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110 4.5 Examination of Footing Size 4.6 In-Depth Re-Examination Effect on the Uncertainty in of the Uncertainty in Bearing Bearing Capacity Analysis Capacity of Footings in/on Granular Soils Under Figure 99 presents the ratio of measured to calculated bear- Vertical-Centric Loading ing capacity (the bias ) versus footing width for vertical- centric loaded footings on/in natural and controlled soils. 4.6.1 Identification of Outliers and Fit Overall, no easily identifiable trend appears in Figure 99 other of Distributions for Calibrations than a general trend of some increase in the bias with the 4.6.1.1 Overview increase in footing size for natural soils, subjected to the pre- sented scatter. The bearing capacity of footings in granular soils is highly Figure 100 shows the mean bias of the bearing resistance ver- controlled by the bearing capacity factor N, in particular for sus the footing size for all the cases in controlled and natural foundations on or near the surface. The factor N is very sen- soil conditions combined. The 95% confidence interval of the sitive to the magnitude of the soil's internal friction angle f as mean bias versus the footing size is also presented for friction expressed by Equation 29, presented in Table 26, and illus- angles less than and greater than 43 (the reason for making trated in Figure 11. Section 4.3 investigated the source of the f = 43 the separator is related to the uncertainty in the factor bias underlying the bearing capacity analysis, demonstrating N presented in Section 4.4). The following observations that the bias increases with the increase in the internal friction related to the database on which Figure 100 was based can be angle (when exceeding 42.5) and is closely associated to the made: smaller footings were tested on soils with larger friction bias in the expression of N as illustrated in Figures 94 to 96. angles, f 43, and larger footings were tested on soils with The varying bias with the soil's internal friction angle sug- smaller friction angles, f < 43. gests that the development of the resistance factors should Overall, it can be concluded that what can be perceived as a follow this trend, unless a correction to the methodology is reduction in the bias with an increase in the foundation size developed and the expression of N is modified. The latter, seems to be more associated with the bias in N associated with although it may have some advantages, is problematic for the internal friction angle. Other conclusions are difficult to several reasons, including the need to change an established derive due to the small number of cases associated with large methodology and modifications of an expression based on a footings (i.e., 1 to 3 cases for footings greater than 1 m) as database that, while extensive, may be modified in the future. compared to 135 cases in the small footing category. As the resistance factors should be developed considering the Footing width, B (ft) 0.1 1 10 3.8 3.4 Natural Soil Condition (n =14) Controlled Soil Condition (n =158) 1s.d. 3 (x) number of cases in each interval 2.6 (2) (3) 2.2 (34) (90) Bias, (4) 1.8 (3) (12) 1.4 (5) (5) 1 0.6 0.2 0.01 0.1 1 10 Footing width, B (m) Figure 99. Variation of the bias in bearing resistance versus footing size for cases under vertical-centric loadings: controlled and natural soil conditions.

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111 B (ft) 1 0.1 10 = 1.81 COV = 0.203 3.5 Mean bias BC (n = 172) s.d. 2 Standard normal quantile (x) no. of cases in each interval 95% confidence interval for f (n = 135) 3 95% confidence interval for f (n = 37) 0 2.5 (2) (3) (90) Vertical-centric loading (5) Bias 2 f = 45 (4) -2 Total data (n = 90) 1.5 Normal distribution (1) Lognormal distribution (4) (5) 1 (1) (34) 0.5 1 1.5 2 2.5 3 3.5 4 (1) (17) Bias (3) (2) 0.5 Figure 101. Standard normal quantile of bias 0.01 0.1 1 data (measured over predicted bearing capacity) Footing width, B (m) for f = 45 + 0.5 and predicted quantiles of Figure 100. Variation of the bias in bearing resistance normal and lognormal distributions. versus footing size for cases under vertical-centric loadings: f > 43 and f < 43. significance level (usually of 1% or 5%), the distribution is rejected. For n = 90, the -squared value for the lognormal dis- bias change with the soil's internal friction angle, f , it is also tribution is 63.0 and the -squared value for the normal distri- reasonable to pursue the identification of data outliers for bution is 228.9, both of which are greater than the -squared subsets based on the magnitude of f. values of 21.66 at the 1% significance level and 16.92 at the 5% significance level, respectively. Hence, both distributions do not fit the data well and are rejected by the -squared GOF test. 4.6.1.2 Outliers and Examination of Fit of The smaller -squared value for the lognormal distribution (in Distributions