Stiffness 6

d1

d2d3

x

yz

d4

d5

d6

x

yz

x

y

z

x

yz

1

AEL

AEL

AEL

AEL

d=1

x

yz

1

AEL

AEL

AEL

AEL

d=1

x

yz

y

z

y6EIzyL2

6EIzyL2

12EIzyL3

12EIzyL3

x

yz

d=1

z

y

x

6EIzL2

12EIzL3

6EIzL2

12EIzL3

x

yz

z

y

6EIzyL2

12EIzyL3

12EIzyL3

6EIzyL2

y

x

yz

d=1

z

y

x

6EIzL2

12EIzL3

6EIzL2

12EIzL3

x

yz

z

z

y

6EIyzL2

12EIyzL3

6EIyzL2

12EIyzL3

x

yz d=1

z

y

x

6EIyL2

12EIyL3 6EIy

L2

12EIyL3

x

yz

z

y

z

6EIyzL2

12EIyzL3

6EIyzL2

12EIyzL3

x

yz

d=1

z

y

x

6EIyL2

12EIyL3 6EIy

L2

12EIyL3

x

yz

z

y z

4 EIzz

L

6 EIzz

L2

6 EIzz

L2

2 EIzz

L

x

yz

d=1

z

y

x

d=1

4 EIz

L

6 EIz

L2

6 EIz

L2

2 EIz

L

x

yz

z

y

z2 EIzz

L

6 EIzz

L2

6 EIzz

L2

4 EIzz

L

x

yz

d=1

z

y

x

d=1

4 EIz

L

6 EIz

L2

6 EIz

L2

2 EIz

L

x

yz

y

6 EIyy

L2

2 EIyy

L

6 EIyy

L2

4 EIyy

L

z

y

x

yz

d=1

z

y

x

6 EIy

L2

2 EIy

L

6 EIy

L2

4 EIy

L

x

yz

y

6 EIyy

L2

4 EIyy

L

2 EIyy

L

6 EIyy

L2

z

y

x

yz

d=1

z

y

x

6 EIy

L2

2 EIy

L

6 EIy

L2 4 EIy

L

x

yz

x

G Ixx

L

G Ixx

L

x

yz

d=1 z

y

x

G Ix

L

G Ix

L

x

yz

x

z

yx

G Ixx

L

G Ixx

L

x

yz

d=1

z

y

x

G Ix

L

G Ix

L

Example 2:Draw all diagrams for the shown space frame where E = 1200 kN/cm2, G = 500 kN/cm2 and the sections are shown in figure

