Civil Engineering (CE) Exam > Civil Engineering (CE) Notes > Structural Analysis > The Direct Stiffness Method: Temperature Changes & Fabrication Errors in Truss Analysis - 2
The Direct Stiffness Method: Temperature Changes & Fabrication Errors in Truss Analysis - 2 | Structural Analysis - Civil Engineering (CE) PDF Download
| Download, print and study this document offline |
Please wait while the PDF view is loading
Page 1
411
3
33
0000
01 0 1
15 10 2 10
0000 3 x 10 10
010 1
k
-
? ?
? ?
-
×××
? ?
?? =
??
? ? ×
? ?
-
? ?
(7)
The global stiffness matrix is of the order 8 8 × ,assembling the three member
stiffness matrices, one gets
[]
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
-
- -
- -
-
- - -
- - -
=
0 0 0 0 0 0 0 0
0 100 0 0 0 0 0 100
0 0 14 . 47 14 . 47 0 0 14 . 47 14 . 47
0 0 14 . 47 14 . 47 0 0 14 . 47 14 . 47
0 0 0 0 100 0 100 0
0 0 0 0 0 0 0 0
0 0 14 . 47 14 . 47 100 0 14 . 147 14 . 47
0 100 14 . 47 14 . 47 0 0 14 . 47 14 . 147
10
3
k (8)
Writing the load displacement equation for the truss
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
8
7
6
5
4
3
2
1
p
p
p
p
p
p
p
p
=
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
-
- -
- -
-
- - -
- - -
0 0 0 0 0 0 0 0
0 100 0 0 0 0 0 100
0 0 14 . 47 14 . 47 0 0 14 . 47 14 . 47
0 0 14 . 47 14 . 47 0 0 14 . 47 14 . 47
0 0 0 0 100 0 100 0
0 0 0 0 0 0 0 0
0 0 14 . 47 14 . 47 100 0 14 . 147 14 . 47
0 100 14 . 47 14 . 47 0 0 14 . 47 14 . 147
10
3
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
8
7
6
5
4
3
2
1
u
u
u
u
u
u
u
u
+
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
-
-
0
0
1
1
0
0
1
1
640
(9)
In the present case, the displacements
1
u and
2
u are not known. All other
displacements are zero. Also 0
2 1
= = p p (as no joint loads are applied).Thus,
Page 2
411
3
33
0000
01 0 1
15 10 2 10
0000 3 x 10 10
010 1
k
-
? ?
? ?
-
×××
? ?
?? =
??
? ? ×
? ?
-
? ?
(7)
The global stiffness matrix is of the order 8 8 × ,assembling the three member
stiffness matrices, one gets
[]
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
-
- -
- -
-
- - -
- - -
=
0 0 0 0 0 0 0 0
0 100 0 0 0 0 0 100
0 0 14 . 47 14 . 47 0 0 14 . 47 14 . 47
0 0 14 . 47 14 . 47 0 0 14 . 47 14 . 47
0 0 0 0 100 0 100 0
0 0 0 0 0 0 0 0
0 0 14 . 47 14 . 47 100 0 14 . 147 14 . 47
0 100 14 . 47 14 . 47 0 0 14 . 47 14 . 147
10
3
k (8)
Writing the load displacement equation for the truss
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
8
7
6
5
4
3
2
1
p
p
p
p
p
p
p
p
=
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
-
- -
- -
-
- - -
- - -
0 0 0 0 0 0 0 0
0 100 0 0 0 0 0 100
0 0 14 . 47 14 . 47 0 0 14 . 47 14 . 47
0 0 14 . 47 14 . 47 0 0 14 . 47 14 . 47
0 0 0 0 100 0 100 0
0 0 0 0 0 0 0 0
0 0 14 . 47 14 . 47 100 0 14 . 147 14 . 47
0 100 14 . 47 14 . 47 0 0 14 . 47 14 . 147
10
3
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
8
7
6
5
4
3
2
1
u
u
u
u
u
u
u
u
+
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
-
-
0
0
1
1
0
0
1
1
640
(9)
In the present case, the displacements
1
u and
2
u are not known. All other
displacements are zero. Also 0
2 1
= = p p (as no joint loads are applied).Thus,
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
8
7
6
5
4
3
2
1
p
p
p
p
p
p
p
p
=
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
-
- -
- -
-
- - -
- - -
=
0 0 0 0 0 0 0 0
0 100 0 0 0 0 0 100
0 0 14 . 47 14 . 47 0 0 14 . 47 14 . 47
0 0 14 . 47 14 . 47 0 0 14 . 47 14 . 47
0 0 0 0 100 0 100 0
0 0 0 0 0 0 0 0
0 0 14 . 47 14 . 47 100 0 14 . 147 14 . 47
0 100 14 . 47 14 . 47 0 0 14 . 47 14 . 147
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
8
7
6
5
4
3
2
1
u
u
u
u
u
u
u
u
+
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
-
-
0
0
1
1
0
0
1
1
640 (10)
Thus unknown displacements are
1
1
3
2
147.14 47.14 0 1
1
(150.82 )
47.14 147.14 0 1 10
u
u
-
- ?????? ??
