stiffness 5
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Lecture No. : 5 الخامسة المحاضرة
Remember :F1
1 2
k1k2
F3
3
k3
F2
d1 d2 d3
k11F1
F2 = k21
F = K D
F3 k31
k12
k22
k32
k13
k23
k33
d1
d2
d32
k11F1
F2 = k21
F3 k31
k12
k22
k32
k13
k23
k33
d1
d2
d3
First column in Stiffness matrix
d1 =1 d2 =0 d3 =0
1 2
k1k2
3
k3
d1=1
k11
k21
k31
=
F1
F2
F3
Remember :
3
k11F1
F2 = k21
F3 k31
k12
k22
k32
k13
k23
k33
d1
d2
d3
Second column in Stiffness matrix
d1 =0 d2 =1 d3 =0
k12
k22
k32
=
F1
F2
F3
1 2
k1k2
3
k3
d2=1
Remember :
4
k11F1
F2 = k21
F3 k31
k12
k22
k32
k13
k23
k33
d1
d2
d3
Third column in Stiffness matrix
d1 =0 d2 =0 d3 =1
k13
k23
k33
=
F1
F2
F3
1 2
k1k2
3
k3
d3=1
Remember :
5
Plan Truss
A
B
C
4
3
F1d1
F2 d2
F = K Dk11F1
F2
=k21
k12
k22
d1
d2
Remember :
6
First column in Stiffness matrixd1 =1 d2 =0
k11F1
F2
=k21
k12
k22
d1
d2
A
B
C
4
3
d1
1
Remember :
7
second column in Stiffness matrixd1 =0 d2 =1
k11F1
F2
=k21
k12
k22
d1
d2
A
B
C
d2
1
Remember :
8
a
b
F1
d1=1
F2
F3
F4
= EA/L cos2
= EA/L cos sin
= - EA/L cos2
= - EA/L cos sin
9
a
b
F1
F2
F3
F4
= EA/L sin2
= EA/L sin cos
= - EA/L sin cos
= - EA/L sin2
d2=1
10
( )cos + ( )sin
bxax
- byay
-AEL
N =
a
b
bx
by
ax
ay
[ ]
For internal forces in truss elementsRemember :
11
4
A
B
E
5 2 4C D
B
Beams
C D
Remember :
12
k11
K = k21
k31
k12
k22
k32
k13
k23
k33
Stiffness matrix
Remember :
13
A
B
E
C D
B
First column in Stiffness matrix
B =1
4 EILAB
4 EILBC
+ 2 EILBC
Remember :
14
A
B
E
C D
C
Second column in Stiffness matrix
C =1
4 EILBC
4 EILCD
+ 2 EILCD
2 EILBC
Remember :
15
A
B
E
C D
D
Third column in Stiffness matrix
D =1
4 EILCD
4 EILDE
+2 EILCD
Remember :
16
A
B
E
C D
60 kNm 50 kNm 30 kNmForce vector
F =
- 60- 50
30
A
B
E
C D
B C D
Remember :
17
Force vectorTransformation from member forces to Joint forces
L
P
8LP
8LP
P a b2
L2
L
P
a b P b a2
L2
Fixed End Reaction (FER) 18
Force vectorTransformation from member forces to Joint forces
L
P
8LP
8LP
P a b2
L2
L
P
a b P b a2
L2
Fixed End Action (FEA) 19
Internal forces in beam elements
MBA=2 EI
L( + 2 ) M(FER) BA +
MAB= ( 2 + ) M(FER) AB +2 EI
L
20
21
I I
2 I
22
d1
d2
d3
23
d1
d2
d3
d11
d4
d5
d6d7
d8
d9
d10 d12
24
25
26
AEL
AEL
AEL
AEL
27
AEL
AEL
AEL
AEL
28
29
6 EI
L2
6 EIL2
12 EIL3
12 EIL3
30
6 EIL2
6 EIL2
12 EIL3
12 EIL3
31
3 EIL2
3 EIL3
3 EIL3
32
3 EIL2
3 EIL3
3 EIL3
33
34
4 EIL
2 EIL
6 EIL2
6 EIL2
35
4 EIL
2 EIL
6 EIL2
6 EIL2
36
3 EIL 3 EI
L2
3 EIL2
37
3 EIL
3 EIL2
3 EIL2
38
Example 1:Construct the stiffness matrix for the shown frame where E = 106 kN/m2
8
A B
5
C
A=0.6 m2
I = 0.02 m4
A=0.4 m2
I = 0.005 m4
39
Modeling
A
B
C
d1
d2
d3
40
k11
K = k21
k31
k12
k22
k32
k13
k23
k33
Stiffness matrix
41
First column in Stiffness matrix
d1 =1A
B
C
d1
42
First column in Stiffness matrix
A B
C
1 AEL 1 6 EI
L2
12 EIL3
B
A B
8
5
C
A=0.6 m2
I = 0.02 m4
A=0.4 m2
I = 0.005 m4
E = 106 kN/m2
43
First column in Stiffness matrix106x.6
8 6X106X.00552
A B
8
5
C
A=0.6 m2
I = 0.02 m4
A=0.4 m2
I = 0.005 m4
E = 106 kN/m2
12X106X.00553
75,0001200
480
44
First column in Stiffness matrix
k11
k21
k31
=
75,480
12000
A B
C
75,0001200
480d1
d2
d3
45
Second column in Stiffness matrix
d2 =1A
B
C
d2
46
A B
C
AEL
6 EIL2
12 EIL3
Second column in Stiffness matrix
11
A B
8
5
C
A=0.6 m2
