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Lecture No. : 9 ال رة ض حا م ل ا عة س ا ت

Stiffness 9

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Page 1: Stiffness 9

Lecture No. : 9 ال تاسعةالمحاضرة

Page 2: Stiffness 9

F = K Dl l l

T K =g K l T

T

m m

Drive the member local stiffness matrix

Obtain the member global stiffness matrix

Drive the member transformation matrixT

Solution Steps of assembly method :Remember

Page 3: Stiffness 9

Make assembly F = K D

Kuu

Kru

Kur

Krr

Fu

Fr

Du

Dr=

Make partition

Kgm K

gm K

gm K

gm

Remember

Page 4: Stiffness 9

Kuu

Kru

Kur

Krr

Fu

Fr

Du

Dr=

Extract the stiffness equation

KuuFu Du= Kur Dr+

KuuDu =-1{ }Fu Kur Dr-

Obtain the deformation

Remember

Page 5: Stiffness 9

Find internal forces in members

Calculate the reactions

KruFr Du= Krr Dr+

gF = K l lm m mT D

T

Remember

Page 6: Stiffness 9
Page 7: Stiffness 9

d1

d2

d3

Page 8: Stiffness 9

d1

d2

Normal Force doesn’t taken

Page 9: Stiffness 9

Drive the member local stiffness matrix

k11F1

F2

=k21

F3 k31

k12

k22

k32

k13

k23

k33

F4k41 k42 k43

k14

k24

k34

k44

d1

d2

d3

d4

d1

d2 d4

d3

Page 10: Stiffness 9

First column inLocal Stiffness matrix

d1 =1

6 EI

L2

6 EIL2

12 EIL3

12 EIL3

d3

d4

d2

d2

Page 11: Stiffness 9

12 EIL3

F2 =

F3 = -

F4 =

F1 =12 EI

L3

6 EIL3

6 EIL3

6 EI

L2

6 EIL2

12 EIL3

12 EIL3

Page 12: Stiffness 9

=k31

12 EIL3

6 EIL2

-12EIL3

6 EIL2

k11

k21

k41

Page 13: Stiffness 9

Second column inLocal Stiffness matrix

d2 =1

4 EIL

2 EIL

6 EIL2

6 EIL2

d3

d4

d1

d2

Page 14: Stiffness 9

F3 =F1 =

F4 =F2 =

6 EIL2

6 EIL2

-

2 EIL

4 EIL

4 EIL

2 EIL

6 EIL2

6 EIL2

Page 15: Stiffness 9

Second column inLocal Stiffness matrix

=

k12

k22

k32

k42

6 EI

L2

4 EIL

6 EIL2-

2 EIL

Page 16: Stiffness 9

Third column inLocal Stiffness matrix

d3 =1

6 EIL2

6 EIL2

12 EIL3

12 EIL3

d3

d4

d1

d2

Page 17: Stiffness 9

12 EIL3

6 EIL2

6 EIL2

12 EIL3F1 = F3 =

F4 =F2 = - -

-

6 EI

L2

6 EIL2

12 EIL3

12 EIL3

Page 18: Stiffness 9

Third column in Local Stiffness matrix

=

k13

k23

k33

k33

12 EIL3

6 EIL2

12 EIL3

6 EIL2

-

-

-

Page 19: Stiffness 9

Fourth column inLocal Stiffness matrix

d4 =1

4 EIL

2 EIL

6 EIL2

6 EIL2

d3

d4

d1

d2

Page 20: Stiffness 9

F1 = F3=

F2 = F4=

6 EIL2

2 EIL

4 EIL

6 EIL2

-

4 EIL

2 EIL

6 EIL2

6 EIL2

Page 21: Stiffness 9

Fourth column inLocal Stiffness matrix

=

k14

k24

k34

k44

6 EI

L2

2 EIL

6 EIL2-

4 EIL

Page 22: Stiffness 9

K l

12 EIL3

6 EIL2

6 EIL2

4 EIL

-12 EIL3

=-12 EI

L3

-6 EIL2

-6 EIL2

12 EIL3

-6 EIL2

6 EIL2

2 EIL

6 EIL2

2 EIL

-6 EIL2

4 EIL

Page 23: Stiffness 9

K l

12 EIL3

6 EI

L2

6 EI

L2

4 EIL

-12 EIL3

=-12 EI

L3

-6 EIL2

-6 EIL2

12 EIL3

-6 EIL2

6 EI

L2

2 EIL

6 EI

L2

2 EIL

-6 EIL2

4 EIL

Page 24: Stiffness 9

K l = EIL3

K l

12 EIL3

6 EIL2

6 EIL2

4 EIL

-12 EIL3

= -12 EIL3

-6 EIL2

-6 EIL2

12 EIL3

-6 EIL2

6 EIL2

2 EIL

6 EIL2

2 EIL

-6 EIL2

4 EIL

12

6 L

-12

6 L

6 L

-6 L

4 L2

2 L2

-12

6 L

12

6 L

-6 L

2 L2

4 L2

6 L

Page 25: Stiffness 9

d1

d2

Normal Force doesn’t taken

Page 26: Stiffness 9

d1

If Shear is omitted

Page 27: Stiffness 9

Drive the member local stiffness matrix

k11F1

F2

=k21

k12

k22

d1

d2

d1 d2

Page 28: Stiffness 9

K l

12 EIL3

6 EIL2

6 EIL2

4 EIL

-12 EIL3

=-12 EI

L3

-6 EIL2

-6 EIL2

12 EIL3

-6 EIL2

6 EIL2

2 EIL

6 EIL2

2 EIL

-6 EIL2

4 EIL

Page 29: Stiffness 9

K l4 EI

L=

2 EIL

2 EIL

4 EIL

K l = 2EIL 1

2

2

1

Page 30: Stiffness 9

Construct the stiffness matrix for the shown beam where EI is

constant for all members

Example 1:

