DHARM336 GEOTECHNICAL ENGINEERINGSince this equation contains F on both sides, the solution should be one by trial and
error.
Bishop and Morgenstern (1960) evolved stability coefficients m and n, which depend
upon c′/γH, φ′, and cot β. In terms of these coefficients,
F = m – nru ...(Eq. 9.34)m is the factor of safety with respect to total stresses and n is a coefficient representing the
effect of the pore pressures on the factor of safety. Bishop and Morgenstern prepared charts of
m and n for sets of c′/γH values and for different slope angles.
If the effect of forces Rn and Rn + 1 is completely ignored, the only vertical force acting on
the slice is W.
Hence P = W cos α∴ F =r
WxclssW ul
ΣΣ[(cos )tan]′+ αφ− ′=r
WclssW ul
ΣΣ
sin[(cos )tan]
α′+ αφ− ′ ...(Eq. 9.35)since r sin α = x.
If u is expressed in terms of pore pressure ratio ru,u = ruγ. z = rW
u b
.But b = ls cos α∴ u =rW
lrW
lu
su
cos s.sec
α= α∴ F =1
ΣΣ
Wcl Wsur
sin[ (cos sec ) tan ]
α′+ ααφ− ′ ...(Eq. 9.36)This is nothing but the Eq. 9.22, obtained by the method of slices and adapted to the
case of steady seepage, pore pressure effects being taken into account. This approximate ap-
proach is the conventional one while Eq. 9.33 represents the vigorous approach.
9.3.3 Friction Circle Method
The friction circle method is based on the fact that the resultant reaction between the two
portions of the soil mass into which the trial slip circle divides the slope will be tangential to a
concentric smaller circle of radius r sin φ, since the obliquity of the resultant at failure is the
angle of internal friction, φ. (This, of course, implies the assumption that friction is mobilised
in full). This can be understood from Fig. 9.19.
This smaller circle is called the ‘friction circle’ or ‘φ-circle’.
The forces acting on the sliding wedge are:
(i) weight W of the wedge of soil
(ii) reaction R due to frictional resistance, and
(iii) cohesive force Cm mobilised along the slip surface. These are shown in Fig. 9.20.