Geotechnical Engineering

DHARM

338 GEOTECHNICAL ENGINEERING

The value of k may be obtained from Fig. 9.22.

120

1.16

1.12

1.08

1.04

1.00

0 20 40 60 80 100 120

Central angle °q

Coefficient k

I

II

[Note: (1) The relationship I is valid for sinusoidal variation of intergranular pressure with

zero values at the ends of the arc, which is considered nearer the actual distribution.

(2) The relationship II is valid for uniform pressure distribution.]

Fig. 9.22 Central angle versus coefficient k for the modified friction circle
Similarly the resultant mobilised cohesive force Cm can be located by equating its mo-
ments and the cohesive forces from elementary or finite lengths, into which the whole arc may
be divided, about the centre. If cm is the mobilised unit cohesion, the total mobilised cohesive
force all along the arc is cm.l; but the resultant total cohesive force Cm can be shown to be cm.lc
only where lc is the chord length since the resultant of an infinite number of small vectors
along the arc is the vector along the chord. Putting it in another way, the components parallel
to the chord add up to one another while those perpendicular to the chord cancel out on the
whole.
Thus, if a is the lever arm of the total cohesive force mobilised, Cm, from the centre of
the circle,

Cm. a = cm. lc.a = cm.l.r

a =

l

l

r

c

. ...(Eq. 9.37)

It may be noted that the line of action of Cm does not depend upon the value of Cm.
Fig. 9.23 illustrates these points clearly in addition to showing all the forces and the
corresponding triangle of forces.

The lines of action of W and Cm are located first. A tangent is drawn to the modified
friction circle from the point of intersection of W and Cm, to give the direction of R. Now the
triangle of forces may be completed as shown in Fig. 9.23 (b) drawing W to a suitable scale.

The factor of safety with respect to cohesion, assuming that friction is mobilised in full,
is given by:

Fc = c/cm ...(Eq. 9.38)