Building Design and Construction Handbook - Merritt - Ventech!

FIGURE 5.50 Statically indeterminate truss.
STRUCTURAL THEORY 5.69
�U � P (5.92)
n
�e n
AE 2 2 3
U � (e � 2e cos �)
2L
which states that the partial derivative of
the strain energy with respect to any
specific deformation gives the corresponding
force.
Suppose, for example, the stress in
the vertical bar in Fig. 5.50 is to be determined.
All bars are made of the same
material and have the same cross section.
If the vertical bar stretches an
amount e under the load P, the inclined
bars will each stretch an amount e cos
�. The strain energy in the system is
[from Eq. (5.30)]
and the partial derivative of this with respect to e must be equal to P; that is
AE 3
P � (2e � 4e cos �)
2L
AEe 3
� (1 � 2 cos �)
L
Noting that the force in the vertical bar equals AEe/L, we find from the above
equation that the required stress equals P/(1 � 2 cos 3 �).
Castigliano’s Theorems. It can also be shown that, if the strain energy is expressed
as a function of statically independent forces, the partial derivative of the
strain energy with respect to one of the forces gives the deformation corresponding
to that force. (See Timoshenko and Young, ‘‘Theory of Structures,’’ McGraw-Hill
Publishing Company, New York.)
�U � e (5.93)
n
�P n
This is known as Castigliano’s first theorem. (His second theorem is the principle
of least work.)
5.10.3 Method of Least Work
If displacement of a structure is prevented, as at a support, the partial derivative of
the strain energy with respect to that supporting force must be zero, according to
Castigliano’s first theorem. This establishes his second theorem:
The strain energy in a statically indeterminate structure is the minimum
consistent with equilibrium.