บทที่ 4 (Slope-Deflection) - Civil Method 3 4.1 สมการที่ใช ในวิธี Slope-Deflection Slope-Deflection Equations แสดงความสัมพันธ

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Text of บทที่ 4 (Slope-Deflection) - Civil Method 3 4.1 สมการที่ใช...

  • Slope-Deflection Method 1

    4 Slope-Deflection Method

    ( ) Force Method

    Slope-Deflection Displacement Method

    Slope-Deflection Method 2

    Slope-Deflection (Beam) (Frame) Prismatic

    Prismatic

    Non-Prismatic

  • Slope-Deflection Method 3

    4.1 Slope-DeflectionSlope-Deflection Equations

    End Distortion + Load

    Mab = f (a , b , , Load)Mba = g (a , b , , Load)

    (+)- (Mab , Mbc) - ( a , b ) - ()

    Mab

    Mba

    ab

    a bLoad

    Slope-Deflection Method 4

    Mab Mba 4 a

    b

    Load Fixed-End Moments

    LEI2

    ML

    EI4M aba

    aab

    =

    = .......

    LEI4

    ML

    EI2M bba

    bab

    =

    = .......

    2ba2ab LEI6

    MLEI6

    M

    =

    = '''''' .......

    Maba

    a b Mba

    b = 0 , = 0

    Mab ba b Mbaa = 0 , = 0

    Mab a b

    Mbaa = 0 , b = 0

    MabF a b MbaFLoad

    a = 0 , b = 0 = 0

  • Slope-Deflection Method 5

    Slope-Deflection

    Fba2

    baFbababababa

    Fab2

    baFababababab

    MLEI6

    LEI4

    LEI2

    MMMMM

    MLEI6

    LEI2

    LEI4

    MMMMM

    +

    +

    =+++=

    +

    +

    =+++=

    '''

    '''

    Fbababa

    Fabbaab

    ML

    32LEI2

    M

    ML

    32LEI2

    M

    +

    +=

    +

    +=

    )(

    )(

    Slope-Deflection Method 6

    Moment 1 ( a )

    - a Hinge - b Fixed EndConjugate Beam Rotation a (a) Condition 1) Moment Conjugate Beam a = 0

    a = 02) Moment Conjugate Beam b = 0

    b = 0

    baMab

    a Mba

    a

    MabEI Mba

    EI

    Elastic Load on Conjugate Beam

  • Slope-Deflection Method 7

    (1) (2)

    ... ...

    =

    =

    =

    =

    20L3L

    LEI

    M21

    3L2

    LEI

    M21

    0M

    03L2

    LEI

    M21

    3L

    LEI

    M21

    0M

    abaab

    b

    baab

    a

    ..........)()(

    ;

    )()(

    ;

    = 1M2M baab .............

    .......L

    EI2M

    LEI4

    M abaa

    ab

    =

    =

    Slope-Deflection Method 8

    Fixed-End MomentsP

    L

    L/28

    PL8

    PL WL12

    WL2

    12WL2

    W

    L

    a

    )( 2222

    a3aL8L6L12

    Wa+ )( a3L4

    L12Wa

    2

    3

    P

    L

    a2

    2

    LPab b

    2

    2

    LbPa

    ML

    a)( a2b

    LMb

    2 b

    )( ab2LMa

    2

    WL20

    WL2

    30WL2

    W

    L96WL5 2

    96WL5 2

    L/2

    1

    3

    5

    2

    4

    6

    7

  • Slope-Deflection Method 9

    EX. Slope-Deflection ABCD

    EI EI/2A800 kg.

    4 m.

    B C D

    300 kg./m.

    2 m. 2 m. 1 m.

    Slope-Deflection Method 10

    1) Fixed End Moments

    .. mkg40012

    430012WL

    M22

    FAB =

    =

    =

    .. mkg8008

    480012

    43008

    PL12WL

    M22

    FBC =

    =

    =

    EI EI/2A

    800 kg.

    4 m.

    B C DW=300 kg./m.

    2 m. 2 m. 1 m.

    MABF MBAF MBCF MCBF

    300 kg./m.A4 m.

