Civil Engineering Reference
In-Depth Information
Figure 19.1
Straining of perfectly plastic soil with an associated flow rule.
envelope as shown. For drained loading the failure envelope is given by Eq. (19.2) and
if the flow rule is associated the angle of dilation at the critical state
ψ
c
is
p
n
−
δε
φ
c
=
tan
ψ
=
tan
(19.3)
c
p
δγ
At the critical state, however, soil strains at a constant state (i.e. at a constant volume)
and so
0, which means that, at failure at the critical state, the flow rule is not
associated and soil in drained loading is not perfectly plastic. This does not actually
matter very much as you can prove that an upper bound for a material with
ψ
=
c
ψ
c
=
φ
c
φ
c
, but you can not do the same for the
lower bound. In practice upper and lower bounds for soil structures calculated with
ψ
c
=
φ
c
give good agreement with experimental observations and, although the lower
bound solution is not absolutely rigorous, the errors seem to be small.
The statements of the bound theorems are simple and straightforward:
is still an upper bound, even if
ψ
c
is less than
1. Upper bound. If you take any compatible mechanism of slip surfaces and consider
an increment of movement and if you show that the work done by the stresses in
the soil equals the work done by the external loads, the structure must collapse
(i.e. the external loads are an upper bound to the true collapse loads).
2. Lower bound. If you can determine a set of stresses in the ground that are in
equilibrium with the external loads and do not exceed the strength of the soil,
