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

Introduction

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

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

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

Master Safe Theorem

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

Discontinuous Edge Displacements

p=0, 4 elements

p=1 p=2

100 elements 2500 elements

Co-Diffusivity of Stresses

Strong Equilibrium

Error in Point Displacement

EFE 1.13%

Abaqus (linear) 10.73%

Abaqus (quadratic) 1.70%

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

Convergence and Bounds

Presentation of Results (Basic)

Equilibrating boundary tractions

Equilibrating model sectioning

Presentation of Results (Advanced)

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 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)

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

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

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

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

Heyman

Pipers Row car parkcollapsed 4th floor slab - 1997

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 EFEdesigned by McAleer & Rushe Group with zones of reinforcement

Reinforced Concrete Flat Slab

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

principal moments

Moments and Shears

elastic deflections

Bending momentsTransverse shear

Elastic Analysis

basic mechanism based on rigid Morley elements

contour lines of a collapse mechanism

yield lines of a collapse mechanism

Yield-Line Analysis

principal moment vectors recovered in Morley elements

(an un-optimised “lower bound” solution)

Equilibrium from Yield Line Solution

Mxx

Myy

Mxy

biconic yield surface for orthotropic reinforcement

Quadratic constraints & a Linearisation

closed star patch of elements

formation of hyperstatic moment fields

Hyperstatic Variables

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

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

Thank you for your Interest

Any Questions