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

## Civil Engineering (CE) : 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.
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.
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.
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.
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.
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

```
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