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y
t
60
60
x
FIGURE P1.23
Triangular element with in-plane stress.
1.24 Develop compatibility differential equations corresponding to three-dimensional stress
functions defined as
2
=
∂
ψ
1
σ
x
∂
z
∂
y
2
σ
y
=
∂
ψ
2
∂
x
∂
z
2
=
∂
ψ
3
σ
z
∂
x
∂
y
∂ψ
1
∂
1
2
∂
∂
x
+
∂ψ
2
y
−
∂ψ
3
τ
xy
=−
z
∂
∂
z
1
2
∂
∂
−
∂ψ
1
∂
x
+
∂ψ
2
y
+
∂ψ
3
τ
yz
=−
x
∂
∂
z
∂ψ
1
2
∂
∂
x
−
∂ψ
y
+
∂ψ
1
2
3
τ
zx
=−
y
∂
∂
∂
z
1.25 Use index notation to verify that Eq. (1.63) represents the displacement formulation
equations for a general, isotropic elastic solid.
1.26 Show that the governing differential equations for the displacement formulation of
the plane stress problem excluding body forces are
∂
y
2
2
u
−
ν
2
∂
2
u
∂
2
v
E
1
E
x
2
+
+
y
=
0
(
−
ν
2
)
∂
∂
(
−
ν)
∂
∂
1
2
1
x
1
2
2
2
u
E
−
ν
2
∂
x
2
+
∂
v
v
E
∂
+
y
=
0
(
1
−
ν
2
)
∂
∂
y
2
2
(
1
−
ν)
∂
x
∂
1.27 Derive the governing differential equations for the displacement formulation of the
plane strain problem.
1.28 Derive the governing differential equations for the mixed formulation of the plane
stress problem.
1.29 Develop as many forms as you can of the governing differential equations for the two-
dimensional polar coordinate element of Fig. P1.10 that is considered in Problems 1.1,
1.2, 1.10, and 1.20. The constitutive relations will be the same as in the rectangular
coordinate case.
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