Civil Engineering Reference
In-Depth Information
so that
F(
{
σ
B
}
)
λ
B
=
(6.60)
T
[
D
e
]
{
a
B
}
{
a
B
}
The change in stress is given by,
{
σ
} =
σ
e
−
F(
{
σ
B
}
)
[
D
e
]
{
a
B
}
(6.61)
T
[
D
e
]
{
a
B
}
{
a
B
}
and final stress by,
{
σ
} = {
σ
X
} +
σ
e
−
λ
B
[
D
e
]
{
a
B
}
(6.62)
Rice and Tracey (1973) advocated a mean normal method for a von Mises yield criterion
so that
T
σ
e
{
a
A
+
a
B
}
=
0
(6.63)
2
In a von Mises yield criterion under plane strain and 3D stress states, the yield surface
appears as a circle on the deviatoric plane. Any “illegal” stress can be corrected along a
radial path directed from the hydrostatic stress axis. The final deviatoric stress at point C is
√
3
c
u
{
s
C
} =
√
3
J
2
{
s
B
}
(6.64)
and the components of
{
σ
C
}
can then be determined by superimposing the hydrostatic
stress from point
.
In practice, it has been found that this method offers no advantages over forward Euler
in constant stiffness algorithms. The same is not true for tangent stiffness methods as is
shown in the next paragraph.
B
6.10 Tangent stiffness approaches
The difference between constant stiffness and tangent stiffness methods was discussed
in Section 6.1. In general, constant stiffness methods can be attractive in displacement-
controlled situations (see Figure 6.20 where the number of iterations per displacement
increment is modest), but in load-controlled situations, particularly close to collapse, large
numbers of iterations tend to arise (see e.g. Figure 6.10). Tangent stiffness approaches
require fewer iterations per load step, however this saving is counterbalanced by the speed
of constant stiffness methods, in which the global stiffness matrix is only factorised once.
If convergence in cases like Figure 6.10 is monitored, it will be found that most Gauss
points have converged to the yield surface, leaving only a few Gauss points responsible for
the lack of convergence. Tangent stiffness methods, with backward Euler integration can
significantly improve the convergence properties of algorithms to the point where the cost
of re-forming and re-factorising the global stiffness can be justified.
