stiffness method

Lecture No. : 13 عشر الثالثة 1المحاضرة

2

Preliminary example :Calculate the deformations of the shown space frame where E = 1200 kN/cm2, G = 500 kN/cm2 and the sections are shown in figure

3

8 m

C

30x80 cm 20x60 cm

100 kN200 kN

10 m

B

A

z

y

xz

y

x30x80 cm

A =2400 cm2

Iz =1,280,000 cm4

Iy =180,000 cm4

Ix =550,180 cm4

20x60 cm

A =1200 cm2

Iz =360,000 cm4

Iy =40,000 cm4

Ix =126,435 cm4

Sections Properties

4

CA

d1d2

d3

d4d5

d6

xy

z

Modeling

5

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

6

C

B

A

d1=1

xy

z

First column in Stiffness matrix

7

z

y

xz

y

x

xy

z

d 1=1

d 1=1

AEL

12EIy

L3

6 EIy

L2

8

1200X1200

80012x1200X180,000

10003

6x1200X180,000

10002

2.592

1,296

1,800

xy

z

2.6

1,296

1,800

d1d2

d3

d4d5

d6

k11

=

k21

k31

k41

k51

k61

1,802.6

0

0

0

0

-1,2969

10

k11

=

k21

k31

k41

k51

k61

1,802.6

0

0

0

0

-1,296

First column in Stiffness matrix

xy

z

Second column in Stiffness matrix

d2=1

11

C

B

A

z

y

x

z

y

x

xy

z

d2=1

d2 =1

AEL

12EIy

L3

6 EIy

L2

12

12x1200X40,000

8003

6x1200X40,000

8002

1200X2400

1000

1.125

450

2,880

xy

z

2,880 1.1

450

k12

=

k22

k32

k42

k52

k62

2,881.1

0

0

0

0

450d1d2

d3

d4d5

d6

13

14

k12

=

k22

k32

k42

k52

k62

2,881.1

0

0

0

0

450

Second column in Stiffness matrix

xy

z

Third column in Stiffness matrix

d3=1

15

C

B

A

z

y

xz

y

x

d3=1d3=1

12EIz

L3

6 EIz

L2

12EIz

L3

6 EIz

L2

16

10.1

4,050

18.4

9,216

xy

z

18.4

9,216 10.1

4,050

k13

=

k23

k33

k43

k53

k63

0

0

28.5

- 4,050

9,216

0d1d2

d3

d4d5

d6

17

18

k13

=

k23

k33

k43

k53

k63

0

0

28.5

- 4,050

9,216

0

Third column in Stiffness matrix

xy

z

Fourth column in Stiffness matrix

d4=1

19

C

B

A

z

y

xz

y

x

xy

z

d 4=1d 4

=1

GIx

L

4 EIz

L6 EIz

L2

20

500X126,435

80079,022

9,216

6,144,000

xy

z

9,216

6,144,000

79,022

k14

=

k24

k34

k44

k54

k64

0

0

9,216

0

6,223,022

0d1d2

d3

d4d5

d6

21

22

k14

=

k24

k34

k44

k54

k64

0

0

9,216

0

0

Fourth column in Stiffness matrix

6,223,022

xy

z

Fifth column in Stiffness matrix

d5=1

23

C

B

A

z

y

x

z

y

x

xy

z

d5 =1

d5 =1

GIx

L

4 EIz

L6 EIz

L2

24

4,050

2,160,000

275,090

xy

z

d1d2

d3

d4d5

d6

275,090

4,050

2,160,000

k15

=

