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# Castigliano’s Theorems (Part - 1) Civil Engineering (CE) Notes | EduRev

## Civil Engineering (CE) : Castigliano’s Theorems (Part - 1) Civil Engineering (CE) Notes | EduRev

Page 1

Instructional Objectives
1. State and prove first theorem of Castigliano.
2. Calculate deflections along the direction of applied load of a statically
determinate structure at the point of application of load.
3. Calculate deflections of a statically determinate structure in any direction at a
point where the load is not acting by fictious (imaginary) load method.
4. State and prove Castigliano’s second theorem.

3.1 Introduction
In the previous chapter concepts of strain energy and complementary strain
energy were discussed. Castigliano’s first theorem is being used in structural
analysis for finding deflection of an elastic structure based on strain energy of the
structure. The Castigliano’s theorem can be applied when the supports of the
structure are unyielding and the temperature of the structure is constant.

3.2 Castigliano’s First Theorem
For linearly elastic structure, where external forces only cause deformations, the
complementary energy is equal to the strain energy. For such structures, the
Castigliano’s first theorem may be stated as the first partial derivative of the
strain energy of the structure with respect to any particular force gives the
displacement of the point of application of that force in the direction of its line of
action.

Page 2

Instructional Objectives
1. State and prove first theorem of Castigliano.
2. Calculate deflections along the direction of applied load of a statically
determinate structure at the point of application of load.
3. Calculate deflections of a statically determinate structure in any direction at a
point where the load is not acting by fictious (imaginary) load method.
4. State and prove Castigliano’s second theorem.

3.1 Introduction
In the previous chapter concepts of strain energy and complementary strain
energy were discussed. Castigliano’s first theorem is being used in structural
analysis for finding deflection of an elastic structure based on strain energy of the
structure. The Castigliano’s theorem can be applied when the supports of the
structure are unyielding and the temperature of the structure is constant.

3.2 Castigliano’s First Theorem
For linearly elastic structure, where external forces only cause deformations, the
complementary energy is equal to the strain energy. For such structures, the
Castigliano’s first theorem may be stated as the first partial derivative of the
strain energy of the structure with respect to any particular force gives the
displacement of the point of application of that force in the direction of its line of
action.

Let

be the forces acting at  from the left end on a simply
supported beam of span
n
P P P ,...., ,
2 1 n
x x x ,......, ,
2 1
L . Let

points  respectively as shown in Fig. 3.1. Now, assume that the
material obeys Hooke’s law and invoking the principle of superposition, the work
done by the external forces is given by (vide eqn. 1.8 of lesson 1)
n
u u u ,..., ,
2 1
n
P P P ,...., ,
2 1

n n
u P u P u P W
2
1
..........
2
1
2
1
2 2 1 1
+ + + =    (3.1)

Page 3

Instructional Objectives
1. State and prove first theorem of Castigliano.
2. Calculate deflections along the direction of applied load of a statically
determinate structure at the point of application of load.
3. Calculate deflections of a statically determinate structure in any direction at a
point where the load is not acting by fictious (imaginary) load method.
4. State and prove Castigliano’s second theorem.

3.1 Introduction
In the previous chapter concepts of strain energy and complementary strain
energy were discussed. Castigliano’s first theorem is being used in structural
analysis for finding deflection of an elastic structure based on strain energy of the
structure. The Castigliano’s theorem can be applied when the supports of the
structure are unyielding and the temperature of the structure is constant.

3.2 Castigliano’s First Theorem
For linearly elastic structure, where external forces only cause deformations, the
complementary energy is equal to the strain energy. For such structures, the
Castigliano’s first theorem may be stated as the first partial derivative of the
strain energy of the structure with respect to any particular force gives the
displacement of the point of application of that force in the direction of its line of
action.

