Dynamic Characteristics of Slender Suspension …characteristics of slender suspension footbridges under human-induced dynamic loads and a footbridge model in full size with pre-tensioned

Dynamic Characteristics of Slender

Suspension Footbridges

By

Ming-Hui Huang

School of Urban Development Faculty of Built Environmental and Engineering

Queensland University of Technology

A THESIS SUBMITTED TO THE SCHOOL OF URBAN DEVELOPMENT QUEENSLAND UNIVERSITY OF TECHNOLOGY

IN PARTIAL FULFILMENT OF REQUIREMENTS FOR THE DEGREE OF

DOCTOR OF PHILOSOPHY

May 2006

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ABSTRACT Due to the emergence of new materials and advanced engineering technology,

slender footbridges are increasingly becoming popular to satisfy the modern

transportation needs and the aesthetical requirements of society. These structures

however are always “lively” with low stiffness, low mass, low damping and low

natural frequencies. As a consequence, they are prone to vibration induced by human

activities and can suffer severe vibration serviceability problems, particularly in the

lateral direction. This phenomenon has been evidenced by the excessive lateral

vibration of many footbridges worldwide such as the Millennium Bridge in London

and the T-Bridge in Japan. Unfortunately, present bridge design codes worldwide do

not provide sufficient guidelines and information to address such vibrations problems

and to ensure safety and serviceability due to the lack of knowledge on the dynamic

performance of such slender vibration sensitive bridge structures.

A conceptual study has been carried out to comprehensively investigate the dynamic

characteristics of slender suspension footbridges under human-induced dynamic

loads and a footbridge model in full size with pre-tensioned reverse profiled cables in

the vertical and horizontal planes has been proposed for this purpose. A similar

physical suspension bridge model was designed and constructed in the laboratory,

and experimental testings have been carried out to calibrate the computer simulations.

The synchronous excitation induced by walking has been modelled as crowd walking

dynamic loads which consist of dynamic vertical force, dynamic lateral force and

static vertical force. The dynamic behaviour under synchronous excitation is

simulated by resonant vibration at the pacing rate which coincides with a natural

frequency of the footbridge structure. Two structural analysis software packages,

Microstran and SAP2000 have been employed in the extensive numerical analysis.

Research results show that the structural stiffness and vibration properties of

suspension footbridges with pre-tensioned reverse profiled cables can be adjusted by

choosing different structural parameters such as cable sag, cable section and pre-

tensions in the reverse profiled cables. Slender suspension footbridges always have

four main kinds of vibration modes: lateral, torsional, vertical and longitudinal

modes. The lateral and torsional modes are often combined together and become

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two kinds of coupled modes: coupled lateral-torsional modes and coupled torsional-

lateral modes. Such kind of slender footbridges also have different dynamic

performance in the lateral and vertical directions, and damping has only a small

effect on the lateral vibration but significant effect on the vertical one.

The fundamental coupled lateral-torsional mode and vertical mode are easily excited

when crowd walking dynamic loads are distributed on full bridge deck. When the

crowd walking dynamic loads are distributed eccentrically on half width of the deck,

the fundamental coupled torsional-lateral mode can be excited and large lateral

deflection can be induced. Higher order vertical modes and coupled lateral-torsional

modes can also be excited by groups of walking pedestrians under certain conditions.

It is found that the coupling coefficient introduced in this thesis to describe the

coupling of a coupled mode, is an important factor which has significant effect on the

lateral dynamic performance of slender suspension footbridges. The coupling

coefficient, however, is influenced by many structural parameters such as cable

configuration, cable section, cable sag, bridge span and pre-tensions, etc. In general,

a large dynamic amplification factor is expected when the fundamental mode of a

footbridge structure is the coupled lateral-torsional mode with a small coupling

coefficient.

The research findings of this thesis are useful in understanding the complex dynamic

behaviour of slender and vibration sensitive suspension footbridges under human-

induced dynamic loads. They are also helpful in developing design guidance and

techniques to improve the dynamic performance of such slender vibration sensitive

footbridges and similar structures and hence to ensure their safety and serviceability.

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KEYWORDS Footbridge, suspension bridge, dynamics, vibration, pedestrian, walking, human-

induced, synchronous excitation, resonance, serviceability, natural frequency,

coupled mode, coupling coefficient, pacing rate, damping, slender, dynamic

amplification factor, dynamic characteristics, pre-tension, reverse profiled cable,

non-linear time history analysis

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ACKNOWLEDGEMENTS I am extremely grateful and deeply indebted to my principal supervisor Professor

David Thambiratnam for his enthusiastic and expertise guidance, constructive

suggestions, encouragements throughout the course of this study and the valuable

assistance in many ways. Without such assistance this study would not have been

what it is. His immense patience and availability for comments whenever approached

even amidst his heavy pressure of work throughout the entire period of study deserve

grateful appreciation. I would like to express my sincere gratitude to my associate

supervisors, Adjunct Professor Nimal Perera and Dr Azhar Nasir, for their stimulating

discussions and suggestions. I would like to thank the Faculty of Built Environmental

and Engineering, Queensland University of Technology for providing International

Research Scholarship (Full-tuition Fee) to carry out this research project. I would

also like to thank the School of Urban development for providing financial support,

necessary facilities and technical support.

I would like to express my thanks to Mr Jim Grandy and Mr Arthur Powell for their

assistances in the manufacture and construction of the physical bridge model and in

the experimental testing. Many thanks also to Mr Craig Windell and Mr Donald Lam

for their help with the computer softwares. It is my pleasure to thank fellow post-

graduate students and friends for their support and contribution to this research.

Finally I wish to express my appreciation to my family for their support,

encouragement, and patience.

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STATEMENT OF ORIGINAL AUTHORSHIP

The work contained in this thesis has not been previously submitted for a

degree or diploma at any other higher education institution. To the best

of my knowledge and belief, the thesis contains no material previously

published or written by another person except where due reference is

made.

Signed: ___________________________

Date: ___________________________

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PUBLICATIONS

Journal Papers:

1. Huang M.-H., Thambiratnam, D.P. and Perera, N.J. (2005), Vibration

characteristics of shallow suspension bridge with pre-tensioned cables,

Engineering Structures, Vol. 27, No. 8, 1220-1233.

2. Huang M.-H., Thambiratnam, D.P. and Perera, N.J. (2005), Resonant

vibration of shallow suspension footbridges, Proceedings of Institute of Civil

Engineering: Bridge Engineering, Vol. 158, Issue BE4, 201-209.

3. Huang M.-H., Thambiratnam, D.P. and Perera, N.J. (2005), Load deformation

characteristics of shallow suspension footbridge with reverse profiled pre-

tensioned cables, Structural Engineering and Mechanics, Vol. 21, No. 4, 375-

392.

4. Huang M.-H., Thambiratnam, D.P. and Perera, N.J., Coupling coefficient and

lateral vibration of slender suspension footbridges, submitted to Computer

and Structures (under review).

5. Huang M.-H., Thambiratnam, D.P. and Perera, N.J., Dynamic performance of

slender suspension footbridges under eccentric walking dynamic loads,

submitted to Journal of Sound and Vibration (under review).

Conference Papers:

6. Huang M.-H., Thambiratnam, D.P. and Perera, N.J. (2005), Vibration of

shallow suspension footbridge under walking dynamic loads, Proceeding of

the Tenth International Conference on Civil, Structural and Environmental

Engineering Computing, Rome, Italy, 30 August – 2 September 2005.

7. Huang M.-H., Thambiratnam, D.P. and Perera, N.J. (2005), Free vibration of

shallow suspension footbridge with pre-tensioned reverse profiled cables,

Australia Structural Engineering Conference 2005 (ASEC 2005), Newcastle,

Australia, 11-14 September 2005.