for f = 45 0.5 comparison to the -squared value of the normal distribution) The largest dataset in the UML-GTR ShalFound07 database for this dataset suggests, however, that the lognormal distribu- is for footings tested under vertical-centric loadings. Subsets tion provides a better fit. of data are formed for each identifiable internal friction angle, It can be seen from the trials outlined in Table 54 that the f ( 0.5). The largest subset is for f = 45 0.5 (90 cases), the removal of outliers from either or both the higher and the mean and COV of the bias for which are found to be 1.81 and lower tails of the bias distribution does not result in an accept- 0.203, respectively. Figure 101 presents a comparison of a able -squared value for either the normal or the lognormal standard normal quantile of the bias data to predicted quan- distribution. Hence, the removal of outliers from the distribu- tiles of the theoretical normal and lognormal distributions. At tion tails does not render normal or lognormal distribution least one possible outlier, a footing with a bias of 3.51, can be acceptable, while a comparatively better fit fluctuates between observed for both the normal and the lognormal distributions. normal and lognormal distribution, based on the -squared Removal of this data point can result in a better fit of the GOF test. Hence, all the available data for the cases in/on soil dataset to the normal distribution, which is further quantified with f = 45 have been used for the resistance factor calibra- by the goodness-of-fit test. In this sense, the outliers identified tion without the identification and removal of outliers and here imply that their removal improves the dataset so it better assumed to follow lognormal distribution. fits a theoretical distribution. In Figure 101, there are four footings with a bias smaller The -squared goodness-of-fit (GOF) tests have been car- than 1.0, the smallest being = 0.82, for which the assumed ried out to test the fit of the theoretical normal and lognormal lognormal distribution overpredicts the bias in the lower tail distributions to follow the bearing resistance bias for n = 90 region, which is more critical than the higher tail region cases, along with the datasets after the removal of some identi- (because bias less than 1.0 means the calculated resistance was fiable outliers. Table 54 lists in detail a number of trials and the more than the actual resistance). This circumstance is exam- corresponding -squared values obtained from the GOF tests. ined in Section 4.6.2.4 following the resistance factor calibra- If the -squared values obtained for an assumed distribution tion in order to ensure that the resistance factor developed for are greater than the acceptable -squared values of a certain f = 45 results in acceptable risk in design.

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112 Table 54. -squared values for the fitted lognormal and normal distributions for vertical-centric loading cases on/in soil with an internal friction angle (f ) of 45. -squared values n Lognormal Normal Comments distribution distribution 90 63.0 228.9 Mean = 1.81, COV = 0.203; all data for f = 45 Mean = 1.79, COV = 0.179; highest bias (3.51) removed (data 89 515.0 60.3 beyond 2s.d.) Mean = 1.822, COV = 0.195; case with 3rd lowest bias (0.87) 89 60.3 428.0 removed; this case is on the lower bias tail and the farthest from theoretical lognormal quantile Mean = 1.83, COV = 0.186; 2 cases with 2nd and 4th lowest 88 57.9 724.0 biases (0.85 and 0.87) removed; in lower bias tail and farthest two from theoretical lognormal quantile Mean = 1.83, COV = 0.185; 2nd and 4th lowest bias cases (0.85 87 805.0 43.6 and 0.87) and the case with the highest bias (3.51) removed Mean = 0.81, COV = 0.161; 2nd and 4th lowest bias cases (0.85 87 62.5 927.0 and 0.87) and the case with the 2nd highest bias (2.37) removed Mean = 1.84, COV = 0.177; 2nd, 3rd and 4th lowest bias cases 87 57.5 1,418.0 (0.85, 0.85 and 0.87) removed Note: Acceptable -squared value for significance level of 1% is 21.666 and for significance level of 5% is 16.919. 4.6.1.3 Outliers and DFs for Internal Friction angles, which is around 0.2. Though the GOF test shows that Angles Other than 45 both normal and lognormal distributions are acceptable, with lognormal being a better fit, the case with the highest bias, Procedures similar to those described in Section 4.6.1.2 = 3.57, has been considered an outlier. The comparison of have been performed for the data subsets of f other than 45. the standard normal quantiles of the dataset and the theo- For f = 44 (n = 30, m = 1.40 and COV = 0.250), both the retical distributions is shown in Figure 102(a). The result- normal and lognormal distributions are accepted by the ing dataset after the removal of this case has a mean of 1.47 -squared GOF test for the 1% significance level. The log- and COV of 0.088 (examination of the database shows that normal