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FIGURE 12.8
Representation of the cross-sectional characteristics for principal axes.
The principal moments of inertia and
I
ω
∗
ω
∗
are shown in Fig. 12.8.
The quantity
J
t
in Eq. (12.17) can be expressed as
y
2
dA
∂ω
∂
2
∂ω
∂
2
dA
z
∂ω
∂
dA
z
∂ω
∂
y
∂ω
∂
y
∂ω
∂
z
2
=
+
+
y
−
+
+
+
y
−
J
t
z
y
z
z
A
A
A
(12.23)
The first integral in Eq. (12.23) is the torsional constant
J
given in Chapter 1, Eq. (1.154).
The second term can be processed as
∂ω
∂
2
∂ω
∂
2
dA
∂
∂
dA
∂
z
2
dA
2
2
ω
∂ω
∂
+
∂
∂
ω
∂ω
∂
y
2
+
∂
ω
ω
+
=
−
A
ω
y
z
y
y
z
z
∂
∂
A
A
From Gauss's integral theorem in Appendix II, Eq. (II.8),
∂
∂
dA
∂ω
∂
z
a
z
dS
ω
∂ω
∂
+
∂
∂
ω
∂ω
∂
+
∂ω
∂
=
S
ω
y
a
y
y
y
z
z
A
For a two-dimensional problem, the volume
V
in Eq. (II.8) becomes the area
A
. The third
integral of Eq. (12.23) can be written as
z
∂ω
∂
dA
∂
∂
dA
y
∂ω
∂
ω)
+
∂
∂
y
−
=
y
(
z
z
(
−
y
ω)
=
S
ω(
za
y
−
ya
z
)
dS
z
A
A
Here Gauss's integral theorem is used again. Thus, the expression of
J
t
appears as
∂
z
2
dA
∂ω
∂
ya
z
dS
2
2
y
2
+
∂
ω
ω
y
a
y
+
∂ω
J
t
=
J
−
A
ω
+
S
ω
z
a
z
+
za
y
−
∂
∂
∂
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