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121
3. The independence of the calculated effective foundation cal, when compared to the case of footings under vertical-
size (B) from the magnitude of the eccentricity and the centric loadings. Figure 110 examines the variations in the
aforementioned recommendations/observations provide bias versus the angle of load inclination (to the vertical),
a solution for the design problems presented by various according to footing sizes. The scatter shows that there is
DOTs (see Section 3.1.7), in which the eccentricity is no clear trend of the bias associated with either the load
unknown at the early design stage. The solution justifies inclination angle or the footing size. All the larger footings
the calculated foundation size during early design to be (B 1.65 ft) were tested under inclined loads with = 0°
referred to as the effective foundation that can then be (inclination along the footing length, see Figure 17), while
modified by twice the eccentricity at the final design stage. the smaller footings were subjected to inclined loads with
4. In light of the presented material, there is no clear evi- = 90° (inclination along the footing width). Although it
dence allowing an increase in the foundation eccentricity appears that the bias increases with an increase in the load
ratio for permanent loading beyond e/B = 1/6. inclination for = 0° while for = 90° the bias decreases
5. For combined loading (permanent and variable), an argu- with an increase in the inclination angle, it is difficult to
ment can be made that the eccentricity ratio can be isolate the effect of the footing size, except in the vicinity of
increased to e/B = 1/3 for which half of the foundation is load inclination of 10°. For the tests with inclination angles
under "tension" conditions. Some performance-based around 10° carried out on different footing sizes, it can be
design codes (e.g., DIN 1054) allow that limit. As no observed that the orientation switched between = 0° and
clear data exists to support such an increase, it is recom- 90° has no effect on the bias, which suggests that no corre-
mended that until further research is carried out in the lation exists with the orientation of the load. This obser-
area, the eccentricity of the combined loading will be vation should be qualified, however, by the fact that the
limited to e/B 1/4, as allowed in the AASHTO standard dataset for loading orientations between 0° and 90° is not
specifications (4.4.8) or recommended in Section 8.4.3.1 sufficiently large to make a general statement. The resis-
of FHWA-NHI-06-089 Soils and Foundation Manual. tance factors can thus be further examined in relation to the
(FHWA, 2006). soil's friction angle.
The total number of data points available for inclined-
4.8 In-Depth Re-Examination centric loading is 39 (bias mean = 1.43 and COV = 0.295),
of the Uncertainty in Bearing while the soil friction angles ranged from 46 (±0.5°) to 38
Capacity of Footings in/on (±0.5°). As a result, the identification of outliers based on the
Granular Soils Under data subset for each f (±0.5°) may not be practical because of
Inclined-Centric Loading the small data subsets. The standard normal quantiles of the
data and those predicted by the developed normal and log-
4.8.1 Examination of the Bias for
normal distributions are presented in Figure 111. A visual
Controlling Parameters
observation clearly shows that the data fits the normal dis-
In the case of footings under inclined-centric loadings, an tribution, while for the data to follow the lognormal distri-
additional factor involved is the load inclination to the verti- bution, some outliers in the lower tail region (especially
2.5
2
Inclined-centric loading
1.5 (Total n = 39)
B 4 in (0.1m); = 90°
Bias,
B = 1.65 ft (0.5m); = 0°
1
B = 3.3 ft (1.0m); = 0°
0.5
0
0 5 10 15 20 25 30
Load inclination to the vertical (deg)
Figure 110. Bias versus load inclination for footings under inclined-
centric loading.

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122
3 3.0
= 1.43 Mean bias, BC 1 S.D.
COV = 0.295 (x) no. of cases in each interval
2
2.5 = 1.25 + 0.0041
Standard normal quantile
BC f
95% confidence interval
1 (6) (10)
2.0
0 (3) (4)
Bias
1.5
-1 (1)
Inclined-centric loading
Total data (n = 39) 1.0
-2 Normal distribution
Lognormal distribution (4)
(11)
0.5
-3
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 n = 39
Bias 0.0
Figure 111. Standard normal quantile of 30 32 34 36 38 40 42 44 46
Friction angle f (deg)
bias data for all data for inclined-centric
loading and predicted quantiles of theoretical Figure 112. Variation of the bias in bearing resistance
distributions. versus soil friction angle for cases under inclined-
centric loadings.
with biases of less than 1.0) need to be removed. However,
lognormal distribution has been assumed to be followed by bias gradually increases with an increase in the soil friction
the resistance bias without removing the outliers because angle. The resistance factor is calibrated using the mean
the lower tail region (where the resistance bias is less than obtained by the best fit line.
1.0) is a critical region for determination of the resistance
factors as it is associated with the area of concern in which
the loading may exceed the resistance. It should be noted
4.8.2 The Statistics of the Bias as a
that in such a case, the use of a lognormal distribution
Function of the Soil's Internal
would result in a more conservative resistance factor eval-
Friction Angle and Resulting
uation than otherwise. Other practices, such as "fitting" the
Resistance Factors
distribution to the tail (ignoring the bulk of the data), The statistics of the bearing resistance bias for the cases
should be discouraged and are not perceived as mathemat- under inclined-centric loadings are presented in Table 62 for
ically or otherwise justifiable. subsets of each f (±0.5°), while the best fit line obtained from
Further examination of the variation of bias with the mag- the regression analysis of the biases available for 38° < 46°
nitude of the soil's friction angle is presented in Figure 112 for in Figure 112, is provided by Equation 124.
cases under inclined-centric loading (each error bar repre-
sents 1 standard deviation). The best fit line suggests that the = 1.25 + 0.0041 f (124)
Table 62. Statistics of bearing resistance bias and the
resistance factors corresponding to soil friction angles in
controlled soil conditions for inclined-centric loading.
Friction angle f Bias Resistance factor ( T = 3)
n
( 0.5 deg) Mean COV MCS Preliminary
46 10 1.81 0.104 1.555 1.00
45 11 1.08 0.376 0.442 0.45
44 4 1.17 0.347 0.520 0.50
43 4 1.43 0.166 1.055 1.00
40 6 1.64 0.217 1.050 1.00
39 3 1.42 0.151 1.088 1.00
38 1 1.14 -- -- --
43 to 46 29 1.39 0.322 0.665 0.65
all angles 39 1.43 0.295 0.737 0.70