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This study examines the potential of two soft computing techniques, namely, support vector machines (SVMs) and genetic programming (GP), to predict ultimate bearing capacity of cohesionless soils beneath shallow foundations. The width of footing (

Design of foundations is performed based on two criteria: ultimate bearing capacity and limiting settlement. The ultimate bearing capacity is governed by shear strength of the soil and is estimated by theories proposed by Terzaghi [

In the recent past, the use of soft computing techniques has attracted many researchers and applied quite successfully for solving many complex geotechnical engineering problems. Artificial neural networks (ANNs) may probably be the most popular among these tools, applied for prediction of bearing capacity of cohesionless soils [

The evolutionary computational techniques may be a better alternative for solving regression problems as they follow an optimization strategy with progressive improvement towards the global optima. They start with possible trial solutions within a decision space, and the search is guided by genetic operators and the principle of “survival of the fittest” [

Support vector machine (SVM) is a relatively recent addition to the family of soft computing techniques evolved from the concept of statistical learning theory explored by Boser et al. [

The intension of SVM is to fit a function that can approximately predict the value of output on supplying a new set of predictors (input variables).

The

The

SVM attempts to find that a function

Consider a linear function of the form,

Some Lagrange multipliers (

Thus, (

Concept of nonlinear regression using SVM.

The functions which satisfy Mercer’s theorem can be used for fitting the data [

Genetic Programming (GP) is an automatic programming technique for evolving computer programs to solve, or approximately solve, problems introduced by Koza [

Flow chart of Genetic Programming.

In the recent past, GP is effectively applied to solve a wide range of geotechnical engineering problems [

The primary step in model development for the estimation of bearing capacity of cohesionless soils underneath shallow foundations is identification of parameters that affect the bearing capacity. The basic form of equation for bearing capacity of cohesionless soil is [

The main factors affecting the bearing capacity are its width (least lateral dimension,

The data used in the present study has been adopted from Padmini et al. [

The data mining software WEKA 3.6.1 proposed by Witten and Frank [

A trial and error approach is followed to find the optimal value of

Scatter plot of SVM model with polykernel for training dataset.

Scatter plot of SVM model with RBF Kernel for training dataset.

The genetic programming software DISCIPULUS [

Scatter plot of GP model for training dataset.

The efficiency of the developed models is analyzed by different statistical performance evaluation criteria such as correlation coefficient (

Performance evaluation criteria.

Evaluation criteria | Equation |
---|---|

Coefficient of correlation ( | |

Coefficient of efficiency ( | |

Root-mean-square error (RMSE) | |

Mean bias error (MBE) | |

Percentage relative error (RE) | |

Percentage mean absolute relative error (MARE) |

Performance evaluation of different models for training dataset.

Performance index |
Meyerhof [ |
Hansen [ |
Vesic [ | SVM* | GP* | |
---|---|---|---|---|---|---|

Polykernel | RBF kernel | |||||

0.9307 | 0.9295 | 0.9408 | 0.9742 | 0.9977 | 0.9923 | |

0.7387 | 0.6908 | 0.7208 | 0.9408 | 0.9948 | 0.9900 | |

RMSE (kPa) | 260.0015 | 282.8295 | 268.7726 | 123.7923 | 36.8280 | 65.0650 |

MBE (kPa) | −78.3587 | −95.8701 | −103.2620 | −9.5919 | −3.5807 | −0.8618 |

MARE | 19.6545 | 20.9268 | 25.6619 | 19.3968 | 1.8036 | 12.9030 |

Predicted bearing capacity of different models for testing dataset.

Sl no. | Observed bearing capacity (kPa) | Meyerhof [ | Hansen [ | Vesic [ | ANN (kPa) | FIS (kPa) | SVM (kPa) | GP | |

RBF Kernel | Polykernel | ||||||||

1 | 1760 | 1174.892 | 1230.649 | 1194.571 | 1753.048 | 1888.48 | 2137.096 | 2005.19 | 1794.292 |

2 | 214 | 163.54 | 164.0827 | 117.9378 | 221.1134 | 195.1252 | 257.952 | 165.809 | 211.9048 |

3 | 681 | 353.1485 | 322.5613 | 372.5071 | 579.7285 | 651.3084 | 999.874 | 676.838 | 639.0338 |

4 | 137 | 114.1407 | 124.0721 | 94.67636 | 161.6463 | 100.6128 | 125.435 | 143.271 | 152.3006 |

5 | 322 | 372.0038 | 365.65 | 329.434 | 226.6236 | 150.2774 | 304.729 | 240.232 | 311.4622 |

6 | 2033 | 1757.721 | 1641.436 | 1510.498 | 2047.19 | 2164.129 | 1863.654 | 2012.747 | 2077.698 |

7 | 464 | 560.7228 | 530.9029 | 542.3884 | 475.0664 | 657.024 | 540.622 | 816.999 | 572.9832 |

