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76
when analyzing data for natural soil deposits result in layer This means that the load inclination was no longer constant
variation (as expressed by the COV) and reduction in the mean during the test but varied from zero up to the maximum
bias. Further investigation as to the source of the obtained load inclination at failure, ult. The step-like load paths were
bias is presented in Section 4.4. applied in tests under inclined-centric and inclined-eccentric
loadings only.
3.6 Uncertainty in the Bearing
3.6.2 Determination of the Measured
Capacity of Footings in/on
Strength Limit State for Foundations
Granular Soils Subjected
Under Inclined Loading
to Vertical-Eccentric,
Inclined-Centric, and The procedure to determine the failure loads from the
Inclined-Eccentric Loading model tests depends on the load paths applied in the tests.
The analysis shows that in the case of a test with a radial load
3.6.1 Scope and Loading Procedures
path it is sufficient to consider only the vertical load versus
of the Case Histories
vertical displacement curve. This curve already includes the
The analysis of failure under vertical-eccentric, inclined- unfavorable effect that a horizontal load or a bending moment
centric, and inclined-eccentric loading is based on test results has on the bearing capacity of a shallow foundation, leading
from DEGEBO, Perau (1995), Montrasio (1994), and Gottardi to smaller vertical failure loads compared to the case of centric
(1992). The test conditions of the various data sources are vertical loading.
summarized in Table 31. The following analysis is based on Figure 65 provides an example using test results with inclined
the loading convention shown in Figure 64(a). loading performed by Montrasio (1994) under different load
The application of loadings in the tests varied. In the tests inclination angles. Both vertical load/vertical displacement
with radial load paths, both the vertical and the horizontal and horizontal load/horizontal displacement curves are shown
loads were increased up to failure, maintaining a constant for each test with inclined load. The load displacement relation-
ratio of F3/F1 during the test, i.e., the load inclination () was ship in Figure 65 indicates that the vertical failure load, F1,ult,
constant (see Figure 64(b)). The same applies to the tests decreases with the increase of the load inclination.
with eccentric loading; the eccentricity, e = M2/F1, was main- Applying the minimum slope criterion to the centric
tained constant during the test, because the vertical load was vertical load test results ( = 0°, MoA2.1) provides the fail-
applied eccentrically at one location. On the other hand, in the ure load F10,ult = 0.956 kip (4.25kN). The failure loads for
tests with step-like load paths, the vertical load was increased the tests with inclined loading decrease to F1,ult = 0.738kip
up to a certain level and then kept constant while the hori- (3.28kN) for a load inclination angle of = 3° (MoD2.1)
zontal load was increased up to failure (see Figure 64(c)). and F1,ult = 0.677 kip (3.01kN) for = 8° (MoD2.2) and further
Table 31. Test data used for failure analysis.
Footing size
Source Soil conditions Footing base Loading1 Load application1
ft2 (m²)
Fine to medium Eccentric radial load path
sand, loose to 1.6 6.6 (0.5 2.0) Inclined radial load path
medium dense, 3.3 3.3 (1.0 1.0) medium rough
DEGEBO
dense; 3.3 9.8 (1.0 3.0) (prefabricated) Inclined-
radial load path
gravel, medium 2.0 6.9 (0.6 2.09) eccentric
dense, dense
Eccentric radial load path
Medium to coarse rough (base
Perau 0.3 0.3 (0.09 0.09) Inclined step-like load path
sand, dense to very glued with
(1995) 0.2 0.2 (0.05 0.15) Inclined- F1-M2: radial load path
dense sand)
eccentric F1-F3: step-like load path
Eccentric radial load path
Medium to coarse 0.3 0.3 (0.08 0.08) rough (base
Montrasio Inclined radial load path
sand (Ticino Sand), 0.5 0.3 (0.16 0.08) glued with
(1994) Inclined- F1-F3: step-like load path
dense 0.8 0.3 (0.24 0.08) sand)
eccentric F3-M2: radial load path
Eccentric radial load path
Medium to coarse rough (base Inclined radial or step-like load path
Gottardi
sand (Adige Sand), 1.6 0.3 (0.5 0.1) glued with F1-M2: radial load path
(1992) Inclined-
dense sand) F1-F3: radial or step-like
eccentric
load path
1
See Figure 64 for details

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77
x2
F1 D
M2
M1
F3
F2
M3 b2
x3
b3 g
, f
x1
(a) Loading convention
F1 F1 F1
F1,const.
= constant arctan e = constant increasing
F3 M2 F3
(b) Radial load path (c) Step-like load path
Figure 64. Loading convention and load paths used during tests.
