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Lecture No. : 12 1
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2
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d1
d2d3x
yz
d4
d5
d6
3
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x
yz
x
z
y
4
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x
z
y
5
d2
d5
d8
d11
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6
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K11 K12 K13 K14 K15 K16 K17 K18 K19 K110 K111 K112
K21 K22 K23 K24 K25 K26 K27 K28 K29 K210 K211 K212
K31 K32 K33 K34 K35 K36 K37 K38 K39 K310 K311 K312
K41 K42 K43 K44 K45 K46 K47 K48 K49 K410 K411 K412
K51 K52 K53 K54 K55 K56 K57 K58 K59 K510 K511 K512
K61 K62 K63 K64 K65 K66 K67 K68 K69 K610 K611 K612
K71 K72 K73 K74 K75 K76 K77 K78 K79 K710 K711 K712
K81 K82 K83 K84 K85 K86 K87 K88 K89 K810 K811 K812
K91 K92 K93 K94 K95 K96 K97 K98 K99 K910 K911 K912
K101 K102 K103 K104 K105 K106 K107 K108 K109 K1010 K1011 K1012
K111 K112 K113 K114 K115 K116 K117 K118 K119 K1110 K1111 K1112
K121 K122 K123 K124 K125 K126 K127 K128 K129 K1210 K1211 K1212
F =l
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x
z
y
9
d2
d5
d8
d11
First column inLocal Stiffness matrix
d1
=1
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AE
L
AEL
AE
L
AE
L
11
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12
K11
K21
K31
K41
K51
K61
K71
K81
K91
K101
K111
K121
=
EAL
0
0
0
0
0
-EAL
0
0
0
0
0
First column inLocal Stiffnessmatrix
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x
z
y
13
d2
d5
d8
d11
Seventh column inLocal Stiffness matrix
d7
=1
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14
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AE
L
AE
L
AE
L
AE
L
15
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16
=
Seventh column inLocal Stiffnessmatrix
-EAL
0
0
0
0
0
EAL
0
0
0
0
0
K17
K27
K37
K47
K57
K67
K77
K87
K97
K107
K117
K127
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17
S d l i
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x
z
y
18
d2
d5
d8
d11
Second column inLocal Stiffness matrix
d2
=1
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19
Shear in Plan
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Dy
z
y6EIzDy
L2
6EIzDy
L2
12EIzDy
L3
12EIzDy
L3 20
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d=1
z
y
x6EIz
L2
12EIz
L3
6EIzL2
12EIzL3
21
S d l i
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22
=
Second column inLocal Stiffnessmatrix
K12
K22
K32
K42
K52
K62
K72
K82
K92
K102
K112
K122
0
12EIzL3
0
0
0
6EIz
L2
0
-12EIz
L3
0
0
0
6EIz
L2
Ei hth l i
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x
z
y
23
d2
d5
d8
d11
Eighth column inLocal Stiffness matrix
d8 =1
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24
z
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x
yz
z
y
6EIzDyL2
12EIzDy
L3
12EIzDyL3
6EIzDyL2
Dy
25
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d=1
z
y
x6EIz
L2
12EIz
L3
6EIzL2
12EIzL3
26
Eighth column in
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27
=
Eighth column inLocal Stiffnessmatrix
0
-12EIz
L3
0
0
0
-6EIz
L2
0
12EIzL3
0
0
0
-6EIz
L2
K18
K28
K38
K48
K58
K68
K78
K88
K98
K108
K118
K128
Third column in
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x
z
y
28
d2
d5
d8
d11
Third column inLocal Stiffness matrix
d3 =1
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29
Shear out of Plan
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Dz
z
y
6EIyDz
L2
12EIyDz
L36EIyDz
L2
12EIyDzL3
30
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d=1
z
y
x
6EIy
L2
12EIy
L3 6EIyL2
12EIyL3
31
Third column in
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32
=
Third column inLocal Stiffnessmatrix
K13
K23
K33
K43
K53
K63
K73
K83
K93
K103
K113
K123
0
0
12EIy
L3
0
-6EIyL2
0
0
0
-
12EIy
L3
0
-6EIyL2
0
Ninth column in
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x
z
y
33
d2
d5
d8
d11
Ninth column inLocal Stiffness matrix
d9 =1
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34
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z
y
Dz
6EIyDz
L2
12EIyDzL3
6EIyDz
L2
12EIyDz
L3
35
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d=1
z
y
x
6EIy
L2
12EIy
