Chapter 8. Design Implications of the Study
8.3 Regression Models for Bearing Pressures
8.3.2 Regression Models
The bearing pressures of different foundation systems, such as homogeneous and layered unreinforced and reinforced systems (i.e., qc, qs, qsg, qsgc, and qsgcg) are considered as the dependent variables for the regression analysis. The influencing parameters, as per the foundation configurations, such as the undrained shear strength of clay (cu), footing settlement levels (s/D), and the thickness of unreinforced and/or reinforced sand layers (H/D) overlying the clay subgrades are considered as the independent variables. The contribution of unreinforced sand is directly computed as the bearing pressures of the homogeneous sand bed as qos. The effect of interfacial resistances in reinforced-systems is considered in terms of friction ratio, f (= s/φ);
where, s is the interfacial frictional angle between sand and geogrid obtained in pull- out tests. Table 8.1 summarizes dependent and independent variables considered for different test series. The regression analysis was performed using built-in data analysis tool pack available in Microsoft Excel®.
Table 8.1 Variables for regression analyses
Test Series Variables
Dependent Independent
A qc cu, s/D
B qs cu, s/D, H/D, qos C qsg cu, s/D, H/D, qos, s/φ D qsgc cu, s/D, H/D, qos, s/φ E qsgcg cu, s/D, H/D, qos, s/φ
Bearing pressure data were obtained from the experimental investigations for the regression analysis. Referring to the experimental data analyses and inferences made, few set of experiments were excluded for the analysis. In the case of unreinforced and geogrid reinforced systems (series B and C), data from the experiments with layer thickness (H) up to 1.67D were considered for regression analysis. While, for geocell and geocell-geogrid reinforced systems (series D and E), test data up to H = 1.15D were used. For each foundation systems, a set of test data corresponding to s/D = 2, 6, 12, 18, and 24% are used to generate the regression coefficients and the remaining data (corresponding to s/D = 0.67, 4, 8, 10, 14, 16, 20, and 22%) were used for validation. For homogeneous clay beds, 20 such data points were used for analysis, while 40 data points were kept for validation.
Homogeneous Beds
In the series A, responses of homogeneous clay beds of different undrained shear strengths (cu) were investigated. In regression analysis, the bearing pressure of the clay beds (qc) are considered as a function of cu and s/D, as presented in Eq. 8.12.
qc = cux
.(s/D)y (8.12) Where, x and y are the regression coefficients. The analysis results are presented in Table 8.2 to Table 8.4. Incorporating the coefficients, the final form of the expression (Eq. 8.12) is Eq. 8.13.
qc = cu0.99
.(s/D)0.49 (8.13)
Table 8.2 Regression statistics (homogeneous clay beds) Regression Statistics
R2 0.9978
R2Adj 0.9421
Standard Error 0.0902
Table 8.3 Analysis of variance (homogeneous clay beds)
df SS MS F
Regression 2 66.87 33.43 4102.53
Residual 18 0.14 0.008
Total 20 67.02
Table 8.4 Summary of t-statistics (homogeneous clay beds)
Coefficients Standard Error t-Stat p-value
c 0.99 0.03 29.61 1.01E-16
s/D 0.49 0.04 11.24 1.43E-09
The overall significance of the regression model is assessed by F-test; whereas, significance of individual regression coefficients are evaluated through t-test. From the F-distribution table, corresponding to the level of significance ( = 0.05), the value of Fcrit, (2, 18) is found as 3.55, as compared to the Fcal = 4102 (Table 8.3). Hence, as Fcal > Fcrit, the null hypothesis is rejected. In a similar way, the null hypothesis for the t-test is rejected for tcrit, (0.025,18) = 2.101. A significantly good agreement is observed between the experimental and predicted bearing pressures, presented in Fig.
8.11.
Fig. 8.11 Observed and predicted bearing pressures for homogeneous beds
In the test program, sand properties were not varied. However, an attempt is made similar to that of Eq. 8.13, where the bearing pressure of homogeneous sand bed (qos) is considered as the function of the footing settlements levels (s/D). Through curve fitting, a relationship as shown in Eq. 8.14 was established.
qos = 40 (s/D)0.52 (8.14)
Layered Foundations
Responses of unreinforced, geogrid, geocell, and geocell-geogrid reinforced foundations in different configurations were investigated in series B to E, respectively.
