | 1980 | 20.84 | 536 |

172 | 165 | 4.7 | 355 | 40.50 | 2476 | 15.01 | 1037 |

173 | 166 | 5 | 287.14 | 34.68 | 3700 | 22.29 | 958.44 |

174 | 127 | 2.413 | 336 | 27.11 | 914 | 7.20 | 627.2 |

175 | 114.3 | 3.1 | 340 | 73.10 | 3370 | 29.48 | 362 |

176 | 165 | 4.7 | 355 | 33.40 | 2475 | 15.00 | 1037 |

177 | 95 | 3.66 | 350 | 31.11 | 1981 | 20.85 | 529 |

178 | 114 | 1.73 | 266 | 40.00 | 1751 | 15.36 | 461 |

179 | 219 | 7 | 273 | 46.50 | 1640 | 7.49 | 2956 |

180 | 95 | 3.4 | 343 | 30.44 | 1980 | 20.84 | 473 |

181 | 210 | 2.5 | 237.2 | 32.93 | 1670 | 7.95 | 1323 |

182 | 95 | 3.78 | 392 | 31.44 | 1980 | 20.84 | 567 |

183 | 114 | 5.99 | 486 | 45.00 | 1750 | 15.35 | 1177 |

184 | 114 | 3.28 | 291 | 43.75 | 2750 | 24.12 | 667 |

185 | 108 | 4 | 338.88 | 35.71 | 5400 | 50.00 | 225.4 |

186 | 114.3 | 3.1 | 340 | 64.56 | 3720 | 32.55 | 305 |

187 | 108 | 4.5 | 259.7 | 25.48 | 1296 | 12.00 | 563 |

188 | 165 | 4.3 | 317.7 | 52.30 | 3640 | 22.06 | 987 |

189 | 114 | 3.29 | 291 | 30.00 | 2250 | 19.74 | 652 |

190 | 166 | 5 | 288.1 | 33.12 | 1040 | 6.27 | 1372 |

191 | 114.3 | 3.1 | 348 | 67.22 | 2040 | 17.85 | 617 |

192 | 168.3 | 4.47 | 302 | 29.33 | 813 | 4.83 | 1744 |

193 | 108 | 4 | 327.1 | 41.55 | 1188 | 11.00 | 686 |

194 | 140 | 5 | 378.3 | 51.25 | 840 | 6.00 | 1539 |

195 | 101.73 | 3.1 | 604.67 | 37.93 | 1524 | 14.98 | 800.1 |

196 | 152.4 | 1.55 | 294 | 43.25 | 914 | 6.00 | 733 |

197 | 219 | 7 | 273 | 46.50 | 1640 | 7.49 | 2956 |

198 | 168.8 | 2.64 | 200.2 | 42.13 | 1830 | 10.84 | 916 |

199 | 108 | 4 | 338.88 | 35.71 | 648 | 6.00 | 828.1 |

200 | 165.2 | 4.1 | 353 | 49.88 | 1322 | 8.00 | 1412 |

201 | 120.9 | 5.54 | 343 | 30.22 | 2311 | 19.11 | 867 |

202 | 114 | 6.14 | 486 | 34.44 | 2250 | 19.74 | 1000 |

203 | 166 | 5 | 313.6 | 51.24 | 1700 | 10.24 | 1460.2 |

204 | 219 | 7 | 273 | 46.50 | 1640 | 7.49 | 2956 |

205 | 76.5 | 1.73 | 364 | 31.11 | 1524 | 19.92 | 245 |

206 | 82.55 | 1.397 | 482.3 | 47.29 | 1117.6 | 13.54 | 356 |

207 | 95 | 12.8 | 283 | 26.22 | 1980 | 20.84 | 886 |

208 | 190.7 | 6 | 505 | 65.44 | 1150 | 6.03 | 3064 |

209 | 95 | 3.4 | 340 | 31.44 | 860 | 9.05 | 656 |

210 | 114 | 5.64 | 486 | 34.44 | 2250 | 19.74 | 902 |

211 | 240 | 10 | 269 | 58.80 | 1440 | 6.00 | 5135 |

212 | 152 | 1.65 | 270 | 83.00 | 900 | 5.92 | 1458 |

213 | 140 | 3 | 426.3 | 40.38 | 840 | 6.00 | 1208 |

214 | 216 | 6.05 | 395 | 29.11 | 2220 | 10.28 | 2462 |

215 | 108 | 4.5 | 348.1 | 46.87 | 4023 | 37.25 | 320 |

216 | 95 | 3.58 | 340 | 31.44 | 1420 | 14.95 | 576 |

217 | 169.3 | 2.62 | 338.1 | 45.13 | 1830 | 10.81 | 756 |

218 | 95 | 12.65 | 275 | 25.89 | 1420 | 14.95 | 930 |

219 | 114 | 6.21 | 486 | 40.00 | 2750 | 24.12 | 941 |

220 | 108 | 4.5 | 259.7 | 25.48 | 972 | 9.00 | 666 |

221 | 121 | 5.56 | 327 | 30.67 | 1050 | 8.68 | 1079 |

222 | 168.1 | 4.52 | 298 | 52.30 | 813 | 4.84 | 2113 |

223 | 216 | 4.06 | 289 | 29.11 | 2220 | 10.28 | 1023 |

224 | 114.3 | 3.1 | 340 | 73.10 | 3370 | 29.48 | 401 |

225 | 165.2 | 4.1 | 353 | 49.88 | 2974 | 18.00 | 1147 |

226 | 140 | 5.3 | 378.3 | 60.56 | 840 | 6.00 | 1664 |

227 | 127 | 2.39 | 289 | 42.75 | 1499 | 11.80 | 623 |

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**Figure 1.**Simple illustration of the of the gene expression programming (GEP) algorithm [22].

**Figure 6.**Evaluation of the concrete-filled steel tubes (CFST) columns experimental and predicted axial bearing capacity.

Parameters | Diameter | Thickness | Yield Stress | Compressive Strength | Length | Length/Diameter | Test |
---|---|---|---|---|---|---|---|

