Transcript Equilibrium Finite Elements

University of Sheffield, 7th September 2009
Angus Ramsay & Edward Maunder
 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
 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
RMA & EFE
Ramsay
Maunder
Turner et al
Constant strain triangle
1950
1960
1970
1980
Robinson
Equilibrium Models
Fraeijs de Veubeke
Equilibrium Formulation
1990
2000
Heyman
Master Safe Theorem
Teixeira de Freitas &
Moitinho de Almeida
Hybrid Formulation
2010
Hybrid equilibrium element
Conventional Displacement element
di
edge
i
gj
node
side/face


Semi-continuous statically admissible
stress fields  = S s
Discontinuous side displacements
=Vv
 Sufficient elements to model geometry
 hp-refinement – local and/or global
 Point displacements/forces inadmissible
 Modelled (more realistically) as line or patch loads
p=1
p=2
p=0, 4 elements
100 elements
2500 elements
Error in Point Displacement
EFE
1.13%
Abaqus (linear)
10.73%
Abaqus (quadratic)
1.70%
axes of symmetry
6. 25
6
5. 75
5. 5
5. 25
5
4. 75
4. 5
4. 25
4
3. 75
3. 5
3. 25
3
2. 75
2. 5
2. 25
2
1. 75
1. 5
1. 25
1
0. 75
0. 5
0. 25
0
0
0. 5
1
1. 5
2
Equilibrating boundary tractions
Equilibrating model sectioning
Stress trajectories
Thrust lines
 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 reanalysis and presentation of results
 Design Optimisation Features
 Model parameters form variables, structural response
forms objectives and constraints
 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
Analyses
• elastic analysis
• upper-bound limit analysis
Demonstrate
• real-time capabilities
Equal isotropic reinforcement top and bottom
• post-processing features
Simply Supported along three edges
Corner column
UDL
• geometric optimisation
Axis of rotation
Axis of symmetry
Analyses
• elastic analysis
Angular velocity
Blade Load
Geometric master variable
Geometric slave variables
Demonstrate
• geometric variables
• design optimisation
Objective – minimise mass
Constraint – burst speed margin
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
•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
 EFE: Equilibrium Finite Elements
 Morley constant moment element to hybrid equilibrium
elements of general degree
Morley
general hybrid
RC flat slab – plan
geometrical model in EFE
designed by McAleer & Rushe Group
with zones of reinforcement
principal moments
principal moment vectors of a linear elastic
reference solution: statically admissible –
elements of degree 4
principal shears
elastic deflections
Bending moments
Transverse shear
basic mechanism based on rigid
Morley elements
contour lines of a
collapse mechanism
yield lines of a collapse
mechanism
principal moment vectors recovered in Morley elements
(an un-optimised “lower bound” solution)
Mxy
Myy
Mxx
biconic yield surface for orthotropic reinforcement
closed star patch of
elements
formation of hyperstatic
moment fields
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
 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