Stiffness 12

8/2/2019 Stiffness 12

1/219

Lecture No. : 12 1


2/219

2


3/219

d1

d2d3x

yz

d4

d5

d6

3


4/219

x

yz

x

z

y

4


5/219

x

z

y

5

d2

d5

d8

d11


6/219

6


7/2197

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


8/2198


9/219

x

z

y

9

d2

d5

d8

d11

First column inLocal Stiffness matrix

d1

=1


10/21910


11/219

AE

L

AEL

AE

L

AE

L

11


12/219

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


13/219

x

z

y

13

d2

d5

d8

d11

Seventh column inLocal Stiffness matrix

d7

=1


14/219

14


15/219

AE

L

AE

L

AE

L

AE

L

15


16/219

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


17/219

17

S d l i


18/219

x

z

y

18

d2

d5

d8

d11

Second column inLocal Stiffness matrix

d2

=1


19/219

19

Shear in Plan


20/219

Dy

z

y6EIzDy

L2

6EIzDy

L2

12EIzDy

L3

12EIzDy

L3 20


21/219

d=1

z

y

x6EIz

L2

12EIz

L3

6EIzL2

12EIzL3

21

S d l i


22/219

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


23/219

x

z

y

23

d2

d5

d8

d11

Eighth column inLocal Stiffness matrix

d8 =1


24/219

24

z


25/219

x

yz

z

y

6EIzDyL2

12EIzDy

L3

12EIzDyL3

6EIzDyL2

Dy

25


26/219

d=1

z

y

x6EIz

L2

12EIz

L3

6EIzL2

12EIzL3

26

Eighth column in


27/219

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


28/219

x

z

y

28

d2

d5

d8

d11

Third column inLocal Stiffness matrix

d3 =1


29/219

29

Shear out of Plan


30/219

Dz

z

y

6EIyDz

L2

12EIyDz

L36EIyDz

L2

12EIyDzL3

30


31/219

d=1

z

y

x

6EIy

L2

12EIy

L3 6EIyL2

12EIyL3

31

Third column in


32/219

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


33/219

x

z

y

33

d2

d5

d8

d11

Ninth column inLocal Stiffness matrix

d9 =1


34/219

34


35/219

z

y

Dz

6EIyDz

L2

12EIyDzL3

6EIyDz

L2

12EIyDz

L3

35


36/219

d=1

z

y

x

6EIy

L2

12EIy

L3 6EIyL2

12EIyL3

36

Ninth column in


37/219

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


38/219

38

Sixth column in


39/219

x

z

y

39

d2

d5

d8

d11

Sixth column inLocal Stiffness matrix

d6 =1

R i i Pl


40/219

40

Rotation in Plan


41/219

z

y qz4 EIzqz

L

6 EIzqzL2

6 EIzqzL2

2 EIzqzL

41


42/219

z

y

x

d=1

4 EIz

L

6 EIz

L2

6 EIz

L2

2 EIz

L

42

Sixth column in


43/219

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


44/219

x

z

y

44

d2

d5

d8

d11

Twelfth column inLocal Stiffness matrix

d12 =1


45/219

45

E


46/219

z

y

qz2 EIzqzL

6 EIzqzL2

6 EIzqzL2

4 EIzqzL

46


47/219

z

y

x

d=1

4 EIz

L

6 EIz

L2

6 EIz

L2

2 EIzL

47

Twelfth column inK


48/219

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


49/219

x

z

y

49

d2

d5

d8

d11

Local Stiffness matrix

d5 =1

Rotation out of Plan


50/219

50

Rotation out of Plan


51/219

qy

6 EIyqyL2

2 EIyqyL

6 EIyq

y

L2

4 EIyq

y

L

z

y

51


52/219

d=1

z

y

x

6 EIy

L2

2 EIy

L

6 EIyL2

4 EIy

L

52

Fifth column inK


53/219

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


54/219

x

z

y

54

d2

d5

d8

d11


d11 =1


55/219

55


56/219

qy

6 EIyqyL2

4 EIyqyL

2 EIyqyL

6 EIyqyL2

z

y

56

6 EI


57/219

d=1

z

y

x

6 EIy

L2

2 EIy

L

6 EIyL2 4 EIy

L

57

Eleventh column in0K


58/219

58

=


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


59/219

59

Fourth column in


60/219

x

z

y

60

d2

d5

d8

d11


d4 =1


61/219

61

G IGM q


62/219

qx

G IxqxL

G IxqxL

62

L

G

I

M

rP

t q


63/219

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


64/219

64

=


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


65/219

x

z

y

65

d2

d5

d8

d11


d10 =1


66/219

66

G I


67/219

x

z

y

qx

G IxqxL

G IxqxL

67


68/219

d=1

z

y

x

G Ix

L

G Ix

L

68

Tenth column inL l S iff 0

K110


69/219

69

=


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


70/219

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


71/219

x

yz

x

z

y

71

TF =g F

l



72/219

72F1

F2

F4

F5

g

g

g

g

q

F3

F6

F2 cos F1 sin


73/219

73

F1

F2g

g

qq

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


74/219

74

F4

F5g

g

qq

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


75/219

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



76/219

x

yz

x

z

y

76TF =g

Fl

lFy

z


77/219

77

x

y

z

qxx

qxyxz

Fx cos qxx

Fx cos qxy

Fx cos qxz

l

l

l

z


78/219

78

x

y

z

qyx

qyyqyz

Fy cos qyx

Fy cos qyy

Fy cos qyz

l

l

l

z


79/219

79

x

y

z

qzx

qzy

qzzFz cos qyx

