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with
=
Du
in
V
and
u
=
u
on
S
u
Some useful theorems related to these principles will be derived now.
3.1.1
Castigliano's Theorem, Part I
A frequently used classical theorem related to virtual work is
Castigliano's first theorem
or
Castigliano's theorem, part I
. This is a global version of the locally applicable relation [Chapter
2, Eq. (2.21)]
The principle of stationary potential energy will be used here to
derive Castigliano's theorem. To begin the process, conside
r
a
ge
nera
l, t
hree
-d
imensional
solid subjected to a system of external forces (or moments)
P
1
, P
2
,
∂
U
0
/∂
=
σ
.
kj
kj
...
P
k
,
...
P
n
,
with c
or
-
responding displace
m
ents
V
1
,V
2
,
...
,V
k
,
...
V
n
.
The term
V
k
is the displacement of
P
k
in the direction of
P
k
.
The total potential energy is given by [Chapter 2, Eqs. (2.64)]
n
=
+
=
−
U
i
U
e
U
i
P
k
V
k
(3.3)
k
=
1
The strain energy
U
i
can be expressed in terms of the displacements
V
k
so that the po-
tential energy is a function of the
V
k
which are often referred to as
generalized coordinates
or
generalized displacements
. If the solid is in equilibrium,
δ
must vanish [Eq. (3.2)]. Thus,
δ
=
∂
U
i
V
1
+
∂
U
i
V
2
+···+
∂
U
i
V
1
δ
V
2
δ
V
n
δ
V
n
−
P
1
δ
V
1
−
P
2
δ
V
2
···−
P
n
δ
V
n
=
0
∂
∂
∂
or
∂
P
1
∂
P
2
∂
P
n
U
i
U
i
U
i
V
1
−
δ
V
1
+
V
2
−
δ
V
2
+···+
V
n
−
δ
V
n
=
0
∂
∂
∂
The variations
V
k
may be considered to be arbitrary, so that the fact
ors
in parentheses m
u
st
each be zero for the complete expression to vanish. Then
δ
∂
U
i
/∂
V
1
=
P
1
,
...
,
∂
U
i
/∂
V
n
=
P
n
,
or
∂
U
i
V
k
=
P
k
(3.4)
∂
This is Castigliano's theorem, part I. This theorem asserts that:
If the strain energy of a body is expressed in terms of displacement components in the
direction of the prescribed forces, then the first partial derivative of the strain energy,
with respect to a displacement, is equal to the corresponding force.
.
The subscript
k
is arbitrary in that it can denote any force from
P
1
to
P
n
If all of the
n
derivatives are taken, then a system of equations
V
n
]
T
V
=
[
V
1
V
2
...
∂
U
i
∂
V
=
P
(3.5)
P
=
[
P
1
P
2
...
P
n
]
T
is obtained. To use these relations, it is necessary to express
U
i
in terms of the displace-
ments
V
.
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