Let

be the forces acting at  from the left end on a simply
supported beam of span
n
P P P ,...., ,
2 1 n
x x x ,......, ,
2 1
L . Let

points  respectively as shown in Fig. 3.1. Now, assume that the
material obeys Hooke’s law and invoking the principle of superposition, the work
done by the external forces is given by (vide eqn. 1.8 of lesson 1)
n
u u u ,..., ,
2 1
n
P P P ,...., ,
2 1

n n
u P u P u P W
2
1
..........
2
1
2
1
2 2 1 1
+ + + =    (3.1)

Work done by the external forces is stored in the structure as strain energy in a
conservative system. Hence, the strain energy of the structure is,

n n
u P u P u P U
2
1
..........
2
1
2
1
2 2 1 1
+ + + =   (3.2)

Displacement

below point  is due to the action of  acting at
distances respectively from left support
.
Hence,  may be expressed
as,
1
u
1
P
n
P P P ,...., ,
2 1
n
x x x ,......, ,
2 1 1
u

n n
P a P a P a u
1 2 12 1 11 1
.......... + + + =    (3.3)

In general,

n i P a P a P a u
n in i i i
,... 2 , 1             ..........
2 2 1 1
= + + + =  (3.4)

where  is the flexibility coefficient at  due to unit force applied at
ij
a i j .
Substituting the values of  in equation (3.2) from equation (3.4), we
get,
n
u u u ,..., ,
2 1

...] [
2
1
....... ...] [
2
1
...] [
2
1
2 2 1 1 2 22 1 21 2 2 12 1 11 1
+ + + + + + + + + = P a P a P P a P a P P a P a P U
n n n
(3.5)

We know from Maxwell-Betti’s reciprocal theorem
ji ij
a a = . Hence, equation (3.5)
may be simplified as,

[]
22 2
11 1 22 2 12 1 2 13 1 3 1 1
1
.... .... ...
2
nn n n n
U a P a P a P a PP a PP a PP ?? =+ ++ + + ++
??
+ (3.6)

Now, differentiating the strain energy with any force  gives,
1
P

n n
P a P a P a
P
U
1 2 12 1 11
1
.......... + + + =
?
?
(3.7)

It may be observed that equation (3.7) is nothing but displacement  at the
1
u
In general,
n
n
u
P
U
=
?
?
(3.8)

Hence, for determinate structure within linear elastic range the partial derivative
of the total strain energy with respect to any external load is equal to the

Page 4

Instructional Objectives
1. State and prove first theorem of Castigliano.
2. Calculate deflections along the direction of applied load of a statically
determinate structure at the point of application of load.
3. Calculate deflections of a statically determinate structure in any direction at a
point where the load is not acting by fictious (imaginary) load method.
4. State and prove Castigliano’s second theorem.

3.1 Introduction
In the previous chapter concepts of strain energy and complementary strain
energy were discussed. Castigliano’s first theorem is being used in structural
analysis for finding deflection of an elastic structure based on strain energy of the
structure. The Castigliano’s theorem can be applied when the supports of the
structure are unyielding and the temperature of the structure is constant.

3.2 Castigliano’s First Theorem
For linearly elastic structure, where external forces only cause deformations, the
complementary energy is equal to the strain energy. For such structures, the
Castigliano’s first theorem may be stated as the first partial derivative of the
strain energy of the structure with respect to any particular force gives the
displacement of the point of application of that force in the direction of its line of
action.

Let

be the forces acting at  from the left end on a simply
supported beam of span
n
P P P ,...., ,
2 1 n
x x x ,......, ,
2 1
L . Let

points  respectively as shown in Fig. 3.1. Now, assume that the
material obeys Hooke’s law and invoking the principle of superposition, the work
done by the external forces is given by (vide eqn. 1.8 of lesson 1)
n
u u u ,..., ,
2 1
n
P P P ,...., ,
2 1

n n
u P u P u P W
2
1
..........
2
1
2
1
2 2 1 1
+ + + =    (3.1)

Work done by the external forces is stored in the structure as strain energy in a
conservative system. Hence, the strain energy of the structure is,

n n
u P u P u P U
2
1
..........
2
1
2
1
2 2 1 1
+ + + =   (3.2)