L3 6EIyL2
12EIyL3
36
Ninth column in
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37
=
Ninth column inLocal Stiffnessmatrix
0
0
-12EIy
L3
0
6EIyL2
0
0
0
12EIy
L3
0
6EIyL2
0
K19
K29
K39
K49
K59
K69
K79
K89
K99
K109
K119
K129
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38
Sixth column in
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x
z
y
39
d2
d5
d8
d11
Sixth column inLocal Stiffness matrix
d6 =1
R i i Pl
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40
Rotation in Plan
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z
y qz4 EIzqz
L
6 EIzqzL2
6 EIzqzL2
2 EIzqzL
41
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z
y
x
d=1
4 EIz
L
6 EIz
L2
6 EIz
L2
2 EIz
L
42
Sixth column in
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43
=
Sixth column inLocal Stiffnessmatrix
K16
K26
K36
K46
K56
K66
K76
K86
K96
K106
K116
K126
0
6EIzL2
0
0
0
4EIz
L
0
-6EIzL2
0
0
0
2EIzL
Twelfth column in
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x
z
y
44
d2
d5
d8
d11
Twelfth column inLocal Stiffness matrix
d12 =1
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45
E
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z
y
qz2 EIzqzL
6 EIzqzL2
6 EIzqzL2
4 EIzqzL
46
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z
y
x
d=1
4 EIz
L
6 EIz
L2
6 EIz
L2
2 EIzL
47
Twelfth column inK
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48
=
e t co uLocal Stiffnessmatrix
0
6EIzL2
0
0
0
2EIz
L0
-6EIzL2
0
0
0
4EIzL
K112
K212
K312
K412
K512
K612
K712
K812
K912
K1012
K1112
K1212
Fifth column in
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x
z
y
49
d2
d5
d8
d11
Local Stiffness matrix
d5 =1
Rotation out of Plan
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50
Rotation out of Plan
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qy
6 EIyqyL2
2 EIyqyL
6 EIyq
y
L2
4 EIyq
y
L
z
y
51
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d=1
z
y
x
6 EIy
L2
2 EIy
L
6 EIyL2
4 EIy
L
52
Fifth column inK
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53
=
Local Stiffnessmatrix
K15
K25
K35
K45
K55
K65
K75
K85
K95
K105
K115
K125
0
0
-6EIy
L2
0
4EIyL
0
0
0
6EIy
L2
0
2EIyL
0
Eleventh column in
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x
z
y
54
d2
d5
d8
d11
Local Stiffness matrix
d11 =1
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55
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qy
6 EIyqyL2
4 EIyqyL
2 EIyqyL
6 EIyqyL2
z
y
56
6 EI
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d=1
z
y
x
6 EIy
L2
2 EIy
L
6 EIyL2 4 EIy
L
57
Eleventh column in0K
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58
=
Local Stiffnessmatrix
0
0
-6EIy
L2
0
2EIyL
0
0
0
6EIy
L2
0
4EIyL
0
K111
K211
K311
K411
K511
K611
K711
K811
K911
K1011
K1111
K1211
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59
Fourth column in
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x
z
y
60
d2
d5
d8
d11
Local Stiffness matrix
d4 =1
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61
G IGM q
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qx
G IxqxL
G IxqxL
62
L
G
I
M
rP
t q
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d=1 z
y
x
G Ix
L
G IxL
63
baa12
b1
a
b
16
36.3
3
1baI
4
43
p
Fourth column inK14
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64
=
Local Stiffnessmatrix
K14
K24
K34
K44
K54
K64
K74
K84
K94
K104
K114
K124
0
0
0GIxL
0
0
0
0
0
-GIxL
0
0
Tenth column in
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x
z
y
65
d2
d5
d8
d11
Local Stiffness matrix
d10 =1
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66
G I
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x
z
y
qx
G IxqxL
G IxqxL
67
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d=1
z
y
x
G Ix
L
G Ix
L
68
Tenth column inL l S iff 0
K110
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69
=
Local Stiffnessmatrix
0
0
0
-GIxL
0
0
0
0
0
GIxL
0
0
K110
K210
K310
K410
K510
K610
K710
K810
K910
K1010
K1110
K1210
EAL
0 0 0 0 0 -EAL
0 0 0 0 0
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70
L L
012EIz
L30 0 0
6EIzL2
0 -12EIz
L30 0 0
6EIzL2
0 012EIy
L30 -
6EIyL2
0 0 0 -12EIy
L30 -
6EIyL2
0
0 0 0GIxL
0 0 0 0 0 -GIxL
0 0
0 0 -6EIyL2
04EIy
L0 0 0
6EIyL2
02EIy
L0
06EIzL2
0 0 04EIz