8 m

10 m

C

30x80 cm

30x4

0 cm

20x60 cm

100 kN200 kN

40 kN

10 m

D

B

A

z

y

xz

y

x

zy

x

30x80 cm

A =2400 cm2

Iz =1,280,000 cm4

Iy =180,000 cm4

Ix =550,180 cm4 30x4

0 cm

20x60 cm

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

Sections Properties

C

D

A

d1d2

d3

d4d5

d6

xy

z

Modeling

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

C

D

B

A

C

D

B

A

d1=1

xy

z

First column in Stiffness matrix

z

y

x

z

y

x

zy

xxy

z

d 1=1

d 1=1

d 1=1

z

y

xz

y

x

z y

x

xy

z

d 1=1

d 1=1

d 1=1

AEL12EIy

L3

6 EIy

L2

12EIy

L3

6 EIy

L2

z

y

xz

y

x

z y

x

xy

z

d 1=1

d 1=1

d 1=1

AEL12EIy

L3

6 EIy

L2

12EIy

L3

6 EIy

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

E = 1200 kN/cm2

z

y

x

xy

z

d 1=1

AEL

A =1200 cm2

Iz =360,000 cm4

Iy =40,000 cm4

Ix =126,435 cm4

E = 1200 kN/cm2

1200X1200

800

1,800

z

y

x

xy

z

d 1=1

12EIy

L3

6 EIy

L2

A =2400 cm2

Iz =1,280,000 cm4

Iy =180,000 cm4

Ix =550,180 cm4

E = 1200 kN/cm2

12x1200X180,000

10003

2.6

6x1200X180,000

10002

1,296

z y

x

xy

z

d 1=1

12EIy

L3

6 EIy

L2

A =1200 cm2

Iz =160,000 cm4

Iy =90,000 cm4

Ix =194,385 cm4

E = 1200 kN/cm2

12x1200X90,000

10003

1.3

6x1200X90,000

10002

648

xy

z

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

-1,296

C

D

B

A

xy

z

Second column in Stiffness matrix

d2=1

z

y

x

z

y

x

z y

x

xy

z

d2=1

d2 =1

d2 =1

z

y

xz

y

x

z y

x

xy

z

d2=1

d2 =1

d2 =1

AEL

12EIz

L3

12EIy

L3

6 EIy

L2

6 EIz

L2

z

y

xz

y

x

z y

x

xy

z

d2=1

d2 =1

d2 =1

AEL

12EIz

L3

12EIy

L3

6 EIy

L2

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

E = 1200 kN/cm2

z

y

x

xy

z

d2=1

12EIy

L3

6 EIy

L2

A =1200 cm2

Iz =360,000 cm4

Iy =40,000 cm4

Ix =126,435 cm4

E = 1200 kN/cm2

12x1200X40,000

8003

1.1

6x1200X40,000

8002

450

z

y

x

xy

z

d2 =1

AEL

A =2400 cm2

Iz =1,280,000 cm4

Iy =180,000 cm4

Ix =550,180 cm4

E = 1200 kN/cm2

1200X2400

1000

2,880

z y

x

xy

z

d2 =1

12EIz

L3

6 EIz

L2

A =1200 cm2

Iz =160,000 cm4

Iy =90,000 cm4

Ix =194,385 cm4

E = 1200 kN/cm2

12x1200X160,000

10003

2.3

6x1200X160,000

10002

1,152

xy

zE = 1200 kN/cm2

2,880

2.31,152

1.1

450

k12

=

k22

k32

k42

k52

k62

2,883.4

0

0

0

1,152

450d1d2

d3

d4d5

d6

C

D

B

A

xy

z

Third column in Stiffness matrix

d3=1

z

y

x

z

y

x

z y

x

xy

z

d3=1

d3=1

d3=1

z

y

x

z

y

x

z y

x

d3=1

d3=1

d3=1

AEL

12EIz

L3

6 EIz

L2

12EIz

L3

6 EIz

L2

xy

z

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

E = 1200 kN/cm2

z

y

x

d3=1

12EIz

L3

6 EIz

L2

xy

zA =1200 cm2

Iz =360,000 cm4

Iy =40,000 cm4

Ix =126,435 cm4

E = 1200 kN/cm2

12x1200X360,000

8003

10.1

6x1200X360,000

8002

4,050

z

y

x

d3=1

12EIz

L3

6 EIz

L2

xy

zA =2400 cm2

Iz =1,280,000 cm4

Iy =180,000 cm4

Ix =550,180 cm4

E = 1200 kN/cm2

12x1200X1,280,000

1000318.4

6x1200X1,280,000

10002 9,216

z y

x

d3=1AEL

xy

z

A =1200 cm2

Iz =160,000 cm4

Iy =90,000 cm4

Ix =194,385 cm4

E = 1200 kN/cm2

1200X1200

1000

1,440

xy

z

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

C

D

B

A

xy

z

Fourth column in Stiffness matrix

d4=1

z

y

x

z

y

x

zy

xxy

z

d 4=1

d 4=1

d 4=1

z

y

x

z

y

x

zy

xxy

z

d 4=1

d 4=1

d 4=1 GIx

L

4 EIz

L6 EIz

L2

4 EIz

L6 EIz