Dependent bearing pressures and their independent influencing parameters are listed in Table 8.1. In layered systems, the bearing pressures are expressed as the function of subgrade strengths (cu), homogeneous sand (qos), footing settlement (s/D), layer thicknesses (H/D), and the friction factor, f (= s/φ). The regression model equation, in general, is shown in Eq. 8.15.
qi = (cu)ai1(s/D)ai2(H/D)ai3(qos)ai4(f)ai5 (8.15) Where, the qi’s are the bearing pressures of different foundation systems and ai’s are the regression coefficients. The results of the regression analysis are summarized in Table 8.5 to Table 8.7. Number of data points used (observations) for each class of dependent parameter is shown in Table 8.5. Regression coefficients for the regression model are shown in Table 8.7. Incorporating the coefficients, the final expressions are shown in Eqs. 8.16 to 8.19 for unreinforced, geogrid reinforced, geocell reinforced, and geocell-geogrid reinforced foundation systems, respectively.
qs = (cu)0.31(s/D)012(H/D)0.35(qos)0.71 (8.16) qsg = (cu)0.32(s/D)0.19(H/D)0.13(qos)0.64(s/φ)-1.13 (8.17) q = (c )0.52(s/D)0.28(H/D)0.50(q )0.55(/φ)-1.05 (8.18)
qsgcbg = (cu)0.42(s/D)0.26(H/D)0.26(qos)0.72(s/φ)-0.59 (8.19) Correlation of experimental and corresponding predicted bearing pressures are presented in Fig. 8.12. The fitted data points are corresponds to s/D = 2, 6, 12, 18, and 24%; whereas, the validated data points are corresponds to 0.67, 4, 8, 10, 14, 16, 20, and 22% of s/D. Depending on the overall performance and scattering of values, the proposed models can be judged good to estimate the foundation behavior.
Fig. 8.12 Variation of observed and predicted bearing pressures
Table 8.5 Regression statistics (layered foundations)
Foundation Condition
Unreinforced Geogrid Geocell Geocell-Geogrid
R2 0.998 0.998 0.999 0.999
R2Adj 0.980 0.981 0.971 0.971
Standard Error 0.085 0.076 0.047 0.057
Observations 60 60 40 40
Table 8.6 Analysis of variance (layered foundations)
Foundation Condition
Unreinforced Geogrid Reinforced Geocell Reinforced Geocell-Geogrid Reinforced
df SS MS F df SS MS F df SS MS F df SS MS F
Regression 4 243.71 60.92 8473 5 261.39 52.27 9152 5 193.00 38.60 17182 5 222.99 44.59 13578
Residual 56 0.403 0.007 55 0.314 0.005 35 0.078 0.002 35 0.115 0.003
Total 60 244.11 60 261.70 40 193.07 40 223.10
Table 8.7 Summary of t-statistics (layered foundations)
Foundation Condition
Unreinforced Geogrid Reinforced Geocell Reinforced Geocell-Geogrid Reinforced
Coefficients Standard Error t-stat p-value Coefficients Standard Error t-stat p-value Coefficients Standard Error t-stat p-value Coefficients Standard Error t-stat p-value
cu 0.31 0.031 10.03 4E-14 0.32 0.028 11.26 6E-16 0.52 0.022 24.19 2E-23 0.42 0.026 15.97 1.2E-17 qos 0.71 0.034 21.10 2E-28 0.64 0.182 3.53 0.001 0.55 0.139 3.91 0.0004 0.72 0.168 4.27 0.0001 s/D 0.12 0.042 2.803 0.034 0.19 0.111 1.758 0.084 0.28 0.084 3.245 0.003 0.26 0.103 2.535 0.016 H/D 0.35 0.063 5.631 6E-07 0.13 0.056 2.398 0.019 0.50 0.50 8.77 2E-10 0.26 0.069 3.705 0.0007
s/φ - - - - -1.13 2.205 -0.511 0.611 -1.05 1.694 -0.623 0.54 -0.59 2.05 -0.286 0.776