Civil Engineering Reference
In-Depth Information
This gives
³
F
1
t
G
u
dS
(16.4)
x
2
x
x
S
Substituting the interpolation for tractions and displacements
2
¦
e
xn
e
xj
(16.5)
t
N
t
;
G
u
N
G
u
N
1
x
n
x
j
j
n
1
(where
j
is 2 for element 2 and 1 for element 3) we obtain
2
2
3
3
³
³
(16.6)
F
1
(
N
t
N
t
)
N
dS
(
N
t
N
t
)
N
dS
x
2
1
x
1
2
x
2
2
2
1
x
1
2
x
2
1
3
S
S
2
3
A second equation can be obtained by applying a virtual displacement in y direction.
Based on this approach a general equation can be derived for computing the equivalent
nodal point force at a point
k
N
¦¦
ee
F
N
t
(16.7)
k
jn
n
Elements
n
1
with k
where the outer sum is over all boundary elements that connect to point
k
, the inner sum
is over all nodes of the element and
j
is the local number of node
k
.
For 2-D problems we have
e
e
(16.8)
N
N
I
jn
jn
where
I
is the unit matrix and
³
e
jn
N
N N dS
(16.9)
j
n
e
S
e
The integration over elements can be conveniently carried out using numerical
integration (Gauss Quadrature) with two points. For 2-D problems this gives
1
2
|
¦
³
N
NNJd
[
NNJW
(16.10)
jn
j
n
j
n
m
m
1
[
1
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