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