Vertical load, F1, F10 (kN) Horizontal load, F3 (kN)
0 1 2 3 4 0 0.2 0.4 0.6 0.8 1 1.2
0 0 0 0
MoD2.1 = 3°
Horizontal displacement, u3 (mm)
Vertical displacement, u1 (mm)
MoD2.2 = 8°
2 MoD2.3 = 14°
0.1 2
0.1
u1 (in)
u3 (in)
4
0.2
MoD2.1 = 3° 4
6 MoD2.2 = 8°
MoD2.3 = 14° 0.2
MoA2.1 = 0°(F10) 0.3
8 6
0 0.2 0.4 0.6 0.8 1 0 0.05 0.1 0.15 0.2 0.25
F1 (kips) F3 (kips)
(a) (b)
Figure 65. Loaddisplacement curves for model tests conducted by Montrasio (1994) with varying
load inclination: (a) vertical load versus vertical displacement and (b) horizontal load versus
horizontal displacement.

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decreases to F1,ult = 0.425kip (1.89kN) when the load inclination could always be used and therefore seemed to have a distinct
increases to = 14° (MoD2.3). Consequently, the correspond- advantage.
ing horizontal component of the failure load, F3,ult, increases
with the increase in the load inclination. Overall, the horizontal
3.6.3 Summary of Mean Bias Statistics
loads are significantly smaller than the vertical failure loads
for Vertical-Eccentric Loading
due to limited soil-foundation frictional resistance. This pro-
cedure results in vertical failure loads (F1,ult) that can be directly Table 32 presents a summary of the statistics of the bias for
related to the theoretical failure loads determined by the calcu- the footings under vertical-eccentric loading. Section G.2 in
lation model for the relevant load inclination, hence making it Appendix G presents the details of the bias calculation for a
possible to obtain the bias of the model for the bearing capacity single relevant case history (ID #471) of database UML-GTR
of foundations under inclined loads. ShalFound07. The total number of cases under vertical-
In the case of a step-like load path, a different procedure has eccentric loading from all sources was 43, including all outliers
to be applied. In these tests, the vertical load was kept constant to be addressed in Chapter 4. Seventeen cases from DEGEBO,
up to failure, hence the vertical load/vertical displacement 14 cases from Montrasio (1994) and Gottardi (1992) and
curves are not meaningful. The failure is analyzed on the basis 12 cases from Perau (1995) could be analyzed. Figure 66 pres-
of the horizontal load/horizontal displacement curves result- ents a histogram and a PDF of the bias as well as the relation-
ing in horizontal failure loads, F3,ult. The vertical failure loads, ship between measured and calculated bearing capacities for
F1,ult, are the ones corresponding to the horizontal failure all vertical, eccentrically loaded foundation cases summarized
loads, F3,ult, and coincide with the constant vertical load in in Table 32. DEGEBO results show the highest mean and COV
each test. As the load inclination is increased during the test, of the bias when using any of the failure criteria. Table 33
the maximum load inclination reached is the load inclination summarizes the statistics of the bias associated with bearing
at failure, tan ult = F3,ult/F1,ult. The theoretical (vertical) failure capacity calculations when using the full geometrical size of
load is then calculated for the load inclination at failure, ult, the foundation width (B). Table 33 was added in order to gain
and compared to the measured vertical failure load, F1,ult, to perspective on the bias in cases where the influence of the
determine the bias. Additionally, the theoretical horizontal effective width is neglected.
failure loads are calculated using the respective load inclina- Comparing Tables 32 and 33, it can be seen that the mean
tion at failure and the theoretical vertical failure loads. It can bias of the ultimate strength estimation decreases and the COV
be shown that the resulting biases of the horizontal failure of the bias increases when full footing geometry (B) is used
loads coincide with the biases of the vertical failure loads and instead of the effective footing dimensions (B). This is an
confirm this procedure. expected outcome considering the larger B would result in
In both procedures, the minimum slope criterion and the a higher bearing capacity (and hence decreased bias) while
two-slope criterion were examined for the failure load inter- the methodology is incorrect, contributing to the increased
pretation. In most cases, the results were found to be com- uncertainty (being represented by the COV). The decreased
parable. However, in some cases, the two-slope criterion was bias and increased COV would necessitate a significant increase
not applicable (FOTIDs #251 and #266, DEGEBO tests on in the resistance to ensure a specified safety, i.e., utilizing
eccentric loading, FOTIDs #301 and #317, and DEGEBO lower resistance factors. For example, considering all cases,
tests on inclined loading) while the minimum slope criterion the resistance factor obtained is 0.60 when B is used and 0.30
Table 32. Summary of the statistics for biases of the test results for
vertical-eccentric loading when using effective foundation width (B).
