Equilibrium Finite Elements for the Safe Limit Analysis of
Plates Angus Ramsay Ramsay Maunder Associates Limited
Slide 2
www.nafems.org 2 Copyright Ramsay Maunder Associates Limited
(2004 2015). All Rights Reserved Formed 2004, Incorporated 2009
Consultants (SimGov, Verification) Research & Development Limit
Analysis of Slabs and Plates
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www.nafems.org RMAs Presentation 3 Beyond FEA! Conventional or
Conforming Elastic and Incremental Plastic Copyright Ramsay Maunder
Associates Limited (2004 2015). All Rights Reserved
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www.nafems.org Finite Element Formulations 4 Copyright Ramsay
Maunder Associates Limited (2004 2015). All Rights Reserved EFE
(Hybrid) Equilibrium Elements Approximate
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www.nafems.org Approximations Converge(CFE) 5 NAFEMS Benchmark
Challenge No. 1 Copyright Ramsay Maunder Associates Limited (2004
2015). All Rights Reserved
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www.nafems.org Weak & Strong Equilibrium 6 Copyright Ramsay
Maunder Associates Limited (2004 2015). All Rights Reserved
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www.nafems.org Nodal Equilibrium (CFE) 7 Copyright Ramsay
Maunder Associates Limited (2004 2015). All Rights Reserved Axial
Stress from CFE Max Stress =9.1MPa FSUM => M=0.6Nm => Max
Stress=10MPa
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www.nafems.org Sleipner A - FE Malpractice? 8 Copyright Ramsay
Maunder Associates Limited (2004 2015). All Rights Reserved Crude
FE model (single element through thickness) led to a near 50%
underestimate of the shear stress which was then used to design the
structure!!! No loss of life thankfully but financial cost >
$400M!
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www.nafems.org Summary Weak equilibrium of CFEs means that
caution is required in interpreting results. Nodal forces are in
equilibrium with applied loads, FE stresses are not in equilibrium
unless the model has been refined sufficiently. 9 Copyright Ramsay
Maunder Associates Limited (2004 2015). All Rights Reserved Strong
equilibrium of EFEs means that stresses are always in equilibrium
with the applied load irrespective of mesh refinement and can
therefore be safely used in the design process.
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www.nafems.org Limit Analysis 10 Copyright Ramsay Maunder
Associates Limited (2004 2015). All Rights Reserved Hence the load
factor is bounded as follows: Plasticity Theorems: Unsafe Safe
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www.nafems.org Incremental Plasticity 11 Rotation Moment
Elastic Partially Plastic Fully Plastic Limit Analysis Solution
Copyright Ramsay Maunder Associates Limited (2004 2015). All Rights
Reserved
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www.nafems.org Limit Analysis for Steel 12 Copyright Ramsay
Maunder Associates Limited (2004 2015). All Rights Reserved Elastic
Solution Plastic Solution
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www.nafems.org Limit Analysis for Reinforced Concrete 13 40%
Unsafe!! Copyright Ramsay Maunder Associates Limited (2004 2015).
All Rights Reserved
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www.nafems.org EFE with Different Yield Criteria 14 Copyright
Ramsay Maunder Associates Limited (2004 2015). All Rights Reserved
15% 42%
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www.nafems.org 15 Copyright Ramsay Maunder Associates Limited
(2004 2015). All Rights Reserved
Slide 16
www.nafems.org NBC No 2 - Results 16 Copyright Ramsay Maunder
Associates Limited (2004 2015). All Rights Reserved
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www.nafems.org Closure Limit Analysis enables the engineer to
quantify the reserve strength and even use some of it! EFE allows
the engineer to do this safely! http://www.ramsay-maunder.co.uk/
Thank you for Listening!! 17 Copyright Ramsay Maunder Associates
Limited (2004 2015). All Rights Reserved