Transcript Slide 1

‫المحاضرة الرابعة عشر‬
‫‪1‬‬
‫‪Lecture No. : 14‬‬
2
3
Example 1:
Calculate the deformations of the shown frame
where E = 106 kN/m2, and the section of all members
is rectangular with dimension 30 cm x 80 cm.
30 kN/m
A
8
B
120 kN
120 kN
D
8
C
4
d2
d3
d5
d6
d1
d4
B
A
D
8
C
5
Stiffness matrix
K=
30,300
0
1,200
-30,000
0
0
0
30,300
1,200
0
-300
1,200
1,200
1,200
12,800
0
-1,200
3,200
-30,000
0
0
30,300
0
1,200
0
-300
-1,200
0
30,300
-1,200
0
1,200
3,200
1,200
-1,200
12,800
6
Force vector
30 kN/m
A
B
120 kN
120 kN
8
D
30x82
12 160 kNm
30 kN/m
A
C
8
160 kNm
B
120 kN
120 kN
8
7
Force vector
30 kN/m
A
120x8
8
60 kN
120 kNm
B
120 kN
120 kN
8
D
B
C
8
120 kN
8
60 kN
C
120 kNm
120x8
8
8
Force vector
30 kN/m
A
120 kNm
120x8
8
A
8
B
120 kN
120 kN
8
D
60 kN
C
8
120 kN
D
60 kN
120 kNm
120x8
8
9
Force vector
160 kNm
A
Fixed End Reaction (FER)
160 kNm
30 kN/m
B
120 kNm
120 kNm
120 kN
A
60 kN
120 kN
60 kN
B
120 kN
120 kN
D
120 kNm
60 kN
60 kN
C
10
120 kNm
Force vector
Fixed End Action (FEA)
160 kNm
30 kN/m
160 kNm
A
B
120 kNm
120 kNm
120 kN
A
120 kN
60 kN
60 kN
d3
120 kN
d2
d1
A
D
120 kNm
60 kN
d6
B
d5
120 kN
d4
B
D
C
60 kN
C
11
120 kNm
Force vector
160 kNm
Fixed End Action (FEA)
30 kN/m
A
B
120 kN
d3
A
60 kN
B
120 kN
60 kN
C
120 kNm
C
F1
60
F2
-120
F3
- 40
F4
120 kNm
d4
B
D
120 kNm
120
D 60 kN
d6
d1
d5
120 kN
120 kNm
A 60 kN
160 kNm
d2
=
- 60
F5
-120
F6
40
12
Stiffness Equation
F= K D
60
30,300
-120
0
- 40
- 60
=
0
30,300 1,200
1,200 1,200 12,800
-30,000 0
0
0
d1
0
-300
1,200
d2
0
-1,200 3,200
d3
1,200 -30,000
0
30,300
-120
0
-300 -1,200
40
0
1,200 3,200 1,200
0
1,200
d4
30,300 -1,200
d5
-1,200 12,800
d6
0
13
Stiffness Equation
D = K-1 F
-1
d1
30,300
d2
0
d3
d4
=
0
1,200 -30,000
30,300 1,200
1,200 1,200 12,800
-30,000 0
0
0
0
60
0
-300
1,200
-120
0
-1,200 3,200
- 40
30,300
d5
0
-300 -1,200
d6
0
1,200 3,200 1,200
0
0
1,200
- 60
30,300 -1,200
-120
-1,200 12,800
40
14
Deformations
d1
0.0010806
d2
- 0.004
d3
- 0.0043017
d4
=
- 0.0010806
d5
- 0.004
d6
0.0043017
15
Another Solution
30 kN/m
A
8
B
120 kN
120 kN
D
8
C
16
Another Solution
d3
d2
d1
d2
d1
d3
B
A
k11
k12
k13
K = k21 k22 k23
k31 k32 k33
D
8
C
17
First column in Stiffness matrix
d1 =1