Training set data | |||||||

Mean | 137.9 | 4.6 | 344.9 | 43.9 | 36.4 | 1651.8 | 13.0 |

Standard error | 4.8 | 0.2 | 5.4 | 1.5 | 1.4 | 81.1 | 0.7 |

Median | 114.3 | 4.1 | 338.9 | 40.5 | 33.4 | 1420.0 | 10.3 |

Mode | 108.0 | 5.0 | 348.0 | 35.7 | 29.0 | 1040.0 | 20.0 |

Standard deviation | 54.4 | 2.5 | 61.2 | 17.3 | 15.9 | 917.2 | 7.9 |

Sample variance | 2956.3 | 6.2 | 3744.4 | 298.6 | 254.3 | 841,300.1 | 62.6 |

Kurtosis | 4.6 | 3.8 | 0.2 | 3.4 | 4.3 | 2.6 | 3.9 |

Skewness | 1.9 | 1.8 | 0.8 | 1.6 | 1.8 | 1.5 | 1.7 |

Range | 279.6 | 11.4 | 271.8 | 91.6 | 86.0 | 4892.0 | 45.5 |

Minimum | 76.0 | 1.4 | 233.2 | 14.4 | 10.0 | 508.0 | 4.5 |

Maximum | 355.6 | 12.8 | 505.0 | 106.0 | 96.0 | 5400.0 | 50.0 |

Sum | 17,646.2 | 592.2 | 44,143.2 | 5615.7 | 4656.1 | 211,432.9 | 1663.8 |

Count | 128.0 | 128.0 | 128.0 | 128.0 | 128.0 | 128.0 | 128.0 |

Testing set data | |||||||

Mean | 127.8 | 4.3 | 340.4 | 39.6 | 32.5 | 1806.0 | 15.0 |

Standard error | 5.2 | 0.3 | 7.8 | 1.9 | 1.7 | 152.4 | 1.4 |

Median | 114.0 | 4.0 | 340.0 | 35.7 | 29.0 | 1648.2 | 11.5 |

Mode | 108.0 | 4.0 | 338.9 | 35.7 | 29.0 | 2700.0 | 6.0 |

Standard deviation | 37.0 | 2.2 | 55.0 | 13.5 | 12.1 | 1077.3 | 9.9 |

Sample variance | 1367.0 | 4.7 | 3021.6 | 182.2 | 147.1 | 1,160,607.4 | 97.2 |

Kurtosis | 1.0 | 7.7 | 1.3 | 1.0 | 1.2 | −0.2 | 0.4 |

Skewness | 1.4 | 2.3 | 0.7 | 1.1 | 1.2 | 0.8 | 1.1 |

Range | 136.5 | 11.3 | 267.8 | 62.8 | 57.2 | 3812.0 | 35.5 |

Minimum | 82.6 | 1.4 | 237.2 | 14.4 | 10.0 | 508.0 | 4.5 |

Maximum | 219.0 | 12.7 | 505.0 | 77.2 | 67.2 | 4320.0 | 40.0 |

Sum | 6388.6 | 214.0 | 17,017.5 | 1982.3 | 1626.9 | 90,302.2 | 751.0 |

Count | 50.0 | 50.0 | 50.0 | 50.0 | 50.0 | 50.0 | 50.0 |

Validation set data | |||||||

Mean | 137.3 | 4.5 | 347.2 | 42.2 | 34.8 | 1755.6 | 13.8 |

Standard error | 5.7 | 0.3 | 10.2 | 1.7 | 1.5 | 127.2 | 1.2 |

Median | 114.3 | 4.1 | 338.9 | 40.2 | 33.0 | 1572.0 | 10.9 |