L2

z

y

x

z

y

x

zy

x

xy

z

d 4=1

d 4=1

d 4=1 GIx

L

4 EIz

L6 EIz

L2

4 EIz

L6 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

E = 1200 kN/cm2

z

y

x

xy

z

d 4=1

GIx

L

A =1200 cm2

Iz =360,000 cm4

Iy =40,000 cm4

Ix =126,435 cm4

G = 500 kN/cm2

500X126,435

80079,021.9

z

y

x

xy

z

d 4=1

4 EIz

L6 EIz

L2

A =2400 cm2

Iz =1,280,000 cm4

Iy =180,000 cm4

Ix =550,180 cm4

E = 1200 kN/cm2

6x1200X1,280,000

10002

9,216

4x1200X1,280,000

1000

6,144,000

zy

x

xy

z

d 4=1

4 EIz

L6 EIz

L2

A =1200 cm2

Iz =160,000 cm4

Iy =90,000 cm4

Ix =194,385 cm4

E = 1200 kN/cm2

6x1200X160,000

10002

1,152

4x1200X160,000

1000

768,000

xy

z

1,152768,0009,216

6,144,000

79,021.9

k14

=

k24

k34

k44

k54

k64

1,152

0

9,216

0

6,991,022

0d1d2

d3

d4d5

d6

C

D

B

A

xy

z

Fifth column in Stiffness matrix

d5=1

z

y

x

z

y

x

zy

xxy

z

d5 =1

d5 =1

d5 =1

z

y

x

z

y

x

zy

xxy

z

d5 =1

d5 =1

d5 =1

GIx

L

4 EIy

L

6 EIy

L2

4 EIz

L6 EIz

L2

z

y

x

z

y

x

zy

x

d5 =1

d5 =1

d5 =1

GIx

L

4 EIy

L

6 EIy

L2

4 EIz

L6 EIz

L2

xy

z

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

E = 1200 kN/cm2

z

y

xd5 =1

4 EIz

L6 EIz

L2

xy

z A =1200 cm2

Iz =360,000 cm4

Iy =40,000 cm4

Ix =126,435 cm4

6x1200X360,000

8002

4,050

4x1200X360,000

800

2,160,000

E = 1200 kN/cm2

z

y

xd

5 =1

GIx

L

xy

z A =2400 cm2

Iz =1,280,000 cm4

Iy =180,000 cm4

Ix =550,180 cm4

G = 500 kN/cm2

500X550,180

1000

275,090

zy

x

d5 =1

4 EIy

L

6 EIy

L2

xy

zA =1200 cm2

Iz =160,000 cm4

Iy =90,000 cm4

Ix =194,385 cm4

E = 1200 kN/cm2

6x1200X90,000

10002

648

4x1200X90,000

1000

432,000

xy

z

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

C

D

B

A

xy

z

Sixth column in Stiffness matrix

d6=1

z

y

x

z

y

x

zy

xxy

z

d6=1

d6=1d6=1

z

y

x

z

y

x

zy

xxy

z

d6=1

d6=1d6=1

GIx

L

4 EIy

L

6 EIy

L2

4 EIy

L

6 EIy

L2

z

y

x

z

y

x

zy

x

d6=1

d6=1d6=1

GIx

L

4 EIy

L

6 EIy

L2

4 EIy

L

6 EIy

L2

xy

z

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

E = 1200 kN/cm2

z

y

xd6=1

4 EIy

L

6 EIy

L2

xy

zA =1200 cm2

Iz =360,000 cm4

Iy =40,000 cm4

Ix =126,435 cm4

E = 1200 kN/cm2

6x1200X40,000

80024x1200X40,000

800450

240,000

z

y

x

d6=1

4 EIy

L

6 EIy

L2

xy

z A =2400 cm2

Iz =1,280,000 cm4

Iy =180,000 cm4

Ix =550,180 cm4

E = 1200 kN/cm2

4x1200X180,000

1000

6x1200X180,000

10002

1,296

864,000

zy

x

d6=1

GIx

L

xy

z

A =1200 cm2

Iz =160,000 cm4

Iy =90,000 cm4

Ix =194,385 cm4

G = 500 kN/cm2

500X194,385

1000

97,192.5

xy

z

1,296

864,000

97,192.5

450240,000

k16

=

k26

k36

k46

k56

k66

450

-1,296

0

0

0

1,201,192.5d1d2

d3

d4d5

d6

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

k13

=

k23

k33

k43

k53

k63

0

0

1,468.5

- 4,050

9,216

0

k12

=

k22

k32

k42

k52

k62

2,883.4

0

0

0

1,152

450

k11

=

k21

k31

k41

k51

k61

1,803.9

0

0

0

- 648

-1,296

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

=K

1,803.9

0

0

0

- 648

-1,296

8 m

10 m

C

30x80 cm

30x4

0 cm

20x60 cm

100 kN200 kN

40 kN

10 mD

B

A

Force vector

200 kN

40 kN

100 kN

20

20

50

250100

100

250

50

50

100

100

Fixed End Reaction(FER)

200 kN

40 kN

100 kN

20

20

50

250100

100

250

50

50

100

100

Fixed End Action(FEA)

