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Frontiers of Structural and Civil Engineering

Front. Struct. Civ. Eng.    2019, Vol. 13 Issue (1) : 103-109     https://doi.org/10.1007/s11709-018-0474-6
RESEARCH ARTICLE |
Multivariable regression model for Fox depth correction factor
Ravi Kant MITTAL1(), Sanket RAWAT1, Piyush BANSAL2
1. Department of Civil Engineering, Birla Institute of Technology & Science, Pilani 333031, India
2. Department of Civil and Environmental Engineering, Virginia Tech, Blacksburg, VA 24061, USA
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Abstract

This paper presents a simple and efficient equation for calculating the Fox depth correction factor used in computation of settlement reduction due to foundation embedment. Classical solution of Boussinesq theory was used originally to develop the Fox depth correction factor equations which were rather complex in nature. The equations were later simplified in the form of graphs and tables and referred in various international code of practices and standard texts for an unsophisticated and quick analysis. However, these tables and graphs provide the factor only for limited values of the input variables and hence again complicates the process of automation of analysis. Therefore, this paper presents a non-linear regression model for the analysis of effect of embedment developed using “IBM Statistical Package for the Social Sciences” software. Through multiple iterations, the value of coefficient of determination is found to reach 0.987. The equation is straightforward, competent and easy to use for both manual and automated calculation of the Fox depth correction factor for wide range of input values. Using the developed equation, parametric study is also conducted in the later part of the paper to analyse the extent of effect of a particular variable on the Fox depth factor.

Keywords settlement      embedment      Fox depth correction factor      regression      multivariable     
Corresponding Authors: Ravi Kant MITTAL   
Just Accepted Date: 03 May 2018   Online First Date: 01 June 2018    Issue Date: 04 January 2019
 Cite this article:   
Ravi Kant MITTAL,Sanket RAWAT,Piyush BANSAL. Multivariable regression model for Fox depth correction factor[J]. Front. Struct. Civ. Eng., 2019, 13(1): 103-109.
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http://journal.hep.com.cn/fsce/EN/10.1007/s11709-018-0474-6
http://journal.hep.com.cn/fsce/EN/Y2019/V13/I1/103
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Fig.1  Fox’s depth correction curves for ν=0.5 [4,6]
Df/B L/B
1.0 1.2 1.4 1.6 1.8 2.0 5.0
Poisson’s ratio= 0.00
0.05 0.950 0.954 0.957 0.959 0.961 0.963 0.973
0.10 0.904 0.911 0.917 0.922 0.925 0.928 0.948
0.20 0.825 0.838 0.847 0.855 0.862 0.867 0.903
0.40 0.710 0.727 0.740 0.752 0.761 0.769 0.827
0.60 0.635 0.652 0.666 0.678 0.689 0.698 0.769
0.80 0.585 0.600 0.614 0.626 0.637 0.646 0.723
1.00 0.549 0.563 0.576 0.587 0.598 0.607 0.686
2.00 0.468 0.476 0.484 0.492 0.499 0.506 0.577
Poisson’s ratio= 0.1
0.05 0.958 0.962 0.965 0.967 0.968 0.970 0.978
0.10 0.919 0.926 0.930 0.934 0.938 0.940 0.957
0.20 0.848 0.859 0.868 0.875 0.881 0.886 0.917
0.40 0.739 0.755 0.786 0.779 0.788 0.795 0.848
0.60 0.665 0.682 0.696 0.708 0.718 0.727 0.793
0.80 0.615 0.630 0.644 0.656 0.667 0.676 0.749
1.00 0.579 0.593 0.606 0.618 0.628 0.637 0.714
2.00 0.496 0.505 0.513 0.521 0.528 0.535 0.606
Poisson’s ratio= 0.3
0.05 0.979 0.981 0.982 0.983 0.984 0.985 0.990
0.10 0.954 0.958 0.962 0.964 0.966 0.968 0.977
0.20 0.902 0.911 0.917 0.923 0.927 0.930 0.951
0.40 0.808 0.823 0.834 0.843 0.851 0.857 0.899
0.60 0.738 0.754 0.767 0.778 0.788 0.796 0.852
0.80 0.687 0.703 0.716 0.728 0.738 0.747 0.813
1.00 0.650 0.665 0.678 0.689 0.700 0.709 0.780
2.00 0.562 0.571 0.580 0.588 0.596 0.603 0.675
Poisson’s ratio= 0.4
0.05 0.989 0.990 0.991 0.992 0.992 0.993 0.995
0.10 0.973 0.976 0.978 0.980 0.981 0.982 0.988
0.20 0.932 0.940 0.945 0.949 0.952 0.955 0.970
0.40 0.848 0.862 0.872 0.881 0.887 0.893 0.927
0.60 0.779 0.795 0.808 0.819 0.828 0.836 0.886
0.80 0.727 0.743 0.757 0.769 0.779 0.788 0.849
1.00 0.689 0.704 0.718 0.730 0.740 0.749 0.818
2.00 0.596 0.606 0.615 0.624 0.632 0.640 0.714
Poisson’s ratio= 0.5
0.05 0.997 0.997 0.998 0.998 0.998 0.998 0.999
0.10 0.988 0.990 0.991 0.992 0.993 0.993 0.996
0.20 0.960 0.966 0.969 0.972 0.974 0.976 0.985
0.40 0.886 0.899 0.908 0.916 0.922 0.926 0.953
0.60 0.818 0.834 0.847 0.857 0.866 0.873 0.917
0.80 0.764 0.781 0.795 0.807 0.817 0.826 0.883
1.00 0.723 0.740 0.754 0.766 0.777 0.786 0.852
2.00 0.622 0.633 0.643 0.653 0.662 0.670 0.747
Tab.1  Fox depth correction factor for different input range [14]
S. No. developed models R2
1 If=0.021L B0.188 DfB+0.306ν+0.796 0.857
2 If=0.6 ( DfB)0.16 (L B)0.075× 1.51ν 0.878
3 If=0.129log L B 0.287logDfB+0.307ν+0.583 0.958
4 If=0.475 (DfB )0.376 (L B) 0.171+ 1.064× 1.271ν 0.9703
5 If=0.626 (DfB )0.331 (L B) 0.184× 0.49ν+1.181 (L B) 0.0103 0.9814
6 If=0.580 (DfB )0.356 (L B) 0.189× 0.544ν+1.14 (L B) 0.008× 1.055ν 0.9820
7 If=0.585 (DfB )0.355 (L B) 0.189× 0.537ν+1.138 (L B) 0.008+ 0.007× 23.78ν 0.9823
8 If=1.001+1.194DfB+0.842 L B+7.63ν1+3.738DfB+0.839 L B+7.3ν 0.9874
Tab.2  Trial Regression Models for Fox depth correction factor
Fig.2  Comparison curve for predicted values relative to the actual ones
Fig.3  Effect of conjoint variation of Df/B and L/B on Fox depth factor
Fig.4  Effect of conjoint variation of Df/B and Poisson’s ratio on Fox depth factor
Fig.5  Effect of conjoint variation of Poisson’s ratio and L/B on Fox depth factor
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