Civil Engineering Reference
In-Depth Information
We propose to evaluate the volume integral numerically with the Gauss Quadrature
method. To apply this method, however, the volume where initial strains are specified
needs to be discretised, i.e., subdivided into cells. We use two-dimensional cells for the
discretisation of 2-D problems and three-dimensional cells for 3-D problems. The cells
have already been introduced in Chapter 3. For the interpolation of the strains inside an
element we have for plane problems with either linear (
N
=4) or quadratic (
N
=8) shape
functions
N
N
¦
e
H
(, )
[K
N
(, )
[K
H
(13.47)
0
n
0
n
n
1
The last integral in Eq. (13.46) is replaced by a sum of integrals over cells
N
N
c
¦¦
³
6
c
PQ
,
H
Q dV
'
6 H
(13.48)
i
0
ni
0
n
c
1
n
1
V
where
c
ni
³
'
6
NPQdV
6
,)
(13.49)
n
i
V
c
The final system of equations will be.
>
^` ^`^`
H
(13.50)
T
u
F
F
This means that the presence of initial strains will result in an additional right hand
side
{}
H
F
where the components of
F
for the
i-th
collocation point are
N
N
c
¦¦
c
ni
F
6 H
(13.51)
i
H
0
n
cn
11
13.5.1
Post-processing
In post-processing the effect of the initial strains has to be included. For calculation of
displacements at point
P
we use Eq. (13.42)
³
³
³
u
P
U
P
,
Q
t
Q dS
ȉ
P
,
Q
u
Q dS
6
P
,
Q
H
Q
dV
(13.52)
a
a
a
a
0
S
S
V
For obtaining strains and stresses we have to take the derivative of the displacement
³
³
u
P
U
P
,
Q
t
Q dS
ȉ
P
,
Q
u
Q dS
,
j
a
,
j
a
,
j
a
(13.53)
S
S
³
6
PQ
,
H
QdV
,
j
a
0
V
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