L0 -
6EIzL2
0 0 02EIz
L
- EAL
0 0 0 0 0 EAL
0 0 0 0 0
0 -12EIz
L30 0 0 -
6EIzL2
012EIz
L30 0 0 -
6EIzL2
0 0 -12EIy
L30
6EIyL2
0 0 012EIy
L30
6EIyL2
0
0 0 0 - GIxL
0 0 0 0 0 GIxL
0 0
0 0 -6EIyL2
02EIy
L0 0 0
6EIyL2
04EIy
L0
06EIzL2
0 0 02EIz
L0 -
6EIzL2
0 0 04EIz
L
F =l
Drive the member transformation matrix
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x
yz
x
z
y
71
TF =g F
l
Drive the member transformation matrix
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72F1
F2
F4
F5
g
g
g
g
q
F3
F6
F2 cos F1 sin
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73
F1
F2g
g
F1 cosq
F2 cosq F1 sinq
F2 sinq
F1 = F1 cos q F2 sin q
F2 = F1 sin q + F2 cos q
g
g
F3
F3 = F3
g
F5 cos F4 sin
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74
F4
F5g
g
F4 cosq
F5 cosq F4 sinq
F5 sinq
F4 = F4 cos q F4 sin q
F5 = F5 sin q + F5 cos q
g
g
F6
F6 = F6
g
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75
F1g
F2g
F3g
F4g
F5g
F6g
=
cos q - sin q 0 0
sin q cos q
0 0
0 0 cos q - sin q
0 0 cos q
0 0
0
1
0
sin q 010 0
0 0 0
000
0
0
000
x
F1
F2
F3
F4
F5
F6
Drive the member transformation matrix
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x
yz
x
z
y
76TF =g
Fl
lFy
z
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77
x
y
z
qxx
qxyxz
Fx cos qxx
Fx cos qxy
Fx cos qxz
l
l
l
z
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78
x
y
z
qyx
qyyqyz
Fy cos qyx
Fy cos qyy
Fy cos qyz
l
l
l
z
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79
x
y
z
qzx
qzy
qzzFz cos qyx
Fz cos qzy
Fz cos qzz
l
l
l
g l l l
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80
Fx =g Fx cos qxxl Fy cos qyxl Fz cos qzxl+ +
Fx = Fx cxxlg Fy cyxl Fz czxl+ +
Fy
=
Fx cxylg Fy cyy
l Fz czyl+ +
Fz = Fx cxzlg Fy cyz
l Fz czzl+ +
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g lc c c
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82
=
Fxg
Fyg
Fzg
Fxl
Fyl
Fzl
cxx cyx czx
cxy cyy czy
cxz cyz czzx
yz
Unit vector of xl
Unit vector of yl
Unit vector of zl
[Fg] = [t] [Fl]
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z
y
xz
y
x
z y
x
83
x
y
z
010001
100
t
010
100
001
t
001
100
010
t
lTransformation matrix
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x
yz
84
TF =g F
l
T
t
=t
tt x
z
y
d2
d5
d8
d11
Example 1:
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p
Draw all diagrams for the shown space frame
where E = 1200 kN/cm2, G = 500 kN/cm2 andthe sections are shown in figure
85
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10 m
C
30x40cm
100 kN200 kN
40 kN
D
B
A
86
Sections Properties
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z
y
xz
y
x
z y
x
A =2400 cm2
Iz =1,280,000 cm4
Iy =180,000 cm4
Ix =550,180 cm
4
30x40cm A =1200 cm2
Iz =360,000 cm4
Iy =40,000 cm4
Ix =126,435 cm4
A =1200 cm2
Iz =160,000 cm4
Iy =90,000 cm4
Ix =194,385 cm4
87
Modeling
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C
D
A
d1
d2
d3
d4d5
d6
xy
z
Modeling
88
Stiffness matrix
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k11
K=
k21
k31
k12
k22
k32
k13
k23
k33
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
89
CAFirst element : ( BA)
E 1200 kN/ 2
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D
A
B
E = 1200 kN/cm2,G = 500 kN/cm2
A =2400 cm2
Iz =1,280,000 cm4
Iy =180,000 cm4
Ix =550,180 cm4
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91
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z
y
x
z
y
x
z y
xxy
z
92
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z
y
xz
y
x
z y
x
xy
z
AE
L12EIy
L36 EIy
L2
12EIy
L36 EIy
L2
93
xzE = 1200 kN/cm2
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z
y
xz
y
x
z y
x
xy
AE
L12EIy
L36 EIy
L2
12EIy
L36 EIy
L2A =2400 cm2
Iz =1,280,000 cm4Iy =180,000 cm
4
Ix =550,180 cm4
A =1200 cm2
Iz
=360,000 cm4
Iy =40,000 cm4
Ix =126,435 cm4
A =1200 cm2
Iz =160,000 cm4
Iy =90,000 cm4
Ix =194,385 cm4
94
xzE = 1200 kN/cm2
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z
y
x
xy
AE
L
A =1200 cm2
Iz
=360,000 cm4
Iy =40,000 cm4
Ix =126,435 cm4
1200X1200
800
1,800
95
xz
A =2400 cm2
E = 1200 kN/cm2
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z
y
x
xy
12EIy
L3
6 EIyL2
A =2400 cm2
Iz =1,280,000 cm4
Iy =180,000 cm4
Ix =550,180 cm4
12x1200X180,00010003
