TENSILE STRUCTURESCABLE-NET METHOD

Sotiris Sotiriou

May 2006

Abstract

Tension Structures - Cable Net Method

by

Sotiris Sotiriou

In recent decades tensile structures have become very popular due to their notable ad-

vantages. Tension Structures are very light, flexible, capable to cover long spans and

efficient in the use of different materials. However, the analysis process is particularly

challenging. The necessity of the form-finding procedure and the need of the large dis-

placement theory make tensile structure analysis a complicated and time-consuming

process. Practical analysis generally does not require accounting for non-linear material

properties.

Finite element method and Cable net method can be used for the analysis of tensile

structures. Cable net models are commonly used in the design of those types of structures

because they are very physical more simple and need less computer power.

This project deals with theoretical and computer programming techniques used in the

numerical analysis of tensile cable structures. The Cable-Net method, Shape finding, and

Geometric non-linear analysis, are explained extensively in terms of three dimensional

application on tensile structures.

Acknowledgments

This project was written at the Department of Civil Engineering at Polytechnic Univer-

sity in New York. The research work was carried under the supervision of Professor J.

Jong Lou.

First of all, I express my gratitude to my supervisor Professor J. Jong Lou, for his

scientific guidance and valuable advice, not only for this project but also for those two

years of being my professor at the Polytechnic University.

Finally, I would like to thank my parents and my sister for their support during my years

of study.

New York, May 2006

Sotiris Sotiriou

Contents

1 Introduction 1

1.1 Scope of Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2 Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2 History and Applications 4

2.1 Historical Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.2 Tensile Structural Systems . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.2.1 Pure Cable net Forms . . . . . . . . . . . . . . . . . . . . . . . . 10

2.2.2 Cable-stayed forms . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.2.3 Tensegrity systems . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.2.3.1 Cable-strut systems . . . . . . . . . . . . . . . . . . . . 13

2.3 Materials and Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.3.1 Cables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.3.1.1 Axial Stiffness . . . . . . . . . . . . . . . . . . . . . . . 16

2.3.2 Glass and Fabric . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3 Analysis Methods 20

3.1 Cable Net Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3.1.1 The Shape Finding . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3.1.1.1 Grid Method. . . . . . . . . . . . . . . . . . . . . . . . . 22

3.2 Non Linear Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3.2.1 Geometric Non-Linear Analysis . . . . . . . . . . . . . . . . . . . 28

