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Page 2
Now horizontal deflection at B, may be calculated as
B
u
( ) ( )
?
= ×
D
A
v B
H
EI
dx x M x M
u
d
) 1 ( (1)
() () ( ) ( ) ( ) ( )
? ? ?
+ + =
D
C
v
C
B
v
B
A
v
EI
dx x M x M
EI
dx x M x M
EI
dx x M x M d d d
()( ) () ( )
0
5 . 2 10 5 . 2 2 5
5 . 2
0
5
0
+
- -
+ =
? ?
EI
dx x x
EI
dx x x
() ()
? ?
-
+ =
5 . 2
0
2 5
0
2
5 . 2 20 5
EI
dx x
EI
dx x
EI EI EI 3
5 . 937
3
5 . 312
3
625
= + =
Hence,
EI
u
A
3
5 . 937
= ( ?) (2)
Example 5.3
Find the rotations of joint B and C of the frame shown in Fig. 5.4a. Assume EI to
be constant for all members.
Page 3
Now horizontal deflection at B, may be calculated as
B
u
( ) ( )
?
= ×
D
A
v B
H
EI
dx x M x M
u
d
) 1 ( (1)
() () ( ) ( ) ( ) ( )
? ? ?
+ + =
D
C
v
C
B
v
B
A
v
EI
dx x M x M
EI
dx x M x M
EI
dx x M x M d d d
()( ) () ( )
0
5 . 2 10 5 . 2 2 5
5 . 2
0
5
0
+
- -
+ =
? ?
EI
dx x x
EI
dx x x
() ()
? ?
-
+ =
5 . 2
0
2 5
0
2
5 . 2 20 5
EI
dx x
EI
dx x
EI EI EI 3
5 . 937
3
5 . 312
3
625
= + =
Hence,
EI
u
A
3
5 . 937
= ( ?) (2)
Example 5.3
Find the rotations of joint B and C of the frame shown in Fig. 5.4a. Assume EI to
be constant for all members.
Rotation at B
Apply unit virtual moment at B as shown in Fig 5.5a. The resulting bending
moment diagram is also shown in the same diagram. For the unit load method, the
relevant equation is,
( ) ( )
?
= ×
D
A
v
B
EI
dx x M x M d
? ) 1 ( (1)
wherein,
B
? is the actual rotation at B, ()
v
M x d is the virtual stress resultant in the
frame due to the virtual load and
()
D
A
M x
dx
EI
?
is the actual deformation of the frame
due to real forces.
Page 4
Now horizontal deflection at B, may be calculated as
B
u
( ) ( )
?
= ×
D
A
v B
H
EI
dx x M x M
u
d
) 1 ( (1)
() () ( ) ( ) ( ) ( )
? ? ?
+ + =
D
C
v
C
B
v
B
A
v
EI
dx x M x M
EI
dx x M x M
EI
dx x M x M d d d
()( ) () ( )
0
5 . 2 10 5 . 2 2 5
5 . 2
0
5
0
+
- -
+ =
? ?
EI
dx x x
EI
dx x x
() ()
? ?
-
+ =
5 . 2
0
2 5
0
2
5 . 2 20 5
EI
dx x
EI
dx x
EI EI EI 3
5 . 937
3
5 . 312
3
625
= + =
Hence,
EI
u
A
3
5 . 937
= ( ?) (2)
Example 5.3
Find the rotations of joint B and C of the frame shown in Fig. 5.4a. Assume EI to
be constant for all members.
Rotation at B
Apply unit virtual moment at B as shown in Fig 5.5a. The resulting bending
moment diagram is also shown in the same diagram. For the unit load method, the
relevant equation is,
( ) ( )
?
= ×
D
A
v
B
EI
dx x M x M d
? ) 1 ( (1)
wherein,
B
? is the actual rotation at B, ()
v
M x d is the virtual stress resultant in the
frame due to the virtual load and
()
D
A
M x
dx
EI
?
is the actual deformation of the frame
due to real forces.
Now, and () ( ) x x M - = 5 . 2 10 ( ) ( ) x x M
v
- = 5 . 2 4 . 0 d
Substituting the values of () M x and ()
v
M x d in the equation (1),
()
?
- =
5 . 2
0
2
5 . 2
4
dx x
EI
B
?
EI
x x
x
EI 3
5 . 62
3 2
5
25 . 6
4
5 . 2
0
3 2
=
?
?
?
?
?
?
