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−
P
5
(
1
−
2
L
3
)
−
(
P
6
−
P
5
)
L
2
−
4
+
6
(
L
2
+
L
3
)
+
r
5
(
1
−
2
L
3
)
−
L
2
(
r
5
+
r
6
)
−
q
5
(
1
−
2
L
3
)
+
(
q
5
+
q
6
)
L
2
L
2
(
P
4
+
P
6
)
N
x, L
3
=
−
L
2
(
r
6
−
r
4
)
(13.135c)
−
L
2
(
q
4
−
q
6
)
P
5
(
1
−
2
L
3
)
−
L
2
(
P
5
+
P
4
)
−
2
+
6
L
3
+
r
5
(
1
−
2
L
3
)
+
L
2
(
r
4
−
r
5
)
−
(
−
)
−
(
−
)
q
5
1
2
L
3
L
2
q
4
q
5
−
t
5
(
1
−
2
L
3
)
−
(
t
6
−
t
5
)
L
2
−
q
5
(
1
−
2
L
3
)
+
L
2
(
q
5
+
q
6
)
−
1
−
t
5
(
1
−
2
L
3
)
+
(
r
5
+
r
6
)
L
2
L
2
(
t
4
+
t
6
)
N
y, L
3
=
−
L
2
(
q
4
−
q
6
)
(13.135d)
−
L
2
(
r
4
−
r
6
)
t
5
(
1
−
2
L
3
)
−
L
2
(
t
5
+
t
4
)
−
q
5
(
1
−
2
L
3
)
−
L
2
(
q
4
−
q
5
)
1
−
r
5
(
1
−
2
L
3
)
−
L
2
(
r
4
−
r
5
)
where
2
ij
2
ij
P
k
=−
6
x
ij
/
=
6
a
k
,
t
k
=−
6
y
ij
/
=
6
d
k
2
ij
3
y
ij
/
2
ij
q
k
=
3
x
ij
y
ij
/
=
4
c
k
,
r
k
=
=−
4
(
2
b
k
+
e
k
)
k
=
4
,
5
,
6
for
ij
=
23
,
31
,
12
,
respectively
Thus, the stiffness matrix can be evaluated.
For the loading vector, a uniform loading can be represented approximately by lumped
concentrated loads applied on the nodes. Then the element loading vector corresponding
to a uniform pressure load
p
0
per unit middle surface area can be written as
A
3
[
p
0
T
p
=
00
p
0
00
p
0
00
(13.136)
where
A
is the area of the element.
Hybrid Stress Model Element (HSM)
In addition to the displacement type of element for the plate bending problem, hybrid
elements have been shown to be viable [Batoz, et al., 1980]. The concept of the hybrid
element is introduced in Chapter 6, Section 6.9. A hybrid element can be based on the
extended principle of complementary virtual work of Eq. (6.150) or Chapter 2, Eq. (2.104).
As is shown in Chapter 6, a typical hybrid element employs two kinds of shape functions.
For example, compatible displacement functions can be assumed along the interelement
boundaries, in addition to an assumed stress field which satisfies the equilibrium conditions
inside the element. As a result, a compatible element is formed. This kind of element is
called the hybrid stress model (HSM). Another type of hybrid element is based on an
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