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116 Table 59. Recommended resistance factors capacity analysis of footings under vertical-eccentric loading. for vertical-centric loading cases. The relations shown in Figure 95 are not similar to those in Figure 94 (investigating footings under vertical-centric load- Soil friction Recommended resistance factor ( T = 3) Soil conditions ing); hence, additional evaluation is required for cases not angle f (deg) Natural Controlled under vertical-centric loading. 3034 0.40 0.50 The variation of the bearing capacity bias with the soil's fric- 3536 0.45 0.60 3739 0.50 0.70 tion angle is presented in Figure 105 for cases under vertical- 4044 0.55 0.75 eccentric loading (each error bar represents one standard 45 0.65 0.80 deviation). It can be observed that for f = 35 0.5 the mean bias of the seven cases is higher than for the other soil friction angles with a relatively lower COV. These seven cases are load factor of 1.75 was presented by Paikowsky et al. (2004) related to a single site and compiled from the DEGEBO litera- and expressed by the following equation: ture. Hence, for the determination of the best fit line of the bias versus the friction angle, these seven cases were excluded. The FS 1.4167 (122) trend in Figure 105 suggests a possible decrease in the bias with the increase in the friction angle, which is contrary to the trend The highest recommended resistance factor in Table 59 established for the case of vertical-centric loading (see Fig- is = 0.80 for f 45, developed assuming the data follow ure 94) or the trend seen in Figure 95 for the soil's friction angles a lognormal distribution. A rough estimate of the equiva- in the range of 43.5 to 46.0. The data in Figure 105 suggest lent factor of safety for this resistance factor is given by Equa- that no clear, unique correlation exists between the bias and tion 122 as 1.77. A safe design requires that the condition in the soil's internal friction angle, and, even upon the exclusion Equation 123 is met: of the aforementioned seven cases, the coefficient of determi- nation (R2) is 0.01, essentially indicating that a correlation does qcalc not exist. The data in Figure 105 may indicate, therefore, that qmeas (123) FS either for the eccentric loading and/or the available data for such cases, factors other than the soil's friction angle contribute where qcalc is calculated bearing capacity and qmeas is measured significantly to the bias. bearing capacity. The minimum allowable bias for the given Figure 106 presents the relationship between the bias of FS is, therefore, the reciprocal of the FS, i.e., the minimum vertical-eccentric loading of foundations and the magnitude bias for which the design will be safe is 1/FS = 0.565. This bias of the eccentricity normalized by the foundation's width, i.e., is much smaller than the smallest bias of the dataset, = 0.82, e/B. Forty-three cases have been tested with load eccentricity for which the standard normal quantile is seen to be over- predicted by the assumed lognormal distribution (see Fig- ure 101). A bias of 0.82, therefore, results in a safe design, and 3.0 n = 43 (4) all the footing cases in the database are safe upon the applica- tion of the developed resistance factor. It can, therefore, be 2.5 (7) concluded that the methodology of utilizing the trend and the (2) (11) assumption of the lognormal distribution for the bias is 2.0 acceptable for resistance factor calibration and is justified by the outcome. Bias 1.5 (6) (4) 4.7 In-Depth Re-Examination 1.0 (9) of the Uncertainty in Bearing Mean bias, BC 1 s.d. Capacity of Footings in/on 0.5 (x) no. of cases in each interval = 2.592exp(-0.01124 f) (R2=0.01) Granular Soils Under BC 95% confidence interval Vertical-Eccentric Loading 0.0 30 32 34 36 38 40 42 44 46 4.7.1 Examination of the Bias for Friction angle f (deg) Controlling Parameters Figure 105. Bearing resistance bias versus soil friction The investigation presented in Section 4.4.3 and Figure 95 angle for cases under vertical-eccentric loadings suggested that the bias in the bearing capacity factor N can (seven cases for f = 35 [all from a single site] have be associated with the general trend of the bias for the bearing been ignored for obtaining the best fit line).