No. of Minimum slope criterion Two-slope criterion
Tests
cases Mean Std. dev. COV Mean Std. dev. COV
DEGEBO radial 17
2.22 0.754 0.340 2.04 0.668 0.328
load path (15)1
Montrasio
(1994)/Gottardi (1992) 14 1.71 0.399 0.234 1.52 0.478 0.313
radial load path
Perau (1995) radial
12 1.43 0.337 0.263 1.19 0.470 0.396
load path
43
All cases 1.83 0.644 0.351 1.61 0.645 0.400
(41)1
1
Number of cases for two-slope criterion

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79
1000
12 Vertical-eccentric loading
Vertical-eccentric loading
using Minimum Slope criterion (Vesic, 1963)
Data (n = 43)
n = 43 0.25
mean = 1.83 Data best fit line
Interpreted bearing capacity, qu,meas
10 COV = 0.351
100 No bias line
0.2
Number of observations
8 lognormal
distribution
Frequency
(ksf)
0.15 10
6
normal
distribution
0.1
4
1
2 0.05
0.1
0 0
0.1 1 10 100
0.4 1.2 2 2.8 3.6 Calculated bearing capacity, qu,calc
Bias, = qu,meas / qu,calc (Vesic, 1975 and modified AASHTO)
(a) (ksf)
(b)
Figure 66. (a) Histogram and probability density function of the bias and (b) relationship between measured
and calculated bearing capacity for all vertical, eccentrically loaded shallow foundations.
when B is used. Thus, Tables 32 and 33 indicate that the bear- presents a histogram and PDF of the bias as well as the relation-
ing capacity obtained using the full footing width (B) is unsafe ship between measured and calculated bearing capacity for all
when compared to the bearing capacity obtained when using inclined, centrically loaded shallow foundations.
the effective width (B). There are no differences in the biases obtained from the
two-slope and the minimum slope failure criteria for the cases
of step-like load paths. Gottardi's tests with radial load paths
3.6.4 Summary of Mean Bias Statistics
sometimes seem to result in smaller biases than the other tests,
for Inclined-Centric Loading
but overall, no significant differences exist in the biases of the
The mean and standard deviation of the calculated biases in step-like and radial load path tests. The biases determined for
the case of inclined loading are summarized in Table 34 for the the DEGEBO tests are also in the same order of magnitude
two failure criteria. Section G.3 of Appendix G presents the as the ones from the small-scale model tests although they
details of the bias calculations for a single relevant case history were carried out on foundations significantly larger in size.
(ID #547) of database UML-GTR ShalFound07. Figure 67 DEGEBO tests were carried out on foundations of 1.6 ft × 3.3 ft
Table 33. Summary of the statistics for biases of the test results for
vertical-eccentric loading when using the full foundation width (B).
No. of Minimum slope criterion Two-slope criterion
Tests
cases Mean Std. dev. COV Mean Std. dev. COV
DEGEBO radial 17
1.30 0.464 0.358 1.20 0.425 0.355
load path (15)1
Montrasio
(1994)/Gottardi (1992) 14 0.97 0.369 0.380 0.86 0.339 0.396
radial load path
Perau (1995) radial
12 0.79 0.302 0.383 0.64 0.296 0.465
load path
43
All cases 1.05 0.441 0.420 0.92 0.423 0.461
(41)1
1
Number of cases for two-slope criterion

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80
Table 34. Summary of the statistics for biases of the test results
for inclined-centric loading when using foundation width (B).
