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orientations of the triangular sides. This happens, for instance, when two sides of the
triangle are parallel to the
x
and
y
axes.
An alternative shape function is to use the two-dimensional natural coordinate system
described in Chapter 6. This is usually considered to be the logical choice for triangles. This
approach will be followed here.
The element displacement field (shape function) can be constructed as the sum of two
parts
w
=
w
+
w
(13.111)
1
2
where
w
2
is the deflection of the element when it is
regarded as being simply supported at the nodes. For plate bending, the rigid body motions
that can occur are
w
1
is the rigid body displacement, and
w
=
1. A rigid body translation in the
z
direction of the form
constant
2. A rigid body rotation about side 1 (Fig. 13.18) of the triangle. It can be observed from
Fig. 13.18b that the rigid body translation in the
z
direction of line
ab,
which is parallel
to side 1, is
1
w
=
h
1
α
,
where
α
is the angle of rotation. Let
A
1
be any triangle with
a vertex on line
ab
.
It is apparent that for points on
ab, L
1
=
A
1
/
A
=
h
1
/
h
.
Then,
L
1
.
3. A rigid body rotation about side 2 (Fig. 13.18) of the triangle of the form
w
1
=
h
1
α
=
h
α
L
1
,
or
w
1
=
(
constant
)
w
1
=
(
constant
)
L
2
It is apparent that the shape function for rigid body motions may be written as
w
=
w
+
w
+
w
1
L
1
2
L
2
3
L
3
(13.112)
1
Assign appropriately the values of
w
1
,
w
2
,
and
w
3
,
and use the condition
L
1
+
L
2
+
L
3
=
1
,
to
w
=
w
=
w
show that Eq. (13.112) satisfies the three rigid body motions. For example, if
3
,
1
2
w
=
w
w
=
w
condition 1 is satisfied, and if
0
,
conditions 2 and 3 are
satisfied, respectively. These three rigid body motions form a rigid body motion plane. It
can be seen from Chapter 6, Section 6.5.6 that a linear combination of the natural coordinates
L
i
,i
=
0 and
=
2
3
1
3
1
,
2
,
3
,
forms a linear function of
x
and
y
which defines a plane.
Since the element has nine DOF, a cubic polynomial can be used as the shape function,
i.e., the shape function can be the linear combination of the cubic terms
=
L
1
L
2
,
L
2
L
3
,
L
3
L
1
,
L
2
L
1
,
L
3
L
2
,
L
1
L
3
,
L
1
L
2
L
3
Substitution of the expressions for
L
1
,L
2
,
and
L
3
of Chapter 6, Eq. (6.75) into the above
relationships will result in cubic polynomials in terms of
x
and
y
. Because the rigid body
movement is already specified in Eq. (13.112), the cubic terms can be used to express the
relative deflection expression
w
2
of Eq. (13.111). It is customary [Zienkiewicz, 1977] to form
these terms using such combinations as
L
2
L
3
1
2
L
1
L
2
L
3
+
.
The first term of this combination
has zero values at the nodes and zero slope along a side and the second term has zero values
and slopes at all three corners. Six of these combinations constitute the relative deflection
part of the displacement shape function.
4
L
3
L
1
2
L
1
L
2
L
3
5
L
3
L
2
2
L
1
L
2
L
3
1
1
w
=
w
+
+
w
+
2
6
L
1
L
3
2
L
1
L
2
L
3
7
L
1
L
2
2
L
1
L
2
L
3
1
1
+
w
+
+
w
+
8
L
2
L
3
2
L
1
L
2
L
3
9
L
2
L
1
2
L
1
L
2
L
3
1
1
+
w
+
w
+
+
(13.113)
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