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69
the load-settlement curve was available with sufficient detail N = unit weight (footing width) term bearing capacity
and extent to be employed. For all other cases, the reported factor as specified in Tables 25 and 26 (dim.);
failure was adopted as the foundation's capacity. 1 = moist or submerged unit weight of soil above the
bearing depth of the footing (kcf);
2 = moist or submerged unit weight of soil below
3.4 Determination of the
the bearing depth of the footing (kcf);
Calculated Strength Limit
Df = footing embedment depth (ft);
States for the Case Histories
B = footing width (ft), equal to the physical footing
(Foundations on Soils)
width (B) in the case of centric loading or effective
3.4.1 Equations for Bearing Capacity footing width (B) in the case of eccentric loading;
(Resistance) Estimation sc, s, sq = footing shape correction factors as specified in
Table 27 (dim.);
The bearing capacity equation specified in AASHTO (2008)
dc, d, dq = depth correction factors to account for the shear-
with minimal necessary adjustment has been used to calculate
ing resistance along the failure surface passing
the bearing capacity of a footing (qn) of length L and width B
through the soil above the bearing elevation as
and supported by a soil with cohesion, c, average friction angle,
specified in Table 28 (dim.); and
f, and average unit weights, 1 and 2, above and below the
ic, i, iq = load inclination factors as specified in Table 29
footing base, respectively. The format presented in Equation 95
(dim.).
is based on the general bearing capacity formulation used by
Vesic´ (1975) as presented in Section 1.5.3 (see Equation 34).
The effective vertical stress calculated at the base of the
The numbering in parentheses represents the proposed num- Df
bering for the modified AASHTO specifications. footing i ( Di+1 - Di ) should be used (where i and Di are
0
qn = c i N cm + 1 i D f i N qm + 0.5 i 2 i B i N m effective unit weight and depth to the ith layer up to a depth of
Df) or alternatively, an average weighted soil unit weight (1)
(10.6.3.1.3a - 1) (95)
should be used above the base. Below the base an average soil
unit weight (2) should be used within a zone of 1.5B. The high-
in which: est anticipated groundwater level should be used in design.
In Tables 27 through 29, B and L are the physical footing
N cm = N c sc dc ic (10.6.3.1.3a - 2) (96) dimensions (in the case of centric loading), or they have to be
substituted with the effective footing dimensions, B and L
N qm = N q sq dq iq (10.6.3.1.3a - 3) (97) (in the case of eccentric loading).
In Table 29, H and V are the unfactored horizontal and
Nm = Ny s y d y i y (10.6.3.1.3a - 4) (98) vertical loads (kips), respectively. The angle is the projected
direction of load in the plane of the footing, measured from
where the side of the footing length, L (deg.). Figure 17 (similar
c = cohesion, taken as undrained shear strength cu to AASHTO Figure 10.6.3.1.3a-1) shows the conventions
in total stress analysis or as cohesion c in effective for determining . The parameter n is defined according to
stress analysis (ksf); Equation 99:
Nc = cohesion term bearing capacity factor as specified
in Tables 25 and 26 (dim.); ( 2 + L B ) 2 ( 2 + B L ) 2
n= cos + sin
Nq = surcharge (embedment) term bearing capacity (1 + L B ) (1 + B L )
factor as specified in Tables 25 and 26 (dim.); (10.6.3.1.3a - 5) (99)
Table 25. Bearing capacity factors Nc (Prandtl, 1921), Nq (Reissner, 1924),
and N (Vesic, 1975) (AASHTO Table 10.6.3.1.3a-1).
Factor Friction angle Cohesion term (Nc) Unit weight term (N) Surcharge term (Nq)
Bearing
f = 0 2+ 0.0 1.0
Capacity
Factors f
f > 0 (N q 1)cot f 2(N q + 1)tan f exp tan f tan 2 45 +
Nc, N , Nq 2

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Table 26. Bearing capacity factors Nc (Prandtl, 1921), Nq (Reissner, 1924),
and N (Vesic, 1975) (AASHTO Table 10.6.3.1.3a-2).
