The yield-line method for concrete slabs: automated at last. Afternoon Workshop Thursday 26 February 2015 IStructE HQ, Bastwick Street, London
Programme 14:00 - 14:15 Arrival / tea and coffee 14:15 - 14:55 Event welcome / How the new automated method Matthew Gilbert University of Sheffield works 14:55 - 15:05 Complementary technology: lower bound Angus Ramsay computational analysis Ramsay Maunder Associates 15:05 - 15:35 Tea / coffee break 15:35 - 15:50 Benefits of plastic analysis methods in practical Jon Shave structural assessment Parsons Brinkerhoff 15:50 - 16:30 Application of the LimitState:SLAB software to slab Tom Pritchard analysis problems LimitState 16:30 - 17:00 Panel discussion. Panel: John Morrison ( Buro Happold ) and workshop speakers.
Welcome!
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
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:
Ensuring research is usable Availability of resource Academic Industry research uptake ‘Valley of death’ Increasing technology readiness
Ensuring research is usable Spinout companies Availability of resource Academic Industry research uptake ‘Valley of death’ Increasing technology readiness
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
Existing LimitState products Masonry arch bridge analysis Geotechnical analysis software: software: Now used by most major UK consultants and contractors, and in over 30 countries worldwide
How the new automated method works Matthew Gilbert University of Sheffield (and founding Director of LimitState Ltd)
Background & motivation
Background & motivation Linear elastic Load Rigid-plastic Deflection
Background & motivation Linear elastic Load Rigid-plastic Deflection
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
Rigid- plastic (‘limit’) analysis • Used to estimate the maximum load sustainable by a body or structure collapse / ‘limit’ load 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
Collapse analysis: existing tools ‘Traditional’ : based on hand (potentially embedded in simple programs / spreadsheets etc.) analysis solutions etc. GAP! More: • complex • time consuming ‘Advanced’ : • input parameters based on non- linear finite • expertise required elements etc. • accurate [potentially at least!]
Collapse analysis: existing tools ‘Traditional’ : based on hand (potentially embedded in simple programs / spreadsheets etc.) analysis solutions etc. ‘Mainstream’ : using numerical rigid-plastic More: analysis? • complex • time consuming ‘Advanced’ : • input parameters based on non- linear finite • expertise required elements etc. • accurate [potentially at least!]
Collapse analysis: existing tools ‘Traditional’ : based on hand (potentially embedded in simple programs / spreadsheets etc.) analysis solutions etc. ‘Mainstream’ : using numerical rigid-plastic More: analysis? • complex • time consuming ‘Advanced’ : • input parameters based on non- linear finite • expertise required elements etc. • accurate [potentially at least!]
Collapse analysis: existing tools ‘Traditional’ : based on hand (potentially embedded in simple programs / spreadsheets etc.) analysis solutions etc. ‘Mainstream’ : using numerical rigid-plastic More: analysis? • complex • time consuming ‘Advanced’ : • input parameters based on non- linear finite • expertise required elements etc. • accurate [potentially at least!]
The yield-line method
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
Calculations (work method) • Equate internal and external work (for chosen yield-line pattern) (from Kennedy & Goodchild, 2004)
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
Renewed interest in the 1990s & 2000s • Middleton and co-workers showed many concrete bridges appeared to have ‘hidden reserves’ of strength: 45 40 35 Capacity (tonnes) 30 25 20 15 10 5 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 Elastic assessment Plastic assessment
Renewed interest in the 1990s & 2000s • The Cardington European Concrete Building Project indicated that yield-line design brought benefits when used in design
‘Yield line design is so easy…. …once you know what you are doing!’ Foreward, Practical Yield Line Design, Kennedy & Goodchild, 2004
‘Yield line design is so easy…. …once you know what you are doing!’ Foreward, Practical Yield Line Design, Kennedy & Goodchild, 2004
Automating the yield-line method
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)
But another method can identify ‘fans’… • Truss ‘layout optimization’ (Dorn et al, 1964) :
But another method can identify ‘fans’… • Modified ‘self - stress’ truss layout optimization:
Truss layout optimization: formulation volum ume length gth/y /yie ield ld stress ss bar ar force rce f nodal al equili ilibrium rium externa nal l force ce
Similarity of f ormulations Truss (‘layout optimization’ Slab (‘discontinuity layout with self-stress): optimization’, DLO):
Similarity of f ormulations Truss (‘layout optimization’ Slab (‘discontinuity layout with self-stress): optimization’, DLO): length gth/y /yie ield ld stress ss volum ume bar ar force rce nodal al equili ilibrium rium impose posed d self-st stre ress ss
Similarity of f ormulations Truss (‘layout optimization’ Slab (‘discontinuity layout with self-stress): optimization’, DLO): length gth x momen ment t energy gy length gth/y /yie ield ld stress ss volum ume capa pacity city rotation ation at bar ar force rce yield ld-line line nodal al equili ilibrium rium nodal al sompa mpatib ibili ility ty impose posed d self-st stre ress ss impose posed d unit displaceme lacement nt Both are simple linear optimization problems
DLO - nodal compatibility constraint • Rotations at nodes must 5 sum to zero: cos 0 node i i 1 i 1 5 sin 0 i i 1 i no node here! • Key feature: compatibility is also implicitly enforced at crossover points
DLO – treating applied loads
Examples
Example 1: Fixed square slab Analytical solution available: = 42.851 • (Fox, Phil. Trans. Roy. Soc, 1974) Best DLO solution: = 42.857, which is • just 0.01% higher (Gilbert et al., Proc. Roy. Soc, 2014)
Example 1: Fixed square slab (cont.) Power law extrapolation gives 5 digit agreement with analytical solution
Example 2: Indented slab Best literature solution: = 29.2 • (Jackson, PhD Thesis, Cambridge University, 2010) Best DLO solution: = 28.988 • (Gilbert et al., Proc. Roy. Soc, 2014)
Example 2: Indented slab (cont.) Simplified collapse patterns can also be obtained, e.g. to facilitate validation via hand calculations: Increasing simplification
Example 3: Apartment
Example 3: Apartment (from Kennedy & Goodchild, 2004)
Example 3: Apartment
Refinements
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