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and
w
=
w
,
w
,a
=
w
,a
on
S
u
(13.143)
These are the equilibrium conditions of Eq. (13.39) (without
p
z
) and the displacement
boundary conditions of Eq. (13.55). These conditions must be satisfied when using the
complementary hybrid model of Eq. (13.137) to establish a finite element formulation.
There are two independent approximate shape functions used for the HSM element. One
is an assumed stress (bending moments) field and the other is the displacement along the
interelement boundaries. Assume that the bending moments vary linearly in the interior of
the element and the displacement
w
varies cubically along the sides, with a linear variation
of
w
,a
. For the bending moment, assume that
s
=
N
σ
σ
(13.144)
where
N
σ
00
0N
σ
N
σ
=
0
00
σ
N
σ
=
[1
x
]
and
1 vector of generalized parameters. Note that Eq. (13.144) satisfies the
equilibrium condition of Eq. (13.142).
Substitute Eq. (13.144) into Eq. (13.139) to obtain
σ
is the 9
×
1
2
1
2
σ
T
B
U
0
(
s
T
E
−
B
s
dA
=−
σ
)
=−
=−
σ
U
1
dV
(13.145)
V
A
where
N
T
σ
E
−
1
B
=
N
σ
dA
B
A
The 9
×
9 matrix
B
can be written as
c
11
φ
c
12
φ
c
13
φ
c
21
φ
c
22
φ
c
23
φ
c
31
φ
c
32
φ
c
33
φ
B
=
(13.146)
where the coefficients
c
ij
are the elements in
E
−
1
and
B
1
xy
x
2
xy
yxy
N
∗
T
σ
N
σ
φ
=
dA
=
dA
(13.147)
A
A
2
Since
E
−
B
is symmetric,
B
is also symmetric. If the centroid of the coordinate system is
located at the centroid of the element,
φ
can be written as
A
00
dA
φ
=
0
I
xx
I
xy
(13.148)
A
0
I
xy
I
yy
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