Civil Engineering Reference
In-Depth Information
}
i
,where
must be solved for the global displacement increments
{
U
i
represents the iter-
}
i
ation number, [
K
] the global stiffness matrix, and
{
F
the global external and internal
m
(body)loads.
The element displacement increments
}
i
}
i
, and these lead to
{
u
are extracted from
{
U
strain increments via the element strain-displacement relationships:
{
}
i
=
}
i
[
B
]
{
u
(6.15)
Assuming the material is yielding, the strains will contain both elastic and (visco) plastic
components, thus
{
}
i
=
e
i
+
p
i
(6.16)
It is only the elastic strain increments
e
i
that generate stresses through the elastic
stress-strain matrix, hence
[
D
e
]
e
i
{
σ
}
i
=
(6.17)
These stress increments are added to stresses already existing from the previous load
step and the updated stresses substituted into the failure criterion (e.g. 6.12). If stress
redistribution is necessary (
}
i
F>
0), this is done by altering the load increment vector
{
F
in equation (6.14). In general, this vector holds two types of load, as given by
}
i
= {
F
b
}
i
{
F
F
a
} + {
(6.18)
F
b
}
i
where
{
F
a
}
is the actual applied external load increment and
{
is the body loads vector
F
b
}
i
vector must be self-equilibrating so
that the net loading on the system is not affected by it. Two simple methods for generating
body loads are now described briefly.
that varies from one iteration to the next. The
{
6.6 Viscoplasticity
In this method (Zienkiewicz and Cormeau, 1974) , the material is allowed to sustain stresses
outside the failure criterion for finite “periods”. Overshoot of the failure criterion, as sig-
nified by a positive value of
F
, is an integral part of the method and is actually used to
drive the algorithm.
Instead of plastic strains, we now refer to viscoplastic strains and these are generated at
a rate that is related to the amount by which yield has been violated through the expression
∂Q
∂
σ
vp
=
F
˙
(6.19)
where
is the plastic potential function.
It should be noted that a pseudo-viscosity property equal to unity is implied on the right
hand side of equation (6.19) from dimensional considerations.
F
is the yield function and
Q