Displacement

below point  is due to the action of  acting at
distances respectively from left support
.
Hence,  may be expressed
as,
1
u
1
P
n
P P P ,...., ,
2 1
n
x x x ,......, ,
2 1 1
u

n n
P a P a P a u
1 2 12 1 11 1
.......... + + + =    (3.3)

In general,

n i P a P a P a u
n in i i i
,... 2 , 1             ..........
2 2 1 1
= + + + =  (3.4)

where  is the flexibility coefficient at  due to unit force applied at
ij
a i j .
Substituting the values of  in equation (3.2) from equation (3.4), we
get,
n
u u u ,..., ,
2 1

...] [
2
1
....... ...] [
2
1
...] [
2
1
2 2 1 1 2 22 1 21 2 2 12 1 11 1
+ + + + + + + + + = P a P a P P a P a P P a P a P U
n n n
(3.5)

We know from Maxwell-Betti’s reciprocal theorem
ji ij
a a = . Hence, equation (3.5)
may be simplified as,

[]
22 2
11 1 22 2 12 1 2 13 1 3 1 1
1
.... .... ...
2
nn n n n
U a P a P a P a PP a PP a PP ?? =+ ++ + + ++
??
+ (3.6)

Now, differentiating the strain energy with any force  gives,
1
P

n n
P a P a P a
P
U
1 2 12 1 11
1
.......... + + + =
?
?
(3.7)

It may be observed that equation (3.7) is nothing but displacement  at the
1
u
In general,
n
n
u
P
U
=
?
?
(3.8)

Hence, for determinate structure within linear elastic range the partial derivative
of the total strain energy with respect to any external load is equal to the

displacement of the point of application of load in the direction of the applied
load, provided the supports are unyielding and temperature is maintained
constant. This theorem is advantageously used for calculating deflections in
elastic structure. The procedure for calculating the deflection is illustrated with
few examples.

Example 3.1
Find the displacement and slope at the tip of a cantilever beam loaded as in Fig.
3.2. Assume the flexural rigidity of the beam EI to be constant for the beam.

Moment at any section at a distance x away from the free end is given by

Px M - =      (1)

Strain energy stored in the beam due to bending is
?
=
L
dx
EI
M
U
0
2
2
(2)

Substituting the expression for bending moment M in equation (3.10), we get,

?
= =
L
EI
L P
dx
EI
Px
U
0
3 2 2
6 2
) (
(3)

Page 5

Instructional Objectives
1. State and prove first theorem of Castigliano.
2. Calculate deflections along the direction of applied load of a statically
determinate structure at the point of application of load.
3. Calculate deflections of a statically determinate structure in any direction at a
point where the load is not acting by fictious (imaginary) load method.
4. State and prove Castigliano’s second theorem.

3.1 Introduction
In the previous chapter concepts of strain energy and complementary strain
energy were discussed. Castigliano’s first theorem is being used in structural
analysis for finding deflection of an elastic structure based on strain energy of the
structure. The Castigliano’s theorem can be applied when the supports of the
structure are unyielding and the temperature of the structure is constant.

3.2 Castigliano’s First Theorem
For linearly elastic structure, where external forces only cause deformations, the
complementary energy is equal to the strain energy. For such structures, the
Castigliano’s first theorem may be stated as the first partial derivative of the
strain energy of the structure with respect to any particular force gives the
displacement of the point of application of that force in the direction of its line of
action.

Let

be the forces acting at  from the left end on a simply
supported beam of span
n
P P P ,...., ,
2 1 n
x x x ,......, ,
2 1
L . Let

points  respectively as shown in Fig. 3.1. Now, assume that the
material obeys Hooke’s law and invoking the principle of superposition, the work
done by the external forces is given by (vide eqn. 1.8 of lesson 1)
n
u u u ,..., ,
2 1
n
P P P ,...., ,
2 1

n n
u P u P u P W
2
1
..........
2
1
2
1
2 2 1 1
+ + + =    (3.1)

