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127
3 4
= 2.93 = 3.78
COV = 0.651 COV = 0.463
2
Standard normal quantile
Standard normal quantile
2
1
0 0
Bearing Capacity of all cases Bearing Capacity of all cases
in rocks using Carter in rocks using Carter
-1 and Kulhawy (1988) and Kulhawy (1988)
RMR 85 -2 65 RMR < 85
Total data (n = 23) Total data (n = 57)
-2 Normal distribution Normal distribution
Lognormal distribution Lognormal distribution
-3 -4
0 1 2 3 4 5 6 7 8 0 1 2 3 4 5 6 7 8 9
Bias Bias
Figure 118. Comparison of the unfiltered bias Figure 119. Comparison of the unfiltered bias
for bearing capacity calculated using the Carter for bearing capacity calculated using the Carter
and Kulhawy (1988) method for all cases in and Kulhawy (1988) method for all cases in
rocks with RMR > 85 and the theoretical normal rocks with 65 <
RMR < 85 and the theoretical
and lognormal distributions. normal and lognormal distributions.
ommended is 0.35. When the rock has RMR 85 the recom- quality (expressed via RMR), and a calibration was required
mended is 0.50. For rocks with RMR lower than 85, = 1.00. following the rock quality designation. The relatively higher
resistance factors are a byproduct of the large bias of the method
and, hence, do not represent efficient design as expressed by the
4.13 Summary of Recommended
low efficiency factor of the method's application compared to
Resistance Factors for Shallow
Goodman's (1989) method.
Foundations in/on Rock
Table 70 summarizes (based on the information presented
in Tables 68 and 69) the recommended resistance factors to be 4.14 Sliding Friction Resistance
used in evaluation of the bearing capacity of shallow founda- 4.14.1 Parametric Study Evaluating the
tions on rock. The resistance factors for both examined meth- Resistance Factor as a Function of
ods are presented along with the efficiency factors providing a the Ratio of Dead to Live Load
measure for the relative efficiency of the methods.
Goodman's (1989) method performed exceptionally well The probabilistic characteristics of the parameter contribut-
consistently, regardless of rock quality. Improvement in the ing directly to the sliding friction resistance, the friction coeffi-
method's performance with an increase in knowledge trans- cient ratio ( fc), have been presented in Section 3.9 and
lates into an increase in the resistance factor and the associated summarized in Table 48. The uncertainties in the friction coef-
method efficiency. ficient ratio ( fc) follow one-to-one transformation to the slid-
The performance of the Carter and Kulhawy (1988) method ing resistance, i.e., the mean of sliding resistance = vertical
has a "built-in" safety that increases as the rock quality load × (mean of fc × tan f ) and the standard deviation (s.d.)
decreases. As such, the method's bias changes with the rock of sliding resistance = vertical load × (s.d. of fc × tan f ). Hence,
Table 69. Calibrated resistance factors for different datasets
of resistance bias obtained using Carter and Kulhawy's
(1988) method.
No. of Bias Resistance factor ( T = 3)
Dataset
cases Mean COV MCS Recommended
All cases 119 8.00 1.240 0.372 0.35
RMR 85 23 2.93 0.651 0.535 0.50
65 RMR < 85 57 3.78 0.463 1.149 1.00
44 RMR < 65 17 8.83 0.651 1.612 1.00
3 RMR < 44 22 23.62 0.574 5.295 1.00

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128
Table 70. Recommended resistance factors for foundations in/on rock
based on T = 3.0 (pf = 0.135%).
Efficiency
Method of
Equation Application factor
analysis
/ (%)
All 0.35 4.4
Carter and RMR 85 0.50 17.1
Kulhawy qult qu m s 65 RMR < 85 26.5
(1988) 44 RMR < 65 1.00 11.3
3 RMR < 44 4.2
For fractured rocks: All 0.30 22.2
qult qu N 1
Goodman Measured f 0.35 24.8
For non-fractured rocks:
(1989) (N 1) N
1 s Measured s 0.40 28.0
qult qu N 1
N 1 B Measured s and f 0.45 29.8
Table 71. Resistance factors obtained from MCS simulations for footings,
either cast in place or prefabricated, in soils with various friction angles,
along with the effect of ratios of lateral dead load to lateral live load.
(a) Cast-in-place footings
Resistance factor from MCS ( MCS)
f
obtained At-rest earth pressure Active earth pressure
from LFD/LFL LFD/LFL LFD/LFL LFD/LFL LFD/LFL LFD/LFL LFD/LFL LFD/LFL
=2 =4 =5 =7 =2 =4 =5 =7
SPT 0.469 0.455 0.452 0.447 0.507 0.498 0.496 0.492
CPT 0.516 0.499 0.494 0.488 0.558 0.545 0.542 0.537
Lab test 0.558 0.535 0.530 0.523 0.603 0.585 0.581 0.576
(b) Prefabricated footings
Resistance factor from MCS ( MCS)
f
obtained At-rest earth pressure Active earth pressure
from LFD/LFL LFD/LFL LFD/LFL LFD/LFL LFD/LFL LFD/LFL LFD/LFL LFD/LFL
=2 =4 =5 =7 =2 =4 =5 =7
SPT 0.195 0.193 0.193 0.191 0.211 0.212 0.211 0.211
CPT 0.217 0.213 0.212 0.210 0.234 0.233 0.232 0.232
Lab test 0.239 0.234 0.232 0.230 0.258 0.256 0.255 0.253
Table 72. Recommended resistance factors for sliding
resistance () for soil friction angles based on different
tests and lateral pressure due to at-rest or active earth
pressure for cast-in-place and prefabricated footings.
Resistance factor for sliding friction ( ) ( T = 3)
f
obtained At-rest earth pressure Active earth pressure
from Cast in- Cast in-
Prefabricated 2 Prefabricated 2
place 1 place 1
SPT 0.40 0.45
CPT 0.45 0.20 0.50 0.20
Lab test3 0.50 0.55
1 2 3
tan s = 0.91 tan f ; tan s = 0.53 tan f , Any laboratory shear strength
measurement of f

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129
the form of the limit state function for sliding resistance is Based on the loadings for the design example bridges consid-
essentially the same as that for the bearing resistance (see Equa- ered in the current research study, it is found that the ratios of
tion 118), which can be expressed as LFD to LFL range from 4 to 7. As a result, the resistance factors
for sliding resistance have been calibrated for LFD to LFL ratios
Z = R - LFD - LFL (125) varying from 2 to 7 and the corresponding results are presented
in Table 71 for cast-in-place and prefabricated footings.
where Z is the load combination for sliding, R is sliding resis-
tance of a footing, LFD is lateral load due to dead load, and LFL
4.14.2 Resistance Factors
is lateral load due to live load. A summary of the uncertainties
in the lateral loads and the load factors as recommended in The calculated resistance factors presented in Table 71 suggest
AASHTO (2007) are presented in Section 4.2.3.3. that the ratio of LFD to LFL does not have a pronounced effect
Analogous to the calibration of resistance factors for the on the magnitude of the resistance factors. As a result, selected
bearing resistance, the influence of the ratio of lateral dead load resistance factors are recommended for use for sliding resistance
to the lateral live load has been studied and presented here. of footings on granular materials as presented in Table 72.