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FRAME ANALYSIS USING THE STIFFNESS METHOD · PDF fileFrame-Member Global Stiffness Matrix FRAME ANALYSIS USING THE STIFFNESS METHOD. 2 Simple Frames. 3 Frame-Member Stiffness Matrix

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Text of FRAME ANALYSIS USING THE STIFFNESS METHOD · PDF fileFrame-Member Global Stiffness Matrix...

  • 1

    ! Simple Frames! Frame-Member Stiffness Matrix! Displacement and Force Transformation Matrices! Frame-Member Global Stiffness Matrix

    ! Special Frames! Frame-Member Global Stiffness Matrix

    FRAME ANALYSIS USING THE STIFFNESSMETHOD

  • 2

    Simple Frames

  • 3

    Frame-Member Stiffness Matrix

    0 0 00 - AE/LAE/L

    4EI/L - 6EI/L2 2EI/L6EI/L2 00

    6EI/L2 - 12EI/L3 6EI/L212EI/L3 00

    0

    2EI/L

    -6EI/L20

    - 6EI/L2

    12EI/L30

    -6EI/L2

    4EI/L

    0

    6EI/L2

    -12EI/L3AE/L

    0

    0

    -AE/L

    0

    0

    m

    xy

    i

    j

    3

    5

    6

    2

    4

    1

    3 62 41 5

    [k]

    456

    12

    3

    6EI/L24EI/L

    6EI/L24EI/L

    AE/L

    AE/L

    6EI/L2

    6EI/L2

    12EI/L312EI/L3

    2EI/L

    d 1 = 1

    AE/L

    AE/L

    d 2 = 1

    12EI/L312EI/L3

    d 3 = 1

    2EI/L

    6EI/L2

    6EI/L2

    6EI/L26EI/L2AE/L

    2EI/L

    6EI/L26EI/L2

    d 6 = 1

    6EI/L26EI/L2

    2EI/L

    6EI/L2

    12EI/L312EI/L3

    4EI/L4EI/L12EI/L3

    12EI/L3

    6EI/L2

    6EI/L2AE/L

    6EI/L2d 4 = 1

    d 5 = 1AE/L

    AE/L

  • 4

    m

    i

    j

    m

    i

    j

    xy

    x

    y

    Displacement and Force Transformation Matrices

    12

    3

    45

    6

    yx

    4

    5

    6

    1

    2

    3

  • 5

    xq4 = q4 cos x - q5 cos yq5 = q4 cos y + q5 cos xq6 = q6

    y

    i

    jy

    x

    y

    x

    45

    6

    1

    2

    3

    y

    x

    m

    i

    j

    x

    y

    12

    3

    45

    6

    Force Transformation

    Lxx ij

    x

    =

    Lyy ij

    y

    =

    =

    6'

    5'

    4'

    6

    5

    4

    10000

    qqq

    qqq

    xy

    yx

    =

    6'

    5'

    4'

    3'

    2'

    1'

    6

    5

    4

    3

    2

    1

    1000000000000000010000000000

    qqqqqq

    qqqqqq

    xy

    yx

    xy

    yx

    [ ] [ ] [ ]'qTq T=

  • 6

    [q] = [T]T[q]

    = [T]T ( [k][d] + [qF] )

    = [T]T [k][d] + [T]T [qF]

    [q] = [T]T [k][T][d] + [T]T [qF] = [k][d] + [qF]

    Therefore, [k] = [T]T [k][T]

    [qF] = [T]T [qF]

    [q] = [T]T[q]

    [d] = [T][d]

    [k] = [T]T [k][T]

  • 7

    [q] = [T]T[q]= [T]T ( [k][d] + [qF] ) = [T]T[k][d] + [T]T[qF] = [T]T [k][T][d] + [T]T [qF]

