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Transcript of Strut and Tie ModelsStrut and Tie Strut and Tie ModelsStrut and Tie Models • Strut •...

  • Strut and Tie ModelsStrut and Tie Models

    • Strut • Strut and Tie SystemsStrut and Tie Systems • Fans - Non-concentric fans - Concentric fans- Concentric fans - Fans with Bond

  • Single StrutSingle Strut x h y−0x 0 0

    0 0

    tan x h y y a x

    θ = = +

    f

    0x

    0y

    2 0 1 1

    2 x a a h h h

    ⎡ ⎤⎛ ⎞⎢ ⎥= + −⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦

    h cf

    2h h h⎢ ⎥⎝ ⎠⎣ ⎦ 21 a aτ ⎡ ⎤⎛ ⎞⎢ ⎥

    θ 1 1 2c

    a a f h h τ ⎡ ⎤⎛ ⎞⎢ ⎥= + −⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦a

    2 0 01 4 1

    2 y yx a a

    h h h h h

    ⎡ ⎤⎛ ⎞ ⎛ ⎞⎢ ⎥= − + −⎜ ⎟⎜ ⎟⎢ ⎥⎝ ⎠⎝ ⎠2h h h h h⎜ ⎟⎜ ⎟⎢ ⎥⎝ ⎠⎝ ⎠⎣ ⎦

  • Single StrutSingle Strut

    0s Y cA f y tf=

    cf

    0 1 2

    s YA f y thf h

    Φ = = ≤h

    cf

    2cthf h

    θ0y

    a0 x

    ( ) 2

    0 1 4 1xP a aτ ⎡ ⎤⎛ ⎞⎢ ⎥= = = Φ −Φ + −⎜ ⎟( )4 12c cf thf h h h ⎢ ⎥= = = Φ Φ + ⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦

  • Fan shaped stress fieldsFan-shaped stress fields

    • Non-concentric fan - Uniform Normal stress type

    D

    Uniform Normal stress type - Uniform Shear stress type

    A

    • Concentric Fan B

    • Concentric Fan

  • C

    T

    Boundary Stress onBoundary Stress on Non-concentric fan-shaped stress field

  • y Py

    Sw(x)

    tcσ ,top

    h

    x

    Z(x) cσ ,botr

    q

    Uniform normal stress on boundary ofUniform normal stress on boundary of Non-concentric fan-shaped stress field

  • d =dx+(dw-dz)k+(w-z)dk

    -dw

    w+dw-[z+dz]

    x+(w-x)k

    k 1

    1

    z

    k+dk

    ddx

    dx

    -dz

    Geometry for Infinitesimal Elements of Non-concentric Fan-shaped stress field (Uniforof Non concentric Fan shaped stress field (Unifor Normal stress type)

  • y σ constτ =

    H

    P

    x

    L Ldx-dz k+( -z)dk 2

    -dz k

    k+dk 1

    1

    k+dk

    Uniform shear stress on boundary of Non-concentric fan-shaped stress field

  • yl/2 y yσ

    0

    2σ =0 xyτ

    yσ y

    yxτ+ y-σ t

    +

    x

    T

    1σ =0

    pl 2-σ x

    +- tτ 1σ =0

    0

    el

    dx

    - y-σ t

    + y-σ t

    +- tτtrT -q

    2-σ

    -- tτ

    y tτ

    2σ =0

    1σ =0

    or

    y- tτtr

    -pt

    os

    p

    r

    s yx- tτ 1σ =0

    2σ 0

    T+dTx

  • l/2l/2

    yy

    ξ ξ

    el

    k k+dk

    11

    k dξ

    θ

    ( ) k+dk 1

    -r

    r

    dr

    Q(x,y)

    b b(x ,y )

    xx

    -s s

    -ds + y-σ t

    + x

    1σ =0

    x

    + y-σ t

    + y- tτ

    + y- tτ

    dr

    2-σ 1σ 0

    -qt x' 2-σw

    +

    dξ R

    R'

    θ

    -ds

    -ptx' 2 +-σ t

    R

    Geometry for Infinitesimal Elements of Non-concentric Fan-shaped stress field (Uniform Shear stress type)

  • θ e

    p

    m

    q

    r

    qq

    Concentric Fan-shaped stress field

  • dx

    2,bottomσσ =

    dx hdK+ 2,

    1 top dK

    dx

    σ η

    = +dx hdK+

    2 tσ dx

    2,topσ

    11 K K dK+

    11

    dx

    K dK+ 2,bottomσ

    dx Stress State in Diagonal Compression Stress FieldsStress State in Diagonal Compression Stress Fields

  • Single Strut actiong 0x

    cf 0y

    hh

    N

    θ a

    0y′ 0C

    a P

    N τN sτ

    P 0 T

  • Strut with Diagonal Compression FieldStrut with Diagonal Compression Field tτ

    cσ cott tσ τ θ=

    f

    N N cf

    θ cf

    PP P 21 1 2s c

    a aP thf h h

    ⎡ ⎤⎛ ⎞⎢ ⎥= + −⎜ ⎟⎢ ⎥⎝ ⎠2s c f

    h h⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦ sP Pτ −=

    ( )0t t h y τ =

    ′−

  • Diagonal Compression FieldDiagonal Compression Field

    cf fσ =

    θ c cfσ =

    2sin 2 f τθ = tanYyrf τ θ= cf

    21 1 a a aτ ⎡ ⎤⎛ ⎞⎢ ⎥= + +Φ⎜ ⎟ ( )1

    τ = Φ −Φ1

    2 ycf h h h ⎢ ⎥= + − +Φ⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦

    ( )1y y cf = Φ Φ

  • Maximum Shear StrengthMaximum Shear Strength τ

    cf

    ( )1y y cf τ = Φ −Φ0.5

    cf 21 1

    2 y a a a

    f h h h τ ⎡ ⎤⎛ ⎞⎢ ⎥= + − +Φ⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦ Φ0.5

    2cf h h h⎢ ⎥⎝ ⎠⎣ ⎦

  • Strut with Fan-Shaped Stress Fieldp