200 kN

40 kN

100 kN

20

20

50

250100

100

250

50

50

100

100

d1d2

d3

d4d5

d6

F1

=

F2

F3

F4

F5

F6

20

0

- 150

- 250

100

- 50

Stiffness Equation F = K D

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

Stiffness Equation D = K-1 F

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

Deformations

0.011027

- 0.000035

- 0.103072

1.0012x10-4

d1

=

d2

d3

d4

d5

d6

-1.0823x10-4

-2.9715x10-5

Internal Forces Normal Force

d1d2

d3

d4d5

d6

0.011027

- 0.000035

- 0.103072

1.0012x10-4

d1

=

d2

d3

d4

d5

d6

-1.0823x10-4

-2.9715x10-5

AEL

N =

Internal Forces Normal Force

d1d2

d3

d4d5

d6

0.011027

- 0.000035

- 0.103072

1.0012x10-4

d1

=

d2

d3

d4

d5

d6

-1.0823x10-4

-2.9715x10-5

z

y

x

d 1=1

AEL

1,800

NBC = 1,800x.0011027 = 2 kN compression

Internal Forces Normal Force

d1d2

d3

d4d5

d6

0.011027

- 0.000035

- 0.103072

1.0012x10-4

d1

=

d2

d3

d4

d5

d6

-1.0823x10-4

-2.9715x10-5

z

y

xd

2 =1AEL

2,880

NBA = 2,880x.000035 = 0.1 kN tension

Internal Forces Normal Force

d1d2

d3

d4d5

d6

0.011027

- 0.000035

- 0.103072

1.0012x10-4

d1

=

d2

d3

d4

d5

d6

-1.0823x10-4

-2.9715x10-5

NBD = 1,440x0.103072 = 148.4 kN compression

z y

x

d3=1AEL

1,440

Internal Forces

B

C

B

A

B

D

148.4

148.4

0.1

0.1

2

2

Normal Force

C

D

A

Internal Forces Normal Force

148.4

0.12

Internal Forces Torsional Moment

d1d2

d3

d4d5

d6

0.011027

- 0.000035

- 0.103072

1.0012x10-4

d1

=

d2

d3

d4

d5

d6

-1.0823x10-4

-2.9715x10-5

IxGL

T t=

Internal Forces Torsional Moment

d1d2

d3

d4d5

d6

0.011027

- 0.000035

- 0.103072

1.0012x10-4

d1

=

d2

d3

d4

d5

d6

-1.0823x10-4

-2.9715x10-5

IxGL

T t=

d 4=1

GIx

L79,021.9

TBC = 79,021.9x1.0012x10-4 = 7.9 kNm

Internal Forces Torsional Moment

d1d2

d3

d4d5

d6

0.011027

- 0.000035

- 0.103072

1.0012x10-4

d1

=

d2

d3

d4

d5

d6

-1.0823x10-4

-2.9715x10-5IxGL

T t=

d5 =1

GIx

L275,090

TBA = 275,090x-1.0823x10-4 = - 29.77 kNm

Internal Forces Torsional Moment

d1d2

d3

d4d5

d6

0.011027

- 0.000035

- 0.103072

1.0012x10-4

d1

=

d2

d3

d4

d5

d6

-1.0823x10-4

-2.9715x10-5

IxGL

T t=

d6=1

GIx

L97,192.5

TBD = 97,192.5x-2.9715x10-5 = - 2.89 kNm

Internal Forces

B

C

B

A

B

D

2.89

2.89

29.77

29.77

7.9

7.9

Torsional Moment

C

D

A

Internal Forces

2.89

29.77 7.9

Torsional Moment

Internal Forces

d1d2

d3

d4d5

d6

0.011027

- 0.000035

- 0.103072

1.0012x10-4

d1

=

d2

d3

d4

d5

d6

-1.0823x10-4

-2.9715x10-5

Bending momentIn Plan

d3=1

6 EIz

L24,050

4,050

Internal Forces

d1d2

d3

d4d5

d6

0.011027

- 0.000035

- 0.103072

1.0012x10-4

d1

=

d2

d3

d4

d5

d6

-1.0823x10-4

-2.9715x10-5

Bending momentIn Plan

d5 =1

4 EIz

L2,160,000

1,080,000

2 EIz

L

Internal Forces

d1d2

d3

d4d5

d6

0.011027

- 0.000035

- 0.103072

1.0012x10-4

d1

=

d2

d3

d4

d5

d6

-1.0823x10-4

-2.9715x10-5

Bending momentIn Plan

100 kN

100

100

Internal Forces0.011027