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117 3.5 3 2.5 Vertical-eccentric loading (n = 43) B 4.0 in (0.1m) Bias, 2 B = 1.65 ft (0.5m) 1.5 B = 3.3 ft (1.0m) 1 0.5 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 Load eccentricity ratio, e/B Figure 106. Bearing resistance bias versus load eccentricity ratio e/B for vertical-eccentric loading. ratios ranging from 0.025 to 0.333 (1/40 to 1/3), with a major- the data related to small footings only are relevant to higher ity of them having an eccentricity ratio of 1/6. It can be seen that friction angles. Figure 107 thus emphasizes that the effect of the while the larger foundations mostly have higher biases, there footing size on the bearing resistance bias when testing eccen- appears to be no correlation between the bearing resistance bias trically loaded foundations is more significant compared to the and the load eccentricity ratio. The large scatter that appears effect of the soil's friction angle. Hence, calibrating resistance for the small foundations may be related to the physical diffi- factors using this dataset, based on f , cannot be justified, as has culties of conducting such tests where eccentric loads need to be been done for the vertical-centric loading cases. applied to a small footing. A closer examination of the relationship between the bias 4.7.2 Identification of Outliers and the magnitude of the eccentricity is presented in Figure 107 for a given eccentricity ratio of e/B = 1/6 versus friction angle The data presented in Figures 105 and 107 lead to the con- f. Cases with various footing widths are available for this clusion that in the absence of a clear underlying factor to eccentricity ratio only (see Figure 106), while tests with other explain the bias, resistance factors may be developed for both load eccentricity ratios mostly utilize footings of widths less natural and controlled soil conditions and a range of f and than or equal to 4 in (0.1 m). While a best fit line for these then compared to the resistance factors developed for vertical- data would show a decrease in the bias with an increase in f , centric loading. 3.5 3 2.5 Vertical-eccentric loading e/B = 1/6 Bias, 2 (Total n = 21) B 4.0 in (0.1m) B = 1.65 ft (0.5m) 1.5 B = 3.3 ft (1.0m) 1 0.5 34 35 36 37 38 39 40 41 42 43 44 45 46 Soil friction angle, f (deg) Figure 107. Change in bearing resistance bias with soil friction angle for tests with a load eccentricity ratio of e/B = 1/6.

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118 3 factors remain essentially at 1.0, with the exception of four = 1.83 cases related to f = 41, for which a large scatter had been COV = 0.351 2 observed (see Figure 105). In addition, the amount of data Standard normal quantile available for some of the f subsets is comparatively small. It 1 has also been concluded in Section 4.5 that using the available data, the effect of footing size on the bias cannot be isolated 0 from the effect of the soil friction angle. All these conditions -1 lead to the issue of whether it is practical and appropriate to Vertical-eccentric loading Total data (n = 43) use the dataset for vertical-eccentric loading conditions alone -2 Normal distribution for the resistance factor calibration of this loading situation. Lognormal distribution Since vertical-centric loading is the simplest loading mode, -3 the uncertainties involved in estimating the resistance of 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 footings under vertical-eccentric loading are assumed to be Bias not less than those involved in the case of footings under Figure 108. Standard normal quantile of the bias vertical-centric loading. This assumption is based on the fol- for all vertical-eccentric loading cases and the lowing: (1) when the source of the lateral load is not perma- predicted quantiles of theoretical distributions. nent, the foundation supports vertical-centric loading only, and (2) very often the magnitudes of the lateral loads (and Figure 108 presents the standard normal quantile of the hence eccentricity) are not known at the bridge foundation dataset with the theoretical predictions of normal and log- design stage (see Section 3.1, in particular, Section 3.1.7). This normal distributions. The presented relations visually suggest means that the resistance factors for vertical-eccentric loading a good match between the lognormal distribution and the conditions have to be either equal to or less than the ones rec- data. The -squared GOF tests verify that the data follow ommended for the vertical-centric loading in Table 59. the lognormal distribution better than the normal distribu- tion (accepted both at the 1% and 5% significance levels), 4.7.4 Examination of the Recommended with -squared values of 8.34 for lognormal distribution ver- Resistance Factors for sus 11.74 for normal distribution. As the data follow the log- Vertical-Eccentric Loading normal distribution, no outliers are identified. The bias mean