The yield-line method for concrete slabs: automated at last. Afternoon Workshop Thursday 26 February 2015 IStructE HQ, Bastwick Street, London

Transcription

1 The yield-line method for concrete slabs: automated at last. Afternoon Workshop Thursday 26 February 2015 IStructE HQ, Bastwick Street, London

2 Programme 14:00-14:15 Arrival / tea and coffee 14:15-14:55 Event welcome / How the new automated method works Matthew Gilbert University of Sheffield 14:55-15:05 Complementary technology: lower bound computational analysis 15:05-15:35 Tea / coffee break 15:35-15:50 Benefits of plastic analysis methods in practical structural assessment 15:50-16:30 Application of the LimitState:SLAB software to slab analysis problems Angus Ramsay Ramsay Maunder Associates Jon Shave Parsons Brinkerhoff Tom Pritchard LimitState 16:30-17:00 Panel discussion. Panel: John Morrison (Buro Happold) and workshop speakers.

3 Welcome!

4 Dept. of Civil & Structural Engineering One of the largest civil engineering departments in the UK Alma mater to many prominent engineers (incl. many past IStructE presidents) Long history of undertaking useful research

5 Dept. of Civil & Structural Engineering Research highly rated in recent REF 2014 quality audit (e.g. 2 nd in the UK for research intensity ) 81M new building will provide state-of-the-art teaching space:

6 Availability of resource Ensuring research is usable Academic research Industry uptake Valley of death Increasing technology readiness

7 Availability of resource Ensuring research is usable Spinout companies Academic research Industry uptake Valley of death Increasing technology readiness

8 LimitState Ltd Spun-out from University in 2006 Commercialising academic research: Providing engineers with powerful software for ultimate limit state analysis & design Taking advantage of state-of-the-art algorithms & optimization technology Ensuring software is robust and well validated Adding value: Ensuring applications are fully supported and are easy to use

9 Existing LimitState products Masonry arch bridge analysis software: Geotechnical analysis software: Now used by most major UK consultants and contractors, and in over 30 countries worldwide

10 How the new automated method works Matthew Gilbert University of Sheffield (and founding Director of LimitState Ltd)

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13 Background & motivation

14 Load Background & motivation Linear elastic Rigid-plastic Deflection

15 Load Background & motivation Linear elastic Rigid-plastic Deflection

16 Background & motivation The finite element method has made linear-elastic analysis convenient and mainstream But to assess collapse rigid-plastic analysis tools are much less well developed

17 Load Rigid-plastic ( limit ) analysis Used to estimate the maximum load sustainable by a body or structure collapse / limit load typical actual response Deflection Benefits (cf. elastic methods for ultimate analysis): Tend to lead to more economic solutions when used in design Can reveal hidden reserves of strength when used in assessment

18 Collapse analysis: existing tools Traditional : based on hand analysis solutions etc. (potentially embedded in simple programs / spreadsheets etc.) More: complex time consuming input parameters expertise required accurate [potentially at least!] GAP! Advanced : based on nonlinear finite elements etc.

19 Collapse analysis: existing tools Traditional : based on hand analysis solutions etc. (potentially embedded in simple programs / spreadsheets etc.) More: complex time consuming input parameters expertise required accurate [potentially at least!] Mainstream : using numerical rigid-plastic analysis? Advanced : based on nonlinear finite elements etc.

20 Collapse analysis: existing tools Traditional : based on hand analysis solutions etc. (potentially embedded in simple programs / spreadsheets etc.) More: complex time consuming input parameters expertise required accurate [potentially at least!] Mainstream : using numerical rigid-plastic analysis? Advanced : based on nonlinear finite elements etc.

21 Collapse analysis: existing tools Traditional : based on hand analysis solutions etc. (potentially embedded in simple programs / spreadsheets etc.) More: complex time consuming input parameters expertise required accurate [potentially at least!] Mainstream : using numerical rigid-plastic analysis? Advanced : based on nonlinear finite elements etc.

