Virtual Work (Part -2) - Energy Methods in Structural Analysis, Structural Analysis - II Civil Engineering (CE) Notes | EduRev

 Page 1


( ) ( )
?
=
L
v
c
EI
dx x M x M
0
) 1 (
d
?                                                (1) 
 
where () x M
v
d and  are the virtual moment resultant and real moment 
resultant at any section
() x M
x . Substituting the value of ( ) x M
v
d and  in the above 
expression, we get  
() x M
 
( ) ( )
? ?
+ =
L
L
L
c
EI
Mdx
EI
Mdx
4 / 3
4 / 3
0
0 1
) 1 ( ? 
   
EI
ML
c
4
3
= ?                     (2) 
 
Vertical deflection at C 
 
To evaluate vertical deflection at C , a unit virtual vertical force is applied ac  as 
shown in Fig 5.3d and the bending moment is also shown in the diagram. 
According to unit load method,  
C
 
( ) ( )
?
=
L
v
A
EI
dx x M x M
u
0
) 1 (
d
                      (3) 
In the present case,  () ?
?
?
?
?
?
- - = x
L
x M
v
4
3
d 
    and  ()
0
M x M + = 
 
?
?
?
?
?
?
?
- -
=
4
3
0
4
3 L
A
dx
EI
M x
L
u 
dx x
L
EI
M
L
?
?
?
?
?
?
?
- - =
4
3
0
4
3
 
4
3
0
2
2 4
3
L
x
x
L
EI
M
?
?
?
?
?
?
- - = 
EI
ML
32
9
2
- = ( ?)               (4) 
 
Example 5.2 
Find the horizontal displacement at joint B of the frame ABCD as shown in Fig. 
5.4a by unit load method. Assume EI to be constant for all members. 
 
 
Page 2


( ) ( )
?
=
L
v
c
EI
dx x M x M
0
) 1 (
d
?                                                (1) 
 
where () x M
v
d and  are the virtual moment resultant and real moment 
resultant at any section
() x M
x . Substituting the value of ( ) x M
v
d and  in the above 
expression, we get  
() x M
 
( ) ( )
? ?
+ =
L
L
L
c
EI
Mdx
EI
Mdx
4 / 3
4 / 3
0
0 1
) 1 ( ? 
   
EI
ML
c
4
3
= ?                     (2) 
 
Vertical deflection at C 
 
To evaluate vertical deflection at C , a unit virtual vertical force is applied ac  as 
shown in Fig 5.3d and the bending moment is also shown in the diagram. 
According to unit load method,  
C
 
( ) ( )
?
=
L
v
A
EI
dx x M x M
u
0
) 1 (
d
                      (3) 
In the present case,  () ?
?
?
?
?
?
- - = x
L
x M
v
4
3
d 
    and  ()
0
M x M + = 
 
?
?
?
?
?
?
?
- -
=
4
3
0
4
3 L
A
dx
EI
M x
L
u 
dx x
L
EI
M
L
?
?
?
?
?
?
?
- - =
4
3
0
4
3
 
4
3
0
2
2 4
3
L
x
x
L
EI
M
?
?
?
?
?
?
- - = 
EI
ML
32
9
2
- = ( ?)               (4) 
 
Example 5.2 
Find the horizontal displacement at joint B of the frame ABCD as shown in Fig. 
5.4a by unit load method. Assume EI to be constant for all members. 
 
 
 
 
The reactions and bending moment diagram of the frame due to applied external 
loading are shown in Fig 5.4b and Fig 5.4c respectively. Since, it is required to 
calculate horizontal deflection at B, apply a unit virtual load at B as shown in Fig. 
5.4d. The resulting reactions and bending moment diagrams of the frame are 
shown in Fig 5.4d. 
 
 
Page 3


( ) ( )
?
=
L
v
c
EI
dx x M x M
0
) 1 (
d
?                                                (1) 
 
where () x M
v
d and  are the virtual moment resultant and real moment 
resultant at any section
() x M
x . Substituting the value of ( ) x M
v
d and  in the above 
expression, we get  
() x M
 
( ) ( )
? ?
+ =
L
L
L
c
EI
Mdx
EI
Mdx
4 / 3
4 / 3
0
0 1
) 1 ( ? 
   
EI
ML
c
4
3
= ?                     (2) 
 
Vertical deflection at C 
 
To evaluate vertical deflection at C , a unit virtual vertical force is applied ac  as 
shown in Fig 5.3d and the bending moment is also shown in the diagram. 
According to unit load method,  
C
 
( ) ( )
?
=
L
v
A
EI
dx x M x M
u
0
) 1 (
d
                      (3) 
In the present case,  () ?
?
?
?
?
?
- - = x
L
x M
v
4
3
d 
    and  ()
0
M x M + = 
 
?
?
?
?
?
?
?
- -
=
4
3
0
4
3 L
A
dx
EI
M x
L
u 
dx x
L
EI
M
L
?
?
?
?
?
?
?
- - =
4
3
0
4
3
 
4
3
0
2
2 4
3
L
x
x
L
EI
M
?
?
?
?
?
?
- - = 
EI
ML
32
9
2
- = ( ?)               (4) 
 
Example 5.2 
Find the horizontal displacement at joint B of the frame ABCD as shown in Fig. 
5.4a by unit load method. Assume EI to be constant for all members. 
 