for vertical-eccentric loading is slightly higher than the bias mean for vertical-centric loading (1.83 ver- 4.7.3 The Statistics of the Bias as a sus 1.58); hence, the same resistance factors used for vertical- Function of the Soil's Internal centric loadings are recommended for vertical-eccentric Friction Angle and Resulting loadings. Resistance Factors Based on Equations 122 and 123, the minimum allowable The bias in the bearing resistance estimation for footings bias for the highest resistance factor of 0.80 is 0.565. The bear- under vertical-eccentric loadings evaluated for subsets of each ing resistance biases of all the cases under vertical-eccentric f (0.5) are presented in Table 60. The associated resistance loading in the database (the minimum being = 0.80) are thus Table 60. Statistics of bearing resistance bias and the resistance factors corresponding to soil friction angles in controlled soil conditions for vertical-eccentric loading. Friction angle f Bias Resistance factor ( T = 3) n ( 0.5 deg) Mean COV MCS Preliminary 46 11 1.80 0.227 1.109 1.00 45 4 1.53 0.199 1.021 1.00 44 9 1.27 0.182 0.889 0.85 43 2 1.88 0.238 1.122 1.00 41 4 2.06 0.604 0.426 0.40 40 6 1.77 0.203 1.168 1.00 35 7 2.69 0.148 2.063 1.00 43 to 46 26 1.58 0.257 0.892 0.85 40 to 46 36 1.67 0.325 0.772 0.75 all angles 43 1.83 0.351 0.783 0.75

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119 safe upon the application of the recommended factors. A dif- An additional examination of the effect of the eccentric- ferent approach may be taken, assuming the eccentric loads are ity ratio (ratio of load eccentricity to footing width) on the permanent, hence, allowing for resistance factors higher than designed footing was carried out. A strip foundation on the those applied for vertical-centric loading. This condition surface of soil with a unit weight of 124.7pcf (19.6 kN/m3) is examined via the effective width (B) versus the actual width was analyzed, hence eliminating the effects of the founda- of footings under vertical-centric loading, i.e., both founda- tion's shape and depth. Bearing resistances of the strip foot- tion sizes are examined (B based on for vertical-centric and ing under an eccentric load with a given eccentricity ratio, B based on for vertical-eccentric) and the larger foundation altered from 1/4 to 1/100, were estimated using the bearing size prevails. Such examination allows review of the recom- capacity equation and expressed as bearing resistance versus mended resistance factors for vertical-centric versus vertical- the effective footing width B (B - 2e), (Meyerhof, 1953). eccentric conditions. A limited examination of this issue Because the effective footing width is used, the effect of follows. eccentricity is "eliminated" and the vertical-eccentric load In Figure 105, the mean bias of vertical-eccentric loading case is essentially transferred to the vertical-centric case, i.e., for friction angles between 40 and 46 is 1.60. Assuming the resulting effective footing width is the same regardless of the mean bias to remain a constant at 1.60 for all friction the load eccentricity ratio. For example, for a required fac- angles and the COV of the bias of the bearing resistance to tored load of 369 ton/ft (1,000 kN/m), the required effective be related to natural and controlled soil conditions, i.e., footing width (B) using the recommended of 0.60 for a f 0.35 and 0.30, respectively, the obtained resistance factors of 35 has been found to be about 6.25 ft (1.90 m) for eccen- are as follows: tricity ratios of 1/4 as well as 1/100. In Figures 109(a) and 109(b), the bearing resistance versus the effective footing Natural soil conditions, for all f : = 0.65 ( obtained width plots have been presented for e/B = 1/4 and 1/100, from MCS = 0.687) respectively, for a frictional soil with an internal friction Controlled soil conditions, for all f : = 0.75 ( obtained angle (f) of 35. from MCS = 0.796) It should be noted that the design (physical) footing width in both cases is different as B = B + 2e and hence depends on Taking these two separate databases, one for vertical-centric the magnitude of the eccentricity. Based on the examination and the other for vertical-eccentric, two sets of resistance fac- above, it can be said that the recommended resistance fac- tors, one for controlled soil conditions and one for natural soil tors using the vertical-centric test data results in an accept- conditions can be obtained, as presented in Table 61. Table 61 able design for vertical-eccentric loading conditions and demonstrates that the recommended resistance factors based that separate sets of resistance factors are not required. The on the extensive data available from vertical-centric load test results in the UML-GTR ShalFound07 database for tests, although they may be conservative, will be also safe vertical-eccentric loadings did not enable evaluation of the when applied for footings designated to be subjected to performance