    BMABF MBAF

    .. mkg40012WL

    M2

    FBA ==

    300 kg./m.B CMBCF MCBF

    800 kg.

    2 m. 2 m... mkg800

    8PL

    12WL

    M2

    FCB =+=

  • Slope-Deflection Method 11

    2) Fixed End Moments Slope-Deflection () = 0

    2 B C

    4002EI

    ML

    32LEI2

    M BF

    ABBAAB =+

    += )(

    400EIML

    32LEI2

    M BF

    BABABA +=+

    += )(

    80024EI

    ML

    32LEI2

    M CBF

    BCCBBC +=+

    += )()(

    80024EI

    ML

    32LEI2

    M CBF

    CBCBCB ++=+

    += )()(

    Slope-Deflection Method 12

    3) (2 )

    MBA+ MBC= 0 MCB= 150 kg.-m.

    4) 2 3

    BMBA MBC

    FBD. B

    A C CMCB 150 kg.-m.

    FBD. C

    B D

    =+++=+ 1080024EI

    400EI0MM CBBBCBA ..........])([][...

    =++= 215080024EI

    150M CBCB .................................])([...

  • Slope-Deflection Method 13

    (2)

    (1)

    (3) (4)

    =+ 3EI6002

    2 CB ...................................................,

    =+ 4EI6001

    6 CB .......................................................,

    ., RadEI11

    8005B = .

    ,Rad

    EI1120017

    C

    =

    Slope-Deflection Method 14

    5) 2

    ..,, mkg115001

    400EI11

    80052EI

    4002EI

    M BAB

    ===

    ..,, mkg1120010

    400EI11

    8005EI400EIM BBA =+=+=

    800EI1120017

    EI118005

    24EI

    80024EI

    M CBBC =+= ),,()(

    .., mkg11

    20010

    =

    800EI1120017

    2EI11

    80054EI

    80024EI

    M CBCB +=++= ),,()(

    .. mkg150 =

  • Slope-Deflection Method 15

    6) FBD.

    MA = 1,500/11 kg.-m. RA = 4,425/11 kg.RB = 43,825/22 kg. RC = 24,325/22 kg.

    300 kg./m.1,500/11 10,200/11

    4,425/11 8,775/11

    B8,775/11 26,275/22

    300 kg./m.10,200/11 150

    26,275/22 17,725/22

    800 kg150

    300A

    1,500/11

    RA RB

    C17,725/22 300

    RC

    Slope-Deflection Method 16

    EX. Slope-Deflection 1. 2. BMD. ABCD ( EI )

    A

    100 kg. B C D300 kg./m.

    2 m. 2 m.

    2 m.

    50 kg.-m.

  • Slope-Deflection Method 17

    1) Fixed End Moments

    ,0MM FBAFAB == 0MM

    FCB

    FBC ==

    .. mkg10012

    230012WL

    M22

    FCD =

    =

    =

    .. mkg10012

    230012WL

    M22

    FDC =

    ==

    300 kg./m.C2 m.

    DMCDF MDCF

    A

    100 kg. B C DW

    2 m. 2 m.

    2 m.

    50 kg.-m. MCDF MDCF

    W = 300 kg./m.

    Slope-Deflection Method 18

    2) Fixed End Moments Slope-Deflection (= 0)

    2 B C

    BF

    ABBAAB EIML32

    LEI2

    M =+

    += )(

    BF

    BABABA EI2ML32

    LEI2

    M =+

    += )(

    )()( CBF

    BCCBBC 2EIML32

    LEI2

    M +=+

    +=

    )()( CBF

    BCCBCB 2EIML32

    LEI2

    M +=+

    +=

    100EI2ML

    32LEI2

    M CF

    CDDCCD =+

    += )(

    100EIML

    32LEI2

    M CF

    CDDCDC +=+

    += )(

  • Slope-Deflection Method 19

    3) (2 ) B C

    MBA+ MBC= 50 MCB + MCD = 0

    CMCB

    FBD. C

    B DMCD

    BMBA MBC

    FBD. B

    A

    C100 kg.50 kg.-m.

