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# Virtual Work (Part -1) - Energy Methods in Structural Analysis, Structural Analysis - II Civil Engineering (CE) Notes | EduRev

## Civil Engineering (CE) : Virtual Work (Part -1) - Energy Methods in Structural Analysis, Structural Analysis - II Civil Engineering (CE) Notes | EduRev

``` Page 1

Instructional Objectives
After studying this lesson, the student will be able to:
1. Define Virtual Work.
2. Differentiate between external and internal virtual work.
3. Sate principle of virtual displacement and principle of virtual forces.
4. Drive an expression of calculating deflections of structure using unit load
method.
5. Calculate deflections of a statically determinate structure using unit load
method.
6. State unit displacement method.
7. Calculate stiffness coefficients using unit-displacement method.

5.1 Introduction
In the previous chapters the concept of strain energy and Castigliano’s theorems
were discussed. From Castigliano’s theorem it follows 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. In this
lesson, the principle of virtual work is discussed. As compared to other methods,
virtual work methods are the most direct methods for calculating deflections in
statically determinate and indeterminate structures. This principle can be applied
to both linear and nonlinear structures. The principle of virtual work as applied to
deformable structure is an extension of the virtual work for rigid bodies. This may
be stated as: if a rigid body is in equilibrium under the action of a system of
forces and if it continues to remain in equilibrium if the body is given a small
(virtual) displacement, then the virtual work done by the
F -
F - system of forces as ‘it
rides’ along these virtual displacements is zero.

5.2 Principle of Virtual Work
Many problems in structural analysis can be solved by the principle of virtual work.
Consider a simply supported beam as shown in Fig.5.1a, which is in equilibrium
under the action of real forces  at co-ordinates  respectively.
Let  be the corresponding displacements due to the action of
forces . Also, it produces real internal stresses
n
F F F ,......., ,
2 1
n ,....., 2 , 1
n
u u u ,......, ,
2 1
n
F F F ,......., ,
2 1 ij
s and real internal
strains
ij
e inside the beam. Now, let the beam be subjected to second system of
forces (which are virtual not real)
n
F F F d d d ,......, ,
2 1
in equilibrium as shown in
Fig.5.1b. The second system of forces is called virtual as they are imaginary and
they are not part of the real loading. This produces a displacement

Page 2

Instructional Objectives
After studying this lesson, the student will be able to:
1. Define Virtual Work.
2. Differentiate between external and internal virtual work.
3. Sate principle of virtual displacement and principle of virtual forces.
4. Drive an expression of calculating deflections of structure using unit load
method.
5. Calculate deflections of a statically determinate structure using unit load
method.
6. State unit displacement method.
7. Calculate stiffness coefficients using unit-displacement method.

5.1 Introduction
In the previous chapters the concept of strain energy and Castigliano’s theorems
were discussed. From Castigliano’s theorem it follows 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. In this
lesson, the principle of virtual work is discussed. As compared to other methods,
virtual work methods are the most direct methods for calculating deflections in
statically determinate and indeterminate structures. This principle can be applied
to both linear and nonlinear structures. The principle of virtual work as applied to
deformable structure is an extension of the virtual work for rigid bodies. This may
be stated as: if a rigid body is in equilibrium under the action of a system of
forces and if it continues to remain in equilibrium if the body is given a small
(virtual) displacement, then the virtual work done by the
F -
F - system of forces as ‘it
rides’ along these virtual displacements is zero.

5.2 Principle of Virtual Work
Many problems in structural analysis can be solved by the principle of virtual work.
Consider a simply supported beam as shown in Fig.5.1a, which is in equilibrium
under the action of real forces  at co-ordinates  respectively.
Let  be the corresponding displacements due to the action of
forces . Also, it produces real internal stresses
n
F F F ,......., ,
2 1
n ,....., 2 , 1
n
u u u ,......, ,
2 1
n
F F F ,......., ,
2 1 ij
s and real internal
strains
ij
e inside the beam. Now, let the beam be subjected to second system of
forces (which are virtual not real)
n
F F F d d d ,......, ,
2 1
in equilibrium as shown in
Fig.5.1b. The second system of forces is called virtual as they are imaginary and
they are not part of the real loading. This produces a displacement

configuration
n
u u u d d d , ,......... ,
2 1
stresses
ij
d s and virtual internal strains
ij
d e inside the beam. Now, apply the
second system of forces on the beam which has been deformed by first system of
forces. Then, the external loads  and internal stresses
i
F
ij
s do virtual work by
moving along
i
u d and
ij
d e . The product
i i
u F d
?
is known as the external virtual
work. It may be noted that the above product does not represent the conventional
work since each component is caused due to different source i.e.
i
u d is not due
to . Similarly the product
i
F
ij ij
s de
?
is the internal virtual work. In the case of
deformable body, both external and internal forces do work. Since, the beam is in
equilibrium, the external virtual work must be equal to the internal virtual work.
