Civil Engineering Reference
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where
'
GG
e
i
(,) ()
PQdSQ
(13.10)
i
S
e
For the three-dimensional case, no singularity occurs as
P
approaches
Q
and,
therefore, the minimum integration order with which we are able to accurately compute
the surface area of the element can be used. The analysis of problems with constant body
forces proceeds the same way as before, except that an additional right hand side term is
assembled. The final system of equations will be.
> @^ ` ^ ` ^ `
b
(13.11)
Tu
F
F
where the components of
F
b
for the
i-th
collocation point are
E
¦
e
(13.12)
F
'
G
ib
i
e
1
13.2.1
Post-processing
When computing internal results the effect of body forces has to be included. For
calculation of displacements
E
N
E
N
E
¦¦
¦¦
¦
11
ee
e e
e
u
P
'
U
t
'
T
u
'
G
(13.13)
a
na
n
na
n
a
en
en
11
e
1
and for computation of stresses
E
N
E
N
E
¦¦
¦¦
¦
11
ˆ
ee
e e
e
V
P
'
St
'
Ru
'
S
(13.14)
a
na
n
na
n
a
en
en
11
e
1
where
'
³
ˆ
ˆ
(,) ()
e
a
(13.15)
S
S
PQdSQ
a
S
e
Matrix
S
is obtained by differentiating (13.6) or (13.8) and multiplying with the
constitutive matrix
D.
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