Multivariable regression model for Fox depth correction factor

MITTAL Ravi Kant¹^,^,, RAWAT Sanket¹, BANSAL Piyush²

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

ravimittal@pilani.bits-pilani.ac.in

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.

Key words: settlement; embedment; Fox depth correction factor; regression; multivariable;

Introduction

Depth correction factor is required to incorporate the effect of depth of embedment on settlement of footing. Its incorporation might lead to up to 50% reduction in the calculated settlement and an increase up to 100% in the calculated allowable pressure, hence results in an economical design of foundation. Various depth factors have been proposed in the existing literature to integrate the effect of embedment [¹–³]. However, these are limited only to sands ( $ν$ =0.3). Conversely, Fox [⁴] recommended a generalized formula valid for a wide range of input variables to calculate the depth correction factor. The equation idea was originally developed by Mindlin [⁵] who tried to extend the classical Boussinesq solutions to a more generalised loading case. Mindlin’s analysis was directly applied to an element of variable load acting at a point to evaluate its vertical displacement. Through applying proper boundary conditions and integration with regard to the coordinate axes, Fox depth correction factor ( $I_{f}$ ) was proposed and expressed as (1) $I f = \frac{\sum_{s = 1}^{5} β s Y s}{(β 1 + β 2) Y 1},$ where(2) $β_{1} = 3 - 4 ν,$ (3) $β_{2} = 5 - 12 ν + 8 ν^{2},$ (4) $β_{3} = - 4 ν (1 - 2 ν),$ (5) $β_{4} = - 1 + 4 ν - 8 ν^{2},$ (6) $β_{5} = - 4 {(1 - 2 ν)}^{2},$ (7) $r = 2 D_{f},$ (8) ${r_{1}}^{2} = L^{2} + r^{2},$ (9) ${r_{2}}^{2} = B^{2} + r^{2},$ (10) $r_{3}^{2} = L^{2} + B^{2} + r^{2}$ (11) $r_{4}^{2} = L^{2} + B^{2},$ (12) $Y_{1} = L \ln \frac{r_{4} + B}{L} + B \ln \frac{r_{4} + L}{B} - \frac{r_{4}^{3} - L^{3} - B^{3}}{3 L B},$ (13) $Y_{2} = L \ln \frac{r_{3} + B}{r_{1}} + B \ln \frac{r_{3} + L}{r_{2}} - \frac{r_{3}^{3} - r_{2}^{3} - r_{1}^{3} + r^{3}}{3 L B},$ (14) $Y_{3} = \frac{r^{2}}{L} \ln \frac{(B + r_{1}) r_{1}}{(B + r_{3}) r} + \frac{r^{2}}{B} \ln \frac{(L + r_{1}) r_{2}}{(L + r_{3}) r},$ (15) $Y_{4} = \frac{r^{2} (r_{1} + r_{2} - r_{3} - r)}{L B},$ (16) $Y_{5} = r \tan^{- 1} \frac{L B}{r r_{3}},$ where D_f is the depth of footing, L is the length of footing, B is the breadth of footing, and $ν$ is Poisson’s ratio.

Fox’s depth correction factor is used worldwide by geotechnical practitioners and has been reported in numerous researchers, code of practices and standard texts [⁶–¹³]. In all the existing literature, its adoption is essentially preferred in the form of simplified graphs and tables since the formulae given by Fox [⁴] are complex which makes the associated calculation time consuming.

Indian Standard IS 8009 [⁶], Hong Kong Code of practice for foundations [⁷], Bowles [⁸] and Das [¹⁰,¹¹] provide the factor in the form of graphs. The use of these complex graphs in the settlement analysis of foundation makes the process tedious as the user has to take the input from the graphs every time to estimate the settlement. Also, the risk of manual error always goes along. Moreover, Bowles [⁸] had used a semi-log graph, and therefore, for intermediate values, simple linear interpolation also does not work. Another drawback of these graphs is that they are valid only for specific values of the Poisson’s ratio (IS 8009 [⁶] is valid for $ν$ = 0.5 only as shown in Fig. 1 and Bowles [⁸] is valid for $ν$ = 0.3 or 0.5). Hence, one has to rely on the intricate equations for the analysis if the Poisson’s ratio is other than 0.3 or 0.5.

