Figure 4.1a shows a shallow rigid rough continuous foundation supported by soil that extends to a great depth. The ultimate bearing capacity of this foundation can be expressed (neglecting the depth factors) as (Chapter 2)
qu=cNc+qNq+12γBNγ (4.1)
The procedure for determining the bearing capacity factors Nc, Nq, and Ng in homogeneous and isotropic soils was outlined in Chapter 2. The extent of the failure zone in soil at ultimate load qu is equal to D. The magnitude of D obtained during the evaluation of the bearing capacity factor Nc by Prandtl1 and Nq by Reissner2 is given in a nondimensional form in Figure 4.2. Similarly, the magnitude of D obtained by Lundgren and Mortensen3 during the evaluation of Ng is given in Figure 4.3.
If a rigid rough base is located at a depth of H < D below the bottom of the founda- tion, full development of the failure surface in soil will be restricted. In such a case, the soil failure zone and the development of slip lines at ultimate load will be as shown in Figure 4.1b. Mandel and Salencon4 determined the bearing capacity factors for such a case by numerical integration using the theory of plasticity. According to Mandel and Salencon’s theory, the ultimate bearing capacity of a rough continuous foundation with a rigid rough base located at a shallow depth can be given by the relation
qu=cNc*+qNq*+12γBNγ* (4.2)
where
N N Nc*, q*, γ* = modified bearing capacity factors B = width of foundation
g = unit weight of soil
FiGure 4.2 Variation of D/B with soil friction angle (for Nc and Nq).
3
2
1
00 10 20 30 40 50
Soil friction angle, φ (deg)
D/B
FiGure 4.1 Failure surface under a rigid rough continuous foundation: (a) homogeneous soil extending to a great depth; (b) with a rough rigid base located at a shallow depth.
45 + φ/2 45 – φ/2
φ γ B
H
q = γDf
q = γDf
D c
φ γ c qu
B qu
(a)
(b) Rough rigid base
Note that for H ≥ D, N*c = Nc, N*q= Nq, and N*g= Ng (Lundgren and Mortensen).
The variations of Nc N*q, and N*gwith H/B and soil friction angle f are given in Figures 4.4, 4.5, and 4.6, respectively.
Neglecting the depth factors, the ultimate bearing capacity of rough circular and rectangular foundations on a sand layer (c = 0) with a rough rigid base located at a shallow depth can be given as
qu=qNq qs* *l +12γBNγ γ* *ls (4.3) where
l*qs, l*gs= modified shape factors
The above-mentioned shape factors are functions of H/B and f. Based on the work of Meyerhof and Chaplin5 and simplifying the assumption that the stresses and shear zones in radial planes are identical to those in transverse planes, Meyerhof6 evaluated the approximate values of l*qsand l*gs as
lqs m B
L
* = -
1 1 (4.4)
and
lγs m B L
* = -
1 2 (4.5)
where
L = length of the foundation
The variations of m1 and m2 with H/B and f are given in Figures 4.7 and 4.8.
FiGure 4.3 Variation of D/B with soil friction angle (for Ng).
3
2
1
00 10 20 30 40 50
Soil friction angle, φ (deg)
D/B
Milovic and Tournier7 and Pfeifle and Das8 conducted laboratory tests to verify the theory of Mandel and Salencon.4 Figure 4.9 shows the comparison of the experi- mental evaluation of N*g for a rough surface foundation (Df= 0) on a sand layer with theory. The angle of friction of the sand used for these tests was 43°. From Figure 4.9 the following conclusions can be drawn:
1. The value of N*g
for a given foundation increases with the decrease in H/B.
2. The magnitude of H/B = D/B beyond which the presence of a rigid rough base has no influence on the N*g value of a foundation is about 50%–75%
more than that predicted by the theory.
3. For H/B between 0.5 to about 1.9, the experimental values of N*g are higher than those predicted theoretically.
4. For H/B < about 0.6, the experimental values of N*g are lower than those predicted by theory. This may be due to two factors: (a) the crushing of sand grains at such high values of ultimate load, and (b) the curvilinear nature of the actual failure envelope of soil at high normal stress levels.
FiGure 4.4 Mandel and Salencon’s bearing capacity factor Nc* [equation (4.2)].
H/B = 0.25
0.33
0.5
1.0
1.6 1.2 0.9
0.7
10 2 5 10 20 50 100 200 500 1000 2000 5000 10,000
10 20 30 40
D/B = 2.4 Nc*
φ (deg)
Cerato and Lutenegger9 reported laboratory model test results on large square and circular surface foundations. Based on these test results they observed that, at about H/B ≥ 3,
Nγ* ≈Nγ
Also, it was suggested that for surface foundations with H/B < 3,
qu=0 4. γBNγ* (square foundation) (4.6) and
qu=0 3. γBNγ* (circular foundation) (4.7) The variation of N*g recommended by Cerato and Lutenegger9 for use in equations (4.6) and (4.7) is given in Figure 4.10.
