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Direct Stiffness Method: Plane Frame - vsb. · PDF fileMember local stiffness matrix: Direct Stiffness Method: Plane Frame Example 1Example 111 Fixed - Fixed Fixed - Hinged Hinged

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Text of Direct Stiffness Method: Plane Frame - vsb. · PDF fileMember local stiffness matrix: Direct...

  • Direct Stiffness Method:Direct Stiffness Method:

    Plane Frame

  • Plane Frame Analysis

    All the members lie in the same plane.

    Members are interconnected by rigid or pin joints.

    The internal stress resultants at a cross-section of member

    consist of bending moment, shear force and an axial force.

    The significant deformations in the plane frame are only

    flexural and axial.

    Stiffness matrix of the member is derived in its local co-

    ordinate axes and then it is transformed to global co-ordinate

    system.

    Members are oriented in different directions and hence before

    forming the global stiffness matrix it is necessary to refer all

    the member stiffness matrices to the same set of axes.

    This is achieved by transformation of forces and

    displacements to global co-ordinate system.

  • Plane Frame, Member Stiffness Matrix

    The frame members have six degrees of freedom

    *au

    a b

    *au

    *

    aw *a

    *

    bu

    *bw

    *b

    =

    *

    *

    *

    *

    *

    b

    b

    b

    a

    a

    a

    ab

    w

    u

    w

    *r

    b b

  • Plane Frame, Member Stiffness Matrix

    The forceforceforceforce displacementdisplacementdisplacementdisplacement relationshiprelationshiprelationshiprelationship can be written:

    **

    aab uX

    *abM

    *baM

    *abX

    *abZ

    *baX

    *baZ

    a b

    ==

    =

    *

    *

    *

    *

    *

    *

    *

    *

    *

    *

    b

    b

    b

    a

    a

    a

    ababab

    ab

    ab

    ab

    ab

    ab

    ab

    ab

    w

    u

    w

    u

    M

    Z

    X

    M

    Z

    X

    **** krkR

    member vector of secondarysecondarysecondarysecondary local forcesforcesforcesforces

    member vector of local joint displacements

    member local stiffness member local stiffness member local stiffness member local stiffness matrixmatrixmatrixmatrix

    *Rab

    *rab*k ab

  • Plane Frame, Member Stiffness Matrix

    Member locallocallocallocal stiffness matrix

    l

    EA

    l

    EA0000

    0000

    l

    EA

    l

    EA

    FixedFixedFixedFixed FixedFixedFixedFixed connectionconnectionconnectionconnection FixedFixedFixedFixed HingedHingedHingedHinged connectionconnectionconnectionconnection

    =

    l

    EI

    l

    EI

    l

    EI

    l

    EIl

    EI

    l

    EI

    l

    EI

    l

    EIl

    EA

    l

    EAl

    EI

    l

    EI

    l

    EI

    l

    EIl

    EI

    l

    EI

    l

    EI

    l

    EIll

    ab

    460

    260

    6120

    6120

    0000

    260

    460

    6120

    6120

    22

    2323

    22

    2323

    *k

    =

    000000

    03

    033

    0

    0000

    03

    033

    0

    03

    033

    0

    0000

    323

    22

    323

    l

    EI

    l

    EI

    l

    EIl

    EA

    l

    EAl

    EI

    l

    EI

    l

    EIl

    EI

    l

    EI

    l

    EIll

    ab*k

    HingedHingedHingedHinged FixedFixedFixedFixed connectionconnectionconnectionconnection HingedHingedHingedHinged HingedHingedHingedHinged connectionconnectionconnectionconnection

    =

    l

    EI

    l

    EI

    l

    EIl

    EI

    l

    EI

    l

    EIl

    EA

    l

    EA

    l

    EI

    l

    EI

    l

    EIl

    EA

    l

    EA

    ab

    3300

    30

    3300

    30

    0000

    000000

    3300

    30

    0000

    22

    233

    233

    *k

    =

    000000

    000000

    0000

    000000

    000000

    0000

    l

    EA

    l

    EA

    l

    EA

    l

    EA

    ab*k

    HingedHingedHingedHinged FixedFixedFixedFixed connectionconnectionconnectionconnection HingedHingedHingedHinged HingedHingedHingedHinged connectionconnectionconnectionconnection

  • Plane Frame, Member Stiffness Matrix

    Member globalglobalglobalglobal stiffness matrix kab

    abababab TkTk* = T abababab TkTk =

    =

    0cossin000

    0sincos000

    000100

    0000cossin

    0000sincos

    abab

    abab

    abab

    ab

    T

    x

    z

    a

    b

    Tab transformation matrix

    100000

    0cossin000 abab z angle of angle of angle of angle of

    transformation transformation transformation transformation

    b

  • Plane Frame, Member vector of primary forces

    Member vector of primary locallocallocallocal forces is corresponding

    to the fixedfixedfixedfixed endendendend rererereactionactionactionaction due to external load.

