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

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Who are we? 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

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Displacement versus Equilibrium Formulation 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

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Turner et al Constant strain triangle Fraeijs de Veubeke Equilibrium Formulation Teixeira de Freitas & Moitinho de Almeida Hybrid Formulation 1950196019701980199020002010 Ramsay Maunder RMA & EFE Robinson Equilibrium Models Heyman Master Safe Theorem

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

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

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p=0, 4 elements p=1 p=2 100 elements 2500 elements

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Error in Point Displacement EFE1.13% Abaqus (linear) 10.73% Abaqus (quadratic) 1.70%

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Equilibrating boundary tractions Equilibrating model sectioning

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Stress trajectories Thrust lines

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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

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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

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

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

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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

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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

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

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

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

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elastic deflections Bending moments Transverse shear

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basic mechanism based on rigid Morley elements contour lines of a collapse mechanism yield lines of a collapse mechanism

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principal moment vectors recovered in Morley elements (an un-optimised “lower bound” solution)

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M xx M yy M xy biconic yield surface for orthotropic reinforcement

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closed star patch of elements formation of hyperstatic moment fields

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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

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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

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