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Slope-Deflection Method 1 บทที4 Slope-Deflection Method วิธีศึกษาที่ผานๆมาจะใชแรง (แรงที่ฐาน หรือ แรงภายในชิ้นสวน) เปนตัวไมทราบคา หรือเรียกวา Force Method สวนวิธี Slope-Deflection นี้จะใชการเคลื่อนที่ของโครงสรางเปนตัวไม ทราบคา หรือเรียกวา Displacement Method Slope-Deflection Method 2 วิธี Slope-Deflection ใชวิเคราะหคาน (Beam) และโครงขอแข็ง (Frame) ไดทุกชนิดแตในชั้นนี้จะศึกษาแตคานและโครงขอแข็งที่ประกอบมาจาก ชิ้นสวน Prismatic คานและโครงขอแข็งประกอบ มาจากชิ้นสวน Prismatic คานและโครงขอแข็งประกอบ มาจากชิ้นสวน Non-Prismatic

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  • 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

    MM2

    MMBB

    DCCDBAAB +=++

    ++ )]()([)()(

    =+++ 30P201

    320200EI

    340200EI

    CC ...................)]()([

    Slope-Deflection Method 32

    (4) , (5) (6)

    6) 3

    Note (Anti-Symmetry) Unknown 1

    ( Yuan-Yu Hsieh)

    .RadEI13P100

    CB == .,

    ftEI39

    P0016=

    13P70

    EI39P0016

    3EI13P100

    20200EI

    MAB

    == ),(

    13P60

    EI39P0016

    3EI13P100

    40200EI

    MBA

    == ),(

    13P60

    EI13P100

    40EI13P100

    80200EI

    MBC =+= )(

    ABDCBACDBCCB MMMMMM === ;;

  • Slope-Deflection Method 33

    EX. Slope-Deflection ( EI )

    A

    300 kg.

    B C

    D

    2.5 m.

    4 m.

    600 kg./m.

    1.5 m.

    1.5 m.

    1 m.

    Slope-Deflection Method 34

    1) Sidesway B C

    2) Fixed End MomentsA

    300 kg.B C

    D2.5 m.

    600 kg./m.

    4 m.1.5 m.

    1.5 m.1 m.

    A300 kg. B

    MABF

    MBAF

    1.5 m.1.5 m.

    C

    600 kg./m.

    BMBCF MCBF

    2.5 m.

    .... mkg531212

    5260012WL

    M22

    FBC =

    =

    =

    ... mkg531212WL

    M2

    FCB ==

    ... mkg51128

    33008PL

    MFAB =

    =

    =

    ... mkg51128

    33008

    PLMFBA =

    ==

    .. mkg0MM FDCFCD ==

  • Slope-Deflection Method 35

    3) Fixed End Moments Slope-Deflection

    4 B , C , D

    51123EI2

    ML

    32LEI2

    M BF

    ABAB

    BAAB .)()( =+

    +=

    511223EI2

    ML

    32LEI2

    M BF

    BAAB

    BABA .)()( +=+

    +=

    531225EI4

    ML

    32LEI2

    M CBF

    BCBC

    CBBC .)()( +=+

    +=

    531225EI4

    ML

    32LEI2

    M CBF

    CBBC

    CBCB .)()( ++=+

    +=

    )()(4

    32

    2EI

    ML

    32LEI2

    M DCF

    CDCD

    DCCD

    +=+

    +=

    )()(4

    32

    2EI

    ML

    32LEI2

    M DCF

    DCCD

    DCDC

    +=+

    +=

    Slope-Deflection Method 36

    4) (4 ) B , C D

    MBA+ MBC= 0 MDC = 0 MCB + MCD = 0

    1

    FBD. BBMBA

    MBC

    A

    C

    03004

    MM3

    450MM DCCDBAAB =++

    ++ )()(

    351300MM BAAB /).( +

    4MM DCCD /)( +

    MDC

    FBD. D

    300 kg.

    MAB MDC

    600 kg./m.