2.6
6x1200X180,000
10002
1,296
96
xzE = 1200 kN/cm2
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z y
x
xy
12EIy
L36 EIy
L2
A =1200 cm2
Iz =160,000 cm4
Iy =90,000 cm4
Ix =194,385 cm4
12x1200X90,000
10
00
3
1.3
6x1200X90,000
10002
648
97
xy
z
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xy
1.3648
2.6
1,296
1,800
d1d2
d3
d4d5
d6
k11
=
k21
k31
k41
k51
k61
1,803.9
0
0
0
- 648
-1,29698
First column in Stiffness matrix
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99
k11
=
k21
k31
k41
k51
k61
1,803.9
0
0
0
- 648
-1,296
Second column in Stiffness matrix
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C
D
B
A
xy
z
d2=1
100
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101
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z
y
x
z
y
x
z y
x
xy
z
102
6 EIy
L2
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z
y
xz
y
x
z y
x
xy
z
AEL
12EIz
L3
12EIy
L3
L
6 EIz
L2
103
x
z 6 EIyL2
E = 1200 kN/cm2
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z
y
xz
y
x
z y
x
xy
AEL
12EIz
L3
12EIy
L3
L
6 EIz
L2
A =2400 cm2
Iz =1,280,000 cm4
Iy =180,000 cm4
Ix =550,180 cm4
A =1200 cm2
Iz =360,000 cm4
Iy =40,000 cm4
Ix =126,435 cm4
A =1200 cm2
Iz =160,000 cm4
Iy =90,000 cm4
Ix =194,385 cm4 104
x
z A =1200 cm2Iz =360,000 cm
4
E = 1200 kN/cm2
8/2/2019 Stiffness 12
105/219
z
y
x
xy
12EIy
L3
6 EIy
L2
Iz =360,000 cmIy =40,000 cm
4
Ix =126,435 cm4
12x1200X40,000
8003
1.1
6x1200X40,000
8002
450
105
x
zE = 1200 kN/cm2
8/2/2019 Stiffness 12
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z
y
x
xy
AEL
A =2400 cm2
Iz =1,280,000 cm4
Iy =180,000 cm4
Ix =550,180 cm4
1200X2400
1000
2,880
106
x
zE = 1200 kN/cm2
8/2/2019 Stiffness 12
107/219
z y
x
xy
12EIz
L3
6 EIz
L2
A =1200 cm2
Iz =160,000 cm4
Iy =90,000 cm4
Ix =194,385 cm4
12x1200X160,000
10003
2.3
6x1200X160,000
10002
1,152
107
xy
zE = 1200 kN/cm2450
8/2/2019 Stiffness 12
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xy
2,880
2.31,152
1.1
k12
=
k22
k32
k42
k52
k62
2,883.4
0
0
0
1,152
450
d1d2
d3
d4d5
d6
108
Second column in Stiffness matrix
8/2/2019 Stiffness 12
109/219
109
k12
=
k22
k32
k42
k52
k62
2,883.4
0
0
0
1,152
450
Third column in Stiffness matrix
8/2/2019 Stiffness 12
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C
D
B
A
xy
z
d3=1
110
8/2/2019 Stiffness 12
111/219
111
yd =1d3=1
8/2/2019 Stiffness 12
112/219
z
y
x
z
y
x
z y
x
xy
z
d3=1
d3=1
112
yd =1d3=1x
zE = 1200 kN/cm2
8/2/2019 Stiffness 12
113/219
z
y
x
z
y
x
z y
x
d3=1
d3=1AEL
12EIz
L3
6 EIz
L2
12EIz
L3
6 EIz
L2
xy
A =2400 cm2
Iz =1,280,000 cm4
Iy =180,000 cm4
Ix =550,180 cm4
A =1200 cm2
Iz =360,000 cm4
Iy =40,000 cm4Ix =126,435 cm
4
A =1200 cm2
Iz =160,000 cm4
Iy =90,000 cm4
Ix =194,385 cm
4
113
xy
zA =1200 cm2
Iz =360,000 cm4
E = 1200 kN/cm2
8/2/2019 Stiffness 12
114/219
z
y
x
d3=1
12EIz
L3
6 EIz
L2
xy Iz 360,000 cmIy =40,000 cm
4
Ix =126,435 cm4
12x1200X360,000
800310.1
6x1200X360,000
8002
4,050
114
x
zA =2400 cm2
I =1 280 000 cm4
E = 1200 kN/cm2
8/2/2019 Stiffness 12
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z
y
x
d3=1
12EIz
L3
6 EIz
L2
xyIz =1,280,000 cm
4
Iy =180,000 cm4
Ix =550,180 cm4
12x1200X1,280,000
10003
18.4
6x1200X1,280,000
100029,216
115
xy
zE = 1200 kN/cm2
8/2/2019 Stiffness 12
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z y
x
d3=1AEL
xy
A =1200 cm2
Iz =160,000 cm4
Iy =90,000 cm4
Ix =194,385 cm
4
1200X1200
1000
1,440
116
xy
8/2/2019 Stiffness 12
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1,440
18.4
9,216 10.1
4,050
k13
=
k23
k33
k43
k53
k63
0
0
1,468.5
- 4,050
9,216
0d1d2
d3
d4d5
d6
117
Third column in Stiffness matrix
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118
k13
=
k23
k33k43
k53
k63
0
0
1,468.5
- 4,050
9,216
0
Fourth column in Stiffness matrix
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119/219
C
D
B
A
xy
z
d4=1
119
8/2/2019 Stiffness 12
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120
y
8/2/2019 Stiffness 12
121/219
z
y
x
z
y
x
z y
xxy
z
121
y
8/2/2019 Stiffness 12
122/219
z
y
x
z
y
x
z y
xxy
z
GIx
L
4 EIz
L
6 EIz
L2
4 EIz
L
6 EIz
L2
122
yxy
zE = 1200 kN/cm2
8/2/2019 Stiffness 12
123/219
z
y
x
z
y
x
z y
x
y
GIx
L
4 EIz
L
6 EIz
L2
4 EIz
L
6 EIz
L2
A =2400 cm2
Iz =1,280,000 cm4
