1. Equilibrium Finite Elements
University of Sheffield, 7th September 2009
Angus Ramsay & Edward Maunder

2. Introduction Who are we? What is our aim? How do we realise our aims?
Partners in RMA
Fellows of University of Exeter
What is our aim?
Safe structural analysis and design optimisation
How do we realise our aims?
EFE an Equilibrium Finite Element system

3. Contents Displacement versus Equilibrium Formulation EFE the Software
Theoretical
Practical
EFE the Software
Features of the software
Live demonstration of software
Design Optimisation - a Bespoke Application
Recent Research at RMA
Plates – upper/lower bound limit analysis

4. A Time Line for Equilibrium Elements
RMA & EFE
Ramsay
Turner et al
Constant strain triangle
Maunder
1950
1960
1970
1980
1990
2000
2010
Heyman
Master Safe Theorem
Robinson
Equilibrium Models
Teixeira de Freitas &
Moitinho de Almeida
Hybrid Formulation
Fraeijs de Veubeke
Equilibrium Formulation

5. Displacement versus Equilibrium Elements
Conventional Displacement element
Hybrid equilibrium element  
Semi-continuous statically admissible stress fields  = S s
Discontinuous side displacements  = V v

6. Master Safe Theorem

7. Modelling with Equilibrium Elements
Sufficient elements to model geometry
hp-refinement – local and/or global
Point displacements/forces inadmissible
Modelled (more realistically) as line or patch loads

8. Discontinuous Edge Displacements
p=0, 4 elements
2500 elements
100 elements

9. Co-Diffusivity of Stresses

10. Error in Point Displacement
Strong Equilibrium
Error in Point Displacement
EFE 1.13%
Abaqus (linear) %
Abaqus (quadratic) 1.70%

11. Computer Aided Catastrophe
Heyman
Sleipner Collapse (1991)
Computer Aided Catastrophe
1.      The offshore platform Sleipner A (1991) is a classic example of what can go wrong when stress equilibrium is violated due to using too coarse a mesh– CAC (computer assisted catastrophe). It is said that shear forces in the tricell walls were underestimated by some 45% - as a result of basing shear forces on samples of shear stresses in a model which used only single elements to represent the thickness of a wall. If only they had used the information available in the form of nodal forces……….!

12. Convergence and Bounds

13. Presentation of Results (Basic)
Equilibrating boundary tractions
Equilibrating model sectioning

14. Presentation of Results (Advanced)
Stress trajectories
Thrust lines

15. (a vehicle for exploiting equilibrium elements)
EFE the software
(a vehicle for exploiting equilibrium elements)
Geometry based modelling
Properties, loads etc applied to geometry rather than mesh
Direct access to quantities of engineering interest
Numerical and graphical
Real-Time Analysis Capabilities
Changes to model parameters immediately prompts re-analysis and presentation of results
Design Optimisation Features
Model parameters form variables, structural response forms objectives and constraints

16. Program Characteristics
Written in Compaq Visual Fortran (F90 + IMSL)
the engineers programming language
Number of subroutines/functions > 4000
each routine approx single A4 page – verbose style
Number of calls per subroutine > 3
non-linear, good utilisation, potential for future development
Number of dialogs > 300
user-friendly
Basic graphics (not OpenGL or similar – yet!)
adequate for current demands

17. Landing Slab Analyses elastic analysis upper-bound limit analysis
Demonstrate
real-time capabilities
post-processing features
geometric optimisation
Equal isotropic reinforcement top and bottom
Simply Supported along three edges
Corner column
UDL

18. Axial Turbine Disc Analyses Axis of rotation elastic analysis
Axis of symmetry
Angular velocity
Blade Load
Geometric master variable
Geometric slave variables
Demonstrate
geometric variables
design optimisation
Objective – minimise mass
Constraint – burst speed margin

19. Bespoke Applications Geometry: Disc outer radius = 0.05m
Disc axial extent = 0.005m
Loading:
Speed = 41,000 rev/min
Number of blades = 21
Mass per blade = 1.03g
Blade radius = .052m
Material = Aluminium Alloy
Results:
Burst margin = 1.41
Fatigue life = 20,000 start-stop cycles

20. Limit Analyses for Flat Slabs
Flat slabs – assessment of ULS
Johansen’s yield line & Hillerborg’s strip methods
Limit analyses exploiting equilibrium models & finite elements
Application to a typical flat slab and its column zones
Future developments

21. collapsed 4th floor slab - 1997
Heyman
Pipers Row car park
collapsed 4th floor slab

22. EFE: Equilibrium Finite Elements
Morley constant moment element to hybrid equilibrium elements of general degree
Morley general hybrid

23. Reinforced Concrete Flat Slab
RC flat slab – plan geometrical model in EFE
designed by McAleer & Rushe Group with zones of reinforcement

24. Moments and Shears principal moments principal shears
principal moment vectors of a linear elastic reference solution: statically admissible – elements of degree 4
principal shears

25. Elastic Analysis
elastic deflections
Transverse shear
Bending moments

26. Yield-Line Analysis basic mechanism based on rigid Morley elements
contour lines of a collapse mechanism
yield lines of a collapse mechanism

27. Equilibrium from Yield Line Solution
principal moment vectors recovered in Morley elements
(an un-optimised “lower bound” solution)

28. Quadratic constraints & a Linearisation
Mxx
Myy
Mxy
biconic yield surface for orthotropic reinforcement

29. Hyperstatic Variables
closed star patch of elements
formation of hyperstatic moment fields

30. Moment redistribution in a column zone
moments direct from yield line analysis: upper bound  = 27.05, “lower bound”  = 9.22
optimised redistribution of moments based on biconic yield surfaces:   21.99

31. Future developments for lower bounds
Refine the equilibrium elements for lower bound optimisation, include shear forces
Initiate lower bound optimisation from an equilibrated linear elastic reference solution & incorporate EC2 constraints e.g.  30% moment redistribution
Use NLP to exploit the quadratic nature of the yield constraints for moments
Extend the basis of hyperstatic moment fields
Incorporate shear into yield criteria
Incorporate flexible columns and membrane forces

32. Thank you for your Interest Any Questions