3.2.1.1 Geometric Stiffness Matrix Method . . . . . . . . . . . . 29

3.2.1.2 Load Step Method . . . . . . . . . . . . . . . . . . . . . 31

iv

CONTENTS

3.3 Detailing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

4 CableNL Software 40

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

4.2 Software Development . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

4.2.1 Gauss Elimination . . . . . . . . . . . . . . . . . . . . . . . . . . 45

4.2.2 Non Linear Analysis Work-Flow . . . . . . . . . . . . . . . . . . . 46

4.2.2.1 LDU Subroutine . . . . . . . . . . . . . . . . . . . . . . 47

4.3 User’s Manual . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

4.4 Verification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

4.5 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

4.5.1 Example. Glass Wall . . . . . . . . . . . . . . . . . . . . . . . . . 67

4.5.2 Example. Fabric Roof . . . . . . . . . . . . . . . . . . . . . . . . 71

5 Conclusions 79

A CableNL – Verification Examples 84

A.1 Shape Finding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

A.1.1 LAYOUT.FOR – Input . . . . . . . . . . . . . . . . . . . . . . . 84

A.1.2 LAYOUT.FOR – Output . . . . . . . . . . . . . . . . . . . . . . . 85

A.1.3 CableNL – Input . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

A.1.4 CableNL – Output . . . . . . . . . . . . . . . . . . . . . . . . . . 91

A.2 Non Linear Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

A.2.1 TR3DNL.FOR – Input . . . . . . . . . . . . . . . . . . . . . . . . 96

A.2.2 TR3DNL.FOR – Output . . . . . . . . . . . . . . . . . . . . . . . 97

A.2.3 CableNL – Input . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

A.2.4 CableNL – Output . . . . . . . . . . . . . . . . . . . . . . . . . . 102

B CableNL – Numerical Examples 104

B.1 Glass wall example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

B.1.1 Input File: UBS Building.cn-in . . . . . . . . . . . . . . . . . . . 104

B.1.2 Analysis Parameters: . . . . . . . . . . . . . . . . . . . . . . . . . 106

B.1.3 Non Linear Analysis Results: . . . . . . . . . . . . . . . . . . . . 110

v

CONTENTS

B.2 Fabric Roof Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

B.2.1 Input File: Roof Example.cn-in . . . . . . . . . . . . . . . . . . . 112

B.2.2 Force Balance and Joint Coordinates: . . . . . . . . . . . . . . . . 115

B.2.3 Non Linear Analysis Results: . . . . . . . . . . . . . . . . . . . . 120

vi

List of Figures

1.1 Membrane Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Cable-Net Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.3 Pneumatic Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2.1 Dorton Arena . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.2 Arena’s structural system . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.3 The German Pavilion in Montreal . . . . . . . . . . . . . . . . . . . . . . 5

2.4 Cable-net cooling tower, Schmehausen 1968 . . . . . . . . . . . . . . . . 6

2.5 Cable Net Tower, Structural System . . . . . . . . . . . . . . . . . . . . 6

2.6 U.S Pavilion Expo 70’ in Osaka . . . . . . . . . . . . . . . . . . . . . . . 7

2.7 The Olympic Stadium of Munich . . . . . . . . . . . . . . . . . . . . . . 8

2.8 Munich Olympic games 1972 . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.9 Georgia Dome, Atlanta 1992 . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.10 Georgia Dome, Roof indoor view . . . . . . . . . . . . . . . . . . . . . . 8

2.11 The Millennium Dome, Aerial view . . . . . . . . . . . . . . . . . . . . . 9

2.12 Millennium Dome, Side view . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.13 The Time Warner, Cable net wall . . . . . . . . . . . . . . . . . . . . . . 10

2.14 Cable net Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.15 Vertical Cable String. Reproduced from [2] . . . . . . . . . . . . . . . . . 11

2.16 Cable State basic unit . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.17 Cable girder dome, Form with opening. Reproduced from [2] . . . . . . 12

2.18 Cable girder dome. Reproduced from Mero Structures. . . . . . . . . . . 12

2.19 Geiber’s Dome. Reproduced from [2] . . . . . . . . . . . . . . . . . . . . 13

vii

LIST OF FIGURES

2.20 Spatially triangulated dome. Reproduced from [2] . . . . . . . . . . . . . 13

2.21 Cable-strut form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.22 Wire Rope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.23 Wire rope, right and left hand lays. . . . . . . . . . . . . . . . . . . . . . 15

2.24 Stress/Strain Curve for Steel and Glass . . . . . . . . . . . . . . . . . . . 18

3.1 The Force Density Method for bar elements . . . . . . . . . . . . . . . . 22

3.2 The Grid Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3.3 Grid Method, Example . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3.4 CableNL, Smoothed Geometry. . . . . . . . . . . . . . . . . . . . . . . . 27

3.5 Force - Displacement curve . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3.6 P −∆ and P − δ effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.7 Member orientation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

3.8 Plane Truss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3.9 Patterning: Triangle Strips . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.10 In plane restraints at bolted connections. Reproduced form [4] . . . . . . 38

3.11 Turnbuckle to swaged eye termination . . . . . . . . . . . . . . . . . . . 38

3.12 Eye and U-clips cable termination . . . . . . . . . . . . . . . . . . . . . . 39

4.1 CableNL flowchart . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

4.2 Shape Finding flowchart . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

4.3 non linear analysis, flowchart . . . . . . . . . . . . . . . . . . . . . . . . . 44

