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where
γ
κ
∂
1
x
=
D
u
=
(2.115)
0
∂
x
If the material laws of Chapter 1, Eqs. (1.106b) and (1.109), are introduced, the component
form becomes
L
M
dx
=
(w
−
w) δ
M
L
0
V
k
s
GA
M
EI
(w
+
θ)
−
θ
−
δ
V
+
δ
V
+
(θ
−
θ)δ
0
on
S
u
(2.116) or (B)
Proper application of integration by parts transforms A and B into the principles of virtual
work and complementary virtual work, respectively.
2.4.3
Principle of Virtual Work (C)
u
T
s
as
In order to transform A into the principle of virtual work, begin by rewriting
δ
δ
u
T
s
L
0
=
V
δθ
L
0
=
V
δθ
L
0
−
V
δθ
L
0
δw
+
δw
+
δw
+
M
M
M
on
S
p
on
S
p
on
S
on
S
u
L
−
V
δθ
L
0
=
[
V
δw
+
M
δθ
]
dx
δw
+
M
(2.117)
0
on
S
u
where
S
=
S
p
+
S
u
.
Insert this expression in the right-hand side of (A), giving
L
V
+
M
−
[
(
p
z
)δw
+
(
V
)δθ
]
dx
0
L
]
0
−
V
δθ
L
0
[
V
δw
+
δw
+
M
δθ
+
δθ
]
dx
=
V
M
−
[
V
δw
+
M
δθ
δw
+
M
0
on
S
u
on
S
p
or
L
=−
M
δw
L
0
−
V
δθ
L
0
δ(θ
+
w
)
−
δθ
+
[
−
V
M
p
z
δw
]
dx
δθ
+
V
δw
+
M
0
on
S
p
on
S
u
(2.118) or (C)
The same result can be derived by applying integration by parts to Eq. (2.111). Since the
classical principle of virtual work utilizes kinematically admissible displacements, the un-
derlined terms would be dropped because they contain displacement boundary conditions
which must be satisfied.
2.4.4 Principle of Complementary Virtual Work (D)
The global form (B) of the strain-displacement relations and the displacement boundary
conditions can be reformed into an integral relationship representing the principle of com-
plementary virtual work. Apply integration by parts to some of the terms of Eq. (2.116),
giving
L
0
w
δ
L
0
wδ
+
wδ
V
L
0
V
dx
Vdx
=−
L
0
θ
δ
L
0
θδ
+
θδ
M
L
0
M
dx
=−
Mdx
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