of Meyerhof's effective width model (1953), load-eccentricity. This is validated when compared to the i.e., the uncertainty in defining B = B - 2e or the ability of resistance factors developed based on vertical-eccentric load the eccentricity ratio to exceed the limiting compression tests (under the aforementioned assumptions) shown in contact value of 1/6. Some discussion of the subject using Table 61 as well. other sources follows. Table 61. Comparison of the recommended resistance factors based on vertical-centric loading to those obtained based on Figure 105 for vertical-eccentric loading. Resistance factor ( T = 3) Controlled soil conditions Natural soil conditions Soil friction Recommended Vertical- Recommended Vertical- angle f Vertical-centric eccentric Vertical-centric eccentric and vertical- based on and vertical- based on eccentric Figure 105 eccentric Figure 105 3034 0.50 0.40 3536 0.60 0.45 3739 0.70 0.75 0.50 0.65 4044 0.75 0.55 45 0.80 0.65

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120 f = 35 f = 35 Eccentricity ratio e/B = 1/4 Eccentricity ratio e/B = 1/100 Unfactored Bearing Capacity Unfactored Bearing Capacity Factored - Natural, = 0.45 (centric) Factored - Natural, = 0.45 (centric) Factored - Natural, = 0.65 (eccentric) Factored - Natural, = 0.65 (eccentric) Factored - Controlled, = 0.60 (centric) Factored - Controlled, = 0.60 (centric) Factored - Controlled, = 0.75 (eccentric) Factored - Controlled, = 0.75 (eccentric) Required vertical-eccentric load Required vertical-eccentric load 1500 1500 Bearing resistance per unit length, Qu (kN/m) Bearing resistance per unit length, Qu (kN/m) 1000 1000 500 500 0 0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 Effective footing width, B' (m) Effective footing width, B' (m) (a) (b) Figure 109. Comparison of the required effective footing widths for different eccentricity ratios: (a) e/B = 1/4 and (b) e/B = 1/100 for a strip footing resting on a soil with internal friction angle (f ) = 35. The limiting eccentricity value of e/B = 1/6 is developed from the effective width rule. The isometric slip lines developed by a theory assuming a linear stress distribution under a rigid Michalowski and You invoking the kinematic approach of footing subjected to eccentric loading (the combination of cen- limit analysis resemble qualitatively the above described exper- tric load and a moment similar to the stress distribution in a imental observations. Michalowski and You concluded that for beam). As such, when the eccentricity ratio is 1/6, the founda- smooth footings, realistic footing models, and cohesive soils, tion is subjected to compression with one edge under no (zero) Meyerhof 's effective width rule is a reasonable account of stress. When the eccentricity ratio exceeds 1/6, the foundation eccentricity in bearing capacity calculations. It is only for sig- is expected to be subjected to "tension," hence the contact area nificant bonding at the soil interface (i.e., no separation or per- between the foundation and the soil decreases. It is well under- fect adhesion) and for large eccentricities (e.g., e/B greater than stood that the load distribution under the foundation depends 0.25) that the effective width rule significantly underestimates on the relative stiffness of the foundation/soil system and, bearing capacity (for clays). For cohesive-frictional soil, this hence, is not necessarily linear. Expected load distributions underestimation decreases with an increase in the internal fric- under vertical-centric loading proposed by Terzaghi and Peck tion angle, becoming more and more "accurate" with limited (1948) were verified experimentally by Paikowsky et al. (2000) eccentricity. using tactile sensor technology and demonstrating concave The examination and discussion presented in Sections 4.6 stress load distribution across a rigid footing in granular soil. and 4.7 lead to the following recommendations: The effect of eccentricity (not presented in Paikowsky et al., 2000) was measured as one side stress concentration support- 1. The use of resistance factors developed and recommended ing the one-sided extensive slip surfaces developing under for vertical-centric loading (see Table 59) could and should an eccentrically loaded foundation as illustrated in Figure F-3 be extended to be used with vertical-eccentric loading. (Appendix F) by Jumikis (1956). 2. The rule of effective foundation size (B = B - 2e) pro- A theoretical study was presented by Michalowski and You posed by Meyerhof (1953) is not overly conservative and (1998) examining Meyerhof 's aforementioned effective width results in realistic bearing capacity predictions for the rule (1953) in calculations of the bearing capacity of shallow foundation-soil conditions expected to be encountered foundations. Michalowski and You developed a limit analysis in bridge construction (rough surface foundations on solution for eccentrically loaded strip footings and assessed granular soils).