Geotechnical Engineering

DHARM

BEARING CAPACITY 559

Considering a unit length of the footing and the equilibrium of wedge ABC, the vertical

components of all forces must sum up to zero.

The weight of the soil in the wedge is given by

W =

γφb^2

4

tan

...(Eq. 14.56)

Hence, for ΣV = 0,

b. qult + γφb

2

4

tan – 2P

p – bc tan φ = 0 ...(Eq. 14.57)

or qult =

2 P

b

p + c tan φ – γφbtan

4

...(Eq. 14.58)

This equation represents the solution to the problem if Pp is known.
For the simpler case of Df = 0 and c = 0, q = 0—that is, if the base of the footing rests on
the horizontal surface of a mass of cohesionless sand, we have

Pp =

1

2

γ^2

δα

H

cos sin

. Kp

In this case, H =

b

2

tan φ, δ = φ, Kp = Kpγ, and α = 180 – φ.

∴ Pp =

1

24

2

2

γφ

φ

b tan

cos. Kpγ ...(Eq. 14.59)

Here Kpγ is the coefficient of passive earth pressure for c = 0, α = 180° – φ, and δ = φ ; that

is, it is the value purely due to the weight of the soil.

Substituting Eq. 14.59 into Eq. 14.58, and putting c = 0,

(qult)c = 0 =

1

4

γb tan φ

Kpγ

cos^2 φ

− 1

F

HG

I

KJ

...(Eq. 14.60)

or (qult)c = 0 =

1

2. γb. Nγ ...(Eq. 14.61)

wherein Nγ =

1

2

tan φ

Kpγ

cos^2 φ

− 1

F

HG

I

KJ ...(Eq. 14.62)

The value of Kpγ is obtained by means of the spiral or the friction circle method. Since

the angle of well friction δ and the slope angle α of the contact face are equal to φ and to

(180° – φ) respectively, the value of Kpγ and hence of Nγ depend only on φ; thus, the values of Nγ

for various values of φ may be established once for all.

Nγ is called the ‘‘bearing-capacity factor’’ expressing the effect of the weight of the soil

wedge, ABC, of a cohesionless soil.

For the calculation of the bearing capacity of a cohesive soil, the computation of Pp

involves a considerable amount of labour. Terzaghi, therefore, advocated a simplified approach,

which is based on the equation

Ppn =

H

sinα (cKpc + qKpq) +

1

2

γH^2.

Kpγ

sinα ...(Eq. 14.63)