I = 0.02 m4
A=0.4 m2
I = 0.005 m4
E = 106 kN/m2
47
A B
C
106x.45
6X106X.0282
12X106X.0283
80,000
1875469
Second column in Stiffness matrix
A B
8
5
C
A=0.6 m2
I = 0.02 m4
A=0.4 m2
I = 0.005 m4
E = 106 kN/m2
48
A B
C
80,000
1875469
k12
k22
k32
=
080,469
-1875
Second column in Stiffness matrix
d1
d2
d3
49
Third column in Stiffness matrix
d3 =1A
B
C
d3
50
Third column in Stiffness matrix
A
B
C
1
1
4 EIL
6 EIL2
4 EIL
6 EIL2
A B
8
5
C
A=0.6 m2
I = 0.02 m4
A=0.4 m2
I = 0.005 m4
E = 106 kN/m2
51
Third column in Stiffness matrix
A
B
C
1
1
6X106X.0282
4X106X.028
6X106X.00552
4X106X.0055
10,0004,000
1200
1875A B
8
5
C
A=0.6 m2
I = 0.02 m4
A=0.4 m2
I = 0.005 m4
E = 106 kN/m2
52
Third column in Stiffness matrix
A
B
C
10,0004,000
1200
1875
d1
d2
d3
k13
k23
k33
=
1200-1875
14,00053
k11
k21
k31
=
75,480
1200
0k12
k22
k32
=
080,469
-1875
k13
k23
k33
=
1200-1875
14,000
K =
75,480
1200
00
80,469
-1875
1200-1875
14,000
54
Example 2:Construct the stiffness matrix for the shown frame where E = 106 kN/m2, and the section of all members is rectangular with dimension 30 cm x 80 cm.
8a
b
8
d
c
55
Modeling
d1
d2
d3
a
b
d
c
d4
d5
d6
56
k11
K =
k21
k31
k12
k22
k32
k13
k23
k33
Stiffness matrix
k41
k51
k61
k42
k52
k62
k43
k53
k63
k14
k24
k34
k15
k25
k35
k16
k26
k36
k44
k54
k64
k45
k55
k65
k46
k56
k66
57
Example 2:
30 cm
80 cmEA= 240,000 kNEI = 12,800 kNm2
Construct the stiffness matrix for the shown frame where E = 106 kN/m2, and the section of all members is rectangular with dimension 30cm x 80 cm.
A = 0.24 m2
I = 0.0128 m4
58
8a
b
8
d
c
EA=240,000 kNEI = 12,800 kNm2
EA/L = 30,0002 EI/L = 3,2004 EI/L = 6,4006 EI/L2 = 1,20012EI/L3= 300
59
First column in Stiffness matrix
d1 =1d1
a
b
d
c
60
First column in Stiffness matrix
b c
a
1 AEL1 6 EI
L2
12 EIL3
b
AEL
d
c
EA/L = 30,0002 EI/L = 3,2004 EI/L = 6,4006 EI/L2 = 1,20012EI/L3= 300
30,000 30,0001,200
300
61
First column in Stiffness matrix
b c
d
c
30,000 30,0001,200
300
a
b
d
c
d1
d2
d3 d4
d5d6
k11
=
k21
k31
k41
k51
k61
30,3000
1,200-30,000
00
62
Second column in Stiffness matrix
d2 =1
d2
a
b
d
c
63
b c
a
AEL
6 EIL2
12 EIL3
Second column in Stiffness matrix
1
1
12 EIL3
6 EIL2
b
d
c
EA/L = 30,0002 EI/L = 3,2004 EI/L = 6,4006 EI/L2 = 1,20012EI/L3= 300
30,0001,200
300
1,200
300
64
Second column in Stiffness matrix
30,0001,200
300
1,200
300
a
b
d
c
d1
d2
d3 d4
d5d6
k12
=
k22
k32
k42
k52
k62
30,3000
1,200
-300
0
1,20065
Third column in Stiffness matrix
d3 =1d3
a
b
d
c
66
Third column in Stiffness matrix
a
1 6 EIL2
4 EIL
6 EIL2
6 EIL2
4 EIL
2 EIL
1
d
c
EA/L = 30,0002 EI/L = 3,2004 EI/L = 6,4006 EI/L2 = 1,20012EI/L3= 300
6,4006,400
3,200
1,2001,200
1,200b c
67
Third column in Stiffness matrix
ad
c
6,400 6,400
3,200
1,200
1,2001,200
b
a
b
d
c
d1
d2
d3 d4
d5d6
k13
=
k23
k33
k43
k53
k63
1,2001,200
12,800
-1,200
0
3,200 68
Fourth column in Stiffness matrix
d4 =1
d4
a
b
d
cc
69
b c
d
1 AEL 1 6 EI
L2
12 EIL3
c
Fourth column in Stiffness matrix
a
b
AEL
EA/L = 30,0002 EI/L = 3,2004 EI/L = 6,4006 EI/L2 = 1,20012EI/L3= 300
30,000 30,000 1,200
300
70
Fourth column in Stiffness matrix
a
b
30,000 30,0001,200
300
a
b
d
c
d1
d2
d3 d4
d5d6
k14
=
k24
k34
k44
k54
k64
0-30,000
0
0
30,300
1,200
71
Fifth column in Stiffness matrix
d5 =1
d5
a
b
d
c
72
b c
d
