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

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

3. www.nafems.org RMA’s Presentation 3 Beyond FEA! Conventional or Conforming Elastic and Incremental Plastic Copyright © Ramsay Maunder Associates Limited (2004 – 2015). All Rights Reserved

4. www.nafems.org Finite Element Formulations 4 Copyright © Ramsay Maunder Associates Limited (2004 – 2015). All Rights Reserved EFE – (Hybrid) Equilibrium Elements Approximate

5. www.nafems.org Approximations Converge(CFE) 5 NAFEMS Benchmark Challenge No. 1 Copyright © Ramsay Maunder Associates Limited (2004 – 2015). All Rights Reserved

6. www.nafems.org Weak & Strong Equilibrium 6 Copyright © Ramsay Maunder Associates Limited (2004 – 2015). All Rights Reserved

7. 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

8. 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!

9. 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.

10. 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

11. 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

12. www.nafems.org Limit Analysis for Steel 12 Copyright © Ramsay Maunder Associates Limited (2004 – 2015). All Rights Reserved Elastic Solution Plastic Solution

13. www.nafems.org Limit Analysis for Reinforced Concrete 13 40% Unsafe!! Copyright © Ramsay Maunder Associates Limited (2004 – 2015). All Rights Reserved

14. www.nafems.org EFE with Different Yield Criteria 14 Copyright © Ramsay Maunder Associates Limited (2004 – 2015). All Rights Reserved 15% 42%

15. www.nafems.org 15 Copyright © Ramsay Maunder Associates Limited (2004 – 2015). All Rights Reserved

16. www.nafems.org NBC No 2 - Results 16 Copyright © Ramsay Maunder Associates Limited (2004 – 2015). All Rights Reserved

17. 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