1. Workshop at Indian Institute of Science 9-13 August, 2010 BangaloreIndia Fire Safety Engineering & Structures in Fire Organisers: CS Manohar and Ananth Ramaswamy Indian Institute of Science Speakers: Jose Torero, Asif Usmani and Martin Gillie The University of Edinburgh Funding and Sponsorship: Basic Structural Mechanics and Modelling in Fire

2. Structural Mechanics at High Temperature The mechanics of restrained heated structures The mechanics of restrained heated structures –Another look at strain –Behaviour of uniformly heated beams –Curvature –Behaviour of beams with thermal gradients –Behaviour of beams heated with thermal gradients

3. Another Look at Strain T+ΔT L ΔLΔL For a rod… …or more generally Thermal strain Mechanical strain Ambient temperature=T P

4. Stresses and Deflections T+ΔT L ΔLΔL Uniformly heated bar L ΔLΔL Bar with end load P In general:remembering

5. Heated Restrained Beam (1) Uniformly heated restrained beam Uniformly heated restrained beam No deflections (unless buckling occurs)… No deflections (unless buckling occurs)… … but compressive stresses … but compressive stresses Thermal effects Mechanical effects T+ΔT T T

6. Heated Restrained Beam (2) Problem: Determine ΔT at failure Assume elastic perfectly plastic material behaviour then either plastic failure will occur at T+ΔT T …or T

7. Heated Restrained Beam (3) T+ΔT T Problem: Determine ΔT at failure T …an Euler buckle will occur at where

8. Thermal Buckling Buckling temperature independent of E Buckling temperature independent of E Buckling expression valid for other end conditions if L interpreted as an effective length Buckling expression valid for other end conditions if L interpreted as an effective length Buckling stable as end displacements defined Buckling stable as end displacements defined Combined yielding-buckling failure possible in reality (as at ambient temperature) Combined yielding-buckling failure possible in reality (as at ambient temperature)

9. Heating of Restrained Beam - Deflections Stocky beam Slender beam Really stocky beam!

10. Heating of Restrained Beam – Axial Force Stocky beam Slender beam Yield

11. Heated Restrained Beam (3) Mechanical strain or temperature Stress Uniform heating then cooling Compression during heating Tension during cooling Finish here! Elastic/plastic

12. Expansion Against Finite Stiffness T+ΔT T Problem: Determine ΔT at failure T K If the stiffness of the support is comparable to the stiffness of the member, the stress produced by thermal expansion will be reduced by a factor of about 2

14. Curvature of Beams - Mechanical R θ d M M Uniform moment, M, produces mechanical curvature Curvature defined as Curvature - a generalised strain

15. Curvature of Beams - Thermal Hot (T 2) Cold (T 1 ) R θ d Thermal gradient produces thermal curvature Uniform thermal gradient in beam with uniform moment Length of hottest fibre Length of coldest fibre

16. Curvature of beams Hot (T 2) Cold (T 2 ) R θ d Uniform thermal gradient in beam with uniform moment Analogous relationship to that for strains

17. Shortening due to Thermal Curvature Interpret shortening due to curvature as a “strain”. From geometry Cold Hot Beam with thermal gradient Note: shortening due to mechanical curvature normally ignored because of high stresses. Large curvature possible with low stresses due to thermal bowing. Problem nonlinear

18. Φ total = 0+Φ thermal Beams with Pure Thermal Gradient Cold Hot Simply supported: Curvature, no moment, contraction, no tension Cold Hot Pin-ended: Deflections, tension, moment P P Φ total = Φ mech +Φ thermal Φ mech -veΦ thermal +ve

19. Beams with Pure Thermal Gradient Cold Hot M M Built-in beam: End moments, moment in beam, no deflections Φ total = Φ mech +Φ thermal =0 Φ mech =-Φ thermal

20. Summary of Results so Far Simple support Pin-supportBuilt-inUniformHeating No (vertical) deflection No force No moment No deflection (or a buckle) Compressive force No moment No deflection (or a buckle) Compressive force No moment Pure thermal gradient Curvature No force No moment Curvature Tensile force Moment No curvature No force Moment

21. Combined Thermal Gradient and Heating Thermal expansion produces expansion strains Thermal expansion produces expansion strains Thermal curvature produces contraction “strains” Thermal curvature produces contraction “strains” Behaviour depends on the interplay between the two effects Behaviour depends on the interplay between the two effects