k25

k35

k45

k55

k65

0

0

- 4,050

0

0

2,435,090

25

26

Fifth column in Stiffness matrix

k15

=

k25

k35

k45

k55

k65

0

0

- 4,050

0

0

2,435,090

xy

z

Sixth column in Stiffness matrix

d6=1

27

C

B

A

z

y

xz

y

x

xy

z

d6=1d6=1

4 EIy

L

6 EIy

L2

4 EIy

L

6 EIy

L2

28

450

240,000

1,296

864,000

xy

z

1,296

864,000

450240,000

k16

=

k26

k36

k46

k56

k66

450

-1,296

0

0

0

1,104,000d1d2

d3

d4d5

d6

29

30

k16

=

k26

k36

k46

k56

k66

450

-1,296

0

0

0

1,104,000

Sixth column in Stiffness matrix

k11

=

k21

k31

k41

k51

k61

1,802.6

0

0

0

0

-1,296

k12

=

k22

k32

k42

k52

k62

2,881.1

0

0

0

0

450

k13

=

k23

k33

k43

k53

k63

0

0

28.5

- 4,050

9,216

0

31

k14

=

k24

k34

k44

k54

k64

0

0

9,216

0

6,223,022

0

k15

=

k25

k35

k45

k55

k65

0

0

- 4,050

0

0

2,435,090

k16

=

k26

k36

k46

k56

k66

450

-1,296

0

0

0

1,104,000

=K

32

1,802.6

0

0

0

0

-1,296

2,881.1

0

0

0

0

450

0

0

28.5

- 4,050

9,216

0

0

0

9,216

0

6,223,022

0

0

0

- 4,050

0

0

2,435,090

450

-1,296

0

0

0

1,104,000

Force vector

33

8 m

C

30x80 cm 20x60 cm

100 kN200 kN

10 m

B

A

200 kN

100 kN

250100

100

250

50

50

100

100

Fixed End Reaction(FER)

34

200 kN

100 kN

250100

100

250

50

50

100

100

Fixed End Action(FEA)

35

200 kN

100 kN

250100

100

250

50

50

100

100

d1d2

d3

d4d5

d6

F1

=

F2

F3

F4

F5

F6

0

0

- 150

- 250

100

0 36

Stiffness Equation F = K D

37

0

0

- 150

- 25000

10000

0

d1

=

d2

d3

d4

d5

d6

1,802.6

0

0

0

0

-1,296

2,881.1

0

0

0

0

450

0

0

28.5

- 4,050

9,216

0

0

0

9,216

0

6,223,022

0

0

0

- 4,050

0

0

2,435,090

450

-1,296

0

0

0

1,104,000

Stiffness Equation

D = K-1 F

d1

=

d2

d3

d4

d5

d6

-1

38

0

0

- 150

- 25000

10000

0

1,802.6

0

0

0

0

-1,296

2,881.1

0

0

0

0

450

0

0

28.5

- 4,050

9,216

0

0

0

9,216

0

6,223,022

0

0

0

- 4,050

0

0

2,435,090

450

-1,296

0

0

0

1,104,000

Deformations

39

0

0

- 11.8714

0.013564

d1

=

d2

d3

d4

d5

d6

- 0.015638

0

40

8 m

C

30x80 cm 20x60 cm

100 kN200 kN

10 m

B

A

41

42

d1

d2d3

d4

d5

d6

d1

d2

d3

d1x

yz

d2

d3

43

Example 1:Draw all diagrams for the shown grid where E = 1200 kN/cm2, G = 500 kN/cm2 and the sections are shown in figure

