Civil Engineering Reference
In-Depth Information
Figure 19.12
Geometrical analysis of Figure 19.11(b).
the stresses on the discontinuities marked
α
and
β
in Fig. 19.13(a). From the geometry
of Figs 19.13(b) and (c),
q
l
+
γ
z
=
4
s
u
+
γ
z
(19.27)
and hence a lower bound for the collapse load is
V
l
=
4
s
u
B
(19.28)
Alternatively, we could consider the rotations of the directions of the major principle
stresses across the discontinuities, making use of Eq. (19.19). For each discontinuity
δθ
=
90
◦
and
2
s
u
; hence, from the geometry of Fig. 19.13(b) and (c) we obtain
Eqs. (19.27) and (19.28). The mean of the upper and lower bound solutions gives
V
c
δ
s
=
20 per cent from this mean. Bearing in
mind the problems in determining true values of
s
u
for natural soils, which may not
be either isotropic or homogeneous, these simple bounds may be adequate for simple
routine designs. However, in order to illustrate the use of slip fans and stress fans we
will examine some alternative solutions.
=
5
s
u
and the bounds differ by about
±
19.9 Upper and lower bound solutions using fans
In Fig. 19.3(f) there is a combination of straight and curved slip surfaces and in order
to have a compatible mechanism it is necessary to have slip fan surfaces as illustrated.
