Virtual Work - 2 Civil Engineering (CE) Notes | EduRev

Structural Analysis

Civil Engineering (CE) : Virtual Work - 2 Civil Engineering (CE) Notes | EduRev

 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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