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TABLE OF CONTENTS Abstract ………………………………………………………………………...… (i)

Keywords ……………………………………………………………………..… (iii)

Acknowledgements …………………………………………………………… (iv)

Statement of original authorship ……………………………………………...… (v)

Publications ……………………………………………………………………... (vi)

Table of content ………………………………………………………………… (vii)

List of figures ……………………………………………………………………. (xi)

List of tables ……………………………………………………………….... (xviii)

Notations …………………………………………………………………...… (xxii)

Abbreviations ……………………………………………………………….. (xxiv)

1 Introduction ………………………………………………………….… (1)

1.1 Background ……………………………………………………………… (1)

1.2 Research objectives ……………………………………………………… (3)

1.3 Methodology …………………………………………………………..… (4)

1.4 Outline of the thesis …………………………………………………… (5)

2 Human-induced dynamic loads and vibration performance of

footbridges ……………………………………………………………… (7)

2.1 Introduction ……………………………………………………………… (7)

2.2 Human-induced dynamic loads ……………………………………..… (7)

2.2.1 Characteristics and measurements of walking and running dynamic

loads induced by a single person ………………………………………… (8)

2.2.2 Modelling of walking and running dynamic loads induced by a

single person ………………………………………………………..….. (16)

2.2.3 Effect of Group of People …………………………………………...…. (21)

2.3 Dynamic performance of footbridges …………………………..…….... (25)

2.3.1 Vibration properties of footbridge structures …………………….……. (25)

2.3.2 Vibration serviceability of footbridges ………………………..……….. (30)

2.3.3 Dynamic performance of footbridges under human-induced loads ….. (36)

2.3.3.1 Dynamic properties of footbridges under moving people ………..…….. (37)

2.3.3.2 Synchronization between walking people in groups …………………. (39)

2.3.3.3 Synchronization excitation and dynamic performance of footbridges .... (42)

2.4 Slender cable supported footbridges and vibration control ………….… (50)

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2.4.1 Slender cable supported footbridges ……………………………...….… (50)

2.4.2 Measures against excessive vibration of slender footbridges ………..… (58)

2.5 Some of existing footbridge design codes regarding to human-

induced vibration ………………………………………………….…… (60)

2.6 Summary ……………………………………………………………..… (63)

3 Suspension footbridge model with pre-tensioned cables …………... (67)

3.1 Introduction ……………………………………………………….…… (67)

3.2 Description of the proposed suspension footbridge model …………..… (68)

3.3 Cable profiles and initial distortions …………………………………… (70)

3.4 Suspension footbridge models for numerical analysis …………………. (73)

3.4.1 Structural analysis softwares …………………………………………… (73)

3.4.2 Bridge models in Microstran and SAP2000 ……………………………. (74)

3.4.3 Finite element modelling of bridge models ……………………………. (79)

4 Load deformation performance and vibration properties ………… (83)

4.1 Introduction ……………………………………………………………. (83)

4.2 Load deformation performance under quasi-static loads ………………. (84)

4.2.1 Applied quasi-static loads ……………………………………………… (84)

4.2.2 Un-pre-tensioned footbridges under symmetric vertical load: effect

of cable sags ……………………………………………………………. (86)

4.2.3 Effect of Pre-tension Forces in the Bottom Cables (Internal Vertical

Forces) ……………………………………………………………......… (89)

4.2.4 The Effects of cable sag and cross sectional area of bottom cables …..... (92)

4.2.5 Effect of pre-tension in the side cables (internal lateral forces) ……….. (95)

4.2.6 Performance under lateral horizontal loads and eccentric vertical

loads ………………………………………………………………….. (95)

4.3 Vibration properties of slender suspension footbridges with pre-

tensioned cables ………………………………………………………… (98)

4.3.1 Vibration Mode Shapes ………………………………………………… (98) 4.3.1.1 Coupled lateral-torsional vibration modes …………………………… (99) 4.3.1.2 Coupled torsional-lateral vibration modes ………………………..…… (102)

4.3.1.3 Vertical vibration modes …………………………………………….… (102)

4.3.1.4 Longitudinal vibration modes ………………………………..…….… (103)

4.3.2 Natural frequencies ……………………………………………………. (104)

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4.3.2.1 Effects of cable sag and cross sectional area of the top supporting

cables (un-pre-tensioned bridge model) …………………………….… (105)

4.3.2.2 Effects of pre-tensions in the reverse profiled bottom and side cables

(extra internal vertical and horizontal Forces) ……………………….. (107)

4.3.2.3 The effects of structural weight and applied loads …………………… (110)

4.3.2.4 Effect of Span Length ………………………………………………… (113)

4.4 Summary …………………………………………………………….… (116)

5 Experimental testing and calibration of physical bridge model …. (119)

5.1 Introduction …………………………………………………………... (119)

5.2 Physical bridge model and experiment system ……………………….. (119)

5.2.1 Design of physical bridge model and experimental system …………... (119)

5.2.2 Construction of physical bridge model ………………………………. (124)

5.3 Experimental testing and calibration of bridge model ………………… (126)

5.3.1 Bridge model cases and testing procedure ……………………………. (126)

5.3.2 Free vibration and natural frequencies ……………………………….. (130)

5.3.3 Load performance under static vertical load ………………………….. (141)

5.4 Comparison of results and discussion …………………………….….. (142)

5.4.1 Results from computer simulations and experimental testing …….….. (142)

5.4.2 Variation in results and discussion …………………………………... (144)

5.5 Modification factor of moment of inertia of cable section properties … (146)

6 Dynamic response of slender suspension footbridges under

crowd walking dynamic loads ………………………………………. (149)

6.1 Introduction …………………………………………………………… (149)

6.2 Crowd walking dynamic loads ……………………………………….. (150)

6.3 Dynamic performance and resonant vibration under crowd walking

dynamic loads …………………………………………………............ (154)

6.3.1 Resonant vibration at the frequencies of first vibration modes ………. (156)

6.3.1.1 Bridge model C120 …………………………………………………… (157)

6.3.1.2 Bridge model C123 …………………………………………………… (166)

6.3.1.3 Bridge model C103 …………………………………………………… (167)

6.3.2 Tension forces in resonant vibrations with first modes …………….… (172)

6.3.3 Resonant vibration at other modes ……………………………………. (176)

6.3.4 Dynamic performance at different pacing rates ………..……………... (178)

6.3.5 Effect of vertical static force on the resonant vibration …………….… (183)

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6.3.6 Effect of damping on the resonant vibration ………………………….. (186)

6.3.7 Resonant vibration under eccentric walking dynamic loads ………….. (188)

6.4 Dynamic characteristics of lateral vibration ………………………..… (196)

6.4.1 Effect of cable section and coupling coefficient ……………………… (196)

6.4.2 Effect of cable sag …………………………………………………..… (202)

6.4.3 Effect of span length ………………………………………………….. (207)

6.4.4 Effect of synchronization ……………………………………………... (209)

6.4.5 Natural frequency and dynamic amplification factors ………………... (209)

6.5 Summary ………………………………………………………………. (217)

7 Conclusions and discussions ………………………………..………. (221)

7.1 Conclusions ……………………………………………………..……. (221)

7.2 Discussions ……………….…………………………………………... (223)

7.3 Contributions to scientific knowledge ………………………………… (227)

7.4 Suggestions for future work ……….…………………………………. (228)

Bibliography …………………………………………………………….….. (231)

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LIST OF FIGURES Figure 2.1 Typical pattern of running and walking forces ……………………. (9)

Figure 2.2 Typical shapes of walking force in (a) vertical, (b) lateral and

(c) longitudinal direction ………………………………………… (10)

Figure 2.3 Force-time functions for various pacing rates, footwear and

surface conditions ………………………………………………... (11)

Figure 2.4 Typical vertical force patterns for different types of human

activities …………………………………………………………. (13)

Figure 2.5 Force function resulting from footfall overlap during walking

with a pacing rate of 2 Hz ………………………………………... (13)

Figure 2.6 Normal distribution of pacing frequencies for normal walking ….. (14)

Figure 2.7 Dependence of stride length, velocity, peak force and contact

time on different pacing rates ……………………………………. (14)

Figure 2.8 Harmonic load components (Fourier amplitudes) of the

direction load-time functions ……………………………………. (16)

Figure 2.9 Forcing function from jumping on the spot with both feet

simultaneously at a jumping rate of 2 Hz ………………………… (20)

Figure 2.10 Vertical and lateral components of the walking loads ……………. (21)

Figure 2.11 DLF for the first harmonic of walking force as a function of a

number of people and walking frequency ……………………….. (24)

Figure 2.12 Footbridge fundamental frequencies as a function of span ………. (27)

Figure 2.13 Lateral natural frequencies of footbridges ……………………….. (27)

Figure 2.14 Different scales of human perceptions …………………………… (32)

Figure 2.15 Acceptability of vibration on footbridge after different scales …... (34)

Figure 2.16 Acceptability of vertical vibration in footbridges after different

scales ……………………………………………………………... (35)

Figure 2.17 Acceptability of horizontal vibration this base curve should be

multiplied by the factor 60 for footbridge ……………………….. (35)

Figure 2.18 Relationship between the bridge capacity, pedestrian density

and their velocity …………………………………………………. (41)

Figure 2.19 Magnification factor for groups of up to 10 peoples …………….. (44)