4.4 Symmetric Members . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

4.5 Geometry Input - Brava Viewer . . . . . . . . . . . . . . . . . . . . . . . 53

4.6 Analysis Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

4.7 Scaled Deformed Shape . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

4.8 Plan View. Use 1/8 of the structure. . . . . . . . . . . . . . . . . . . . . 61

4.9 Joint and members labels of 1/8 of the structure . . . . . . . . . . . . . . 61

4.10 CableNL, Input Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . 61

4.11 CableNL, Output Geometry . . . . . . . . . . . . . . . . . . . . . . . . . 61

4.12 CableNL, Joint Labels . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

4.13 Plane Cable-net . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

4.14 Non-linear analysis, deformed shape . . . . . . . . . . . . . . . . . . . . . 66

viii

LIST OF FIGURES

4.15 Maximum displacements . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

4.16 UBS Tower. Cable net glass wall. . . . . . . . . . . . . . . . . . . . . . . 67

4.17 Transform load to joints . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

4.18 Elevation of a Typical Bay. . . . . . . . . . . . . . . . . . . . . . . . . . . 70

4.19 CableNL, Deformed Shape. Scale factor equals four. . . . . . . . . . . . . 71

4.20 Horizontal maximum displacements . . . . . . . . . . . . . . . . . . . . . 71

4.21 Vertical maximum displacements . . . . . . . . . . . . . . . . . . . . . . 71

4.22 Joint Loads. Triangular panels . . . . . . . . . . . . . . . . . . . . . . . . 73

4.23 Roof - Geomerty Input . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

4.24 New Shape after grid method . . . . . . . . . . . . . . . . . . . . . . . . 75

4.25 Roof. Analysis Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . 76

4.26 Deformed shape. Scale Factor two . . . . . . . . . . . . . . . . . . . . . . 77

4.27 Roof, side view . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

4.28 Vertical maximum displacements (ft) . . . . . . . . . . . . . . . . . . . . 78

4.29 Horizontal maximum displacements (ft) . . . . . . . . . . . . . . . . . . . 78

ix

List of Tables

2.1 Mechanical Properties of Steel Cables . . . . . . . . . . . . . . . . . . . . 16

2.2 Comparison of Polyester and Fiberglass characteristics . . . . . . . . . . 18

2.3 Mechanical Properties of Glass. Pilkington Technical Information ATS-

129 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

4.1 Shape Finding. Free joints’ coordinates . . . . . . . . . . . . . . . . . . . 62

4.2 Joint Force Balance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

4.3 Shape Finding. Member Forces . . . . . . . . . . . . . . . . . . . . . . . 63

4.4 Total displacements (in) . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

4.5 Member Forces (lbs) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

x

Chapter 1

Introduction

Every part of a tensile structure is loaded only in tension, with no requirement to resist

compression or bending forces. The basic types of tension structures are:

• Membranes: The structural membrane acts also as the weather shield. Fig:1.1

• Cable Nets: A separate grid of structural cables supports a nonstructural weather

shield. Fig:1.2

• Pneumatics: The tension force is created by an interior positive pressure and the

membrane acts as the weather shield. Fig:1.3

Figure 1.1: Membrane Structure

1

CHAPTER 1. INTRODUCTION

Tensile structures have always fascinated architects and engineers, mainly because of

their special features. Aesthetic shapes, light weight and flexibility, combined with new

materials, make tension structures the new trend in architectural design.

Figure 1.2: Cable-Net Structure Figure 1.3: Pneumatic Structure

However, these special features require special design, that makes the analysis much more

complicated. Due to absence of flexural stiffness of cables and membranes, the initial

configuration of these structures must be stressed. Thus, before the analysis, the initial

geometry configuration must be found.

The shape of a tensile structure, governs the load-bearing capacity of the structure.

Therefore, the process of determining the initial equilibrium configuration, calls the

designers ability to find an optimum compromise between shape, load capacity and

constructional requirements. After the shape finding procedure, geometrical nonlinear

analysis is required due to a high degree of flexibility of the structure.

1.1 Scope of Work

The scope of this work is to familiarize the reader, with the challenging design of tension

Structures. According to Campbel 1, “no other class of architectural structural systems

1D. Campbel. The Unique Role of Computing in the Design and Construction of Tensile MembraneStructures.

2

CHAPTER 1. INTRODUCTION

is as dependent upon the use of digital computers as are tensile structures”. The project

goes beyond the theoretical approach with the development of a computer software.

1.2 Organization

The project consists of three major parts. The first part will be a general overview about

history, applications and materials that are used for tension structures. The second part

will focus on the analysis procedures using theCable net method. The third part will be

a brief description about the developed software “CableNL”.

3