Yield Lines and Tensile Membrane Action - a New Look. Ian Burgess University of Sheffield

Transcription

1 Yield Lines and Tensile Membrane Action - a New Look Ian Burgess University of Sheffield

2 Why did TMA become an issue?

3 Cardington Max beam temperature ~1150 C cf. Code critical temperature ~ 680 C

4 BRE Membrane Test Compression failure Fracture Fracture

5 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.

6 TSLAB v3.0 BRE (770 C) TSLAB 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 burnout.

7 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.

8 The basis: Hayes (1968)

9 Where are we now?

10 Small-deflection yield-line mechanism rl nl Hogging rotations about edges of panel Tensile crack across short mid-span g Sagging rotations about internal yield lines l Large-deflection failure crack observed experimentally and used in Bailey/BRE, FRACOF and NZ SPM Yield-line pattern is optimized for minimum failure load.

11 Current simplified methods of representing TMA S C T 2 T 1 1. Large deflection progressively enhances yield-line load capacity 2. Limiting value of deflection due to integrity failure given by through-depth tensile cracking.

12 Problems with current simplified methods 1.1KT 0 L/2 (Ultimate strength of reinforcement across Fracture) S C T 2 T 1 Basic mechanics of enhancement: Kinematics doesn t work. Why elastic tension distributions? Illogical aggregation of enhancements. e 1 = e 1m + e 1b e 2 = e 2m + e 2b e = e 1 e 1 e μa 2

13 Problems with current simplified methods For the heated case:. Mechanically illogical superposition of incompatible cases for limiting deflection. w ε = 0.5f sy 3L 2 E s 8 w θ = α T 2 T 1 l 2 16h

14 First thoughts on enhancement

15 Kinematically consistent large-deflection behaviour N.B. Considering only the mechanics of TMA modelling without temperature distribution. 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.

16 Yield lines at small deflection (usual mechanism) rl nl Hogging rotations about edges of panel g Sagging rotations about internal yield lines A l Angle g, or intersection length nl, optimised for minimum failure load.

17 Slab facets at high deflection (usual mechanism) Crack a Crack b 1 Crack b 3 Crack b 2 2b Crack b 1 Crack a Crack b A 3 b a D A f A l/2 2b A Crack a Crack b 2 Crack a d A Crack b 1 Crack b 1 D A q d A A A nl

18 Plastic theory External work= S(force resultant x vertical movement) External work= Internal plastic work from stretching rebars

19 Rebar extensions Crack opening at rebar level. x h s mt t Assumed reduced effective depth. t mt z Anchor points Assumed gauge length h Assumed debonded rebar length

20 Internal plastic work in rebars Maximum Intact rebar length/l if rebar bars fractured length/l Factor for whole slab Crack Rebar direction a x 4 y 4 y 4c* x 2 y 2 Short edge x Internal plastic energy 0 * c=0 for isolated slab, c=1 for continuous slab. 2c*

21 Capacity enhancement factor Enhancement of yield-line capacity 9m x 6m slab, rebar fracture ductility 5% Initial yield line capacity Slab capacity 1.5 b2 fracture start ay fracture start b3 fracture start b3 fracture complete ax fracture start ay fracture complete ax fracture complete d A /d 1

22 Enhancement factor Sequence of rebar fractures 2.5 9m x 6m slab: C30, S500, 5% rebar fracture ductility b3 fracture start d A /d 1

23 Enhancement factor Sequence of rebar fractures 2.5 9m x 6m slab: C30, S500, 5% rebar fracture ductility. 2.0 b 1.5 b2 fracture start d A /d 1

24 Enhancement factor Sequence of rebar fractures 2.5 9m x 6m slab: C30, S500, 5% rebar fracture ductility. 2.0 b 1.5 b3 fracture complete d A /d 1

25 Enhancement factor Sequence of rebar fractures 2.5 9m x 6m slab: C30, S500, 5% rebar fracture ductility ay fracture start b d A /d 1

26 Enhancement factor Sequence of rebar fractures 2.5 9m x 6m slab: C30, S500, 5% rebar fracture ductility ax fracture start b d A /d 1

27 Enhancement factor Sequence of rebar fractures 2.5 9m x 6m slab: C30, S500, 5% rebar fracture ductility ay fracture complete ax fracture complete b d A /d 1

28 Capacity enhancement factor 2.5 Enhancement of yield-line capacity 9m x 6m slab, effect of ductility Ductility 10% 1.0 Ductility 7.5% Ductility 5% d A /d 1

29 Capacity enhancement factor Different plate aspect ratios (10% ductility) 2.5 r=1.0 r=1.5 r= r= r=10.0 r=10.0 r= r= d A /d 1

30 Q & A Q: Isn t this counter-intuitive? Surely a long plate effectively just folds about its central hinge A1: No A2: Also There s still considerable change of geometry to stretch the rebar, apart from folding, away from the central zone. Plastic mechanism theory is inherently upper-bound - we have to find the right mechanism.