Figure 2.20 T-bridge in Japan: (a) – layout; (b) – girder section …………….. (47)

Figure 2.21 The Maximum span length of bridges ………………………….... (52)

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Figure 2.22 The M-bridge in Japan ………………………………………….... (53)

Figure 2.23 Macintosh Island Park Suspension Bridge, Gold Coast,

Australia (a) – layout; (b) – additional cables ……………………. (54)

Figure 2.24 Stress ribbon bridge, Prague-Troja, Czechoslovakia ……………. (56)

Figure 2.25 the Millennium Footbridge in London, U.K. …………………….. (57)

Figure 2.26 Dynamic amplitude limits for pedestrian bridges ………………... (63)

Figure 3.1 Pre-tensioned cable supported bridge model: (a) – elevation; (b)

– top view; (c) – middle transverse bridge frame …………..…….. (69)

Figure 3.2 A typical cable profile ……………………………………………. (70)

Figure 3.3 Extra internal forces in cables ……………………………………. (72)

Figure 3.4 Hollow sections of bridge members (HSB): (a) – member of

bridge frame; (b) – supporting beams; (c) – deck units …………. (75)

Figure 3.5 Footbridge model C123 in SAP2000 …………………………….. (76)

Figure 3.6 Footbridge model C120 in SAP2000 ……………….……...…….. (77)

Figure 3.7 Footbridge model C103 in SAP2000 …………………………….. (77)

Figure 3.8 Footbridge model in Microstran …………..…………………….. (78)

Figure 3.9 Displacement degrees of freedom in the joint local coordinate

system …………..…………………………………………….….. (80)

Figure 3.10 Beam/frame element and corresponding coordinate systems …….. (80)

Figure 4.1 Applied loads: (a) – symmetric vertical loads; (b) – eccentric

vertical loads; (c) – lateral horizontal loads ……………………... (85)

Figure 4.2 Deflections and deformed bridge frame ………………………….. (86)

Figure 4.3 Maximum vertical deflections under symmetric applied vertical

loads with different cable sags …………………………………... (87)

Figure 4.4 Maximum tension forces in top cables under applied vertical

loads with different cable sags …………………………………… (87)

Figure 4.5 Maximum vertical deflections under applied vertical loads with

different top cable cross sectional area (diameter) ……………….. (88)


loads with different top cable cross sections …………………….. (88)

Figure 4.7 Maximum deflections under applied vertical loads with

different internal vertical forces ………………………………….. (90)


load with different internal vertical forces ………………………. (91)

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Figure 4.9 Maximum tension force in bottom cables under applied vertical

load with different internal vertical Forces ………………………. (91)

Figure 4.10 Sum of total horizontal forces of top and bottom cables under

applied vertical load with different internal vertical forces ……… (92)

Figure 4.11 Maximum deflection under applied load with different bottom

cable sags ……………………………………………………….... (93)

Figure 4.12 Tension forces in bottom cables under applied load with

different bottom cable sags ………………………………………. (93)

Figure 4.13 Maximum deflections under applied load with different bottom

cable sections …………………………………………………….. (94)

Figure 4.14 Tension forces in bottom cables under applied load with

different bottom cable sections …………………………………... (94)

Figure 4.15 Maximum vertical deflections with pre-tensioned bottom and

side cables ………………………………………………………... (95)

Figure 4.16 Horizontal deflections under lateral horizontal applied loads ……. (96)

Figure 4.17 Lateral horizontal deflections along bridge under eccentric

vertical loads ……………………………………………………... (97)

Figure 4.18 Vertical deflections along bridge under eccentric vertical loads …. (97)

Figure 4.19 Coupled lateral-torsional vibration modes – elevation; (b) – top

view; (c) – side view ……………………………………………. (100)

Figure 4.20 Coupled torsional-lateral vibration modes (a) – elevation; (b) –

top view; (c) – side view ……………………………………… (101)

Figure 4.21 Vertical vibration modes ……………………………………….. (102)

Figure 4.22 Longitudinal swaying vibration modes …………………………. (103)

Figure 5.1 The experimental bridge model and support system ……………. (120)

Figure 5.2 Detail of the transverse bridge frame ………………………….... (120)

Figure 5.3 Elevation of headstock …………………………………………... (122)

Figure 5.4 Detail of cable clamp ……………………………………………. (123)

Figure 5.5 Connection of support system …………………………………... (123)

Figure 5.6 The physical bridge model constructed in laboratory …………… (124)

Figure 5.7 The support system of physical bridge model …………………... (125)

Figure 5.8 The experimental bridge model in Microstran …………….…... (127)

Figure 5.9 The experimental bridge model in SAP2000 ……….…………. (127)

Figure 5.10 Distribution of transducers on physical bridge model …………... (128)

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Figure 5.11 Acceleration transducers installed on the middle bridge frame … (128)

Figure 5.12 Data acquisition and analysis system …………………………… (129)

Figure 5.13 Applied static vertical loading system …………………………... (129)

Figure 5.14 Case 1: accelerations at point 1 (initial lateral excitation) ……… (131)

Figure 5.15 Case 1: accelerations at point 1 (initial torsional excitation) …… (131)

Figure 5.16 Case 1: accelerations at point 1 (initial vertical excitation) …….. (131)

Figure 5.17 Case 1: vertical accelerations at point 1 and point 4 (initial

torsional excitation) ……………………………………………... (132)

Figure 5.18 Case 1: vertical accelerations at point 1 and point 4 (initial

vertical excitation) ………………………………………………. (132)

Figure 5.19 Case 1: spectra of accelerations at point 2 (initial lateral

excitation) ……………………………………………………….. (133)

Figure 5.20 Case 1: spectra of accelerations at point 2 (initial torsional

excitation) ……………………………………………………….. (134)

Figure 5.21 Case 1: spectra of accelerations at point 2 (initial vertical

excitation) ……………………………………………………….. (134)


excitation) ……………………………………………………….. (136)


excitation) ………………………………………………………. (136)


excitation) ……………………………………………………….. (137)


excitation) ……………………………………………………….. (138)


excitation) ………………………………………………………. (139)


excitation) ……………………………………………………….. (139)


excitation) ………………………………………………………. (140)


excitation) ……………………………………………………….. (140)


excitation) ……………………………………………………….. (141)

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Figure 6.1 Vertical force-time functions of a footfall ……………………… (151)

Figure 6.2 Vertical force-time function of normal walk ………………….... (152)

Figure 6.3 A typical continuous vertical force-time function …………….... (153)

Figure 6.4 A typical continuous lateral force-time function ……………….. (153)

Figure 6.5 Bridge model C120: lateral dynamic deflection at pacing rate

of 1.5 Hz (ζ=0.01) ………………………………………………. (158)

Figure 6.6 Bridge model C120: vertical dynamic deflection at pacing rate

of 1.5 Hz (ζ=0.01) ………………………………………………. (158)

Figure 6.7 Bridge model C120: steady dynamic deflections in details at

pacing rate of 1.5 Hz (ζ=0.01) ………………………………….. (159)

Figure 6.8 Bridge model C120: lateral dynamic acceleration at pacing rate

of 1.5 Hz (ζ=0.01) ………………………………………………. (160)

Figure 6.9 Bridge model C120: vertical dynamic acceleration at pacing

rate of 1.5 Hz (ζ=0.01) ………………………………………….. (160)

Figure 6.10 Bridge model C120: dynamic lateral deflection at pacing rate

of 1.0943 Hz (ζ=0.01) …………………………………………... (162)

Figure 6.11 Bridge model C120: dynamic vertical deflection at pacing rate

of 1.0943 Hz (ζ=0.01) …………………………………………... (162)

Figure 6.12 Bridge model C120: dynamic lateral acceleration at pacing rate

of 1.0943 Hz (ζ=0.01) …………………………………………... (163)

Figure 6.13 Bridge model C120: dynamic vertical acceleration at pacing

rate of 1.0943 Hz (ζ=0.01) ……………………………………… (163)

Figure 6.14 Bridge model C120: dynamic lateral deflection at pacing rate

of 1.1949 Hz (ζ=0.01) …………………………………………... (165)

Figure 6.15 Bridge model C120: dynamic vertical deflection at pacing rate

of 1.1949 Hz (ζ=0.01) …………………………………………... (165)

Figure 6.16 Bridge model C103: dynamic lateral deflection under crowd

walking dynamic loads at pacing rate of 1.5 Hz (ζ=0.01) …….... (168)