8 | 461 | 270 | 295.4085 | 214.7597 | 348.0043 | 274.1106 | 409.256 | 368.095 | 484.8436 |

9 | 1140 | 865.35 | 841.3887 | 780.5128 | 1064.874 | 1097.022 | 1023.901 | 1059.188 | 1118.019 |

10 | 630 | 516.9667 | 565.6161 | 411.1984 | 512.1459 | 541.044 | 667.666 | 632.805 | 498.9244 |

11 | 1540 | 697.3765 | 608.3824 | 664.3541 | 1626.086 | 1683.99 | 1691.924 | 1271.259 | 1558.777 |

12 | 180.5 | 139.4636 | 134.7036 | 115.906 | 269.2338 | 241.4368 | 188.525 | 175.087 | 217.0617 |

13 | 91.5 | 101.6777 | 88.88982 | 80.81764 | 99.8448 | 107.0916 | 93.131 | 77.511 | 88.75574 |

14 | 244.6 | 290.428 | 279.7286 | 249.5037 | 233.3973 | 230.3643 | 242.907 | 250.901 | 250.7569 |

15 | 143.3 | 138.7347 | 138.3666 | 119.3544 | 108.4208 | 151.7547 | 139.657 | 138.799 | 127.4984 |

16 | 131.5 | 144.9232 | 135.2796 | 162.6324 | 130.6124 | 156.5113 | 129.475 | 109.495 | 153.612 |

17 | 253.6 | 283.7822 | 270.4981 | 333.1746 | 226.1098 | 275.0038 | 251.614 | 255.153 | 266.6882 |

18 | 135.2 | 125.613 | 121.5267 | 136.3814 | 128.8321 | 178.5992 | 135.021 | 131.044 | 133.1664 |

19 | 264.5 | 329.4678 | 314.9475 | 383.6036 | 198.7189 | 347.2885 | 272.425 | 386.591 | 245.1108 |

Performance evaluation of different models for testing dataset.

Performance index |
Meyerhof [ |
Hansen [ |
Vesic [ |
ANN [ |
FIS [ | SVM* | GP* | |
---|---|---|---|---|---|---|---|---|

Polykernel | RBF Kernel | |||||||

0.9410 | 0.9366 | 0.9456 | 0.9951 | 0.9899 | 0.9775 | 0.9806 | 0.9972 | |

0.7863 | 0.7583 | 0.7321 | 0.9942 | 0.9858 | 0.9541 | 0.9504 | 0.9965 | |

RMSE (kPa) | 269.947 | 287.099 | 302.269 | 62.620 | 98.002 | 125.087 | 130.102 | 44.967 |

MBE (kPa) | −127.724 | −139.611 | −158.552 | −21.89 | 13.92 | 34.11 | 4.75 | 4.019 |

MARE | 22.0679 | 20.0152 | 28.515 | 13.314 | 19.456 | 14.7278 | 9.4528 | 7.6817 |

Further the scatter plots between observed and predicted values of UBC for SVM models are presented in Figure

Scatter plot and 5% error bar lines for SVM model with polykernel (testing dataset).

Scatter plot and 5% error bar lines for SVM model with RBF Kernel (testing dataset).

Scatter plot and 5% error bar lines for GP model (testing dataset).

A statistical evaluation of the predictions by the different soft computing models for the testing dataset is performed and presented in Table

Statistical properties of values predicted by different models.

Statistical Measure | Observed | ANN | FIS | SVM (Poly)* | SVM (RBF)* | GP* |
---|---|---|---|---|---|---|

Maximum | 2033 | 2047.19 | 2164.13 | 2137.1 | 2077.7 | 2077.7 |

Minimum | 91.5 | 99.84 | 100.61 | 93.13 | 77.51 | 88.75 |

Average | 569.83 | 547.93 | 583.7 | 603.9 | 574.58 | 573.84 |

Average deviation | 459.48 | 455.92 | 488.4 | 501.1 | 468.73 | 454.59 |

Standard deviation | 600.02 | 609.3 | 645.8 | 641.0 | 608.60 | 607.91 |

Coefficient of variation | 1.052 | 1.112 | 1.106 | 1.059 | 1.059 | 1.059 |

The different performance evaluation measures of SVM-based modelling (in Tables

In this paper the application of two relatively recent soft computing techniques—SVM and GP—is investigated for the prediction of ultimate bearing capacity of cohesionless soils beneath shallow foundations. SVM results are competent and demand the optimal selection of only a few number of control parameters when compared with ANN. Performance evaluation based on multiple error criteria shows that error is the least and correlation coefficient (

The C++ Program to predict the ultimate bearing capacity of cohesionless soils is given here. V[0] to V[4] represent the input parameters width of footing (