Hence, one needs to consider both internal and external virtual work to establish
equations of equilibrium.

5.3 Principle of Virtual Displacement
A deformable body is in equilibrium if the total external virtual work done by the
system of true forces moving through the corresponding virtual displacements of
the system i.e.  is equal to the total internal virtual work for every
kinematically admissible (consistent with the constraints) virtual displacements.
i i
u F d
?

Page 3

Instructional Objectives
After studying this lesson, the student will be able to:
1. Define Virtual Work.
2. Differentiate between external and internal virtual work.
3. Sate principle of virtual displacement and principle of virtual forces.
4. Drive an expression of calculating deflections of structure using unit load
method.
5. Calculate deflections of a statically determinate structure using unit load
method.
6. State unit displacement method.
7. Calculate stiffness coefficients using unit-displacement method.

5.1 Introduction
In the previous chapters the concept of strain energy and Castigliano’s theorems
were discussed. From Castigliano’s theorem it follows 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. In this
lesson, the principle of virtual work is discussed. As compared to other methods,
virtual work methods are the most direct methods for calculating deflections in
statically determinate and indeterminate structures. This principle can be applied
to both linear and nonlinear structures. The principle of virtual work as applied to
deformable structure is an extension of the virtual work for rigid bodies. This may
be stated as: if a rigid body is in equilibrium under the action of a system of
forces and if it continues to remain in equilibrium if the body is given a small
(virtual) displacement, then the virtual work done by the
F -
F - system of forces as ‘it
rides’ along these virtual displacements is zero.

5.2 Principle of Virtual Work
Many problems in structural analysis can be solved by the principle of virtual work.
Consider a simply supported beam as shown in Fig.5.1a, which is in equilibrium
under the action of real forces  at co-ordinates  respectively.
Let  be the corresponding displacements due to the action of
forces . Also, it produces real internal stresses
n
F F F ,......., ,
2 1
n ,....., 2 , 1
n
u u u ,......, ,
2 1
n
F F F ,......., ,
2 1 ij
s and real internal
strains
ij
e inside the beam. Now, let the beam be subjected to second system of
forces (which are virtual not real)
n
F F F d d d ,......, ,
2 1
in equilibrium as shown in
Fig.5.1b. The second system of forces is called virtual as they are imaginary and
they are not part of the real loading. This produces a displacement

configuration
n
u u u d d d , ,......... ,
2 1
stresses
ij
d s and virtual internal strains
ij
d e inside the beam. Now, apply the
second system of forces on the beam which has been deformed by first system of
forces. Then, the external loads  and internal stresses
i
F
ij
s do virtual work by
moving along
i
u d and
ij
d e . The product
i i
u F d
?
is known as the external virtual
work. It may be noted that the above product does not represent the conventional
work since each component is caused due to different source i.e.
i
u d is not due
to . Similarly the product
i
F
ij ij
s de
?
is the internal virtual work. In the case of
deformable body, both external and internal forces do work. Since, the beam is in
equilibrium, the external virtual work must be equal to the internal virtual work.
Hence, one needs to consider both internal and external virtual work to establish
equations of equilibrium.

5.3 Principle of Virtual Displacement
A deformable body is in equilibrium if the total external virtual work done by the
system of true forces moving through the corresponding virtual displacements of
the system i.e.  is equal to the total internal virtual work for every
kinematically admissible (consistent with the constraints) virtual displacements.
i i
u F d
?

That is virtual displacements should be continuous within the structure and also it
must satisfy boundary conditions.

dv u F
ij ij i i
de s d
?
?
=                                              (5.1)
where
ij
s are the true stresses due to true forces  and
i
F
ij
de are the virtual strains
due to virtual displacements
i
u d .

5.4 Principle of Virtual Forces
For a deformable body, the total external complementary work is equal to the total
internal complementary work for every system of virtual forces and stresses that
satisfy the equations of equilibrium.

dv u F
ij ij i i
e ds d
?
?
=                                              (5.2)

where
ij
d s are the virtual stresses due to virtual forces
i
F d and
ij
e are the true
strains due to the true displacements .
i
u
As stated earlier, the principle of virtual work may be advantageously used to
calculate displacements of structures. In the next section let us see how this can
be used to calculate displacements in a beams and frames. In the next lesson, the
truss deflections are calculated by the method of virtual work.

