(Slope Deflection Method – Displacement Method) ??아대학교 토목공학과 구조역학 2 강의: 박현우 Lecture 4 - 1 4. 처짐각법 – 변위법 (Slope Deflection Method – Displacement Method) Objective of this

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  • 2 :

    Lecture 4 - 1

    4. (Slope Deflection Method Displacement Method)

    Objective of this chapter:

    .

    What will be presented:

    Theoretical background

    George A. Maney 1915 .

    .

    (Fixed end moment)

    .

    .

  • 2 :

    Lecture 4 - 2

    4.1

    (1) (Fixed-end moment; FEM): AB

    (P3, P4, w) A B

    FEMABM , FEMBAM

    (2) A B A B

    BAAB MM ,

    (3) A B

    BAAB MM ,

    LA B

    P1 P2 P5 P3 P4

    w

    A B FEMABM

    FEMBAM

    A B

    B A ABM

    BAM

    A B

    ABMBAM =/L

  • 2 :

    Lecture 4 - 3

    (4) A B

    .

    A: )( ABABFEMABAB MMMM

    B: )( BABAFEMBABA MMMM

  • 2 :

    Lecture 4 - 4

    A,B ABFEMABM FEM ,

    BAFEMBAM FEM :

    A B FEMABM

    FEMBAM

  • 2 :

    Lecture 4 - 5

    A, B A B

    BAAB MM ,

    - A, B A B

    A B .

    -

    BAABBAABA MMEILL

    EIMLL

    EIML

    L

    2

    63232

    21

    -

    ABBAABBAB MMEILL

    EIMLL

    EIML

    L

    2

    63232

    21

    )2(2),2(2 BABABAAB LEIM

    LEIM

    EIABM

    EIBAM

    L

    A B

    +

    A B

    B A ABM

    BAM

  • 2 :

    Lecture 4 - 6

    A B

    BAAB MM ,

    - A, B -

    A B .

    BAAB

    BAABBAAB

    A

    MMEIL

    MMEILL

    EIMLL

    EIML

    L

    26

    26323

    22

    1

    ABBA

    ABBAABBA

    B

    MMEIL

    MMEILL

    EIMLL

    EIML

    L

    26

    26323

    22

    1

    LEIMM BAAB

    6

    EIABM

    EIBAM

    ML

    A=- B=-

    +

    A B

    ABM

    BAM =/L

  • 2 :

    Lecture 4 - 7

    A, B

    .

    - A:

    )32(2FEM

    6)2(2FEM

    )(

    BAAB

    BAAB

    ABABFEMABAB

    LEI

    LEI

    LEI

    MMMM

    - B:

    )32(2FEM

    6)2(2FEM

    )(

    ABBA

    ABBA

    BABAFEMBABA

    LEI

    LEI

    LEI

    MMMM

    ():

    ( ) .

    (matrix analysis) .

  • 2 :

    Lecture 4 - 8

    4.2

    :

    - 12

    FEM2wL

    AB , 12FEM

    2wLBA

    - 8

    FEM PLBC , 8FEM PLCB

    : 00 BCABCA

    (Slope deflection equation):

    - 12

    2212

    2 22 wLLEIMwL

    LEIM BBABAB

    - 8

    28

    22 PLLEIMPL

    LEIM BCBBBC

    B :

    12880

    2wLPLLEIMM BBCBA EI

    PLEI

    wLB 6496

    23

    EI = A B C

    w

    L L

    P

  • 2 :

    Lecture 4 - 9

    :

    - 3248

    5 2 PLwLM AB 1624

    2 PLwLM BA

    - 1624

    2 PLwLM BC 325

    48

    2 PLwLMCB

    :

    - 02 1

    2

    LBMwLM yBAAB 323

    167

    1PwLBy

    - 323

    169

    1PwLBwLA yy

    - 02

    LCMPLM yCBBC 3219

    16PwLCy

    - 32

    13162

    PwLCPB yy

    - 2221PwLBBB yyy

    - ..