    The fixed end reactions were derived by forceforceforceforce methodmethodmethodmethod The fixed end reactions were derived by forceforceforceforce methodmethodmethodmethod

    (as well as secondary forces for member stiffness matrix).

    a b*abX

    *baX

    =*

    *

    *

    ab

    ab

    ab

    M

    Z

    X

    *R

    a b*abZ

    *abM *

    baZ

    *baM

    =

    *

    *

    *

    ab

    ab

    ab

    abab

    M

    Z

    X*R

  • Plane Frame, Member vector of primary forces

    a) Continuous load

    Member vector of primary locallocallocallocal forces

    1q 2qa) Continuous load

    1n2n

    Member connection

    +

    +

    20/)37(

    6/)2(

    21

    21

    lqq

    lnn

    +

    +

    40/)916(

    6/)2(

    21

    21

    lqq

    lnn

    +

    +

    40/)411(

    6/)2(

    21

    21

    lqq

    lnn

    ( )

    +

    +

    6/2

    6/)2(

    21

    21

    lqq

    lnn

    *

    *

    ab

    ab

    Z

    X

    +

    +

    +

    +

    60/)32(

    20/)73(

    6/)2(

    60/)23(

    221

    21

    21

    221

    21

    lqq

    lqq

    lnn

    lqq

    +

    +

    +

    0

    40/)114(

    6/)2(

    120/)78(

    21

    21

    221

    lqq

    lnn

    lqq

    +

    +

    +

    120/)87(

    40/)169(

    6/)2(

    0

    221

    21

    21

    lqq

    lqq

    lnn

    ( )

    +

    +

    0

    6/2

    6/)2(

    0

    21

    21

    21

    lqq

    lnn

    =

    *

    *

    *

    *

    ba

    ba

    ba

    ab

    ab

    M

    Z

    X

    M

    Z

    *abR

  • Plane Frame, Member vector of primary forces

    a) Loading by force

    Member vector of primary locallocallocallocal forces

    F

    ba) Loading by force

    a ba b

    Member connection

    ( )

    322 2/)3(

    /

    lblbF

    lbF

    z

    x

    /

    /

    lbF

    lbF

    z

    x

    ( )

    2/)3(

    /

    32 lblbF

    lbF

    z

    x

    +

    32 /)2(

    /

    lalbF

    lbF

    z

    x

    *

    *

    ab

    ab

    Z

    X( )

    ( )

    ( ) ( )

    +

    2

    32

    2/

    2/)3(

    /

    0

    2/)3(

    lalabF

    lalaF

    laF

    lblbF

    z

    z

    x

    z

    0

    /

    /

    0

    laF

    laF

    z

    x

    z( )

    ( ) ( )

    ( )

    +

    0

    2/)3(

    /

    2/

    2/)3(

    322

    2

    lalaF

    laF

    lblabF

    lblbF

    z

    x

    z

    z

    +

    +

    22

    32

    22

    /

    /)2(

    /

    /

    /)2(

    lbaF

    lblaF

    laF

    labF

    lalbF

    z

    z

    x

    z

    z

    =

    *

    *

    *

    *

    ba

    ba

    ba

    ab

    ab

    M

    Z

    X

    M

    Z

    *abR

  • Plane Frame, Member vector of primary forces

    a) Loading by bending moment

    Member vector of primary locallocallocallocal forces

    Ma ba) Loading by bending moment a ba b

    Member connection

    3/6

    0

    lMab ( ) ( )

    2/3

    0

    322 lblM ( ) ( )

    322 2/3

    0

    lalM

    /

    0

    lM

    *

    *

    ab

    ab

    Z

    X

    2

    3

    2

    /)32(

    /6

    0

    /)32(

    /6

    lalMa

    lMab

    lblMb

    lMab ( ) ( )( )

    ( ) ( )

    0

    2/3

    0

    2/)3(

    2/3

    322

    222

    lblM

    lblM

    lblM ( ) ( )

    ( ) ( )( )

    222

    322

    2/)3(

    2/3

    0

    0

    2/3

    lalM

    lalM

    lalM

    0

    /

    0

    0

    /

    lM

    lM

    =

    *

    *

    *

    *

    ba

    ba

    ba

    ab

    ab

    M

    Z

    X

    M

    Z

    *abR

  • Plane Frame, Member vector of primary forces

    Member vector of primary globalglobalglobalglobal forces

    *Tababab RTR = ababab RTR =

    =

    0cossin000

    0sincos000

    000100

    0000cossin

    0000sincos

    abab

    abab

    abab

    ab

    T

    x

    z

    a

    b

    100000

    0cossin000 abab z angle of angle of angle of angle of

    transformation transformation transformation transformation

    b

  • Plane Frame, The load displacement equation

    After establishing the globalglobalglobalglobal stiffnessstiffnessstiffnessstiffness matrixmatrixmatrixmatrix and loadloadloadload

    vectorvectorvectorvector, the load displacement relationship can be

    written:written:

    Global stiffness matrix K is established by the

    localization of member global stiffness matrixes kab Global load vector F is taken as difference between

    vector of joint loads S and vector of primary global

    FKrFrK == 1

    v

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