Structural Analysis Ch1...آ  2019-05-03آ  Analysis of Statically Indeterminate Frames 7. Influence Lines

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  • Dept. of Civil and Environmental Eng., SNU

    Structural Analysis Lab. Prof. Hae Sung Lee, http://strana.snu.ac.kr

    Structural Analysis I

    Spring Semester, 2018

    Hae Sung Lee

    Dept. of Civil and Environmental Engineering Seoul National University

    y δ

    y f

    z δ zf

    x δ

    x f

    y M yθ

    z M zθ xM xθ

  • Dept. of Civil and Environmental Eng., SNU

    Structural Analysis Lab. Prof. Hae Sung Lee, http://strana.snu.ac.kr

    This page is intentionally left blank.

  • Dept. of Civil and Environmental Eng., SNU

    Structural Analysis Lab. Prof. Hae Sung Lee, http://strana.snu.ac.kr

    Contents

    1. Introduction

    2. Reactions & Internal Forces by Free Body Diagra

    ms

    3. Principle of Virtual Work

    4. Analysis of Statically Indeterminate Beams

    5. Analysis of Statically Indeterminate Trusses

    6. Analysis of Statically Indeterminate Frames

    7. Influence Lines for Determinate Structures

    8. Influence Lines for Indeterminate Structures

  • Dept. of Civil and Environmental Eng., SNU

    Structural Analysis Lab. Prof. Hae Sung Lee, http://strana.snu.ac.kr

    This page is intentionally left blank.

  • Dept. of Civil and Environmental Eng., SNU

    Structural Analysis Lab. Prof. Hae Sung Lee, http://strana.snu.ac.kr

    1

    Chapter 1

    Introduction

  • Dept. of Civil and Environmental Eng., SNU

    Structural Analysis Lab. Prof. Hae Sung Lee, http://strana.snu.ac.kr

    2

    1.1 Mechanics of Material - Structural Mechanics

     Problem

    Calculate the reaction force at each support and draw the moment and shear force dia

    gram for the two-span beam shown in the figure.

     Solution

     Equilibrium Equation

    qLRRRF cbay 20 =++→=

    qLRRLRLRLqLM cbcba 220220 =+→=×−×−×→=

    00 22

    0 =−→=×−×+×+×−→= cacab RRLRLR L

    qL L

    qLM

    qLRRLRLRLqLM babac 220220 =+→=×+×+×−→=

    Since there are three unknowns in two independent equations, we cannot determine a unique

    solution for the given structure, and thus we need one more equation to solve this problem.

    The main issue of this class is how to build additional equations to analyze statically inde-

    terminate structures.

    EI EI

    q

    Ra Rb Rc

    L L

    q

  • Dept. of Civil and Environmental Eng., SNU

    Structural Analysis Lab. Prof. Hae Sung Lee, http://strana.snu.ac.kr

    3

    Fundamentals of Differential Equations

     Governing Equations

    The governing equations of a (engineering) system are usually defined by a system of diffe-

    rential equations, which governs behaviors of the given system within a domain. Notice that

    the domain does not include boundaries.

    11 qwEI =′′′′ for lx

  • Dept. of Civil and Environmental Eng., SNU

    Structural Analysis Lab. Prof. Hae Sung Lee, http://strana.snu.ac.kr

    4

    1.2 Mechanics of Material

     Governing Equation

    – Left span

    11 2

    1 3

    1

    4

    1 ''''

    1 24 dxcxbxa

    EI

    qx wqEIw ++++=→=

    – Right Span

    22 2

    2 3

    2

    4

    2 ''''

    2 24 dxcxbxa

    EI

    qx wqEIw ++++=→=

     Boundary Conditions

    – Left support

    0)0()0( , 0)0( 111 =′′−== wEIMw

    – Center support

    )()( , )()( , 0)()( 212121 LwLwLwLwLwLw ′′=′′′−=′==

    – Right support

    0)0()0( , 0)0( 222 =′′−== wEIMw

    Since there are eight unknowns with eight conditions, we can solve this problem.