Iy =180,000 cm4
Ix =550,180 cm4
A =1200 cm2
Iz =360,000 cm4
Iy =40,000 cm4
Ix =126,435 cm4
A =1200 cm2
Iz =160,000 cm4
Iy =90,000 cm4
Ix
=194,385 cm4
123
xy
z A =1200 cm2Iz =360,000 cm
4
Iy =40,000 cm4
G = 500 kN/cm2
bb36314
8/2/2019 Stiffness 12
124/219
z
y
x
y
GIx
L
Iy 40,000 cmIx =126,435 cm
4
500X126,435
800
79,022
124
baa12
b1
a
b
16
36.3
3
1baI
4
43
p
xy
z A =2400 cm2Iz =1,280,000 cm
4
I 180 000 4
E = 1200 kN/cm2
8/2/2019 Stiffness 12
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z
y
x
y
4 EIz
L
6 EIz
L2
Iy =180,000 cm4
Ix =550,180 cm4
6x1200
X1
,280,000
10002
9,216
4x1200X1,280,000
1000
6,144,000
125
xy
z A =1200 cm2Iz =160,000 cm
4
I =90 000 cm4
E = 1200 kN/cm2
8/2/2019 Stiffness 12
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z y
x
y
4 EIz
L
6 EIz
L2
Iy =90,000 cmIx =194,385 cm
4
6x1200X160,000
10002
1,152
4x1200X160,000
1000
768,000
126
xy
z
8/2/2019 Stiffness 12
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y
768,0009,216
6,144,000
79,022
k14
=
k24
k34
k44
k54
k64
1,152
0
9,216
0
6,991,022
0
d1d2
d3
d4d5
d6
127
Fourth column in Stiffness matrix
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128
k14
=
k24
k34
k44
k54
k64
1,152
0
9,216
0
6,991,022
0
A
Fifth column in Stiffness matrix
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129/219
C
D
B
A
xy
z
d5=1
129
8/2/2019 Stiffness 12
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130
y
8/2/2019 Stiffness 12
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z
y
x
z
y
x
z y
xxy
z
131
y
8/2/2019 Stiffness 12
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z
y
x
z
y
x
z y
xxy
z
GIx
L
4 EIy
L
6 EIy
L2
4 EIz
L
6 EIz
L2
132
yxyzE = 1200 kN/cm2
8/2/2019 Stiffness 12
133/219
z
y
x
z
y
x
z y
x
GIx
L
4 EIy
L
6 EIy
L2
4 EIz
L
6 EIz
L2
y
A =2400 cm2
Iz =1,280,000 cm4
Iy
=180,000 cm4
Ix =550,180 cm4
A =1200 cm2
Iz =360,000 cm4
Iy
=40,000 cm4
Ix =126,435 cm4
A =1200 cm2
Iz =160,000 cm4
Iy =90,000 cm4
Ix =194,385 cm4
133
xyz A =1200 cm
2
Iz =360,000 cm4
Iy =40,000 cm4
I =126 435 cm4
E = 1200 kN/cm2
8/2/2019 Stiffness 12
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z
y
x
4 EIz
L6 EIz
L2
y Ix =126,435 cm
6x1200X360,000
8002
4,050
4x1200X360,000
800
2,160,000
134
xyz A =2400 cm
2
Iz =1,280,000 cm4
Iy =180,000 cm4
I =550 180 cm4
G = 500 kN/cm2
8/2/2019 Stiffness 12
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z
y
x
GIx
L
Ix =550,180 cm
500X550,180
1000
275,090
135
xy
z A =1200 cm2Iz =160,000 cm
4
Iy =90,000 cm4
E = 1200 kN/cm2
8/2/2019 Stiffness 12
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z y
x 4 EIyL
6 EIy
L2
y ,Ix =194,385 cm
4
6x1200X90,000
10002
648
4x1200X90,000
1000
432,000
136
xy
z
8/2/2019 Stiffness 12
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d1d2
d3
d4d5
d6
275,090
4,050
2,160,000
648 432,000
k15
=
k25
k35
k45
k55
k65
0
- 648
- 4,050
0
0
2,867,090
137
Fifth column in Stiffness matrix
8/2/2019 Stiffness 12
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138
k15
=
k25
k35
k45
k55
k65
0
- 648
- 4,050
0
0
2,867,090
A
Sixth column in Stiffness matrix
d 1
8/2/2019 Stiffness 12
139/219
C
D
B
A
xy
z
d6=1
139
8/2/2019 Stiffness 12
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140
y
8/2/2019 Stiffness 12
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z
x
z
y
x
z y
xxy
z
d6=1
d6=1d6=1
141
y
8/2/2019 Stiffness 12
142/219
z
x
z
y
x
z y
xxy
z
d6=1
d6=1
d6=1GIx
L
4 EIy
L
6 EIy
L2
4 EIy
L
6 EIyL2
142
yxy
zE = 1200 kN/cm2
8/2/2019 Stiffness 12
143/219
z
x
z
y
x
z y
x
d6=1
d6=1
d6=1GIx
L
4 EIy
L
6 EIy
L2
4 EIy
L
6 EIyL2
y
A =2400 cm2
Iz =1,280,000 cm4
Iy =180,000 cm4
Ix =550,180 cm4
A =1200 cm2
Iz =360,000 cm4
Iy =40,000 cm4
Ix =126,435 cm4
A =1200 cm2
Iz =160,000 cm4
Iy =90,000 cm4
Ix =194,385 cm4
143
xy
z A =1200 cm2Iz =360,000 cm
4
Iy =40,000 cm4
E = 1200 kN/cm2
8/2/2019 Stiffness 12
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z
y
x
d6=1
4 EIy
L
6 EIy
L2
y ,Ix =126,435 cm
4
6x1200X40,000
80024x1200X40,000
800450
240,000
144
xy
z A =2400 cm2Iz =1,280,000 cm
4
Iy =180,000 cm4
I 550 180 cm4
E = 1200 kN/cm2
8/2/2019 Stiffness 12
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z
y
x
d6=1
4 EIy
L
6 EIy
L2
Ix =550,180 cm4
4x1200X180,000
1000
6x1200X180,000