=-
?? ? ? ? ?
??
-
???? ?? ??
(11)
4
1
4
2
7.763 10
7.763 10
um
um
-
-
=×
=×
Now reactions are calculated as
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
+
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
+
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
-
- -
- -
-
=
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
0
0
1
1
0
0
640
0
0
0
0
0
0
0 0 0 0 0 0
0 100 0 0 0 0
0 0 14 . 47 14 . 47 0 0
0 0 14 . 47 14 . 47 0 0
0 0 0 0 100 0
0 0 0 0 0 0
0 0
0 100
14 . 47 14 . 47
14 . 47 14 . 47
100 0
0 0
10
2
1 3
8
7
6
5
4
3
u
u
p
p
p
p
p
p
3
4
5
6
7
8
0
77.63
77.63
77.63
77.63
0
p
p
p
p
p
p
??
??
??
??
-
??
??
??
??
?? ? ?
=
?? ? ?
?? ? ?
?? ? ?
-
?? ? ?
??
????
??
kN (12)
Page 3
411
3
33
0000
01 0 1
15 10 2 10
0000 3 x 10 10
010 1
k
-
? ?
? ?
-
×××
? ?
?? =
??
? ? ×
? ?
-
? ?
(7)
The global stiffness matrix is of the order 8 8 × ,assembling the three member
stiffness matrices, one gets
[]
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
-
- -
- -
-
- - -
- - -
=
0 0 0 0 0 0 0 0
0 100 0 0 0 0 0 100
0 0 14 . 47 14 . 47 0 0 14 . 47 14 . 47
0 0 14 . 47 14 . 47 0 0 14 . 47 14 . 47
0 0 0 0 100 0 100 0
0 0 0 0 0 0 0 0
0 0 14 . 47 14 . 47 100 0 14 . 147 14 . 47
0 100 14 . 47 14 . 47 0 0 14 . 47 14 . 147
10
3
k (8)
Writing the load displacement equation for the truss
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
8
7
6
5
4
3
2
1
p
p
p
p
p
p
p
p
=
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
-
- -
- -
-
- - -
- - -
0 0 0 0 0 0 0 0
0 100 0 0 0 0 0 100
0 0 14 . 47 14 . 47 0 0 14 . 47 14 . 47
0 0 14 . 47 14 . 47 0 0 14 . 47 14 . 47
0 0 0 0 100 0 100 0
0 0 0 0 0 0 0 0
0 0 14 . 47 14 . 47 100 0 14 . 147 14 . 47
0 100 14 . 47 14 . 47 0 0 14 . 47 14 . 147
10
3
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
8
7
6
5
4
3
2
1
u
u
u
u
u
u
u
u
+
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
-
-
0
0
1
1
0
0
1
1
640
(9)
In the present case, the displacements
1
u and
2
u are not known. All other
displacements are zero. Also 0
2 1
= = p p (as no joint loads are applied).Thus,
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
8
7
6
5
4
3
2
1
p
p
p
p
p
p
p
p
=
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
-
- -
- -
-
- - -
- - -
=
0 0 0 0 0 0 0 0
0 100 0 0 0 0 0 100
0 0 14 . 47 14 . 47 0 0 14 . 47 14 . 47
0 0 14 . 47 14 . 47 0 0 14 . 47 14 . 47
0 0 0 0 100 0 100 0
0 0 0 0 0 0 0 0
0 0 14 . 47 14 . 47 100 0 14 . 147 14 . 47
0 100 14 . 47 14 . 47 0 0 14 . 47 14 . 147
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
8
7
6
5
4
3
2
1
u
u
u
u
u
u
u
u
+
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
-
-
0
0
1
1
0
0
1
1
640 (10)
Thus unknown displacements are
1
1
3
2
147.14 47.14 0 1
1
(150.82 )
47.14 147.14 0 1 10
u
u
-
- ?????? ??
