The yield-line method for concrete slabs: automated at last. · The yield-line method for concrete...
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The yield-line method for concrete slabs: automated at last. Afternoon Workshop Thursday 26 February 2015 IStructE HQ, Bastwick Street, London
Transcript of The yield-line method for concrete slabs: automated at last. · The yield-line method for concrete...
The yield-line method for concrete slabs: automated at last.
Afternoon WorkshopThursday 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 works
Matthew GilbertUniversity of Sheffield
14:55 - 15:05 Complementary technology: lower boundcomputational analysis
Angus Ramsay Ramsay Maunder Associates
15:05 - 15:35 Tea / coffee break
15:35 - 15:50 Benefits of plastic analysis methods in practical structural assessment
Jon ShaveParsons Brinkerhoff
15:50 - 16:30 Application of the LimitState:SLAB software to slab analysis problems
Tom PritchardLimitState
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 IStructEpresidents)
• Long history of undertaking ‘useful’ research
Dept. of Civil & Structural Engineering
• Research highly rated in recent ‘REF 2014’ quality audit (e.g. 2nd in the UK for ‘research intensity’)
• £81M new building will provide state-of-the-art teaching space:
Ensuring research is usable
‘Valley of death’
Academic research
Industry uptake
Increasing technology readiness
Ava
ilab
ility
of
reso
urc
e
Ensuring research is usable
‘Valley of death’
Academic research
Industry uptake
Increasing technology readiness
Ava
ilab
ility
of
reso
urc
e
Spinout companies
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 software:
Geotechnical analysis software:
Now used by most major UK consultants and contractors, and in over 30 countries worldwide
How the new automated method works
Matthew GilbertUniversity of Sheffield
(and founding Director of LimitState Ltd)
Background & motivation
Background & motivation
Rigid-plastic
Linear elastic
Deflection
Lo
ad
Background & motivation
Rigid-plastic
Linear elastic
Deflection
Lo
ad
Background & motivation
• The finite element method has made linear-elastic analysis convenient and mainstream
But to assess collapse rigid-plasticanalysis tools are much less well developed
Rigid-plastic (‘limit’) analysis
• Used to estimate the maximum load sustainable by a body or structure
• 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 / ‘limit’ load
Deflection
Load
typical actual
response
Collapse analysis: existing tools
More:
• complex
• time consuming
• input parameters
• expertise required
• accurate [potentially at least!]
‘Traditional’:
based on hand
analysis solutions
etc.
‘Advanced’:
based on non-
linear finite
elements etc.
(potentially embedded in simple programs /
spreadsheets etc.)
GAP!
Collapse analysis: existing tools
More:
• complex
• time consuming
• input parameters
• expertise required
• accurate [potentially at least!]
‘Traditional’:
based on hand
analysis solutions
etc.
‘Advanced’:
based on non-
linear finite
elements etc.
(potentially embedded in simple programs /
spreadsheets etc.)
‘Mainstream’:
using numerical
rigid-plastic
analysis?
Collapse analysis: existing tools
More:
• complex
• time consuming
• input parameters
• expertise required
• accurate [potentially at least!]
‘Traditional’:
based on hand
analysis solutions
etc.
‘Advanced’:
based on non-
linear finite
elements etc.
(potentially embedded in simple programs /
spreadsheets etc.)
‘Mainstream’:
using numerical
rigid-plastic
analysis?
Collapse analysis: existing tools
More:
• complex
• time consuming
• input parameters
• expertise required
• accurate [potentially at least!]
‘Traditional’:
based on hand
analysis solutions
etc.
‘Advanced’:
based on non-
linear finite
elements etc.
(potentially embedded in simple programs /
spreadsheets etc.)
‘Mainstream’:
using numerical
rigid-plastic
analysis?
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:
0
5
10
15
20
25
30
35
40
45
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21
Cap
acit
y (t
on
nes
)
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:
• 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)
e.g. H.S.L. Chan, 1972
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
volume
nodal equilibriumbar force
length/yield stress
f
external force
Similarity of formulations
Truss (‘layout optimization’ with self-stress):
Slab (‘discontinuity layout optimization’, DLO):
Similarity of formulations
Truss (‘layout optimization’ with self-stress):
Slab (‘discontinuity layout optimization’, DLO):
volume
nodal equilibrium
bar force
imposed self-stress
length/yield stress
Similarity of formulations
Truss (‘layout optimization’ with self-stress):
Both are simple linear optimization problems
Slab (‘discontinuity layout optimization’, DLO):
volume
nodal equilibrium
bar force
imposed self-stress
length/yield stress
imposed unit displacement
energy
rotation at yield-linenodal sompatibility
length x moment capacity
DLO - nodal compatibility constraint
5
1
0cosi
ii 1
• Rotations at nodes must sum to zero:
• Key feature: compatibility is also implicitly enforced at crossover points
5
1
0sini
ii
node
no node here!
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
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:
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%):
Conclusions
Conclusions
• The yield-line method provides a powerful means of analysingthe 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): www.limitstate.com