     Determination of Integration Constant – Left Support

    xcxa EI

    qx wbwdw 1

    3 1

    4

    11111 24 02)0( , 0)0( ++=→==′′==

    – Right Support

    xcxa EI

    qx wbwdw 2

    3 2

    4

    22222 24 02)0( , 0)0( ++=→==′′==

    x x

    y

    z

    z

    y

    q

    w1 w2

  • Dept. of Civil and Environmental Eng., SNU

    Structural Analysis Lab. Prof. Hae Sung Lee, http://strana.snu.ac.kr

    5

    – Center Support

      

     

    ==

    −== →

        

       

    +=+

    −−−=++

    =++=

    =++=

    EI

    qL cc

    EI

    qL aa

    La EI

    qL La

    EI

    qL

    cLa EI

    qL cLa

    EI

    qL

    LcLa EI

    qL Lw

    LcLa EI

    qL Lw

    48

    48

    3

    6 2

    6 2

    3 6

    3 6

    0 24

    )(

    0 24

    )(

    3

    21

    21

    2

    2

    1

    2

    2 2

    2

    3

    1 2

    1

    3

    2 3

    2

    4

    2

    1 3

    1

    4

    1

    )32( 48

    334 21 xLLxxEI

    q ww +−=≡

    8 3

    , 8

    3 2 11

    2 11

    qL qxwEIVx

    qL x

    q wEIM +−=′′′−=+−=′′−=

     Moment Diagram

     Shear Diagram

     Reactions

    0.375qL

    L 8 3

    +

    -

    +

    0.625qL

    -

    0.375qL 0.375qL 1.25qL

    0.125qL2

    0.070qL2

    L 8

    3

    +

    -

    +

  • Dept. of Civil and Environmental Eng., SNU

    Structural Analysis Lab. Prof. Hae Sung Lee, http://strana.snu.ac.kr

    6

    1.3 Mechanics of Material + α

    1.3.1 Main idea

     Original Problem

     Case I (Removal of the center support)

     Case II (Application of the reaction force)

     Original Problem = Case I + Case II (compatibility condition)

    δ0+ δR=0

    1.3.2 Calculation of δ0

     Bending Moment

    qLx qx

    M +−= 2

    2

    q

    q

    δ0

    δR

    Rb

    q

    qL2/2

  • Dept. of Civil and Environmental Eng., SNU

    Structural Analysis Lab. Prof. Hae Sung Lee, http://strana.snu.ac.kr

    7

     Governing Equation

    bax qLxqx

    EIwMwEI ++−=→−=′′ 624

    34

    00

     Boundary (support) Conditions

    – Left Support : 0)0(0 =w

    – Right Support : 0)2(0 =Lw

     Determination of Integration Constant

    – Left Support

    0 0)0(0 =→= bEIw

    – Right Support

    EI

    Lq aLa

    LqLq LEIw

    24

    )2( 0)2(

    6

    )2(

    24

    )2( )2(

    334

    0 =→=+−=

     Deflection

    ))2()2(2( 24

    334 0 LxLxxEI

    q w +−=

    EI

    Lq LLLLL

    EI

    q Lw

    384 )2(5

    ))2()2(2( 24

    )( 4

    334 00 =+−==δ

    1.3.3 Calculation of δR

     Bending Moment

    22 ,

    2 2

    2 1

    1

    LRxR M

    xR M bbb −=−=

     Governing Equation

      

     

    +++−=

    ++= →

      

    −=′′ −=′′

    222

    2 2

    3 2

    2

    111

    3 1

    1

    22

    11

    412

    12

    bxa LxRxR

    EIw

    bxa xR

    EIw

    MwEI

    MwEI

    bb R

    b R

    R

    R

    1x 2x

    RbL/2

    Rb

  • Dept. of Civil and Environmental Eng., SNU

    Structural Analysis Lab. Prof. Hae Sung Lee, http://strana.snu.ac.kr

    8

     Boundary (support and mid-span) Conditions

    – Left Support & Right Support

    0)( , 0)0( 21 == Lww RR

    – Mid-span

    )( )0()()( , )0()( 221121 LwLwLwLw RRRRRR θ=′=′=θ=

     Determination of Integration Constant

    – Left Support

    0 0)0( 11 =→= bwR

    – Right Support & Mid-span

      

     

    −=

    =

    −=

      

      

    ′==+=′

    ==+=

    =+++−=

    6

    0 4

    (0) 4

    )(

    (0) 12

    )(

    0 412

    )(

    3

    2

    2

    2

    1

    221

    2

    1

    221

    3

    1

    22

    33

    2

    LR b

    a

    LR a

    wEIaa LR

    LwEI

    EIwbLa LR

    LEIw

    bLa LRLR

    LEIw

    b

    b

    R b

    R

    R b

    R

    bb R

     Deflection

    )23( 12

    )3( 12

    32 2

    3 22

    1 23

    11

    LLxx EI

    R w

    xLx EI

    R w

    b R

    b R

    +−−=

    −=

    EI

    RL wLw bRRR 48

    )2( )0()(

    3

    21 −===δ

     Compatibility Condition

    δ0+ δR=0 → 0 48

    )2(