8 10

A CB

Page 31: Stiffness 9

First element : (A-B )Start Joint : A End Joint : B

K l = EIL3

12

6 L

-12

6 L

6 L

-6 L

4 L2

2 L2

-12

6 L

12

6 L

-6 L

2 L2

4 L2

6 L

Page 32: Stiffness 9

K l K g=

K l

12 EIL3

6 EI

L2

6 EI

L2

4 EIL

-12 EIL3

=-12 EI

L3

-6 EIL2

-6 EIL2

12 EIL3

-6 EIL2

6 EI

L2

2 EIL

6 EI

L2

2 EIL

-6 EIL2

4 EIL

Page 33: Stiffness 9

K l = EIL3

12

6 L

-12

6 L

6 L

-6 L

4 L2

2 L2

-12

6 L

12

6 L

-6 L

2 L2

4 L2

6 L

Page 34: Stiffness 9

K l = EI

.023

.0937

.-023

.0937

.0937

.5

.0937

.25

.-023

.0937

.023

.-0937

.0937

.25

.-0937

.5

A

B

�ِA

B

Page 35: Stiffness 9

Second element : ( B-c)Start Joint : B End Joint : c

K l

12 EIL3

6 EI

L2

6 EI

L2

4 EIL

-12 EIL3

=-12 EI

L3

-6 EIL2

-6 EIL2

12 EIL3

-6 EIL2

6 EI

L2

2 EIL

6 EI

L2

2 EIL

-6 EIL2

4 EIL

Page 36: Stiffness 9

K l = EI

.012

.06

.-012

.06

.06

.4

.-06

.2

.-012

.-06

.012

.-0937

.06

.20

.-06

.4

B C

C

B

Page 37: Stiffness 9

Assembly :

K =g1 EI

.023

.0937

.-023

.0937

.0937

.5

.0937

.25

.-023

.0937

.023

.-0937

.0937

.25

.-0937

.5

A B

B

A

Page 38: Stiffness 9

EI .06

.-012

.06

.06

.4

.2

.-012

.-06

.012

.-06

.06

.2

.-06

.012

.-06

K =g2

.4

B C

B

C

Page 39: Stiffness 9

Ks = EI

.023

.0937

.-023

.093700

.0937

.5

.0937

.2500

.-023

.0937

.035

.-0337.-012.06

.0937

.25

.-0337

.9

.-06

.2

0

0

.-012

.-06

.012.-06

0

0

.06

.2

.-06

.4

A B C

B

A

C

Page 40: Stiffness 9

Partition

KuuK =Kru

Kur

Krr

u ru

r

Kuu = EI .90

Page 41: Stiffness 9

Example 2:Construct the stiffness matrix for the shown beam where EI is constant for all members

4254

ECB DA

Page 42: Stiffness 9

= E I

12/L^3

6/L^2

-12/L^3

6/L^2

6/L^2

2/L

6/L^2-

4/L

12/L^3-

6/L^2-

12/L^3

6/L^2-

6/L^2

-6/L^2

4/L

2/L

K gK l =

K l

Page 43: Stiffness 9

First element : (A-B )Start Joint : A

End Joint : BAngle : 0s = sin = 0c = cos = 1

Is conastant EA

LAB = 400 cm

Page 44: Stiffness 9

Kl= EI

.1875

.375

.-1875

.375

.375

1

.-375

.5

.-1875

.-375

.1875

.-375

.375

.5

.-375

1

A B

B

A

Page 45: Stiffness 9

Second element : B-C( Start Joint : B

End Joint : cAngle : 0s = sin = 0c = cos = 1

Is conastant EA

LBC = 500 cm

Page 46: Stiffness 9

EI .24

.-096

.24

.24

.8

.4

.-096

.-24

.096

.-24

.24

.4

.-24

.096

.-24

.8

B C

B

C

K =g1

Page 47: Stiffness 9

End Joint : D

Third element : (C-D )

Start Joint : C

Angle : 0s = sin = 0c = cos = 1

LCD = 200 cm

Is conastant EA

Page 48: Stiffness 9

= EI

1.5

1.5

-1.5

1.5

1.5

2

-1.5

.2

-1.5

-1.5

1.5

-1.5

1.5

1

-1.5

.2

C D

D

CK l

Page 49: Stiffness 9

Fourth element : (D-E )

