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124
as could be observed in all other cases of loading, the recom- Table 67. Recommended resistance factors for
mended resistance factor may be conservatively reduced to 0.80. shallow foundations on natural deposited granular
soil conditions.
4.10 Summary of Recommended Soil friction
Loading conditions
Resistance Factors for Footings angle f Vertical-centric
Inclined-centric
Inclined-eccentric
or -eccentric Positive Negative
in/on Granular Soils
30°34° 0.40 0.65
0.35
Tables 66 and 67 present the resistance factors recom- 35°36° 0.45 0.40
0.70
mended for use in the design of shallow foundations in/on 37°39° 0.50
0.40
granular soils (controlled soil conditions and natural soil con- 40°44° 0.55 0.45
0.75
45° 0.65 0.50 0.45
ditions, respectively) with soil friction angles (f) in the range
of 30° to 45° and relative density (DR) 35%. The resistance Notes:
(1) f determined from Standard Penetration Test results.
factors for controlled soil conditions are to be used when (2) Granular material is assumed to extend below the base of the footing at least
the foundations are placed in/on compacted engineering fills two (2.0) times the width of the foundation.
(3) The resistance factors were evaluated for a target reliability (T) = 3.0.
extending to a depth of no less than two (2.0) times the foun-
dation width below the foundation base. The internal friction
angle in such cases is to be determined by laboratory testing.
Use of the resistance factors for natural soil conditions is rec- measured. Figure 113 presents the standard normal quantile of
ommended when the foundations are placed on/in the in situ the unfiltered bias data for all cases with the theoretical normal
soil, and the soil's internal friction angle is assumed to be eval- and lognormal distributions based on the calculated mean and
uated from correlations with Standard Penetration Testing. standard deviation. The -squared values of the normal and
lognormal distributions are found to be 121.28 and 18.79,
respectively. The match observed in Figure 113 and the GOF
4.11 Goodman's (1989) test results indicate that the lognormal distribution is the
Semi-Empirical Bearing matching underlying distribution for the data with an accep-
Capacity Method for Footings tance level of the GOF test at 1% (for which the acceptable
in/on Rock highest -squared value is 21.67). These results also mean that
4.11.1 Identification of Outliers no outliers need to be identified for the dataset of all cases.
Figures 114 and 115 present the standard normal quantiles
The -squared GOF tests have been carried out on the of the unfiltered bias data for the cases with measured rock
datasets containing all the cases and subsets: (1) cases with friction angle and measured rock discontinuity spacing, respec-
measured friction angle, (2) cases with measured rock discon- tively, along with the relations predicted from the theoretical
tinuity spacing s, and (3) cases with both friction angle and s
4
Table 66. Recommended resistance factors for = 1.35
COV = 0.535
shallow foundations on granular soils placed under
Standard normal quantile
controlled conditions. 2
Loading conditions
Soil friction
Vertical-centric Inclined-eccentric
angle f Inclined-centric 0
or -eccentric Positive Negative
30°34° 0.50 Bearing Capacity using
0.40 0.40 0.70
35°36° 0.60 Goodman (1989)
-2 All data
37°39° 0.70 0.45 0.45 0.75 Total data (n = 119)
40°44° 0.75 0.50 Normal distribution
0.50 0.80
45° 0.80 0.55 Lognormal distribution
-4
Notes:
(1) f determined by laboratory testing. 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5
(2) Compacted controlled fill or improved ground are assumed to extend below the Bias
base of the footing to a distance to at least two (2.0) times the width of the
foundation (B). If the fill is less than 2B thick, but overlays a material equal or Figure 113. Comparison of the unfiltered bias for
better in strength than the fill itself, then the recommendation stands. If not, bearing capacity calculated using the Goodman
then the strength of the weaker material within a distance of 2B below the
(1989) method for all data and the theoretical
footing prevails.
(3) The resistance factors were evaluated for a target reliability (T) = 3.0. normal and lognormal distributions.

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125
4 4
= 1.41 = 1.51
COV = 0.541 COV = 0.459
Standard normal quantile
Standard normal quantile
2 2
0 0
Bearing Capacity using
Bearing Capacity using Goodman (1989)
Goodman (1989) All cases with measured
All cases with discontinuity spacing and
measured friction angle measured friction angle
-2 Total data (n = 98) -2
Total data (n = 67)
Normal distribution Normal distribution
Lognormal distribution Lognormal distribution
-4 -4
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5
Bias Bias
Figure 114. Comparison of the unfiltered Figure 116. Comparison of the unfiltered
bias for bearing capacity calculated using the bias for bearing capacity calculated using the
Goodman (1989) method for all data on rocks Goodman (1989) method for all data on rocks
with measured friction angles and the theoretical with measured discontinuity spacing and friction
normal and lognormal distributions. angle and the theoretical normal and lognormal
distributions.
normal and lognormal distributions. For the dataset of cases
with measured friction angle presented in Figure 114, the Figure 116 examines the standard normal quantile for the
-squared value for the normal distribution is found to be resistance bias dataset of cases with both friction angle and dis-
64.35 while that for the lognormal distribution is 15.60, continuity spacing measured along with the predicted rela-
which is accepted with a significance level of 5%. For the tions for the theoretical normal and lognormal distributions.
dataset of cases with measured rock discontinuity spacing The -squared value from the GOF tests obtained for the nor-
presented in Figure 115, the -squared value for the normal mal distribution is 66.27 while that for the lognormal distri-
distribution is found to be 113.92 while that for the lognor- bution is 11.77.
mal distribution is 11.99, which is also accepted with a sig- Based on the data and analyses of Figures 113 to 116, it can
nificance level of 5%. be concluded that the bias associated with Goodman's (1989)
analysis of shallow foundations on rock as an entire set and
its subsets match the lognormal distribution, and no outliers
4 exist for the examined datasets.
= 1.43
COV = 0.461
4.11.2 Calibration of Resistance Factors
Standard normal quantile
2
Table 68 shows the resistance factors () obtained from the
MCS using one million samples for each dataset considered. As
0
Bearing Capacity using
can be expected, the uncertainties in the estimated bearing
Goodman (1989) resistance decrease with the increase in the available reliable
All cases with
measured discontinuity spacing information, thereby increasing the confidence of the estimated
-2 Total data (n = 83)
Normal distribution
resistances, and thus resulting in higher resistance factors. When
Lognormal distribution all data are used, without differentiating between data for which
-4 the rock properties information is available from the field and
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 testing and data for which rock properties information is esti-
Bias mated by the outlined procedure, the recommended resistance
Figure 115. Comparison of the unfiltered factor is 0.30. The resistance factor can be increased to 0.45
bias for bearing capacity calculated using the when the relevant rock properties, i.e., rock friction angle and
Goodman (1989) method for all data on rocks rock discontinuity spacing, are measured values.
with measured discontinuity spacing s' and the Figures 113 and 114 indicate that the assumed lognormal
theoretical normal and lognormal distributions. distribution overpredicts the bias in the lower tail regions of the