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Now apply Green's theorem in the form of [Appendix II, Eq. (II.6)], as in Example 2.8.
x φ
dA d x
τ xy δ ∂ω
y + τ xz δ ∂ω
z
A
A δω φ ∂τ xy
dA dx
y + ∂τ xz
δω φ xz a z + τ xy a y )
=−
ds
(3)
z
x
x δφ
dA d x
xy ∂ω
xz ∂ω
τ
y + τ
z
A
A ωδφ ∂τ
dA dx
y + ∂τ
xy
xz
ωδφ
=−
xz a z
+ τ
xy a y
)
ds
(4)
z
x
Furthermore [Chapter 1, Eq. (1.148)],
x δφ
L
0
A xy z
τ xz y
)
dA d x
=
M t δφ |
=
M t δφ | L
M t δφ | 0
(5)
Substitution of (3), (4), and (5) into (2) gives
∂τ
dA dx
y + ∂τ
xy
xz
δ(ωφ )(τ
A δ(ωφ )
xz a z
+ τ
xy a y
)
ds
+
z
x
+
[
(
M t
M t
)δφ
] L
=
0
(A) or (6)
Note that (6) contains the equilibrium conditions and the statical boundary conditions of
Eq. (6) of Example 2.8. Thus (6) is identical to principle A.
Consider now this same torsion bar starting from the principle of complementary virtual
work. For torsion, Eq. (2.78) can be expressed as
V xy δτ xy + γ xz δτ xz )
dV
φδ
M t =
0
(7)
Extend this relation by including the force boundary condition at x
=
L
(
M t =
M t or
δ
M t
=
0
)
, the surface force condition
τ
xz a z
+ τ
xy a y
=
0or
δτ
xz a z
+ δτ
xy a y
=
0on S p , and
the conditions of equilibrium
0in V . As explained in
Chapter 1, Section 1.9.3, these are the static admissibility requirements.
V xy δτ xy + γ xz δτ xz )
τ
+
τ
=
0or
δ∂
τ
+ δ∂
τ
=
y
xy
z
xz
y
xy
z
xz
dV
+
[
φδ
M t ] 0
[
φδ
M t ] L
dA dx
δ ∂τ
y + δ ∂τ
xy
xz
φ ω(δτ
A φ ω
+
xz a z
+ δτ
xy a y
)
ds
+
=
0
(D) or (8)
z
x
The extended terms are underlined. This can be rewritten by noting that
x φ
[
φδ
M t ] L =
A ( δτ xy z
+ δτ xz y
)
dA d x
+
[
φδ
M t ] 0
(9)
Then
φ
φ ω(δτ xz a z + δτ xy a y )
A xy δτ xy + γ xz δτ xz )
dA
A ( δτ xy z
+ δτ xz y
)
dA
+
ds
x
dA dx
δ ∂τ xy
y + δ ∂τ xz
A φ ω
+
+
[
φ)δ
M t ] 0 =
0
(10)
z
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