- 0.000035

- 0.103072

1.0012x10-4

d1

=

d2

d3

d4

d5

d6

-1.0823x10-4

-2.9715x10-5

Bending momentIn Plan

d3=1

4,050

4,050

d5 =1 2,160,000

1,080,000

100

100FER

B

C

MBC = 100+4,050 d3 – 2,160,000 d5

MBC = 100+4,050x-.103072– 2,160,000x-1.0823x10-4

MBC = - 83.66 kNm

Internal Forces0.011027

- 0.000035

- 0.103072

1.0012x10-4

d1

=

d2

d3

d4

d5

d6

-1.0823x10-4

-2.9715x10-5

Bending momentIn Plan

d3=1

4,050

4,050

d5 =1 2,160,000

1,080,000

100

100FER

B

C

MCB = -100+4,050 d3 – 1,080,000 d5

MCB = -100+4,050x-.103072– 1,080,000x-1.0823x10-4

MCB = - 400.55 kNm

Internal Forces Bending momentIn Plan

83.66

400.55B

C100 kN

50 + (400.55+83.66)/8110.53

50 - (400.55+83.66)/810.53

Internal Forces

d1d2

d3

d4d5

d6 Bending momentIn Plan

d3=1

6 EIz

L2

9,216

9,216

d 4=14 EIz

L6,144,000

3,072,000

200 kN

250

250FER

MBA = 250+9,216 d3 + 6,144,000 d4

MBA = 250+9,216x-.103072+ 6,144,000x1.0012x10-4

MBA = - 84.77 kNm

B

A

0.011027

- 0.000035

- 0.103072

1.0012x10-4

d1

=

d2

d3

d4

d5

d6

-1.0823x10-4

-2.9715x10-5

Internal Forces

d1d2

d3

d4d5

d6 Bending momentIn Plan

d3=1

6 EIz

L2

9,216

9,216

d 4=14 EIz

L6,144,000

3,072,000

200 kN

250

250FER

MAB = -250+9,216 d3 + 3,072,000 d4

MAB = -250+9,216x-.103072+ 3,072,000x1.0012x10-4

MAB = - 892.3 kNm

B

A

0.011027

- 0.000035

- 0.103072

1.0012x10-4

d1

=

d2

d3

d4

d5

d6

-1.0823x10-4

-2.9715x10-5

Internal Forces

d1d2

d3

d4d5

d6 Bending momentIn Plan

84.77

892.3B

A200 kN

100 + (892.3+84.77)/10197.71 100 - (892.3+84.77)/10

2.29

Internal Forces

d1d2

d3

d4d5

d6

0.011027

- 0.000035

- 0.103072

1.0012x10-4

d1

=

d2

d3

d4

d5

d6

-1.0823x10-4

-2.9715x10-5

Bending momentIn Plan

d2 =1

6 EIz

L2

1,152

1,152

d 4=14 EIz

L

768,000

384,000

B

D

MBD = 1,152 d2 + 768,000 d4

MBD = 1,152x-.000035+ 768,000x1.0012x10-4

MBD = 76.85 kNm

Internal Forces

d1d2

d3

d4d5

d6

0.011027

- 0.000035

- 0.103072

1.0012x10-4

d1

=

d2

d3

d4

d5

d6

-1.0823x10-4

-2.9715x10-5

Bending momentIn Plan

d2 =1

6 EIz

L2

1,152

1,152

d 4=14 EIz

L

768,000

384,000

B

D

MDB = 1,152 d2 + 384,000 d4

MDB = 1,152x-.000035+ 384,000x1.0012x10-4

MDB = 38.41 kNm

Internal Forces

d1d2

d3

d4d5

d6

0.011027

- 0.000035

- 0.103072

1.0012x10-4

d1

=

d2

d3

d4

d5

d6

-1.0823x10-4

-2.9715x10-5

Bending momentIn Plan

B

D

76.85

38.41

(76.85+38.41)/10

11.53

11.53

Internal Forces Bending momentIn Plan

83.66

400.55B

C100 kN

110.53

10.53

84.77

892.3B

A200 kN

197.71

2.29 B

D

76.85

38.41

11.53

11.53

C

D

A

C

D

A

Internal Forces

d1d2

d3

d4d5

d6 Bending momentOut of Plan

d2 =1

6 EIy

L2

450

450

d6=1

4 EIy

L240,000

120,000

0.011027

- 0.000035

- 0.103072

1.0012x10-4

d1

=

d2

d3

d4

d5

d6

-1.0823x10-4

-2.9715x10-5

B

C

MBC = 450 d2 + 240,000 d6

MBC = 450x-.000035 + 240,000x-2.9715x10-5

MBC = - 7.15 kNm

Internal Forces

d1d2

d3

d4d5

d6 Bending momentOut of Plan

d2 =1

6 EIy