    Slope-Deflection Method 20

    4) 2 3

    (1) (2)

    .RadEI3

    20B = .RadEI3

    70C =

    =++=+ 1502EIEI250MM CBBBCBA ....................)(...

    =++=+ 20100EI22EI0MM CCBCDCB ..............)(...

  • Slope-Deflection Method 21

    5) B C 2

    ../)/( mkg320EI320EIEIM BAB ===

    ../)()( mkg3110EI3

    70EI3

    202EI2EIM CBBC =+=+=

    ../)/( mkg340EI320EI2EI2M BBA ===

    ../)()( mkg3160EI3

    702

    EI320

    EI2EIM CBCB =+=+=

    ../ mkg3160100EI3

    70EI2100EI2M CCD ===

    ../ mkg3370100EI3

    70EI2100EIM CDC =+=+=

    Slope-Deflection Method 22

    6) FBD.

    MA = 20/3 kg.-m. RA,x = 10 kg.RA,Y = 45 kg. RC = 310 kg.MD = 370/3 kg.-m. RA,x = 110 kg.RD,Y = 335 kg. 1

    A

    20/345

    10

    1045 40/3

    B 10100 kg.

    50 kg.-m.

    40/3

    110/3110

    45

    45

    110/3110

    45 160/3110

    45C

    RC=310

    11011045 265 300 kg./m.

    160/3 370/3265110

    335110 D

  • Slope-Deflection Method 23

    7) (BMD.)

    2

    A

    C DB

    110/3

    160/320/3

    40/3

    370/3

    185/3

    BMD(kg.-m.)

    ++

    --

    +-

    Slope-Deflection Method 24

    4.2 Slope-Deflection Joint Translation

    Joint Translation

    ......

    ...Frame ...

    =0

    =0

  • Slope-Deflection Method 25

    Frame Symmetry () Load ( Sway )

    =0

    Centerline

    Centerline

    Centerline

    Slope-Deflection Method 26

    Joint Translation

    Frame Symmetry Load Symmetry

    ...Frame ...

    1

    2

  • Slope-Deflection Method 27

    EX. Slope-Deflection ( EI )

    A

    P B C

    D

    10 ft.

    20 ft.

    Slope-Deflection Method 28

    1) Side Sway - 1 B C (1 DOF Joint Translation)

    2) Fixed End Moments

    0MM0MM0MM FDCFCD

    FCB

    FBC

    FBA

    FAB ====== ,

    A

    P B C

    D10 ft.

    20 ft.

  • Slope-Deflection Method 29

    3) Fixed End Moments Slope-Deflection

    3 B , C

    )()( =++= 320200EI

    ML

    32LEI2

    M BF

    ABAB

    BAAB

    )()( =++= 3400200EI

    ML

    32LEI2

    M BF

    BAAB

    BABA

    )()( CBF

    BCBC

    CBBC 4080200EI

    ML

    32LEI2

    M +=+

    +=

    )()( CBF

    CBBC

    CBCB 8040200EI

    ML

    32LEI2

    M +=+

    +=

    )()( =++= 340200EI

    ML

    32LEI2

    M CF

    CDCD

    DCCD

    )()( =++= 320200EI

    ML

    32LEI2

    M CF

    DCCD

    DCDC

    Slope-Deflection Method 30

    4) (3 ) B C

    MBA+ MBC= 0 MCB + MCD = 0 1

    FBD. B

    BMBA

    MBC

    A

    CP

    0P2

    MM2

    MM DCCDBAAB =++

    ++ )()(20MM BAAB /)( + 20MM DCCD /)( +

    A

    P B C

    DMAB MDC

    FBD. C

    BMCD

    MCB

    D

    C

  • Slope-Deflection Method 31

    5) 3 4

    =+

    =+=+

    6EI

    P4001126060

    5031204040340120

    CB

    CB

    CB

    .................................................,......................................................................................................................

    =++=+ 104080200EI

    340200EI

    0MM CBBBCBA .......)()(...

    =++=+ 20340200EI

    8040200EI

    0MM CCBCDCB .......)()(...

    201

    340200EI

    320200EI

    0P2