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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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