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FIGURE P6.15
6.17 Show that a stiffness matrix for a homogeneous, isotropic plane strain solid can be
changed to one for plane stress by replacing
E
by
E
2
(
1
−
ν
)
and then
ν
by
ν/(
1
+
ν)
.
6.18 For the tetrahedron of Section 6.5.8, show that
γ
β
1
γ
β
γ
β
γ
β
4
1
1
2
1
3
1
γ
β
1
γ
β
γ
β
γ
β
4
E
ij
36
V
2
2
2
2
3
2
N
T
y
∂
∂
=
E
ij
x
N
dV
γ
β
1
γ
β
γ
β
γ
β
4
V
3
3
2
3
3
3
γ
4
β
1
γ
4
β
2
γ
4
β
3
γ
4
β
4
6.19 Derive the interpolation functions of Eq. (6.147) beginning with the trial function
2
2
2
2
u
x
=
u
1
+
u
2
ξ
+
u
3
η
+
u
4
ξη
+
u
5
ξ
+
u
6
η
+
u
7
ηξ
+
u
8
η
ξ
v
i
.
6.20 Derive the interpolation functions for the element shown in Fig. P6.20. This type of
element has been useful in the analysis of stress concentration and crack problems.
Let
u
y
be of the same form with
u
i
replaced by
Hint:
Begin with the same trial functions used in Problem 6.19.
6.21 Find an interpolation function for a one-dimensional element with nodes at
x
i
and
x
j
, and 1 DOF per node. Begin with
u
=
u
1
+
u
2
x
.
=
x
j
−
x
x
j
−
x
i
−
x
x
i
Answer:
N
x
i
x
j
−
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