=-
?? ? ? ? ?
??
-
???? ?? ??
(11)
4
1
4
2
7.763 10
7.763 10
um
um
-
-
=×
=×
Now reactions are calculated as
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
+
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
+
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
-
- -
- -
-
=
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
0
0
1
1
0
0
640
0
0
0
0
0
0
0 0 0 0 0 0
0 100 0 0 0 0
0 0 14 . 47 14 . 47 0 0
0 0 14 . 47 14 . 47 0 0
0 0 0 0 100 0
0 0 0 0 0 0
0 0
0 100
14 . 47 14 . 47
14 . 47 14 . 47
100 0
0 0
10
2
1 3
8
7
6
5
4
3
u
u
p
p
p
p
p
p
3
4
5
6
7
8
0
77.63
77.63
77.63
77.63
0
p
p
p
p
p
p
??
??
??
??
-
??
??
??
??
?? ? ?
=
?? ? ?
?? ? ?
?? ? ?
-
?? ? ?
??
????
??
kN (12)
The support reactions are shown in Fig.26.2c.The member forces can be easily
calculated from reactions. The member end forces can also be calculated by
using equation (26.10a) and (26.10b). For example, for member (1),
o
0 = ?
100 10
3 '
2
× = p [] 0 1 0 1 - 4
4
0
0
7.763 10
7.763 10
-
-
? ?
? ?
? ?
? ?
×
? ?
? ?
×
??
(13)
= 77.763 kN. Thus the member (1) is in tension.
Member (2)
o
45 = ?
281 . 94 10
3 '
2
× = p [-0.707 -0.707 0.707 0.707]
?
?
?
?
?
?
?
?
?
?
?
?
?
?
×
×
-
-
3
3
10 2942 . 3
10 2942 . 3
0
0
kN. 78 . 109
2
- = ' p
Thus member (2) is in compression
Page 4
411
3
33
0000
01 0 1
15 10 2 10
0000 3 x 10 10
010 1
k
-
? ?
? ?
-
×××
? ?
?? =
??
? ? ×
? ?
-
? ?
