Tension Structures Form and Behaviour

Contents

Preface ix

Acknowledgements xiii

1. Introduction 1

1.1 Definitions and classifications 3

1.1.1 Boundary tensioned membranes 5

1.1.2 Pneumatic structures 10

1.1.3 Pre-stressed cable nets and beams 12

1.2 Design process of tension membranes 15

1.3 Main features of tension membranes 17

1.4 Conventional roofing forms versus tension membranes 17

1.4.1 Common misconceptions relating to tension membrane design 19

1.5 Closing remarks 20

References 20

2. Form-finding 23

2.1 General concepts. Nature’s ‘secrets’ 23

2.2 Concept of a ‘minimal surface’ – historical background 26

2.3 Form-finding methodologies 28

2.3.1 Physical models 28

2.3.2 Computational models 29

2.3.3 ‘Soap-film debate’ 31

2.3.4 Form-finding, or form-dictating? 35

References 36

3. Geometrically non-linear behaviour. Solutions by commonly used numerical methods 39

3.1 Geometric non-linearity 39

3.2. Commonly used computational methods for the analysis of geometrically non-linear behaviour 41

3.3 Transient stiffness method 41

3.3.1 Static analysis of skeletal, ‘linear’ structures 42

3.3.2 Static analysis of skeletal, ‘non-linear’ structures – why iterative computations are necessary 45

3.3.3 Transient stiffness method applied to static analysis of tension structures. Role of geometric stiffness 46

3.3.4 Application of the transient stiffness method to form-finding of tension structures 48

3.3.5 Evaluation of the transient stiffness method 50

3.4 Force density method 51

3.4.1 Application of the force density method 52

3.4.2 Numerical example 53

3.4.3 Matrix formulation of the force density method 54

3.4.4 Evaluation of the force density method 57

3.5 Dynamic relaxation method 57

3.6 Computational static analysis versus form-finding 58

3.6.1 Out-of-balance forces in static analysis and in form-finding 58

3.6.2 Elastic and geometric effects in static analysis and in form-finding 58

References 59

4. Dynamic relaxation method 61

4.1 Dynamic relaxation method with viscous damping 62

4.1.1 Stability of the iterative solution 63

4.1.2 Critical viscous damping coefficient 65

4.2 Dynamic relaxation method with kinetic damping 67

4.2.1 Iterative process 68

4.2.1.1 Location of the point of maximum kinetic energy: correction for displacements 69

4.2.2 Application of the dynamic relaxation method with kinetic damping to static analysis of tension cable nets 70

4.3 Efficiency of the dynamic relaxation relative to the transient stiffness method 72

4.3.1 Simple net 72

4.3.2 (2×1) cable net 72

4.3.3 (2×2) cable net 72

4.3.4 Hypar net 73

4.3.5 Spatial net 74

4.3.6 Saddle net 75

4.3.7 Results 75

4.3.8 Discussion 77

4.4 Evaluation of the dynamic relaxation method 79

References 80

5. Case studies of cable roof structures. Design issues: form-finding and patterning 81

5.1 Introduction 81

5.2 Case studies 83

5.2.1 Spatial net 83

5.2.1.1 Geometry of the geodesic net and differentially stressed net 84

5.2.1.2 Behaviour of the geodesic and differentially stressed nets under imposed loading 85

5.2.1.3 Geodesic net versus differentially stressed form and the ‘soap-film debate’ 94

5.2.2 Saddle net 95

5.2.3 Poskitt truss 97

5.2.3.1 Load Case 1 99

5.2.3.2 Load Case 2 100

5.2.3.3 Accuracy in model manufacture/patterning 103

5.2.3.4 Conclusions 106

References 106

6. Tension cables in suspension bridges. A case of form-finding 109

6.1 ‘Shape equation’ for an inextensible suspension cable 110

6.1.1 The parabolic approximation 112

6.1.2 The catenary 113

6.1.3 Constant stress cable 114

6.2 ‘Shape equation’ for an extensible suspension cable 115

6.2.1 Extensible (elastic) cable under self-weight 115

6.2.2 Extensible cable under self-weight and deck weight 117

6.3 Numerical modelling of shape of suspension bridge cables 119

6.3.1 ‘Geometric’ case. Form-finding of an inextensible suspension cable under dead weight of the bridge 119

6.3.1.1 Inextensible cable. Form-finding for Load Case 1 – deck weight only 122

6.3.1.1.1 Hanger arrangements 1(a) and 1(b) -uneven number of hangers 122

6.3.1.1.2 Finite difference analysis. Deck

weight only. Hanger arrangement 1(a) – centre hanger present and end spacing halved 125

6.3.1.1.3 Finite difference analysis. Deck weight only. Hanger arrangement 1(b) – centre hanger present and constant spacing throughout 126

6.3.1.1.4 Analysis of results – Load Case 1 127

6.3.1.2 Inextensible cable. Numerical form-finding for Load Case 2 – deck weight plus cable weight 128

6.3.2 ‘Elastic’ case. Form-finding of an extensible cable under dead weight of the bridge 129

6.3.2.1 Extensible cable. Numerical form-finding for Load Cases 1 and 2 129

6.3.3 Comparisons between the ‘elastic’ and ‘geometric’ cases 136

6.3.3.1 Comparison of geometry 136

6.3.3.2 Comparison of load transfer 136

6.3.3.3 Practical aspects 138

6.3.3.3.1 Estimate of temperature effect 138

6.4 General comments 140

6.4.1 Form-finding of suspension bridge cables 140

References 141

7. Modelling of tension membranes 143

7.1 Introduction 143

7.2 Surface discretization 143

7.3 Surface discretization for use with the transient stiffness method: limitations of the approach 144

7.3.1 Line elements 144

7.3.1.1 Elastic stiffness formulation of a line element for use with the transient stiffness method 145

7.3.1.2 Geometric stiffness formulation of a line element for use with the transient stiffness method 149

7.3.2 Limitations of the approach 152

7.4 Surface discretizations used with the dynamic relaxation method 153

7.4.1 Line elements 153

7.4.2 Triangular, ‘triple force’ elements 153

7.4.3 High-order elements 155

7.4.4 Non-finite element formulations 156

7.4.4.1 Cubic spline method 156

7.4.4.2 Triangular mesh method 161

7.5 Mesh control – implications for design 163

7.5.1 Case study: mesh control in form-finding and patterning 163

7.5.1.1 Analysis of results and consequences for design 165

7.6 Patterning of membranes 167

7.7 Line elements in modelling of stable minimal surface membranes 169

7.8 Triangular elements in modelling of stable minimal surface membranes 170

7.9 Numerical accuracy and criteria used for convergence 174

7.10 Pre-processing 175

References 177

Appendix I. Materials used for tension structures 181

AI.1 Fabric membranes 181

AI.2 Cable structures 182

Appendix II. Minimal surfaces 185

AII.1 Calculus of variations 185

AII.2 Variational problem: Euler-Lagrange equation for a minimal surface 187

Appendix III. Viscous damping in dynamic relaxation 193

Index 195