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FIGURE 13.19
Edges of triangular elements.
configuration. The deformation throughout the plate is expressed solely in terms of the
middle surface deflection
w
. The fact that the plate behavior is characterized by the single
variable
has considerable advantages, but it also means that the nature of classical plate
theory differs significantly from that of two- or three-dimensional elasticity theory. As
far as the finite element displacement approach is concerned,
C
1
w
continuity (Chapter 6,
Section 6.5), i.e., continuity of
and its first derivatives, on interelement boundaries is
required for compatibility. The requirement for
C
1
continuity does complicate element
development considerably. This requirement cannot, in general, be satisfied.
Use the triangular elements in Fig. 13.19 to illustrate the problem of
C
1
w
continuity not
being satisfied. The displacement
w
of Eq. (13.114) along side 2-3 of element 1, where
L
1
=
0
,
of Fig. 13.19 can be written as a function of
x,
i.e.,
1
2
x
2
3
x
3
w
=
α
+
α
1
x
+
α
+
α
0
This expression can be obtained by substituting
L
1
0 and the relationships for
L
2
and
L
3
of Chapter 6, Eq. (6.75) into Eq. (13.114). Note that along this boundary, the
y
coordinate is
constant. Similarly, the displacement along side 2-3 of element 2 is
=
2
2
x
2
3
x
3
w
=
β
+
β
1
x
+
β
+
β
0
The parameters
α
0
,
α
1
,
α
2
,
and
α
3
can be uniquely expressed in terms of the four DOF
(w
2
,
w
3
,
θ
x
2
,
and
θ
x
3
)
at nodes 2 and 3, and since these DOF are shared by the adjacent
1
2
elements, we have
w
=
w
.
This means that the displacement and the rotation
θ
x
=
−
∂w/∂
x
are continuous along side 2-3 for elements 1 and 2. For the slope
∂w/∂
y
in the
y
direction, however, the situation is quite different. Let
L
1
=
0 and take the derivative of
w
of Eq. (13.114) with respect to
y
. A polynomial
∂w
∂
+
γ
2
x
2
y
=
γ
0
+
γ
1
x
(13.119)
is obtained for side 2-3. Since there are only two DOF (
θ
y
at nodes 2 and 3) related to
∂w/∂
y
along this side, the parameters
3
,
cannot be uniquely expressed in terms of these
two DOF. On the other hand, if the expression of Eq. (13.119) is to be obtained from Eq.
(13.115), the parameters
γ
i
,i
=
0
,
...
y
may be different for elements 1 and 2 of Fig. 13.19 along side 2-3. Therefore, discontinuities
of normal slope, or kinks, will generally occur and the requirement of continuity of the first
derivatives of the shape function is violated. This kind of element with the discontinuity
of shapes at the boundary is referred to as a
nonconforming element.
The present element
γ
i
must involve the nodal variables at node 1. Thus,
θ
=−
∂w/∂
y
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