distribution provides a better fit, with a -squared the remaining three cases are from the same site, hence value of 13.74 versus 17.82 for the normal distribution. explaining the very small COV). Comparison of the stan- For f = 43, 42, 38, 36, and 32, although the normal dis- dard normal quantiles of the filtered dataset and the theo- tributions provide better fits than the lognormal distributions, retical distributions is shown in Figure 102(b). Lognormal lognormal distributions have been considered. This is done distribution is considered for this dataset also. Hence, only because lognormal distribution is naturally expected to better one outlier was removed from the total dataset, resulting in represent the dataset of a ratio (i.e., bias) restricted by values 172 cases used for the resistance factor calibration for vertical- greater than zero or due to similar behavior, small dataset, and centric loading. so forth as further detailed. For f = 43 (n = 14, m = 1.34, and COV = 0.283), the -squared value is 18.53 for normal versus 22.69 for lognormal. For f = 42 (n = 4, m = 1.60, and COV = 4.6.2 The Statistics of the Bias as a 0.416), the -squared value is 12.20 for normal versus 12.74 for Function of the Soil's Internal lognormal. For f = 38 (n = 12, m = 1.26, and COV = 0.215), Friction Angle and Resulting the -squared value is 16.75 for normal versus 74.62 for log- Resistance Factors normal. The minimum bias of 0.55, which is overpredicted by 4.6.2.1 In-Depth Examination of Subsets Based the lognormal distribution for this dataset, will be examined on Internal Friction Angle following the resistance factor calibration. For f = 36 (n = 4, m = 1.20, and COV = 0.233), the -squared value is 19.78 for Tables 55 through 57 present the biases evaluated for the normal versus 21.61 for lognormal, and, for f = 32 (n = 4, bearing capacity estimation according to the soil's friction m = 1.25, and COV = 0.347), the -squared value is 10.77 for angles. The corresponding resistance factors have been a normal distribution versus 11.15 for lognormal. obtained for a target reliability index T of 3.0 (exceedance For f = 35 (n = 4), the mean bias is found to be 2.00 and probability of 0.135%). Table 55 presents the cases in con- the bias COV is 0.528, which is exceptionally high compared to trolled soil conditions while Table 56 shows the cases in nat- the COVs for the datasets of the closer-in-magnitude friction ural soil conditions. Table 57 presents all the cases in the

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113 2 1.5 = 2.00 = 1.47 COV = 0.528 COV = 0.088 1 Standard normal quantile 1 Standard normal quantile 0.5 0 0 Vertical-centric loading f = 35 Vertical-centric loading Total data (n = 4) -0.5 f = 35 -1 Identified outlier (n = 1) Filtered data (n = 3) Normal distribution -1 Normal distribution Lognormal distribution Lognormal distribution -2 -1.5 0.5 1 1.5 2 2.5 3 3.5 4 4.5 1 1.5 2 2.5 Bias Bias (a) (b) Figure 102. Standard normal quantile of bias data for f = 35 + 0.5 and predicted quantiles of normal and lognormal distributions (a) for all data and (b) with the outlier removed. database, both controlled and natural soil conditions, under number in parentheses represents the number of cases in each vertical-centric loadings. All the cases in the controlled soil of the friction angles' subsets. conditions are in soils with relative densities above 35%. Graphical presentation of the bias in bearing resistance esti- 4.6.2.2 Factor Development Based on Data Trend mation versus soil friction angle is shown in Figure 103. The error bars represent one standard deviation of the mean bias The bias in bearing resistance estimation for the cases under for each friction angle, taken as a range of f 0.5, and the vertical-centric loading, both in/on controlled and natural soil Table 55. Statistics of bearing resistance bias and the resistance factors corresponding to soil friction angles in controlled soil conditions for vertical-centric loading. Friction angle f Bias Resistance factor ( T = 3) n ( 0.5 deg) Mean COV MCS Preliminary 46 2 1.81 0.071 1.655 1.00 45 90 1.81 0.203 1.194 1.00 44 30 1.40 0.250 0.807 0.80 43 14 1.34 0.283 0.700 0.70 42 4 1.60 0.416 0.700 0.70 39 1 1.02 -- -- -- 38 11 1.32 0.122 1.081 1.00 36 3 1.34 0.079 1.206 1.00 35 3 1.47 0.088 1.300 1.00 43 to 46 136 1.67 0.247 0.971 0.95 38 3 22 1.38 0.225 0.855 0.85 all angles 158 1.63 0.252 0.934 0.90 Table 56. Statistics of bearing resistance bias and the resistance factors corresponding to soil friction angles in natural soil conditions for vertical-centric loading. Bias Resistance factor ( T = 3) Friction angle f n Mean COV MCS Preliminary 33 2.5 (all angles) 14 1.00 0.329 0.457 0.45

