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Equilibrium Finite Elements for the Safe Limit Analysis of Plates Angus Ramsay Ramsay Maunder Associates Limited

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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

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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