d1
d1
18
First column in Stiffness matrix
2EA
1 6 EI L
1,200
L2
2E A
L
2
60,000
60,000
300
12 EI
L3
EA/L = 30,000
2 EI/L = 3,200
4 EI/L = 6,400
6 EI/L2 = 1,200
12EI/L3= 300
19
First column in Stiffness matrix
60,000
60,000
1,200
300
d2
d3
d1
k11
k21 =
k31
60,300
0
1,200
20
Second column in Stiffness matrix
d2 =1
d2
d2
21
Second column in Stiffness matrix
EA/L = 30,000
EA
30,000 L
1
d2
d3
d1
k12
0
k22 = 30,000
k32
0
22
Third column in Stiffness matrix
d3 =1
d3
d3
23
Third column in Stiffness matrix
4 EI
6,400 L
1
1,200
6 EI
L2
a
24
Third column in Stiffness matrix
d3 =1
d3
d3
25
1
4 EI
L
2 EI
L
1
4 EI
L
2 EI
L
1
2 EI
L
1
2 EI
L
26
Third column in Stiffness matrix
1
4 EI
6,400 L
1,200
2 EI
L
1
3,200
2 EI 1
L
3,200
6 EI
L2
EA/L = 30,000
2 EI/L = 3,200
4 EI/L = 6,400
6 EI/L2 = 1,200
12EI/L3= 300
27
Third column in Stiffness matrix
4 EI
6,400 L
2 EI
L
1,200
6 EI
L2
2 EI
L
3,200
3,200
d2
d3
d1
k13
k23 =
k33
1,200
0
9,600
28
Stiffness matrix
k11
k21 =
k31
60,300
0
1,200
K=
k12
k13
0
k22 = 30,000
k23 =
k32
k33
0
60,300
0
1,200
0
30,000
0
1,200
0
9,600
1,200
0
9,600
29
Force vector
30 kN/m
A
B
120 kN
120 kN
8
D
30x82
12 160 kNm
30 kN/m
A
C
8
160 kNm
B
120 kN
120 kN
8
30
Force vector
30 kN/m
A
120 kNm
120x8
8
B
8
B
120 kN
120 kN
8
D
60 kN
C
8
120 kN
C
60 kN
120 kNm
120x8
8
31
Force vector
160 kNm
A
Fixed End Reaction (FER)
160 kNm
30 kN/m
B
120 kNm
120 kN
120 kN
B
B
60 kN
120 kN
120 kN
C
120 kNm
60 kN
C
32
Force vector
160 kNm
A
Fixed End Action (FEA)
160 kNm
30 kN/m
B
120 kNm
120 kN
B
120 kN
B
60 kN
120 kN
120 kN
C
C
120 kNm
60 kN
33
Force vector
Fixed End Action (FEA)
d2
160 kNm
30 kN/m
A
160 kNm
d3
d1
B
120 kN
120 kN
120 kNm
B
60 kN
F1
B
F2 = -120
120 kN
120
60 kN
C
120 kNm
60
F3
- 40
C
34
Stiffness Equation
F= K D
60
-120
- 40
60,300
0
1,200
d1
0
30,000
0
d2
1,200
0
9,600
d3
D=
-1
K F
=
-1
d1
d2
d3
=
60,300
0
1,200
60
0
30,000
0
-120
1,200
0
9,600
- 40
35
Deformations
d1
d2 =
d3
0.0010806
- 0.004
- 0.0043017
d1
0.0010806
d2
- 0.004
d3
- 0.0043017
d4
=
- 0.0010806
d5
- 0.004
d6
0.0043017
36
Example 2:
Calculate the deformations of the shown frame
where E = 106 kN/m2, and the section of all members
is rectangular with dimension 30 cm x 80 cm.