Figure 6.17 Bridge model C103: dynamic vertical deflection under crowd

walking dynamic loads at pacing rate of 1.5 Hz (ζ=0.01) ……... (168)


walking dynamic loads at pacing rate of 0.7531 Hz (ζ=0.01) ….. (169)

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Figure 6.24 Tension forces of bridge model C123 in resonant vibration

with the mode L1T1 at pacing rate of 1.5 Hz (ζ=0.01) ……….... (173)

Figure 6.25 Tension forces of bridge model C123 in resonant vibration

with the mode V1 at pacing rate of 0.9046 Hz (ζ=0.01) ………... (173)

Figure 6.26 Detail of tension forces in bridge model C123 in resonant

vibration with the mode V1 at pacing rate of 0.9046 Hz

(ζ=0.01) …………………………………………………………. (174)

Figure 6.27 Bridge model C120: lateral deflection at pacing rate of 2.0 Hz

………………………………………………………………….. (180)

Figure 6.28 Bridge model C120: vertical deflection at pacing rate of 2.0 Hz

………………………………………………………………….. (180)

Figure 6.29 DAF of vertical deflection …………………………………….... (181)

Figure 6.30 DAF of lateral deflection ……………………………………….. (181)

Figure 6.31 Lateral acceleration …………………………………………….. (182)

Figure 6.32 Vertical acceleration ……………………………………………. (182)

Figure 6.33 Bridge model C120: dynamic lateral deflection under eccentric

walking loads at pacing rate of 1.5 Hz …………………………. (190)

Figure 6.34 Bridge model C120: dynamic vertical deflection under

eccentric walking loads at pacing rate of 1.5 Hz ……………….. (190)


walking loads at pacing rate of 1.0943 Hz …………………….... (192)


eccentric walking loads at pacing rate of 1.0943 Hz ………….... (192)

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eccentric walking loads at pacing rate of 1.1949 Hz ……………. (193)




walking loads at pacing rate of 0.8982 Hz ……………………... (195)

Figure 6.41 Lateral deflections of footbridges under static lateral force …… (199)

Figure 6.42 Coupling coefficients of the first coupled mode L1T1 with lateral

natural frequency ………………………………………………… (216)

Figure 6.43 Dynamic amplification factors with the natural frequency

(ζ=0.01) ………………………………………………………... (216)

Figure 6.44 Dynamic amplification factors with the natural frequency

(ζ=0.05) ……………………………………………………..…. (217)

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LIST OF TABLES Table 2.1 Typical pacing and jumping frequency in Hz ……………………. (15)

Table 2.2 DLFs for single person force models after different authors …….. (19)

Table 2.3 Common value of damping ratio for beam-type footbridge ……... (28)

Table 2.4 Measured damping ratios (for vertical ζv, horizontal ζh and

torsional ζt) for some footbridges ………………………………... (30)

Table 2.5 Case reports of excessive vibrations in footbridges ……………. (43)

Table 2.6 The Leading 10 long-span bridges worldwide by the year 2005

……………………………………………………………………. (51)

Table 4.1 Natural frequencies and corresponding modes with the cable

sag ………………………………………………………………. (106)

Table 4.2 Natural frequencies and corresponding modes with the cross

sectional area (diameter) ………………………………………... (107)

Table 4.3 Internal vertical forces and the natural frequencies and their

corresponding modes …………………………………………… (108)

Table 4.4 Internal horizontal forces and the natural frequencies and their

corresponding modes ………………………………………….... (109)

Table 4.5 Cross sectional area (diameter) of the pre-tensioned bottom

cables and the vibration properties ……………………………... (110)

Table 4.6 Effects of additional weight on the natural frequencies of un-

pre-tensioned suspension footbridges …………………………... (111)

Table 4.7 Effects of additional weight on the vibration properties of pre-

tensioned suspension footbridges ……………………………….. (112)

Table 4.8 Effects of applied load on the vibration properties of un-pre-


Table 4.9 Effects of applied load on the vibration properties of pre-


Table 4.10 Effect of internal forces on the vibration properties of

footbridge with span length of 40 m ………………………….. (115)

Table 4.11 Effect of internal forces on the vibration properties of

footbridges with span length of 120 m …………………………. (115)

Table 5.1 Material properties of stainless steel wires ……………………… (125)

Table 5.2 Natural frequencies from experimental testing ………………… (133)

- xix -

Table 5.3 Static vertical deflections measured from experimental testing … (141)

Table 5.4 Natural frequencies of physical bridge model in case 1 and

case 2 …………………………………………………………… (142)

Table 5.5 Natural frequencies of physical bridge model in case 3 and

case 4 …………………………………………………………… (143)

Table 5.6 Static vertical deflection of physical bridge model in case 1 and

case 2 (in mm) ………………………………………………….. (144)

Table 5.7 Static vertical deflection of physical bridge model in case 3 and

case 4 (in mm) …………………………………………………... (144)

Table 5.8 Effect of modification factors of cable section properties ……… (148)

Table 6.1 Vibration properties of different Bridge models ………………... (156)

Table 6.2 Bridge model C120: Dynamic deflections of the first vibration

modes excited by pedestrians …………………………………... (161)

Table 6.3 Bridge model C120: Dynamic accelerations of the first

vibration modes …………………………………………………. (161)

Table 6.4 Bridge model C123: Dynamic deflections of the first vibration

modes …………………………………………………………… (166)

Table 6.5 Bridge model C123: Dynamic accelerations of the first

vibration modes …………………………………………………. (167)

Table 6.6 Tension forces in bridge model C120 …………………………... (174)

Table 6.7 Tension forces in bridge model C123 …………………………... (175)

Table 6.8 Bridge model C120: resonant deflections of higher vibration

modes …………………………………………………………... (177)

Table 6.9 Resonant deflections of higher vibration modes (C123) ……….. (177)

Table 6.10 Bridge model C120: Resonant deflections of the coupled mode

L1T1 under different load cases …………………………….…. (183)

Table 6.11 Bridge model C120: resonant deflections of the vertical mode

V1 under different load cases …………………………………... (184)

Table 6.12 Bridge model C123: Resonant deflections of the coupled mode

L1T1 under different load cases ………………………………... (185)

Table 6.13 Bridge model C123: resonant deflections of the vertical mode

V1 under different load cases …………………………………... (185)

Table 6.14 Bridge model C120: effect of damping on the coupled mode

L1T1 …………………………………………………………..... (186)

- xx -

Table 6.15 Bridge model C120: effect of damping on the vertical mode V1

…….…………………………………………………………… (186)

Table 6.16 Bridge model C123: effect of damping on the coupled mode

L1T1 ……………………………………………………………. (187)


…………………………………………………………………. (188)

Table 6.18 Bridge model C120: Dynamic deflections under eccentric

walking dynamic loads …………………………………………. (189)

Table 6.19 Bridge model C123: Dynamic deflections under eccentric

walking dynamic loads …………………………………………. (194)

Table 6.20 Vibration properties of footbridges with different cable section

…………………………………………………………………... (198)

Table 6.21 Resonant lateral deflection of footbridges with different cable

section …………………………………………………………... (198)

Table 6.22 Coupling coefficients of coupled vibration modes of

footbridges with different cable sections ……………………….. (200)

Table 6.23 Vibration properties and coefficients with cable sag …………... (202)

Table 6.24 Dynamic lateral deflection of footbridge models with cable sag

…………………………………………………………………... (204)

Table 6.25 Natural frequencies and coupling coefficients with span length

………………………………………………………………….. (206)

Table 6.26 Resonant lateral deflection with span length …………………… (207)

Table 6.27 Effect of synchronization on the dynamic response of the

footbridge with cable configuration C120 …………………….... (208)

Table 6.28 Effect of synchronization on the dynamic response of the

footbridge with cable configuration C123 …………………….... (210)

Table 6.29 Vibration properties and coupling coefficients of bridge model

C120 with span length of 40 m …………………………………. (210)






C120 with span length of 120 m ………………………………... (214)

- xxi -


C123 with span length of 120 m ……………………………….. (215)

