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This same equation can be obtained more directly using the requirement for the warping
to be single-valued and continuous
∂
2
ω
y
=
∂
2
ω
(1.160)
∂
z
∂
∂
y
∂
z
Insertion of Eq. (1.158) into Eq. (1.160) leads to Eq. (1.159). A partial differential equation
of the form of Eq. (1.159) is called
Poisson's equation
.
Upon introduction of Eq. (1.155), the boundary condition of Eq. (1.145) becomes
∂ψ
∂
ds
+
∂ψ
dy
dz
ds
=
0
(1.161)
y
∂
z
By the chain rule of differentiation,
d
is constant along the boundary of the
cross-section. Since the stresses are defined in terms of derivatives of
ψ/
ds
=
0or
ψ
ψ
rather than
ψ
itself,
the magnitude of the constant
ψ
is arbitrary. Therefore, without loss of generality, it is
common to assume that
0 along the boundary.
The resultant moment condition of Eq. (1.148) now appears as
ψ
=
z
dy dz
−
∂ψ
∂
−
∂ψ
∂
M
t
=
y
(1.162)
y
z
A
where Eq. (1.155) has been employed. Integration by parts and invoking
ψ
=
0onthe
boundaries yields
2
M
t
=
A
ψ
dy dz
(1.163)
and hence the torsional constant is
A
ψ
M
t
G
2
G
J
=
φ
=
dy dz
(1.164)
φ
EXAMPLE 1.7 Bar of Elliptical Cross-Section
The equation for the ellipse of Fig. 1.19 is
y
2
a
2
+
z
2
b
2
=
1
(1)
y
b
a
z
FIGURE 1.19
Elliptical cross-section of bar.
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