Work done by the external forces is stored in the structure as strain energy in a
conservative system. Hence, the strain energy of the structure is,

n n
u P u P u P U
2
1
..........
2
1
2
1
2 2 1 1
+ + + =   (3.2)

Displacement

below point  is due to the action of  acting at
distances respectively from left support
.
Hence,  may be expressed
as,
1
u
1
P
n
P P P ,...., ,
2 1
n
x x x ,......, ,
2 1 1
u

n n
P a P a P a u
1 2 12 1 11 1
.......... + + + =    (3.3)

In general,

n i P a P a P a u
n in i i i
,... 2 , 1             ..........
2 2 1 1
= + + + =  (3.4)

where  is the flexibility coefficient at  due to unit force applied at
ij
a i j .
Substituting the values of  in equation (3.2) from equation (3.4), we
get,
n
u u u ,..., ,
2 1

...] [
2
1
....... ...] [
2
1
...] [
2
1
2 2 1 1 2 22 1 21 2 2 12 1 11 1
+ + + + + + + + + = P a P a P P a P a P P a P a P U
n n n
(3.5)

We know from Maxwell-Betti’s reciprocal theorem
ji ij
a a = . Hence, equation (3.5)
may be simplified as,

[]
22 2
11 1 22 2 12 1 2 13 1 3 1 1
1
.... .... ...
2
nn n n n
U a P a P a P a PP a PP a PP ?? =+ ++ + + ++
??
+ (3.6)

Now, differentiating the strain energy with any force  gives,
1
P

n n
P a P a P a
P
U
1 2 12 1 11
1
.......... + + + =
?
?
(3.7)

It may be observed that equation (3.7) is nothing but displacement  at the
1
u
In general,
n
n
u
P
U
=
?
?
(3.8)

Hence, for determinate structure within linear elastic range the partial derivative
of the total strain energy with respect to any external load is equal to the

displacement of the point of application of load in the direction of the applied
load, provided the supports are unyielding and temperature is maintained
constant. This theorem is advantageously used for calculating deflections in
elastic structure. The procedure for calculating the deflection is illustrated with
few examples.

Example 3.1
Find the displacement and slope at the tip of a cantilever beam loaded as in Fig.
3.2. Assume the flexural rigidity of the beam EI to be constant for the beam.

Moment at any section at a distance x away from the free end is given by

Px M - =      (1)

Strain energy stored in the beam due to bending is
?
=
L
dx
EI
M
U
0
2
2
(2)

Substituting the expression for bending moment M in equation (3.10), we get,

?
= =
L
EI
L P
dx
EI
Px
U
0
3 2 2
6 2
) (
(3)

Now, according to Castigliano’s theorem, the first partial derivative of strain
energy with respect to external force P gives the deflection  at A in the
direction of applied force. Thus,
A
u

EI
PL
u
P
U
A
3
3
= =
?
?
(4)

To find the slope at the free end, we need to differentiate strain energy with
respect to externally applied momentM atA . As there is no moment atA , apply
a fictitious moment  at
0
M A. Now moment at any section at a distance x away
from the free end is given by

0
M Px M - - =

Now, strain energy stored in the beam may be calculated as,

?
+ + =
+
=
L
EI
L M
EI
PL M
EI
L P
dx
EI
M Px
U
0
2
0
2
0
3 2 2
0
2 2 6 2
) (
(5)

Taking partial derivative of strain energy with respect to , we get slope at
0
M A .

2
0
0
2
A
M L UPL
M EI EI
?
?
== +
?
(6)

But actually there is no moment applied atA. Hence substitute in
equation (3.14) we get the slope at A.
0
0
= M

EI
PL
A
2
2
= ?      (7)

Example 3.2
A cantilever beam which is curved in the shape of a quadrant of a circle is loaded
as shown in Fig. 3.3. The radius of curvature of curved beam isR, Young’s
modulus of the material is E and second moment of the area is I about an axis
perpendicular to the plane of the paper through the centroid of the cross section.
Find the vertical displacement of point A on the curved beam.

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## Structural Analysis

30 videos|122 docs|28 tests

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