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  • STRUCTURAL OPTIMISATION

    IN BUILDING DESIGN PRACTICE:

    CASE-STUDIES IN TOPOLOGY OPTIMISATION

    OF BRACING SYSTEMS

    Robert Baldock

    Corpus Christi College

    June 2007

    A dissertation submitted for the Degree of Doctor of Philosophy

    Cambridge University Engineering Department

  • Declaration

    Except where otherwise stated, this thesis is the result of my own research and does not

    include the outcome of work done in collaboration.

    This thesis has not been submitted in whole or in part for consideration for any other

    degree of qualification at the University or any other institute of learning.

    The thesis contains 49 figure, 14 tables and less than 42,000 words.

    Robert Baldock

    Corpus Christi College

    Cambridge

    June 2007

  • Abstract

    Keywords: structural topology optimisation, structural design practice, bracing

    design, Evolutionary Structural Optimisation, Pattern Search, Optimality Criteria,

    Genetic Programming, computer-aided design, large-scale structural size optimisation

    This thesis aims to contribute to the reduction of the significant gap between the state-

    of-the-art of structural design optimisation in research and its practical application in

    the building industry. The research has focused on structural topology optimisation,

    investigating three distinct methods through the common example of bracing design

    for lateral stability of steel building frameworks. The research objective has been

    aided by collaboration with structural designers at Arup.

    It is shown how Evolutionary Structural Optimisation can be adapted to improve

    applicability to practical bracing design problems by considering symmetry

    constraints, rules for element removal and addition, as well as the definition of element

    groups to enable inclusion of aesthetic requirements. Size optimisation is added in the

    optimisation method to improve global optimality of solutions.

    A modified Pattern Search algorithm is developed, suitable for the parameterised,

    grid-based, topological design problem of a live, freeform tower design project. The

    alternative objectives of minimising bracing member piece count or bracing volume

    are considered alongside an efficient simultaneous size and topology optimisation

    approach, through integration of an Optimality Criteria method. A range of alternative

    optimised designs, suitable for assessment according to unmodelled criteria, are

    generated by stochastic search, parametric studies and changes in the initial design.

    This study is significant in highlighting practical issues in the application of structural

    optimisation in the building industry.

    A Genetic Programming formulation is presented, using design modification operators

    as modular "programmes", and shown to be capable of synthesising a range of novel,

    optimally-directed designs. The method developed consistently finds the global

    optimum for a small 2D planar test problem, generates high-performance designs for

    larger scale tasks and shows the potential to generate designs meeting user-defined

    aesthetic requirements.

    The research and results presented contribute to establishing a structural optimisation

    toolbox for design practice, demonstrating necessary method extensions and

    considerations and practical results that are directly applicable to building projects.

  • Acknowledgements

    I wish to thank my academic supervisor, Kristina Shea, for her dedicated support,

    guidance and encouragement throughout the course of this project. I am also greatly

    indebted to Geoff Parks for his efficiency and advice in the role of advisor and

    subsequently as administrative supervisor. Thanks are due to Marina Gourtovaia and

    Andrew Flintham for their valuable assistance in computing matters and to all my

    friends and colleagues in the Engineering Design Centre, Cambridge for many

    stimulating discussions.

    The collaboration with Arup has been fundamental to this research. I therefore wish to

    express my sincere gratitude to Ed Clark and Alvise Simondetti, as industrial

    supervisors, as well as Damian Eley, Chris Neighbour, Steve McKechnie, Martin Holt,

    Colin Jackson, Jan-Peter Koppitz, Chris Carroll, Pat Dallard and Peter Young, all of

    whom generously gave time to aid me in various aspects of this project. Additionally,

    the support of Chris Kaethner and Stephen Hendry, in relation to Oasys GSA, has been

    very beneficial.

    I could not have completed this thesis without the fantastic friends who have inspired,

    distracted and kept me sane.

    I have been blessed with loving and loyal parents who have supported me from my

    first steps to the conclusion of this thesis. I owe them the greatest thanks of all.

    This research has been made possible through funding by the Engineering and Physical

    Sciences Research Council and an Industrial CASE studentship from Arup. Additional

    financial support from Cambridge University Engineering Department, Corpus Christi

    College, Cambridge and the Royal Commission for the Exhibition of 1851 is also

    gratefully acknowledged.