Start Joint :D

End Joint : EAngle : 0s = sin = 0c = cos = 1

Is conastant EA

LD-E = 400 cm

Page 50: Stiffness 9

Kl= EI

.1875

.375

.-1875

.375

.375

1

.-375

.5

.-1875

.-375

.1875

.-375

.375

.5

.-375

1

D "ُE

D

E

Page 51: Stiffness 9

K =s

.1875 .375.375 1

.-1875 .-375

.375 .5

0 0

0 0

0 0

0 0

.-096 .-24.24 .4

0 00 0

0

0

0 00

00

0 0

0 0

0

0

0

0

0

0 00

0.-1875 .375.-375 .5

.2835 .-135.-135 1.8

0000

.-096

.-24

.24.4

1.596 1.261.26 2.8

-1.5 1.51.5 1

0 00 0

00

-1.5 1.51.5 1

1.687 -1.125-1.125 3.-1875.375

.-375.5

0

00

0

0

0

.-1875 .375.-375 .5

.1875.-375 1

.-375

A B C D E

AB

CDE

Page 52: Stiffness 9

KUU1.8 EI0.4 EI

0

0.4 EI2.8 EI

EI

0EI

3 EI=

1.80.4

0

0.42.8

1

01

3EI=

Page 53: Stiffness 9

Example 3:Calculate the deformation of the shown beam where EI = 105 kN.m2 for all members

4254DCB

EA 30 kNm50 kNm60 kNm

Page 54: Stiffness 9

KUU

1.8 EI0.4 EI

0

0.4 EI2.8 EI

EI

0EI

3 EI=

1.80.4

0

0.42.8

1

01

3EI

The stiffness equation

F = K D

Stiffness matrix From Exampl (2)

Page 55: Stiffness 9

F1

F2

=-60

-50

F3 30

= d1

d2

-60

-50

1.8

.4

.4

2.8

F = K D

30 d3

0

1

0 1 3

D =

B

C

D

Page 56: Stiffness 9

D = K-1 F- 60- 50

30=

1.80.4

0

0.42.8

1

01

3

B

C

D

1EI

-1

- 60- 50

30=

7.4-1.2

.4

-1.25.4

-1.8

.4-1.8

4.88

B

C

D

1105

112.84

=

- 0.290- 0.196

0.165

X10-3

rad

Page 57: Stiffness 9

Example 4:Draw B.M.D for the shown beam where EI = 105 kN.m2 for all members

4

A

B

E

5 2 4C D

60 kNm 50 kNm 30 kNm

Page 58: Stiffness 9

From the previous example

=

B

C

D

- 0.290

- 0.196

0.165

X10-3

rad

Page 59: Stiffness 9

4

AB

E

5 2 4C D

=

B

C

D

- 0.290- 0.196

0.165

X10-3

rad

EI = 105 kN.m2

For member AB

LAB

MAB = == - 14.5 kN.m

2 EILAB

MBA= (2 B + A ) = 5x104(2X-0.029)x10-3

5x104(0-0.29)x10-3(2 A + B )2 EI

= - 29 kN.m

Page 60: Stiffness 9

4

AB

E

5 2 4C D

=C

D

- 0.290- 0.196

0.165

X10-3

rad

B

EI = 105 kN.m2

For member BC

LBC

MBC = (2 B + C ) =

2 EI

LBC

MCB= (2 C + B ) =

4x104(2x-0.29-0.196)x10-3

= - 31 kN.m

4x104(2x-0.196-0.029)x10-3

= - 27.3 kN.m

Page 61: Stiffness 9

MAB = - 14.5 kN.mMBA= - 29 kN.m

MBC = - 31 kN.mMCB= - 27.3 kN.m

MCD = - 22.7 kN.mMDC= 13.5 kN.m

MDE = 16.5 kN.mMED= 8.3 kN.m

A B B C C D D E

14.5 29 31 27.3 22.7 13.5 16.5 8.3

Page 62: Stiffness 9

A B B C C D D E

14.5 29 31 27.3 22.7 13.5 16.5 8.3

14.531

27.3

22.713.5

16.5

B.M.D

Page 63: Stiffness 9

14.5 13.5

16.527.3

22.731

29

A

B

E

C D

60 kNm 50 kNm 30 kNm

Page 64: Stiffness 9

Example 5:Draw B.M.D for the shown beam where EI is constant for all members

2

A B C D

1 2 2 1 2

100 kN 200 kN 150 kN

Page 65: Stiffness 9

Solution Steps of assembly method :