AEL
6 EIL2
12 EIL3
11
Fifth column in Stiffness matrix
a
b
c
6 EIL2
12 EIL3
EA/L = 30,0002 EI/L = 3,2004 EI/L = 6,4006 EI/L2 = 1,20012EI/L3= 300
30,000
1,200
1,200300
300
73
Fifth column in Stiffness matrix
a
b
30,000
1,200
1,200300
300
a
b
d
c
d1
d2
d3 d4
d5d6
k15
=
k25
k35
k45
k55
k65
-3000
-1,200
30,300
0
-1,20074
Sixth column in Stiffness matrix
d6 =1
d6
a
b
d
c
75
b c
d
1
1
4 EIL
6 EIL2
4 EIL
6 EIL2
Sixth column in Stiffness matrix
a
b
c
2 EIL
6 EIL2
EA/L = 30,0002 EI/L = 3,2004 EI/L = 6,4006 EI/L2 = 1,20012EI/L3= 300
3,200 6,4006,400
1,2001,200
1,200
76
b c
d
Sixth column in Stiffness matrix
a
bc
3,200 6,400
6,400
1,200 1,200 1,200
a
b
d
c
d1
d2
d3 d4
d5d6
k16
=
k26
k36
k46
k56
k66
1,2000
3,200
-1,200
1,200
12,80077
k11
=
k21
k31
k41
k51
k61
30,3000
1,200-30,000
00
k12
=
k22
k32
k42
k52
k62
30,3000
1,200
-300
0
1,200
k13
=
k23
k33
k43
k53
k63
1,2001,200
12,800
-1,200
0
3,200
k14
=
k24
k34
k44
k54
k64
0-30,000
0
0
30,300
1,200
k15
=
k25
k35
k45
k55
k65
-3000
-1,200
30,300
0
-1,200
k16
=
k26
k36
k46
k56
k66
1,2000
3,200
-1,200
1,200
12,80078
=K
30,3000
1,200-30,000
00
30,3000
1,200
-300
0
1,200
1,2001,200
12,800
-1,200
0
3,200
0-30,000
0
0
30,300
1,200
-3000
-1,200
30,300
0
-1,200
1,2000
3,200
-1,200
1,200
12,800
79
Example 3:Construct the stiffness matrix for the shown frame where E = 106 kN/m2
8
A B
5
C
A=0.6 m2
I = 0.02 m4
A=0.4 m2
I = 0.005 m4
80
Modeling
A
B
C
d1
d2
d3
First Model No DOF at C
81
k11
K = k21
k31
k12
k22
k32
k13
k23
k33
Stiffness matrix
82
First column in Stiffness matrix
d1 =1A
B
C
d1
83
First column in Stiffness matrix
A B
C
1 AEL 1 3 EI
L2
3 EIL3
B
A B
8
5
C
A=0.6 m2
I = 0.02 m4
A=0.4 m2
I = 0.005 m4
E = 106 kN/m2
84
First column in Stiffness matrix106x.6
8 3X106X.00552
A B
8
5
C
A=0.6 m2
I = 0.02 m4
A=0.4 m2
I = 0.005 m4
E = 106 kN/m2
3X106X.00553
75,000600
120
85
First column in Stiffness matrix
k11
k21
k31
=
75,120
6000
A B
C
75,000600
120d1
d2
d3
86
Second column in Stiffness matrix
d2 =1A
B
C
d2
87
A B
C
AEL
6 EIL2
12 EIL3
Second column in Stiffness matrix
11
A B
8
5
C
A=0.6 m2
I = 0.02 m4
A=0.4 m2
I = 0.005 m4
E = 106 kN/m2
88
A B
C
106x.45
6X106X.0282
12X106X.0283
80,000
1875469
Second column in Stiffness matrix
A B
8
5
C
A=0.6 m2
I = 0.02 m4
A=0.4 m2
I = 0.005 m4
E = 106 kN/m2
89
A B
C
80,000
1875469
k12
k22
k32
=
080,469
-1875
Second column in Stiffness matrix
d1
d2
d3
90
Third column in Stiffness matrix
d3 =1A
B
C
d3
91
Third column in Stiffness matrix
A
B
C
1
1
4 EIL
6 EIL2
3 EIL
3 EIL2
A B
8
5
C
A=0.6 m2
I = 0.02 m4
A=0.4 m2
I = 0.005 m4
E = 106 kN/m2
92
Third column in Stiffness matrix
A
B
C
1
1
6X106X.0282
4X106X.028
3X106X.00552
3X106X.0055
10,0003,000
600
1875A B
8
5
C
A=0.6 m2
I = 0.02 m4
A=0.4 m2
I = 0.005 m4
E = 106 kN/m2
93
Third column in Stiffness matrix
A
B
C
10,0003,000
600
1875
d1
d2
d3
k13
k23
k33
=
600-1875
13,00094
k11
k21
k31
=
75,120
600
0k12
k22
k32
=
080,469
-1875
k13
k23
k33
=
600-1875
13,000
K =
75,120
600
00
80,469
-1875
600-1875
13,000
95
Modeling
A
B
C
d1
d2
d3
Second Model Take DOF at C
d4
96
k11
K =k21
k31
k12
k22
k32
k13
k23
k33
Stiffness matrix
k41 k42 k43
k14
k24
k34
k44
97
First column in Stiffness matrix
d1 =1A
B
C
d1
98
First column in Stiffness matrix
A B
C
1 AEL 1 6 EI
