Yield Lines and Tensile Membrane Action - a New Lookfire- 2013. 9. 25.¢  Yield Lines and Tensile Membrane

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  • Yield Lines and Tensile Membrane Action - a New Look

    Ian Burgess

    University of Sheffield

  • Why did TMA become an issue?

  • Cardington

    Max beam temperature ~1150C

    cf. Code critical temperature ~ 680C

    Max beam temperature ~1150C

    cf. Code critical temperature ~ 680C

  • Fracture

    Fracture

    Compression failure

    BRE Membrane Test

  • UK post-Cardington design guidance

    Fire Safe Design (SCI P288) A new approach to multi-storey steel framed buildings

    • Based on observations of Cardington tests.

    • Checked using simplified (BRE-Bailey) design method.

    Fire Safe Design (SCI P288) A new approach to multi-storey steel framed buildings

    • Based on observations of Cardington tests.

    • Checked using simplified (BRE-Bailey) design method.

  • TSLAB v3.0

    TSLAB v3.0 (SCI) Excel-based free software updating BRE method:

    • Thermal analysis to give weighted average reinforcement mesh temperature at any time.

    • Optimises mesh size for FLS loading at required fire resistance time – or for burn- out.

    TSLAB v3.0 (SCI) Excel-based free software updating BRE method:

    • Thermal analysis to give weighted average reinforcement mesh temperature at any time.

    • Optimises mesh size for FLS loading at required fire resistance time – or for burn- out.

    BRE (770°C) TSLAB BRE (770°C) TSLAB

  • FRACOF

    FRACOF

    • Based on a European project

    • Outcome almost identical to UK simplified (BRE-Bailey) design method. A few changes to safety factors, extra deflection check.

    FRACOF

    • Based on a European project

    • Outcome almost identical to UK simplified (BRE-Bailey) design method. A few changes to safety factors, extra deflection check.

  • The basis: Hayes (1968)

  • Where are we now?

  • Small-deflection yield-line mechanism

    rl

    nl

    l

    g

    Hogging rotations about edges of panel

    Sagging rotations about internal yield lines

    Tensile crack across short mid-span

    Large-deflection failure crack observed experimentally and used in Bailey/BRE, FRACOF and NZ SPM

    Yield-line pattern is optimized for minimum failure load. Yield-line pattern is optimized for minimum failure load.

  • Current simplified methods of representing TMA

    C S

    T2

    T1

    1. Large deflection progressively enhances yield-line load capacity

    2. Limiting value of deflection due to integrity failure given by through-depth tensile cracking.

    1. Large deflection progressively enhances yield-line load capacity

    2. Limiting value of deflection due to integrity failure given by through-depth tensile cracking.

  • 1.1KT0L/2

    (Ultimate strength of reinforcement across Fracture)

    C S

    T2

    T1

    Basic mechanics of enhancement:

    • Kinematics doesn’t work.

    • Why elastic tension distributions?

    • Illogical aggregation of enhancements .

    Basic mechanics of enhancement:

    • Kinematics doesn’t work.

    • Why elastic tension distributions?

    • Illogical aggregation of enhancements .

    Problems with current simplified methods

    𝑒1 = 𝑒1𝑚 + 𝑒1𝑏

    𝑒2 = 𝑒2𝑚 + 𝑒2𝑏

    𝑒 = 𝑒1 − 𝑒1 − 𝑒2 1 + 2𝜇𝑎2

    𝑒1 = 𝑒1𝑚 + 𝑒1𝑏

    𝑒2 = 𝑒2𝑚 + 𝑒2𝑏

    𝑒 = 𝑒1 − 𝑒1 − 𝑒2 1 + 2𝜇𝑎2

  • For the heated case:.

    Mechanically illogical superposition of incompatible cases for limiting deflection.

    For the heated case:.

    Mechanically illogical superposition of incompatible cases for limiting deflection.

    Problems with current simplified methods

    𝑤𝜀 = 0.5𝑓𝑠𝑦

    𝐸𝑠

    3𝐿2

    8 𝑤𝜀 =

    0.5𝑓𝑠𝑦

    𝐸𝑠

    3𝐿2

    8 𝑤𝜃 =

    𝛼 𝑇2 − 𝑇1 𝑙 2

    16ℎ 𝑤𝜃 =

    𝛼 𝑇2 − 𝑇1 𝑙 2

    16ℎ

  • First thoughts on enhancement

  • Kinematically consistent large-deflection behaviour

    Principles:

    • Based on small-deflection yield-line mechanism (flat facets) plus any necessary pure membrane tension cracks.