(7)
The global stiffness matrix is of the order 8 8 × ,assembling the three member
stiffness matrices, one gets
[]
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
-
- -
- -
-
- - -
- - -
=
0 0 0 0 0 0 0 0
0 100 0 0 0 0 0 100
0 0 14 . 47 14 . 47 0 0 14 . 47 14 . 47
0 0 14 . 47 14 . 47 0 0 14 . 47 14 . 47
0 0 0 0 100 0 100 0
0 0 0 0 0 0 0 0
0 0 14 . 47 14 . 47 100 0 14 . 147 14 . 47
0 100 14 . 47 14 . 47 0 0 14 . 47 14 . 147
10
3
k (8)
Writing the load displacement equation for the truss
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
8
7
6
5
4
3
2
1
p
p
p
p
p
p
p
p
=
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
-
- -
- -
-
- - -
- - -
0 0 0 0 0 0 0 0
0 100 0 0 0 0 0 100
0 0 14 . 47 14 . 47 0 0 14 . 47 14 . 47
0 0 14 . 47 14 . 47 0 0 14 . 47 14 . 47
0 0 0 0 100 0 100 0
0 0 0 0 0 0 0 0
0 0 14 . 47 14 . 47 100 0 14 . 147 14 . 47
0 100 14 . 47 14 . 47 0 0 14 . 47 14 . 147
10
3
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
8
7
6
5
4
3
2
1
u
u
u
u
u
u
u
u
+
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
-
-
0
0
1
1
0
0
1
1
640
(9)
In the present case, the displacements
1
u and
2
u are not known. All other
displacements are zero. Also 0
2 1
= = p p (as no joint loads are applied).Thus,
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
8
7
6
5
4
3
2
1
p
p
p
p
p
p
p
p
=
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
-
- -
- -
-
- - -
- - -
=
0 0 0 0 0 0 0 0
0 100 0 0 0 0 0 100
0 0 14 . 47 14 . 47 0 0 14 . 47 14 . 47
0 0 14 . 47 14 . 47 0 0 14 . 47 14 . 47
0 0 0 0 100 0 100 0
0 0 0 0 0 0 0 0
0 0 14 . 47 14 . 47 100 0 14 . 147 14 . 47
0 100 14 . 47 14 . 47 0 0 14 . 47 14 . 147
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
8
7
6
5
4
3
2
1
u
u
u
u
u
u
u
u
+
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
-
-
0
0
1
1
0
0
1
1
640 (10)
Thus unknown displacements are
1
1
3
2
147.14 47.14 0 1
1
(150.82 )
47.14 147.14 0 1 10
u
u
-
- ?????? ??
=-
?? ? ? ? ?
??
-
???? ?? ??
(11)
4
1
4
2
7.763 10
7.763 10
um
um
-
-
=×
=×
Now reactions are calculated as
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
+
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
+
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
-
- -
- -
-
=
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
0
0
1
1
0
0
640
0
0
0
0
0
0
0 0 0 0 0 0
0 100 0 0 0 0
0 0 14 . 47 14 . 47 0 0
0 0 14 . 47 14 . 47 0 0
0 0 0 0 100 0
0 0 0 0 0 0
0 0
0 100
14 . 47 14 . 47
14 . 47 14 . 47
100 0
0 0
10
2
1 3
8
7
6
5
4
3
u
u
p
p
p
p
p
p
3
4
5
6
7
8
0
77.63
77.63
77.63
77.63
0
p
p
p
p
p
p
??
??
??
??
-
??
??
??
??
?? ? ?
=
?? ? ?
?? ? ?
?? ? ?
-
?? ? ?
??
????
??
kN (12)
The support reactions are shown in Fig.26.2c.The member forces can be easily
calculated from reactions. The member end forces can also be calculated by
using equation (26.10a) and (26.10b). For example, for member (1),
o
0 = ?
100 10
3 '
2
× = p [] 0 1 0 1 - 4
4
0
0
7.763 10
7.763 10
-
-
? ?
? ?
? ?
? ?
×
? ?
? ?
×
??
(13)
= 77.763 kN. Thus the member (1) is in tension.
Member (2)
o
45 = ?
281 . 94 10
3 '
2
× = p [-0.707 -0.707 0.707 0.707]
?
?
?
?
?
?
?
?
?
?
?
?
?
?
×
×
-
-
3
3
10 2942 . 3
10 2942 . 3
0
0
kN. 78 . 109
2
- = ' p
Thus member (2) is in compression
Example 26.2
Analyze the truss shown in Fig.26.3a, if the member BC is made 0.01m too short
before placing it in the truss. Assume AE=300 kN for all members.
Page 5
411
3
33
0000
01 0 1
15 10 2 10
0000 3 x 10 10
010 1
k
-
? ?
? ?
-
×××
? ?
?? =
??
? ? ×
? ?
-
? ?