44

8 m

C

30x80 cm 20x60 cm

100 kN200 kN

10 m

B

A

z

y

xz

y

x30x80 cm

Iz =1,280,000 cm4

Ix =550,180 cm4

20x60 cm

Iz =360,000 cm4

Ix =126,435 cm4

Sections Properties

45

CA d1

d2d3

xy

z

Modeling

46

k11

K = k21

k31

k12

k22

k32

k13

k23

k33

Stiffness matrix

47

xy

z

d1=1

48

C

B

A

First column in Stiffness matrix

z

y

xz

y

x

d1=1d1=1

12EIz

L3

6 EIz

L2

12EIz

L3

6 EIz

L2

49

10.1

4,050

18.4

9,216

xy

z

18.4

9,216 10.1

4,050

k11

=k21

k31

28.5

- 4,050

9,216

50

d1

d2d3

51

k11

=k21

k31

28.5

- 4,050

9,216

First column in Stiffness matrix

xy

z

d2=1

52

C

B

A

Second column in Stiffness matrix

z

y

xz

y

x

xy

z

d 2=1d 2

=1

GIx

L

4 EIz

L6 EIz

L2

53

500X126,435

80079,022

9,216

6,144,000

xy

z

9,216

6,144,000

79,022

k12

=k22

k32

9,216

0

6,223,022

54

d1

d2d3

55

k12

=k22

k32

9,216

0

6,223,022

Second column in Stiffness matrix

xy

z

Third column in Stiffness matrix

d3=1

56

C

B

A

z

y

x

z

y

x

xy

z

d3 =1

d3 =1

GIx

L

4 EIz

L6 EIz

L2

57

4,050

2,160,000

275,090

xy

z

275,090

4,050

2,160,000

k13

=k23

k33

- 4,050

0

2,435,090

58

d1

d2d3

59

k13

=k23

k33

- 4,050

0

2,435,090

Third column in Stiffness matrix

60

k11

=k21

k31

28.5

- 4,050

9,216

k12

=k22

k32

9,216

0

6,223,022

k13

=k23

k33

- 4,050

0

2,435,090

61

=

28.5

- 4,050

9,216

9,216

0

6,223,022

- 4,050

0

2,435,090

K

=K

1,802.6

0

0

0

0

-1,296

2,881.1

0

0

0

0

450

0

0

28.5

- 4,050

9,216

0

0

0

9,216

0

6,223,022

0

0

0

- 4,050

0

0

2,435,090

450

-1,296

0

0

0

1,104,000

Force vector

62

8 m

C

30x80 cm 20x60 cm

100 kN200 kN

10 m

B

A

200 kN

100 kN

250100

100

250

50

50

100

100

Fixed End Reaction(FER)

63

200 kN

100 kN

250100

100

250

50

50

100

100

Fixed End Action(FEA)

