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Integration by parts gives
+
φ
∂ω
∂
z
+
φ
∂ω
∂
y
xz
dV
γ
y
+
δτ
+
γ
z
−
δτ
+
(φ
−
φ)δ
=
[
M
t
]
0
0
xy
xy
xz
V
(B) or (11)
which is identical to form
B
and from which the strain-displacement relations and displace-
ment boundary conditions follow.
Now that
A, B , C ,
and
D
have been defined for the torsion problem, combinations of these
as outlined in Table 2.3 can be formed as generalized variational principles. For example,
AB
B
can provide all the fundamental equations. Form
AB
corresponds to Chapter
1, Eq. (1.168). To show this, multiply
A
by
=
A
+
−
G
, use the constitutive relations of Chapter 1,
Eq. (1.143), and form
A
+
B
=
AB,
giving
φ
∂ω
∂
z
φ
ω)(∂
y
τ
xy
+
∂
z
τ
xz
)
+
−
δ(
G
τ
xy
+
G
y
+
δτ
xy
V
φ
∂ω
∂
y
δτ
xz
dV
φ
ω)(τ
xz
a
z
+
τ
xy
a
y
)
+
τ
xz
+
G
z
−
+
δ(
G
ds dx
x
−
[
(
M
t
−
M
t
)δφ
]
L
G
+
[
(φ
−
φ)δ
M
t
]
0
G
=
0
(12)
In matrix notation,
φ
τ
xz
τ
xy
0
∂
z
∂
y
ω
G
0
−
dV
G
φ
ωτ
xz
φ
ωτ
xz
φ
dV
V
δ
[
G
τ
xy
]
∂
z
10
+
V
δ
[
G
τ
xy
]
y
∂
y
01
z
φ
0
a
z
a
y
00 0
00 0
ω
G
ds dx
φ
ω
xz
+
δ
[
G
τ
xy
]
τ
xz
x
τ
xy
−
[
(
M
t
−
M
t
)δφ
]
L
G
+
[
(φ
−
φ)δ
M
t
]
0
G
=
0
(AB) or (13)
Note the lack of symmetry in one of the matrix expressions. Euler's equations for the volume
terms are the mixed governing differential equations of Chapter 1, Eq. (1.168).
Other combinations of
A, B , C ,
and
D
lead to different generalized variational principles.
Computational considerations can make it advantageous to employ a generalized principle
that results in symmetric equations. One such combination is
AD
=
A
+
D
.
To derive this
−
+
=
principle, multiply
D
by
1 and form
A
D
AD,
giving
A
ωφ
∂τ
xy
dA d x
y
+
∂τ
xz
ωφ
(τ
xz
a
z
+
τ
xy
a
y
)
−
δ
ds dx
+
δ
∂
∂
z
x
x
x
φ
−
A
(γ
xy
δτ
xy
+
γ
xz
δτ
xz
)
dA d x
+
A
(
−
δτ
xy
z
+
δτ
xz
y
)
dA d x
x
+
[
(
M
−
M
t
)δφ
]
L
−
[
(φ
−
φ)δ
M
t
]
0
=
0
(AD) or (14)
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