Fig.1 Fox’s depth correction curves for $ν$ =0.5 [⁴,⁶]

Bowles [¹⁴] solved the Fox equation [⁴] for different combinations of the inputs and provided the factor in the form of a table (Table 1).

Tab.1 Fox depth correction factor for different input range [¹⁴]

D_f/B	L/B
D_f/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]

Though the table is easy to read as compared to the graphs, however linear interpolation would be required for several inputs variables. In those cases, it becomes inconvenient to incorporate the tables in the automated design process. Instead, a simple model with fewer terms based on the values given by Fox [⁴] will make the computation of the same much faster and easier. The accuracy in the estimation of the factor will further improve as compared to the manual estimation of the value from the graphs and tables. Furthermore, it will also improve the computational efficiency of the optimization study. Therefore, this paper presents a convenient equation to estimate the Fox depth correction factor; valid for a broad range of input variables, using IBM Statistical Package for the Social Sciences (SPSS) software.

Regression model

Fox equations [⁴] were incorporated in MATLAB routine and 19992 data points of Fox depth correction factor were developed with the use of different combinations of input variables i.e., varying D_f/B at interval of 0.01 within range 0.05 to 2; L/B at an interval of 0.25 within 1 to 5 range and Poisson’s ratio at an interval of 0.1 within 0 to 0.5 range. Numerous regression models were attempted for their accuracy to reach out to the most accurate ones in IBM SPSS software. For each cycle, the standard regression procedure was repeated. Different combinations in terms of arithmetic operations were tried along with separate combinations for curve fitting for each case. Some of the examples of the attempted regression models have been described in Table 2.

Tab.2 Trial Regression Models for Fox depth correction factor

S. No.	developed models	R²
1	$I f = 0.021 L B − 0.188 D f B + 0.306 ν + 0.796$	0.857
2	$I f = 0.6 (D f B) − 0.16 (L B) 0.075 × 1.51 ν$	0.878
3	$I f = 0.129 log L B − 0.287 log D f B + 0.307 ν + 0.583$	0.958
4	$I f = − 0.475 (D f B) 0.376 (L B) − 0.171 + 1.064 × 1.271 ν$	0.9703
5	$I f = − 0.626 (D f B) 0.331 (L B) − 0.184 × 0.49 ν + 1.181 (L B) − 0.0103$	0.9814
6	$I f = − 0.580 (D f B) 0.356 (L B) − 0.189 × 0.544 ν + 1.14 (L B) − 0.008 × 1.055 ν$	0.9820
7	$I f = − 0.585 (D f B) 0.355 (L B) − 0.189 × 0.537 ν + 1.138 (L B) − 0.008 + 0.007 × 23.78 ν$	0.9823
8	$I f = 1.001 + 1.194 D f B + 0.842 L B + 7.63 ν 1 + 3.738 D f B + 0.839 L B + 7.3 ν$	0.9874

Tab.2 Trial Regression Models for Fox depth correction factor

Results and discussion

Equation (17) depicts the finalized model of the non-linear regression analysis performed using IBM SPSS Statistics for the correction factor for depth of embedment.

Proposed equation of Fox depth correction factor:(17) $I_{f} = \frac{1.001 + 1.194 \frac{D_{f}}{B} + 0.842 \frac{L}{B} + 7.63 ν}{1 + 3.738 \frac{D_{f}}{B} + 0.839 \frac{L}{B} + 7.3 ν} .$

To illustrate the accuracy of the developed equation, data mentioned in Bowles [¹⁴] table (Table 1) is compared with the values predicted by the developed equation. This regression model shows a good agreement between the predicted values and the actual values (Fig. 2) since its coefficient of determination is reported as 0.9874. It has been found that more than 98.6% of the data has a percentage error within ± 5%. The generalized equation for calculating the Fox depth correction factor is uncomplicated, straightforward and valid for D_f/B = 0.05 to 2, L/B = 1 to 5 and $ν$ = 0 to 0.5. The equation can be easily used by the practitioners to automate the process of analysis of settlement of foundation.