No. of Minimum slope criterion Two-slope criterion
Tests
cases Mean Std. dev. COV Mean Std. dev. COV
DEGEBO/
Montrasio 26
1.56 0.346 0.222 1.35 0.452 0.334
(1994)/Gottardi (1992) (24)1
radial load path
Perau (1995)/Gottardi
(1992) step-like load 13 1.17 0.537 0.459 1.17 0.537 0.459
path
39
All cases 1.43 0.422 0.295 1.29 0.455 0.353
(37)1
1
Number of cases for two-slope criterion
(0.5 m × 1.0 m) to 3.3 ft × 9.8 ft (1 m × 3 m) versus the small scale shallow foundation cases. As in the inclined-centric loading
models having foundation sizes of 2 in. × 6 in. (5 cm × 15 cm) cases, there is no significant difference in the tests results
to 4 in. × 20 in. (10 cm × 50 cm). between the radial and the step-like load paths. The bearing
capacity calculations of these case histories were noticeably
affected by using the effective foundation width (B) versus
3.6.5 Summary of Mean Bias Statistics
the geometrical actual foundation width (B). Table 36 sum-
for Inclined-Eccentric Loading
marizes the statistics associated with the bearing capacity
Table 35 presents a summary of the statistics of the bias for calculations using the full geometrical foundation width (B)
footings subjected to inclined-eccentric loadings, with both in order to gain perspective on the bias in cases where the influ-
radial and step-like load paths and including the effective ence of the effective width is neglected. The biases presented
foundation width, B. Figure 68 presents a histogram and PDF in Table 36 indicate that for the examined case histories the
of the bias as well as the relationship between measured and calculated bearing capacity using the effective width resulted in
calculated bearing capacity for all inclined, eccentrically loaded a bias about two times larger (i.e., a bearing capacity two times
100
Inclined-centric loading
Inclined-centric loading Data (n = 39)
using Minimum Slope criterion (Vesic, 1963)
12 n = 39 0.3 Data best fit line
mean = 1.43
Interpreted bearing capacity, qu,meas
No bias line
COV = 0.295
10
Number of observations
8 0.2
Frequency
(ksf)
lognormal normal
distribution distribution
1
4 0.1
0.1
0 0
0.1 1 10 100
0.2 0.6 1 1.4 1.8 2.2 2.6 Calculated bearing capacity, qu,calc
Bias, = qu,meas / qu,calc
(Vesic, 1975 and modified AASHTO)
(a) (ksf)
(b)
Figure 67. (a) Histogram and probability density function of the bias and (b) relationship between measured
and calculated bearing capacity for all inclined, centrically loaded shallow foundations.

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Table 35. Summary of the statistics for biases of the test results
for inclined-eccentric loading when using effective foundation width (B).
No. of Minimum slope criterion Two-slope criterion
Tests
cases Mean Std. dev. COV Mean Std. dev. COV
DEGEBO/Gottardi (1992)
8 2.06 0.813 0.394 1.78 0.552 0. 310
radial load path
Montrasio (1994)/
6 2.13 0.496 0.234 2.12 0.495 0.233
Gottardi (1992)
Perau (1995)
positive 8 2.16 1.092 0.506 2.15 1.073 0.500
Step-like eccentricity
load path Perau (1995)
negative 7 3.43 1.792 0.523 3.39 1.739 0.513
eccentricity
All step-like load
21 2.57 1.352 0.526 2.56 1.319 0.516
cases
All cases 29 2.43 1.234 0.508 2.34 1.201 0.513
100
9
Inclined-eccentric loading 0.3
using Minimum Slope criterion (Vesic, 1963)
n = 29
8
Interpreted bearing capacity, qu,meas
mean = 2.43
COV = 0.508 0.25
7
10
Number of observations
6
0.2
Frequency
(ksf)
5
0.15
4
lognormal 1
3 0.1
distribution Inclined-eccentric loading
2 normal Data (n = 29)
distribution Data best fit line
0.05
1 No bias line
0.1
0 0
0.1 1 10 100
1.2 1.8 2.4 3 3.6 4.2 4.8 5.4 6 6.6 7.2 Calculated bearing capacity, qu,calc
Bias, = qu,meas / qu,calc (Vesic, 1975 and modified AASHTO)
(a) (ksf)
(b)
Figure 68. (a) Histogram and probability density function of the bias and (b) relationship between measured
and calculated bearing capacity for all inclined, eccentrically loaded shallow foundations.
Table 36. Summary of the statistics for biases of the test results
for inclined-eccentric loading when using foundation width (B).
No. of Minimum slope criterion Two-slope criterion
Tests
cases Mean Std. dev. COV Mean Std. dev. COV
DEGEBO/Gottardi (1992)
8 1.07 0.448 0.417 0.94 0.365 0. 387
radial load path
Montrasio (1994)/
6 1.18 0.126 0.106 1.18 0.125 0.106
Gottardi (1992)
Perau (1995)
positive 8 0.70 0.136 0.194 0.70 0.135 0.194
Step-like eccentricity
load path Perau (1995)
negative 7 1.09 0.208 0.191 1.08 0.208 0.193
eccentricity
All step-like load
21 0.97 0.267 0.276 0.96 0.267 0.277
cases
All cases 29 1.00 0.322 0.323 0.96 0.290 0.303