Mode | 108.0 | 4.0 | 486.0 | 35.7 | 29.0 | 1640.0 | 6.0 |

Standard deviation | 43.9 | 2.3 | 79.1 | 13.3 | 11.9 | 985.3 | 9.0 |

Sample variance | 1927.1 | 5.1 | 6256.5 | 175.8 | 141.8 | 970,885.0 | 81.2 |

Kurtosis | 0.4 | 4.5 | 1.0 | 0.5 | 0.9 | 2.3 | 3.8 |

Skewness | 1.1 | 1.8 | 1.0 | 1.0 | 1.1 | 1.4 | 1.7 |

Range | 190.9 | 11.4 | 404.5 | 57.5 | 52.8 | 4892.0 | 45.2 |

Minimum | 76.5 | 1.4 | 200.2 | 25.5 | 20.2 | 508.0 | 4.8 |

Maximum | 267.4 | 12.8 | 604.7 | 83.0 | 73.0 | 5400.0 | 50.0 |

Sum | 8236.8 | 271.1 | 20,831.6 | 2534.8 | 2089.2 | 105,337.0 | 830.1 |

Count | 60 | 60 | 60 | 60 | 60 | 60 | 60 |

Variable | Diameter | Thickness | Steel Yield Strength | Compressive Strength | Length | Length/Diameter |
---|---|---|---|---|---|---|

Diameter | 1 | 0.367 | −0.197 | 0.123 | 0.246 | −0.293 |

Thickness | 0.367 | 1 | 0.031 | −0.041 | 0.091 | −0.102 |

Steel yield strength | −0.197 | 0.031 | 1 | 0.088 | −0.028 | 0.075 |

Compressive strength | 0.123 | −0.041 | 0.088 | 1 | −0.016 | −0.102 |

Length | 0.246 | 0.091 | −0.028 | −0.016 | 1 | 0.813 |

Length/diameter | −0.293 | −0.102 | 0.075 | −0.102 | 0.813 | 1 |

Parameter | Settings | |
---|---|---|

General | ||

Chromosomes | 30 | |

Genes | 3 | |

Head size | 8 | |

Gene size | 26 | |

Linking function | Addition | |

Function set | +, −, ×, ÷, √,^{3}√ | |

Genetic operators | ||

Mutation rate | 0.0138 | |

Inversion rate | 0.00546 | |

IS Transposition rate | 0.00546 | |

RIS transposition rate | 0.00546 | |

One-point recombination rate | 0.00277 | |

Two-point recombination rate | 0.00277 | |

Gene recombination rate | 0.00755 | |

Gene transposition rate | 0.00277 | |

Numerical constants | ||

Constants per gene | 10 | |

Data type | Floating Point | |

Lower bound | −10 | |

Upper bound | 10 |

Model | Experimental Axial Capacity vs. Predicted Axial Capacity | ||
---|---|---|---|

