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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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FAQs on Engesser Theorem & Truss Deflections by Virtual Work Principles - 2 - Structural Analysis - Civil Engineering (CE)
| 1. What is Engesser's Theorem? |  |
Engesser's Theorem is a principle in structural engineering that allows the calculation of deflections in trusses using virtual work principles. It provides a method to determine the displacement of individual members in a truss structure under external loads.
| 2. How does Engesser's Theorem work? | |
Engesser's Theorem applies the principle of virtual work, which states that the virtual work done by external forces on a system in equilibrium is equal to the virtual work done by internal forces. By considering the deflection of each member as a virtual displacement, the theorem allows the calculation of internal forces and deflections in a truss.
| 3. What are the advantages of using Engesser's Theorem in truss deflection analysis? | |
Engesser's Theorem provides a simplified approach to determine truss deflections compared to more complex methods such as the flexibility matrix method. It is particularly useful for trusses with relatively simple geometry and loads. Additionally, Engesser's Theorem can be easily applied to solve deflection problems by hand calculations.
| 4. Are there any limitations to using Engesser's Theorem for truss deflection analysis? | |
Yes, Engesser's Theorem has some limitations. It assumes that the truss structure is statically determinate, meaning that the number of unknown forces and displacements is equal to the number of equilibrium equations. It also assumes that the truss members behave linearly elastically and that the deflections are small.
| 5. Can Engesser's Theorem be applied to more complex structures beyond trusses? | |
Engesser's Theorem is specifically formulated for truss structures, which are composed of straight members connected at joints. While it may not directly apply to more complex structures, the principles of virtual work can be extended to analyze other types of structures, such as beams, frames, and even three-dimensional structures, using appropriate modifications and assumptions.
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