    Frame Member Global Stiffness Matrix

    [k] [qF][ k ] = [ T ]T[ k ][T] =

    Ui

    Vi

    Mi

    Uj

    Vj

    Mj

    Vj Mj

    - iy6EIL2

    ix6EIL2

    2EIL

    jy6EIL2

    - jx6EIL2

    4EIL

    Ui Vi Mi

    - iy6EIL2

    ix6EIL2

    4EIL

    jy6EIL2

    jx6EIL2

    -

    2EIL

    Uj

    AEL

    - ixiy)(12EIL3

    AEL

    iy2 + 12EIL3

    ix2 )(

    ix6EIL2

    ix6EIL2

    AEL

    iyjx - 12EI

    L3ixjy)-(

    AEL

    ixjx + 12EIL3

    iyjy)-(

    jy6EIL2

    AEL

    jx2 + 12EIL3

    jy2 )(

    jy6EIL2

    jx6EIL2

    -

    AEL

    - jxjy)(12EIL3

    - jx6EIL2

    AEL

    ixjy -12EIL3

    iyjx)-(

    AEL

    - ixiy)(12EIL3

    - iy6EIL2

    - iy6EIL2

    AEL

    ixjx + 12EIL3

    iy jy)-(

    )(AEL

    ix2 + 12EIL3

    iy2

    AEL

    iyjy + 12EIL3

    ix jx )-(

    AEL

    iyjy +12EIL3

    ixjx)-(

    12EIL3

    jx2 )AEL

    - ( ixjy- iyjx )12EIL3

    AEL

    iyjx - 12EIL3

    ixjy)-(

    AEL

    - jxjy)(12EIL3

    AEL jy

    2 + (

  • 8

    5 kN

    6 m

    6 m

    AB

    C

    Example 1

    For the frame shown, use the stiffness method to:(a) Determine the deflection and rotation at B.(b) Determine all the reactions at supports.(c) Draw the quantitative shear and bending moment diagrams.E = 200 GPa, I = 60(106) mm4, A = 600 mm2

  • 9

    5 kN

    6 m

    6 m

    AB

    C kN/m666.667(6m)

    )m10)(60mkN1012(20012

    3

    462

    6

    3 =

    =

    LEI

    kN/m200006m

    )mkN10)(200m10(600 2

    626

    =

    =

    LAE

    kN2000(6m)

    )m10)(60mkN106(2006

    2

    462

    6

    2 =

    =

    LEI

    mkN80006m

    )m10)(60mkN104(2004

    462

    6

    =

    =

    LEI

    mkN40006m

    )m10)(60mkN102(2002

    462

    6

    =

    =

    LEI

    Global :

    AB

    C

    1

    2

    78 9

    4

    6 5

    12 3

  • 10

    Global :

    AB

    C

    1

    2

    78 9

    4

    6 5

    12 3

    A B14

    5 6

    1

    2 3

    Local :

    5

    2

    1 3

    4

    6

    2

    Using Transformation Matrix:

    Member Stiffness Matrix

    [ ]

    =

    LEILEILEILEILEILEILEILEI

    LAELAELEILEILEILEILEILEILEILEI

    LAEAE/L

    /4/60/2/60/6/120/6/120

    00/00//2/60/4/60/6/120/6/120

    00/00

    k'

    22

    2323

    22

    2323

    Mi

    VjMj

    Vi

    Nj

    Ni

    i j ji ji

  • 11

    A B14

    5 6

    1

    2 3

    Local :

    [q] = [q]

    -> [k]1 = [k]1

    Stiffness Matrix: Member 1

    Global:

    AB

    C

    1

    2

    78 9

    4

    6 5

    12 3

    4 6 5 1 2 34

    6

    5

    1

    2

    3

    20000

    0

    0

    -20000

    0

    0

    0

    666.667

    2000

    0

    -666.667

    2000

    0

    2000

    8000

    0

    -2000

    4000

    -20000

    0

    0

    20000

    0

    0

    0

    -666.667

    -2000

    0

    666.667

    -2000

    0

    2000

    4000

    0

    -2000

    8000

    [k]1 =

  • 12

    Local:

    5

    2

    1 3

    4

    6

    2

    [q]2 = [ T ]T[ q]2

    q1

    q3

    q2

    q4q5q6

    q1

    q3

    q2

    q7q8q9

    [T]T

    Stiffness Matrix: Member 2

    =

    123789

    40000

    0-1

    5000100

    6000001

    10

    0-1

    000

    2100000

    3001000

    90o

    jx = cos (-90o) = 0jy = sin (-90o) = -1

    ix = cos (-90o) = 0iy = sin (-90o) = -1

    Global:

