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# Theorem of Least Work - 1 Civil Engineering (CE) Notes | EduRev

## Civil Engineering (CE) : Theorem of Least Work - 1 Civil Engineering (CE) Notes | EduRev

``` Page 1

Instructional Objectives
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
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)

Page 2

Instructional Objectives
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
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)

Page 3

Instructional Objectives
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
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)

In the first case (4.2a), obtain deflection below A due to applied loads .
This can be easily accomplished through Castigliano’s first theorem as discussed
in Lesson 3. Since there is no load applied at
n
P P P ,...., ,
2 1
A , apply a fictitious load at Q A as in
Fig. 4.2. Let be the deflection below
a
u A .
Now the strain energy
s
U stored in the determinate structure (i.e. the support A
removed) is given by,

a n n S
Qu u P u P u P U
2
1
2
1
..........
2
1
2
1
2 2 1 1
+ + + + =  (4.1)

It is known that the displacement

below point  is due to action of
acting at respectively and due to Q at
1
u
1
P
12
, ,....,
n
PP P
n
x x x ,......, ,
2 1
A .

Hence,  may be
expressed as,
1
u

Page 4

Instructional Objectives
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
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)

In the first case (4.2a), obtain deflection below A due to applied loads .
This can be easily accomplished through Castigliano’s first theorem as discussed
in Lesson 3. Since there is no load applied at
n
P P P ,...., ,
2 1
A , apply a fictitious load at Q A as in
Fig. 4.2. Let be the deflection below
a
u A .
Now the strain energy
s
U stored in the determinate structure (i.e. the support A
removed) is given by,

a n n S
Qu u P u P u P U
2
1
2
1
..........
2
1
2
1
2 2 1 1
+ + + + =  (4.1)

It is known that the displacement

below point  is due to action of
acting at respectively and due to Q at
1
u
1
P
12
, ,....,
n
PP P
n
x x x ,......, ,
2 1
A .

Hence,  may be
expressed as,
1
u

(4.2)
1111 122 1 1
..........
nn a
uaP aP aP aQ =+ + + +

where, is the flexibility coefficient at  due to unit force applied at
ij
a i j . Similar
equations may be written for . Substituting for
in equation (4.1) from equation (4.2), we get,
23
, ,...., and
n
uu u u
a 23
, ,...., and
na
uu u u

1 11 1 12 2 1 1 2 21 1 22 2 2 2
1 1 22 1 1 22
11
[ ... ] [ ... ] .......
22
11
[ ... ] [ .... ]
22
Snna nn
n n n nn n na a a an n aa
U PaP a P a P aQ P a P a P a P a Q
Pa P a P a P a Q Qa P a P a P a Q
=+ ++ + + + + + +
++ + + + + ++ +
a
(4.3)

Taking partial derivative of strain energy
s
U with respect to Q, we get deflection
atA .

11 2 2
........
s
aa ann aa
U
aP a P a P a Q
Q
?
=+ + + +
?
(4.4)

Substitute  as it is fictitious in the above equation,  0 Q =

11 2 2
........
s
aa a an
U
uaP aP aP
Q
?
== + + +
?
n
(4.5)

Now the strain energy stored in the beam due to redundant reaction
A
R is,

23
6
a
r
R L
U
EI
=     (4.6)

Now deflection at A due to
a
R is

3
3
a r
a
a
R L U
u
R EI
?
=- =
?
(4.7)

The deflection due to should be in the opposite direction to one caused by
superposed loads , so that the net deflection at
a
R
12
, ,....,
n
PP P A is zero. From
equation (4.5) and (4.7) one could write,

r
a
a
U Us
u
QR
? ?
==-
? ?
(4.8)
Since is fictitious, one could as well replace it by Q
a
R . Hence,

Page 5

Instructional Objectives
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
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)

In the first case (4.2a), obtain deflection below A due to applied loads .
This can be easily accomplished through Castigliano’s first theorem as discussed
in Lesson 3. Since there is no load applied at
n
P P P ,...., ,
2 1
A , apply a fictitious load at Q A as in
Fig. 4.2. Let be the deflection below
a
u A .
Now the strain energy
s
U stored in the determinate structure (i.e. the support A
removed) is given by,

a n n S
Qu u P u P u P U
2
1
2
1
..........
2
1
2
1
2 2 1 1
+ + + + =  (4.1)

It is known that the displacement

below point  is due to action of
acting at respectively and due to Q at
1
u
1
P
12
, ,....,
n
PP P
n
x x x ,......, ,
2 1
A .

Hence,  may be
expressed as,
1
u

(4.2)
1111 122 1 1
..........
nn a
uaP aP aP aQ =+ + + +

where, is the flexibility coefficient at  due to unit force applied at
ij
a i j . Similar
equations may be written for . Substituting for
in equation (4.1) from equation (4.2), we get,
23
, ,...., and
n
uu u u
a 23
, ,...., and
na
uu u u

1 11 1 12 2 1 1 2 21 1 22 2 2 2
1 1 22 1 1 22
11
[ ... ] [ ... ] .......
22
11
[ ... ] [ .... ]
22
Snna nn
n n n nn n na a a an n aa
U PaP a P a P aQ P a P a P a P a Q
Pa P a P a P a Q Qa P a P a P a Q
=+ ++ + + + + + +
++ + + + + ++ +
a
(4.3)

Taking partial derivative of strain energy
s
U with respect to Q, we get deflection
atA .

11 2 2
........
s
aa ann aa
U
aP a P a P a Q
Q
?
=+ + + +
?
(4.4)

Substitute  as it is fictitious in the above equation,  0 Q =

11 2 2
........
s
aa a an
U
uaP aP aP
Q
?
== + + +
?
n
(4.5)

Now the strain energy stored in the beam due to redundant reaction
A
R is,

23
6
a
r
R L
U
EI
=     (4.6)

Now deflection at A due to
a
R is

3
3
a r
a
a
R L U
u
R EI
?
=- =
?
(4.7)

The deflection due to should be in the opposite direction to one caused by
superposed loads , so that the net deflection at
a
R
12
, ,....,
n
PP P A is zero. From
equation (4.5) and (4.7) one could write,

r
a
a
U Us
u
QR
? ?
==-
? ?
(4.8)
Since is fictitious, one could as well replace it by Q
a
R . Hence,

()
sr
a
UU
R
0
?
+ =
?
(4.9)

or,

0
a
U
R
?
=
?
(4.10)

This is the statement of theorem of least work. Where U is the total strain energy
of the beam due to superimposed loads  and redundant reaction .
12
, ,....,
n
PP P
a
R

Example 4.1
Find the reactions of a propped cantilever beam uniformly loaded as shown in Fig.
4.3a. Assume the flexural rigidity of the beam EI to be constant throughout its
length.

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## Structural Analysis

30 videos|122 docs|28 tests

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