+ - = (2)
Rotation at C
For evaluating rotation at C by unit load method, apply unit virtual moment at C as
shown in Fig 5.5b. Hence,
( ) ( )
?
= ×
D
A
v
C
EI
dx x M x M d
? ) 1 ( (3)
() ( )
?
-
=
5 . 2
0
4 . 0 5 . 2 10
dx
EI
x x
C
?
EI
x x
EI 3
25 . 31
3 2
5 . 2 4
5 . 2
0
3 2
=
?
?
?
?
?
?
- = (4)
5.6 Unit Displacement Method
Consider a cantilever beam, which is in equilibrium under the action of a system of
forces . Let be the corresponding displacements and
and be the stress resultants at section of the beam. Consider a second
system of forces (virtual)
n
F F F ,....., ,
2 1 n
u u u ,....., ,
2 1
M P, V
n
F F F d d d ,....., ,
2 1
causing virtual
displacements
n
u u u d d d ,....., ,
2 1
. Let
v v
M P d d , and
v
V d be the virtual axial force,
bending moment and shear force respectively at any section of the beam.
Apply the first system of forces on the beam, which has been
previously bent by virtual forces
n
F F F ,....., ,
2 1
n
F F F d d d ,....., ,
2 1
. From the principle of virtual
displacements we have,
( ) ( )
?
?
=
=
n
j
v
j j
EI
ds x M x M
u F
1
d
d
(5.11)
?
=
V
T
v d de s
Page 5
Now horizontal deflection at B, may be calculated as
B
u
( ) ( )
?
= ×
D
A
v B
H
EI
dx x M x M
u
d
) 1 ( (1)
() () ( ) ( ) ( ) ( )
? ? ?
+ + =
D
C
v
C
B
v
B
A
v
EI
dx x M x M
EI
dx x M x M
EI
dx x M x M d d d
()( ) () ( )
0
5 . 2 10 5 . 2 2 5
5 . 2
0
5
0
+
- -
+ =
? ?
EI
dx x x
EI
dx x x
() ()
? ?
-
+ =
5 . 2
0
2 5
0
2
5 . 2 20 5
EI
dx x
EI
dx x
EI EI EI 3
5 . 937
3
5 . 312
3
625
= + =
Hence,
EI
u
A
3
5 . 937
= ( ?) (2)
Example 5.3
Find the rotations of joint B and C of the frame shown in Fig. 5.4a. Assume EI to
be constant for all members.
Rotation at B
Apply unit virtual moment at B as shown in Fig 5.5a. The resulting bending
moment diagram is also shown in the same diagram. For the unit load method, the
relevant equation is,
( ) ( )
?
= ×
D
A
v
B
EI
dx x M x M d
? ) 1 ( (1)
wherein,
B
? is the actual rotation at B, ()
v
M x d is the virtual stress resultant in the
frame due to the virtual load and
()
D
A
M x
dx
EI
?
is the actual deformation of the frame
due to real forces.
Now, and () ( ) x x M - = 5 . 2 10 ( ) ( ) x x M
v
- = 5 . 2 4 . 0 d
Substituting the values of () M x and ()
v
M x d in the equation (1),
()
?
- =
5 . 2
0
2
5 . 2
4
dx x
EI
B
?
EI
x x
x
EI 3
5 . 62
3 2
5
25 . 6
4
5 . 2
0
3 2
=
?
?
?
?
?
?
+ - = (2)
Rotation at C
For evaluating rotation at C by unit load method, apply unit virtual moment at C as
shown in Fig 5.5b. Hence,
( ) ( )
?
= ×
D
A
v
C
EI
dx x M x M d
? ) 1 ( (3)
() ( )
?
-
=
5 . 2
0
4 . 0 5 . 2 10
dx
EI
x x
C
?
EI
x x
EI 3
25 . 31
3 2
5 . 2 4
5 . 2
0
3 2
=
?
?
?
?
?
?
- = (4)
5.6 Unit Displacement Method
Consider a cantilever beam, which is in equilibrium under the action of a system of
forces . Let be the corresponding displacements and
and be the stress resultants at section of the beam. Consider a second
system of forces (virtual)
n
F F F ,....., ,
2 1 n
u u u ,....., ,
2 1
M P, V
n
F F F d d d ,....., ,
2 1
causing virtual
displacements
n
u u u d d d ,....., ,
2 1
. Let
v v
M P d d , and
v
V d be the virtual axial force,
bending moment and shear force respectively at any section of the beam.