(7)
The global stiffness matrix is of the order 8 8 × ,assembling the three member
stiffness matrices, one gets
[]
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
-
- -
- -
-
- - -
- - -
=
0 0 0 0 0 0 0 0
0 100 0 0 0 0 0 100
0 0 14 . 47 14 . 47 0 0 14 . 47 14 . 47
0 0 14 . 47 14 . 47 0 0 14 . 47 14 . 47
0 0 0 0 100 0 100 0
0 0 0 0 0 0 0 0
0 0 14 . 47 14 . 47 100 0 14 . 147 14 . 47
0 100 14 . 47 14 . 47 0 0 14 . 47 14 . 147
10
3
k (8)
Writing the load displacement equation for the truss
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
8
7
6
5
4
3
2
1
p
p
p
p
p
p
p
p
=
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
-
- -
- -
-
- - -
- - -
0 0 0 0 0 0 0 0
0 100 0 0 0 0 0 100
0 0 14 . 47 14 . 47 0 0 14 . 47 14 . 47
0 0 14 . 47 14 . 47 0 0 14 . 47 14 . 47
0 0 0 0 100 0 100 0
0 0 0 0 0 0 0 0
0 0 14 . 47 14 . 47 100 0 14 . 147 14 . 47
0 100 14 . 47 14 . 47 0 0 14 . 47 14 . 147
10
3
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
8
7
6
5
4
3
2
1
u
u
u
u
u
u
u
u
+
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
-
-
0
0
1
1
0
0
1
1
640
(9)
In the present case, the displacements
1
u and
2
u are not known. All other
displacements are zero. Also 0
2 1
= = p p (as no joint loads are applied).Thus,
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
8
7
6
5
4
3
2
1
p
p
p
p
p
p
p
p
=
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
-
- -
- -
-
- - -
- - -
=
0 0 0 0 0 0 0 0
0 100 0 0 0 0 0 100
0 0 14 . 47 14 . 47 0 0 14 . 47 14 . 47
0 0 14 . 47 14 . 47 0 0 14 . 47 14 . 47
0 0 0 0 100 0 100 0
0 0 0 0 0 0 0 0
0 0 14 . 47 14 . 47 100 0 14 . 147 14 . 47
0 100 14 . 47 14 . 47 0 0 14 . 47 14 . 147
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
8
7
6
5
4
3
2
1
u
u
u
u
u
u
u
u
+
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
-
-
0
0
1
1
0
0
1
1
640 (10)
Thus unknown displacements are
1
1
3
2
147.14 47.14 0 1
1
(150.82 )
47.14 147.14 0 1 10
u
u
-
- ?????? ??
=-
?? ? ? ? ?
??
-
???? ?? ??
(11)
4
1
4
2
7.763 10
7.763 10
um
um
-
-
=×
=×
Now reactions are calculated as
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
+
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
+
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
-
- -
- -
-
=
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
0
0
1
1
0
0
640
0
0
0
0
0
0
0 0 0 0 0 0
0 100 0 0 0 0
0 0 14 . 47 14 . 47 0 0
0 0 14 . 47 14 . 47 0 0
0 0 0 0 100 0
0 0 0 0 0 0
0 0
0 100
14 . 47 14 . 47
14 . 47 14 . 47
100 0
0 0
10
2
1 3
8
7
6
5
4
3
u
u
p
p
p
p
p
p
3
4
5
6
7
8
0
77.63
77.63
77.63
77.63
0
p
p
p
p
p
p
??
??
??
??
-
??
??
??
??
?? ? ?
=
?? ? ?
?? ? ?
?? ? ?
-
?? ? ?
??
????
??
kN (12)
The support reactions are shown in Fig.26.2c.The member forces can be easily
calculated from reactions. The member end forces can also be calculated by
using equation (26.10a) and (26.10b). For example, for member (1),
o
0 = ?
100 10
3 '
2
× = p [] 0 1 0 1 - 4
4
0
0
7.763 10
7.763 10
-
-
? ?
? ?
? ?
? ?
×
? ?
? ?
×
??
(13)
= 77.763 kN. Thus the member (1) is in tension.