For saturated clay (that is, f = 0), equation (4.2) will simplify to the form
qu=c Nu c*+q (4.8)
Mandel and Salencon10 performed calculations to evaluate N*c for continuous foundations. Similarly, Buisman11 gave the following relationship for obtaining the FiGure 4.5 Mandel and Salencon’s bearing capacity factor Nq* [equation (4.2)].
H/B = 0.2
0.4 0.6 1.0
1.6 1.9
2.4
1.2 1.4
120 2 5 10 20 50 100 200 500 1000 2000 5000 10,000
25 30 35 40 45
D/B =
N*q 3.0
φ (deg)
FiGure 4.6 Mandel and Salencon’s bearing capacity factor Ng* [equation (4.2)].
H/B = 0.2
0.4 0.6
1.0
0.8 1.0
1.2
0.6 0.5
120 2 5 10 20 50 100 200 500 1000 2000 5000 10,000
25 30 35 40 45
D/B =
N*γ 1.5
φ (deg)
FiGure 4.7 Variation of m1 (Meyerhof’s values) for use in the modified shape factor equation [equation (4.4)].
020 0.2 0.4 m1
0.6 0.8 1.0
25 30 40
2.0 0.6 1.0
0.4 0.2 H/B = 0.1
35 45
φ (deg)
ultimate bearing capacity of square foundations:
q B
H c q B
u(square)= + + - u for H
+ - ≥
π 2 2
2
2 2
2
2 0
(4.9)
where
cu = undrained shear strength
FiGure 4.8 Variation of m2 (Meyerhof’s values) for use in equation (4.5).
020 0.2 0.4 m2
0.6 0.8 1.0
25 30 40
0.6 1.0 0.4
0.2 H/B = 0.1
35 45
φ (deg)
FiGure 4.9 Comparison of theory with the experimental results of Ng* (Note: f = 43°, c = 0).
Theory [4]
Experiment [8]
Scale change
0 0.4 0.8 1.2 1.6 2.0 4.0 5.0
H/B 5000
2000
1000
500
200
100 Nγ*
Equation (4.9) can be rewritten as
qu( ) . . BH . cu q
square = + .-
+ = 5 14 1 0 5 0 707
5 14 Nc*(square)cu+q (4.10)
Table 4.1 gives the values of N*cfor continuous and square foundations.
Equations (4.8) and (4.9) assume the existence of a rough rigid layer at a limited depth. However, if a soft saturated clay layer of limited thickness (undrained shear strength = cu(1)) is located over another saturated clay with a somewhat larger shear strength cu(2) [Note: cu(1) < cu(2); Figure 4.11], the following relationship suggested by Vesic12 and DeBeer13 may then be used to estimate the ultimate bearing capacity:
q B
L
c
u c
u u
B
= + H
+ -
1 0 2 5 14 1 1
2
. . ( )
( )
--
( + )
+ 2
2 BL 1 cu( )1 q (4.11) where
L = length of the foundation
FiGure 4.10 Cerato and Lutenegger’s bearing capacity factor Ng* for use in equations (4.6) and (4.7). Source: Cerato, A. B., and A. J. Lutenegger. 2006. Bearing capacity of square and circular footings on a finite layer of granular soil underlain by a rigid base. J. Geotech.
Geoenv. Eng., ASCE, 132(11): 1496.
2000
1500
1000
500
0 Nγ*
20 25 30 35 40 45
Friction angle, φ (deg) H/B = 0.5
1.0 2.0
3.0
table 4.1
Values of N*c for Continuous and Square Foundations (f = 0 Condition)
B H
N*c
Squarea Continuousb
2 5.43 5.24
3 5.93 5.71
4 6.44 6.22
5 6.94 6.68
6 7.43 7.20
8 8.43 8.17
10 9.43 9.05
a Buisman’s analysis. Source: From Buisman, A. S.
K. 1940. Grond-mechanica. Delft: Waltman.
b Mandel and Salencon’s analysis. Source: From Mandel, J., and J. Salencon. 1969. Force portante d’un sol sur une assise rigide, in Proc., VII Int. Conf.
Soil Mech. Found Engg., Mexico City, 2, 157.
FiGure 4.11 Foundation on a weaker clay underlain by a stronger clay layer (Note: cu(1)< cu(2)).
B
H
qu
cu(1)
q = γDf
Weaker clay layer φ1 = 0
cu(2)
Stronger clay layer φ2 = 0