Drive the member local stiffness matrixLocal

k11F1

F2

=k21

F3 k31

k12

k22

k32

k13

k23

k33

F4k41 k42 k43

k14

k24

k34

k44

d1

d2

d3

d4

Page 66: Stiffness 9

d3

d1

d2 d4

First column inLocal Stiffness matrix

6 EI

L2

6 EI

L212 EI

L3

Page 67: Stiffness 9

F1 = 12 EIL3

F2 = 6 EI

L2

6 EI

L2

F3 = 12 EIL3

-

F4 =

first column in Local Stiffness matrix

Page 68: Stiffness 9

=

k31

k41

12 EIL3

6 EIL2

-12 EIL3

6 EIL2

k11

k21

Page 69: Stiffness 9

Second column inLocal Stiffness matrix

d2 =1

4 EIL

2 EIL

6 EIL2

6 EIL2

Page 70: Stiffness 9

F3 =F1 =

F4 =F2 =

6 EIL2

6 EIL2

-

2 EIL

4 EIL

Page 71: Stiffness 9

Second column inLocal Stiffness matrix

=

k12

k22

k32

k42

6 EI

L2

4 EIL

6 EIL2

- 2 EIL

Page 72: Stiffness 9

Third column in Local Stiffness matrix

d3 =1

12 EIL3

12 EIL3

6 EIL2

6 EIL2

Page 73: Stiffness 9

Third column in Local Stiffness matrix

=

k13

k23

k33

k33

-12 EIL3

-6 EIL2

12 EIL3

-6 EIL2

Page 74: Stiffness 9

12 EIL3

-6 EIL2

-6 EIL2

F4 =

F3 = F1 = -12 EIL3

F2 =

Page 75: Stiffness 9

Fourth column inLocal Stiffness matrix

6 EIL2

6 EIL2

4 EIL

2 EIL

Page 76: Stiffness 9

F1 = F3=

F2 = F4 =

6 EIL2

2 EIL

-6 EIL2

4 EIL

Page 77: Stiffness 9

K l

12 EIL3

6 EIL2

6 EIL2

4 EIL

-12 EIL3

=

-12 EIL3

-6 EIL2

-6 EIL2

12 EIL3

-6 EIL2

6 EIL2

2 EIL

6 EIL2

2 EIL

-6 EIL2

4 EIL

Page 78: Stiffness 9

First element : (A-B )

Start Joint : A

End Joint : BAngle : 0s = sin = 0c = cos = 1EA Is conastant

LAB = 300 cm

Page 79: Stiffness 9

K l K g=

K l = E I

12/L^3

6/L^2

-12/L^3

6/L^2

6/L^2

2/L

6/L^2-

4/L

12/L^3-

6/L^2-

12/L^3

6/L^2-

6/L^2

-6/L^2

4/L

2/L

Page 80: Stiffness 9

K l = EI

0.44

0.67

-0.44

0.67

0.67

1.33

-0.67

0.67

-0.44

-0.67

0.44

-0.67

0.67

0.67

-0.67

1.33

A B

"ِ"ِA

B

Page 81: Stiffness 9

Second element : ( B-c)Start Joint : B End Joint : c

= E I

12/L^3

6/L^2

-12/L^3

6/L^2

6/L^2

2/L

6/L^2-

4/L

12/L^3-

6/L^2-

12/L^3

6/L^2-

6/L^2

-6/L^2

4/L

2/LK l

Page 82: Stiffness 9

K l = EI

0.1875

0.375

-0.1875

0.375

0.375

1

-0.375

0.5

-0.1875

-0.375

0.1875

-0.375

0.375

0.5

-0.375

1

B C

C

B

Page 83: Stiffness 9

Second element : ( C-D )Start Joint : C End Joint : D

= E I

12/L^3

6/L^2

-12/L^3

6/L^2

6/L^2

2/L

6/L^2-

4/L

12/L^3-

6/L^2-

12/L^3

6/L^2-

6/L^2

-6/L^2

4/L

2/LK l

Page 84: Stiffness 9

K l = EI

0.44

0.67

-0.44

0.67

0.67

1.33

-0.67

0.67

-0.44

-0.67

0.44

-0.67

0.67

0.67

-0.67

1.33

C D

�ِC

D

Page 85: Stiffness 9

K

g = EI

0.44

0.67

-0.44

0.67

0.67

1.33

-0.67

0.67

-0.44

-0.67

0.44

-0.67

0.67

0.67

-0.67

1.33

A B

�Aِ

B

Page 86: Stiffness 9

K g= EI

0.1875

0.375

-0.1875

0.375

0.375

1

-0.375

0.5

-0.1875

-0.375

0.1875

-0.375

0.375

0.5

-0.375

1

B C

C

B

Page 87: Stiffness 9

K g= EI

0.44

0.67

-0.44

0.67

0.67

1.33

-0.67

0.67

-0.44

-0.67

0.44

-0.67

0.67

0.67

-0.67

1.33

C D

�ِC

D

Page 88: Stiffness 9

Ks = EI

0.44

.0937

0.67

-0.44

0.67

0

0

0.67

1.33

-0.67

0.67

0

0

0.67

0.67

-.0.295

2.33

-0.375

0.5

0

0

-0.1875

.-375

.2525

0.295

0

0

0.375

0.5

0.295

2.33

0

0

.-44

.-67

A B C

B

A

C

K l K g=

00 0 .-44 .-67 .-67

0 0 0 .67 0.67 1.33

D

D

-0.44

-0.67

.0.6275

-0.295

-0.1875

0.375

0

0

0

0

.67

0.67

0

0

0

0

.-67

0.44

Page 89: Stiffness 9

Partition

KuuK =Kru

Kur

Krr

u ru

r

Kuu = EI 2.33

2.33

0.5

0.5

Page 90: Stiffness 9

Force vectorTransformation from member forces to Joint forces

L

P

8LP

8LP

90

P a b2

L2

L

P

a b P b a2

L2

Page 91: Stiffness 9

2

A B C D

1 2 2 1 2

100 kN 200 kN 150 kN

100 kN 150 kN

200 kN

100 kNm100 kNm

44.4 kNm

22.2 kNm

66.7 kNm

33.3 kNm

Fixed End Reaction

(FER)

Page 92: Stiffness 9

2

A B C D

1 2 2 1 2

100 kN 200 kN 150 kN

100 kN 150 kN

200 kN

100 kNm100 kNm

44.4 kNm

22.2 kNm

66.7 kNm

33.3 kNm

Fixed End Action (FEA)