- xxii -

NOTATIONS

a horizontal distance of two adjacent bridge frames

A acceleration

Al, Av lateral and vertical accelerations

ALk, AVk lateral and vertical accelerations at point k

Amax, Amin maximum and minimum accelerations in entire vibration

Astdmax, Astdmin maximum and minimum accelerations in steady vibration

Aumax, Aamax amplitudes of deflection and acceleration in entire vibration

Austd, Aastd amplitudes of deflection and acceleration in steady vibration

D1, D2, D3 diameters of top, bottom and side cables

fn pacing rate of normal walk

fp pacing rate of walking load

E1, E2, E3 Young’s modulus of top, bottom and side cables

F1, F2, F3 cable sags of top, bottom and side cables

Fn[t] force function of normal walk

Fnl(t) continuous lateral force function

Fnv(t) continuous vertical force function

G gravity of one bridge frame and deck units between two adjacent

bridge frames

i,j,k integer numbers

K middle node of a cable profile

L span length

LmTn coupled lateral-torsional modes

LSWm longitudinal swaying modes

m, n number of half wave

M mass density

N number of segments of a cable profile

Mul, Muv mean value of lateral and vertical deflections

Mal, Mav mean value of lateral and vertical accelerations

qnv(t) vertical dynamic force

qnl(t) lateral dynamic force

qsv(t) vertical ramped static force

Qint internal horizontal force

- xxiii -

t time

T, Tji tension force

T1, T2, T3 tension forces in top, bottom and side cables

Tstdmax, Tstdmin maximum and minimum tension force in steady vibration

ATstd amplitude of tension force in steady vibration

MTstd mean value of tension force in steady vibration

Tn period of normal walk

Tnc contact time

Tp period of walking load

TmLn coupled torsional-lateral modes

U deflection

Ul, Uv lateral and vertical deflections

Vm vertical modes

W equal concentrated load on a cable

Wint internal vertical force

xji,yji coordinates of ith node on jth cable

∆Lji tensile deformation of the i th segment, jth cable

∆L1i, ∆L2i, ∆L3i initial distortions in the i th cable segment of top, bottom and

side cables

∆T1i, ∆T2i, ∆T3i temperature loads in the i th cable segment of top, bottom and

side cables

αi, βji, γji coefficients

η time factor

f degree of synchronization

y coupling coefficient

ζ damping ratio

ζv, ζh, ζt damping ratios of vertical, horizontal and torsional modes

- xxiv -

ABBREVIATIONS

C100 bridge model with slack bottom and side cables

C103 bridge model with top supporting cables and pre-tensioned side

reverse profiled cables

C120 bridge model with top supporting cables and pre-tensioned bottom

reverse profiled cables

C123 bridge model with top supporting cables and pre-tensioned bottom as

well as side pre-tensioned cables

DAF dynamic amplification factor

DLF dynamic load factor

HSB bridge model with hollow section members

SSB bridge model with solid section members

LD load case consisting of only lateral dynamic force

LDF lateral dynamic force

LVD dynamic loads consisting of lateral and vertical dynamic forces

LVS load case consisting of lateral, vertical dynamic force and vertical

static force

UPTB un-pre-tensioned bridge model

VD load case with only vertical dynamic load

VDF vertical dynamic force

VSF vertical (ramped) static force

VDS load case consisting of vertical static force and quasi-static vertical

dynamic force

- 1 -

Introduction

1.1 Background

Aesthetically pleasing slender bridges, both pedestrian and road bridges, have gained

popularity in recent times to meet modern transportation needs. Due to the

application of high strength and light weight material, modern footbridges can cross

longer spans and be constructed much lighter than ever before. As a consequence,

these footbridge structures are slender and flexible with low stiffness, low structural

mass and damping ratio. Such slender footbridges are prone to vibration and can

become dangerously “live” and experience excessive vibration induced by

pedestrians and other human activities.

In 2000, the Millennium Bridge in London experienced unexpected excessive lateral

vibration when crowd pedestrians walked across the new and attractive footbridge

during its opening day. The footbridge was closed two days later in order to fully

investigate the cause of the movements and the procedure for retrofitting. It was

found that this vibration problem arose as the design did not account for the human-

induced synchronous lateral excitation for which there was no code provision. To

suppress the excessive human-induced lateral vibration, a large number of viscous

and tuned mass dampers were installed to increase damping [Dallard et al. 2001c].

The retrofitting was expensive and inconvenient costing nearly £5 million and lasting

18 months. It was also found that many other bridges, with different structural forms,

had also been vulnerable to synchronous lateral excitation and they include the

Harbour Road Bridge in Auckland, Tom Holliday Bridge in London, Alexandra

Bridge in Ottawa, the “T” bridge in Tokyo and others, indicating that synchronous

lateral excitation is not limited to any particular structural form of the bridge

structure. As this vibration problem was becoming alarming and could occur on a

range of different structural types of footbridges, it attracted more than 1000 press

articles and over 150 broadcasts in the media around the world after the Millennium

1

- 2 -

Bridge saga. A major international conference, entitled as Footbridge 2002, was held

in Paris, in November, 2002, to discuss the problem of design and dynamic

behaviour of footbridges worldwide. This was followed by the 2nd international

conference, Footbridge 2005, held recently in Venice, Italy, in December, 2005.

It has been established that the synchronous lateral excitation experienced by the

Millennium Bridge could occur on any footbridge with a lateral frequency less than

1.3 Hz under a certain loading pattern. The excessive vibrations of footbridges were

found to be caused by resonance of one or more vibration modes induced by walking

pedestrians when the footbridges have the natural frequencies coinciding with the

dominant frequencies of human-induced dynamic loads. However, most footbridges

will have natural frequencies within the frequency range of human activities when

the bridge span increases to more than 50 m [Bachmann and Ammann 1987, Dallard

et al. 2001b]. This will make more long span flexible footbridges, particular slender

cable supported footbridges, with such low frequencies vulnerable to the same

problem experienced by the Millennium Bridge in London.

Although some bridge design codes such as BS 5400 [2002] and AS 5100-2004 [SAI

2004] have been updated in recent years, they do not give adequate information to

address the synchronous excitation on footbridges and the complexity of multi-modal

vibration. For example, the updated version of BS 5400: BD 37/01 [2002] requires a

check on the vibration serviceability in the lateral direction and a detailed dynamic

analysis for all footbridges having fundamental lateral natural frequency below 1.5

Hz. However, the checking procedure is not given, and the code does not cover

synchronous vertical and lateral excitation arising from groups or crowds of

pedestrians under normal usage. Such loadings can be significantly greater than

normal code provisions. The new Australia standard for bridge design AS 5100.2-

2004 (Part 2: bridge loads) [SAI 2004] provides a clause for the vibration

serviceability of footbridges. It requires that the vibration of superstructures of

pedestrian bridges with resonant frequencies for vertical vibration in the range 1.5 Hz

to 3.5 Hz should be investigated as a serviceability limit state. For the pedestrian

bridges with the fundamental frequency of horizontal vibration below 1.5 Hz, a

special consideration is required to treat the possibility of lateral movements of

unacceptable magnitude excited by pedestrians. The Code also mentions that bridges

- 3 -

with low mass and damping that are expected to be used by crowds of people, are

particularly susceptible to human-induced vibrations and specialist literature should

be referred to. However there is no detail information about the special consideration

and the specialist literature mentioned in the code.

Cable supported footbridges are becoming the main structural form of bridges to

cross large spans due to the ease of construction and low material consumption.

However, they are always slender and flexible with low stiffness, low mass and low

natural frequencies. Such slender footbridge structures are often weak in the lateral

direction and are easy to suffer excessive lateral vibration induced by walking

pedestrians. This has been shown by the poor dynamic performance of several cable

supported footbridges such as the Millennium Bridge in London (suspension)

[Dallard et al. 2000], M-Bridge (suspension) [Nakamura 2003] and T-bridge (cable-

stayed) [Fujino et al. 1993] in Japan, etc. It is evidently important that research is

required to investigate and understand the structural behaviour of such footbridge

under human-induced dynamic loads.

The main concern of the conceptual study carried out in this thesis will be on the

static and dynamic performance of slender suspension footbridges with pre-tensioned

reverse profiled cables. Extensive numerical analysis will be conducted on such

slender footbridges to investigate the load deformation performance under quasi-

static loading, the vibration properties and dynamic characteristics under human-

induced walking dynamic loads. The research findings will expand the knowledge

base of the performance of suspension footbridges and be helpful to better

understand their structural behaviour and to ensure the safety and efficient

serviceability in practical bridge designs.

1.2 Research objectives

A suspension footbridge model with pre-tensioned reverse profiled cables in vertical

and horizontal planes is proposed to investigate the dynamic characteristics of

slender footbridges under human-induced dynamic loads. The proposed suspension

footbridge model was chosen due to its range of low natural frequencies and

- 4 -

especially due to the ease of modifying the dynamic properties by varying the cable

profiles and cable forces, and hence obtaining a range of natural frequencies which

coincide with the frequency range of human-induced dynamic loads.