A
8
B
120 kN
120 kN
D
8
C
37
d2
d3
d5
d6
d1
d4
B
A
D
8
C
38
Stiffness matrix
K=
30,300
0
1,200
-30,000
0
0
0
30,300
1,200
0
-300
1,200
1,200
1,200
12,800
0
-1,200
3,200
-30,000
0
0
30,300
0
1,200
0
-300
-1,200
0
30,300
-1,200
0
1,200
3,200
1,200
-1,200
12,800
39
Force vector
120 kNm
A
120x8
8
B
8
B
120 kN
120 kN
8
D
C
8
60 kN
120 kN
C
60 kN
120 kNm
120x8
8
40
A
Force vector
B
120 kN
120 kNm
120x8
8
A
8
120 kN
8
D
C
8
60 kN
120 kN
D
60 kN
120 kNm
120x8
8
41
Force vector
A
120 kNm
A
B
120 kNm
B
60 kN
60 kN
120 kN
120 kN
D
120 kNm
Fixed End Reaction (FER)
60 kN
C
120 kNm
60 kN
42
Force vector
Fixed End Action (FEA)
A
B
120 kNm
120 kNm
B
B
60 kN
d3
120 kN
d2
d1
A
C
120 kNm
60 kN
d6
d5
d4 120
60 kN
kN
B
D
C
C
120 kNm
60 kN
43
Force vector
Fixed End Action (FEA)
d3
A
B
B
60 kN
60 kN
D60 kN
120 kNm
C
120 kNm
60 kN
d4
C
F1
60
F2
0
F3
120
F4
d5
B
D
120 kN
120
d6
d1
A
120 kNm
120 kNm
A
d2
=
60
F5
0
F6
120
44
Stiffness Equation
F= K D
60
30,300
0
0
120
60
=
0
30,300 1,200
1,200 1,200 12,800
-30,000 0
0
0
d1
0
-300
1,200
d2
0
-1,200 3,200
d3
1,200 -30,000
0
30,300
0
0
-300 -1,200
120
0
1,200 3,200 1,200
0
1,200
d4
30,300 -1,200
d5
-1,200 12,800
d6
0
45
Stiffness Equation
D = K-1 F
-1
d1
30,300
d2
0
d3
d4
=
0
1,200 -30,000
30,300 1,200
1,200 1,200 12,800
-30,000 0
0
0
0
60
0
-300
1,200
0
0
-1,200 3,200
30,300
d5
0
-300 -1,200
d6
0
1,200 3,200 1,200
0
0
120
1,200
60
30,300 -1,200
0
-1,200 12,800
120
46
Deformations
d1
0.2436
d2
0.009
d3
- 0.0109
d4
=
0.2436
d5
- 0.009
d6
- 0.0109
47
Another Solution
A
8
B
120 kN
120 kN
D
8
C
48
Another Solution
d3
d2
d1
d3
k12
d1
B
A
k11
d2
k13
K = k21 k22 k23
k31 k32 k33
D
8
C
49
First column in Stiffness matrix
d1 =1
d1
d1
50
First column in Stiffness matrix
1 6 EI
1,200
L2
300
12 EI
L3
EA/L = 30,000
2 EI/L = 3,200
4 EI/L = 6,400
6 EI/L2 = 1,200
12EI/L3= 300
51
First column in Stiffness matrix
1,200
300
d2
d3
d1
k11
k21 =
k31
300
0
1,200
52
Second column in Stiffness matrix
d2 =1
d2
d2
53
Second column in Stiffness matrix
E A 2,400
30,000 L
1
12 EI
L2
2
12 EI 24 EI
3
2
L
L
600
2,400
24 EI
L3 600
b
c
d2
d3
d1
k12
0
k22 = 30,600
k32
2,400
54
Third column in Stiffness matrix
d3 =1
d3
d3
55
Third column in Stiffness matrix
4 EI
6,400 L
1
1,200
6 EI
L2
a
56
Third column in Stiffness matrix
d3 =1
d3
d3
57
1
4 EI
L
2 EI
L
1
4 EI
L
2 EI
L
1
6 EI
L
1
6 EI
L
58
Third column in Stiffness matrix
1
4 EI
6,400 L
1,200
6 EI
L2
6 EI
L
1
9,600
6 EI
L
9,600
2,400
2,400
EA/L = 30,000
2 EI/L = 3,200
4 EI/L = 6,400
6 EI/L2 = 1,200
12EI/L3= 300
59