Page 93: Stiffness 9

2

A B C D

1 2 2 1 2

100 kN 200 kN 150 kN

100 kNm

44.4 kNm 66.7 kNm

100 kNm

33.3 kNm55.6 kNm

Page 94: Stiffness 9

2

A B C D

1 2 2 1 2

100 kN 200 kN 150 kN

33.3 kNm55.6 kNm

F1

F2

=-55.6

33.3

Page 95: Stiffness 9

F = K Dk11F1

F2

=k21

k12

k22

d1

d2

-55.6

33.3=

7/3

0.5 7/3

0.5EI

B

C

B

C=

1EI

7/3

0.5 7/3

0.5-1

-55.6

33.3

Page 96: Stiffness 9

B

C=

1EI

7/3

0.5 7/3

0.5-1

-55.6

33.3

B

C=

1EI

-28.18

20.31

Page 97: Stiffness 9

Internal forces in beam elements2 EI

LMAB= ( 2 + )

MBA=2 EI

L( + 2 )

MBA=2 EI

L( + 2 ) M(FER) BA +

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

L

Page 98: Stiffness 9

2

A B CD

1 2 2 1 2

100 kN 200 kN 150 kN

100 kN 150 kN

200 kN

100 kNm100 kNm

44.4 kNm

22.2 kNm

66.7 kNm

33.3 kNm

Fixed End Reaction

(FER)

Page 99: Stiffness 9

B

C=

1EI

-28.18

20.31

2 EILMAB= ( 2 + ) M(FER) AB +

MBA=2 EI

L( + 2 ) M(FER) BA +

100 kN

44.4 kNm22.2 kNm

= 22.2 + 2/3 (-28.18) = 3.4

= -44.4 + 2/3 (2x-28.18) = - 82

A B

Page 100: Stiffness 9

B

C=

1EI

-28.18

20.31

2 EILMBC= ( 2 + ) CM(FER) BC +

MCB=2 EI

L( + 2 ) CM(FER) CB +

= 100 + 2/4 (2x-28.18+20.31) = 82

= -100 + 2/4 (-28.18+2x20.31) = - 93.8

200 kN

100 kNm100 kNm

B C

Page 101: Stiffness 9

B

C=

1EI

-28.18

20.31

2 EILMCD= ( 2 + )C DM(FER) CD +

MDC=2 EI

L( + 2 )C DM(FER) DC +

= 66.7 + 2/3 (2x20.31) = 93.8

= -33.3 + 2/3 (20.31) = - 19.8

150 kN

33.3 kNm66.7 kNm

C D

Page 102: Stiffness 9

MAB= 3.4MBA= -82

MBC= 82MCB= -93.8

MCD= 93.8MDC= -19.8

3.4

82

19.8

93.8

B.M.D

Page 103: Stiffness 9

3.4

82

19.8

93.8

B.M.D

2

A B C D

1 2 2 1 2

100 kN 200 kN 150 kN

55.887.9

69.1

66.7

10.9

200

112.1

100

30.9

Page 104: Stiffness 9

Example 6:Draw B.M.D for the shown beam where EI is shown in figure

3

A B C D

5 5 2

100 kN 200 kN 100 kN

3 2

250 kN

III2 I2 I2 I

E

Page 105: Stiffness 9

Solution Steps of assembly method :

Drive the member local stiffness matrixLocal

k11F1

F2

=k21

F3 k31

k12

k22

k32

k13

k23

k33

F4k41 k42 k43

k14

k24

k34

k44

d1

d2

d3

d4

Page 106: Stiffness 9

d3

d1

d2 d4

First column inLocal Stiffness matrix

6 EI

L2

6 EI

L212 EI

L3

Page 107: Stiffness 9

F1 = 12 EIL3

F2 = 6 EI

L2

6 EI

L2

F3 = 12 EIL3

-

F4 =

first column in Local Stiffness matrix

Page 108: Stiffness 9

=

k31

k41

12 EIL3

6 EIL2

-12 EIL3

6 EIL2

k11

k21

Page 109: Stiffness 9

Second column inLocal Stiffness matrix

d2 =1

4 EIL

2 EIL

6 EIL2

6 EIL2

Page 110: Stiffness 9

F3 =F1 =

F4 =F2 =

6 EIL2

6 EIL2

-

2 EIL

4 EIL

Page 111: Stiffness 9

Second column inLocal Stiffness matrix

=

k12

k22

k32

k42

6 EI

L2

4 EIL

6 EIL2

- 2 EIL

Page 112: Stiffness 9

Third column in Local Stiffness matrix

d3 =1

12 EIL3

12 EIL3

6 EIL2

6 EIL2

Page 113: Stiffness 9

Third column in Local Stiffness matrix

=

k13

k23

k33

k33

-12 EIL3

-6 EIL2

12 EIL3

-6 EIL2

Page 114: Stiffness 9

12 EIL3

-6 EIL2

-6 EIL2

F4 =

F3 = F1 = -12 EIL3

F2 =

Page 115: Stiffness 9

Fourth column inLocal Stiffness matrix

6 EIL2

6 EIL2

4 EIL

2 EIL

Page 116: Stiffness 9

F1 = F3=

F2 = F4 =

6 EIL2

2 EIL

-6 EIL2

4 EIL

Page 117: Stiffness 9

K l

12 EIL3

6 EIL2

6 EIL2

4 EIL

-12 EIL3

=

-12 EIL3

-6 EIL2

-6 EIL2

12 EIL3

-6 EIL2

6 EIL2

2 EIL

6 EIL2

2 EIL

-6 EIL2

4 EIL

Page 118: Stiffness 9

First element : (B-C )