10002
1,296
864,000145
xy
zG = 500 kN/cm2
8/2/2019 Stiffness 12
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z y
x
d6=1
GIx
L
A =1200 cm2
Iz =160,000 cm4
Iy =90,000 cm4
Ix =194,385 cm4
500X194,385
1000
97,192.5
146
xy
8/2/2019 Stiffness 12
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1,296
864,000
97,192.5
450
240,000
k16
=
k26
k36
k46
k56
k66
450
-1,296
0
0
0
1,201,192.5d1d2
d3
d4d5
d6
147
Sixth column in Stiffness matrix
8/2/2019 Stiffness 12
148/219
148
k16
=
k26
k36
k46
k56
k66
450
-1,296
0
0
0
1,201,192.5
k13
k23 0
0k12
k22 2,883.4
0k11
k21
1,803.9
0
8/2/2019 Stiffness 12
149/219
k16
=
k26
k36
k46
k56
k66
450
-1,296
0
0
0
1,201,192.5
k15
=
k25
k35
k45
k55
k65
0
- 648
- 4,050
0
0
2,867,090
k14
=
k24
k34
k44
k54
k64
1,152
0
9,216
0
6,991,022
0
=
k33
k43
k53
k63
1,468.5
- 4,050
9,216
0
=
k32
k42
k52
k62
0
0
1,152
450
=
k31
k41
k51
k61
0
0
- 648
-1,296
149
8/2/2019 Stiffness 12
150/219
450-1,296
0
0
0
1,201,192.5
0
- 648
- 4,050
0
0
2,867,090
1,1520
9,216
0
6,991,022
0
0
0
1,468.5
- 4,050
9,216
0
2,883.4
0
0
0
1,152
450
=K
1,803.9
0
0
0
- 648
-1,296
150
100 kN200 kNA
Force vector
8/2/2019 Stiffness 12
151/219
10 m
C
30x40cm
40 kN
D
B
151
200 kN 100 kN250 100
8/2/2019 Stiffness 12
152/219
40 kN
20
20
50
250100
100 50
50
100
100
Fixed End Reaction(FER)
152
50
200 kN 100 kN250 100
8/2/2019 Stiffness 12
153/219
40 kN
20
20
50
250100
100 50
50
100
100
Fixed End Action(FEA)
153
50
200 kN 100 kN250 100
d3
d6
8/2/2019 Stiffness 12
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40 kN
20
20
50
250100
100 50
50
100
100
d1
d2d4d5
F1
=
F2
F3
F4
F5
F6
20
0
- 150
- 250
100
- 50154
Stiffness EquationF = K D
8/2/2019 Stiffness 12
155/219
20
0
- 150
- 250
100
- 50
450
-1,296
0
0
0
1,201,192.5
0
- 648
- 4,050
0
0
2,867,090
1,152
0
9,216
0
6,991,022
0
0
0
1,468.5
- 4,050
9,216
0
2,883.4
0
0
0
1,152
450
1,803.9
0
0
0
- 648
-1,296
d1
=
d2
d3
d4
d5
d6
155
Stiffness Equation
D = K-1 F
8/2/2019 Stiffness 12
156/219
20
0
- 150
- 250
100
- 50
450
-1,296
0
0
0
1,201,192.5
0
- 648
- 4,050
0
0
2,867,090
1,152
0
9,216
0
6,991,022
0
0
0
1,468.5
- 4,050
9,216
0
2,883.4
0
0
0
1,152
450
1,803.9
0
0
0
- 648
-1,296
d1
=
d2
d3
d4
d5
d6
-1
156
Deformations
8/2/2019 Stiffness 12
157/219
0.011027
- 0.000035
- 0.103072
1.0012x10-4
d1
=
d2
d3
d4
d5
d6
-1.0823x10-4
-2.9715x10-5
157
Internal Forces Normal Forced3
d4d
d6
8/2/2019 Stiffness 12
158/219
d1d2d4d5
AEL
ND
=
158
Internal Forces Normal Forced3
d4d
d6
0.011027d1
8/2/2019 Stiffness 12
159/219
d1d2d4d5
- 0.000035
- 0.103072
1.0012x10-4
1
=
d2
d3
d4
d5
d6
-1.0823x10-4
-2.9715x10-5
z
y
x
AE
L1
,800
NBC = 1,800x.0011027 = 2 kN compression
159
Internal Forces Normal Force
d
d3
d4d
d6
0.011027d1
8/2/2019 Stiffness 12
160/219
d1d2d4d5
- 0.000035
- 0.103072
1.0012x10-4=
d2
d3
d4
d5
d6
-1.0823x10-4
-2.9715x10-5
z
y
x
AE
L
2,880
NBA = 2,880x.000035 = 0.1 kN tension
160
Internal Forces Normal Force
dd
d3
d4d
d6
0.011027d1d 1AE
1,440
8/2/2019 Stiffness 12
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d1d2d4d5
- 0.000035
- 0.103072
1.0012x10-4=
d2
d3
d4
d5
d6
-1.0823x10-4
-2.9715x10-5
NBD = 1,440x0.103072 = 148.4 kN compression
z y
x
d3=1AE
L
161
Internal ForcesC
A 2
Normal Force
8/2/2019 Stiffness 12
162/219
BB
A
B
D
148.4
148.4
0.1
0.1
2
162
A
Internal Forces Normal Force
8/2/2019 Stiffness 12
163/219
C
D
A
148.4
0.12
163
Internal Forces Torsional Moment
dd
d3
d4d5
d6
8/2/2019 Stiffness 12
164/219
d1d24d5
IxGL
T qt=
164
Internal Forces Torsional Moment
dd
d3
d4d5
d6
0.011027d1IGT
8/2/2019 Stiffness 12
165/219
d1d24d5
- 0.000035
- 0.103072
1.0012x10-4=
d2
d3
d4
d5
d6
-1.0823x10-4
-2.9715x10-5
IxG
L
T qt=
GIx
L79,021.9
TBC = 79,021.9x1.0012x10-4 = 7.9 kNm
165
Internal Forces Torsional Moment
dd
d3
d4d5
d6
0.011027d1
8/2/2019 Stiffness 12
166/219
d1d24d5
- 0.000035
- 0.103072
1.0012x10-4=
d2
d3
d4
d5
d6
-1.0823x10-4
-2.9715x10-5IxG
LT qt=
GIx
L275,090
TBA = 275,090x-1.0823x10-4 = - 29.77 kNm
166
Internal Forces Torsional Moment
dd
d3
d4d5
d6
0.011027d1GIx
L
8/2/2019 Stiffness 12
167/219
d1d24d5
- 0.000035
- 0.103072
1.0012x10-4=
d2
d3
d4
d5
d6
-1.0823x10-4
-2.9715x10-5
IxG
LT qt=
d6=1
L
97,192.5
TBD = 97,192.5x-2.9715x10-5 = - 2.89 kNm
167
Internal ForcesC
A 7.9
