Stiffness 3

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Lecture No. : 3

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Remember :F1

1 2

k1 k2

F3

3

k3

F2

d1 d2 d3

k11F1

F2 = k21

F = K D

F3 k31

k12

k22

k32

k13

k23

k33

d1

d2

d3

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k11F1

F2 = k21

F3 k31

k12

k22

k32

k13

k23

k33

d1

d2d3

First column in Stiffness matrix

d1 =1 d2 =0 d3 =0

1 2

k1 k2

3

k3

d1=1

k11

k21

k31

=

F1

F2

F3

Remember :

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k11F1

F2 = k21

F3 k31

k12

k22

k32

k13

k23

k33

d1

d2

d3

Second column in Stiffness matrix

d1 =0 d2 =1 d3 =0

k12

k22

k32

=

F1

F2

F3

1 2

k1 k2

3

k3

d2=1

Remember :

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k11F1

F2 = k21

F3 k31

k12

k22

k32

k13

k23

k33

d1

d2

d3

Third column in Stiffness matrix

d1 =0 d2 =0 d3 =1

k13

k23

k33

=

F1

F2

F3

1 2

k1 k2

3

k3

d3=1

Remember :

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a

b

D

q

D1= D cos q

D1 D2

D2= D sin q

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D

P

P

AE

LPD=

AE

L

P D=

Relation between force andDisplacement in truss element

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Degrees of freedom (DOF)

d1

d2

d1

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AB

C 4 m

3

Example 1:

Construct the stiffness matrix for the shown

truss where EA is constant for all members

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Modeling

A

B

C

4

3F1d1

F2 d2

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F = K D

k11F1

F2=

k21

k12

k22

d1

d2

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First column in Stiffness matrix

d1 =1 d2 =0k11F1

F2

=k21

k12

k22

d1

d2

A

B

C

4

3F1d

1

F2

d2

k11F1

F2= k21

k12

k22

1

0

k11 F1

F2

=k21

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First column in Stiffness matrix

d1 =1 d2 =0k11F1

F2

=k21

k12

k22

d1

d2

A

B

C

4

3d1

1

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A

B

C

4

3

1A

1

q

A cos qEL

AE

LAE

L

A cos qEL

AB

C 4 m3

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A

B

C

4

3

1A

1

q

A x 0.8E

5

AE

4AE

4

A x 0.8E

5

AB

C 4 m3

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4

3q

0.16 EA

0.25 EA 0.25 EA

0.16 EA

AE

4AE

4

A x 0.8E

5

A x 0.8E5

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A

B

C

4

30.25 EA

0.16 EA

q

0.16 EA

0.25 EA 0.25 EA

0.16 EA

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A

B

C

4

30.25 EA

0.16 EA

q 0.16 EA cos q

0.16 EA sin q

cos q=0.8sin q=0.6

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A

B

C

4

30.25 EA

0.16 EA cos q

0.16 EA sin q

0.16 EA x 0.8

0.16 EA x 0.6

0.128 EA

0.096 EA

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A

B

C

4

30.25 EA

0.128 EA

0.096 EA

0.378 EA

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A

B

C

4

3

0.096 EA

0.378 EA

A

B

C

F1

F2

F1 = 0.378 EAF2 = 0.096 EA

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A

B

C

4

3

0.096 EA

0.378 EA

F1 = 0.378 EAF2 = 0.096 EA

k11 F1

F2

=k21

k11 =k21

0.378 EA

0.096 EA

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second column in Stiffness matrix

d1 =0 d2 =1k11F1

F2

=k21

k12

k22

d1

d2

A

B

C

4

3F1d

1

F2

d2

k11F1

F2

= k21

k12

k22

0

1

k12 F1

F2

=k22

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second column in Stiffness matrix

d1 =0 d2 =1k11F1

F2

=k21

k12

k22

d1

d2

A

B

C

4

3

d2

1

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A

B

C

4

3

1

A

1

a

A cos aEL

0

A cos aEL

0

cos a = 0.6L = 5

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A

B

C

4

3

1

A

1

a

A x 0.6E5

0

A x 0.6E

5

0

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4

3

0.12 EA

0 0

0.12 EA

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A

B

C

4

30

0.12 EA

q

0.12 EA

0 0

0.12 EA

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A

B

C

4

30

0.12 EA

q 0.12 EA cos q

0.12 EA sin q

cos q=0.8sin q=0.6

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A

B

C

4

30

0.12 EA cos q

0.12 EA sin q

0.12 EA x 0.8

0.12 EA x 0.6

0.096 EA

0.072 EA

cos q=0.8sin q=0.6

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A

B

C

4

3

0.096 EA

0.072 EA

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A

B

C

4

3

0.072 EA

A

B

C

F1

F2

F1 = 0.096 EAF2 = 0.072 EA

0.096 EA

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A

B

C

4

3 k12 F1F2=k22k12 =k22

0.096 EA

0.072 EA

F1 = 0.096 EAF2 = 0.072 EA

0.072 EA

0.096 EA

S

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Summary

k11F1

F2=

k21

F = K D

k12

k22

d1

d2

k11 =k21

0.378 EA

0.096 EA

k12 =k22

0.096 EA

0.072 EA

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k11 =k21

0.378 EA

0.096 EA

k12 =k22

0.096 EA

0.072 EA

K =

0.378 EA

0.096 EA

0.096 EA

0.072 EA

K =0.378

0.096

0.096

0.072

EA

E l 2

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AB

C 4 m

3

Example 2:

Calculate the joint displacements for the

shown truss due to the shown force whereEA = 105 kN for all members

12 kN

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k11F1

F2=

k21

F = K D

k12

k22

d1

d2

From the previous example

K =0.378

0.096

0.096

0.072

EA

F th i l

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From the previous example

K =

0.378

0.096

0.096

0.072

EA

EA = 105 kN

K = 0.378

0.096

0.096

0.072

105 = 378

96

96

72

100

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AB

C 4 m

312 kN

A

B

C

F1

F2

F1

F2=

0

-12

Force vector

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=378

96

96

72

100d1

d2

0

-12

d1

d2= 100

378

96

96

72

0

-12

1

-1

d1

d2 =105

4

-5.33

-5.33

21

0

-12

1

=

0.0006

-0.0025=

0. 6

-2.5m mm

E l 3

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AB

C 4 m

3

Example 3:

Calculate the joint displacements for the

shown truss due to the shown force whereE = 2000 kN/cm2, ABA= 100 cm

2, ACA= 200 cm2

12 kN

M d li

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Modeling

A

B

C

4

3F1d1

F2 d2

EABA= 2x105 kN

EACA= 4x105

kN

E = 2000 kN/cm2

, ABA= 100 cm2

, ACA= 200 cm2

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k11F1

F2=

k21

F = K D

k12

k22

d1

d2

First column in Stiffness matrix k k d

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First column in Stiffness matrix

d1 =1 d2 =0k11F1

F2

=k21

k12

k22

d1

d2

A

B

C

4

3F1d1

F2 d2

k11F1

F2

=k21

k12

k22

1

0

k11 F1

F2

=k21

First column in Stiffness matrix k k d

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First column in Stiffness matrix

d1 =1 d2 =0k11F1

F2

=k21

k12

k22

d1

d2

A

B

C

4

3d1

1

1

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A

B

C

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