  • Contents

    1. INTRODUCTION .. 1

    1.1. The nature of design optimisation . 1

    1.2. Optimisation of structures . 3

    1.3. The design process for building structures ... 5

    1.4. Drivers and barriers for structural optimisation in the building industry 7

    1.5. Summary of research contributions .. 10

    1.6. Thesis structure . 11

    2. STATE-OF-THE-ART: RESEARCH AND PRACTICE OF DESIGN

    OPTIMISATION IN STRUCTURAL ENGINEERING ... 12

    2.1. Structural design optimisation research 12

    2.1.1. Section-size optimisation ... 14

    Optimality Criteria 14

    Mathematical Programming . 14

    Fully Stressed Design 15

    Additional considerations .. 15

    2.1.2. Discrete topology optimisation methods 16

    Ground structure approach 16

    Ruled-based approaches 18

    2.1.3. Evolutionary Algorithms in topology optimisation 19

    Genetic Algorithms ....... 19

    Genetic Programming 20

    Evolutionary Strategies .. 21

    Evolutionary Programming .... 21

    2.1.4. Continuum-based optimisation methods . 21

    Homogenisation .. 22

    Bubble method 22

  • Evolutionary Structural Optimisation 22

    2.1.5. Computer-based conceptual design methods .. 24

    2.2. Optimisation in building engineering design practice 25

    2.2.1. Comparison of structural design in the automotive and aeronautical

    industries versus the building industry 25

    2.2.2. Commercial optimisation software .. 27

    2.2.3. Published literature on industrial applications . 29

    Section-size optimisation 29

    Evolutionary Structural Optimisation (ESO) .. 30

    Parametric optimisation .. 31

    Non-parametric optimisation .. 31

    2.2.4. Facilitating structural optimisation .. 32

    Software .. 32

    Parametric optimisation case studies .. 33

    Non-parametric discrete optimisation and design generation case studies

    . 33

    2.3. Conclusions .. 34

    2.4. Justification of case study 35

    2.5. Context of research contributions ... 37

    Evolutionary Structural Optimisation . 37

    Pattern Search and Optimality Criteria ... 37

    Genetic Programming using design modification operators ... 38

    3. CONTINUUM TOPOLOGY OPTIMISATION OF BRACED STEEL FRAMES

    .. 40

    3.1. Introduction .. 40

    3.2. Background .. 40

    4.7.1 Method overview .. 40

    3.2.1. Addition considerations and extensions ... 41

  • 3.2.2. Evolutionary Structural Optimisation (ESO) for stiffness and

    displacement constraints .. 43

    3.2.3. Bi-directional Evolutionary Structural Optimisation (BESO) 45

    3.3. Benchmark problem: structural model specifications 45

    3.4. Optimisation for minimal mean compliance .. 46

    3.5. Optimisation for displacement constraint .. 47

    3.6. Including optimisation of domain thickness ... 53

    3.7. Including architectural requirements and pattern definition ... 58

    3.8. Discrete interpretation of continuum topologies . 60

    3.9. Conclusions . 64

    3.10. Guidelines for practical use 64

    4. BRACING TOPOLOGY AND SECTION-SIZE OPTIMISATION BY A HYBRID

    ALGORITHM: AN INDUSTRIAL CASE-STUDY ... 67

    4.1. Introduction 67

    4.2. Background 68

    4.4.1. Overview of studies 69

    4.3. Design task definition . 69

    4.3.1. Structural models 69

    4.3.2. Topology optimisation models 72

    Optimisation model A 72

    Optimisation model B 72

    4.4. Pattern Search method 73

    4.5. Live project optimisation 75

    4.5.1. Topology optimisation by Modified Pattern Search ... 75

    4.5.2. Parametric studies 76

    4.5.3. Outline proposals . 77

    4.6. Characterisation of design space . 78

    4.7. Topology optimisation method development . 80

    4.7.1. Objective function formulation ... 83

  • Formulation 1 . 83

    Formulation 2 . 83

    4.7.2. Comparative investigation ... 84

    Evolving designs from fully-braced initial configuration .. 85

    Alternative objective function formulations .. 86

    Scheduling of exploratory moves .. 86

    Performance of designs evolved from randomly generated initial

    configurations 86

    Use of pattern moves . 87

    4.8. Topology optimisation: structural model B 87

    4.8.1. Results . 89

    4.8.2. Observations 91

    4.8.3. Diversity .. 91

    4.9. Size optimisation 92

    4.9.1 Overview . 92

    4.9.2. Derivation of iterative approach from Optimality Criteria .. 92

    4.9.3. Pitfalls . 97

    Complex values of Ai.. 97

    Convergence failure ... 97

    Negative values of Cj* and Cj. . 97

    4.9.4. Assignment of discrete sections .. 98

    4.9.5. Size optimisation of fully braced configuration .. 99

    4.9.6. Size optimisation by Optimality Criteria with bending moments ... 102

    4.10. Integration of topology and size optimisation.... 102

    4.10.1. Results . 106

    4.10.2. Observations 107

    4.11. Summary of results from optimisation model B 108

    4.12. Conclusions. 112

  • 5. STRUCTURAL TOPOLOGY OPTIMISATION OF BRACED STEEL

    FRAMEWORKS USING GENETIC PROGRAMMING .. 114

    5.1. Introduction 114

    5.2. Background 115

    5.3. Genetic Programming method ... 115

    5.3.1. Introduction 115

