Civil Engineering Reference
In-Depth Information
Initialisation, e.g.
.
..
{U
0
}={U
0
}={U
0
}={0}
For all time steps
.
..
Replace {U
1
} by {U
0
}+
∆
t{U
0
}+0.5
∆
t{U
0
}
For all elements
Retrieve element geometry.
Find nodal displacments {u}.
For all Gauss points
Find the [B] and [D] matrices.
Find the element strains {
e
}=[B]{u}
Find the element stresses {
s
} =[D]{
e
}
Find the element internal forces
{bload}=
∫
[B]
T
{s
}dV
Assemble global internal forces
{bdylds}
Add external loads to internal loads
..
Compute {U
1
}= [M]
−
1
*{bdylds}
.
.
..
..
Replace {U
1
} by {U
0
}+ 0.5
∆
t({U
0
}+{U
1
}).
..
..
Replace {U
1
} by {U
0
}
Figure 11.18
Structure chart for Program 11.7
was described in Chapter 6. The nomenclature used is therefore drawn from the earlier
programs in this chapter and from those in Chapter 6, particularly Program 6.1 which dealt
with von Mises solids. The structure chart for the program is shown in Figure 11.18 and
the problem layout and data in Figure 11.19.
Turning to the program code, after input and initialisation, the von Mises plastic stress-
strain matrix [
D
p
] (called
pl
in the program, see Appendix C) is formed by subroutine
vmdpl
. The remainder of the program is a large explicit integration time-stepping loop. The
displacements (called
x1
) are updated and then, scanning all elements and Gauss points,
new strains can be computed. The constitutive relation then determines the appropriate level
of stress and hence whether the yield stress has been violated or not. If the yield stress has
not been violated, the material remains elastic, otherwise the constitutive matrix is updated
by subtracting a proportion of the plastic matrix
pl
from the elastic matrix
dee
(see Figure
6.7). The corrected stresses are then redistributed as “body loads”
bdylds
, whence the
new accelerations (
d2x1
) can be found and integrated to find the new velocities (
d1x1
).
The next cycle of displacements can then be updated. The load weightings are read as data,
and the loading function is held in function subprogram called
load
.
The example problem shown in Figure 11.19, is of a simply supported plane strain
slab subjected to a sudden application of a uniformly distributed load of 180 kN/m
2
which