R^{2} | MAE | RMSE | |

Learning | 0.97 | 134.8 | 210.3 |

Validation | 0.98 | 153.9 | 226.1 |

Testing | 0.99 | 124.3 | 173.7 |

Equation No: | Code Specification | Ultimate Axial Moment Capacity (N_{U}) | Limitations |
---|---|---|---|

1 | AS5100.6 (2004) | ${N}_{u}={\alpha}_{c}\left[{\eta}_{a}{A}_{s}{f}_{y}+\left(1+\frac{{\eta}_{c}t{f}_{y}}{{d}_{o}{f}_{c}^{\prime}}\right){A}_{c}{f}_{c}^{\prime}\right]$ ${\alpha}_{c}=\xi \left(1-\sqrt{1-{\left(\frac{90}{\xi \lambda}\right)}^{2}}\right);\xi =\frac{{\left(\frac{\lambda}{90}\right)}^{2}+1+\eta}{2{\left(\frac{\lambda}{90}\right)}^{2}}$ $\lambda ={\lambda}_{n}+{\alpha}_{a}{\alpha}_{b};\eta =0.00326\left({\lambda}_{n}-13.5\right)\ge 0;{\lambda}_{n}=90{\lambda}_{r}$ ${\lambda}_{r}=\sqrt{\frac{{N}_{s}}{{N}_{cr}}};{N}_{s}={A}_{s}{f}_{y}+{A}_{c}{f}_{c}^{\prime}$ ${N}_{cr}=\frac{{\pi}^{2}{\left(EI\right)}_{eff}}{{l}^{2}};{\left(EI\right)}_{eff}={E}_{s}{I}_{s}+{E}_{c}{I}_{c}$ ${\alpha}_{a}=\frac{2100\left({\lambda}_{n}-13.5\right)}{{\lambda}_{n}^{2}-13.5{\lambda}_{n}+2050};{\alpha}_{b}=Presentedincode$ ${\eta}_{2}=0.25\left(3+2{\lambda}_{r}\right)\ge 0;{\eta}_{1}=4.9-18.5{\lambda}_{r}+17.5{\lambda}_{r}^{2}\ge 0$ | |

2 | AISC (2005) | ${N}_{u}={\varphi}_{c}{N}_{n};{\varphi}_{c}=0.75\left(LRFD\right)$ $If{N}_{e}\ge 0.44{N}_{o};{N}_{n}={N}_{o}\left[{0.658}^{\left(\frac{{N}_{o}}{{N}_{e}}\right)}\right]$ $If{N}_{e}0.44{N}_{o};{N}_{n}=0.877{N}_{e}$ ${N}_{o}={A}_{s}{f}_{y}+0.95{A}_{c}{f}_{c}^{\prime}$ ${N}_{e}=\frac{{\pi}^{2}{\left(EI\right)}_{eff1}}{{\left(KL\right)}^{2}};E{I}_{eff1}={E}_{s}{I}_{s}+{C}_{1}{E}_{c}{I}_{c}$ ${C}_{1}=0.1+2\left(\frac{{A}_{s}}{{A}_{c}+{A}_{s}}\right)\le 0.3;{E}_{c}={\left({f}_{c}^{\prime}\right)}^{\frac{1}{2}}\left(MPa\right)$ | $21MPa\le {f}_{c}^{\prime}\le 70MPa$ ${f}_{y}\le 525MPa$ ${A}_{s}\ge 0.01{A}_{g}$ $\frac{D}{t}\le \sqrt{\frac{8{E}_{s}}{{f}_{y}}}$ |

3 | BS5400 | ${\alpha}_{c}=\frac{0.45{f}_{cc}{A}_{c}}{{N}_{u}};0.1{\alpha}_{c}0.8$ ${N}_{u}=0.91{f}_{y}^{\prime}{A}_{s}+0.45{f}_{cc}{A}_{c}$ ${f}_{y}^{\prime}={C}_{2}{f}_{y};{f}_{cc}={f}_{cu}+{C}_{1}\frac{t}{D}{f}_{y}$ ${C}_{1}and{C}_{2}areconstantsdependson\frac{{l}_{e}}{D}$ | ${f}_{cu}\ge 20MPa$ ${f}_{y}=Grade43or50$ $\frac{D}{t}\le \sqrt{\frac{8{E}_{s}}{{f}_{y}}}$ $Nominalaggregatesize\le 20mm$ |