The main aim of this thesis is to generate fundamental knowledge and contribute to a

better understanding of dynamic performance of such slender suspension footbridges

under human-induced dynamic loads, and to investigate the effects of structural

parameters on the vibration properties and structural behaviour. The specific research

objectives are as follows:

• To study the effects of structural parameters on structural behaviour and on

vibration properties under different quasi-static loads;

• To design and test an experimental bridge model with low natural frequencies

and calibrate computer models;

• To model the crowd walking dynamic loads and use this model to simulate

synchronous excitations on footbridge structures;

• To carry out extensive numerical study (Finite element analysis) on the

dynamic characteristics of slender footbridge structures with low natural

frequencies under crowd walking dynamic loads;

• To evaluate the research results and generate information for providing design

guidance for vibration sensitive footbridge structures.

1.3 Methodology

In order to achieve the research aims and pursue the research objectives, extensive

numerical analysis using Finite Element techniques supported by experimental

testing is carried out on the proposed suspension footbridge model. Research results

are mainly generated by the computer simulation. A similar physical experimental

bridge model is designed and constructed, and experimental testing is conducted to

calibrate the numerical bridge model.

- 5 -

To carry out the computer simulations, the computer model of the proposed slender

suspension bridge in full size is established. The span length and cable sag

considered varies from 40 m to 120 m and from 1.2 m to 2.4 m respectively. The

entire computer simulation is divided into two stages. In the first stage, the finite

element packages Microstran (V8) [Engineering Systems 2002] is adopted to

investigate the static performance and vibration properties as well as the effects of

important structural parameters. In the second stage, the finite element package

SAP2000 (V9) [CSI 2004] is employed to study the dynamic characteristics of the

slender suspension footbridge model under human-induced dynamic loads. Resonant

vibration using non-linear time history analysis is used to simulate the dynamic

behaviour of the proposed footbridge model under different synchronous excitations

induced by walking pedestrians. Research results are evaluated to generate

information to better understand the dynamic performance and to improve the

vibration serviceability of slender footbridge structures.

To ensure that the computer simulations are correct and efficient, a physical bridge

model was designed for the purpose of calibration and it has similar features to the

proposed suspension footbridge. The experimental model is 4.5 m in span length and

its fundamental natural frequencies can vary from about 1.0 Hz to 4.0 Hz. Results

from experimental testings are compared with those from computer simulations

using Microstran and SAP2000 to calibrate the computer models.

1.4 Outline of the thesis

This thesis consists of seven chapters and one bibliography. A general introduction to

this conceptual study on slender suspension footbridges is described in chapter 1, and

the relevant references are listed in bibliography.

A comprehensive literature review is carried out in chapter 2 to provide detailed

background knowledge on human-induced dynamic loads and the dynamic

performance of some footbridge structures. The main topics reviewed include: the

measuring and modelling of human-induced loads; vibration properties and dynamic

- 6 -

performance of footbridges; some existing design guidelines related to the vibration

serviceability of footbridges.

Chapter 3 describes the proposed slender suspension footbridge model with pre-

tensioned reverse profiled cables in details. The descriptions include the cable

systems and their cable profiles; pre-tensions and internal forces; footbridge models

with different cable configurations considered in the numerical analysis.

Chapter 4 details the investigation into the load-deformation performance and

vibration properties of the proposed slender suspension footbridge. In this chapter,

the static behaviour of the bridge model under different quasi-static loads is studied

and the effects of some important parameters on the structural stiffness are discussed.

The vibration modes and their corresponding natural frequencies are described in

details and a series of numerical analyses are carried out to investigate the effects of

important parameters on the vibration properties.

Chapter 5 describes the design of the physical bridge model and the experimental

testing. In this chapter, the physical bridge model is also modelled and simulated

using Microstran and SAP2000, and all the results are compared with those from

experimental testing to modify the computer models and ensure that the numerical

analysis is correct.

Chapter 6 details the extensive numerical analyses on the dynamic performance of

slender suspension footbridges with different cable configurations under human-

induced loads. A force model for crowd walking dynamic loads is proposed to model

the human-induced synchronous excitations and resonant vibration is used to

simulate the dynamic performance when pedestrians walk across the footbridge at a

pacing rate coinciding with one of the natural frequencies. The dynamic performance

in the lateral direction and the effects of structural parameters are the main concern.

Here the influence of damping on the dynamic response is also investigated.

Conclusions and discussions are presented in Chapter 7. Some suggestions for future

research work are also offered briefly in this chapter.

- 7 -

Human-induced dynamic loads and vibration performance of footbridges

2.1 Introduction

As the primary purpose of footbridges is to carry pedestrians and cyclists, the bridge

structures need to be designed and constructed to be safe and exhibit satisfactory

serviceability for the users. However, due to the application of light weight and high

strength materials, larger and larger spans are chosen while the thicknesses of beams

and slabs have kept decreasing which result in bridge structures being more slender

and flexible. Such slender bridge structures with small stiffness and low mass are

more prone to vibrations induced by pedestrians and other human activities than

those stiffer structures with larger masses. This has been shown by the dynamic

behaviours of many footbridges around the world such as the T-bridge in Japan, and

the Millennium Bridge in London, which have experienced excessive lateral

vibrations caused by pedestrians. As a consequence, the serviceability has become a

major concern of bridge engineers and researchers and governmental design

consideration of slender footbridges.

In this chapter, the features of dynamic loads induced by human activities will be

described first and the dynamic behaviours of slender and flexible footbridges under

such dynamic loads will then be reviewed.

2.2 Human-induced dynamic loads

As footbridges are mainly designed for the conveyance of pedestrians and cyclists,

the dominant design loads are those induced by pedestrians. Since the pedestrian

loads are produced by different kinds of human activities such as walking, running,

jumping and bouncing as well as other human movements, these loads are in fact

human-induced dynamic loads and it was noted very early that these dynamic

excitations could cause excessive vibration, and in extreme cases even a collapse of

2

- 8 -

the structure. For example, the oldest case reported of footbridge failure due to these

human-induced dynamic loads in detail was probably the collapse of Broughton

Bridge in UK. The bridge structure collapsed in 1831 when 60 soldiers crossed the

bridge marching in unison. It was this event that prompted the placement of some

notices on several bridges with a warning to troops to break steps when crossing

[Tilly et al. 1984]. For modern footbridges, these human-induced dynamic

excitations could cause serious serviceability problems rather than problems of

safety, especially for the slender footbridges with natural frequencies lower than 5.0

Hz, independent of the structural form. These human-induced dynamic excitations

could also cause excessive vibration and serviceability problems for other kinds of

slender and flexible structures such as floors, staircases, gymnasium and stadiums.

2.2.1 Characteristics and measurements of walking and running dynamic

loads induced by a single person

For footbridge structures, walking and running are more important human activities

than others. Walking is the most common activity considered in design since it is

related to the normal use of footbridges, while running, on the other hand, could

generally be seen as a peripheral activity. Other human activities such as jumping,

bouncing, lateral body swaying and rhythmic body movements are related principally

to deliberate excitations and they are more important and occur more often in other

structures such as concert halls and stadiums.

Walking is considered as a motion with medium pacing rate. To describe this kind of

motion, Inman et al. [1994] explained that there are two basic requisites to

characterising the act of walking and the dynamic forces induced by walking:

continuous reaction forces to support the body (i.e., there is at least one foot in

contact with the ground) and periodic movement of each foot from one supporting

point to the next in the walking direction. This involves a short period when both feet

are on the ground and during this period the support of the body is transferred from

one leg to another. While running is always considered as a motion of higher pacing

rate than walking and the most important characteristic is that there is a

discontinuous ground contact when the support of body is transferred from one foot

to another (Figure 2.1). These requisites and characteristic are always observable and

- 9 -

are the key points in investigating and modelling the dynamic forces induced by

walking and running.

Figure 2.1 Typical pattern of running and walking forces [Galbraith and Barton 1970]

The measurements of human-induced dynamic forces were carried out for different

purposes other than investigating their effect on structures in early research. One of

the first measurements was conducted by Harper et al. [1961] and Harper [1962].

Using an instrumented force plate installed in a rigid platform, they measured the

vertical and longitudinal forces generated by a footstep with the aim to investigate

the friction and slipperiness of floors. Andriacchi et al. [1977] used a force plate to

measure the single step walking forces in all three directions with the aim to

investigate the difference in the step patterns between patients who were healthy and

those with abnormalities. They reported that increasing walking velocity led to

increasing step length and peak force magnitude. The typical shapes of force-time

functions are presented in Figure 2.2. In order to get data for a study to improve

detection of intruders by sensing signal from footstep, Galbraith and Barton [1970]

conducted a more comprehensive study and measured the vertical force induced by

human activities ranging from slow walking to running on an aluminium plate. They

carried out a very detailed investigation on the effects of many parameters such as

pacing rate, subject weight, footwear and surface condition on the dynamic loads.