64

200 kN

100 kN

250100

100

250

50

50

100

100

F1

=F2

F3

- 150

- 250

100

65

d1

d2d3

F1

=

F2

F3

F4

F5

F6

0

0

- 150

- 250

100

0

Stiffness Equation

F = K D

66

d1

d2

d3

- 150

- 25000

10000

=

28.5

- 4,050

9,216

9,216

0

6,223,022

- 4,050

0

2,435,090

Stiffness Equation

D = K-1 F

-1

67

d1

d2

d3

- 150

- 25000

10000

=

28.5

- 4,050

9,216

9,216

0

6,223,022

- 4,050

0

2,435,090

Deformations

68

d1

=d2

d3

- 11.8714

0.013564

- 0.015638

- 11.8714

0.013564- 0.015638

d1

d2d3

0

0

- 11.8714

0.013564

d1

=

d2

d3

d4

d5

d6

- 0.015638

0

Internal Forces Torsional Moment

IxGL

T t=

d 2=1

GIx

L79,021.9

TBC = 79,021.9x0.013564 = 1072 kNcm = 10.72 kNm

69

- 11.8714

0.013564- 0.015638

d1

d2d3

Internal Forces Torsional Moment

IxGL

T t=

d3 =1

GIx

L275,090

TBA = 275,090x-0.015638= 4302 kNcm = 43.02 kNm

70

- 11.8714

0.013564- 0.015638

d1

d2d3

Internal Forces

B

C

B

A

43.02

43.02

10.72

10.72

Torsional Moment

71

CA

Internal Forces

43.02

10.72

Torsional Moment

72

43.02

43.0210.72

10.72

Internal Forces Bending momentIn Plan

d1=1

6 EIz

L24,050

4,050

73

d3 =1

4 EIz

L2,160,000

1,080,000

2 EIz

L

100 kN

100

100

Internal Forces Bending momentIn Plan

d1=1

4,050

4,050

d3 =1 2,160,000

1,080,000

100

100FER

B

C

MBC = -100 - 4,050 d1 + 2,160,000 d3

MBC = -100+(-4,050x-11.8714+2,160,000x-0.015638)/100

MBC = 43.01 kNm 74

- 11.8714

0.013564- 0.015638

d1

d2d3

Internal Forces Bending momentIn Plan

d1=1

4,050

4,050

d3 =1 2,160,000

1,080,000

100

100FER

B

C

MCB = 100 - 4,050 d1 +1,080,000 d3

MCB = 100+(- 4,050x-11.8714+1,080,000x-0.015638)/100

MCB = 411.9 kNm 75

- 11.8714

0.013564- 0.01564

d1

d2d3

Internal Forces Bending momentIn Plan

43.01

411.9B

C100 kN

50 + (411.9 + 43.01)/8106.86

50 – (411.9 + 43.01)/8- 6.86

76

z

y

x

d1=1

12EIz

L310.1

77

z

y

xd3 =1

6 EIz

L2

4,050

100 kN

50

50

B

C

yBC = 50 + 10.1 d1 – 4,050 d3

yBC = 50 + 10.1x-11.8714 – 4,050x-0.015638

yBC = - 6.57 kN

- 11.8714

0.013564- 0.015638

d1

d2d3

z

y

x

d1=1

12EIz

L310.1

78

z

y

xd3 =1

6 EIz

L2

4,050

100 kN

50

50

B

C

yCB = 50 - 10.1 d1 + 4,050 d3

yCB = 50 - 10.1x-11.8714+ 4,050x-0.015638

yCB = 106.57 kN

10.1

4,050

- 11.8714

0.013564- 0.01564

d1

d2d3

Internal Forces Bending momentIn Plan

43.01

411.88B

C100 kN

106.57

- 6.57

79

Internal Forces Bending momentIn Plan

d1=1

6 EIz

L2

9,216

9,216

d 2=1

4 EIz

L6,144,000

3,072,000

200 kN

250

250FER

MBA = 250+9,216 d1 + 6,144,000 d2

MBA = 250+(9,216x-11.8714+ 6,144,000x 0.013564)/100

MBA = - 10.7 kNm

B

A

80

- 11.8714

0.013564- 0.01564

d1

d2d3

Internal Forces Bending momentIn Plan

d1=1

6 EIz

L2

9,216

9,216

d 2=14 EIz

L6,144,000

3,072,000

200 kN

250

250FER

MAB = -250+9,216 d1 + 3,072,000 d2

MAB = -250+(9,216x-11.8714+ 3,072,000x0.013564)/100

MAB = - 927.38 kNm

B

A

81

- 11.8714

0.013564- 0.015638

d1

d2d3

Internal Forces Shear ForceIn Plan

d1=1

d 2=1

200 kN

FER

yBA = 100+18.4 d2 + 9,216 d2

yBA = 100+18.4x-11.8714+ 9,216x0.013564

yBA = 6.57 kN

B

A

82

- 11.8714

0.013564- 0.015638

d1

d2d3

100

10018.4

18.4

9,216

9,216

Internal Forces Shear ForceIn Plan

d1=1

d 2=1

200 kN

FER

yAB = 100-18.4 d2 - 9,216 d2

yAB = 100-18.4x-11.8714- 9,216x0.013564

yAB = 193.43 kN

B

A

83

- 11.8714

0.013564- 0.015638

d1

d2d3

100

10018.4

18.4

9,216

9,216

Internal Forces Bending momentIn Plan

10.7

927.38B

A200 kN

193.43

6.57

84

Bending moment In Plan

85

CA

927.38

458.34

41.66

43.0

1411.9

184.45

15.55

200

500

10.7

43.01

411.9B

C100 kN

10.7

927.38B

A200 kN

Shear Force In Plan

86

CA

193.43

106.57

6.57

6.57

B

B

C100 kN

106.576.57

B

A200 kN

193.43

6.57

87

B

C100 kN

106.57

6.57

B

A200 kN

193.43

6.57

10.7 43.01

43.02

10.72

Summary

88

89

d1

d2d3

d4

d5

d6

d1

d2

d3

Questions

90