L2
12 EIL3
B
6 EIL2
A B
8
5
C
A=0.6 m2
I = 0.02 m4
A=0.4 m2
I = 0.005 m4
E = 106 kN/m2
99
First column in Stiffness matrix106x.6
8 6X106X.00552
12X106X.00553
75,0001200
480
6X106X.00552
1200
A B
8
5
C
A=0.6 m2
I = 0.02 m4
A=0.4 m2
I = 0.005 m4
E = 106 kN/m2
100
First column in Stiffness matrix
k11
k21
k31
=
75,480
12000
A B
C
75,0001200
480d1
d2
d3
d4
1200k41 1200
101
Second column in Stiffness matrix
d2 =1A
B
C
d2
102
A B
C
AEL
6 EIL2
12 EIL3
Second column in Stiffness matrix
11
A B
8
5
C
A=0.6 m2
I = 0.02 m4
A=0.4 m2
I = 0.005 m4
E = 106 kN/m2
103
A B
C
106x.45
6X106X.0282
12X106X.0283
80,000
1875469
Second column in Stiffness matrix
A B
8
5
C
A=0.6 m2
I = 0.02 m4
A=0.4 m2
I = 0.005 m4
E = 106 kN/m2
104
A B
C
80,000
1875469
k12
k22
k32
=
080,469
-1875
Second column in Stiffness matrix
d1
d2
d3
d4 k42 0105
Third column in Stiffness matrix
d3 =1A
B
C
d3
106
Third column in Stiffness matrix
A
B
C
1
1
4 EIL
6 EIL2
4 EIL
6 EIL2
A B
8
5
C
A=0.6 m2
I = 0.02 m4
A=0.4 m2
I = 0.005 m4
E = 106 kN/m2
2 EIL 107
Third column in Stiffness matrix
A
B
C
1
1
6X106X.0282
4X106X.028
6X106X.00552
4X106X.0055
10,0004,000
1200
1875A B
8
5
C
A=0.6 m2
I = 0.02 m4
A=0.4 m2
I = 0.005 m4
E = 106 kN/m2
2X106X.0055 2,000
108
Third column in Stiffness matrix
A B
C
10,0004,000
1200
1875
k13
k23
k33
=
1200-1875
14,000
d1
d2
d3
d4 2,000k43 2,000
109
Fourth column in Stiffness matrix
d4 =1A
B
Cd4
110
A B
C
1
2 EIL
6 EIL2
A B
8
5
C
A=0.6 m2
I = 0.02 m4
A=0.4 m2
I = 0.005 m4
E = 106 kN/m2
4 EIL
Fourth column in Stiffness matrix
111
A
B
C
1
6X106X.00552
2X106X.00552,000
1200
A B
8
5
C
A=0.6 m2
I = 0.02 m4
A=0.4 m2
I = 0.005 m4
E = 106 kN/m2
4X106X.0055 4,000
Fourth column in Stiffness matrix
112
A B
C
2,000
1200
k14
k24
k34
=
12000
2,000
d1
d2
d3
d4 4,000k44 4,000
Fourth column in Stiffness matrix
113
K =
k11
k21
k31
=
75,480
12000
k41 1200
k12
k22
k32
=
080,469
-1875k42 0
k13
k23
k33
=
1200-1875
14,000k43 2,000
k14
k24
k34
=
12000
2,000k44 4,000
75,480
12000
1200
080,469
-18750
1200-1875
14,000
2,000
12000
2,0004,000
114
Example 4:Construct the stiffness matrix for the shown frame where E = 106 kN/m2
8
A B
5
C
A=0.6 m2
I = 0.02 m4
A=0.4 m2
I = 0.005 m4
115
Modeling
A
B
C
d1
d2
d3
d5
d4
116
k11
K =
k21
k31
k12
k22
k32
k13
k23
k33
Stiffness matrix
k41 k42 k43
k14
k24
k34
k44
k51 k52 k53 k54
k15
k25
k35
k45
k55
117
First column in Stiffness matrix
d1 =1A
B
C
d1
118
First column in Stiffness matrix
A B
C
1 AEL 1 6 EI
L2
12 EIL3
B
6 EIL2
12 EIL3
A B
8
5
C
A=0.6 m2
I = 0.02 m4
A=0.4 m2
I = 0.005 m4
E = 106 kN/m2
119
First column in Stiffness matrix106x.6
8 6X106X.00552
12X106X.00553
75,0001200
480
1200
A B
8
5
C
A=0.6 m2
I = 0.02 m4
A=0.4 m2
I = 0.005 m4
E = 106 kN/m2
480
120
First column in Stiffness matrix
k11
k21
k31 =
75,480
12000
A B
C
75,0001200
480
1200
k41 - 480
d1
d2
d3
d5
d4
480
k51 1200 121
Second column in Stiffness matrix
d2 =1A
B
C
d2
122
A B
C
AEL
6 EIL2
12 EIL3
Second column in Stiffness matrix
11
A B
8
5
C
A=0.6 m2
I = 0.02 m4
A=0.4 m2
I = 0.005 m4
E = 106 kN/m2
123
A B
C
106x.45
6X106X.0282
12X106X.0283
80,000
1875469
Second column in Stiffness matrix
A B
8
5
C
A=0.6 m2
I = 0.02 m4
A=0.4 m2
I = 0.005 m4
E = 106 kN/m2
124
AB
C
80,000
1875469
k12
k22
k32
=
080,469