This research revealed that the pacing rate and subject weight were identified as

halla

This figure is not available online. Please consult the hardcopy thesis available from the QUT Library

- 10 -

important parameters which increase led to higher peak force value, while the

footwear and walking surface were of minor importance (Figure 2.3). Similar

measurements were carried out by many other researchers. In order to get more

advanced and informative measurements of continuous walking time histories

comprising several steps, Blanchard et al. [1977] used a gait machine to measure the

individual step forces. Ohlsson [1982] measured the vertical walking force to study

the vibration of flexible floor and human discomfort. Ebrahimpour et al. [1992;

1994; 1996] used a platform instrumented with several force plates to measure single

step forces as well as crowd walking dynamic forces. Kerr [1997] measured the step

forces induced by pedestrians on flexible staircases.

Figure 2.2 Typical shapes of walking force in (a) – vertical, (b) – lateral and (c) – longitudinal direction [Andriacchi et al. 1977]

Regarding the dynamic loading of footbridges, Wheeler [1980; 1982] conducted a

very comprehensive research into the dynamic forces caused by the activities from

slow walking to running (Figure 2.4). He systematised the previous work of other

researchers and investigated the effect of pacing rate on the other parameters such as

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- 11 -

step length, moving velocity, peak force and contact time (the time while one foot is

in contact with the ground). It should be noted that in shapes of force-time functions,

the pacing rate of normal walk is higher than that of brisk walk. Rainer et al. [1988]

measured the continuous dynamic forces during walking, running and jumping by

using an instrumented floor strip as platform to investigate the dynamic loading and

response of footbridges. Fourier amplitude spectrum of results showed that the force

produced by one person walking consists of distinct frequency components at integer

multiples (harmonics) of the footstep rate, with amplitudes that decrease with

increasing frequency.

Figure 2.3 Force-time functions for various pacing rates, footwear and surface conditions [Bachmann and Ammann 1987]

In general, dynamic loads induced by human activities are different from person to

person, and they are affected by many parameters such as pacing rate, stepping

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- 12 -

particularities (heel/ball contribution), person’s weight, person’s gender, type of

footwear and surface condition of structure [Bachmann and Ammann 1987], as well

as the measuring procedures and the interaction of pedestrian and structure.

However, measurements of dynamic forces in the many references confirmed that

the shape of vertical force-time function for walking with a medium pacing rate

always has one saddle and two peaks (Figure 2.5) [Bachmann and Ammann 1987].

For different test person with a given pacing rate, the shape is similar, but the

amplitude increases with the weight. This similarity is due to the basic mechanism of

walking common for all persons. The first observable peak corresponds to the heel

strike, while the second to the pushing off with the ball of the foot. This feature

disappears with increasing pacing rate and degenerates to a single peak of sharp rise

and descent when the person is running from low to high pacing rate, the contact

time of foot on the surface decreases while the load maximum increases. For

strolling with a frequency below 1.0 Hz, the maximum load hardly exceeds the

weight of the person, it increases by about 25% to 30% for 2.0 Hz and by 50%

around 2.5 Hz, and at about 3.5 Hz it reaches about double of the weight of the test

person [Bachmann and Ammann 1987].

Although the human-induced dynamic loads are affected by many parameters, pacing

rate is the most important one. This has been shown by Figure 2.3 and mentioned

before. A comprehensive study into pacing rate of normal walking was conducted

by Matsumoto et al. [1972]. They investigated more than 500 test subjects randomly

chosen and found that the pacing rate follows an almost normal distribution with a

mean value of 2.0 Hz and a standard deviation of 0.173 Hz and they suggested that a

practical range for pacing rate can be set from 1.6 Hz to 2.4 Hz (Figure 2.6). Similar

results from other studies were cited by Eyre and Cullington [1985] as well as others.

Bachmann and Ammann [1987] mentioned that Schulze [1980] got a normal

distribution of walking pacing rate with the mean value of 2 Hz and a standard

derivation of 0.13 Hz, while Kramer and Kebe [1979] got one with a mean value of

2.2 Hz and a standard derivation of 0.3 Hz. Another comprehensive study into

vertical footfall forces was carried out by Kerr [1997] and Kerr and Bishop [2001],

who measured over 1000 individual footfall forces time histories from 40 subjects

walking between 1 and 3 Hz on a level surface. Based on the observations, they

- 13 -

concluded that the range of “comfortable” walking pace is between 1.7 and 2.1 Hz

with a mean value of approximately 1.9 Hz.

Figure 2.4 Typical vertical force patterns for different types of human activities [Wheeler 1982]

Figure 2.5 Force function resulting from footfall overlap during walking with a pacing rate of 2 Hz [Bachmann and Ammann 1987]

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- 14 -

t

Figure 2.6 Normal distribution of pacing frequencies for normal walking [Matsumoto et al. 1972]

Figure 2.7 Dependence of stride length, velocity, peak force and contact time on different pacing rates [Wheeler 1982]

Unfortunately, there is no comprehensive investigation similar to the one given by

Matsumoto et al. [1972] into the dynamic forces induced by other types of human

activities such as running and jumping et al., though much research was carried out.

For example, Wheeler [1980; 1982] presented dependence of many parameters for

human activities from walking to running as a function of pacing rate (Figure 2.7).

These parameters include step length, moving velocity, peak force and contact time.

He reported that pacing rate as high as 5.0 Hz, related to forward speeds of 7 to 8

halla


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m/s, are unlikely to be attained by an untrained runner. Although running is

considered as a motion with higher pacing rate than walking, there is no precise

pacing rate to mark the transferring from walking to of running, since brisk and fast

walking could be developed at the same pacing rate as jogging (slow running).

Baumann and Bachmann [1987] suggested the range from 2.4 to 2.7 Hz for normal

running and up to 3.2 Hz for intensive running, while Tilly et al. [1984] reported that

pacing rates higher than 3.5 Hz are rare on public footpaths.

Based on the research described above, there are some proposals as to the frequency

ranges for different human activities from walking to running, as well as others such

as jumping and bouncing. For example, Bachmann et al. [1997] defined typical

pacing rate ranges of 1.6 – 2.4 Hz for walking, 2.0 – 3.5 Hz for running, 1.8 – 3.4 Hz

for jumping, 1.5 – 3.0 Hz for bouncing and 0.4 – 0.7 Hz for horizontal body swaying

while stationary. Later, Bachmann [2002] changed these ranges as 1.4 – 2.4 Hz for

walking, 1.9 – 3.3 Hz and 1.3 – 3.4 Hz for running and jumping (Table 2.1), as many

footbridges were reported experiencing excessive lateral vibration at lateral

frequencies below 0.8 Hz.

Table 2.1 Typical pacing and jumping frequency in Hz [Bachmann 2002]

In general, the dynamic forces induced by human activities are generated in three

axial directions. The vertical force is considered the main component of dynamic

loads and is produced due to the up and down movements as the body passes over a

supporting leg and descends as this leg moves behind. In the lateral direction, the

forces are caused by the periodic sway of the body weight from one leg to the other.

These two forces are of most interest as sources of excitation in footbridges although

there are forces which are also applied in the longitudinal direction [Pimental 1997].

Though many measurements were carried out, they were mainly on the investigation

of vertical force. The literature contains relatively little data on the horizontal and

longitudinal forces due to human activities such walking, running and jumping.

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This table is not available online. Please consult the hardcopy thesis available from the QUT Library

- 16 -

Figure 2.2 shows a typical force function in three directions at the pacing rate of 2.0

Hz given by Andriacchi et al. [1977]. Bachmann and Ammann [1987] cited an

investigation into the dynamic force in three directions conducted by Schulze [1980].

The dynamic forces measured were induced by a relatively light person of 587 N at

pacing rate of 2.0 Hz and their Fourier amplitudes are given by Figure 2.8. It can be

seen that the lateral and longitudinal forces are much smaller than the vertical one.

The lateral sway of the person’s centre of gravity occurs with half the pacing rate

(1.0 Hz) whereas the longitudinal one is dominated by the full pacing rate as in the

vertical direction. Based on an investigation on the harmonic analysis of lateral force

traces of 40 subjects walking on a stationary platform, Willford [2002] stated that the

first harmonic DLF (dynamic load factor, i.e., the ratio of Fourier amplitude to the

static weight) ranges between 0.03 and 0.07.

