Bearing Capacity and Shallow Foundations - Elements of Soil Mechanics - page 265

9.6 Developments in bearing capacity equations

Terzaghi's bearing capacity equations have been successfully used in the design of numerous shallow
foundations throughout the world and are still in use. However, they are viewed by many to be conserva-
tive as they do not consider factors that affect bearing capacity such as inclined loading, foundation depth
and the shear resistance of the soil above the foundation. This section describes developments that have
been made to the original equations.

9.6.1 General form of the bearing capacity equation

Meyerhof ( 1963 ) proposed the following general equation for q u :

q = cN s i d + γ zN s i d + 0 5 . γ BN s i d

(10) u c c c c q q q q γ γ γ γ

where

s c , s q and s γ are shape factors
i c , i q and i γ are inclination factors
d c , d q and d γ are depth factors.

Other factors, G c , G q and G γ to allow for a sloping ground surface, and B c , B q and B γ to allow for any
inclination of the base, can also be included when required.
It must be noted that the values of N c , N q and N γ used in the general bearing capacity equation are
not the Terzaghi values. The values of N c and N q are now obtained from Meyerhof's equations ( 1963) , as
they are recognised as probably being the most satisfactory.

φ π

N = ( N − 1 )cot φ , N = tan

2

45 °+ e

tan φ

c q q

2

Unfortunately there is not the same agreement about the remaining factor N γ and the following expres-
sions all have their supporters:

N γ = (N q − 1) tan 1.4 φ Meyerhof ( 1963 )
N γ = 1.5(N q − 1) tan φ Hansen ( 1970)
N γ = 2(N q + 1) tan φ

Vesic ( 1973)

N γ = 2(N q − 1) tan φ where friction between foundation base and soil, δ ≥ φ /2 (Chen, 1975 ;

EN 1997-1: 2004)

It should be noted that Hansen suggested that the operating value of φ should be that corresponding
to plane strain, which is some 10% greater than the value of φ obtained from the triaxial test and normally
used. With this approach Hansen's expression for N γ = 1.5(N q − 1) tan 1.1 φ , which applies to a continuous
footing but is probably not so relevant to other shapes of footings.
It seems that the expression offered in EN 1997-1:2004 Annex D has become the preferred option.
Further examples in this chapter will therefore use the following expressions for the bearing capacity
coefficients:

N = ( N − 1 )cot φ c q

φ

2 π tan φ

N q = tan 45 °+ e

2

N = 2 ( N q − 1 ) tan φ γ

Typical values are shown in Table 9.1.