Member (2)
o
45 = ?
281 . 94 10
3 '
2
× = p [-0.707 -0.707 0.707 0.707]
?
?
?
?
?
?
?
?
?
?
?
?
?
?
×
×
-
-
3
3
10 2942 . 3
10 2942 . 3
0
0
kN. 78 . 109
2
- = ' p
Thus member (2) is in compression
Example 26.2
Analyze the truss shown in Fig.26.3a, if the member BC is made 0.01m too short
before placing it in the truss. Assume AE=300 kN for all members.
Solution
A similar truss with different boundary conditions has already been solved in
example 25.1. For the sake of completeness the member of nodes and members
are shown in Fig.26.3b.The displacements
6 5 4 3
, , , u u u u ,
7
u and
8
u are zero due to
boundary conditions. For the present problem the unconstrained degrees of
freedom are
1
u and
2
u .The assembled stiffness matrix is of the order 8 8 × and is
available in example 25.1.
In the given problem the member (2) is short by 0.01m.The forces developed in
member (2) in the global coordinate system due to fabrication error is
()
()
()
()
()
?
?
?
?
?
?
?
?
?
?
?
?
?
?
-
-
-
=
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
sin
cos
sin
cos
4
01 . 0
0 2
0 1
0 4
0 3
AE
p
p
p
p
=
?
?
?
?
?
?
?
?
?
?
?
?
?
?
-
75 . 0
0
75 . 0
0
kN (1)
Now force-displacement relations for the truss are
Read More
|
34 videos|140 docs|31 tests
|
FAQs on The Direct Stiffness Method: Temperature Changes & Fabrication Errors in Truss Analysis - 2 - Structural Analysis - Civil Engineering (CE)
| 1. How does temperature changes affect truss analysis using the Direct Stiffness Method? |  |
Ans. Temperature changes can cause thermal expansion or contraction of the truss elements, leading to changes in their lengths. This can result in changes in the internal forces and displacements of the truss members, which need to be considered in the analysis using the Direct Stiffness Method. The temperature effects can be incorporated by modifying the stiffness matrix of the truss elements based on the coefficient of thermal expansion and the temperature change.
| 2. What are fabrication errors and how do they impact truss analysis using the Direct Stiffness Method? | |
Ans. Fabrication errors refer to the deviations or inaccuracies in the physical construction of the truss members compared to their ideal design. These errors can occur due to manufacturing tolerances, welding or assembly imperfections, or any other factors that affect the actual dimensions and alignment of the truss elements. Fabrication errors can lead to variations in the stiffness and geometry of the truss, which can affect the calculated internal forces and displacements using the Direct Stiffness Method.
| 3. How can temperature changes be accounted for in the Direct Stiffness Method analysis of trusses? | |
Ans. Temperature changes can be accounted for in the Direct Stiffness Method analysis of trusses by introducing additional terms in the stiffness matrix of the truss elements. These additional terms represent the thermal expansion or contraction effects and are determined based on the coefficient of thermal expansion and the temperature change. By including these temperature-related terms in the stiffness matrix, the analysis can accurately capture the changes in the truss behavior due to temperature variations.
| 4. What are the potential challenges in incorporating temperature changes and fabrication errors in truss analysis using the Direct Stiffness Method? | |
Ans. Incorporating temperature changes and fabrication errors in truss analysis using the Direct Stiffness Method can pose several challenges. Firstly, accurately determining the coefficient of thermal expansion and the temperature change can be difficult, especially in real-world scenarios. Secondly, quantifying the extent and nature of fabrication errors can be challenging, as they can vary for different truss members. Additionally, modifying the stiffness matrix to account for temperature and fabrication effects can increase the complexity of the analysis, requiring additional computational efforts.
| 5. How can the Direct Stiffness Method be used to analyze trusses with temperature changes and fabrication errors? | |
Ans. The Direct Stiffness Method can be used to analyze trusses with temperature changes and fabrication errors by considering the effects of these factors in the stiffness matrix of the truss elements. By accurately incorporating the thermal expansion or contraction effects and accounting for fabrication errors, the modified stiffness matrix can provide more realistic results for the internal forces and displacements of the truss members. This allows engineers to assess the structural behavior of trusses under various temperature conditions and fabrication tolerances, aiding in the design and evaluation process.