-1875
Second column in Stiffness matrix
k42 0
d1
d2
d3
d5
d4
k52 0125
Third column in Stiffness matrix
d3 =1A
B
C
d3
126
Third column in Stiffness matrix
A
B
C
1
1
4 EIL
6 EIL2
4 EIL
6 EIL2
2 EIL
A B
8
5
C
A=0.6 m2
I = 0.02 m4
A=0.4 m2
I = 0.005 m4
E = 106 kN/m2 6 EIL2
127
Third column in Stiffness matrix
A
B
C
1
1
6X106X.0282
4X106X.028
6X106X.00552
4X106X.0055
10,0004,000
1200
1875
2X106X.0055 2,000
A B
8
5
C
A=0.6 m2
I = 0.02 m4
A=0.4 m2
I = 0.005 m4
E = 106 kN/m2 6X106X.00552
1200
128
Third column in Stiffness matrix
A B
C
10,0004,000
1200
1875k13
k23
k33
=
1200-1875
14,000
2,000
k43 -1,200
d1
d2
d3
d5
d4
1200
k53 2,000
129
Fourth column in Stiffness matrix
d4 =1A
B
Cd4
130
A B
A B
8
5
C
A=0.6 m2
I = 0.02 m4
A=0.4 m2
I = 0.005 m4
E = 106 kN/m2
Fourth column in Stiffness matrix
1
6 EIL212 EI
L3
6 EIL2
12 EIL3
131
A
B
A B
8
5
C
A=0.6 m2
I = 0.02 m4
A=0.4 m2
I = 0.005 m4
E = 106 kN/m2
Fourth column in Stiffness matrix6X106X.005
52
12X106X.00553
1200480
12X106X.00553
1200
480
6X106X.00552 132
A B
k14
k24
k34 =
- 4800
-1,200k44 480
d1
d2
d3
d5
d4
Fourth column in Stiffness matrix
480
1200
480
1200
k54 -1,200133
Fifth column in Stiffness matrix
d5 =1A
B
Cd5
134
A B
C
1
2 EIL
6 EIL2
4 EIL
A B
8
5
C
A=0.6 m2
I = 0.02 m4
A=0.4 m2
I = 0.005 m4
E = 106 kN/m2
Fifth column in Stiffness matrix
6 EIL2
135
A
B
C
1
6X106X.00552
2X106X.00552,000
1200
4X106X.0055 4,000
A B
8
5
C
A=0.6 m2
I = 0.02 m4
A=0.4 m2
I = 0.005 m4
E = 106 kN/m2
Fifth column in Stiffness matrix
6X106X.00552 1200
136
A B
C
2,000
1200
k15
k25
k35 =
12000
2,000
4,000
k45 -1,200d1
d2
d3
d5
d4
Fifth column in Stiffness matrix
1200
k55 4,000
137
k11
k21
k31 =
75,480
12000
k41 - 480k51 1200
k12
k22
k32 =
080,469
-1875k42 0k52 0
k13
k23
k33 =
1200-1875
14,000k43 -1,200k53 2,000
k15
k25
k35 =
12000
2,000k45 -1,200
k55 4,000
k14
k24
k34 =k44
k54
- 4800
-1,200480
-1,200138
K =
75,480
12000
- 4801200
080,469
-187500
1200-1875
14,000-1,2002,000
12000
2,000
-1,200
4,000
- 4800
-1,200480
-1,200
139
Example 5:Construct the stiffness matrix for the shown frame where E = 106 kN/m2
8
A
B
6
C
A=0.6 m2
I = 0.02 m4
A=0.4 m2
I = 0.005 m4
140
Modeling
B
C
d1
d2
d3
A
141
k11
K = k21
k31
k12
k22
k32
k13
k23
k33
Stiffness matrix
142
First column in Stiffness matrix
d1 =1B
C
d1
A
143
First column in Stiffness matrix
C
1 6 EIL2
12 EIL3
B
A
B
8
6
C
A=0.6 m2
I = 0.02 m4
A=0.4 m2
I = 0.005 m4
E = 106 kN/m2
144
First column in Stiffness matrix
6X106X.00562
12X106X.00563
833.33
277.78
A
B
8
6
C
A=0.6 m2
I = 0.02 m4
A=0.4 m2
I = 0.005 m4
E = 106 kN/m2
145
First column in Stiffness matrixsin 6EI sin
L212EI sin
L3 B
A
1
sin
cos
EA cos L
146
First column in Stiffness matrix 6EI sinL2
12EI sinL3
B
A
EA cos L
147
First column in Stiffness matrix 6EI sinL2
12EI sinL3
BA
EA cos L
EA cos2
L
EA cos sin
L
12EI sin2L3
12EI sincos
L3
EA cos2 L
12EI sin2L3+
EA cos sin
L-12EI sincos
L3
148
First column in Stiffness matrix
6EI sinL2
B
A
EA cos2 L
12EI sin2L3+
EA cos sin
L-12EI sincos
L3
A
B
8
6
C
A=0.6 m2
I = 0.02 m
4
A=0.4 m2
I = 0.005 m4
E = 106 kN/m2
106x.610
= 60,000AEL
=6 EI
L2
12 EIL3
6X106X.02102
12X106X.02103
1200
240
=
= =
=
cos = 0.8sin = 0.6
149
First column in Stiffness matrix
6EI sinL2
B
A
EA cos2 L
12EI sin2L3+
EA cos sin