    A BwP

    B

    MAB MBA MBC MCB Ay By1 By2 Cy

    By

    By1 By2

  • 2 :

    Lecture 4 - 10

    4.3

    (1) Sidesway

    P D E . sidesway . .

    ( C) sidesway .

  • 2 :

    Lecture 4 - 11

    (axis of symmetry) sidesway .

    (2) Sidesway

    sidesway

    L

    L EI = constnat

    w

    A

    B C

    D

  • 2 :

    Lecture 4 - 12

    12,

    12

    022 wLFEMwLFEM

    FEMFEMFEMFEM

    CBBC

    DCCDBAAB

    0,0 DACDBCAB

    CB , 2.

    DCCDDCDC

    CDCDDCCD

    CBBCCBCB

    BCBCCBBC

    BAABBABA

    ABABBAAB

    FEMLEIM

    FEMLEIM

    FEMLEIM

    FEMLEIM

    FEMLEIM

    FEMLEIM

    322

    322

    322

    322

    322

    322

    ,

    CDCCCDCBCB

    CBBCBBABAB

    LEIM

    LEIMwL

    LEIM

    wLLEIM

    LEIM

    LEIM

    2,4,12

    2212

    22,4,2

    2

    2

  • 2 :

    Lecture 4 - 13

    B C

    00

    CDCB

    BCBA

    MMMM

    0

    1282

    012

    28

    2

    2

    wLLEI

    LEI

    wLLEI

    LEI

    CB

    CB

    0 CB

    EI

    wLEI

    wLCB 72

    ,72

    33

    :

    36,

    18

    18,

    18

    18,

    36

    22

    22

    22

    wLMwLM

    wLMwLM

    wLMwLM

    DCCD

    CBBC

    BAAB

  • 2 :

    Lecture 4 - 14

    :

    w

    36

    2wL

    18

    2wL 18

    2wL

    36

    2wL

    18

    2wL 18

    2wL

    2wL

    2wL

    12wL

    12wL

    2wL

    12wL

    12wL

    2wL

    2wL

    2wL

    12wL

    12wL

    2wL

    12wL

    [A.F.D.]

    +

    +

    12wL

    2wL

    12wL

    2wL

    [S.F.D.]

    36

    2wL

    18

    2wL 72

    5 2wL

    [B.M.D.]

  • 2 :

    Lecture 4 - 15

    (3) Sidesway

    0 DCCDCBBCBAAB FEMFEMFEMFEMFEMFEM

    CDABBC

    DA

    ,00

    ,, CB 3.

    .

    32,322

    22,22

    322,32

    CDCCCD

    CBCBCBBC

    BBABAB

    LEIM

    LEIM

    LEIM

    LEIM

    LEIM

    LEIM

    L

    L EI = constnat

    A

    B C

    D

    P

  • 2 :

    Lecture 4 - 16

    B C

    00

    CDCB

    BCBA

    MMMM

    3( ,, CB ) 2??

    Sidesway

    , sidesway

    .

    :

    0 DA HHP

    - B : 0 BAAAB MLHM LMMH BAABA

    - , L

    MMH DCCDB

    PLMMMM DCCDBAAB

    MAB

    MBA

    HA HD MDC

    MCD

  • 2 :

    Lecture 4 - 17

    ,

    PLLEI

    LEI

    LEI

    LEI

    LEI

    LEI

    LEI

    LEI

    LEI

    CB

    CB

    CB

    2466

    0682

    0628

    53

    CB

    EI

    PLEI

    PLEI

    PLCB 84

    52828

    222

    :

    72,

    143

    143,

    143

    143,

    72

    PLMPLM

    PLMPLM

    PLMPLM

    DCCD

    CBBC

    BAAB

  • 2 :

    Lecture 4 - 18

    :

    72PL

    143PL 14

    3PL

    72PL

    143PL

    143PL

    73P

    73P

    2P

    2P

    73P

    2P

    2P

    73P

    2P

    2P

    P

    73P

    73P

    +

    73P

    2P

    [A.F.D.]

    +

    +

    2P

    73P

    2P

    73P

    [S.F.D.]

    72PL

    143PL

    [B.M.D.] +

    + 14

    3PL

    72PL