f Nc Nq N f Nc Nq N
0 5.14 1.0 0.0 23 18.1 8.7 8.2
1 5.4 1.1 0.1 24 19.3 9.6 9.4
2 5.6 1.2 0.2 25 20.7 10.7 10.9
3 5.9 1.3 0.2 26 22.3 11.9 12.5
4 6.2 1.4 0.3 27 23.9 13.2 14.5
5 6.5 1.6 0.5 28 25.8 14.7 16.7
6 6.8 1.7 0.6 29 27.9 16.4 19.3
7 7.2 1.9 0.7 30 30.1 18.4 22.4
8 7.5 2.1 0.9 31 32.7 20.6 26.0
9 7.9 2.3 1.0 32 35.5 23.2 30.2
10 8.4 2.5 1.2 33 38.6 26.1 35.2
11 8.8 2.7 1.4 34 42.2 29.4 41.1
12 9.3 3.0 1.7 35 46.1 33.3 48.0
13 9.8 3.3 2.0 36 50.6 37.8 56.3
14 10.4 3.6 2.3 37 55.6 42.9 66.2
15 11.0 3.9 2.7 38 61.4 48.9 78.0
16 11.6 4.3 3.1 39 67.9 56.0 92.3
17 12.3 4.8 3.5 40 75.3 64.2 109.4
18 13.1 5.3 4.1 41 83.9 73.9 130.2
19 13.9 5.8 4.7 42 93.7 85.4 155.6
20 14.8 6.4 5.4 43 105.1 99.0 186.5
21 15.8 7.1 6.2 44 118.4 115.3 224.6
22 16.9 7.8 7.1 45 133.9 134.9 271.8
Table 27. Shape correction factors sc, s, sq (Vesic
´, 1975)
(AASHTO Table 10.6.3.1.3a-3).
Factor Friction angle Cohesion term (sc) Unit weight term (s ) Surcharge term (sq)
B
f =0 1 0.2 1.0 1.0
Shape Factors L
sc, s , sq B Nq B B
f >0 1 1 0.4 1 tan f
L Nc L L
Table 28. Depth correction factors dc, d, dq (Brinch Hansen, 1970)
(AASHTO Table 10.6.3.1.3a-4).
Friction Cohesion term Unit weight term Surcharge term
Factor (dc) (dq)
angle (d )
for Df B:
Df
1 0.4
B
f =0 for Df > B: 1.0 1.0
Depth Df
1 0.4 arctan
Correction B
Factors for Df B:
d c, d , d q Df
2
1 2 tan f 1 sin f
1 dq B
>0 dq 1.0
f
Nq 1 for Df > B:
2 Df
1 2 tan f 1 sin f arctan
B
Table 29. Load inclination factors ic, i, iq (Vesic
´, 1975) (AASHTO Table 10.6.3.1.3a-5).
Factor Friction angle Cohesion term (ic) Unit weight term (i ) Surcharge term (iq)
n H
Load f =0 1 1.0 1.0
Inclination c B L Nc
Factors 1 iq n 1 n
iq H H
ic, i , iq f >0 1 1
Nq 1 V c B L cot f V c B L cot f

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71
Table 30. Summary of equations correlating internal friction angle (f)
to corrected SPT N value (N1)60.
Equation
Reference Correlation equation
no.
Peck, Hanson, and Thornburn (PHT) (1974)
(100)
as mentioned in Kulhawy and Mayne (1990) f 54 27.6034 exp 0.014 N1 60
f 20 N1 60
20
Hatanaka and Uchida (1996) (101)
for 3.5 N1 60
30
PHT (1974)
2
as mentioned by Wolff (1989) f 27.1 0.3 N1 60
0.00054 N1 60
(102)
Mayne et al. (2001) based on data from
Hatanaka and Uchida (1996) f 15.4 N1 60
20 (103)
Specifications for Highway Bridges (SHB) f 15 N1 60
15
(104)
Japan, JRA (1996) for N1 5 and 45
60 f
Note: pa is the atmospheric pressure and v is effective overburden pressure in the same units. For English units, pa = 1
and v is expressed in tsf at the depth N60 is observed. (N1)60 is the corrected SPT N value corrected using the
correction given by Liao and Whitman (1986):
pa
(N1)60 = N60 (105)
v
The depth correction factor should be used only when the B in the depth factor expressions results in a more conservative
soils above the footing bearing elevation are competent and evaluation as discussed by Paikowsky et al. (2009a).
there is no danger of their removal over the foundation's
lifetime; otherwise, the depth correction factor should be taken
as 1.0, or Df should be reduced to include the competent, 3.4.2 Estimation of Soil Parameters
secured depth only. Based on Correlations
The depth correction factors presented in Table 28 refer, 3.4.2.1 Correlations Between Internal Friction
when applicable, to the effective foundation width B. Some de-
Angle (f) and SPT N
sign practices use the physical footing width (B) for evaluating
the depth factors under eccentric loading as well. The calibra- Table 30 summarizes various correlations between SPT N
tion presented in this study was conducted using B. The use of and the soil's internal friction angle (see Equations 100 to 105).
