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notation, e.g.,
u x . For
u x , let
u x 4 ] T
u x =
[
u x 1
u x 2
u x 3
(6.166a)
and form the unknown state vector at nodes k as
σ xy ] k
z k
=
[
u x
u y
σ x
σ y
(6.166b)
Element Matrices
Substitute Eqs. (6.165) into (6.163) to obtain the two-dimensional equivalent of Eq. (6.157).
We choose to neglect the effect of the surface loads and to use trial functions that fulfill the
geometric boundary conditions. Then we obtain the simplified form
0
N u
D T N
z T t
σ
δ
z dA
N T
N T
σ
E 1 N
σ (
DN u )
A i
σ
i
(6.167)
N u p
0
dS
0
N T
σ
dS
z T
z T
δ
+ δ
=
0
A u
S pi
S ui
With
N ux, x
0
−−−
0
|
(
DN u
) =
B u
=
N uy, y
−−−
N ux, y
|
|
N uy, x
the operator matrix can be written more explicitly as
[
u x
u y
σ x
σ y
σ xy
]
N ux, x N
0
0
0
ux, y N
σ
x
σ
xy
N uy, y N
N uy, x N
0
0
0
σ
y
σ
xy
t
1
E N T
N T
σ
x N ux, x
0
E N T
x N σ x
x N σ y
0
a
·
b
·
d
ξ
d
η
(6.168)
σ
σ
ξ
η
E N T
T
σ
1
E N T
0
y N uy, y
y N σ x
y N σ y
0
σ
σ
N T
σ
N T
σ
G N T
1
xy N ux, y
xy N uy, x
0
0
xy N
σ
xy
σ
Choose the same trial function for the displacements and stresses
=
=
=
=
N ux
N uy
N σ x
N σ y
N σ xy
ξη
(6.169)
=
[
(
1
ξ)(
1
η)
ξ(
1
η)
(
1
ξ)η
]
The derivatives needed in Eq. (6.168) are
N ux, x =
N uy, x =
[
(
1
η)/
a
(
1
η)/
a
η/
a
η/
a ]
(6.170)
N ux, y =
N uy, y =
[
(
1
ξ)/
b
ξ/
b
ξ/
b
(
1
ξ)/
b ]
Insertion of these trial functions into Eq. (6.168) leads to the element matrix for the CB
functional, which has the same structure as Eq. (6.168).
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