 
 
 
The reactions and bending moment diagram of the frame due to applied external 
loading are shown in Fig 5.4b and Fig 5.4c respectively. Since, it is required to 
calculate horizontal deflection at B, apply a unit virtual load at B as shown in Fig. 
5.4d. The resulting reactions and bending moment diagrams of the frame are 
shown in Fig 5.4d. 
 
 
 
 
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 4


( ) ( )
?
=
L
v
c
EI
dx x M x M
0
) 1 (
d
?                                                (1) 
 
where () x M
v
d and  are the virtual moment resultant and real moment 
resultant at any section
() x M
x . Substituting the value of ( ) x M
v
d and  in the above 
expression, we get  
() x M
 
( ) ( )
? ?
+ =
L
L
L
c
EI
Mdx
EI
Mdx
4 / 3
4 / 3
0
0 1
) 1 ( ? 
   
EI
ML
c
4
3
= ?                     (2) 
 
Vertical deflection at C 
 
To evaluate vertical deflection at C , a unit virtual vertical force is applied ac  as 
shown in Fig 5.3d and the bending moment is also shown in the diagram. 
According to unit load method,  
C
 
( ) ( )
?
=
L
v
A
EI
dx x M x M
u
0
) 1 (
d
                      (3) 
In the present case,  () ?
?
?
?
?
?
- - = x
L
x M
v
4
3
d 
    and  ()
0
M x M + = 
 
?
?
?
?
?
?
?
- -
=
4
3
0
4
3 L
A
dx
EI
M x
L
u 
dx x
L
EI
M
L
?
?
?
?
?
?
?
- - =
4
3
0
4
3
 
4
3
0
2
2 4
3
L
x
x
L
EI
M
?
?
?
?
?
?
- - = 
EI
ML
32
9
2
- = ( ?)               (4) 
 
Example 5.2 
Find the horizontal displacement at joint B of the frame ABCD as shown in Fig. 
5.4a by unit load method. Assume EI to be constant for all members. 
 
 
 
 
The reactions and bending moment diagram of the frame due to applied external 
loading are shown in Fig 5.4b and Fig 5.4c respectively. Since, it is required to 
calculate horizontal deflection at B, apply a unit virtual load at B as shown in Fig. 
5.4d. The resulting reactions and bending moment diagrams of the frame are 
shown in Fig 5.4d. 
 
 
 
 
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 5


( ) ( )
?
=
L
v
c
EI
dx x M x M
0
) 1 (
d
?                                                (1) 
 
where () x M
v
d and  are the virtual moment resultant and real moment 
resultant at any section
() x M
x . Substituting the value of ( ) x M
v
d and  in the above 
expression, we get  
() x M
 
( ) ( )
? ?
+ =
L
L
L
c
EI
Mdx
EI
Mdx
4 / 3
4 / 3
0
0 1
) 1 ( ? 
   
EI
ML
c
4
3
= ?                     (2) 
 
Vertical deflection at C 
 
To evaluate vertical deflection at C , a unit virtual vertical force is applied ac  as 
shown in Fig 5.3d and the bending moment is also shown in the diagram. 
According to unit load method,  
C
 
( ) ( )
?
=
L
v
A
EI
dx x M x M
u
0
) 1 (
d
                      (3) 
In the present case,  () ?
?
?
?
?
?
- - = x
L
x M
v
4
3
d 
    and  ()
0
M x M + = 
 
?
?
?
?
?
?
?
- -
=
4
3
0
4
3 L
A
dx
EI
M x
L
u 
dx x
L
EI
M
L
?
?
?
?
?
?
?
- - =
4
3
0
4
3
 
4
3
0
2
2 4
3
L
x
x
L
EI
M
?
?
?
?
?
?
- - = 
EI
ML
32
9
2
- = ( ?)               (4) 
 
Example 5.2 
Find the horizontal displacement at joint B of the frame ABCD as shown in Fig. 
5.4a by unit load method. Assume EI to be constant for all members. 
 
 
 
 
The reactions and bending moment diagram of the frame due to applied external 
loading are shown in Fig 5.4b and Fig 5.4c respectively. Since, it is required to 
calculate horizontal deflection at B, apply a unit virtual load at B as shown in Fig. 
5.4d. The resulting reactions and bending moment diagrams of the frame are 
shown in Fig 5.4d. 
 
 
 
 
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