45
Soil friction angle, f (deg)
40
35 (PHT 1974) (Wolff 1989)
(PHT 1974) (Kulhawy & Mayne, 1990)
(Hatanaka & Uchida 1996)
30 (Hatanaka & Uchida 1996)
(Mayne et al. 2001)
(JRA 1996)
25
0 10 20 30 40 50 60 70
Corrected SPT count, (N1)60
Figure 56. Comparison of various correlations between granular
soil friction angle and corrected SPT blow counts using the
overburden correction proposed by Liao and Whitman (1986).

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72
Figure 56 presents a comparison of the different correlations "natural soil condition" cases) was therefore evaluated using
listed in Table 30. The graph in Figure 56 suggests that in the Equation 100 relationship.
the range of about (N1)60 = 27 to 70, the Peck, Hanson, and
Thornburn (PHT) (1974) correlation (modified by Kulhawy
3.4.2.2 Correlations Between and SPT N
and Mayne, 1990, see Equation 100) provides the most conser-
vative yet realistic estimate of the soil's friction angle. The following equation was established by Paikowsky et al.
The use of Equations 100 and 101 is examined in Figure 57, (1995) for estimation of the unit weight of granular soils from
where the bias (measured over calculated bearing capacity) SPT blow counts:
when using both equations is presented. The use of Equation 100
resulted in the increase of the bias mean from 0.32 to 0.97 and = 0.88 ( N1 )60 + 99 ( pcf ) for 146pcf (106)
COV improved from 0.454 to 0.362 compared to that when
using Equation 101. Using Equation 101, the bias mean was The unit weights for the footing cases (for which soil unit
0.32 and the COV was 0.454; however, using Equation 100, the weight was not specified and SPT blow counts are available)
have been estimated through an iteration process, as shown
bias mean increased to 0.97 and the COV improved, becom-
in the flowchart presented in Figure 58. For an ith layer of
ing 0.32. For example, for the footing cases with Footing IDs
thickness (Di+1 - Di), as shown, the unit weight of soil is esti-
(FOTIDs) of #46, #49, and #77, the friction angles obtained
mated through an iteration until a precision of a small error
using Equation 101 are 41.0°, 33.9°, and 35.9°, and those using
() is obtained.
Equation 100 are 33.75°, 29.8°, and 32.3°. The resulting biases
were found to be 0.41, 0.39, and 0.77, in the previous case,
and 1.20, 0.69, and 1.30 in the latter, respectively. 3.4.2.3 Correlation Between f and
The correlation proposed by PHT (1974) as modified by
For the unique set of tests conducted at the University of
Kulhawy and Mayne (1990) was adopted for the friction angle
Duisburg-Essen (UDE), soil friction angles were estimated
evaluation. The PHT (1974) correlation has been found to give
using locally developed correlation with soil bulk density. The
more reasonable soil friction angles based on SPT N counts
soil friction angle used in these laboratory tests was exten-
than other correlations. The same correlation was also used sively tested, and Figure 59 shows the results of 52 direct shear
in NCHRP Project 24-17 (published as NCHRP Report 507: tests carried out on dry Essen sand with a dry unit weight in the
Load and Resistance Factor Design (LRFD) for Deep Foundations) range of 15.46 17.54 kN/m3 (98.5 111.75 pcf ).
and NCHRP Project 12-66 "AASHTO LRFD Specifications The tests were carried out with normal stresses between 50
for Serviceability in the Design of Bridge Foundations." The 200 kPa (0.52 2.09 tsf). Essen sand is a medium-to-
friction angle of the soils for the footings for which SPT N was coarse, sharp-edged silica sand. The sand has a specific gravity
available (typically field tests, categorized in later sections as of Gs 2.693 ± 0.004 and minimum and maximum porosities
of nmin 0.330 ± 0.012 and nmax 0.443 ± 0.006, respectively.
The correlation was revised after identifying outlier(s). The
1 best fit lines are as shown in Figure 59. Perau (1995) used all
Bias using Hatanaka and Uchida (1996)
n = 15 52 test data. The revised correlation is the best fit line obtained
0.8 from linear regression on 51 samples, with the circled test result
considered as an outlier.
0.6
The correlation given by Perau (1995) is the following:
Mean bias
f = 3.9482 - 23.492 (n = 52, R 2 = 0.771) (107)
0.4
The revised correlation is the following:
0.2
f = 3.824 - 21.527 (n = 51, R 2 = 0.804 ) (108)
0
0 0.4 0.8 1.2 1.6 2
It was found that the difference between the ultimate
Bias using Peck, Hanson and Thornburn (1974) bearing capacities obtained for a square footing (1.0 m2)
as mentioned in Kulhawy and Mayne (1990) using the friction angles obtained from the original correla-
Figure 57. Comparison of biases for the cases in tion, Equation 107 (Perau, 1995), and the revised correlation
natural soil conditions when using Equations 100 (Equation 108) is 10% to 18% for the range of friction angles
and 101. between 40° and 47°.