Courses

# Engesser’s Theorem & Truss Deflections by Virtual Work Principles - 2 Civil Engineering (CE) Notes | EduRev

## Civil Engineering (CE) : Engesser’s Theorem & Truss Deflections by Virtual Work Principles - 2 Civil Engineering (CE) Notes | EduRev

``` Page 1

Example 6.2
Compute the vertical deflection of joint  and horizontal displacement of joint
of the truss shown in Fig. 6.3a due to
b D
b) Increase in temperature of  in the top chord BD. Assume
0
25 C
1
C
75000
per a=° , . The cross sectional areas of the
members in square centimeters are shown in parentheses.
2 5
/ 10 00 . 2 mm N E × =

Page 2

Example 6.2
Compute the vertical deflection of joint  and horizontal displacement of joint
of the truss shown in Fig. 6.3a due to
b D
b) Increase in temperature of  in the top chord BD. Assume
0
25 C
1
C
75000
per a=° , . The cross sectional areas of the
members in square centimeters are shown in parentheses.
2 5
/ 10 00 . 2 mm N E × =

Page 3

Example 6.2
Compute the vertical deflection of joint  and horizontal displacement of joint
of the truss shown in Fig. 6.3a due to
b D
b) Increase in temperature of  in the top chord BD. Assume
0
25 C
1
C
75000
per a=° , . The cross sectional areas of the
members in square centimeters are shown in parentheses.
2 5
/ 10 00 . 2 mm N E × =

The complete calculations are shown in the following table.

Table 6.3 Computational details for example 6.2

Mem
i
L

i i i
E A L /

i
P
i
v
v
P ) ( d
i
H
v
P ) ( d

i ti
tL a = ?

i i
i i i
v
v
A E
L P P ) ( d

i i
i i i
H
v
A E
L P P ) ( d

ti i
v
v
P ? ) ( d

ti i
H
v
P ? ) ( d

units m (10
-5
)
m/kN
kN kN kN m (10
-3
)
kN.m
(10
-3
)
kN.m
(10
-3
)
kN.m
(10
-3
)
kN.m
aB 5 1.0 -112.5 -0.937 +0.416 0 1.05 -0.47 0 0
ab 3 1.0 +67.5 +0.562 +0.750 0 0.38 0.51 0 0
bc 3 1.0 +67.5 +0.562 +0.750 0 0.38 0.51 0 0
Bc 5 1.0 +37.5 -0.312 -0.416 0 -0.12 -0.16 0 0
BD 6 2.0 -67.5 -0.562 +0.500 0.002 0.76 -0.68 -1.13 1
cD 5 1.0 +37.5 +0.312 +0.416 0 0.12 0.16 0 0
cd 3 1.0 +67.5 +0.187 +0.250 0 0.13 0.17 0 0
de  3 1.0 +67.5 +0.187 +0.250 0 0.13 0.17 0 0
De 5 1.0 -112.5 -0.312 -0.416 0 0.35 0.47 0 0
Bb 4 2.0 +60.0 1 0 0 1.2 0 0 0
Dd 4 2.0 +60.0 0 0 0 0 0 0 0

?
4.38 0.68 -1.13 1

Page 4

Example 6.2
Compute the vertical deflection of joint  and horizontal displacement of joint
of the truss shown in Fig. 6.3a due to
b D
b) Increase in temperature of  in the top chord BD. Assume
0
25 C
1
C
75000
per a=° , . The cross sectional areas of the
members in square centimeters are shown in parentheses.
2 5
/ 10 00 . 2 mm N E × =

The complete calculations are shown in the following table.

Table 6.3 Computational details for example 6.2

Mem
i
L

i i i
E A L /

i
P
i
v
v
P ) ( d
i
H
v
P ) ( d

i ti
tL a = ?

i i
i i i
v
v
A E
L P P ) ( d

i i
i i i
H
v
A E
L P P ) ( d

ti i
v
v
P ? ) ( d

ti i
H
v
P ? ) ( d

units m (10
-5
)
m/kN
kN kN kN m (10
-3
)
kN.m
(10
-3
)
kN.m
(10
-3
)
kN.m
(10
-3
)
kN.m
aB 5 1.0 -112.5 -0.937 +0.416 0 1.05 -0.47 0 0
ab 3 1.0 +67.5 +0.562 +0.750 0 0.38 0.51 0 0
bc 3 1.0 +67.5 +0.562 +0.750 0 0.38 0.51 0 0
Bc 5 1.0 +37.5 -0.312 -0.416 0 -0.12 -0.16 0 0
BD 6 2.0 -67.5 -0.562 +0.500 0.002 0.76 -0.68 -1.13 1
cD 5 1.0 +37.5 +0.312 +0.416 0 0.12 0.16 0 0
cd 3 1.0 +67.5 +0.187 +0.250 0 0.13 0.17 0 0
de  3 1.0 +67.5 +0.187 +0.250 0 0.13 0.17 0 0
De 5 1.0 -112.5 -0.312 -0.416 0 0.35 0.47 0 0
Bb 4 2.0 +60.0 1 0 0 1.2 0 0 0