Related Exams
About this Document
|
4.98/5 Rating |
|
Nov 22, 2024 Last updated |
Document Description: The Direct Stiffness Method: Temperature Changes & Fabrication Errors in Truss Analysis - 2 for Civil Engineering (CE) 2024 is part of Structural Analysis preparation.
The notes and questions for The Direct Stiffness Method: Temperature Changes & Fabrication Errors in Truss Analysis - 2 have been prepared according to the Civil Engineering (CE) exam syllabus. Information about The Direct Stiffness Method: Temperature Changes & Fabrication Errors in Truss Analysis - 2 covers topics
like and The Direct Stiffness Method: Temperature Changes & Fabrication Errors in Truss Analysis - 2 Example, for Civil Engineering (CE) 2024 Exam. Find important definitions, questions, notes, meanings, examples, exercises and tests below for The Direct Stiffness Method: Temperature Changes & Fabrication Errors in Truss Analysis - 2.
Introduction of The Direct Stiffness Method: Temperature Changes & Fabrication Errors in Truss Analysis - 2 in English is available as part of our Structural Analysis
for Civil Engineering (CE) & The Direct Stiffness Method: Temperature Changes & Fabrication Errors in Truss Analysis - 2 in Hindi for Structural Analysis course.
Download more important topics related with notes, lectures and mock test series for Civil Engineering (CE)
Exam by signing up for free. Civil Engineering (CE): The Direct Stiffness Method: Temperature Changes & Fabrication Errors in Truss Analysis - 2 | Structural Analysis - Civil Engineering (CE)
Description
Full syllabus notes, lecture & questions for The Direct Stiffness Method: Temperature Changes & Fabrication Errors in Truss Analysis - 2 | Structural Analysis - Civil Engineering (CE) - Civil Engineering (CE) | Plus excerises question with solution to help you revise complete syllabus for Structural Analysis | Best notes, free PDF download
Information about The Direct Stiffness Method: Temperature Changes & Fabrication Errors in Truss Analysis - 2
In this doc you can find the meaning of The Direct Stiffness Method: Temperature Changes & Fabrication Errors in Truss Analysis - 2 defined & explained in the simplest way possible. Besides explaining types of
The Direct Stiffness Method: Temperature Changes & Fabrication Errors in Truss Analysis - 2 theory, EduRev gives you an ample number of questions to practice The Direct Stiffness Method: Temperature Changes & Fabrication Errors in Truss Analysis - 2 tests, examples and also practice Civil Engineering (CE)
tests
|
Explore Courses for Civil Engineering (CE) exam
|
|
Signup for Free!
Signup to see your scores go up within 7 days! Learn & Practice with 1000+ FREE Notes, Videos & Tests.
Related Searches
Semester Notes
,MCQs
,The Direct Stiffness Method: Temperature Changes & Fabrication Errors in Truss Analysis - 2 | Structural Analysis - Civil Engineering (CE)
,study material
,The Direct Stiffness Method: Temperature Changes & Fabrication Errors in Truss Analysis - 2 | Structural Analysis - Civil Engineering (CE)
,Previous Year Questions with Solutions
,Objective type Questions
,Important questions
,ppt
,Exam
,Summary
,The Direct Stiffness Method: Temperature Changes & Fabrication Errors in Truss Analysis - 2 | Structural Analysis - Civil Engineering (CE)
,Free
,video lectures
,mock tests for examination
,shortcuts and tricks
,past year papers
,Viva Questions
,Extra Questions
,practice quizzes
,Sample Paper
;
Additional Information about The Direct Stiffness Method: Temperature Changes & Fabrication Errors in Truss Analysis - 2 for Civil Engineering (CE) Preparation
The Direct Stiffness Method: Temperature Changes & Fabrication Errors in Truss Analysis - 2 Free PDF Download
The The Direct Stiffness Method: Temperature Changes & Fabrication Errors in Truss Analysis - 2 is an invaluable resource that delves deep into the core of the Civil Engineering (CE) exam.