Third column in Stiffness matrix
4 EI
6,400 L
9,600
1,200
6 EI
L2
2,400
2,400
d2
d3
9,600
d1
k13
1,200
k23 =
2,400
k33
16,000
60
Stiffness matrix
k11
k21 =
k31
300
0
1,200
K=
k12
0
k22 = 30,600
k32
2,400
300
0
1,200
0
30,600
2,400
1,200
k13
1,200
k23 =
2,400
k33
16,000
2,400 16,000
61
Force vector
120 kNm
A
120x8
8
B
8
B
120 kN
120 kN
8
D
C
8
60 kN
120 kN
C
60 kN
120 kNm
120x8
8
62
A
Force vector
B
120 kN
120 kNm
120x8
8
A
8
120 kN
8
D
C
8
60 kN
120 kN
D
60 kN
120 kNm
120x8
8
63
Force vector
A
120 kNm
A
B
120 kNm
B
60 kN
60 kN
120 kN
120 kN
D
120 kNm
Fixed End Reaction (FER)
60 kN
C
120 kNm
60 kN
64
Force vector
Fixed End Action (FEA)
A
B
120 kNm
120 kNm
B
B
60 kN
d2
d1
d3
120 kN
d2
d3
C
120 kNm
kN
B
A
60 kN
d120
1
60 kN
D
C
C
120 kNm
60 kN
65
Force vector
Fixed End Action (FEA)
d2
d2
d1
d3
d3
B
A
A
B
C
D
120 kNm
120 kNm
A
B
60 kN
60 kN
F1
120 kN
120
F2 =
F3
D60 kN
120 kNm
d1
C
120 kNm
60
0
120
60 kN
66
Stiffness Equation
F= K D
60
300
0
1,200
d1
0
30,600
2,400
d2
2,400 16,000
d3
=
0
120
1,200
D=
-1
K F
-1
d1
d2
d3
=
300
0
1,200
60
0
30,600
2,400
0
2,400 16,000
120
1,200
67
Deformations
d1
0.2436
d2
0.009
d3
- 0.0109
d1
0.2436
d2
0.009
d4
d3
- 0.0109
d5
- 0.009
d6
- 0.0109
=
0.2436
68
69
A
B
50 kN
D
8
C
70
d2
d3
d5
d6
d1
d4
B
A
D
8
C
71
Another solution
d1
d1
B
A
D
8
C
72
First column in Stiffness matrix
c
b
30,000
1,200
d2
d3
300
d5
d1
b
a
30,000
d6
c
d
d4
k11
30,300
k21
0
k31
1,200
k41
=
c
-30,000
k51
0
k61
0
d
73
Another solution
d2
d3
d1
d1
B
A
D
8
C
74
Example 3:
Calculate the deformations of the shown frame
where E = 106 kN/m2, and the section of all members
is rectangular with dimension 30 cm x 80 cm.
30 kN/m
A
8
B
60 kN
120 kN
D
8
C
75
d2
d3
d5
d6
d1
d4
B
A
D
8
C
76
Stiffness matrix
K=
30,300
0
1,200
-30,000
0
0
0
30,300
1,200
0
-300
1,200
1,200
1,200
12,800
0
-1,200
3,200
-30,000
0
0
30,300
0
1,200
0
-300
-1,200
0
30,300
-1,200
0
1,200
3,200
1,200
-1,200
12,800
77
Force vector
30 kN/m
A
B
120 kN
120 kN
8
D
30x82
12 160 kNm
30 kN/m
A
C
8
160 kNm
B
120 kN
120 kN
8
78
Force vector
30 kN/m
A
120 kNm
120x8
8
A
8
B
120 kN
120 kN
8
D
60 kN
C
8
120 kN
D
60 kN
120 kNm
120x8
8
79
Force vector
30 kN/m
A
60x8
8
30 kN
60 kNm
B
120 kN
120 kN
8
D
B
C
8
60 kN
8
30 kN
C
60 kNm
60x8
8
80
Force vector
160 kNm
A
Fixed End Reaction (FER)
160 kNm
30 kN/m
B
120 kNm
120 kN
A
60 kN
60 kNm
120 kN
30 kN
B
120 kN
60 kN
C
120 kNm
60 kN
30 kN
C
60 kNm
81
Force vector
Fixed End Action (FEA)
160 kNm
30 kN/m
160 kNm
A
B
120 kNm
120 kN
A
60 kN
30 kN
d3
120 kN
d2
d1
A
D