4 | DBJ (1999) | ${N}_{u}={\gamma}_{m}{f}_{scy}{W}_{scm}$ ${f}_{scy}=\left(1.18+0.85\xi \right){f}_{ck}$ ${W}_{scm}=\frac{\pi}{32}{D}^{3}$ $\xi =\frac{{A}_{s}{f}_{yk}}{{A}_{c}{f}_{ck}}$ ${\gamma}_{m}=1.04+0.48\mathrm{ln}\left(\xi +0.1\right)$ | $100mm\le D\le 2000mm$ $200MPa\le {f}_{scy}\le 500MPa$ $20MPa\le {f}_{ck}\le 80MPa$ |

5 | AIJ (2001) | ${N}_{u1}=0.85{A}_{c}{f}_{c}^{\prime}+\left(1+\eta \right){A}_{s}{f}_{y}\text{};\left(\frac{l}{D}\le 4\right)$ ${N}_{u2}={N}_{u1}-0.125\left\{{N}_{u1}-{N}_{u3}\right\}\left(\frac{l}{D}-4\right)\text{};\left(4\frac{l}{D}\le 12\right)$ ${N}_{u3}={A}_{c}{\sigma}_{cr}+{N}_{crs}\text{};\left(\frac{l}{D}12\right)$ ${\sigma}_{cr}=\frac{1.7{f}_{c}^{\prime}}{1+\sqrt[]{{\lambda}_{1}^{4}+1}}\text{};\text{}{\lambda}_{1}\le 1.0$ ${\sigma}_{cr}=0.83\mathrm{exp}\left\{\left(0.568+0.00612{f}_{c}^{\prime}\right)\left(1-{\lambda}_{1}\right)\right\}0.85{f}_{c}^{\prime}\text{};\text{}{\lambda}_{1}1$ ${\lambda}_{1}=\frac{\lambda}{\pi}\sqrt[]{0.93{\left(0.85{f}_{c}^{\prime}\right)}^{\frac{1}{4}}\times {10}^{-3}}$ ${N}_{crs}={A}_{s}{f}_{y}\text{};\text{}{\lambda}_{1}\le 0.3$ ${N}_{crs}=1-0.545\left({\lambda}_{1}-0.3\right)\text{};\text{}0.3\le {\lambda}_{1}1.3$ ${N}_{crs}=\frac{{N}_{Es}}{1.3}\text{};\text{}{\lambda}_{1}\ge 1.3$ ${\lambda}_{1}=\frac{\lambda}{\pi}\sqrt[]{\frac{{f}_{y}}{{E}_{s}}}$ ${N}_{Es}=\frac{{\pi}^{2}{E}_{s}{I}_{s}}{{l}^{2}}$ $\lambda =slenderness\text{}ratio\text{}of\text{}concrete\text{}column$ | $\frac{D}{t}\le \frac{35250}{{f}_{y}}$ |

6 | EC4 (2004) | ${N}_{u}={\eta}_{a}{A}_{s}{f}_{y}+\left(1+{\eta}_{c}\frac{t}{D}\frac{{f}_{y}}{{f}_{c}^{\prime}}\right){A}_{c}{f}_{c}^{\prime}$ ${\eta}_{a}=0.25\left(3+2\overline{\lambda}\right)\le 1.0\text{};\text{}{\eta}_{c}=4.9-18.5\overline{\lambda}+17{\overline{\lambda}}^{2}\ge 0$ $\overline{\lambda}=\frac{{N}_{pIR}}{{N}_{cr}};{N}_{pIR}={A}_{s}{f}_{y}+{A}_{c}{f}_{c}^{\prime}$ ${N}_{cr}=\frac{{\pi}^{2}\left(EI\right){}_{eff2}}{{l}_{2}};\left(EI\right){}_{eff2}={E}_{s}{I}_{s}+{K}_{c}{E}_{c}{I}_{c};{K}_{c}=0.6$ ${E}_{c}=\mathrm{22,000}{\left[\frac{\left({f}_{c}^{\prime}+8\right)}{10}\right]}^{0.3}\left(MPa\right)$ | $20MPa\le {f}_{c}^{\prime}\le 60MPa$ ${f}_{y}\le 460MPa$ $\frac{D}{t}\le \frac{0.15{E}_{s}}{{f}_{y}}$ |