Dd 4 2.0 +60.0 0 0 0 0 0 0 0

?
4.38 0.68 -1.13 1

a) Vertical deflection of joint  b
Applying principle of virtual work as applied to an ideal pin jointed truss,

1
11
()
m
vij i i
jj
ji
ii
PPL
Fu
EA
d
d
==
=
??
(1)

For calculating vertical deflection at b , apply a unit virtual load 1
b
F d = . Then the
above equation may be written as,

?
= ×
i i
i i i
v
v v
b
A E
L P P
u
) (
1
d
(2)

? =
=
+
?=
mm
m
KN
KNm
u
b
38 . 4
00438 . 0
1
00438 . 0

2) Due to change in temperature

?
? = ?
ti i
v
v
t
b
P u ) ( ) )( 1 ( d

m
KN
m KN
u
t
b
00113 . 0
1
. 001125 . 0
- =
-
?=
? = mm u
t
b
13 . 1

b) Horizontal displacement of joint ‘D’

1) Due to externally applied loads

?
= ×
i i
i i i
H
v H
b
A E
L P P
u
) (
1
d

? =
=
+
?=
mm
m
KN
KNm
u
H
D
68 . 0
00068 . 0
1
00068 . 0

Page 5

Example 6.2
Compute the vertical deflection of joint  and horizontal displacement of joint
of the truss shown in Fig. 6.3a due to
b D
b) Increase in temperature of  in the top chord BD. Assume
0
25 C
1
C
75000
per a=° , . The cross sectional areas of the
members in square centimeters are shown in parentheses.
2 5
/ 10 00 . 2 mm N E × =

The complete calculations are shown in the following table.

Table 6.3 Computational details for example 6.2

Mem
i
L

i i i
E A L /

i
P
i
v
v
P ) ( d
i
H
v
P ) ( d

i ti
tL a = ?

i i
i i i
v
v
A E
L P P ) ( d

i i
i i i
H
v
A E
L P P ) ( d

ti i
v
v
P ? ) ( d

ti i
H
v
P ? ) ( d

units m (10
-5
)
m/kN
kN kN kN m (10
-3
)
kN.m
(10
-3
)
kN.m
(10
-3
)
kN.m
(10
-3
)
kN.m
aB 5 1.0 -112.5 -0.937 +0.416 0 1.05 -0.47 0 0
ab 3 1.0 +67.5 +0.562 +0.750 0 0.38 0.51 0 0
bc 3 1.0 +67.5 +0.562 +0.750 0 0.38 0.51 0 0
Bc 5 1.0 +37.5 -0.312 -0.416 0 -0.12 -0.16 0 0
BD 6 2.0 -67.5 -0.562 +0.500 0.002 0.76 -0.68 -1.13 1
cD 5 1.0 +37.5 +0.312 +0.416 0 0.12 0.16 0 0
cd 3 1.0 +67.5 +0.187 +0.250 0 0.13 0.17 0 0
de  3 1.0 +67.5 +0.187 +0.250 0 0.13 0.17 0 0
De 5 1.0 -112.5 -0.312 -0.416 0 0.35 0.47 0 0
Bb 4 2.0 +60.0 1 0 0 1.2 0 0 0
Dd 4 2.0 +60.0 0 0 0 0 0 0 0

?
4.38 0.68 -1.13 1

a) Vertical deflection of joint  b
Applying principle of virtual work as applied to an ideal pin jointed truss,

1
11
()
m
vij i i
jj
ji
ii
PPL
Fu
EA
d
d
==
=
??
(1)

For calculating vertical deflection at b , apply a unit virtual load 1
b
F d = . Then the
above equation may be written as,

?
= ×
i i
i i i
v
v v
b
A E
L P P
u
) (
1
d
(2)

? =
=
+
?=
mm
m
KN
KNm
u
b
38 . 4
00438 . 0
1
00438 . 0

2) Due to change in temperature

?
? = ?
ti i
v
v
t
b
P u ) ( ) )( 1 ( d

m
KN
m KN
u
t
b
00113 . 0
1
. 001125 . 0
- =
-
?=
? = mm u
t
b
13 . 1

b) Horizontal displacement of joint ‘D’

1) Due to externally applied loads

?
= ×
i i
i i i
H
v H
b
A E
L P P
u
) (
1
d

? =
=
+
?=
mm
m
KN
KNm
u
H
D
68 . 0
00068 . 0
1
00068 . 0

2) Due to change in temperature

?
? = ?
ti i
H
v
Ht
D
P u ) ( ) )( 1 ( d

m
KN
m KN
u
Ht
D
001 . 0
1
. 001 . 0
= ?=
? = mm u
Ht
D
00 . 1

Summary
In this chapter the Crotti-Engessor’s theorem which is more general than the
Castigliano’s theorem has been introduced. The unit load method is applied
statically determinate structure for calculating deflections when the truss is

```
Offer running on EduRev: Apply code STAYHOME200 to get INR 200 off on our premium plan EduRev Infinity!

## Structural Analysis

30 videos|122 docs|28 tests

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

;