Fig.2 Comparison curve for predicted values relative to the actual ones

Parametric study

The parametric studies are critical in quantifying the extent of effect of a particular parameter on the output [¹⁵]. The main variables in the developed equation are D_f/B, L/B and Poisson’s ratio. Therefore, to quantify the exact effect of different variables on Fox depth correction factor, the parametric study becomes an important basis.

Fig.3 Effect of conjoint variation of D_f/B and L/B on Fox depth factor

Effect of D_f/B and L/B factors conjointly on Fox depth factor

Figure 3(a) depicts the change in the Fox depth correction factor with respect to D_f/B for different L/B ratio. The Poisson’s ratio is assumed to a constant value of 0.3 here. It can be clearly observed that on increase in the D_f/B ratio, the depth factor decreases considerably as increase from 0.05 to 2 leads to a reduction of about 43.01% in the Fox depth correction factor for L/B =1. As the L/B ratio is increased, the percentage reduction decreases since, at L/B = 5 it is found to be 33.35% in comparison to 43.01% at L/B = 1. To get a clear picture of effect of L/B effect on Fox depth factor, effect of variation of L/B factor is studied for specific D_f/B ratios (Fig. 3(b)). It can be clearly observed that the variation of L/B does not influence the Fox depth factor much for low D_f/B ratio (only 0.41% increase after increasing L/B ratio from 1 to 5). However, at higher D_f/B ratios, this effect can be enhanced since 17.40% increase in Fox depth correction factor was observed on increasing the L/B ratio from 1 to 5 at D_f/B = 2. Overall, it can be concluded that at a constant Poisson’s ratio, influence of D_f/B factor on Fox depth factor is very high in comparison to the influence of L/B factor. However, a high L/B factor at high D_f/B ratio can intensify the effect.

Effect of D_f/B and Poisson’s ratio conjointly on Fox depth factor

The effect of variation of D_f/B ratio at different Poisson’s ratio (L/B fixed as constant to 2) follows the same pattern as the behaviour at different L/B ratio (Fig. 3). Figure 4(a) clearly depicts that on increase in the D_f/B ratio at specific Poisson’s ratio, the Fox depth factor reduces substantially. This behaviour is slightly enhanced than the one observed at specific L/B ratio (Fig. 3(a)). Increase in D_f/B ratio from 0.05 to 2 leads to a reduction of about 45.04% in the Fox depth correction factor for Poisson’s ratio= 0.1. Figure 4(b) separately depicts the effect of Poisson’s ratio on Fox depth correction factor and proves the significance of the enhanced behaviour observed in the present case in comparison to the previous section. At low D_f/B ratio, the effect of variation of Poisson’s ratio on Fox depth factor follows the same pattern as Fig. 3, since only 3.19% increase is observed on increasing the Poisson’s ratio to 5 times. However, this behaviour is enhanced at higher D_f/B ratio as the percentage increase goes up to 20.08%. Therefore, it can be concluded that the influence of D_f/B factor on Fox depth factor is high than the Poisson’s ratio at a constant L/B ratio. However, in comparison to L/B ratio (Fig. 3), the effect of Poisson’s ratio on Fox depth factor is slightly extensive.

Fig.4 Effect of conjoint variation of D_f/B and Poisson’s ratio on Fox depth factor

Effect of L/B and Poisson’s ratio conjointly on Fox depth factor

Now, as it has been already substantiated that the effect of D_f/B factor is most significant on Fox depth correction factor. Also, Poisson’s ratio (at constant L/B) affects the Fox depth factor somewhat considerably as compared to L/B factor (at constant Poisson’s ratio). Moreover, this behaviour is more evident at high D_f/B ratio. Figure 5 further validates the effect of mutual variation of L/B and Poisson’s ratio at constant D_f/B ratio (D_f/B =1). It can be clearly observed from Fig. 5 that the effect of variation of Poisson’s ratio on Fox depth factor at a specific L/B follows the same pattern as the reverse variation. In both the cases, the increase of 5 times in the abscissa value lead to approx. 23% increase in the Fox depth factor. At higher L/B ratio (Fig. 5(a)) or higher Poisson’s ratio (Fig. 5(b)), the percentage increase is observed to be reduced but is essentially the same for both the cases. Therefore, it can be concluded that though the mutual variation of L/B and Poisson’s ratio affect the Fox depth factor, this effect tends to reduce at their higher ratios.