L-12EI sincos
L3
cos = 0.8sin = 0.6
= 60,000AEL
6 EIL2
12 EIL3
= 1200 = 240
60,000x0.64+240x0.36
60,000x0.48- 240x0.48
38,486.4
28,684.8
1200x0.6 720
150
First column in Stiffness matrix
B
A
38,486.428,684.8
720
38,764.18
28,684.8
1553.33
k11
k21
k31
=
38,764.18
1553.3328,684.8
833.33
277.78
151
Second column in Stiffness matrix
d2 =1
B
C
d2
A
152
C
AEL
Second column in Stiffness matrix
1
106x.46
A
B
8
6
C
A=0.6 m2
I = 0.02 m
4
A=0.4 m2
I = 0.005 m4
E = 106 kN/m2
66,666.67
153
6EI cosL2
12EI cosL3
B
A
1
cos
sin EA sin
Lcos sin
Second column in Stiffness matrix
154
6EI cosL2
12EI cosL3
B
A
EA sin L
Second column in Stiffness matrix
155
6EI cosL2
12EI cosL3
BA
EA sin L
EA sin2 L
EA sin cos
L
12EI cos2L3
12EI cossin
L3
EA sin2 L
12EI cos2L3+
EA sincos
L-12EI cossin
L3
Second column in Stiffness matrix
156
6EI cosL2
B
A A
B
8
6
C
A=0.6 m2
I = 0.02 m
4
A=0.4 m2
I = 0.005 m4
E = 106 kN/m2
106x.610
= 60,000AEL
=6 EI
L2
12 EIL3
6X106X.02102
12X106X.02103
1200
240
=
= =
=
EA sin2 L
12EI cos2L3+
EA sincos
L-12EI cossin
L3
Second column in Stiffness matrix
cos = 0.8sin = 0.6
157
6EI cosL2
B
A
cos = 0.8sin = 0.6
= 60,000AEL
6 EIL2
12 EIL3
= 1200 = 240
60,000x0.48-240x0.48
60,000x0.36+240x0.64
28,684.8
21,753.6
1200x0.8 960
EA sin2 L
12EI cos2L3+
EA sincos
L-12EI cossin
L3
Second column in Stiffness matrix
158
B
A
28,684.8
88,420.27
960
k12
k22
k32
=
28,684.8
- 96088,420.27
28,684.821,753.6
960
Second column in Stiffness matrix
C
66,666.67
159
Third column in Stiffness matrix
d3 =1
B
C
d3
A
160
Third column in Stiffness matrix
C
1A
B 1
4 EIL
6 EIL2
4 EIL
6 EIL2
161
Third column in Stiffness matrix
C
1
4 EIL
6 EIL2
A
B
8
6
C
A=0.6 m2
I = 0.02 m4
A=0.4 m2
I = 0.005 m4
E = 106 kN/m2
6X106X.00562
4X106X.00563,333.33
833.33
162
Third column in Stiffness matrix
A
B
4 EIL
6 EIL2
163
Third column in Stiffness matrix
A
4 EIL
6 EIL2
6 EI sin L2
6 EI cos L2
164
Third column in Stiffness matrix
A
4 EIL
6 EI sin L2
6 EI cos L2
A
B
8
6
C
A=0.6 m2
I = 0.02 m4
A=0.4 m2
I = 0.005 m4
E = 106 kN/m2
4X106X.0210
6X106X.02102
cos = 0.8sin = 0.6
6 EIL2
= 1,200=
720
960
8,000
165
Third column in Stiffness matrixA
B8,000
C
3,333.33
833.33960
720
k13
k23
k33
=
1553.33-960
11,333.33
11,333.33
960
1553.33
166
K =
k13
k23
k33
=
1553.33
-960
11,333.33
k12
k22
k32
=
28,684.8
- 960
88,420.27
k11
k21
k31
=
38,764.18
1553.33
28,684.8
1553.33
-960
11,333.33
28,684.8
- 960
88,420.27
38,764.18
1553.33
28,684.8
167
168
Bd4
d5
d6
A
d1
d2
d3
169
sin 6EI sin
L212EI sin
L3 B
A
1
sin
cos
EA cos L
Horizontal Deformation
170
6EI sinL2
12EI sinL3
B
A
EA cos L
171
6EI sinL2
12EI sinL3
BA
EA cos L
EA cos2
L
EA cos sin
L
12EI sin2L3
12EI sincos
L3
EA cos2 L
12EI sin2L3+
EA cos sin
L-12EI sincos
L3
172
6EI sinL2
EA cos2 L
12EI sin2L3+
EA cos sin
L-12EI sincos
L3
6EI sinL2
6EI sinL2
EA cos2 L
12EI sin2L3+
EA cos sin
L-12EI sincos
L3
173
6EI cosL2
12EI cosL3
B
A
1
cos
sin EA sin
Lcos sin
Vertical Deformation
174
6EI cosL2
12EI cosL3
B
A
EA sin L
175
6EI cosL2
12EI cosL3
BA
EA sin L
EA sin2 L
EA sin cos
L
12EI cos2L3
12EI cossin
L3
EA sin2 L
12EI cos2L3+
EA sincos
L-12EI cossin
L3 176
6EI cosL2
EA sin2 L
12EI cos2L3+
EA sincos
L-12EI cossin
L3
6EI cosL2
EA sincos
L-12EI cossin
L3
EA sin2 L
12EI cos2L3+