31 Alternative kinematically admissible mechanisms

32 TMA Enhancements from other mechanisms

33 Capacity enhancement factor Comparison of Mechanisms A and B 9m x 6m slab, rebar fracture ductility 5% d A /d1

34 Capacity enhancement factor Comparison of Mechanisms A and C 9m x 6m slab, rebar fracture ductility 5% d A /d1

35 Capacity enhancement factor Comparison of Mechanisms A and C 9m x 6m slab, rebar fracture ductility 5% d A /d1

36 Capacity enhancement factor Comparison of all mechanisms 600m (!!) x 6m slab, rebar fracture ductility 5% d A /d1

37 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.

38 How does the mid-span crack happen?

39 Increasing deflection of yield-line mechanism 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. So the initial large-deflection mechanism is Mechanism B.

40 Force equilibrium of Mechanism B M 1 M 2 S C 1 T 2 C 2 T 1 Shape of concrete compression blocks is dictated by compatibility and equilibrium: Initially tension and compression at every point of yieldlines. As deflection increases concrete compression blocks concentrate towards slab corners, rebar fractures when it exceeds ductility. No tension within compression blocks M 3

41 Change of stress blocks ductile y-reinforcement z x Compression y y Shape of concrete compression blocks is dictated by compatibility and equilibrium: Initially tension and compression at every point of yieldlines. As deflection increases concrete compression blocks concentrate towards slab corners, rebar fractures when it exceeds ductility. No tension within compression blocks Tension

42 Change of stress blocks ductile y-reinforcement z x Compression y y Shape of concrete compression blocks is dictated by compatibility and equilibrium: Initially tension and compression at every point of yieldlines. As deflection increases concrete compression blocks concentrate towards slab corners, rebar fractures when it exceeds ductility. No tension within compression blocks Tension

43 Change of stress blocks ductile y-reinforcement z x Compression y y Shape of concrete compression blocks is dictated by compatibility and equilibrium: Initially tension and compression at every point of yieldlines. As deflection increases concrete compression blocks concentrate towards slab corners, rebar fractures when it exceeds ductility. No tension within compression blocks Tension

44 Change of stress blocks ductile y-reinforcement z x Compression y y Shape of concrete compression blocks is dictated by compatibility and equilibrium: Initially tension and compression at every point of yieldlines. As deflection increases concrete compression blocks concentrate towards slab corners, rebar fractures when it exceeds ductility. No tension within compression blocks Tension

45 Change of stress blocks ductile y-reinforcement z x Compression y y Shape of concrete compression blocks is dictated by compatibility and equilibrium: Initially tension and compression at every point of yieldlines. As deflection increases concrete compression blocks concentrate towards slab corners, rebar fractures when it exceeds ductility. No tension within compression blocks Tension

46 Change of stress blocks ductile y-reinforcement z x Compression y y Shape of concrete compression blocks is dictated by compatibility and equilibrium: Initially tension and compression at every point of yieldlines. As deflection increases concrete compression blocks concentrate towards slab corners, rebar fractures when it exceeds ductility. No tension within compression blocks Tension

47 Change of stress blocks ductile y-reinforcement z x Compression y y Shape of concrete compression blocks is dictated by compatibility and equilibrium: Initially tension and compression at every point of yieldlines. As deflection increases concrete compression blocks concentrate towards slab corners, rebar fractures when it exceeds ductility. No tension within compression blocks Tension

48 Change of stress blocks possibilities with less ductility z x y y With different rebar ductility, mesh can either fracture abruptly or progressively at any stage. Tension

49 Change of stress blocks possibilities with less ductility z x y y With different rebar ductility, mesh can either fracture abruptly or progressively at any stage. Tension

50 Change of stress blocks possibilities with less ductility z x y y With different rebar ductility, mesh can either fracture abruptly or progressively at any stage. Tension

51 Stress (N/mm2) Maximum concrete tensile stress - Mechanism B M 1 M 2 T 2 C 2 T 1 S C 1 Slab 9m x 6m Concrete Grade 30 Mesh A142, S500 Thickness 120mm Mesh depth 60mm Ductile rebar no fractures M 1 M 2 T 1 C 1 S EC3 tensile strength for C30 [0.3f c 0.67 ] M Deflection/slab thickness M 3

52 Stress (N/mm2) Maximum concrete tensile stress - Mechanism B M T 1 S C 1 Slab 6m x 6m Concrete Grade 30 Mesh A142, S500 Thickness 120mm Mesh depth 60mm Ductile rebar no fractures 3.5 C 1 S M 1 M 2 T EC3 tensile strength for C30 [0.3f c 0.67 ] M 3 M Deflection/slab thickness

53 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.

54 Thank you