Virtual Work - 1 | Structural Analysis - Civil Engineering (CE) PDF Download

 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
. The virtual loading system produces virtual internal 
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
. The virtual loading system produces virtual internal 
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. 
 
 
5.5 Unit Load Method 
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
. The virtual loading system produces virtual internal 
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. 
 
 
5.5 Unit Load Method 
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
. The virtual loading system produces virtual internal 
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. 
 
 
5.5 Unit Load Method 
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