Fz cos qzy

Fz cos qzz

l

l

l

g l l l


80/219

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+ +


81/219

g lc c c


82/219

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]


83/219

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


84/219

x

yz

84

TF =g F

l

T

t

=t

tt x

z

y

d2

d5

d8

d11

Example 1:


85/219

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


86/219

10 m

C

30x40cm

100 kN200 kN

40 kN

D

B

A

86

Sections Properties


87/219

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


88/219

C

D

A

d1

d2

d3

d4d5

d6

xy

z

Modeling

88

Stiffness matrix


89/219

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


90/219

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


91/219

91


92/219

z

y

x

z

y

x

z y

xxy

z

92


93/219

z

y

xz

y

x

z y

x

xy

z

AE

L12EIy

L36 EIy

L2

12EIy

L36 EIy

L2

93

xzE = 1200 kN/cm2


94/219

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


95/219

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


96/219

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


97/219

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


98/219

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


99/219

99

k11

=

k21

k31

k41

k51

k61

1,803.9

0

0

0

- 648

-1,296

Second column in Stiffness matrix


100/219

C

D

B

A

xy

z

d2=1

100


101/219

101


102/219

z

y

x

z

y

x

z y

x

xy

z

102

6 EIy

L2


103/219

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


104/219

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


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


106/219

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


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


108/219

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


109/219

109

k12

=

k22

k32

k42

k52

k62

2,883.4

0

0

0

1,152

450

Third column in Stiffness matrix


110/219

C

D

B

A

xy

z

d3=1

110


111/219

111

yd =1d3=1


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


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


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


115/219

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


116/219

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


117/219

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


118/219

118

k13

=

k23

k33k43

k53

k63

0

0

1,468.5

- 4,050

9,216

0

Fourth column in Stiffness matrix


119/219

C

D

B

A

xy

z

d4=1

119


120/219

120

y


121/219

z

y

x

z

y

x

z y

xxy

z

121

y


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


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


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


125/219

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


126/219

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


127/219

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


128/219

128

k14

=

k24

k34

k44

k54

k64

1,152

0

9,216

0

6,991,022

0

A

Fifth column in Stiffness matrix


129/219

C

D

B

A

xy

z

d5=1

129


130/219

130

y


131/219

z

y

x

z

y

x

z y

xxy

z

131

y


132/219

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


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


134/219

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


135/219

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


136/219

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


137/219

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


138/219

138

k15

=

k25

k35

k45

k55

k65

0

- 648

- 4,050

0

0

2,867,090

A

Sixth column in Stiffness matrix

d 1


139/219

C

D

B

A

xy

z

d6=1

139


140/219

140

y


141/219

z

x

z

y

x

z y

xxy

z

d6=1

d6=1d6=1

141

y


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


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


144/219

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


145/219

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


146/219

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


147/219

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


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


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


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


151/219

10 m

C

30x40cm

40 kN

D

B

151

200 kN 100 kN250 100


152/219

40 kN

20

20

50

250100

100 50

50

100

100

Fixed End Reaction(FER)