These study notes are curated by experts and cover all the essential topics and concepts, making your preparation more efficient and effective.
With the help of these notes, you can grasp complex subjects quickly, revise important points easily,
and reinforce your understanding of key concepts. The study notes are presented in a concise and easy-to-understand manner,
allowing you to optimize your learning process. Whether you're looking for best-recommended books, sample papers, study material,
or toppers' notes, this PDF has got you covered. Download the The Direct Stiffness Method: Temperature Changes & Fabrication Errors in Truss Analysis - 2 now and kickstart your journey towards success in the Civil Engineering (CE) exam.
Importance of The Direct Stiffness Method: Temperature Changes & Fabrication Errors in Truss Analysis - 2
The importance of The Direct Stiffness Method: Temperature Changes & Fabrication Errors in Truss Analysis - 2 cannot be overstated, especially for Civil Engineering (CE) aspirants.
This document holds the key to success in the Civil Engineering (CE) exam.
It offers a detailed understanding of the concept, providing invaluable insights into the topic.
By knowing the concepts well in advance, students can plan their preparation effectively.
Utilize this indispensable guide for a well-rounded preparation and achieve your desired results.
The Direct Stiffness Method: Temperature Changes & Fabrication Errors in Truss Analysis - 2 Notes
The Direct Stiffness Method: Temperature Changes & Fabrication Errors in Truss Analysis - 2 Notes offer in-depth insights into the specific topic to help you master it with ease.
This comprehensive document covers all aspects related to The Direct Stiffness Method: Temperature Changes & Fabrication Errors in Truss Analysis - 2.
It includes detailed information about the exam syllabus, recommended books, and study materials for a well-rounded preparation.
Practice papers and question papers enable you to assess your progress effectively.
Additionally, the paper analysis provides valuable tips for tackling the exam strategically.
Access to Toppers' notes gives you an edge in understanding complex concepts.
Whether you're a beginner or aiming for advanced proficiency, The Direct Stiffness Method: Temperature Changes & Fabrication Errors in Truss Analysis - 2 Notes on EduRev are your ultimate resource for success.
The Direct Stiffness Method: Temperature Changes & Fabrication Errors in Truss Analysis - 2 Civil Engineering (CE) Questions
The "The Direct Stiffness Method: Temperature Changes & Fabrication Errors in Truss Analysis - 2 Civil Engineering (CE) Questions" guide is a valuable resource for all aspiring students preparing for the
Civil Engineering (CE) exam. It focuses on providing a wide range of practice questions to help students gauge
their understanding of the exam topics. These questions cover the entire syllabus, ensuring comprehensive preparation.
The guide includes previous years' question papers for students to familiarize themselves with the exam's format and difficulty level.
Additionally, it offers subject-specific question banks, allowing students to focus on weak areas and improve their performance.
Study The Direct Stiffness Method: Temperature Changes & Fabrication Errors in Truss Analysis - 2 on the App
Students of Civil Engineering (CE) can study The Direct Stiffness Method: Temperature Changes & Fabrication Errors in Truss Analysis - 2 alongwith tests & analysis from the EduRev app,
which will help them while preparing for their exam. Apart from the The Direct Stiffness Method: Temperature Changes & Fabrication Errors in Truss Analysis - 2,
students can also utilize the EduRev App for other study materials such as previous year question papers, syllabus, important questions, etc.
The EduRev App will make your learning easier as you can access it from anywhere you want.
The content of The Direct Stiffness Method: Temperature Changes & Fabrication Errors in Truss Analysis - 2 is prepared as per the latest Civil Engineering (CE) syllabus.
|
© EduRev
|
Education Revolution
|
Follow Us
|
Signup to see your scores
go up within 7 days!
Access 1000+ FREE Docs, Videos and Tests
Takes less than 10 seconds to signup