Geotechnical Engineering

DHARM

560 GEOTECHNICAL ENGINEERING

where Ppn = normal component of the passive earth pressure on a plane contact face with a

height H,

α = slope angle of the contact face, and

Kpc, Kpq, and Kpγ = coefficients whose values are indpendent of H and γ. In the present case,

H =

b

2

tan φ, α = 180° – φ, δ = φ.

Also the total passive earth pressure Pp on the contact face is equal to

Ppn

cosδ

.

∴ Pp =

Ppn

cosδ

=

Ppn

cosφ

=

b

2 cos^2 φ

(cKpc + qKpq) +

1

24

2

γ 2

φ

φ

..

tan

cos

b

. Kpγ ...(Eq. 14.64)

where (cKpc + qKpq) = Ppn is the normal component of the passive earth pressure comprehend-
ing the effect of cohesion and surcharge.

Combining this equation with Eq. 14.58, we have

qult = c

Kpc

cos

2 tan

φ

φ+

F

HG

I

KJ

+ q

Kpq

cos^2 φ

+ γb

4

tan φ

Kpγ

cos^2 φ

− 1

F

HG

I

KJ

...(Eq. 14.65)

wherein Kpc, Kpq, and Kpγ are pure numbers whose values are independent of b.

If the soil wedge, ABC, is assumed weightless (γ = 0) (Prandtl, 1920), Eq. (14.65) takes

the form

qqultcq+ ult = c

Kpc

cos

2 tan

φ

φ+

F

HG

I

KJ

+ q.

Kpq

cos^2 φ

= cNc + q. Nq ...(Eq. 14.66)

The factors Nc and Nq are pure numbers whose values depend only on the value φ in

Coulomb’s equation. The value qultcrepresents the bearing capacity of the weightless soil, if

the surcharge q were equal to zero (γ = 0 and q = 0), and qultqis the bearing capacity exclusively
due to the surcharge q(γ = 0 and c = 0).
On the other hand, if c = 0 and q = 0, γ being greater than zero, the bearing capacity is
given by Eqs. 14.61 and 14.62:

qult =^1

2

1

4

γγφ 2 1

γ φ

bN =−b F Kpγ

HG

I

KJ

tan

cos

(However, it should be noted that the failure surface for this condition is somewhat

above that for γ = 0, and also the exact mathematical shape is not known).

If the values c, Df,and γ are greater than zero,

qult = qultc + qultq+ qultr = cNc + γDf Nq +

1

2

γb Nγ ...(Eq. 14.67)
This is called ‘‘Terzaghi’s general bearing capacity formula’’. (The discrepancy arising
out of the difference in failure surfaces for the two conditions—γ = 0 and γ > 0—is considered
inconsequential).
The coefficients Nc, Nq, and Nγ are called ‘‘bearing capacity factors’’ for shallow continu-
ous footings. Since their values depend only on the angle of shearing resistance φ they can be
computed once and for all.