Start Joint : B

End Joint : CAngle : 0s = sin = 0c = cos = 1EA Is conastant

LAB = 1000 cm

Page 119: Stiffness 9

K l K g=

K l =E I

12/L^3

6/L^2

-12/L^3

6/L^2

6/L^2

2/L

6/L^2-

4/L

12/L^3-

6/L^2-

12/L^3

6/L^2-

6/L^2

-6/L^2

4/L

2/L

Page 120: Stiffness 9

K l = EI

0.024

0.12

-0.024

0.12

0.12

0.8

-0.12

0.4

-0.024

-0.12

0.024

-0.12

0.12

0.4

-0.12

0.8

B C

�ِB

C

Page 121: Stiffness 9

First element : (C-D )

Start Joint : C

End Joint : DAngle : 0s = sin = 0c = cos = 1EA Is conastant

LAB = 500 cm

Page 122: Stiffness 9

K l K g=

K l = E I

12/L^3

6/L^2

-12/L^3

6/L^2

6/L^2

2/L

6/L^2-

4/L

12/L^3-

6/L^2-

12/L^3

6/L^2-

6/L^2

-6/L^2

4/L

2/L

Page 123: Stiffness 9

K l = EI

0.096

0.24

-0.096

0.24

0.24

0.8

-0.24

0.4

-0.096

-0.24

0.096

-0.24

0.24

0.4

-024

0.8

C D

C

D

Page 124: Stiffness 9

Assembly :

K g= EI

0.024

0.12

-0.024

0.12

0.12

0.8

-0.12

0.4

-0.024

-0.12

0.024

-0.12

0.12

0.4

-0.12

0.8

B C

�ِB

C

Page 125: Stiffness 9

K g= EI

0.096

0.24

-0.096

0.24

0.24

0.8

-0.24

0.4

-0.096

-0.24

0.096

-0.24

0.24

0.4

-024

0.8

C D

C

D

Page 126: Stiffness 9

Ks = EI

0.024

.0937

0.12

-0.024

0.12

0

0

0.12

0.8

-0.12

0.4

0

0

-0.024

-0.12

0.12

0.12

-0.096

0.024

0.12

0.4

0.12

1.6

-0.024

0.4

0

0

-0.096

-0.24

0.096

-0.24

0

0

0.24

0.4

-0.24

0.8

B C D

C

B

D

K l K g=

Page 127: Stiffness 9

Partition

KuuK =Kru

Kur

Krr

u ru

r

Kuu = EI 0.8

0.4

0

0.4

1.60.4 0.8

0.4

0

Page 128: Stiffness 9

Force vectorTransformation from member forces to Joint forces

L

P

8LP

8LP

128

P a b2

L2

L

P

a b P b a2

L2

Page 129: Stiffness 9

100 kN

100 kN200 kN

250 kNm250 kNm

300 kNm

200 kNm

Fixed End Reaction (FER)

3

A B C D

5 5 2

100 kN 200 kN 100 kN

3 2

250 kN

E

250 kN180 kNm 120 kNm

Page 130: Stiffness 9

100 kN

100 kN200 kN

250 kNm250 kNm

300 kNm

200 kNm

Fixed End Action (FEA)

3

A B C D

5 5 2

100 kN 200 kN 100 kN

3 2

250 kN

E

250 kN180 kNm 120 kNm

Page 131: Stiffness 9

250 kNm250 kNm

300 kNm

200 kNm

3

A B C D

5 5 2

100 kN 200 kN 100 kN

3 2

250 kN

E

180 kNm 120 kNm

50 kNm 70 kNm 80 kNm

Page 132: Stiffness 9

3

A B C D

5 5 2

100 kN 200 kN 100 kN

3 2

250 kN

E

50 kNm 70 kNm 80 kNm

F1

F2 =

50

70

F3 - 80

Page 133: Stiffness 9

F = K DThe stiffness equation

=

50

70- 80

0.80.4

0

0.41.6

0.4

00.4

0.8EI

B

C

D

=

50

70- 80

0.80.4

0

0.41.6

0.4

00.4

0.8

B

C

D

1EI

-1

=1EI

27.08

70.83- 135.42

Page 134: Stiffness 9

=

B

C

D

1EI

27.08

70.83- 135.42

Page 135: Stiffness 9

Internal forces in beam elements

2 EILMAB= ( 2 + )

MBA=2 EI

L( + 2 )

M(FER) AB +

M(FER) BA +

Page 136: Stiffness 9

100 kN

100 kN200 kN

250 kNm250 kNm

300 kNm

200 kNm

Fixed End Reaction (FER)

3

A B C D

5 5 2

100 kN 200 kN 100 kN

3 2

250 kN

E

250 kN180 kNm 120 kNm

Page 137: Stiffness 9

2 E(2I)LMBC= ( 2 + ) CM(FER) BC +

MCB=2 E(2I)

L( + 2 ) CM(FER) CB +

= 250 + 4/10 (2x27.08+70.83) = 300

= -250 + 4/10 (27.08+2x70.83) = - 182.5

200 kN

250 kNm250 kNm

B C

=

B

C

D

1EI

27.08

70.83- 135.42

Page 138: Stiffness 9

2 E(2I)LMBC= ( 2 + ) CM(FER) BC +

MCB=2 E(2I)

