University of Sheffield, 7 th September 2009 Angus Ramsay & Edward Maunder.
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Transcript of University of Sheffield, 7 th September 2009 Angus Ramsay & Edward Maunder.
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Equilibrium Finite Elements
University of Sheffield, 7th 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
Introduction
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Contents 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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A Time Line for Equilibrium Elements
Turner et alConstant strain triangle
Fraeijs de VeubekeEquilibrium Formulation
Teixeira de Freitas &Moitinho de Almeida Hybrid Formulation
1950 1960 1970 1980 1990 2000 2010
Ramsay
Maunder
RMA & EFE
RobinsonEquilibrium Models
HeymanMaster Safe Theorem
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Displacement versus Equilibrium Elements
Discontinuous side displacements = V v
Semi-continuous statically admissible stress fields = S s
edge
node gj
di
i
side/face
Hybrid equilibrium elementConventional Displacement element
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Master Safe Theorem
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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
Modelling with Equilibrium Elements
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Discontinuous Edge Displacements
p=0, 4 elements
p=1 p=2
100 elements 2500 elements
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Co-Diffusivity of Stresses
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Strong Equilibrium
Error in Point Displacement
EFE 1.13%
Abaqus (linear) 10.73%
Abaqus (quadratic) 1.70%
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Heyman
0
0.25
0.5
0.75
1
1.25
1.5
1.75
2
2.25
2.5
2.75
3
3.25
3.5
3.75
4
4.25
4.5
4.75
5
5.25
5.5
5.75
6
6.25
0 0.5 1 1.5 2
axes of symmetry
Sleipner Collapse (1991)Computer Aided Catastrophe
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Convergence and Bounds
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Presentation of Results (Basic)
Equilibrating boundary tractions
Equilibrating model sectioning
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Presentation of Results (Advanced)
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
EFE the software(a vehicle for exploiting equilibrium elements)
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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
Program Characteristics
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Landing Slab
Demonstrate
• real-time capabilities
• post-processing features
• geometric optimisation
Analyses
• elastic analysis
• upper-bound limit analysis
Equal isotropic reinforcement top and bottomSimply Supported along three edges
Corner column
UDL
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Axial Turbine Disc
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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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
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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
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Heyman
Pipers Row car parkcollapsed 4th floor slab - 1997
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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 EFEdesigned by McAleer & Rushe Group with zones of reinforcement
Reinforced Concrete Flat Slab
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principal moment vectors of a linear elastic reference solution: statically admissible – elements of degree 4 principal shears
principal moments
Moments and Shears
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elastic deflections
Bending momentsTransverse shear
Elastic Analysis
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basic mechanism based on rigid Morley elements
contour lines of a collapse mechanism
yield lines of a collapse mechanism
Yield-Line Analysis
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principal moment vectors recovered in Morley elements
(an un-optimised “lower bound” solution)
Equilibrium from Yield Line Solution
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Mxx
Myy
Mxy
biconic yield surface for orthotropic reinforcement
Quadratic constraints & a Linearisation
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closed star patch of elements
formation of hyperstatic moment fields
Hyperstatic Variables
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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
Moment redistribution in a column zone
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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
Future developments for lower bounds
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Thank you for your Interest
Any Questions