6EI cosL2
-177
1
4 EIL
6 EIL2
2 EIL
Rotational Deformation
178
A
B
4 EIL
6 EIL2
2 EIL
179
4 EIL
6 EIL2
6 EI sin L2
6 EI cos L2
2 EIL
180
4 EIL
6 EI sin L2
6 EI cos L2
2 EIL
-
6 EI sin L2
6 EI cos L2
4 EIL
181
Example 6:Construct the stiffness matrix for the shown frame where E = 106 kN/m2, A = 0.6 m2, I = .02 m4
A B
O
5
C D E
150 135 60 45
182
Modeling
A B
O
5
C D E
150 135 60 45
d1
d2
d3
183
k11
K = k21
k31
k12
k22
k32
k13
k23
k33
Stiffness matrix
184
6EI sinL2
EA cos2 L
12EI sin2L3+
EA cos sin
L-12EI sincos
L3
Horizontal Deformation
185
EA sincos
L-12EI cossin
L3
EA sin2 L
12EI cos2L3+
6EI cosL2
-
Vertical Deformation
186
-
6 EI sin L2
6 EI cos L2
4 EIL
Rotational Deformation
187
EA cos2 L
12EI sin2
L3+
EA cossin
L-12EI sincos
L3
6EI sin
L2
EA sincos
L-12EI cossin
L3
EA sin2 L
12EI cos2
L3+
6EI cos
L2-
6EI sin
L2
6EI cos
L2-
4 EI
L
K =
188
member-
-
L--
EA/L-
4EI/L-
6 EI / L2
_12 EI / L3
_
OA 150 10.02 59890.5 7985.4 1195.6 238.7
OB 135 7.08 84772.3 11303.0 2395.4 676.9
OC 90 5.00 120000.0 16000.0 4800.0 1920.0
OD 60 5.77 103948.3 13859.8 3601.8 1248.0
OE 45 7.07 84879.6 11317.3 2401.5 679.5
A B
O
5
C D E
150 135 60 45
189
Member-
cos
sin
cos2
sin2
sin cos
OA 150 -0.867 0.499 0.751 0.249 -0.432
OB 135 -0.708 0.706 0.501 0.499 -0.500
OC 90 -0.001 1.000 0.000 1.000 -0.001
OD 60 0.500 0.866 0.250 0.750 0.433
OE 45 0.707 0.707 0.500 0.500 0.500
A B
O
5
C D E
150 135 60 45
190
EA cos2 L
12EI sin2
L3+
EA cossin
L-12EI sincos
L3
6EI sin
L2
EA sincos
L-12EI cossin
L3
EA sin2 L
12EI cos2
L3+
6EI cos
L2-
6EI sin
L2
6EI cos
L2-
4 EI
L
k11
k21
k31
k22
k32 k33
191
member k11 k21 k31 k22 k32 k33
OA 45031.92 -25798.5 297.8152 15097.23 1036.069 7985.394
OB 42804.35 -42047.6 1195.452 42644.84 1695.443 11302.97
OC 1920.046 -74.6554 4799.996 119999.9 3.034773 16000
OD 26885.59 44448.89 2702.628 78310.72 -1799.56 13859.78
OE 42752.93 42100.08 1201.518 42806.17 -1697.59 11317.28
159394.8 18628.19 10197.41 298858.9 -762.607 60465.43
159394.8
K = 18628.19
10197.41
18628.19
298858.9
762.607
10197.41
762.607
60465.43
192
( )cos + ( )sin
bxax
- byay
-AEL
N =
a
b
bx
by
ax
ay
[ ]
Internal forces
Normal Force
193
Internal forces
Bending momentDue to rotational deformation
MBA=2 EI
L( + 2 ) M(FER) BA +
MAB= ( 2 + ) M(FER) AB +2 EI
L
194
a
b
bx
by
ax
ay
( )cos
- ( )sin
bxax
-byay
- =
Internal forces
Bending momentDue to shear deformation
6EI L2
6EI
L2
195
Internal forces
Bending moment
MBA=2 EI
L( + 2 )
M(FER) BA +
MAB= ( 2 + ) M(FER) AB +2 EI
L6EI
L2-
6EI L2
-
MAB= ( 2 + ) M(FER) AB +2 EI
L3 L-
MBA=2 EI
L( + 2 )
M(FER) BA +3 L-
196
Example 7:Draw N.F, S.F & B.M.Ds for the shown frame where E = 106 kN/m2
8
A B
3
C
A=0.6 m2
I = 0.02 m4
A=0.4 m2
I = 0.005 m450 kN
12 kN/m
2
197
Modeling
A
B
C
d1
d2
d3
d5
d4
198
K =
75,480
12000
- 4801200
080,469
-187500
1200-1875
14,000-1,2002,000
12000
2,000
-1,200
4,000
- 4800
-1,200480
-1,200
From Example 4 :
199
Force vector
8
A B
C
A=0.6 m2
I = 0.02 m4
A=0.4 m2
I = 0.005 m450 kN
12 kN/m
3
2
200
Force vector
8
A
12 kN/m
B
64 kNm12x82
12 64 kNm
48 kN 48 kN
201
Force vector
B
C
50 kN
3
2
24 kNm
50x2x32
52
36 kNm
32.4 kN
50x3x22
52
17.6 kN
202
Force vector
A B
64 kNm 64 kNm
48 kN 48 kN B
C
24 kNm
36 kNm
32.4 kN
17.6 kN
Fixed End Reaction