Torsional Moment
8/2/2019 Stiffness 12
168/219
BB
A
B
D
2.89
2.89
29.77
29.77
7.9
168
A
Internal Forces Torsional Moment
8/2/2019 Stiffness 12
169/219
C
D
A
2.89
29.77 7.9
169
Internal Forces
dd
d3
d4d5
d6 Bending momentIn Plan
8/2/2019 Stiffness 12
170/219
d1d25
0.011027
- 0.000035
- 0.103072
1.0012x10-4
d1
=
d2
d3
d4
d5
d6
-1.0823x10-4
-2.9715x10-5
d3=1
6 EIz
L24,050
4,050
170
Internal Forces
dd
d3
d4d5
d6 Bending momentIn Plan
8/2/2019 Stiffness 12
171/219
d1d25
0.011027
- 0.000035
- 0.103072
1.0012x10-4
d1
=
d2
d3
d4
d5
d6
-1.0823x10-4
-2.9715x10-5
4 EIz
L2,160,000
1,080,000
2 EIz
L
171
Internal Forces
dd
d3
d4d5
d6 Bending momentIn Plan
8/2/2019 Stiffness 12
172/219
d1d25
0.011027
- 0.000035
- 0.103072
1.0012x10-4
d1
=
d2
d3
d4
d5
d6
-1.0823x10-4
-2.9715x10-5
100 kN
100
100
172
Internal Forces0.011027
- 0.000035
d1
d2
Bending momentIn Plan
d 1
C
8/2/2019 Stiffness 12
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- 0.103072
1.0012x10-4=
d3
d4
d5
d6
-1.0823x10-4
-2.9715x10-5
d3=1
4,050
4,050
2,160,000
1,080,000
100
100
B
C
MBC = 100+4,050 d3 2,160,000 d5MBC = 100+4,050x-.103072 2,160,000x-1.0823x10
-4
MBC
= - 83.66 kNm 173
Internal Forces0.011027
- 0.000035
d1
d2
Bending momentIn Plan
d 1
C
8/2/2019 Stiffness 12
174/219
- 0.103072
1.0012x10-4=
d3
d4
d5
d6
-1.0823x10-4
-2.9715x10-5
d3=1
4,050
4,050
2,160,000
1,080,000
100
100
B
C
MCB = -100+4,050 d3 1,080,000 d5MCB = -100+4,050x-.103072 1,080,000x-1.0823x10
-4
MCB
= - 400.55 kNm 174
8/2/2019 Stiffness 12
175/219
Internal Forces
dd
d3
d4d5
d6 Bending momentIn Plan
d 1 200 kNA
8/2/2019 Stiffness 12
176/219
d1d2d3=1
6 EIz
L2
9,216
9,216
4 EIz
L6,144,000
3,072,000
200 kN
250
250
MBA = 250+9,216 d3 + 6,144,000 d4MBA = 250+9,216x-.103072+ 6,144,000x1.0012x10
-4
MBA = - 84.77 kNm
B
0.011027
- 0.000035
- 0.103072
1.0012x10-4
d1
=
d2
d3
d4
d5
d6
-1.0823x10-4
-2.9715x10-5
176
Internal Forces
dd
d3
d4d5
d6 Bending momentIn Plan
d 1 200 kNA
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177/219
d1d2d3=1
6 EIz
L2
9,216
9,216
4 EIz
L6,144,000
3,072,000
200 kN
250
250
MAB = -250+9,216 d3 + 3,072,000 d4MAB = -250+9,216x-.103072+ 3,072,000x1.0012x10
-4
MAB = - 892.3 kNm
B
0.011027
- 0.000035
- 0.103072
1.0012x10-4
d1
=
d2
d3
d4
d5
d6
-1.0823x10-4
-2.9715x10-5
177
Internal Forces
dd2
d3
d4d5
d6 Bending momentIn Plan
8/2/2019 Stiffness 12
178/219
d1d2
84.77
892.3B
A200 kN
100 + (892.3+84.77)/10
197.71 100 - (892.3+84.77)/10
2.29
178
Internal Forces
dd2
d3
d4d5
d6 Bending momentIn Plan
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d1d20.011027
- 0.000035
- 0.103072
1.0012x10-4
d1
=
d2
d3
d4
d5
d6
-1.0823x10-4
-2.9715x10-5
6 EIz
L2
1,152
1,152
4 EIz
L
768,000
384,000
B
D
MBD = 1,152 d2 + 768,000 d4MBD = 1,152x-.000035+ 768,000x1.0012x10
-4
MBD = 76.85 kNm
179
Internal Forces
d1d2
d3
d4d5
d6 Bending momentIn Plan
8/2/2019 Stiffness 12
180/219
d1d20.011027
- 0.000035
- 0.103072
1.0012x10-4
d1
=
d2
d3
d4
d5
d6
-1.0823x10-4
-2.9715x10-5
6 EIz
L2
1,152
1,152
4 EIz
L
768,000
384,000
B
D
MDB = 1,152 d2 + 384,000 d4MDB = 1,152x-.000035+ 384,000x1.0012x10
-4
MDB = 38.41 kNm
180
Internal Forces
d1d2
d3
d4d5
d6 Bending momentIn Plan
8/2/2019 Stiffness 12
181/219
d1d20.011027
- 0.000035
- 0.103072
1.0012x10-4
d1
=
d2
d3
d4
d5
d6
-1.0823x10-4
-2.9715x10-5
B
D
76.85
38.41
(76.85+38.41)/10
11.53
11.53
181
Internal Forces Bending momentIn PlanC
100 kN892 3
A200 kN
8/2/2019 Stiffness 12
182/219
83.66
400.55B
100 kN
110.53
10.53
84.77
892.3B
197.71
2.29 B
D
76.85
38.41
11.53
11.53182
CA
8/2/2019 Stiffness 12
183/219
C
D
183
CA
8/2/2019 Stiffness 12
184/219
C
D
184
Internal Forces
d1d2
d3
d4d5
d6 Bending momentOut of Plan
4500.011027d1
d
8/2/2019 Stiffness 12
185/219
d1d2
6 EIy
L2
450
450
d6=1
4 EIy
L
240,000
120,000
- 0.000035
- 0.103072
1.0012x10-4=
d2
d3
d4
d5
d6
-1.0823x10-4
-2.9715x10-5
B
C
MBC = 450 d2 + 240,000 d6MBC = 450x-.000035 + 240,000x-2.9715x10
-5
MBC = - 7.15 kNm
185
Internal Forces
d1d2
d3
d4d5
d6 Bending momentOut of Plan
4500.011027d1
d
8/2/2019 Stiffness 12
186/219
d12
6 EIy
L2
450
450
d6=1
4 EIy
L
240,000
120,000
- 0.000035
- 0.103072
1.0012x10-4=
d2
d3
d4
d5
d6
-1.0823x10-4
-2.9715x10-5
B
C
MCB = 450 d2 + 120,000 d6MCB = 450x-.000035 + 120,000x-2.9715x10