L2

450

450

d6=1

4 EIy

L240,000

120,000

0.011027

- 0.000035

- 0.103072

1.0012x10-4

d1

=

d2

d3

d4

d5

d6

-1.0823x10-4

-2.9715x10-5

B

C

MCB = 450 d2 + 120,000 d6

MCB = 450x-.000035 + 120,000x-2.9715x10-5

MCB = - 3.58 kNm

7.15

3.58

B

C

(7.15+3.58)/8

1.34

1.34

Internal Forces Bending momentOut of Plan

Internal Forces

d1d2

d3

d4d5

d6 Bending momentOut of Plan

0.011027

- 0.000035

- 0.103072

1.0012x10-4

d1

=

d2

d3

d4

d5

d6

-1.0823x10-4

-2.9715x10-5

MBA = 50-1,296 d1 + 864,000 d6

MBA = 50-1,296x.011027 + 864,000x-2.9715x10-5

MBA = 10.04 kNm

d 1=1

6 EIy

L2

1,296

1,296

d6=1

4 EIy

L

864,000

432,000

B

A

40 kN

50

50

Internal Forces

d1d2

d3

d4d5

d6 Bending momentOut of Plan

0.011027

- 0.000035

- 0.103072

1.0012x10-4

d1

=

d2

d3

d4

d5

d6

-1.0823x10-4

-2.9715x10-5

MAB = -50-1,296 d1 + 432,000 d6

MAB = -50-1,296x.011027 + 432,000x-2.9715x10-5

MAB = - 77.13 kNm

d 1=1

6 EIy

L2

1,296

1,296

d6=1

4 EIy

L

864,000

432,000

B

A

40 kN

50

50

77.13

10.04

B

C

26.71

40 kN

20+(77.13-10.04)/10

13.29

20-(77.13-10.04)/10

Internal Forces Bending momentOut of Plan

Internal Forces

d1d2

d3

d4d5

d6

0.011027

- 0.000035

- 0.103072

1.0012x10-4

d1

=

d2

d3

d4

d5

d6

-1.0823x10-4

-2.9715x10-5

Bending moment

d 1=1

6486 EIy

L2

648

d5 =1

4 EIy

L432,000

216,000

B

D

MBD = 648 d1 - 432,000 d5

MBD = 648x.011027 - 432,000x-1.0823x10-4

MBD = 53.9 kNm

Out of Plan

Internal Forces

d1d2

d3

d4d5

d6

0.011027

- 0.000035

- 0.103072

1.0012x10-4

d1

=

d2

d3

d4

d5

d6

-1.0823x10-4

-2.9715x10-5

Bending moment

d 1=1

6486 EIy

L2

648

d5 =1

4 EIy

L432,000

216,000

B

D

MDB = 648 d1 - 216,000 d5

MDB = 648x.011027 - 216,000x-1.0823x10-4

MDB = 30.52 kNm

Out of Plan

53.9

30.52

8.44(53.9+30.52)/10

8.44

Internal Forces Bending momentOut of PlanB

D

Internal Forces Bending moment

B

C

B

A

B

D

Out of Plan

53.9

30.528.44

8.44

77.13

10.04

26.71

40 kN

13.29

7.15

3.58

1.34

1.34

C

D

A

C

D

A

C

D

A

Internal Forces Bending moment

53.9

8.44

10.04

13.29

7.15

1.34

84.77

2.29

76.85

11.53

10.53

83.660.1

148.4

2

29.77

2.89

7.9

Internal Forces

B

C

B

A

B

D

2.89

2.89

29.77

29.77

7.9

7.9

Torsional Moment

Summary

d1

d2

d3

AEL

AEL

AEL

AEL

6 EI

L2

6 EIL2

12 EIL3

12 EIL3

6 EIL2

6 EIL2

12 EIL3

12 EIL3

3 EIL2

3 EIL3

3 EIL3

3 EIL2

3 EIL3

3 EIL3

4 EIL

2 EIL

6 EI

L2

6 EI

L2

4 EIL

2 EIL

6 EI

L2

6 EI

L2

3 EIL 3 EI

L2

3 EI

L2

3 EIL

3 EI

L2

3 EI

L2

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

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

4 EIL

6 EI sin L2

6 EI cos L2

2 EIL

-

6 EI sin L2

6 EI cos L2

4 EIL

Rotational Deformation

( )cos + ( )sin

bxax

- byay

-AEL

N =

a

b

bx

by

ax

ay

[ ]

Internal forces

Normal Force

Internal forces

Bending moment

MAB= ( 2 + ) M(FER) AB +2 EI

L

3 L

-

MBA=2 EI

L( + 2 )

M(FER) BA +3 L

-

Questions