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

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

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

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

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

Master Safe Theorem

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

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

Co-Diffusivity of Stresses

Error in Point Displacement Strong Equilibrium Error in Point Displacement EFE 1.13% Abaqus (linear) 10.73% Abaqus (quadratic) 1.70%

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……….!

Convergence and Bounds

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

Presentation of Results (Advanced) Stress trajectories Thrust lines

(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

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

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

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

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

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

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

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

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

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

Elastic Analysis elastic deflections Transverse shear Bending moments

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

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

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

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

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

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

Thank you for your Interest Any Questions