    FBD. CB

    MCD

    MCB

    D

    C

  • Slope-Deflection Method 37

    5) 3 4

    =+++ 10531225EI4

    511223EI2

    CBB ...........)(.)(

    =

    ++++ 204

    32

    2EI

    531225EI4

    DCCB ............)(.)(

    31

    450511223EI2

    51123EI2

    BB ++ ].)(.)([

    =+

    ++

    ++ 3030041

    43

    22EI

    43

    22EI

    DCDC .......)]()([

    =

    + 404

    32

    2EI

    DC .........................................................)(

    Slope-Deflection Method 38

    (5) (8)

    =+ 50003EI10EI12EI44 CB ...............................................,

    =++ 650012EI20EI15EI104EI32 DCB .............................,

    =++ 760021EI54EI91EI54EI96 DCB ...............................,

    =+ 80EI8EI3EI4 DC ...........................................

    .. RadEI

    8244B =

    .. mEI

    3572=

    .. RadEI

    6170C

    =

    .RadEI

    300D =

  • Slope-Deflection Method 39

    6) B , C , D 3

    MAB = -330.9 kg.-m. MBA = 57.34 kg.-m. MBC = -57.34 kg.-m. MCB = 235.3 kg.-m. MCD = -235.3 kg.-m. MDC = 0 kg.-m.

    Slope-Deflection Method 40

    EX. Slope-Deflection ( EI )

    A

    BC

    D

    1.5 T.

    /m.5 m.

    10 m.E

    12 m. 12 m.

  • Slope-Deflection Method 41

    1) Sidesway 1 2

    2) Fixed End Moments

    A

    BC

    D

    1.5 T.

    /m.

    5 m.

    10 m.E

    12 m. 12 m.

    1 2

    A1.5 T.

    /m.

    B

    MABF

    MBAF

    10 m..... mT512

    121051

    12WL

    M22

    FAB =

    =

    =

    ... mT51212WL

    M2

    FBA ==

    ...sin. mT12531213

    13551

    M2

    FBC =

    =

    B1.5

    T./m.

    MABF

    MBAF

    13 m.C

    ...sin. mT12531213

    13551

    M2

    FCB =

    =

    Slope-Deflection Method 42

    3) Rotation 1 2

    Rotation AB = Rotation BC =

    Rotation CD = Rotation DE =

    101 /

    1013SIN21221 )( =

    1013SIN22121 )( =

    102 /

    A

    BC

    D

    1.5 T.

    /m.

    5 m.

    10 m.E

    12 m. 12 m.

    1 2

    1/10 2/10

  • Slope-Deflection Method 43

    4) Fixed End Moments Slope-Deflection

    51210

    35EI

    ML

    32LEI2

    M 1BF

    ABAB

    BAAB .)()(

    =+

    +=

    51210

    32

    5EI

    ML

    32LEI2

    M 1BF

    BAAB

    BABA .)()( +

    =+

    +=

    125310

    32

    13EI2

    ML

    32LEI2

    M 12CBF

    BCBC

    CBBC .])([)( +=++=

    125310

    32

    13EI2

    ML

    32LEI2

    M 12CBF

    CBBC

    CBCB .])([)( ++=++=

    ])([)(10

    32

    13EI2

    ML

    32LEI2

    M 21DCF

    CDCD

    DCCD

    +=+

    +=

    ])([)(10

    32

    13EI2

    ML

    32LEI2

    M 21DCF

    DCCD

    DCDC

    +=+

    +=

    Slope-Deflection Method 44

    5 B , C , D , 1 2

    5) (5 ) - B , C D

    ][)(10

    32

    10EI2

    ML

    32LEI2

    M 2EDF

    DEDE

    EDDE

    +=+

    +=

    ][)(10

    32

    10EI2

    ML

    32LEI2

    M 2EDF

    EDDE

    EDED

    +=+

    +=

    =+= 10MM0M BCBAB ..........................................................;

    =+= 20MM0M CDCBC ..........................................................;

    =+= 30MM0M DEDCD ..........................................................;

  • Slope-Deflection Method 45

    HA HE

    E

    D

    MED

    MDEVD

    VE

    HE

    HD010HMM0M EEDDED =++= ;

    )( EDDEE MM101

    H +=

    1551HH0F EAX =+= .;

    A1.5 T.

    /m.

    B

    MAB

    MBAVB

    VA

    HA

    HB 05105110HMM0M ABAABB =++= .;

    )(. BAABA MM101

    57H +=

    522MM101

    MM101

    57 EDDEBAAB .)()(. =++

    =+++ 4150MMMM EDDEBAAB ....................................