The principle of virtual force leads to unit load method. It is assumed throughout
our discussion that the method of superposition holds good. For the derivation of
unit load method, we consider two systems of loads. In this section, the principle of
virtual forces and unit load method are discussed in the context of framed
structures. Consider a cantilever beam, which is in equilibrium under the action of
a first system of forces  causing displacements as shown in
Fig. 5.2a. The first system of forces refers to the actual forces acting on the
structure. Let the stress resultants at any section of the beam due to first system of
forces be axial force (
n
F F F ,....., ,
2 1 n
u u u ,....., ,
2 1
P ), bending moment (M ) and shearing force (V ). Also the
corresponding incremental deformations are axial deformation ( ), flexural
deformation (
? d
? d ) and shearing deformation ( ? d ) respectively.
For a conservative system the external work done by the applied forces is equal to
the internal strain energy stored. Hence,

Page 4

Instructional Objectives
After studying this lesson, the student will be able to:
1. Define Virtual Work.
2. Differentiate between external and internal virtual work.
3. Sate principle of virtual displacement and principle of virtual forces.
4. Drive an expression of calculating deflections of structure using unit load
method.
5. Calculate deflections of a statically determinate structure using unit load
method.
6. State unit displacement method.
7. Calculate stiffness coefficients using unit-displacement method.

5.1 Introduction
In the previous chapters the concept of strain energy and Castigliano’s theorems
were discussed. From Castigliano’s theorem it follows 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. In this
lesson, the principle of virtual work is discussed. As compared to other methods,
virtual work methods are the most direct methods for calculating deflections in
statically determinate and indeterminate structures. This principle can be applied
to both linear and nonlinear structures. The principle of virtual work as applied to
deformable structure is an extension of the virtual work for rigid bodies. This may
be stated as: if a rigid body is in equilibrium under the action of a system of
forces and if it continues to remain in equilibrium if the body is given a small
(virtual) displacement, then the virtual work done by the
F -
F - system of forces as ‘it
rides’ along these virtual displacements is zero.

5.2 Principle of Virtual Work
Many problems in structural analysis can be solved by the principle of virtual work.
Consider a simply supported beam as shown in Fig.5.1a, which is in equilibrium
under the action of real forces  at co-ordinates  respectively.
Let  be the corresponding displacements due to the action of
forces . Also, it produces real internal stresses
n
F F F ,......., ,
2 1
n ,....., 2 , 1
n
u u u ,......, ,
2 1
n
F F F ,......., ,
2 1 ij
s and real internal
strains
ij
e inside the beam. Now, let the beam be subjected to second system of
forces (which are virtual not real)
n
F F F d d d ,......, ,
2 1
in equilibrium as shown in
Fig.5.1b. The second system of forces is called virtual as they are imaginary and
they are not part of the real loading. This produces a displacement

configuration
n
u u u d d d , ,......... ,
2 1
stresses
ij
d s and virtual internal strains
ij
d e inside the beam. Now, apply the
second system of forces on the beam which has been deformed by first system of
forces. Then, the external loads  and internal stresses
i
F
ij
s do virtual work by
moving along
i
u d and
ij
d e . The product
i i
u F d
?
is known as the external virtual
work. It may be noted that the above product does not represent the conventional
work since each component is caused due to different source i.e.
i
u d is not due
to . Similarly the product
i
F
ij ij
s de
?
is the internal virtual work. In the case of
deformable body, both external and internal forces do work. Since, the beam is in
equilibrium, the external virtual work must be equal to the internal virtual work.
Hence, one needs to consider both internal and external virtual work to establish
equations of equilibrium.

5.3 Principle of Virtual Displacement
A deformable body is in equilibrium if the total external virtual work done by the
system of true forces moving through the corresponding virtual displacements of
the system i.e.  is equal to the total internal virtual work for every
kinematically admissible (consistent with the constraints) virtual displacements.
i i
u F d
?

That is virtual displacements should be continuous within the structure and also it
must satisfy boundary conditions.

dv u F
ij ij i i
de s d
?
?
=                                              (5.1)
where
ij
s are the true stresses due to true forces  and
i
F
ij
de are the virtual strains
due to virtual displacements
i
u d .

5.4 Principle of Virtual Forces
For a deformable body, the total external complementary work is equal to the total
internal complementary work for every system of virtual forces and stresses that
satisfy the equations of equilibrium.

dv u F
ij ij i i
e ds d
?
?