    5.3.2. GP for bracing design . 116

    Creating initial designs .. 118

    Analysis and fitness ... 120

    Generating subsequent populations ... 120

    Handling geometrically infeasible designs 121

    5.4. Bracing design for a 2x6 framework .. 125

    5.5. Bracing design for a 6x30 framework 130

    5.6. Defining aesthetic style .. 138

    5.7. Further work .. 139

    5.8. Conclusions 139

    6. CONCLUDING REMARKS .. 141

    6.1. Review of contributions . 141

    6.2. Recommendations for future work . 145

    Evolutionary Structural Optimisation .... 145

    Pattern Search - Optimality Criteria .. 145

    Genetic Programming 146

    6.3. Application of structural optimisation in practice .. 147

    6.4. Projected trends in structural design automation and optimisation in practice

    148

    6.5. Closing notes .. 149

    APPENDIX 1. STRUCTURAL ANALYSIS . 151

  • APPENDIX 2. SOFTWARE DEVELOPMENT AND PROTOTYPING . 152

    REFERENCES . 153

  • List of Figures

    Figure 1.1: Structural optimisation tasks illustrated through the example of the design

    of a simply-supported, centrally point-loaded structure .. 5

    Figure 2.1: Michell truss subjected to load F at point A and fixed at a circular support

    at point B, after Michell (1904) ... 13

    Figure 2.2: Optimal self-adjoint cantilever trusses with six and eleven joints, subjected

    to load F at point A and fixed at support points B (Prager 1977) 13

    Figure 2.3: Fully-connected ground structure for a relatively simple (3x6) grid . 17

    Figure 2.4: Concept sketches for bracing design of 122 Leadenhall St. Building

    (reproduced by kind permission of Chris Neighbour, Arup) 34

    Figure 3.1: Weighting factors used for averaging sensitivity numbers across elements

    to avoid checkerboarding .. 42

    Figure 3.2: Real loads (left), including member groupings and geometric

    specifications, and virtual load (right). ASCE standard section specifications

    (below) .. 46

    Figure 3.3: Design topology of Liang et al. (2000): =0.024, element retention = 22%

    (left) Comparative result to Liang et al. (2000) topology: =0.024, element

    retention = 23% (right) . 47

    Figure 3.4: Elements removed in the top left bay unit in the first iteration, based on

    maximum cross strain energy (left) and sum of strain energies (right) in pairs of

    elements grouped by the horizontal symmetry condition . 48

    Figure 3.5: ESO results with element removal determined by cross-strain energy.

    25.4mm designable domain, 8 elements removed per iteration (left: maximum

    of sensitivity number in pairs of elements, right: sum of sensitivity number in

    pairs of elements) .. 49

    Figure 3.6: Minimum volume designs satisfying the displacement constraint derived by

    BESO for varying domain thickness and starting configuration .. 51

    Figure 3.7: Process flowchart for ESO with domain thickness optimisation ... 54

    Figure 3.8: Flowchart for domain thickness optimisation loop 55

    Figure 3.9: Process history for simultaneous topology and domain thickness

    optimisation with a single thickness group .. 57

    Figure 3.10: Best designs derived by simultaneous thickness and topology

    optimisation, with one, three and six thickness groups 58

    Figure 3.11: Evolving topologies with prescribed symmetry, using simultaneous

    thickness optimisation of appropriate groups ... 60

  • Figure 3.12: Discrete bracing topologies (with circular solid sections) optimised for

    minimum mass satisfaction of displacement constraint 63

    Figure 4.1: Fully-braced analysis model (left to right): plan view; side elevation;

    isometric view (shown with two spirals highlighted); isometric split sections

    70

    Figure 4.2: Split elevation view of the upper section of structural model 1, with spiral

    numbering and bracing members at the tip of each element highlighted . 75

    Figure 4.3: Parametric Studies .. 77

    Figure 4.4: Designs generated for consideration for outline proposal .. 78

    Figure 4.5: 2D simplified representation of design domain, model A .. 80

    Figure 4.6. A sample exploratory move 81

    Figure 4.7: Pattern Search topology optimisation flowchart 82

    Figure 4.8: Design solutions from topology optimisation of structural model 2 .. 89

    Figure 4.9: Size optimisation flowchart 99

    Figure 4.10: Convergence of size optimisation algorithm from maximum section sizes

    in fully braced design 101

    Figure 4.11: Convergence of size optimisation algorithm from minimum section sizes

    in fully braced design 101

    Figure 4.12: Flowchart for combined size and topology optimisation algorithm . 105

    Figure 4.13: Arup design proposal, without requirement for bracing members to be

    grouped in continuous spirals 109

    Figure 4.14: Volume reduction by simultaneous versus sequential topology and size

    optimisation routines . 111

    Figure 5.1: Tree representation of mathematical equation: y = 4/(X*X) + 5*(7-X) 116

    Figure 5.2: Function set for GP trees...

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