    Slope-Deflection Method 46

    VA VE

    EDDECDE M241

    M81

    M121

    V =

    0VV0F EAY =+= ;

    057155112V15HMM0M AACBABC =+++= ..;

    CBBAABA M121

    M81

    M241

    68754V ++= .

    0M241

    M81

    M121

    M121

    M81

    M241

    68754 EDDECDCBBAAB =+++.

    =+++ 505112MM3M2M2M3M EDDECDCBBAAB .............

    A1.5

    T./m. B

    MAB

    MCB

    VA

    HA

    HCC

    VC

    12V15HMM0M EEEDCDC =++= ;

    E

    D

    MED

    MCD VC

    VE

    HE

    HC C

  • Slope-Deflection Method 47

    6) Slope-Deflection 5

    7) (6) (10)

    =+ 638609EI3EI90EI10EI46 21CB .....................................

    =++ 73120EIEI4EI DCB .........................................................

    =+ 80EI90EI3EI46EI10 21DC .............................................

    =+ 9250EI20EI20EIEI 21DB ............................................

    =+ 1016120EI390EI390EIEI 21DB .....................................

    .. RadEI3289

    B =

    .. mEI5612

    1 =

    .. RadEI

    8655C

    = .. Rad

    EI8113

    D =

    .. mEI1010

    2 =

    Slope-Deflection Method 48

    8) 7 Slope-Deflection ( 4) MAB = -69.99 T.-m. MBA = -27.12 T.-m. MBC = 27.12 T.-m. MCB = 11.04 T.-m. MCD = -11.04 T.-m. MDC = 15.06 T.-m. MDE = -15.06 T.-m. MED = -37.82 T.-m.

  • Slope-Deflection Method 49

    EX. Slope-Deflection 1. B C 2. B 3. BMD ( EI )

    A B

    C D

    9 m.2I

    100 kN./m.

    4 m.

    6 m.

    2I I

    Slope-Deflection Method 50

    1) Sidesway

    - AB CD AB = -CD = -vB

    - BC BC = 0

    2) 2

    - B (B) - C (C)

    A B

    C D

    9 m.2I

    100 kN./m.

    4 m.

    6 m.

    2I I

    -AB

    -CD

    A B

    C D

    9 m.2I

    100 kN./m.

    4 m.

    6 m.

    2I I

    B

    C

  • Slope-Deflection Method 51

    3) Fixed End Moments

    4) Slope-Deflection ( 6 )

    A D = 0

    D

    100 kg./m.C

    MCDF MDCF

    9 m.

    .. mkg67512

    910012WL

    M22

    FCD =

    =

    =

    .. mkg67512WL

    M2

    FDC ==

    .. mkg0MMMM FCBFBC

    FBA

    FAB ====

    0EIv3330EI6670EI3331M BBAAB +++= ...0EIv3330EI3331EI6770M BBABA +++= ...

    0EIv0000EI0001EI002M BCBBC +++= ...0EIv0000EI0002EI001M BCBCB +++= ...

    675EIv0740EI2220EI4440M BDCCD += ...675EIv0740EI4440EI2220M BDCCD ++= ...

    Slope-Deflection Method 52

    5) (3 )

    MBA+ MBC= 0 MCB + MCD = 0

    FBD. C

    CMCB

    MCD

    B

    D

    FBD. BA

    MCBMBA

    C

    B

    A B

    C D

    9 m.2I

    100 kN./m.

    4 m.

    6 m.

    2I ICDDCCDABBAAB LMMLMM /)(/)( +++

    0L1002L100 CDCD =+ )/(

  • Slope-Deflection Method 53

    6) 4 5

    7) (1) , (2) (3)

    1,2

    =+++ 100EIv3330EI0001EI3333 BCB .................................

    =+ 20675EIv0740EI4442EI0001 BCB ..............................

    =+ 30450EIv1280EI0740EI3330 BCB ...........................

    .. RadEI

    1519B =

    .., mEI

    49334vB

    =

    .. RadEI

    785C

    =

    Slope-Deflection Method 54

    8) 7 Slope-Deflection 4 MAB = -1,298.4 kN.-m. MBA = -952.4 kN.-m. MBC = 952.4 kN.-m.MCB = 347.7 kN.-m. MCD = -347.7 kN.-m. MDC = 1,021.4 kN.-m.

