Yield Lines and Tensile Membrane Action - a New Lookfire-research.group.shef.ac.uk/steelinfire/downloads/IWB... · 2013. 9. 25. · Yield Lines and Tensile Membrane Action - a New

Yield Lines and Tensile Membrane Action - a New Look

Ian Burgess

University of Sheffield

Why did TMA become an issue?

Cardington

Max beam temperature ~1150C

cf. Code critical temperature ~ 680C

Max beam temperature ~1150C

cf. Code critical temperature ~ 680C

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 − 𝑒21 + 2𝜇𝑎2

𝑒1 = 𝑒1𝑚 + 𝑒1𝑏

𝑒2 = 𝑒2𝑚 + 𝑒2𝑏

𝑒 = 𝑒1 −𝑒1 − 𝑒21 + 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 𝑙2

16ℎ 𝑤𝜃 =

𝛼 𝑇2 − 𝑇1 𝑙2

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

Cra

ck b2

Cra

ck b2

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

Cap

acit

y en

han

cem

ent

fact

or

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

Enh

ance

men

t fa

cto

r

dA/d1

b3 fracture

start

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


0.0

0.5

1.0

1.5

2.0

2.5

0.0 2.0 4.0 6.0 8.0 10.0

Enh

ance

men

t fa

cto

r

dA/d1

b

b2 fracture

start



0.0

0.5

1.0

1.5

2.0

2.5

0.0 2.0 4.0 6.0 8.0 10.0

Enh

ance

men

t fa

cto

r

dA/d1

b

b3 fracture

complete



0.0

0.5

1.0

1.5

2.0

2.5

0.0 2.0 4.0 6.0 8.0 10.0

Enh

ance

men

t fa

cto

r

dA/d1

b ay fracture

start



0.0

0.5

1.0

1.5

2.0

2.5

0.0 2.0 4.0 6.0 8.0 10.0

Enh

ance

men

t fa

cto

r

dA/d1

b ax fracture

start



0.0

0.5

1.0

1.5

2.0

2.5

0.0 2.0 4.0 6.0 8.0 10.0

Enh

ance

men

t fa

cto

r

dA/d1

b ay fracture

complete

ax fracture

complete


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

Cap

acit

y en

han

cem

ent

fact

or

dA/d1

Enhancement of yield-line capacity 9m x 6m slab, effect of ductility

Ductility 5%

Ductility 7.5%

Ductility 10%

Different plate aspect ratios (10% ductility)

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

Cap

acit

y en

han

cem

ent

fact

or

dA/d1

r=2.0 r=1.5 r=1.0

r=1.0

r=2.0

r=1.5

r=10.0

r=10.0

Q & A

Q: Isn’t this counter-intuitive?

Surely a long plate effectively just folds about its central hinge ……

Q: Isn’t this counter-intuitive?

Surely a long plate effectively just folds about its central hinge ……

A2: Also …

Plastic mechanism theory is inherently upper-bound - we have to find the right mechanism.

A2: Also …

Plastic mechanism theory is inherently upper-bound - we have to find the right mechanism.

A1: No …

There’s still considerable change of geometry to stretch the rebar, apart from folding, away from the central zone.

A1: No …

There’s still considerable change of geometry to stretch the rebar, apart from folding, away from the central zone.

Alternative kinematically admissible mechanisms

TMA Enhancements from other mechanisms

Comparison of Mechanisms A and B 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

Cap

acit

y en

han

cem

ent

fact

or

dA/d1

Comparison of Mechanisms A and C 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

Cap

acit

y en

han

cem

ent

fact

or

dA/d1

Comparison of Mechanisms A and C 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

Cap

acit

y en

han

cem

ent

fact

or

dA/d1

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

Cap

acit

y en

han

cem

ent

fact

or

dA/d1

Comparison of all mechanisms 600m (!!) x 6m slab, rebar fracture ductility 5%

What else can this approach do?