22. An equivalent effective strain to combine the two thermal effects

23. Combined Thermal Gradient and Heating Increasing Thermal gradient Pin-ended beam Constant centroidal temperature Varying thermal gradient

24. Combined Thermal Gradient and Heating Increasing Thermal gradient Pin-ended beam Constant centroidal temperature Varying thermal gradient

25. Runaway in simple beams  Unrestrained as in furnace tests  Restrained as in large framed structures  Large displacement effects important

26. Runaway in beams

27. Runaway temperatures vs loads

28. Composite beam-slab moment-resisting connections Mean temperature Thermal gradient C C T C T y T C Gravity load EI   T M load  vent 1: local buckling of beam bottom flange 

29. Numerical Modelling of Heated Structures Needed for all but simple structures Needed for all but simple structures Finite element models normal Finite element models normal Some “intermediate” analysis methods exist but limited Some “intermediate” analysis methods exist but limited Challenging! Challenging!

30. Types of Analysis – Heat Transfer Specify temperature of the surface Specify temperature of the surface –Numerically simple – conduction only –Does not require estimates of emissivity and heat transfer coefficient –Useful for modelling experiments Model radiation and convection Model radiation and convection –Numerically complex –Need to estimate parameters – tricky –Normally required for design Model heat flux Model heat flux –Can be useful if using input from a CFD code

31. Descretization – Heat Transfer

32. Types of Analysis - Structural Static Quasi-static Dynamic – implicit or explicit schemes Coupled thermo-mechanical Plasticity Buckling Geometric nonlinearity Creep Inertia effects – e.g collapse Numerically more stable Effects such as spalling Currently a research area INCREASINGCOMPLEXITYINCREASINGCOMPLEXITY

33. Geometric Non-Linearity P If deflections are large, axial forces produced In the beam due to deflection “Catenary” action “Tensile membrane action” in 3-d Geometric non-linearity must be modelled to capture this effect Tension due to deflections

34. Geometric Non-linearity Many numerical codes allows for this Many numerical codes allows for this Must be used for accurate results at high temperature Must be used for accurate results at high temperature Means analyses must be solved incrementally Means analyses must be solved incrementally – therefore take longer and are more demanding

35. Material Behaviour - Ambient Stress Strain Linear or Elastic plastic Often assumed at ambient temperature

36. Material Behaviour – High Temperature Stress Strain Full non-linearity needed Temperature dependence T+

37. Von Mises Yield Surface - Steel

38. Drucker-Prager Yield Surface – Concrete Compression

39. Element Choice Detailed models computationally expensive Detailed models computationally expensive Simply models may miss phenomena Simply models may miss phenomena How to model a beam How to model a beam –Beam elements? –Shell elements –Solid elements? It depends! It depends!

40. Benchmark 1 T T t σ T / σ A σ ε T+ Uniform load 4250N/m Heating 800C 1000C Elastic-plastic material 1m 35mm 75% axial stiffness of beam

41. Purpose of Benchmark 1 Model not “real” but… Model not “real” but… … shows if complex phenomena captured … shows if complex phenomena captured –Non-linear material behaviour  Temperature dependent  Plastic  Thermal expansion –Non-linear geometric behaviour –Boundary conditions important Can be used for demonstrating Can be used for demonstrating –Software capability –Appropriate modelling techniques

42. Benchmark 1 - Deflections Simply-supported (Standard Fire Test) “Runaway”

43. Benchmark 1 - Axial Force Simply-supported (Standard Fire Test) Buckling

44. Effect of BCs on Deflections Simply-supported (Standard Fire Test) “Runaway”

45. Effect of BCs on Axial Force Simply-supported (Standard Fire Test)

46. Effect of Non-linear Geometry on Deflections

47. Aside – Cardington Tests

48. Aside - Cardington Test 1

49. Example Real structure Real structure Based on Cardington test 1 Based on Cardington test 1 –Carefully conducted test on real structure (v. rare) –Has been extensively modelled –Experimental data available Simplified so Simplified so –Precisely defined –Practical to model As challenging as many larger structures As challenging as many larger structures

50. Example

51. Model

52. Example Deflections

53. Example Axial Force