120 kNm
60 kN
60 kNm
120 kN
d6
B
d5
d4
60 kN
B
D
C
30 kN
C
60 kNm
82
Force vector
160 kNm
Fixed End Action (FEA)
30 kN/m
A
B
120 kN
d3
A
B
30 kN
60 kN
120
D
120 kNm
30 kN
C
60 kNm
C
F1
60
F2
-120
F3
- 40
F4
60 kN
d4
B
D
60 kNm
60 kN
d6
d1
d5
120 kN
120 kNm
A
160 kNm
d2
=
- 30
F5
-120
F6
100
83
Stiffness Equation
F= K D
60
30,300
-120
0
- 40
- 30
=
0
30,300 1,200
1,200 1,200 12,800
-30,000 0
0
0
d1
0
-300
1,200
d2
0
-1,200 3,200
d3
1,200 -30,000
0
30,300
-120
0
-300 -1,200
100
0
1,200 3,200 1,200
0
1,200
d4
30,300 -1,200
d5
-1,200 12,800
d6
0
84
Stiffness Equation
D = K-1 F
-1
d1
30,300
d2
0
d3
d4
=
0
1,200 -30,000
30,300 1,200
1,200 1,200 12,800
-30,000 0
0
0
0
60
0
-300
1,200
-120
0
-1,200 3,200
- 40
30,300
d5
0
-300 -1,200
d6
0
1,200 3,200 1,200
0
0
1,200
- 30
30,300 -1,200
-120
-1,200 12,800
100
85
Deformations
d1
0.0618
d2
- 0.0038
d3
- 0.0101
d4
=
0.0600
d5
- 0.0042
d6
0.0047
86
Another solution
d2
d3
d1
B
A
k11
k12
d1
k13
K = k21 k22 k23
k31 k32 k33
D
8
C
87
First column in Stiffness matrix
d1 =1
d1
d1
88
First column in Stiffness matrix
1 6 EI
1,200
L2
1,200
300
12 EI
L3
300
EA/L = 30,000
2 EI/L = 3,200
4 EI/L = 6,400
6 EI/L2 = 1,200
12EI/L3= 300
89
First column in Stiffness matrix
1,200
300
d2
d1
d3
d1
1,200
300
k11
600
k21 =
1,200
k31
1,200
90
Second column in Stiffness matrix
d2 =1
d2
91
Second column in Stiffness matrix
2 EI
3,200 L
1
4 EI
6,400 L
4 EI
L
1 6,400
6 EI
L2
1,200
6 EI
L2
6 EI
L2
EA/L = 30,000
2 EI/L = 3,200
4 EI/L = 6,400
6 EI/L2 = 1,200
12EI/L3= 300
92
Second column in Stiffness matrix
3,200
6,400
6,400
1,200
d2
d1
d3
k12
1,200
k32
3,200
d1
k22 = 12,800
93
Third column in Stiffness matrix
d3 =1
d3
94
Sixth column in Stiffness matrix
2 EI
L 3,200
6,400 4
EI
L
4 EI
6,400L
1,200
1,200
6 EI
L2
6 EI
L2
1
1,200
1
6 EI
L2
EA/L = 30,000
2 EI/L = 3,200
4 EI/L = 6,400
6 EI/L2 = 1,200
12EI/L3= 300
95
Sixth column in Stiffness matrix
6,400
3,200
6,400
1,200
d2
d1
1,200
1,200
d3
d1
k13
1,200
k23 =
3,200
k33
12,800
96
k11
600
k12
1,200
k13
1,200
3,200
k21 =
1,200
k22 = 12,800
k23 =
k31
1,200
k32
k33
K=
3,200
600
1,200
1,200
1,200
12,800
3,200
1,200
3,200
12,800
12,800
97
Force vector
160 kNm
A
Fixed End Reaction (FER)
160 kNm
30 kN/m
B
120 kNm
120 kN
A
60 kN
60 kNm
120 kN
30 kN
B
120 kN
60 kN
C
120 kNm
60 kN
30 kN
C
60 kNm
98
Force vector
160 kNm
Fixed End Action (FEA)
160 kNm
30 kN/m
A
B
120 kNm
120 kN
A
60 kN
120 kN
d1
d3
30 kN
B
d1
60 kN
D
120 kNm
d2
60 kNm
120 kN
60 kN
30 kN
C
60 kNm
99
Force vector
160 kNm
30 kN/m
A
B
120 kN
120 kN