152

50

200 kN 100 kN250 100


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


154/219

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


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


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


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


158/219

d1d2d4d5

AEL

ND

=

158

Internal Forces Normal Forced3

d4d

d6

0.011027d1


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


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


dd

d3

d4d

d6

0.011027d1d 1AE

1,440


161/219

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


162/219

BB

A

B

D

148.4

148.4

0.1

0.1

2

162

A



163/219

C

D

A

148.4

0.12

163

Internal Forces Torsional Moment

dd

d3

d4d5

d6


164/219

d1d24d5

IxGL

T qt=

164


dd

d3

d4d5

d6

0.011027d1IGT


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


dd

d3

d4d5

d6

0.011027d1


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


dd

d3

d4d5

d6

0.011027d1GIx

L


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


168/219

BB

A

B

D

2.89

2.89

29.77

29.77

7.9

168

A



169/219

C

D

A

2.89

29.77 7.9

169

Internal Forces

dd

d3

d4d5

d6 Bending momentIn Plan


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



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



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


173/219

- 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

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


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


175/219

Internal Forces

dd

d3

d4d5


d 1 200 kNA


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


d 1 200 kNA


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



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



179/219

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

MBD = 1,152 d2 + 768,000 d4MBD = 1,152x-.000035+ 768,000x1.0012x10

-4

MBD = 76.85 kNm

179

Internal Forces

d1d2

d3

d4d5



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



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


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


183/219

C

D

183

CA


184/219

C

D

184

Internal Forces

d1d2

d3

d4d5

d6 Bending momentOut of Plan

4500.011027d1

d


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


4500.011027d1

d


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


187/219

7.15

3.58

B

C

(7.15+3.58)/8

1.34

1.34

187

Internal Forces

d1d2

d3

d4d5


0.011027

0 000035

d1

d

A


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


0.011027

0 000035

d1

d

A


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



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


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


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


193/219

53.9

30.52

8.44(53.9+30.52)/10

8.44D

193


77.13 3.58


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


195/219

C

D

195

CA


196/219

C

D

196

CA


197/219

D

A

197

Internal Forces Bending moment

10 04 2 2910.53


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


199/219

BB

B

D

2.89

2.89

29.77

29.77

7.9

199


200/219

200


201/219

201


202/219

202


203/219

Summary

203

d2


204/219

d1

d2d3

204

D1AEAE


205/219

AE

L

AE

L

D1

AELAEL

205

D6 EI


206/219

DD6 EI

L2

L2

D12 EI

L3

D12 EI

L3

206

D6 EI

L2


207/219

D

L

D6 EI

L2

D12 EI

L3D12 EI

L3

207

D3 EI

L2


208/219

D

L

D3 EI

L3

D3 EI

L3

208

D3 EI


209/219

D

L2

D3 EI

L3D

3 EIL3

209


210/219

q

q4 EIL

q2 EI

L

q6 EI

L2

q6 EI

L2

210


211/219

q

q4 EI

Lq2 EI

Lq6 EI

L2

q6 EI

L2

211

q


212/219

q3 EI

L q3 EIL2

q3 EIL2

212


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


214/219

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


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


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


217/219

( )cos q+ ( )sin qbx Dax- Dby Day-AELN=

a

q

DaxD

ay

[ ]

217

Internal forces

Bending moment


218/219

MAB= ( 2 + )qA qBM(FER) AB +2 EI

L

3DL

-

MBA=2 EI

L( + 2 )q

AqBM(FER) BA +

3DL

-

218


219/219

Questions

Documents

Stiffness 12