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FIGURE P6.6
=
ξ
+
η
+
ξ
2
+
η
2
3.
u
u
2
u
3
u
4
u
5
(
π
2
(
π
2
(
π
2
(
π
2
4.
u
=
u
1
+
u
2
sin
)
+
u
3
sin
)
+
u
4
sin
)
sin
)
N
3
]
T
6.7 Find the interpolation functions
N
=
[
N
1
N
2
for a three node extension bar
2
.
6.8 Use the formulation of Section 6.5.4 to find the expression for the stiffness matrix
of Eq. (6.56) for the element of Fig. 6.10. Also, show that an incompatibility occurs
between elements 1 and 2 of Fig. 6.14b.
element, using the trial function
u
=
u
1
+
u
2
ξ
+
u
3
ξ
Hint:
Use the shape functions
N
1
and
N
2
of Eq. (6.55). Then
u
x
u
y
N
1
v
x
N
2
=
N
u
=
=
v
N
1
N
2
u
2
where
N
isa2
6. Use Eq. (6.31)
to find
k
i
. To check compatibility, calculate the displacement of the mid-span point
of the common edge of element 1 and element 2. Compare the results.
×
6 matrix. From Eq. (6.25),
B
=
DN
, where
B
is 3
×
6.9 Find expressions for the coefficients
α
2
,
β
2
,
γ
2
,
α
3
,
β
3
,
and
γ
3
of Eq. (6.75). Also find
the same quantities for Eq. (6.82).
6.10 Derive an interpolation
N
of
u
=
Nv
for the 8 node cubic element of Fig. P6.10.
FIGURE P6-10
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