Civil Engineering Reference
In-Depth Information
y
3
2
b
∂
w
∂
x
2
w
∂
∂
∂
x
y
1
x
∂
w
∂
y
4
a
w
Figure 2.8 Rectangular plate bending element
Consider, for example, the rectangular element shown in Figure 2.8. If there are assumed
to be four degrees of freedom per node, namely
2
∂w
∂x
,θ
y
=
∂w
∂y
w
∂x∂y
∂
w,
θ
x
=
and
θ
xy
=
then the appropriate element shape functions can be shown to be products of the beam
shape functions already described, that is,
w
=
[
N
]
{
w
}
(2.89)
where, if the freedoms
{
w
}
are numbered
(w, θ
x
,θ
y
,θ
xy
)
node
=
1
,
2
,
3
,
4
, the “first” shape func-
tionwouldbe,
1
a
1
b
3
2
3
3
2
3
N
1
=
3
(a
−
3
ax
+
2
x
)
3
(b
−
3
by
+
2
y
)
(2.90)
Defining,
1
a
3
2
3
P
1
=
3
(a
−
3
ax
+
2
x
)
1
b
3
2
3
Q
1
=
3
(b
−
3
by
+
2
y
)
1
a
2
2
3
P
2
=
2
(a
x
−
2
ax
+
x
)
1
b
2
2
3
Q
2
=
2
(b
y
−
2
by
+
y
)
(2.91)
1
a
2
3
P
3
=
3
(
3
ax
−
2
x
)