Other capabilities:

• Continuity across slab edges (if bisymmetric),

Asymmetric boundary conditions makes optimizing yield-line pattern complex.

• Orthotropic mesh

Can even rotate the yield-line pattern.

• Non-rectangular slabs

Isosceles triangles are easy, trapezia are harder to optimize.

Other capabilities:

• Continuity across slab edges (if bisymmetric),

Asymmetric boundary conditions makes optimizing yield-line pattern complex.

• Orthotropic mesh

Can even rotate the yield-line pattern.

• Non-rectangular slabs

Isosceles triangles are easy, trapezia are harder to optimize.

How does the mid-span crack happen?

Increasing deflection of yield-line mechanism

Yield-line mechanism is a plastic bending mechanism at small deflections.

Yield-line mechanism is a plastic bending mechanism at small deflections.

As deflections start to increase the yield-line pattern increases the rotations of its flat facets, with the rebar yielding until it fractures.

As deflections start to increase the yield-line pattern increases the rotations of its flat facets, with the rebar yielding until it fractures.

So the initial large-deflection mechanism is Mechanism B. So the initial large-deflection mechanism is Mechanism B.

Force equilibrium of Mechanism B

S

C1

T1 T2 C2

M1 M2

M3

Shape of concrete compression blocks is dictated by compatibility and equilibrium:

• Initially tension and compression at every point of yield-lines.

• As deflection increases concrete compression blocks concentrate towards slab corners, rebar fractures when it exceeds ductility.

• No tension within compression blocks













Change of stress blocks – ductile y-reinforcement

Compression

Tension

z

y

x

y










Compression

Tension

z

y

x

y










Compression

Tension

z

y

x

y










Compression

Tension

z

y

x

y










Compression

Tension

z

y

x

y










Compression

Tension

z

y

x

y










Compression

Tension

z

y

x

y

With different rebar ductility, mesh can either fracture abruptly or progressively at any stage.


Change of stress blocks – possibilities with less ductility

Tension

z

y

x

y




Tension

x

y

z

y




Tension

x

y

z

y

Maximum concrete tensile stress - Mechanism B

S

C1

T1 T2 C2

M1 M2

M3

C1

T1

S

M1

M2

M3

0 6 3 1 2 4 5 0

6

3

1

2

4

5

Stre

ss (

N/m

m2

)

Deflection/slab thickness

• Slab 9m x 6m • Concrete Grade 30 • Mesh A142, S500 • Thickness 120mm • Mesh depth 60mm • Ductile rebar – no

fractures


fractures

EC3 tensile strength for C30

[0.3fc0.67]


[0.3fc0.67]

Maximum concrete tensile stress - Mechanism B

S

C1

T1

M

C1

T1

S

M1

M2

M3


fractures


fractures

M

0 6 3 1 2 4 5

Stre

ss (

N/m

m2

)

Deflection/slab thickness

0

3.0

1.5

0.5

1.0

2.0

3.5

2.5


[0.3fc0.67]


[0.3fc0.67]

What’s still to do?

• Complete cracking + enhancement for all ductility cases,

• Composite action with beams – complete and partial,

• A better rebar debonding approach,

………………………..

• The effect of thermal displacements,

• Temperature distribution through the slab,

………………………..

• Asymmetric continuity – edge panels,

• Different panel shapes,

• Integration with transverse edge beam bending.

• Complete cracking + enhancement for all ductility cases,

• Composite action with beams – complete and partial,

• A better rebar debonding approach,

………………………..

• The effect of thermal displacements,

• Temperature distribution through the slab,

………………………..

• Asymmetric continuity – edge panels,

• Different panel shapes,

• Integration with transverse edge beam bending.

Thank you

Documents

Yield Lines and Tensile Membrane Action - a New Lookfire-research.group.shef.ac.uk/steelinfire/downloads/IWB... · 2013. 9. 25. · Yield Lines and Tensile Membrane Action - a New