torsion vlasov muy buena

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  • 8/12/2019 Torsion Vlasov Muy Buena

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    4 Prof. Ing. Josef Machek, DrSc.

    OK3 1

    4. TorsionOpen and closed cross sections, simple St. Venant and warping torsion,

    interaction of bending and torsion, Eurocode approach.

    Common is elastic solution (nonlinear plastic analysis e.g. Strelbickaja)Eurocode 3 enables combination of plastic bending moment and elastic torsion.

    Should be distinguished:

    - simple torsion: only shear stresses arise,

    - warping torsion: both shear and direct (normal) stresses arise.

    1. Open cross sections (e.g. I, U, L)

    a) Simple (Saint Venants) torsion (occurs only exceptionally, see later)

    t

    iTt

    t

    b i= 1

    i= 2

    i= 3

    Only shear arises:

    3

    / 0My(i)

    t

    tt(i)

    ftI

    T (maximal in tmax)

    i

    3iit

    31 tbI

    (influence of rounding of rolled sections,

    otherwise = 1)

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    4 Prof. Ing. Josef Machek, DrSc.

    OK3 2

    b) Warping tors ion (according to Vlasovs theory)

    one part of a torsion moment Tis transmitted by simple torsion Tt , other part by bending torsion Tw : T= Tt + Tw

    Assumptions:

    1. Rigid cross section,

    2. Null shear deformation

    (shear lag ignored).

    moment of simple

    torsion

    moment of bending

    torsion

    bimoment

    bending torsion

    T

    S shear centre

    (bending centre)

    wt w

    = ++

    +

    +

    -

    -

    in warping torsion everything

    is related to central line

    internal forces: Tt Tw B

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    4 Prof. Ing. Josef Machek, DrSc.

    OK3 3

    Shear stresses:

    simple torsion t

    bending torsion

    e.g.:

    Direct (normal) stress:

    Result ing stresses:

    ww

    wW

    B

    I

    wB =

    w

    www

    It

    ST

    =first sectorial moment

    warping constant

    t

    w,max tt

    stress through th ickness t

    sectorial section modulus

    Applies bending analogy : B M or w Tw V or w

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    4 Prof. Ing. Josef Machek, DrSc.

    OK3 4

    Sectorial characteristics

    Rolled sections see tables.

    In general from sectorial coordinate:

    I cross section:

    4ds

    s

    hbrw =

    z

    2

    A

    2w

    4dA I

    hwI =U cross section:

    a = a

    w Swin this position

    no torsion !!

    GS

    The main sectorial coordinate:

    First sectorial moment:

    16dA

    2

    Aw

    tbhwS =

    Second sectorial moment(warping constant):

    w, Sw, Iw ... see tables

    0dAdAAA

    =wzwyPosition of S:

    (product sectorial moments)

    t

    h

    b

    wSw

    r

    G S

    z

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    4 Prof. Ing. Josef Machek, DrSc.

    OK3 5

    Determination of internal forces due to torsion:

    solution of Vlasovs differential equations, or directly from formulas,

    based on bending analogy.

    Distribution of torsion moment:

    eF

    V

    simple support in torsion (couple of forces)

    M

    Abending

    torsion

    T = Ve

    B

    Distribution of torsional moment due to eccentrical force

    corresponds to dis tribution of transverse force at

    eccentricity.Part transmitted by simple torsion is set aside:

    eVTt 1w eVT1eMBMe

    Superposition for more complex loading is necessary:

    ... see table of EurocodeCzech NA

    e2

    e1

    - e3

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    4 Prof. Ing. Josef Machek, DrSc.

    OK3 6

    Simplified (conservative) solution neglects simple torsion:

    bending of flanges only = bimoment

    (often adequate: it is conservative

    from w point of view)

    Important notes:

    1. Large direct stresses, they can not be ignored!!

    2. Direct stresses (warping torsion) do not arise:

    a) for loading by stresses t, roughly also due to endTloading (simple torsion arises only):

    b) in sections composed of radiating outstands

    (because of w= Iw = 0):

    3. In practice usually occurs torsion about enforced axis (V):

    T

    0

    T T/h

    T/h

    h

    (shear centre Sis in cross point)

    moreover, torsion is usually restricted by cladding rigidity torsion may often be ignored.

    analysis about original shear centre S

    is uneconomical !!!S

    S V e

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    4 Prof. Ing. Josef Machek, DrSc.

    OK3 7

    a) Simple torsion (shear stresses only, usual design)

    2. Closed cross sections (e.g. )

    Bredts shear flow (t t ) = const .

    (i)t(i)

    t

    Tt

    =

    b) Warping torsion: - Umanskijs theory (rigid cross section),- Vlasovs theory with non rigid cross section,

    - FEM (including influence of bevelled cross section,

    gives also transversal bending moments in plates).

    The stresses are the same as in open cross sections: t, w, w.However, w, w are very small, commonly ignored even for br idges.

    = 2As

    As

    t

    tidi

    Tt

    Contrary to open cross section the maximal sheart

    is in the thinnest plate and along thickness constant !!

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    4 Prof. Ing. Josef Machek, DrSc.

    OK3 8

    3. Interaction of bending and torsion ( My + T)e

    Direct stresses (open cross sections only):

    M1ywMy /fM1y

    w

    Ed

    yLT

    Edy, /fWB

    W

    M Shear stresses:

    i.e.

    Rdpl,My Edw,M0y Edt,RdT,pl, V//f//f,V

    0332511

    1RdT,pl,

    Ed

    V

    VVpl,T,Rd is design plastic shear resistance of the cross section.

    For open sections I and U

    For closed sections Rdpl,My Edt,RdT,pl, V//fV

    031

    in U sections only

    In general, bending and torsion stresses may be summed and von Mises criterion

    applied:13

    2

    000

    22

    0

    My

    Ed

    My

    Edz,

    My

    Edx,

    M0y

    Edz,

    My

    Edx,

    /f/f/f/f/f