New design equations for assessment of load carrying capacity of castellated steel beams: a machine learning approach


This paper presents an innovative machine learning approach for the formulation of load carrying capacity of castellated steel beams (CSB). New design equations were developed to predict the load carrying capacity of CSB using linear genetic programming (LGP), and an integrated search algorithm of genetic programming and simulated annealing, called GSA. The load capacity was formulated in terms of the geometrical and mechanical properties of the castellated beams. An extensive trial study was carried out to select the most relevant input variables for the LGP and GSA models. A comprehensive database was gathered from the literature to develop the models. The generalization capabilities of the models were verified via several criteria. The sensitivity of the failure load of CSB to the influencing variables was examined and discussed. The employed machine learning systems were found to be effective methods for evaluating the failure load of CSB. The prediction performance of the optimal LGP model was found to be better than that of the GSA model.

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Correspondence to Pejman Aminian.


Appendix 1: The BEST LGP solution for the prediction of the load carrying capacity of CSB

The optimum LGP program can be compiled in any C++ environment. (Note: v[0], …, v[3], respectively, represent d g, h, t w × F yw, and e. f[0] is the output parameter.)


double f[8];
double tmp = 0;
f[1] = f[2] = f[3] = f[4] = f[5] = f[6] = f[7] = 0;
f[0] = v[0];
l0: f[0]/ = v[3];
l1: tmp = f[1]; f[1] = f[0]; f[0] = tmp;
l2: f[0]+ =f[1];
l3: f[0] = sqrt(f[0]);
l4: tmp = f[1]; f[1] = f[0]; f[0] = tmp;
l5: f[0]− = 8;
l6: f[0]− = 4;
l7: f[0]* = 3;
l8: f[0]* = f[1];
f[1]/ = f[0];
l9: f[0]− = 9;
l10: f[0]+ = v[0];
l11: f[0]− = v[1];
l12: f[0]/ = 5;
l13: f[0]+ = v[2];
l14: f[0]/ = 9;
l15: f[0]* = f[0];
f[0]− = f[1];
l16: f[0]− = v[1];
l17: f[0]− = v[1];
l18: f[0]− = 5;
l19: f[0]+ = v[0];
l20: f[0]− = v[1];
l21: f[0]/ = 5;
return f[0];

Appendix 2: The best GSA solution for the prediction of the load carrying capacity of CSB

The optimum GSA program can be compiled in any C++ environment.


double f[8];
double tmp = 0;
f[1] = f[2] = f[3] = f[4] = f[5] = f[6] = f[7] = 0;
f[0] = v[0];
l0: f[0]− = v[1];
l1: f[0]+ = 4;
l2: f[0]+ = v[2];
l3: f[0]− = v[1];
l4: f[0]+ = v[2];
l5: f[0]* = f[0];
tmp = f[1]; f[1] = f[0]; f[0] = tmp;
l6: f[0]− = 8;
l7: f[0]− = 5;
l8: f[0]+ = 1;
l9: f[0]* = 7;
l10: f[1]/ = f[0];
f[0]− = f[1];
l11: f[0]− = v[3];
l12: f[0]+ = v[2];
l13: f[0]/ = 5;
l14: f[0]/ = 5;
return f[0];

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Aminian, P., Niroomand, H., Gandomi, A.H. et al. New design equations for assessment of load carrying capacity of castellated steel beams: a machine learning approach. Neural Comput & Applic 23, 119–131 (2013).

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  • Castellated beam
  • Load carrying capacity
  • Linear genetic programming
  • Simulated annealing
  • Formulation