Apply the first system of forces on the beam, which has been
previously bent by virtual forces
n
F F F ,....., ,
2 1
n
F F F d d d ,....., ,
2 1
. From the principle of virtual
displacements we have,
( ) ( )
?
?
=
=
n
j
v
j j
EI
ds x M x M
u F
1
d
d
(5.11)
?
=
V
T
v d de s
The left hand side of equation (5.11) refers to the external virtual work done by the
system of true/real forces moving through the corresponding virtual displacements
of the system. The right hand side of equation (5.8) refers to internal virtual work
done. The principle of virtual displacement states that the external virtual work of
the real forces multiplied by virtual displacement is equal to the real stresses
multiplied by virtual strains integrated over volume. If the value of a particular force
element is required then choose corresponding virtual displacement as unity. Let
us say, it is required to evaluate , then choose
1
F 1
1
= u d and n i u
i
,....., 3 , 2 0 = = d .
From equation (5.11), one could write,
()
?
=
EI
ds M M
F
v 1
1
) (
1
d
(5.12)
where, (
1 v
M) d is the internal virtual stress resultant for 1
1
= u d . Transposing the
above equation, we get
?
=
EI
Mds M
F
v 1
1
) ( d
(5.13)
The above equation is the statement of unit displacement method. The above
equation is more commonly used in the evaluation of stiffness co-efficient .
ij
k
Apply real displacements in the structure. In that set and the other
all displacements . For such a case the quantity in
equation (5.11) becomes i.e. force at 1 due to displacement at 2. Apply virtual
displacement
n
u u ,.....,
1
1
2
= u
) ,......, 3 , 1 ( 0 n i u
i
= =
j
F
ij
k
1
1
= u d . Now according to unit displacement method,
()
?
=
EI
ds M M
k
v 2 1
12
) (
1
d
(5.14)
Summary
In this chapter the concept of virtual work is introduced and the principle of virtual
work is discussed. The terms internal virtual work and external virtual work has
been explained and relevant expressions are also derived. Principle of virtual
forces has been stated. It has been shown how the principle of virtual load leads to
unit load method. An expression for calculating deflections at any point of a
structure (both statically determinate and indeterminate structure) is derived. Few
problems have been solved to show the application of unit load method for
calculating deflections in a structure.
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FAQs on Virtual Work - 2 - Structural Analysis - Civil Engineering (CE)
| 1. What is virtual work in civil engineering? |  |
Virtual work in civil engineering is a method used to analyze the structural behavior of a system or component under external loads. It involves calculating the work done by internal forces within the structure when it undergoes virtual displacements. This method is particularly useful in determining the equilibrium state and deformations of structures subjected to various loads.
| 2. How is virtual work used in civil engineering? | |
Virtual work is used in civil engineering to analyze and design structures. It provides a mathematical framework to calculate internal forces, deformations, and stresses within a structure. By applying the principle of virtual work, engineers can determine the equilibrium conditions, assess structural stability, and optimize designs. It is commonly used in the analysis of beams, trusses, frames, and other structural components.
| 3. What are the advantages of using virtual work in civil engineering analysis? | |
There are several advantages of using virtual work in civil engineering analysis. Firstly, it provides a systematic and rigorous approach to analyze complex structural systems. Secondly, it allows engineers to consider the effects of both external and internal forces on the behavior of the structure. Additionally, virtual work helps in identifying critical areas prone to excessive deformations or stresses. Lastly, it enables engineers to optimize structural designs by assessing different load scenarios and identifying the most efficient solutions.
| 4. Are there any limitations to using virtual work in civil engineering analysis? | |
While virtual work is a powerful tool in civil engineering analysis, it has some limitations. One limitation is that it assumes linear behavior of materials, neglecting the effects of nonlinearity. Additionally, it requires knowledge of the internal forces within the structure, which may not always be readily available. Furthermore, the method may become computationally intensive for large and complex structures. Despite these limitations, virtual work remains an essential technique in structural analysis.
| 5. How does virtual work differ from actual work in civil engineering? | |
Virtual work and actual work differ in their definitions and applications in civil engineering. Actual work refers to the physical work done by external forces on a structure, such as the energy expended to lift a load or the force applied to deform a material. On the other hand, virtual work is a mathematical concept that accounts for the work done by internal forces within a structure when it undergoes virtual displacements. Virtual work is used to analyze the equilibrium and deformations of structures, while actual work is concerned with the physical energy transfer.
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Dec 19, 2024 Last updated |
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