Geotechnical Engineering

DHARM

BEARING CAPACITY 555

(ii) The Mohr-Coulomb equation for failure envelope τ = c + σ tan φ is valid for the soil,
as shown in Fig. 14.7 (b).
(iii) Wedges I and III act as rigid bodies. The zones in Sectors II deform plastically. In
the plastic zones all radius vectors or planes through A and B are failure planes and the curved
boundary is a logarithmic spiral.

(iv) Wedge I is elastically pushed down, tending to push zones III upward and outward,
which is resisted by the passive resistance of soil in these zones.

(v) The stress in the elastic zone I is transmitted hydrostatically in all directions.

qultb B(pole of logarithmic spiral)

III

F A

G

a

ro

II

a yy

P+Pat

r (^1) I
aa II
aa
III
D
Tangent to the spiral C (90° – )f
E
Logarithmic/spiral (r = r eoqftan)
Tangent to the spiral
= 45° + /2
= 45° – /2
r=re
yf
af
1o

• tan^12 pf

s

f

sfi(= c cot ) sa

qult

c

2 qfcr= 90° +

s=c+

sftan

s

(a) Prandtl’s system

(b) Mohr’s circle for active zone

Fig. 14.7 Prandtl’s method of determining bearing capacity of a c – φ soil

It may be noted that the section is symmetrical up to the point of failure, with an equal

chance of failure occurring to either side. (That is why the section to one side, say to the left, is

shown by dashed lines). The equilibrium of the plastic sector is considered by Prandtl.

Let BC be r 0. The equation to a logarithmic spiral is:

r = r 0 eθ tan φ, where θ is the spiral angle.

Then BD = r 0 e(π/2) tan φ, since ∠CBD = 90° = π/2 rad.

From the Mohr’s circle for c – φ soil, Fig. 14.7 (b), the normal stress corresponding to the

cohesion intercept is:

σi = c cot φ ...(Eq. 14.42)

This is termed the ‘initial stress’, which acts normally to BC in view of assumption (v);

also qult, the applied pressure is assumed to be transferred normally on to BC. Thus the force

on BC is