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114 Table 57. Statistics of bearing resistance bias and the resistance factors corresponding to soil friction angles in controlled and natural soil conditions combined, for vertical-centric loading. Friction angle f Bias Resistance factor ( T = 3) n ( 0.5 deg) Mean COV MCS Preliminary 46 2 1.81 0.071 1.655 1.00 45 90 1.81 0.203 1.194 1.00 44 30 1.40 0.250 0.807 0.80 43 14 1.34 0.283 0.700 0.70 42 4 1.60 0.416 0.700 0.70 39 2 0.83 0.330 0.378 0.35 38 12 1.26 0.215 0.804 0.80 36 4 1.20 0.233 0.727 0.70 35 3 1.47 0.088 1.300 1.00 34 2 1.09 0.135 0.865 0.85 33 3 1.03 0.126 0.836 0.80 32 4 1.25 0.347 0.542 0.50 30.5 2 0.98 0.423 0.339 0.30 43 to 46 136 1.67 0.247 0.971 0.95 36 3 36 1.23 0.296 0.619 0.60 all angles 172 1.58 0.278 0.838 0.80 conditions, can be expressed by the best fit line in Figure 103 with higher friction angles compared to that for natural and in Equation 121, for which the coefficient of determina- soil conditions. The bias expressed by Equation 121 has tion is 0.200. This line shows that the bearing resistance bias been used to develop resistance factors for the whole range (BC) increases with an increase in the soil friction angle: of soil friction angles for both controlled and natural soil conditions. BC = 0.308 exp ( 0.0372 f ) (121) Based on Tables 55 and 56, the COVs of the bias for all the controlled and natural soil condition cases are 0.252 and 0.329, The details provided in Tables 55 and 56 indicate that the respectively. Hence, COV of 0.25 and 0.35 may be taken to data available for controlled soil conditions relate to soils represent the COVs of the biases for the controlled soil and nat- 3.5 Mean bias, BC (n = 172) 1 s.d. (x) no. of cases in each interval 3.0 BC = 0.308 exp(0.0372f) (R2=0.200) 2.5 95% confidence interval 2.0 (2) (4) (30) Bias (3) (12) 1.5 (2) (2) (90) 1.0 (3) (4) (4) (14) 0.5 (2) 0.0 30 32 34 36 38 40 42 44 46 Friction angle f (deg) Figure 103. Bearing resistance bias versus average soil friction angle (taken f + 0.5) including 95% confidence interval for all cases under vertical-centric loading.

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115 Table 58. Resistance factors for vertical-centric loading bias with the soil friction angle. The cases for which a small cases based on the bias versus f best fit line of resistance factor was developed based on a very small sub- Equation 121 and the COV of natural versus controlled set (two cases each) could be justifiably overruled in the soil conditions. context of the established trend and the large datasets sup- Resistance factor ( T = 3) porting it. Mean bias Soil Conditions Soil friction angle Natural Controlled f (deg) (Equation 121) (COV = 0.35) (COV = 0.25) 4.6.2.3 Recommended Resistance Factors MCS Rec.* MCS Rec. 30 0.94 0.403 0.40 0.542 0.50 The recommended resistance factors for vertical-centric 35 1.13 0.485 0.45 0.652 0.60 loading cases are presented for different friction angles in 37 1.22 0.524 0.50 0.703 0.70 Table 59 based on the values calculated and recommended 38 1.27 0.545 0.50 0.732 0.70 in Table 58. The values in Table 59 are applicable for soils 40 1.36 0.584 0.55 0.784 0.75 45 1.64 0.704 0.65 0.946 0.80 with relative densities greater than 35%. Further consider- ation is necessary for soils with friction angles less than *Rec. = Recommended 30 combined with relative densities less than 35%. For these soils, which are in a very loose state, it is recom- ural soil conditions, respectively. Table 58 presents the resis- mended either to consider ground improvement to a depth tance factors calculated using these statistics for friction angles of at least twice the footing width (subjected to a settlement ranging from 30 to 45, on foundations in/on natural and criterion), ground replacement, or an alternative founda- controlled soil conditions. tion type. Figure 104 presents the recommended resistance factors for controlled and natural soil conditions detailed in Table 58. Fig- 4.6.2.4 Examination of the Recommended ure 104 also presents a comparison of the recommended resis- Resistance Factors tance factors to those obtained in Table 57 (based on the database) and the 95% confidence interval of the bearing A rough estimate of the equivalent factor of safety (FS) for resistance bias. It can be observed that the recommended a resistance factor of , developed using a ratio of dead load to resistance factors follow the trend in the bearing resistance live load of 2.0; dead-load, load factor of 1.25; and live-load, 3.0 95% confidence interval for Resistance factor based on database (x) no. of cases in each interval 2.5 Recommended f for Controlled soil conditions Recommended f for Natural soil conditions (90) (3) (2) 2.0 1.0 (2) (30) (3) (12) Bias 0.8 1.5 (4) (4) (14) Resistance factor, 0.6 (4) 1.0 (2) 0.4 (2) 0.5 0.2 n = 172 0.0 0.0 30 32 34 36 38 40 42 44 46 Friction angle f (deg) Figure 104. Recommended resistance factors for soil friction angles (taken f + 0.5) between 30 and 46, with comparisons to 95% confidence interval and resistance factors obtained for the cases in the database; the bubble size represents the number of data cases in each subset.