Civil Engineering Reference
In-Depth Information
from Eq. (19.39) noting that the fan angle
θ
f
=
π
/2
s
b
s
a
=
exp
π
φ
tan
(19.52)
and the lower bound collapse load is:
pB
tan
2
4
+
φ
exp
π
φ
V
l
=
tan
(19.53)
2
The upper bound given by Eq. (19.49) is the same as the lower bound given by
Eq. (19.53) so we have an exact solution. This solution is however for the artificial
case of a foundation on a weightless soil. The bearing capacity arises from the stresses
p
on the surface outside the foundation. Bearing capacity of foundations for both the
drained and the undrained cases will be considered further in Chapter 22.
19.12 Summary
1. Estimates of the collapse of structures can be found from relatively simple upper
and lower bound calculations. An upper bound solution gives an unsafe load and
if this load is applied the structure must collapse; a lower bound gives a safe load
and with this load the structure cannot collapse.
2. To calculate an upper bound you have to choose a compatible mechanism of
collapse and equate the work done by the external loads with the work done by
the internal stresses. Mechanisms consist of slip surfaces that have circular arcs,
logarithmic spirals or straight lines and may be arranged as fan zones.
3. To calculate a lower bound you need to find a distribution of stress that is in
equilibrium with the external loads and does not exceed the appropriate failure
criterion. An equilibrium state of stress may have strong discontinuities or stress
fans.
The cases discussed in this chapter have been relatively simple and were intended
simply to illustrate the basic principles of the upper and lower bound calculations.
Other, more complicated, cases are given by Atkinson (1981).
Worked examples
Example 19.1: Loads on trench struts for undrained soil
The trench shown in
Fig. 19.23 is supported by smooth sheet piles held apart by struts, 1 m apart out
of the page, placed so that the piles do not rotate.
(a) Upper bound. Figure 19.24(a) shows a collapse mechanism and Fig. 19.24(b) is
the corresponding displacement diagram. The forces acting on the moving block