L( + 2 ) CM(FER) CB +

= 250 + 4/10 (2x27.08+70.83) = 300

= -250 + 4/10 (27.08+2x70.83) = - 182.5

200 kN

250 kNm250 kNm

B C

=

B

C

D

1EI

27.08

70.83- 135.42

Page 139: Stiffness 9

MBC= 300MCB= -182.5

MCD= 182.5MDC= -200

300200

B.M.D

A

B C D

E

182.5

Page 140: Stiffness 9

B.M.D

3

A B C D

5 5 2

100 kN 200 kN 100 kN

3 2

250 kN

E

300200

A

B C D

E

182.5241.25

500

258.75

189.5

300

110.5

Page 141: Stiffness 9

Example 7:Draw B.M.D for the shown beam where EI is constant for all members

A B

5 5

240 kN

C

5 5

120 kN

Page 142: Stiffness 9

K l

12 EIL3

6 EIL2

6 EIL2

4 EIL

-12 EIL3

=

-12 EIL3

-6 EIL2

-6 EIL2

12 EIL3

-6 EIL2

6 EIL2

2 EIL

6 EIL2

2 EIL

-6 EIL2

4 EIL

4 EIL

Page 143: Stiffness 9

First element : (A-B )

EA conastant

LAB = 10 m

12 EIL3

=0.012EI

6 EIL2

4 EIL

2 EIL

=0.06 EI

=0.4 EI

=0.2 EI

Page 144: Stiffness 9

K l =

0.012 EI

0.012 EI 0.012 EI

0.012 EI

0.06 EI

0.06 EI

0.06 EI0.06 EI

0.06 EI

0.06 EI

0.06 EI

0.06 EI

0.4 EI

0.4 EI

0.2 EI

0.2 EI

A

A

B

B

Page 145: Stiffness 9

Second element : ( B-c)

EI conastant LAB = 10 m

12 EIL3

=0.012EI

6 EIL2

4 EIL

=0.06 EI

=0.4 EI

=0.2 EI2 EI

L

Page 146: Stiffness 9

K l =

0.012 EI

0.012 EI 0.012 EI

0.012 EI

0.06 EI

0.06 EI

0.06 EI0.06 EI

0.06 EI

0.06 EI

0.06 EI

0.06 EI

0.4 EI

0.4 EI

0.2 EI

0.2 EI

A

A

B

B

Page 147: Stiffness 9

K gK l =

Assembly :

=

K gg1

0.012 EI 0.06 EI 0.012 EI 0.06 EI

0.06 EI 0.4 EI 0.06 EI 0.2 EI

0.012 EI 0.06 EI 0.06 EI0.012 EI

0.06 EI 0.2 EI 0.4 EI0.06 EI

Page 148: Stiffness 9

2=

K gg

0.012 EI

0.012 EI 0.012 EI

0.012 EI

0.06 EI

0.06 EI

0.06 EI0.06 EI

0.06 EI

0.06 EI

0.06 EI

0.06 EI

0.4 EI

0.4 EI

0.2 EI

0.2 EI

A B

A

B

Page 149: Stiffness 9

=

0.012

-0.012

-0.012

0.06

0.06

-0.06 0.06

0.06

-0.06

0.06

-0.06

0.4

0.4

0.2

0.2

A

A

B

B

K s

0.0

0.8

0.024

0.0

0.0

0.0

0.0 0.0

0.0

0

0.0

0.0

0.012

-0.012

-0.012

0.06

-0.06

0.2

0.2

-0.06

-0.06

EI

c

c

Page 150: Stiffness 9

Partition

KuuK =Kru

Kur

Krr

u ru

r

Page 151: Stiffness 9

Force vectorTransformation from member forces to Joint forces

L

P

8LP

8LP-

Page 152: Stiffness 9

240 kN300 kNm300 kNm

Fixed End Reaction

(FER)

A B

5 5

240 kNC

5 5

120 kN

120 kN150 kNm150 kNm

Page 153: Stiffness 9

300 kNm300 kNm

A B

5 5

240 kNC

5 5

120 kN

150 kNm150 kNm

120120 6060

Page 154: Stiffness 9

F1

F2

=

120

-300

F3

F6

F5

F4

180

150

60

150

Page 155: Stiffness 9

120

-300

180

150

60

150

0.012

-0.012

-0.012

0.06

0.06

-0.06 0.06

0.06

-0.06

0.06

-0.06

0.4

0.4

0.2

0.2

0.0

0.8

0.024

0.0

0.0

0.0

0.0 0.0

0.0

0

0.0

0.0

0.012

-0.012

-0.012

0.06

-0.06

0.2

0.2

-0.06

-0.06

d1

d2

d3

d4

d5

d6

= EI

Page 156: Stiffness 9

120

-300

180

150

60

150

-0.012 0.06

-0.06

0.06

-0.06

0.06

-0.06

0.4

0.4

0.2

0.2

0.0

0.8

0.024

0.0

0.0

0.0

0.0 0.0

0.0

0.0

0.0

0.0

0.012

-0.012

-0.012

0.06

-0.06

0.2

0.2

-0.06

-0.06

d1

d2

d3

d4

d5

d6

=EI

Page 157: Stiffness 9

=B

C

150

150

0.8

0.2 0.4

0.2EI

B

C=

1EI

-1150

150

0.8

0.2 0.4

0.2

Page 158: Stiffness 9

B

C=

1EI

107.14

321.43

B

C=

1EI

-1150

150

0.8

0.2 0.4

0.2

Page 159: Stiffness 9

Internal forces in beam elements

2 EILMAB= ( 2 + )

MBA=2 EI

L( + 2 )

