Civil Engineering Reference
In-Depth Information
The first theorem of Castigliano is also applicable to rotational displace-
ments. In the case of rotation, the partial derivative of strain energy with respect
to a rotational displacement is equal to the moment/torque applied at the point of
concern in the sense of the rotation. The following example illustrates the appli-
cation in terms of a simple torsional member.
EXAMPLE 2.5
A solid circular shaft of radius
R
and length
L
is subjected to constant torque
T
. The shaft
is fixed at one end, as shown in Figure 2.9. Formulate the elastic strain energy in terms of
the angle of twist
at
x
=
L
and show that Castigliano's first theorem gives the correct
expression for the applied torque.
■
Solution
From strength of materials theory, the shear stress at any cross section along the length of
the member is given by
Tr
J
where
r
is radial distance from the axis of the member and
J
is polar moment of inertia of
the cross section. For elastic behavior, we have
=
G
=
Tr
JG
where
G
is the shear modulus of the material, and the strain energy is then
=
d
x
Tr
J
Tr
JG
d
A
L
1
2
1
2
U
e
=
d
V
=
V
0
A
L
T
2
2
J
2
G
T
2
L
2
JG
r
2
d
A
d
x
=
=
0
A
where we have used the definition of the polar moment of inertia
r
2
d
A
J
=
A
R
T
L
Figure 2.9
Example 2.5:
Circular cylinder subjected to
torsion.
