Theorem of Least Work - 1 | Structural Analysis - Civil Engineering (CE) PDF Download

 Page 1


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) 
 
 
Page 2


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) 
 
 
 
 
Page 3


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) 
 
 
 
 
 
 
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 
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) 
 
 
 
 
 
 
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 
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) 
 
 
 
 
 
 
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.