-5
MCB = - 3.58 kNm
186
3 58
Internal Forces Bending momentOut of Plan
8/2/2019 Stiffness 12
187/219
7.15
3.58
B
C
(7.15+3.58)/8
1.34
1.34
187
Internal Forces
d1d2
d3
d4d5
d6 Bending momentOut of Plan
0.011027
0 000035
d1
d
A
8/2/2019 Stiffness 12
188/219
12 - 0.000035
- 0.103072
1.0012x10-4=
d2
d3
d4
d5
d6
-1.0823x10-4
-2.9715x10-5
MBA = 50-1,296 d1 + 864,000 d6MBA = 50-1,296x.011027 + 864,000x-2.9715x10
-5
MBA = 10.04 kNm
6 EIy
L2
1,296
1,296
d6=1
4 EIy
L
864,000
432,000
B
40 kN
50
50
188
Internal Forces
d1d2
d3
d4d5
d6 Bending momentOut of Plan
0.011027
0 000035
d1
d
A
8/2/2019 Stiffness 12
189/219
12 - 0.000035
- 0.103072
1.0012x10-4=
d2
d3
d4
d5
d6
-1.0823x10-4
-2.9715x10-5
MAB = -50-1,296 d1 + 432,000 d6MAB = -50-1,296x.011027 + 432,000x-2.9715x10
-5
MAB = - 77.13 kNm
6 EIy
L2
1,296
1,296
d6=1
4 EIy
L
864,000
432,000
B
40 kN
50
50
189
77.1320+(77.13-10.04)/10
Internal Forces Bending momentOut of Plan
8/2/2019 Stiffness 12
190/219
10.04
B
C
26.71
40 kN
13.29
20-(77.13-10.04)/10
190
Internal Forces
d1d2
d3
d4d5
d6
0 011027d
Bending moment
B
Out of Plan
8/2/2019 Stiffness 12
191/219
120.011027
- 0.000035
- 0.103072
1.0012x10-4
d1
=
d2
d3
d4
d5
d6
-1.0823x10-4
-2.9715x10-5
6486 EIy
L2
648
4 EIy
L
432,000
216,000
B
D
MBD
= 648 d1
- 432,000 d5
MBD = 648x.011027 - 432,000x-1.0823x10-4
MBD = 53.9 kNm
191
Internal Forces
d1d2
d3
d4d5
d6
0 011027d
Bending moment
B
Out of Plan
8/2/2019 Stiffness 12
192/219
10.011027
- 0.000035
- 0.103072
1.0012x10-4
d1
=
d2
d3
d4
d5
d6
-1.0823x10-4
-2.9715x10-5
6486 EIy
L2
648
4 EIy
L
432,000
216,000
B
D
MDB
= 648 d1
- 216,000 d5
MDB = 648x.011027 - 216,000x-1.0823x10-4
MDB = 30.52 kNm
192
Internal Forces Bending momentOut of PlanB
8/2/2019 Stiffness 12
193/219
53.9
30.52
8.44(53.9+30.52)/10
8.44D
193
Internal Forces Bending momentOut of Plan
77.13 3.58
8/2/2019 Stiffness 12
194/219
B
C
B
A
B
D
53.9
30.528.44
8.44
10.04
26.7140 kN
13.29
7.15
1.34
1.34
194
CA
8/2/2019 Stiffness 12
195/219
C
D
195
CA
8/2/2019 Stiffness 12
196/219
C
D
196
CA
8/2/2019 Stiffness 12
197/219
D
A
197
Internal Forces Bending moment
10 04 2 2910.53
8/2/2019 Stiffness 12
198/219
53.9
8.44
10.04
13.29
7.15
1.34
84.77
2.29
76.85
11.53
83.660.1
148.4
2
29.77
2.89
7.9
198
Internal ForcesC
A 7.9
Torsional Moment
8/2/2019 Stiffness 12
199/219
BB
B
D
2.89
2.89
29.77
29.77
7.9
199
8/2/2019 Stiffness 12
200/219
200
8/2/2019 Stiffness 12
201/219
201
8/2/2019 Stiffness 12
202/219
202
8/2/2019 Stiffness 12
203/219
Summary
203
d2
8/2/2019 Stiffness 12
204/219
d1
d2d3
204
D1AEAE
8/2/2019 Stiffness 12
205/219
AE
L
AE
L
D1
AELAEL
205
D6 EI
8/2/2019 Stiffness 12
206/219
DD6 EI
L2
L2
D12 EI
L3
D12 EI
L3
206
D6 EI
L2
8/2/2019 Stiffness 12
207/219
D
L
D6 EI
L2
D12 EI
L3D12 EI
L3
207
D3 EI
L2
8/2/2019 Stiffness 12
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D
L
D3 EI
L3
D3 EI
L3
208
D3 EI
8/2/2019 Stiffness 12
209/219
D
L2
D3 EI
L3D
3 EIL3
209
8/2/2019 Stiffness 12
210/219
q
q4 EIL
q2 EI
L
q6 EI
L2
q6 EI
L2
210
8/2/2019 Stiffness 12
211/219
q
q4 EI
Lq2 EI
Lq6 EI
L2
q6 EI
L2
211
q
8/2/2019 Stiffness 12
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q3 EI
L q3 EIL2
q3 EIL2
212
8/2/2019 Stiffness 12
213/219
q
q3 EI
L
q3 EI
L2
q3 EI
L2
213
6EI sin EA cos2q 12EI sin2q
+
EA cos q sin qL
-12EI sinq cosqL3
HorizontalDeformation
8/2/2019 Stiffness 12
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6EI sinqL2
q
L L3+
6EI sinq
L2 SSS 6EI sinq
L2
EA cos2qL
12EI sin2qL3
+EA cos q sin q
L-12EI sinq cosq
L3
214
EA sin2qL
12EI cos2qL3
+
EA sinq cosq 12EI cosq sinq
VerticalDeformation
8/2/2019 Stiffness 12
215/219
6EI cosqL2
q
L -
L3
6EI cosqL2 S
SS
EA sinq cosqL
-12EI cosq sinqL3
EA sin2qL
12EI cos2qL3
+6EI cosq
L2-
215
4 EI
6 EI sin qL2
RotationalDeformation
8/2/2019 Stiffness 12
216/219
L6 EI cos q
L2
2 EI
L
SSS
-
6 EI sin qL2
6 EI cos qL2
4 EI
L216
bDbx
DbyInternal forces
Normal Force
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( )cos q+ ( )sin qbx Dax- Dby Day-AELN=
a
q
DaxD
ay
[ ]
217
Internal forces
Bending moment
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MAB= ( 2 + )qA qBM(FER) AB +2 EI
L
3DL
-
MBA=2 EI
L( + 2 )q
AqBM(FER) BA +
3DL
-
218
8/2/2019 Stiffness 12
219/219
Questions