M(FER) AB +

M(FER) BA +

Page 160: Stiffness 9

240 kN300 kNm300 kNm

Fixed End Reaction

(FER)

A B

5 5

240 kNC

5 5

120 kN

120 kN150 kNm150 kNm

Page 161: Stiffness 9

2 EILMAB= ( 2 + )

MBA=2 EI

L( + 2 )

M(FER) AB +

M(FER) BA +

= 300 + 2/10 (107.14) = 321.4

= -300 + 2/10 (2x107.14) = - 257.1

B

C=

1EI

107.14

321.43

240 kN

300 kNm300 kNm

A B

Page 162: Stiffness 9

2 EILMBC= ( 2 + ) C

MCB=2 EI

L( + 2 ) C

M(FER) BC +

M(FER) CB +

= 150 + 2/10 (2x107.14+321.43) = 257.1

= -150 + 2/10 (107.14+2x321.43) = 0

120 kN

150 kNm150 kNm

B C B

C=

1EI

107.14

321.43

Page 163: Stiffness 9

MAB= 321.4MBA= -257.1

MBC= 257.1MCB= 0

321.4257.1

B.M.D

Page 164: Stiffness 9

93.8

128.55

300

171.45

A B

5 5

240 kNC

5 5

120 kN

321.4257.1

B.M.D

289.25

600

310.75

Page 165: Stiffness 9

Beams with settlement

6 EIL2

6 EIL2 12 EI

L3

12 EIL3

Fixed End Reaction

(FER)

Page 166: Stiffness 9

Beams with settlement

6 EIL2

6 EIL2

12 EIL3

12 EIL3

Fixed End Reaction

(FER)

Page 167: Stiffness 9

Beams with settlement

3 EIL2 3 EI

L3

3 EIL3

Fixed End Reaction

(FER)

Page 168: Stiffness 9

Beams with settlement

3 EIL2

3 EIL3

3 EIL3

Fixed End Reaction

(FER)

Page 169: Stiffness 9

169

Page 170: Stiffness 9

Example 8:Draw B.M.D for the shown beam due to the shown loads and vertical downward settlement at support B (2000/EI) and at support C (1000/EI) where EI is constant for all members

A B

5 5

240 kN

C

5 5

120 kN

Page 171: Stiffness 9

=0.8

0.2 0.4

0.2K EI

From example 7 :

Page 172: Stiffness 9

240 kN300 kNm300 kNm

Fixed End Reaction

(FER)

A B

5 5

240 kNC

5 5

120 kN

120 kN150 kNm150 kNm

ForLoads

Page 173: Stiffness 9

Fixed End Reaction (FER)

A B

5 5

C

5 5

Forsettlement

1000EI

2000EI

6 EIx1000102 EI

6 EIx1000102 EI

6 EIx2000102 EI

6 EIx2000102 EI

120

120 6060

Page 174: Stiffness 9

300 kNm300 kNm

A B

5 5

240 kNC

5 5

120 kN

150 kNm150 kNmFor

Loads

Forsettlement 120 120 60 60

Fixed End Reaction

(FER)

210 kNmTotal 90 kNm

420 180 90 210

Page 175: Stiffness 9

A B

5 5

240 kNC

5 5

120 kN

Fixed End Reaction

(FER)210 kNm90 kNm

Fixed End Action (FEA)

210 kNm90 kNm

Page 176: Stiffness 9

A B

5 5

240 kNC

5 5

120 kN

Fixed End Action (FEA)

210 kNm90 kNm

F1

F2

=90

210

Page 177: Stiffness 9

F = K Dk11F1

F2

=k21

k12

k22

d1

d2

=B

C

B

C=

1EI

-1

90

210

90

210

0.8

0.2 0.4

0.2EI

0.8

0.2 0.4

0.2

Page 178: Stiffness 9

B

C=

1EI

-21.43

535.71

B

C=

1EI

-190

210

0.8

0.2 0.4

0.2

Page 179: Stiffness 9

Internal forces in beam elements

2 EILMAB= ( 2 + )

MBA=2 EI

L( + 2 )

M(FER) AB +

M(FER) BA +

Page 180: Stiffness 9

300 kNm300 kNm

A B

5 5

240 kNC

5 5

120 kN

150 kNm150 kNmFor

Loads

Forsettlement 120 120 60 60

Fixed End Reaction

(FER)

210 kNmTotal 90 kNm

420 180 90 210

Page 181: Stiffness 9

2 EILMAB= ( 2 + )

MBA=2 EI

L( + 2 )

M(FER) AB +

M(FER) BA +

= 420 + 2/10 (-21.43) = 415.7

= -180 + 2/10 (2x-21.43) = - 188.6

180 kNm420 kNm

A B B

C=

1EI

-21.43

535.71

Page 182: Stiffness 9

2 EILMBC= ( 2 + ) C

MCB=2 EI

L( + 2 ) C

M(FER) BC +

M(FER) CB +

= 90 + 2/10 (2x-21.43+535.7) = 188.6

= -210 + 2/10 (-21.43+2x535.71) = 0

210 kNm90 kNm

B C B

C=

1EI

-21.43

535.71

Page 183: Stiffness 9

MAB= 321.4MBA= -257.1

MBC= 257.1MCB= 0

415.7188.6

B.M.D

Page 184: Stiffness 9

93.8

94.3

300

205.7

A B

5 5

240 kNC

5 5

120 kN

415.7188.6

B.M.D

302.15

600

297.85

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