=                                              (5.2)

where
ij
d s are the virtual stresses due to virtual forces
i
F d and
ij
e are the true
strains due to the true displacements .
i
u
As stated earlier, the principle of virtual work may be advantageously used to
calculate displacements of structures. In the next section let us see how this can
be used to calculate displacements in a beams and frames. In the next lesson, the
truss deflections are calculated by the method of virtual work.

The principle of virtual force leads to unit load method. It is assumed throughout
our discussion that the method of superposition holds good. For the derivation of
unit load method, we consider two systems of loads. In this section, the principle of
virtual forces and unit load method are discussed in the context of framed
structures. Consider a cantilever beam, which is in equilibrium under the action of
a first system of forces  causing displacements as shown in
Fig. 5.2a. The first system of forces refers to the actual forces acting on the
structure. Let the stress resultants at any section of the beam due to first system of
forces be axial force (
n
F F F ,....., ,
2 1 n
u u u ,....., ,
2 1
P ), bending moment (M ) and shearing force (V ). Also the
corresponding incremental deformations are axial deformation ( ), flexural
deformation (
? d
? d ) and shearing deformation ( ? d ) respectively.
For a conservative system the external work done by the applied forces is equal to
the internal strain energy stored. Hence,

1
11 1 1
d ? d ? d ?
22 2 2
n
ii
i
Fu P M V
=
=+ +
?
? ??

? ? ?
+ + =
L L L
AG
ds V
EI
ds M
EA
ds P
0
2
0
2
0
2
2 2 2
(5.3)

Now, consider a second system of forces
n
F F F d d d ,....., ,
2 1
, which are virtual and
causing virtual displacements
n
u u u d d d ,....., ,
2 1
respectively (see Fig. 5.2b). Let the
virtual stress resultants caused by virtual forces be
v v
M P d d , and
v
V d at any cross
section of the beam. For this system of forces, we could write

? ? ?
?
+ + =
=
L
v
L
v
L
v
n
i
i i
AG
ds V
EI
ds M
EA
ds P
u F
0
2
0
2
0
2
1
2 2 2 2
1 d d d
d d       (5.4)

where
v v
M P d d , and
v
V d are the virtual axial force, bending moment and shear force
respectively. In the third case, apply the first system of forces on the beam, which
has been deformed, by second system of forces
n
F F F d d d ,....., ,
2 1
as shown in Fig
5.2c. From the principle of superposition, now the deflections will be
()( ) (
n n
u u u u u u ) d d d + + + ,......, ,
2 2 1 1
respectively

Page 5

Instructional Objectives
After studying this lesson, the student will be able to:
1. Define Virtual Work.
2. Differentiate between external and internal virtual work.
3. Sate principle of virtual displacement and principle of virtual forces.
4. Drive an expression of calculating deflections of structure using unit load
method.
5. Calculate deflections of a statically determinate structure using unit load
method.
6. State unit displacement method.
7. Calculate stiffness coefficients using unit-displacement method.

5.1 Introduction
In the previous chapters the concept of strain energy and Castigliano’s theorems
were discussed. From Castigliano’s theorem it follows 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. In this
lesson, the principle of virtual work is discussed. As compared to other methods,
virtual work methods are the most direct methods for calculating deflections in
statically determinate and indeterminate structures. This principle can be applied
to both linear and nonlinear structures. The principle of virtual work as applied to
deformable structure is an extension of the virtual work for rigid bodies. This may
be stated as: if a rigid body is in equilibrium under the action of a system of
forces and if it continues to remain in equilibrium if the body is given a small
(virtual) displacement, then the virtual work done by the
F -
F - system of forces as ‘it
rides’ along these virtual displacements is zero.

5.2 Principle of Virtual Work
Many problems in structural analysis can be solved by the principle of virtual work.
Consider a simply supported beam as shown in Fig.5.1a, which is in equilibrium
under the action of real forces  at co-ordinates  respectively.
Let  be the corresponding displacements due to the action of
forces . Also, it produces real internal stresses
n
F F F ,......., ,
2 1
n ,....., 2 , 1
n
u u u ,......, ,
2 1
n
F F F ,......., ,
2 1 ij
s and real internal
strains
ij
e inside the beam. Now, let the beam be subjected to second system of
forces (which are virtual not real)
n
F F F d d d ,......, ,
2 1
in equilibrium as shown in
Fig.5.1b. The second system of forces is called virtual as they are imaginary and
they are not part of the real loading. This produces a displacement

configuration
n
u u u d d d , ,......... ,
2 1
stresses
ij
d s and virtual internal strains
ij
d e inside the beam. Now, apply the
second system of forces on the beam which has been deformed by first system of
forces. Then, the external loads  and internal stresses
i
F
ij
s do virtual work by
moving along
i
u d and
ij
d e . The product
i i
u F d
?
is known as the external virtual
work. It may be noted that the above product does not represent the conventional
work since each component is caused due to different source i.e.
i
u d is not due
to . Similarly the product
i
F
ij ij
s de
?
is the internal virtual work. In the case of
deformable body, both external and internal forces do work. Since, the beam is in
equilibrium, the external virtual work must be equal to the internal virtual work.