#define TRUNC(x)(((x)>=0) ? floor(x): ceil(x)) #define C_FPREM (_finite(f[0]/f[1]) ? f[0]-(TRUNC(f[0]/f[1]) #define C_F2XM1 (((fabs(f[0])<=1) && float DiscipulusCFunction(float v[])
{
long double f[8]; long double tmp = 0; int cflag = 0; f[0]=f[1]=f[2]=f[3]=f[4]=f[5]=f[6]=f[7]=0; L0: f[0]/=-1.364008665084839f; L1: f[0]+=f[1]; L2: f[0]=−f[0]; L3: f[0]−=v[0]; L4: f[0]+=v[4]; L5: f[0]+=v[4]; L6: f[0]=cos(f[0]); L7: f[0]+=f[0]; L8: f[0]+=f[0]; L9: f[0]+=f[0]; L10: f[0]*=v[1]; L11: f[0]+=v[4]; L12: f[0]−=1.252994060516357f; L13: f[0]*=pow(2,TRUNC(f[1])); L14: cflag=(f[0] < f[1]); L15: f[0]=sqrt(f[0]); L16: f[0]*=0.2877938747406006f; L17: tmp=f[1]; f[1]=f[0]; f[0]=tmp; L18: f[0]−=v[3]; L19: f[0]*=−0.494312047958374f; L20: f[0]*=0.7790718078613281f; L21: f[0]*=f[1]; L22: f[0]=fabs(f[0]); L23: f[0]=cos(f[0]); L24: f[0]=−f[0]; L25: f[0]=sqrt(f[0]); L26: if (cflag) f[0] = f[1]; L27: f[0]+=v[4]; L28: f[0]*=0.9955191612243652f; L29: f[0]*=0.4281637668609619f; L30: f[0]−=f[0]; L31: f[0]−=v[3]; L32: f[0]/=v[0]; L33: f[0]−=0.9955191612243652f; L34: f[0]+=v[4]; L35: cflag=(f[0] < f[1]); L36: f[0]−=f[1]; L37: f[0]/=f[0]; L38: f[0]*=pow(2,TRUNC(f[1])); L39: f[0]−=f[1]; L40: f[0]=−f[0]; L41: f[0]=fabs(f[0]); L42: f[0]=−f[0]; L43: f[0]*=0.4281637668609619f; L44: f[0]*=f[0]; L45: f[0]*=f[0]; L46: f[0]/=f[0]; L47: f[0]*=−0.7297487258911133f; L48: f[0]/=1.084159851074219f; L49: f[0]+=v[4]; L50: tmp=f[0]; f[0]=f[0]; f[0]=tmp; L51: f[0]/=f[1]; L52: f[0]+=0.7790718078613281f; L53: f[0]+=v[4]; L54: f[0]−=f[1]; L55: f[0]*=f[1]; L56: f[0]*=0.4281637668609619f; L57: f[0]−=v[3]; L58: f[0]−=−0.9486191272735596f; L59: if (cflag) f[0] = f[1]; L60: f[0]/=v[2]; L61: f[0]+=v[4]; L62: f[0]−=v[2]; L63: f[0]−=v[2]; L64: f[0]−=v[2]; L65: f[0]*=v[1]; L66: f[0]+=v[4]; L67: f[0]*=f[1]; L68: f[0]*=0.4281637668609619f; L69: f[0]−=v[3]; L70: f[1]*=f[0]; L71: f[0]*=f[0]; L72: f[0]−=f[1]; L73: f[1]+=f[0]; L74: if (!cflag) f[0] = f[1]; L75: f[0]−=1.987620830535889f; L76: f[0]−=v[4]; L77: f[0]−=1.987620830535889f; L78: f[0]−=v[4]; L79: f[0]−=v[4]; L80: f[0]−=0.7361507415771484f; L81: f[0]−=1.987620830535889f; L82: f[0]−=v[4]; L83: f[0]−=1.987620830535889f; L84: f[0]−=v[4]; L85: f[0]−=1.501374244689941f; L86: f[0]+=v[2]; L87: f[0]−=v[4]; L88: f[0]−=1.530829906463623f; L89: if (!cflag) f[0] = f[1]; L90: f[0]+=v[3]; L91: f[0]−=v[4]; L92: f[0]+=−1.907608032226563f; L93: if (!cflag) f[0] = f[1]; L94: f[0]+=v[3]; L95: f[0]+=v[3]; L96: f[0]+=v[3]; L97: f[0]+=v[3]; L98: f[0]+=v[3]; L99: f[0]+=v[3]; L100: f[0]+=v[3]; L101: f[0]+=v[3]; L102: f[0]+=v[3]; L103: f[0]+=v[3]; L104: f[0]+=v[3]; L105: f[0]+=v[3]; L106: f[0]+=v[3]; L107: f[0]+=v[3]; L108: f[0]+=v[2]; L109: f[0]+=v[3]; L110: if (!_finite(f[0])) f[0]=0; return f[0]; }.

This paper is a part of a research work carried out at the Department of Civil Engineering, TKM College of Engineering Kollam, Kerala, India, in 2010. The authors thank the Department of Civil Engineering, TKM College of Engineering Kollam for providing all necessary help. They also thank the anonymous reviewer/s who helped to improve the quality of the paper.