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Seeing and TouchingStructural Concepts

Tianjian Ji and Adrian Bell

Taylor . FrancisTaylor Francis Cr ou p

LONDON AND NEW YORK

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Contents

Preface xiiAcknowledgements xvi

PARTIStatics 1

1 Equilibrium 3

1.1 Definitions and concepts 31.2 Theoretical background 31.3 Model demonstrations 5

1.3.1 Action and reaction forces 51.3.2 Stable and unstable equilibrium 61.3.3 A plate-bottle system 71.3.4 A magnetic float model 7

1.4 Practical examples 81.4.1 A barrier 81.4.2 A footbridge 91.4.3 An equilibrium kitchen scale 101.4.4 Stage performance 101.4.5 Magnetic float train 111.4.6 A dust tray 11

Centre of mass 13

2.1 Definitions and concepts 132.2 Theoretical background 132.3 Model demonstration 18

2.3.1 Centre of mass of a piece of cardboard of arbitrary shape 18

2.3.2 Centre of mass and centroid of a body 192.3.3 Centre of mass of a body in a horizontal plane 192.3.4 Centre of mass of a body in a vertical plane 212.3.5 Centre of mass and stability 222.3.6 Centre of mass and motion 24

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2.4 Practical examples 242.4.1 Cranes on construction sites 242.4.2 The Eiffel Tower 252.4.3 A display unit 262.4.4 The Kio Towers 26

3 Effect of different cross sections 28

3.1 Definitions and concepts 283.2 Theoretical background 283.3 Model demonstrations 33

3.3.1 Two rectangular beams and an I-section beam 333.3.2 Lifting a book using a bookmark 34

3.4 Practical examples 35

3.4.1 A steel-framed building 353.4.2 A railway bridge 363.4.3 I-section members with holes (cellular beams and

columns) 36

4 Bending 38

4.1 Definitions and concepts 384.2 Theoretical background 384.3 Model demonstration 42

4.3.1 Assumptions in beam bending 424.4 Practical examples 43

4.4.1 Profiles of girders 434.4.2 Reducing bending mom ents using overhangs 434.4.3 Failure due to bending 444.4.4 Deformation of a staple due to bending 45

5 Shear and torsion 47

5.1 Definitions and concepts 475.2 Theoretical background 475.2.1 Shear stresses due to bending 475.2.2 Shear stresses due to torsion 49

5.3 Model demonstrations 535.3.1 Effect of torsion 535.3.2 Effect of shear stress 535.3.3 Effect of shear force 555.3.4 Open and closed sections subject to torsion with w arping 56

5.3.5 Open and closed sections subject to torsion without warping 575.4 Practical examples 58

5.4.1 Composite section of a beam 585.4.2 Shear walls in a building 585.4.3 Opening a drinks bottle 59

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6 Stress distribution 61

6.1 Concept 616.2 Theoretical background 616.3 Model demonstrations 62

6.3.1 Balloons on nails 626.3.2 Uniform and non-uniform stress distributions 636.4 Practical examples 64

6.4.1 Flat shoes vs. high-heel shoes 646.4.2 The Leaning Tower of Pisa 65

7 Span and deflection 67

7.1 Concepts 67

7.2 Theoretical background 677.3 Model demonstrations 717.3.1 Effect of spans 717.3.2 Effect of boundary conditions 727.3.3 The bending mom ent at one fixed end of a beam 73

7.4 Practical examples 747.4.1 Column supports 747.4.2 The phenomenon of prop roots 757.4.3 Metal props used in structures 75

8 Direct force paths 77

8.1 Definitions, concepts and criteria 778.2 Theoretical background 77

8.2.1 Introduction 778.2.2 Concepts for achieving a stiffer structure 788.2.3 Implementation 828.2.4 Discussion 87

8.3 Model demonstrations 908.3.1 Experimental verification 908.3.2 Direct and zigzag force paths 92

8.4 Practical examples 928.4.1 Bracing systems of tall buildings 928.4.2 Bracing systems of scaffolding structures 93

9 Smaller internal forces 96

9.1 Concepts 969.2 Theoretical background 969.2.1 Introduction 969.2.2 A ring and a tied ring 97

9.3 Model demonstrations 105

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9.3.1 A pair of rubber rings 1059.3.2 Post-tensioned p lastic beams 106

9.4 Practical examples 1079.4.1 R aleigh Arena 1079.4.2 Zhejiang Dragon Sports Centre 1089.4.3 A cable-stayed bridge 1119.4.4 A floor structure experiencing excessive vibration 111

10 Buckling 113

10.1 Definitions and concepts 11310.2 Theoretical background 113

10.2.1 Buckling of a column with different boundary conditions 11310.2.2 Lateral torsional buckling of beams 116

10.3 Model demonstrations 11910.3.1 Buckling shapes of plastic columns 11910.3.2 Buckling loads and boundary conditions 12010.3.3 Lateral buckling of beams 12210.3.4 Buckling of an empty aluminium can 123

10.4 Practical examples 12410.4.1 Buckling of bracing mem bers 12410.4.2 Buckling of a box girder 12510.4.3 Prevention of lateral buckling of beams 125

11 Prestress 127

11.1 Definitions and concepts 12711.2 Theoretical background 12711.3 Model demonstrations 133

11.3.1 Prestressed wooden blocks forming a beam and a column 13311.3.2 A toy using prestressing 134

11.4 Practical examples 13411.4.1 A centrally post-tensioned column 13411.4.2 An eccentrically post-tensioned beam 13511.4.3 Spider s web 13511.4.4 A cable-net roof 137