    9)

    3-1,298.4

    -1,021.4

    -347.7-347.7 952.4

    952.4

    356

    BMD.(kN.-m.)

    A B

    C D

  • Slope-Deflection Method 55

    EX. Slope-Deflection 1. C D 2. C D 3. D ( EI )

    A BC D

    8 m.

    60 kN.

    4 m.

    6 m. 6 m.E

    Slope-Deflection Method 56

    1) Fixed End Moments

    .. mkN0MMMM FCEFEC

    FDC

    FCD ====

    .. mkN908

    12608PL

    MFAC =

    ==

    MCAFMACFA B

    C D

    8 m.

    60 kN.

    4 m.

    6 m. 6 m.E

    60 kNA

    12 m.C

    MACF MCAFB .. mkN908PL

    MFCA ==

  • Slope-Deflection Method 57

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

    2 C D

    90EI1670EI3330ML

    32LEI2

    M CAF

    ACCAAC +=+

    += ..)(

    90EI3330EI1670ML

    32LEI2

    M CAF

    CACACA ++=+

    += ..)(

    DCF

    CDDCCD EI250EI50ML32

    LEI2

    M +=+

    += ..)(

    DCF

    DCCBDC EI50EI250ML32

    LEI2

    M +=+

    += ..)(

    ECF

    CEECCE EI50EIML32

    LEI2

    M +=+

    += .)(

    ECF

    ECECEC EIEI50ML32

    LEI2

    M +=+

    += .)(

    Slope-Deflection Method 58

    3) (2 ) B C

    MCA+ MCD+ MCE= 0 MDC = 0

    4) 2 3

    FBD. D

    DMDC

    FBD. C

    MCA MCDC

    MCE

    =+ 190EI250EI8331 DC ................................................

    =+ 20EI50EI250 DC ......................................................

  • Slope-Deflection Method 59

    5) (1) (2) 4

    1

    6) Slope-Deflection MAC = -98.8 kN.-m. MDC = 0 kN.-m. MCA = 72.4 kN.-m. MCE = -52.7 kN.-m. MCD = -19.8 kN.-m. MEC = -26.3 kN.-m. 2

    7) D

    3

    .. RadEI

    752C

    = .. Rad

    EI326

    D =

    CDYC M8D0M ==

    .. kN472D Y =FBD. CD

    DMCD CVCD

    NCDDY

    DX

    Slope-Deflection Method 60

    EX. Slope-Deflection 1. B 2. B 3. A ( EI )

    AB C

    4 m.

    10 kN./m.

    3I

    6 m.

    I

  • Slope-Deflection Method 61

    1) Sidesway

    - AB CD AB = BC = -

    2) 1

    - B (B)

    A B C

    4 m.

    10 kN./m.

    3I

    6 m.

    I

    A B C

    4 m.

    10 kN./m.

    3I

    6 m.

    I

    Slope-Deflection Method 62

    3) Fixed End Moments

    4) Slope-Deflection ( 4 )

    B

    10 kN./m.A

    MABF MBCF

    6 m.

    .. mkN3012

    61012WL

    M22

    FAB =

    =

    =

    .. mkN3012WL

    M2

    FBA ==

    .. mkN0MM FCBFBC ==

    30EI50EIML32LEI2M BFABABABBAABABAB =++= .)/(/

    30EI50EI2ML32LEI2M BFBAABABBAABABBA +=++= .)/(/

    0EI3750EIML32LEI2M BFBCBCBCCBBCBCBC ++=++= .)/(/

    0EI3750EI50ML32LEI2M BFCBBCBCCBBCBCCB ++=++= ..)/(/

  • Slope-Deflection Method 63

    5) (2 )

    MBA+ MBC= 0 VAB - VCB = (10)(6)

    6) 4 5

    = 130EI1250EI3 B ........................................................

    FBD. BA

    MBCMBACB

    FBD.A

    VCBVABCB

    =+ 230EI3540EI1250 B ..................................................

    BCCBBCCBABBAABAB LMMVL3610MMV /)(;/)( +=+=

    Slope-Deflection Method 64

    7) (1) (2)

    1,2

    8) 7 Slope-Deflection ( 3) MAB = -77.8 kN-m. 3

    .. RadEI

    66B

    = .. m

    EI482

    =