    AB

    C

    1

    2

    78 9

    4

    6 5

    12 3

  • 13

    [k]2 = [ T ]T[ k ]2[ T ]

    1 2 3 4 5 61

    2

    3

    4

    5

    6

    20000

    0

    0

    -20000

    0

    0

    0

    666.667

    2000

    0

    -666.667

    2000

    0

    2000

    8000

    0

    -2000

    4000

    -20000

    0

    0

    20000

    0

    0

    0

    -666.667

    -2000

    0

    666.667

    -2000

    0

    2000

    4000

    0

    -2000

    8000

    [k]2 =

    1 2 3 7 8 9

    666.667 20001

    2

    3

    7

    8

    9

    2000

    0

    0

    -666.667

    -666.667

    0

    0

    2000

    2000

    20000 0

    0

    0

    0

    -20000

    -20000

    0

    0

    8000 -2000

    -2000

    0

    0

    4000

    4000

    666.667 0

    0

    -2000

    -2000

    20000 0

    0 8000

    [k]2 =

  • 14

    [k]14 6 5 1 2 3

    4

    6

    5

    1

    2

    3

    20000

    0

    0

    -20000

    0

    0

    0666.667

    20000

    -666.667

    2000

    02000

    80000

    -2000

    4000

    -200000

    020000

    0

    0

    0-666.667

    -20000

    666.667

    -2000

    02000

    40000

    -2000

    8000

    1 2 3 7 8 9666.667 20001

    2

    3

    7

    8

    9

    2000

    00

    666.667

    -666.667

    0

    0

    2000

    2000

    20000 0

    0

    0

    0

    -20000

    -20000

    0

    0

    8000 -2000-2000

    0

    0

    4000

    4000

    666.667 0

    0

    -2000

    2000

    20000 0

    0 8000

    [k]2

    Global Stiffness Matrix:

    20000

    0

    -20000

    0

    0

    0

    8000

    0

    -2000

    4000

    0

    -2000

    0

    4000

    -20000

    0

    0

    20666.667

    -2000

    2000

    -2000

    16000

    20666.667

    0

    2000

    [K]

    4

    5

    1

    2

    3

    4 5 1 2 3

    Global:

    AB

    C

    1

    2

    78 9

    4

    6 5

    12 3

  • 15

    AB

    C

    Q4 = 0Q5 = 0

    Q1 = 5Q2 = 0

    Q3 = 0

    D4D5D1D2D3

    +

    0 0

    0 0

    0

    D4D5D1D2D3

    =

    0.01316 m

    0.01316 m9.199(10-4) rad

    -9.355(10-5) m

    -1.887(10-3) rad

    5 kN

    6 m

    6 m

    AB

    C

    1

    2

    Global:

    1

    2

    78 9

    4

    6 5

    12 35 kN

    =

    4

    5

    1

    2

    3

    4 5 1 2 3-20000 0 0

    20666.667 00

    2000

    2000

    20666.667 -2000

    -2000 16000

    0-2000

    4000

    08000 0 -2000 4000

    -200000

    20000

    0

    0

    [Q] = [K][D] + [QF]

  • 16

    D4 = 0.01316

    D5 = 9.199(10-4)

    D6 = 0

    D1 = 0.01316

    D2 = -9.355(10-5)

    D3 = -1.887(10-3)

    15 kN

    6 m

    6 m

    AB

    C

    2

    q4

    q5

    q6

    q1

    q2

    q3

    0

    0

    -1.87

    0

    1.87

    -11.22

    Member 1

    A B11

    2 3

    4

    6 5

    A B1

    1.87 kN 11.22 kNm1.87 kN

    4 6 5 1 2 3

    4

    6

    5

    1

    2

    3

    20000

    0

    0

    -20000

    0

    0

    0

    666.667

    2000

    0

    -666.667

    2000

    0

    2000

    8000

    0

    -2000

    4000

    -20000

    0

    0

    20000

    0

    0

    0

    -666.667

    -2000

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