(FER)
203
Force vector
A B
64 kNm 64 kNm
48 kN 48 kN B
C
24 kNm
36 kNm
32.4 kN
17.6 kN
Fixed End Action (FEA)
AB
C
d1
d2
d3
d5 d4
F =
F1 -17.6
=
F2 - 48F3 40F4 -32.4
F5 36204
F = K D
d1
d2
-17.6
=- 4840
-32.4
36
75,480
12000
- 4801200
080,469
-187500
1200-1875
14,000-1,2002,000
12000
2,000
-1,200
4,000
- 4800
-1,200480
-1,200
d3
d4
d5
205
D = K-1 F
d1
d2
-17.6
=- 4840
-32.4
36
75,480
12000
- 4801200
080,469
-187500
1200-1875
14,000-1,2002,000
12000
2,000
-1,200
4,000
- 4800
-1,200480
-1,200
d3
d4
d5
-1
206
D = K-1 F
d1
d2
- 0.0007
=- 0.0008- 0.0088- 0.2244
- 0.0538
d3
d4
d5
207
Internal forcesd1d2
- 0.0007
=
- 0.0008
- 0.0088
- 0.2244
- 0.0538
d3
d4
d5
AB
C
d1
d2
d3
d5 d4
Bending moment
A=0.6 m2
I = 0.02 m4
A=0.4 m2
I = 0.005 m4
E = 106 kN/m2
EIAB = 2x104
EIBC = 5x103
208
Force vector
A B
64 kNm 64 kNm
48 kN 48 kN B
C
24 kNm
36 kNm
32.4 kN
17.6 kN
Fixed End Reaction
(FER)
209
= 64 + 2x2x104/8 (-.0088-3x-.0008/8) = 21.75
A B
64 kNm 64 kNm
48 kN 48 kN
d1d2
- 0.0007
=
- 0.0008
- 0.0088
- 0.2244
- 0.0538
d3
d4
d5
MAB= ( 2 + ) M(FER) AB +2 EI
L3 L-
MBA=2 EI
L( + 2 )
M(FER) BA +3 L-
= -64 + 2x2x104/8 (2x-.0088-3x-.0008/8) = -150
d1
d2d3
d5d4
210
= 24 + 2x5x103/5 (2x-.0088-.0538-3x-.2237/5) = 150
d1d2
- 0.0007
=
- 0.0008
- 0.0088
- 0.2244
- 0.0538
d3
d4
d5
MBC= ( 2 + )B CM(FER) AB +2 EI
L3 L-
MCB=2 EI
L( + 2 )B CM(FER) BA +
3 L-
= -36 + 2x5x103/5 (-.0088+2x-.0538-3x-.2237/5) = 0
B
C
24 kNm
36 kNm
17.6 kN
=d4- d1=-0.2237
d1
d2d3
d5d4
211
A
12 kN/m
B
21.75 kNm 150 kNm
48 kN 48 kN
16 kN 16 kN
32 kN 64 kN
B
C
50 kN
150 kNm
0 kN
50 kN3
28
212
Internal forcesd1d2
- 0.0007
=
- 0.0008
- 0.0088
- 0.2244
- 0.0538
d3
d4
d5
AB
C
d1
d2
d3
d5 d4
A=0.6 m2
I = 0.02 m4
A=0.4 m2
I = 0.005 m4
E = 106 kN/m2
EAAB = 6x105
EABC = 4x105
Normal Force AEL
N =
NAB = 6x105 /8 x -.0007 = - 50NBC = 4x105 /5 x -.0008 = - 64
213
A B 64 kN
B
C
50 kN
214
A
12 kN/m
B
21.75 kNm 150 kNm
32 kN 64 kN B
C
50 kN
150 kNm
50 kN
64 kN50 kN
215
A
12 kN/m
B
21.75 kNm 150 kNm
32 kN 64 kN B
C
50 kN
150 kNm
50 kN
64 kN50 kN
N.F.D
50
64
-
-
216
A
12 kN/m
B
21.75 kNm 150 kNm
32 kN 64 kN B
C
50 kN
150 kNm
50 kN
64 kN50 kN
S.F.D
64
50
50
32
-
+
+
217
A
12 kN/m
B
21.75 kNm 150 kNm
32 kN 64 kNB
C
50 kN
150 kNm
50 kN
64 kN
50 kN
B.M.D
-
15021.75
15085.875
96
10.125
--+
218
Summary
219
d1
d2
d3
220
AEL
AEL
AEL
AEL
221
6 EI
L2
6 EIL2
12 EIL3
12 EIL3
222
6 EIL2
6 EIL2
12 EIL3
12 EIL3
223
3 EIL2
3 EIL3
3 EIL3
224
3 EIL2
3 EIL3
3 EIL3
225
4 EIL
2 EIL
6 EIL2
6 EIL2
226
4 EIL
2 EIL
6 EIL2
6 EIL2
227
3 EIL 3 EI
L2
3 EIL2
228
3 EIL
3 EIL2
3 EIL2
229
6EI sinL2
EA cos2 L
12EI sin2L3+
EA cos sin
L-12EI sincos
L3
6EI sinL2
6EI sinL2
EA cos2 L
12EI sin2L3+
EA cos sin
L-12EI sincos
L3
Horizontal Deformation
230
6EI cosL2
EA sin2 L
12EI cos2L3+
EA sincos
L-12EI cossin
L3
6EI cosL2
EA sincos
L-12EI cossin
L3
EA sin2 L
12EI cos2L3+
6EI cosL2
-
Vertical Deformation
231
4 EIL
6 EI sin L2
6 EI cos L2
2 EIL
-
6 EI sin L2
6 EI cos L2
4 EIL
Rotational Deformation
232
( )cos + ( )sin
bxax
- byay
-AEL
N =
a
b
bx
by
ax
ay
[ ]
Internal forces
Normal Force
233
Internal forces
Bending moment
MAB= ( 2 + ) M(FER) AB +2 EI
L3 L-
MBA=2 EI
L( + 2 )
M(FER) BA +3 L-
234
Questions
235
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