Shear Strength Applications in Plastic and Limit Equilibrium - Unsaturated Soil Mechanics in Engineering Practice - page 627

12.4.1 Terzaghi Bearing Capacity Theory
The Terzaghi bearing capacity formulation assumes that the
base of the footing is rough and the slip surface is bounded
by a straight line and a logarithmic spiral (Fig. 12.63a). The
load Q at failure is calculated by considering the forces on
the sliding mass of soil (Terzaghi, 1943).
The soil above the footing base was treated as a sur-
charge pressure for foundations embedded at some depth
(Fig. 12.63b). The bearing capacity equation consists of three
components: The first component, q γ , takes into account the
weight of the soil and the passive earth pressure block. This
portion of bearing capacity can be written as follows:

where:

N q = surcharge bearing capacity factor.

The ultimate bearing capacity of a soil can be expressed
as the sum of the above-mentioned components:

1

q f =

2 ρgBN γ +

cN c + ρgD f N q (12.66)

where:

N γ ,N c ,N q = dimensionless coefficients related to angle

of internal friction.

1

q γ =

2 ρgBN γ

(12.63)

The bearing capacity factors computed by Terzaghi (1943)
are shown in Fig. 12.64. The bearing capacity equation 12.66
was derived for strip footings but was further refined to
accommodate various shapes of the footing:

where:

N γ =

proportionally bearing capacity factor.

λ γ ρg B

The second component is related to the cohesion of the
soil:

q f =

2 N γ +

λ c cN c + ρgD f N q (12.67)

q c = cN c (12.64)

where:

where:
N c =

λ γ ,λ c = shape factors.

cohesion bearing capacity factor and

c = total cohesion of the soil. The shape factors for circular footings are λ γ = 0 . 6 and

λ c = 1 . 3 (Terzaghi, 1943). The shape factors for rectangu-

lar footings are λ γ

The cohesion value used in the analysis is dependent on
how the shear strength of the soil is defined. The third
component takes into account the surcharge effect of the
soil above the base of the footing:

0 . 2 (B/L)
(Skempton, 1951), where B is the width of the footing and
L the length of the footing.
The ultimate bearing capacity can be reduced to an allow-
able bearing capacity through use of an appropriate factor
of safety. Shallow foundations are of significant interest in

= −

0 . 2 (B/L) and λ c =

+

1 1

q q = ρgD f N q (12.65)

Q

Figure 12.63 Boundaries of zone of plastic equilibrium after failure of soil beneath continuous
footing: (a) rough-based footing; (b) rough-based footing with surcharge (from Terzaghi, 1943).