The yield-line method for concrete slabs: automated at last. · The yield-line method for concrete slabs: automated at last. ... 2. Lower bound solutions • The yield-line ... •

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