Geotechnical Engineering

DHARM

548 GEOTECHNICAL ENGINEERING

The bearing capacity may be determined from the relation between the principal stresses

at failure. The pertinent methods are those of Rankine, Pauker and Bell.

Rankine’s Method

This method, based on Rankine’s earth pressure theory, is too approximate and conservative

for practical use. However, it is given just as a matter of academic interest.

Rankine uses the relationship between principal stresses at limiting equilibrium condi-

tions of soil elements, one located just beneath the footing and the other just outside it as

shown in Fig. 14.1.

Df

b

qult

q= Dg f

s

qult

III

Fig. 14.1 Rankine’s method for bearing capacity of a footing

In element I, just beneath the footing, at the base level of the foundation, the applied

pressure qult is the major principal stress; under its influence, the soil adjacent to the element

tends get pushed out, creating active conditions. The active pressure is σ on the vertical faces

to the element. From the relationship between the principal stresses at limiting equilibrium

relating to the active state, we have:

σ = qult · KA = qult 1

1

−

+

F

HG

I

KJ

sin

sin

φ

φ

...(Eq. 14.5)

In element II, just outside the footing, at the base level of the foundation, the tendency
of the soil adjacent to the element is to compress, creating passive conditions. The pressure σ
on the vertical faces of the element will thus be the passive resistance. This will thus be the
major principal stress and the corresponding minor principal stress is q(= γDf), the vertical
stress caused by the weight of a soil column on it, or the surcharge dut to the depth of the
foundation. From the relationship between the principal stresses at limiting equilibrium re-
lating to the passive state, we have,

σ = q · Kp = γDf · Kp = γDf

1

1

+

−

F

HG

I

KJ

sin

sin

φ

φ ...(Eq. 14.6)

The two values of σ may be equated from Eqs. 14.5 and 14.6 to get a relationship for qult:

qult^ = γDf 1

1

2

+

−

F

HG

I

KJ

sin

sin

φ

φ

...(Eq. 14.7)

This gives the bearing capacity of the footing. It does not appear to take into account the
size of the footing. Further the bearing capacity reduces to zero for Df = 0 or for a footing
founded at the surface. This is contrary to facts.