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For the element in Fig. 6.10, with
ξ
=
x
/
a
and
η
=
y
/
b
, the displacements of Eqs. (6.11)
and (6.12) lead to the strains
x
=
∂
u
x
∂
x
=
∂
u
x
∂ξ
∂ξ
∂
x
+
∂
u
x
∂η
∂η
∂
1
a
(
x
=
u
3
η
+
u
2
)
y
=
∂
u
y
∂
y
=
∂
u
y
∂ξ
∂ξ
∂
y
+
∂
u
y
∂η
∂η
∂
1
b
(
y
=
u
7
ξ
+
u
8
)
(1)
=
∂
u
x
∂
y
+
∂
u
y
∂
x
=
∂
u
x
∂ξ
∂ξ
∂
y
+
∂
u
x
∂η
∂η
∂
y
+
∂
u
y
∂ξ
∂ξ
∂
x
+
∂
u
y
∂η
∂η
∂
γ
xy
x
1
b
(
1
a
(
ξ
+
η
+
=
u
3
u
4
)
+
u
7
u
6
)
It is apparent that if
u
2
is non-zero, a constant
x
can be obtained. Similar reasoning holds
for
xy
. Thus, it can be seen that the displacement functions are complete.
Now, consider the compatibility. From Eqs. (6.11) and (6.12) the displacements
u
x
and
u
y
are continuous throughout the element. From Eq. (6.30), the highest derivative appearing
in the principle of virtual work is 1, so that the displacement function must be continuous
on the interelement boundary in order for the element to be compatible. Note that the trial
functions for both
u
x
and
u
y
have continuous first derivatives, with respect to
y
and
γ
ξ
and
η
throughout the element.
Consider elements 1 and 2 in Fig. 6.14. These two elements have a common boundary
connecting node 2 and node 5. Assume that nodes 2 and 5 have displacements
u
2
and
u
5
in the
x
direction. Using the shape functions given in Eq. (6.19), the displacement at the
common boundary for element 1, where
u
2
=
=
ξ
=
u
x
2
,u
5
u
x
3
, and
1, is
u
x
=
(
1
−
η)
u
2
+
η
u
5
(2)
and that for element 2, where
u
2
=
u
x
1
,u
5
=
u
x
4
, and
ξ
=
0, is
u
x
=
(
1
−
η)
u
2
+
η
u
5
(3)
It is seen that
u
x
and
u
x
are the same, so that the displacements are continuous at this
boundary. We conclude that the element is compatible.
6.5.3
Test of Convergence and Accuracy
The Patch Test
A useful test of acceptability of particular elements in practice is the patch test. A patch
test is used to check the completeness of a group of elements. It extends the philosophy of
the constant strain requirement from the individual element to a group of elements. In this
test, a small field (a patch) of elements, with at least one being completely surrounded by
elements, is assembled, and a set of displacements or forces are applied at the boundary of
the structure such that the constant strain state should occur inside the structure. For certain
elements, e.g., quadrilateral and solid elements, some standard finite element meshes and
boundary conditions for patch tests have been proposed [MacNeal and Harder, 1985].
Typical proposed meshes are shown in Fig. 6.23, and the boundary conditions along with
the theoretical solutions of the problems are shown in Table 6.4. The strains corresponding to
these boundary conditions are constant throughout the elements. The elements in Fig. 6.23
are distorted intentionally, as this is essential to the success of the test. The rectangular
exterior shapes of the elements ease the task of applying boundary conditions that should
lead to the constant strains in the elements.
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