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Then Eq. (2.116) becomes
L
M
dx
V
k
s
GA
δ
M
EI
δ
V
+
θδ
M
−
−
wδ
V
−
V
−
θδ
0
=−
wδ
V
L
0
−
θδ
M
L
0
+
(w
−
w) δ
M
L
0
V
+
(θ
−
θ)δ
on
S
on
S
on
S
u
or
V
k
s
GA
δ
M
dx
L
L
M
EI
δ
V
w
−
δ
M
θ)
−
V
+
−
(δ
V
θ
+
δ
dx
0
0
=−
wδ
M
L
0
−
wδ
M
L
0
V
+
θδ
V
+
θδ
(2.119) or (D)
on
S
p
on
S
u
The classical theorem would require the underlined term to be dropped because they rep-
resent statically admissible boundary conditions and equilibrium.
2.4.5
Generalized Principles
The classical variational principles for a beam are summarized in Table 2.5. These funda-
mental forms can be combined to form the generalized principles as follows:
A
+
B
=
AB
C
+
D
=
CD
A
+
D
=
AD
C
+
B
=
CB
For example, consider the
C
+
B
combination,
V
dx
L
V
k
s
GA
−
V
δ
M
M
EI
δ
(δθ
+
δw
)
+
δθ
+
θ
δ
(w
+
θ)
−
M
M
+
δ
V
0
L
−
V
δθ
L
0
−
p
z
δw
dx
δw
+
M
0
on
S
p
−
δ
V
(θ
−
θ)
L
(w
−
w)
+
M
0
=
0
(2.120) or (CB)
on
S
u
In matrix notation this symmetric principle is the same as Eq. (7) of Example 2.11. Other
generalized forms can be derived in a similar fashion.
2.5
Structure of the Differential and Integral Forms
of the Governing Equations
The general form of the basic equations given in Chapter 1 and in this chapter is shown in
Table 2.6. The equations are arranged such that the dual character of the principle of virtual
work
C
and the principle of complementary virtual work
D
is evident. The local equations
corresponding to principle
C
are the equilibrium conditions
A
and the force boundary con-
ditions
A
B
, with the kinematic equations
B
and the displacement boundary conditions
B
B
as side conditions to be satisfied a priori. The reverse holds for principle
D
in a fully dual
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