120 kNm
A
Fixed End Action (FEA)
d3
d2
d1
d1
160 kNm
60 kNm
F1
B
60 kN
30 kN
F2 = - 40
60 kN
120
60 kN
D
120 kNm
30
F3
100
30 kN
C
60 kNm
100
Stiffness Equation
F= K D
30
600
1,200
1,200
d1
- 40
= 1,200
12,800
3,200
d2
100
1,200
3,200
12,800
d3
D=
d1
-1
600
d2 = 1,200
d3
-1
K F
1,200
1,200
1,200
30
12,800
3,200
- 40
3,200
12,800
100
101
Deformations
d1
0.0618
d1
0.0607
d2
- 0.0038
d2
-0.0100
d3
- 0.0101
d3
0.0046
d4
=
0.0600
d5
- 0.0042
d6
0.0047
102
103
Example 4:
Draw N.F, S.F & B.M.Ds for the shown frame where
E = 106 kN/m2
12 kN/m
A
B
m2
A=0.6
I = 0.02 m4
3
m2
A=0.4
I = 0.005 m4
50 kN
2
C
8
60
104
Modeling
A
d2
d3
d1
B
d5
d4
C
105
Stiffness matrix
k11 k12 k13
k14 k15
k21
k22 k23
k24
k25
K = k31
k32 k33
k34
k35
k41
k42 k43
k44
k45
k51
k52 k53
k54
k55
106
First column in Stiffness matrix
d =1
1
A
d1
B
C
107
First column in Stiffness matrix
1
A
A
B B
2
E = 106 kN/mB
A=0.6 m2
I = 0.02 m4
5
A=0.4 m2
I = 0.005 m4
8
C
C
EA
L
1
6 EI
L2
12 EI
L3
12 EI
L3
6 EI
L2
108
First column in Stiffness matrix
75,000
1200
106x.6
6X106X.005
8
52
12X106X.005
A
53
480
2
E = 106 kN/mB
A=0.6 m2
I = 0.02 m4
5
A=0.4 m2
I = 0.005 m4
8
480
1200
C
109
First column in Stiffness matrix
75,000
1200
A
B
d2
d3
d1
d4
d5
k11
75,480
k21
0
k31 =
1200
k41
- 240
k51
1200
480
C
480
60
240
1200
110
Second column in Stiffness matrix
d2 =1
d2
A
B
C
111
Second column in Stiffness matrix
12 EI
L3
A
A
6 EI
L2
EA
L
1
1
B
2
E = 106 kN/mB
A=0.6 m2
I = 0.02 m4
5
A=0.4 m2
I = 0.005 m4
8
C
C
112
Second column in Stiffness matrix
6X106X.02
469
12X106X.02
82
1875
80,000
106x.4
5
83
A
A
B
2
E = 106 kN/mB
A=0.6 m2
I = 0.02 m4
5
A=0.4 m2
I = 0.005 m4
8
C
C
80,000
113
Second column in Stiffness matrix
1875
80,000
469
B
A
k12
d2
d3
d1
d4
d5
0
k22 = 80,469
k32
-1875
k42
-69,282
k52
0
C
80,000 sin 60
69,282
80,000
114
Third column in Stiffness matrix
d3 =1
A
d3
B
C
115
Third column in Stiffness matrix
4 EI
L
A
6 EI
L2
A
4 EI
L
5
A=0.4 m2
I = 0.005 m4
8
1
6 EI
L2
6 EI
L2
2
E = 106 kN/mB
A=0.6 m2
I = 0.02 m4
B
1
C
2 EI
L
C
116
Third column in Stiffness matrix
4X106X.02
4,000 5
8
10,000
A
6X106X.02
82
A
4X106X.005
E=
106
1
6X106X.005
52
1875
6X106X.005
kN/m2
B
52 1200
A=0.6 m2
I = 0.02 m4
5
A=0.4 m2
I = 0.005 m4
8
B
1
1200
C
2X106X.005
C
5
2,000
117
Third column in Stiffness matrix
A
B
4,000
10,000
1200
1875
d2
d3
d1
k13
1200
k23
-1875
k33 = 14,000
k43
d4
d5
k53
1200
60
- 600
2,000
600
C
2,000
118
Fourth column in Stiffness matrix
d4 =1
A
B
C
119
Fourth column in Stiffness matrix
A
A
B
6 EI
L3
2
E = 106 kN/mB
