Instructional Objectives
After reading this lesson, the reader will be able to:
1. State and prove theorem of Least Work.
2. Analyse statically indeterminate structure.
3. State and prove Maxwell-Betti’s Reciprocal theorem.
4.1
Introduction
In the last chapter the Castigliano’s theorems were discussed. In this chapter
theorem of least work and reciprocal theorems are presented along with few
selected problems. We know that for the statically determinate structure, the
partial derivative of strain energy with respect to external force is equal to the
displacement in the direction of that load at the point of application of load. This
theorem when applied to the statically indeterminate structure results in the
theorem of least work.
4.2
Theorem of Least Work
According to this theorem, the partial derivative of strain energy of a statically
indeterminate structure with respect to statically indeterminate action should
vanish as it is the function of such redundant forces to prevent any displacement
at its point of application. The forces developed in a redundant framework are
such that the total internal strain energy is a minimum.
This can be proved as
follows. Consider a beam that is fixed at left end and roller supported at right end
as shown in Fig. 4.1a. Let
be the forces acting at distances
from the left end of the beam of span
n
P
P
P
,....
,
,
2
1
n
x
x
x
,......
,
,
2
1
L
. Let
be the
displacements at the loading points
respectively as shown in Fig. 4.1a.
This is a statically indeterminate structure and choosing
n
u
u
u
,...,
,
2
1
n
P
P
P
,....
,
,
2
1
a
R
as the redundant
reaction, we obtain a simple cantilever beam as shown in Fig. 4.1b. Invoking the
principle of superposition, this may be treated as the superposition of two cases,
viz, a cantilever beam with loads
and a cantilever beam with redundant
force
n
P
P
P
,....
,
,
2
1
a
R
(see Fig. 4.2a and Fig. 4.2b)
Version 2 CE IIT, Kharagpur