22 The yield-line method

23 The yield-line method of analysis The term yield-line was first coined by Ingerslev, in the first ever paper to appear in The Structural Engineer Johansen then developed the theory underpinning the method Later shown that the yield-line method is an upper bound plastic analysis method

24 Calculations (work method) Equate internal and external work (for chosen yield-line pattern) (from Kennedy & Goodchild, 2004)

25 Pros and cons of the yield-line method Pros: Simple, direct, estimate of the collapse load Leads to economical designs (and/or realistic assessments of capacity of existing slabs) Cons: Non-conservative (unsafe) if incorrect mechanism chosen Only considers flexural failure

26 Capacity (tonnes) Renewed interest in the 1990s & 2000s Middleton and co-workers showed many concrete bridges appeared to have hidden reserves of strength: Elastic assessment Plastic assessment

27 Renewed interest in the 1990s & 2000s The Cardington European Concrete Building Project indicated that yield-line design brought benefits when used in design

28 Yield line design is so easy. once you know what you are doing! Foreward, Practical Yield Line Design, Kennedy & Goodchild, 2004

29 Yield line design is so easy. once you know what you are doing! Foreward, Practical Yield Line Design, Kennedy & Goodchild, 2004

30 Automating the yield-line method

31 Automating the yield-line method Element based formulations have been tried: e.g. H.S.L. Chan, 1972 But solutions highly dependent on element topology! Better solutions via geometry optimization (moving nodes), but e.g. fan mechanisms could still not be identified (e.g. Johnson 1994)

32 But another method can identify fans Truss layout optimization (Dorn et al, 1964):

33 But another method can identify fans Modified self-stress truss layout optimization:

34 Truss layout optimization: formulation volume length/yield stress nodal equilibrium f bar force external force

35 Similarity of formulations Truss ( layout optimization with self-stress): Slab ( discontinuity layout optimization, DLO):

36 Similarity of formulations Truss ( layout optimization with self-stress): Slab ( discontinuity layout optimization, DLO): volume nodal equilibrium length/yield stress bar force imposed self-stress

37 Similarity of formulations Truss ( layout optimization with self-stress): Slab ( discontinuity layout optimization, DLO): volume nodal equilibrium length/yield stress energy bar force nodal sompatibility length x moment capacity rotation at yield-line imposed self-stress imposed unit displacement Both are simple linear optimization problems

38 DLO - nodal compatibility constraint Rotations at nodes must sum to zero: node 1 5 i 1 i cos i 0 5 i 1 i sin i 0 Key feature: compatibility is also implicitly enforced at crossover points no node here!

39 DLO treating applied loads

40 Examples

41 Example 1: Fixed square slab Analytical solution available: = (Fox, Phil. Trans. Roy. Soc, 1974) Best DLO solution: = , which is just 0.01% higher (Gilbert et al., Proc. Roy. Soc, 2014)

42 Example 1: Fixed square slab (cont.) Power law extrapolation gives 5 digit agreement with analytical solution

43 Example 2: Indented slab Best literature solution: = 29.2 (Jackson, PhD Thesis, Cambridge University, 2010) Best DLO solution: = (Gilbert et al., Proc. Roy. Soc, 2014)

44 Example 2: Indented slab (cont.) Simplified collapse patterns can also be obtained, e.g. to facilitate validation via hand calculations: Increasing simplification

45 Example 3: Apartment

46 Example 3: Apartment (from Kennedy & Goodchild, 2004)

47 Example 3: Apartment

48 Refinements

49 1. Tidying up yield-line patterns In the automated method, yield-lines must terminate at nodes on a predefined grid Post-processing the solution using geometry optimization gives even clearer yield-line patterns, e.g:

50 2. Lower bound solutions The yield-line method provides upper bound solutions The automated yield-line method provides solutions which are for engineering purposes exact However, for completeness, a lower bound solution can be obtained (e.g. see Ramsay presentation) Example problem, from Benchmark article (gap = 0.2%):

51 Conclusions

52 Conclusions The yield-line method provides a powerful means of analysing the ultimate (collapse) limit state However, the lack of a general implementation has limited usage in recent years The yield-line method has been automated via discontinuity layout optimization (DLO) : Typically involves linear optimization (easy to solve) Fan type mechanisms (and others) identified automatically Automated yield-line analysis software is now available for use in industry (and free for academic use):