Statistical Parameters | GEP | AS5100.6 | EC4 | AISC | BS | DBJ | AIJ |
---|---|---|---|---|---|---|---|

Rsq | 0.98 | 0.98 | 0.98 | 0.97 | 0.96 | 0.97 | 0.97 |

MAE | 138.7 | 249.4 | 220.6 | 333.5 | 205.0 | 228.0 | 194.4 |

RMSE | 258.0 | 484.7 | 452.9 | 701.4 | 352.9 | 512.0 | 408.4 |

$\mathbf{Row}\text{}\left(\mathit{\rho}\right)$ | 0.1 | 0.2 | 0.1 | 0.2 | 0.3 | 0.2 | 0.2 |

Average | 1.2 | 1.1 | 1.2 | 1.0 | 1.0 | 0.9 | 1.2 |

Maximum | 1.2 | 1.6 | 1.7 | 2.0 | 1.7 | 1.5 | 1.2 |

Minimum | 0.7 | 0.7 | 0.8 | 0.6 | 0.5 | 0.6 | 0.8 |

SD | 0.1 | 0.1 | 0.1 | 0.1 | 0.3 | 0.2 | 0.1 |

COV | 0.1 | 0.1 | 0.1 | 0.1 | 0.2 | 0.1 | 0.1 |

Sr. No | Formula | Condition | GEP |
---|---|---|---|

1 | Equation (5) | R > 0.8 | 0.973 |

2 | $K=\frac{{\sum}_{i=1}^{n}\left({x}_{i}\times {y}_{i}\right)}{{x}_{i}^{2}}$ | 0.85 < K < 1.15 | 0.983 |

3 | ${K}^{\prime}=\frac{{\sum}_{i=1}^{n}\left({x}_{i}\times {y}_{i}\right)}{{y}_{i}^{2}}$ | 0.85 < K′ < 1.15 | 1.003 |

4 | ${R}_{m}={R}^{2}\times \left(1-\sqrt{\left|{R}^{2}-{R}_{0}^{2}\right|}\right)$ | R_{m} > 0.5 | 0.838 |

${R}_{0}^{2}$ is squared correlation coefficient between predicted and experimental values | 0.999 |

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## Share and Cite

**MDPI and ACS Style**

Javed, M.F.; Farooq, F.; Memon, S.A.; Akbar, A.; Khan, M.A.; Aslam, F.; Alyousef, R.; Alabduljabbar, H.; Rehman, S.K.U.
New Prediction Model for the Ultimate Axial Capacity of Concrete-Filled Steel Tubes: An Evolutionary Approach. *Crystals* **2020**, *10*, 741.
https://doi.org/10.3390/cryst10090741

**AMA Style**

Javed MF, Farooq F, Memon SA, Akbar A, Khan MA, Aslam F, Alyousef R, Alabduljabbar H, Rehman SKU.
New Prediction Model for the Ultimate Axial Capacity of Concrete-Filled Steel Tubes: An Evolutionary Approach. *Crystals*. 2020; 10(9):741.
https://doi.org/10.3390/cryst10090741

**Chicago/Turabian Style**

Javed, Muhammad Faisal, Furqan Farooq, Shazim Ali Memon, Arslan Akbar, Mohsin Ali Khan, Fahid Aslam, Rayed Alyousef, Hisham Alabduljabbar, and Sardar Kashif Ur Rehman.
2020. "New Prediction Model for the Ultimate Axial Capacity of Concrete-Filled Steel Tubes: An Evolutionary Approach" *Crystals* 10, no. 9: 741.
https://doi.org/10.3390/cryst10090741