Fig.5 Effect of conjoint variation of Poisson’s ratio and L/B on Fox depth factor

Advantages of the developed equation

Optimization studies

Economy in the design of structural members has gain significant attention recently along with the earlier objectives of safety and serviceability. However, this design process is primarily a trial and error procedure depending upon the load combinations obtained during analysis. Multiple trials are performed to reach to a design that incurs the less construction cost. Since, performing these multiple iterations manually, is a tedious and time-consuming process, hence optimization studies started gaining immense attention to lower down the computational cost. Numerous studies have been performed to optimize the foundation while successively improving either the type of constraints functions or number of design variables [¹⁶–²²]. On the whole. it can be believed that settlement of footing is an important constraint in the optimization study of foundations. Moreover, the depth of embedment is also found to be a controlling design variable in existing literature [¹⁹,²⁰,²²]. To account the actual behaviour of the depth of embedment, it becomes necessary to incorporate the depth correction factor into optimization algorithm. Fox depth correction is widely recognised correction factor to consider the depth of embedment and has been advocated by many international codes of practices and text books as already discussed [⁶–¹³]. Most of the existing literature recommends the use of Fox depth correction curves since the original Fox depth equation contains large number of intricate terms and it becomes unfeasible to use the equation. Moreover, in the iterative optimization process, it also becomes necessary to increase the number of constraints for the actual Fox equation as it contains many logarithmic terms which has a particular range of functioning. Also, the large number of terms present in Fox depth equation can considerably reduce the computational efficiency of an optimization process. On the contrary, the developed equation contains few simple terms which has no likelihood of being in the zone of indeterminacy like the logarithmic term in the original Fox equation. Furthermore, the developed equation can be easily incorporated within any program or software to reflect the effect of embedment.

Other domains

Apart from the optimization studies, the developed equation is very much suitable for every other area where the settlement of coarse or fine-grained soils need to be evaluated. Indian Standard IS 8009 [⁶] provides the graph of Fox depth correction factor for Poisson’s ratio= 0.5 only. As the Poisson’s ratio is found to be the second most effective input parameter in the parametric study, the designer is bound to take the original value of poison’s ratio and cannot use the depth correction factor at default value of 0.5. However, it becomes difficult to assess the factor at any other value of Poisson’s ratio as Indian Standard IS 8009 [⁶] does not specify any way to calculate the values at other Poisson’s ratio. The same problem persists in Hong Kong code of practice for foundations [⁷]. Though multiple graphs are referred in the code, yet for intermediate values of Poisson’s ratio, inter-graphs interpolation would be required. Moreover, it is also complicated to read the accurate value from graph. Therefore, for a prompt and accurate measurement of the depth correction factor during practical execution and hand calculation during third party inspection, the developed equation demonstrates its worth. Furthermore, the equation is equally beneficial for the academic organizations in teaching applications and the associated future researchers.

Conclusions

An efficient and straightforward equation valid for a wide range of input variables (D_f/B = 0.05 to 2, L/B = 1 to 5 and $ν$ = 0 to 0.5) is proposed to estimate the Fox depth correction factor. This factor is widely used in incorporating the effect of depth of embedment on the settlement of a footing. The developed multivariable non-linear regression model of the data given by Fox [⁴] shows a good agreement between the predicted values and the values reported by Bowles [¹⁴], making it suitable for design engineers to automate the analysis for the realistic prediction of settlement of footing. Moreover, integration of the developed equation will also make the process rapid and efficient as compared to incorporating graphs or tables in the optimization study of the foundation.

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