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The equation of equilibrium of (6a) reduces immediately with the assistance of Chapter 1,
Eqs. (1.142) and (1.143), to the displacement form of the governing differential equation:
2
2
∂
y
2
+
∂
ω
ω
2
z
2
=
0or
∇
ω
=
0
(7)
∂
∂
2
y
z
where
This is the Laplace equation. As indicated in Chapter 1, Section 1.9.4,
(6c) can be reduced to the governing equation along the bar by introducing the torsional
constant of Chapter 1, Eq. (1.150). Thus
∇
=
∂
+
∂
.
φ
=
GJ
M
t
(8)
2.2.2
Principle of Stationary Potential Energy
The principle of virtual work can be specialized for systems for which a potential exists
for both the internal and external forces. In the case of internal forces, it follows from Eqs.
(2.18b) and (2.22) that
=
∂
U
0
()
∂
ij
δ
U
0
δ
=
σ
δ
(2.60)
ij
ij
ij
Then the left-hand integral of the first equation in Eq. (2.54) can be written as
V
σ
ij
δ
ij
dV
=
V
δ
U
0
dV
=
δ
U
0
dV
=
δ
U
i
(2.61)
V
Now the first relation in Eq. (2.54) appears as
V
δ
−
p
Vi
δ
−
p
i
δ
=
U
0
dV
u
i
dV
u
i
dS
0
(2.62)
V
S
p
Suppose that in the final two integrals of Eq. (2.62) the surface tractions and body forces do
not alter their magnitudes and directions during deformation, i.e., they are derivable from
a potential, so that we can write
p
i
u
i
dS
δ
U
0
dV
−
p
Vi
u
i
dV
−
=
0
(2.63)
V
V
S
p
or
δ
=
0 where
=
U
0
dV
−
p
Vi
u
i
dV
−
p
i
u
i
dS
=
U
i
+
U
e
(2.64)
V
V
S
p
with
U
e
defined by Eq. (2.32b). It is known from the calculus of variations that a zero first
variation
assumes an extremal
value. We have thus arrived at the
principle of a stationary value of the total potential energy
or
simply the
principle of stationary potential energy.
Of all kinematically admissible deformations, the actual deformations (those which correspond
to stresses which satisfy equilibrium) are the ones for which the total potential energy assumes a
stationary value, i.e., an extremal value.
The principle of stationary potential energy and the principle of virtual work appear to
be the same in form. The difference between them is that for the principle of stationary
δ
is equivalent to a stationary value
. In other words,
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