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

Structural Analysis

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
a) Applied loading as shown in figure. 
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
a) Applied loading as shown in figure. 
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
a) Applied loading as shown in figure. 
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
a) Applied loading as shown in figure. 
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) 
 
1) Due to external loads 
 
  
? =
=
+
?=
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
a) Applied loading as shown in figure. 
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) 
 
1) Due to external loads 
 
  
? =
=
+
?=
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 
subjected to various types of loadings such as: mechanical loading, temperature 
loading and fabrication errors. 
 
 
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