A=0.6 m2
I = 0.02 m4
5
A=0.4 m2
I = 0.005 m4
8
C
0.5
6 EI
L3
3 EI
L2
3 EI
L2
120
Fourth column in Stiffness matrix
3X106X.005
52
A
600
240
B
6X106X.005
53
A
E=
106
6X106X.005
kN/m2
B
53
m2
A=0.6
I = 0.02 m4
5
A=0.4 m2
I = 0.005 m4
8
240
3X106X.005
C
52
600
121
A
EA
Sin 60
L
B
80,000 sin 60
69,282
C
C
Sin 60
60
EA
Sin 60
L 69,282
122
69,282
Fourth column in Stiffness matrix
600
240
A
B
d2
d3
d1
d4
d5
k14
- 240
k24
-69,282
k34 =
- 600
k44
60,120
k54
- 600
240
600
240 cos 60
120
69,282
69,282 sin 60
60,000
123
Fifth column in Stiffness matrix
d5 =1
A
B
d5
C
124
Fifth column in Stiffness matrix
A
2 EI
L
B
6 EI
L2
A
6 EI
L2
2
E = 106 kN/mB
A=0.6 m2
I = 0.02 m4
5
A=0.4 m2
I = 0.005 m4
8
1
C
4 EI
L
C
125
Fifth column in Stiffness matrix
2X106X.005
2,000 5
A
1200
B
6X106X.005
52
A
6X106X.005
2
E = 106 kN/mB
52 1200
A=0.6 m2
I = 0.02 m4
5
A=0.4 m2
I = 0.005 m4
8
4X106X.005
C
5
1
C
4,000
126
Fifth column in Stiffness matrix
A
2,000
B
1200
d2
d3
d1
d4
d5
k15
1200
k25
0
k35 =
2,000
k45
- 600
k55
4,000
1200
60
C
600
4,000
127
k11
75,480
k12
0
k13
1200
k21
0
k22
80,469
k23
-1875
k31
1200
k32
-1875
k41
- 240
k42
-69,282
k33 = 14,000
k43
- 600
k51
1200
k52
0
=
=
k53
2,000
k14
- 240
k15
1200
k24
-69,282
k25
0
k34 = - 600
k35 =
2,000
k44
60,120
k45
- 600
k54
- 600
k55
4,000
128
Stiffness matrix
75,480
0
K = 1200
- 240
1200
- 240
1200
80,469 -1875
-69,282
0
-1875 14,000
- 600
2,000
0
1200
-69,282 - 600
0
2,000
60,120 - 600
- 600 4,000
129
Force vector
12 kN/m
A
B
m2
A=0.6
I = 0.02 m4
3
A=0.4 m2
I = 0.005 m4
8
50 kN
2
C
130
Force vector
12x82
64 kNm
12
12 kN/m
A
64 kNm
B
48 kN
48 kN
8
131
Force vector
50x3x22 24 kNm
52
3
17.6 kN
B
50 kN
2
32.4 kN
C
36 kNm
50x2x32
52
132
Fixed End
Reaction
(FER)
Force vector
64 kNm
64 kNm
24 kNm
A
48 kN
B
48 kN
17.6 kN
32.4 kN
B
C
36 kNm 133
Fixed End Action (FEA)
Force vector
64 kNm
64 kNm
A
B
48 kN
d1
F
d4
d5
17.6 kN
B
48 kN
d2
d3
24 kNm
F1
-17.6
F2
- 48
40
= F3 =
F4
- 16.2
F5
32.4 kN
C
16.2 kN
36 kNm
36
134
F= K D
-17.6
75,480
- 48
0
40
= 1200
- 16.2
- 240
36
1200
- 240
1200
80,469 -1875
-69,282
0
-1875 14,000
- 600
2,000
0
1200
-69,282 - 600
0
2,000
60,120 - 600
- 600 4,000
d1
d2
d3
d4
d5
135
D = K-1 F
-1
- 240 1200
0
1200
d1 75,480
d2
0
80,469 -1875 -69,282
0
d3 = 1200 -1875 14,000 - 600 2,000
d4
- 240 -69,282 - 600 60,120 - 600
d5
1200
0
2,000 - 600 4,000
-17.6
- 48
40
- 16.2
36
136
- 0.0007
D = K-1 F
- 0.0008
- 0.0088
- 0.2244
- 0.0538
A
B
C
d1
- 0.0002
- 0.43796
d2
d3 = - 0.07314
- 0.50601
d4
d5
- 0.03027
A
B
C
60
137
Questions
138