Geotechnical Engineering

DHARM

BEARING CAPACITY 561

The problem of Nc and Nq has been rigorously solved by means of Airy’s stress function

(Prandtl 1920, Reissner, 1924), for the condition γ = 0:

Nc = cot φ

aθ

φ

2

2452 cos (^2 °+ / )−^1

L

N

M

M

O

Q

P

P

...(Eq. 14.68)

and Nq =

aθ

φ

2

2452cos (^2 °+ / ) ...(Eq. 14.69)

wherein aθ = e(3π/4–φ/2) tan φ ...(Eq. 14.70)

The values Nc and Nq depend only on the value of φ.

The critical load per unit length of the strip footing is given by

Qult = b. qult ...(Eq. 14.71)

Also, Nc = cot φ (Nq – 1) ...(Eq. 14.72)

For a purely cohesive soil, φ = 0

Nc =

3

2

π + 1 = 5.7 (obtained by applying L’ Hospital’s rule, since
Nc = ∞ × 0 for φ = 0) ...(Eq. 14.73)
Nq = 1 ...(Eq. 14.74)
and Nγ = 0 ...(Eq. 14.75)
Thus, the bearing capacity of a strip footing with a rough base on the ground surface is
given by

qult = 5.7c ...(Eq. 14.76)
This compares very well with the corresponding value from Prandtl’s equation for a
continuous footing with a smooth base.

For strip footing at a depth Df in a purely cohesive soil

qult = 5.7c + γDf. ...(Eq. 14.77)

Equation 14.67, along with the bearing capacity factors Nc, Nq and Nγ are valid for

‘general shear failure’. An explanation of ‘general shear failure’ and ‘local shear failure’, as

given by Terzaghi, is set out below:

Before the load is applied, the soil beneath the base of the footing is in a state of elastic

equilibrium. When the load is increased beyond a certain critical value, the soil gradually

passes into a state of plastic equilibrium. During this process of transition both the distribu-

tion of soil reactions over the base of the footing and the orientation of the principal stresses in

the soil beneath the footing change. The transition starts at the outer edges of the base and

spreads outwards. If the mechanical properties of the soil are such that the strain which pre-

cedes the failure of the soil by plastic flow is very small, the footing does not sink into the

ground until a state of plastic equilibrium indicated in Fig. 14.8 (a) has been reached. The

failure occurs by sliding in the two outward directions. The corresponding relation between

load and settlement is shown by the solid curve C 1 in Fig. 14.9. This type of failure is called

‘general shear failure’. This is applicable to dense and stiff soils.

On the other hand, if the mechanical properties of the soil are such that the plastic flow

is preceded by a very important strain, the approach to general shear failure is associated with