    • Internal work is done in plastic stretching of rebar mesh.

    • Rebar ductility to fracture can be defined. Provisionally, rebar gauge length for x or y bars is spacing of transverse bars (weld points as anchors).

    • Pattern of yield lines and tension cracks must be kinematically consistent as a mechanism.

    Principles:

    • Based on small-deflection yield-line mechanism (flat facets) plus any necessary pure membrane tension cracks.

    • Internal work is done in plastic stretching of rebar mesh.

    • Rebar ductility to fracture can be defined. Provisionally, rebar gauge length for x or y bars is spacing of transverse bars (weld points as anchors).

    • Pattern of yield lines and tension cracks must be kinematically consistent as a mechanism.

    N.B. Considering only the mechanics of TMA modelling – without temperature distribution.

  • g

    Hogging rotations about edges of panel

    Sagging rotations about internal yield lines

    A

    Yield lines at small deflection (usual mechanism)

    rl

    nl

    l

    Angle g, or intersection length nl, optimised for minimum failure load.

  • Crack b1 Crack b1

    Crack b1 Crack b1

    C ra

    ck b 2

    C ra

    ck b 2

    Crack b3 Crack b3

    Crack a Crack a

    Crack a Crack a

    b

    2b

    2b

    a

    A

    Slab facets at high deflection (usual mechanism)

    nl

    q

    dA

    DA

    A A’

    l/2 f

    dA

    D’A

    A

    A’

  • Plastic theory

    External work= S(force resultant x vertical movement)

    External work= Internal plastic work from stretching rebars

  • Rebar extensions

    Crack opening at rebar level.

    h

    x

    mt t

    s

    mt t z

    Assumed reduced effective depth.

    Assumed debonded rebar length

    h

    “Anchor” points Assumed gauge length

  • Internal plastic work in rebars

    Crack Rebar

    direction

    Maximum rebar

    length/l

    Intact rebar length/l if bars fractured

    Internal plastic energy Factor for

    whole slab

    a x 4

    y 4

    y 4c*

    x 2

    y 2

    Short

    edge x 0 2c*

    * c=0 for isolated slab, c=1 for continuous slab.

  • Enhancement of yield-line capacity 9m x 6m slab, rebar fracture ductility 5%

    0.0

    0.5

    1.0

    1.5

    2.0

    2.5

    0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0

    C ap

    ac it

    y en

    h an

    ce m

    en t

    fa ct

    o r

    dA/d1

    Initial yield line capacity

    Slab capacity

    b3 fracture start

    b3 fracture complete

    b2 fracture start

    ay fracture start

    ax fracture start

    ay fracture complete

    ax fracture complete

  • Sequence of rebar fractures

    0.0

    0.5

    1.0

    1.5

    2.0

    2.5

    0.0 2.0 4.0 6.0 8.0 10.0

    En h

    an ce

    m en

    t fa

    ct o

    r

    dA/d1

    b3 fracture start

    9m x 6m slab: C30, S500, 5% rebar fracture ductility.

  • Sequence of rebar fractures

    0.0

    0.5

    1.0

    1.5

    2.0

    2.5

    0.0 2.0 4.0 6.0 8.0 10.0

    En h

    an ce

    m en

    t fa

    ct o

    r

    dA/d1

    b

    b2 fracture start

    9m x 6m slab: C30, S500, 5% rebar fracture ductility.

  • Sequence of rebar fractures

    0.0

    0.5

    1.0

    1.5

    2.0

    2.5

    0.0 2.0 4.0 6.0 8.0 10.0

    En h

    an ce

    m en

    t fa

    ct o

    r

    dA/d1

    b

    b3 fracture complete

    9m x 6m slab: C30, S500, 5% rebar fracture ductility.

  • Sequence of rebar fractures

    0.0

    0.5

    1.0

    1.5

    2.0

    2.5

    0.0 2.0 4.0 6.0 8.0 10.0

    En h

    an ce

    m en

    t fa

    ct o

    r

    dA/d1

    b ay fracture

    start

    9m x 6m slab: C30, S500, 5% rebar fracture ductility.

  • Sequence of rebar fractures

    0.0

    0.5

    1.0

    1.5

    2.0

    2.5

    0.0 2.0 4.0 6.0 8.0 10.0

    En h

    an ce

    m en

    t fa

    ct o

    r

    dA/d1

    b ax fracture

    start

    9m x 6m slab: C30, S500, 5% rebar fracture ductility.

  • Sequence of rebar fractures

    0.0

    0.5

    1.0

    1.5

    2.0

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