Hence, one needs to consider both internal and external virtual work to establish
equations of equilibrium.

5.3 Principle of Virtual Displacement
A deformable body is in equilibrium if the total external virtual work done by the
system of true forces moving through the corresponding virtual displacements of
the system i.e.  is equal to the total internal virtual work for every
kinematically admissible (consistent with the constraints) virtual displacements.
i i
u F d
?

That is virtual displacements should be continuous within the structure and also it
must satisfy boundary conditions.

dv u F
ij ij i i
de s d
?
?
=                                              (5.1)
where
ij
s are the true stresses due to true forces  and
i
F
ij
de are the virtual strains
due to virtual displacements
i
u d .

5.4 Principle of Virtual Forces
For a deformable body, the total external complementary work is equal to the total
internal complementary work for every system of virtual forces and stresses that
satisfy the equations of equilibrium.

dv u F
ij ij i i
e ds d
?
?
=                                              (5.2)

where
ij
d s are the virtual stresses due to virtual forces
i
F d and
ij
e are the true
strains due to the true displacements .
i
u
As stated earlier, the principle of virtual work may be advantageously used to
calculate displacements of structures. In the next section let us see how this can
be used to calculate displacements in a beams and frames. In the next lesson, the
truss deflections are calculated by the method of virtual work.

The principle of virtual force leads to unit load method. It is assumed throughout
our discussion that the method of superposition holds good. For the derivation of
unit load method, we consider two systems of loads. In this section, the principle of
virtual forces and unit load method are discussed in the context of framed
structures. Consider a cantilever beam, which is in equilibrium under the action of
a first system of forces  causing displacements as shown in
Fig. 5.2a. The first system of forces refers to the actual forces acting on the
structure. Let the stress resultants at any section of the beam due to first system of
forces be axial force (
n
F F F ,....., ,
2 1 n
u u u ,....., ,
2 1
P ), bending moment (M ) and shearing force (V ). Also the
corresponding incremental deformations are axial deformation ( ), flexural
deformation (
? d
? d ) and shearing deformation ( ? d ) respectively.
For a conservative system the external work done by the applied forces is equal to
the internal strain energy stored. Hence,

1
11 1 1
d ? d ? d ?
22 2 2
n
ii
i
Fu P M V
=
=+ +
?
? ??

? ? ?
+ + =
L L L
AG
ds V
EI
ds M
EA
ds P
0
2
0
2
0
2
2 2 2
(5.3)

Now, consider a second system of forces
n
F F F d d d ,....., ,
2 1
, which are virtual and
causing virtual displacements
n
u u u d d d ,....., ,
2 1
respectively (see Fig. 5.2b). Let the
virtual stress resultants caused by virtual forces be
v v
M P d d , and
v
V d at any cross
section of the beam. For this system of forces, we could write

? ? ?
?
+ + =
=
L
v
L
v
L
v
n
i
i i
AG
ds V
EI
ds M
EA
ds P
u F
0
2
0
2
0
2
1
2 2 2 2
1 d d d
d d       (5.4)

where
v v
M P d d , and
v
V d are the virtual axial force, bending moment and shear force
respectively. In the third case, apply the first system of forces on the beam, which
has been deformed, by second system of forces
n
F F F d d d ,....., ,
2 1
as shown in Fig
5.2c. From the principle of superposition, now the deflections will be
()( ) (
n n
u u u u u u ) d d d + + + ,......, ,
2 2 1 1
respectively

Since the energy is conserved we could write,

22 2 2
11 1
00 00
22
00 0 0 0
11
22 2 2 2 2
22
LL LL
nn n
vv v
jj j j jj
jj j
LL L L L
vv
Pds M ds V ds Pds
Fu F u Fu
v
EAEI AG E
Mds Vds
Pd M d Vd
EI AG
dd d
dd d
A
d d? d
== =
++ = + + +
++ ?+ +
?? ?
?? ??
?? ? ? ?
?
+
(5.5)

In equation (5.5), the term on the left hand side ( )
? j j
u F d , represents the work
done by virtual forces moving through real displacements. Since virtual forces act

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