12 Ho rizon tal movem ents of structures induced by vertical loads 139

12.1 Concepts 13912.2 Theoretical background 139

12.2.1 Static response 14012.2.2 Dynam ic response 149

12.3 Model demonstrations 15112.3.1 A symm etric frame 15112.3.2 An anti-symm etric frame 152

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12.3.3 An asymm etric frame 15212.4 Practical examples 153

12.4.1 A grandstand 15312.4.2 A building floor 15312.4.3 Rail bridges 156

PART IIDynamics 157

13 Energy exchange 159

13.1 Definitions and concepts 15913.2 Theoretical background 15913.3 Model demonstrations 164

13.3.1 A moving wheel 16413.3.2 Collision balls 16513.3.3 Dropping a series of balls 167

13.4 Practical examples 16813.4.1 Roller coasters 16813.4.2 A torch without a battery 169

14 Pendulums 17

14.1 Definitions and concepts 17014.2 Theoretical background 170

14.2.1 A simple pendulum 17014.2.2 A generalised suspended system 17214.2.3 Translational and rotational systems 176

14.3 Model demonstrations 17614.3.1 Natural frequency of suspended systems 17614.3.2 Effect of added masses 17814.3.3 Static behaviour of an outward inclined suspended system

18014.4 Practical examples 18214.4.1 An inclined suspended wooden bridge in a playground 18214.4.2 Seismic isolation of a floor 18214.4.3 The Foucault pendulum 182

15 Free vibration 185

15.1 Definitions and concepts 18515.2 Theoretical background 186

15.2.1 A single-degree-of-freedom system 18615.2.2 A generalised single-degree-of-freedom system 19115.2.3 A multi-degrees-of-freedom (M DO F) system 19515.2.4 Relationship between the fundamental natural frequency

and the maximum displacement of a beam 196

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15.2.5 Relationship between the fundamental natural frequencyand the tension force in a straight string 198

15.3 Model demonstrations 19915.3.1 Free vibration of a pendulum system 19915.3.2 Vibration decay and natural frequency 2 0015.3.3 An overcritically-damped system 20115.3.4 Mode shapes of a discrete model 20215.3.5 Mode shapes of a continuous model 20215.3.6 Tension force and natural frequency of a straight tension

bar 20315.4 Practical examples 204

15.4.1 A musical box 20415.4.2 Measurement o f the fundamental natural frequency of a

building through free vibration generated using vibrators 20615.4.3 Measurement of the natural frequencies of a stack throughvibration generated by the environment 207

15.4.4 The tension forces in the cables in the London Eye 208

16 Resonance 21

16.1 Definitions and concepts 21016.2 Theoretical background 210

16.2.1 A SD OF system subjected to a harmonic load 21116.2.2 A SD OF system subject to a harmonic support movem ent

21716.2.3 Resonance frequency 219

16.3 Model demonstrations 2 2116.3.1 Dynamic response of a SDO F system subject to harmonic

support movements 22116.3.2 Effect of resonance 222

16.4 Practical examples 222

16.4.1 The London Millennium Footbridge 22316.4.2 Avoidance of resonance: design of structures used for popconcerts 225

16.4.3 Measurement of the resonance frequency of a building 22716.4.4 An entertaining resonance phenomenon 228

17 Damping in structures 231

17.1 Concepts 23117.2

Theoretical background 231

17.2.1 Evaluation of viscous-damping ratio from free vibrationtests 231

17.2.2 Evaluation of viscous-damping ratio from forced vibrationtests 233

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17.3 Model demonstrations 23417.3.1 Observing the effect of damping in free vibrations 23417.3.2 Hearing the effect of damping in free vibrations 234

17.4 Practical examples 23517.4.1 Damping ratio obtained from free vibration tests 23517.4.2 Damping ratio obtained from forced vibration tests 23717.4.3 Reducing footbridge vibrations induced by walking 23717.4.4 Reducing floor vibration induced by walking 238

18 Vibration reduction 241

18.1 Definitions and concepts 24118.2 Theoretical background 241

18.2.1 Change of dynamic properties of systems 242

18.2.2 Tuned mass dampers 24418.3 Model demonstrations 246

18.3.1 A tuned mass damper (TMD ) 24618.3.2 A tuned liquid damper (TLD) 24718.3.3 Vibration isolation 248

18.4 Practical examples 24818.4.1 Tyres used for vibration isolation 24818.4.2 The London Eye 24918.4.3 The London Millennium Footbridge 249

19 Human body models in structural vibration 252

19.1 Concepts 25219.2 Theoretical background 252

19.2.1 Introduction 25219.2.2 Identification of human body models in structural vibration

25419.3 Demonstration tests 257

19.3.1 The body model of a standing person in the verticaldirection 257

19.3.2 The body model of a standing person in the lateraldirection 259

19.4 Practical examples 26119.4.1 The effect of stationary spectators on a grandstand 26119.4.2 Calculation of the natural frequencies of a grandstand 26319.4.3 Dynamic response of a structure u sed at pop concerts 263

19.4.4 Indirect measurement of the fundamental natural frequencyof a standing person 26319.4.5 Indirect measurement of the fundamental natural frequency

of a chicken 264

Index 266