Figure 2.8 Harmonic load components (Fourier amplitudes) of the direction load-time

functions [Bachmann and Ammann 1987]

2.2.2 Modelling of walking and running dynamic loads induced by a single

person

It is necessary to model the human-induced forces analytically in order to apply them

into the dynamic analysis of structures. However this is a complicated task, since the

footfall forces induced by human activities are affected by many factors such as

pacing rate, subject’s weight, footwear and surface condition of the dynamic loads

etc [Galbraith and Barton 1970], and they are different from person to person, and

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change not only in time but also in space, as well as with human-structure

interaction. Considering the common features of human-induced forces, the force

models based on some justifiable assumptions do exist and are used in contemporary

design.

It is believed that dynamic force induced by footfalls is a summation of the forces

produced by continuous paces and may be simulated by pulse trains created by a

single footstep force. This assumes that the force from each footstep is approximately

the same and that the time the feet overlap is kept constant for a given pacing rate,

i.e. the force has a periodic nature. For this reason, the dynamic force induced by a

single person is always represented by means of Fourier series which decomposes

the periodic force into distinct harmonic components. The predominant first

harmonic is supposed to have a frequency equal to the pacing rate and it is found that

a few additional higher harmonics, whose frequencies are multiples of this pacing

rate, are sufficient for an accurate representation. Therefore the nature of the periodic

force Fp(t), being a summation of a static component, which is the weight of the

person G, plus a fluctuating component, enables it to be expressed in terms of a

Fourier series as [Bachmann et al. 1995]:

∑=

−+=n

iipip tfiGGtF

1

)2sin()( ϕπα (2.1)

where t is time, fp is the pacing rate, n is the number of harmonics, αi is the Fourier

coefficient of the ith harmonic, φi is a phase angle that can be seen as representing a

time shift with respect to the first harmonic.

Since Gαi represent the amplitudes of the dynamic components, the coefficients αi

are also called dynamic load factors (DLFs). Most research has been carried out

based on the Fourier decomposition to quantify the DLFs for the dynamic forces

caused by different human activities such as walking, running and jumping etc.

Blanchard et al. [1977] proposed a simple walking force model for the dynamic

analysis of footbridges with fundamental vertical frequency up to 4.0 Hz. In this

model, only the first harmonic was adopted and the corresponding DLF and

pedestrian weight were set to 0.257 and 700 N. Some reductions were applied for the

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fundamental vertical frequency between 4.0 Hz and 5.0 Hz to consider the lower

amplitude of second harmonic as the frequency in this range couldn’t be excited by

the first harmonic of walking. In 1982, Kajikawa [Yoneda 2002] proposed a vertical

force model for walking and running using “correction coefficients” (i.e. DLF). In

this model, the DLF as well as the person’s velocity depended on the walk pacing

rate. Bachmann and Ammann [1987] described an investigation into the walking

force for a pacing rate of 2.0 Hz. After harmonic analysis, the walking force with

three components in vertical, lateral and longitudinal directions was expressed by the

first five harmonics. The DLFs are shown in Figure 2.8 and the weight of person is

587 N. They suggested the DLF values for the first harmonic of the vertical force

between 0.4 and 0.5 at 2.0 Hz and 2.4 Hz, with linear interpolation for the frequency

between 2.0 Hz and 2.4 Hz. For the second and third harmonics, the DLFs were

suggested equal to 0.1 for walking frequency near 2.0 Hz.

Rainer et al. [1986; 1988] conducted a series measurements of continuous forces

induced by human activities not only from walking, but also from running and

jumping, and modelled the forces using the first four harmonics. In the force models,

the DLFs strongly depended on the frequency of activity (Figure 2.9). Kerr [1998]

also conducted a series of measurement of walking force on floor and stair. About

1000 force records from slow walking at 1.0 Hz to fast walking at 3.0 Hz generated

by 40 subjects were studied and characterised statistically by the mean value and the

coefficient of variation. Based on the work of Kerr and many others, Young [2001]

considered the stochastic nature of human walking and proposed the DLFs for the

first four harmonics as functions of walking frequency f which was assumed to vary

from 1.0 Hz to 2.8 Hz. The mean values are shown in table 2.1 and design values of

DLFs are expressed as

Hz 2.110.4 0065.0013.0

Hz 4.80.3 0064.0033.0

Hz 6.50.2 0056.0069.0

Hz 8.20.1 56.0)95.0(41.0

4

3

2

1

−=+=−=+=−=+=−=≤−=

ff

ff

ff

ff

αααα

(2.2)

Zivanovic et al. [2005] comprehensively reviewed the force models proposed by

different researchers. Table 2.2 shows the DLFs for different force models including

walking, running and jumping forces.

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It seems that a disagreement exits on the evaluation of the phase angles φ2 and φ3.

Rainer and Pernica [1986] evaluated these angles as 90o and 0o, respectively. On the

other hand, Bachmann and Ammann [1987] reported the phase angle exhibiting a

large scatter and suggested a value of 90o for each as a reference for computations.

Table 2.2 DLFs for single person force models after different authors [Zivanovic et al. 2005]

Although running can be modelled as sinusoidal functions by the Fourier series,

higher number harmonics are needed since running activity is a movement with

discontinuous ground contact [Rainer and Pernica 1986]. Considering the

characteristic of discontinuous ground contact of running and jumping (Figure 2.9), a

more simple force model was always used by adopting a half-sine sinusoidal

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function to model this kind of discontinuous forces (Galbraith and Barton 1970;

Wheeler 1982; Bachmann and Ammann 1987) in the form:

≤<≤

=pp

ppp

p Ttt

ttttGktF

for 0

for )/sin()(

π (2.3)

where kp is a dynamic impact factor, G is the weight of the pedestrian, tp is the

contact duration, Tp is the period being equal to the inverse of the pacing rate.

Figure 2.9 Forcing function from jumping on the spot with both feet

simultaneously at a jumping rate of 2 Hz [Bachmann et al. 1995]

Wheeler [1982] presented a chart (Figure 2.7) to show the relationship of dynamic

impact factor and the contact duration with the pacing rate. Bachmann and Ammann

[1987] also developed a graphically relation between the dynamic impact factor and

the ratio tp/Tp.

Generally the force generated by human activity has components in the vertical,

lateral and longitudinal directions, as mentioned before, but there is only one model

proposed for the lateral and longitudinal components and the DLFs are shown in

Figure 2.8 [Bachmann and Ammann 1987]. Although the magnitude of load in the

lateral direction is much smaller than that in the vertical direction, it is supposed that

a person with an average weight of 750 N may create about 25 N lateral force during

the body swaying from one leg to the other (Figure 2.10).

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It should also be noticed that the DLFs in the described models were obtained by

direct or indirect force measurements on rigid surfaces. However, it was found that

the force induced by human activity measured on slender structures was smaller than

that measured on rigid surface. Pimental [1997] investigated analytically and

experimentally the vibration of two full-scale footbridge under human activities and

found that the DLFs for the first two resonant vertical harmonics were considerably

lower than those reported in literature. It seemed that the human-induced force on

low frequency structures such as slender footbridges differed from the force

measured on rigid structures and was affected by the human-structure interaction.

Figure 2.10 Vertical and lateral components of the walking loads [http://www.arup.com/millenniumbridge/]

2.2.3 Effect of Group of People

The presence of groups of pedestrians on a footbridge is a common situation. Early

studies on the effects of crowd pedestrians showed that it was a rather conservative

approach to evaluate the total loads as the load of a single person multiplied by the

number of people involved.

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There are many factors which would affect the crowd loads: variability in the

pedestrian weight and pacing rate, phase difference among the pedestrians, density

effect and synchronisation among the pedestrians. The complexity of the problem is

usually tackled by taking simplifications and adopting a stochastic analysis as some

of the factors can be described in terms of probabilities.

The investigation into crowd loading usually departed from the loads due to an

individual and attempted to relate to the loads produced by crowd. Matsumoto et al.

[1978] published one of the first results of practical significance. The probability of

pedestrians arriving at a footbridge was investigated as a first step to a stochastic

analysis of the response of the structure to several pedestrians. The number of people

who crossed a prototype footbridge during one day was taken as a basis for

considering the arrival as random phenomenon, following a Poisson distribution. It

was found that, for a common given pacing rate and pedestrian weight, the

theoretical mean vibration amplitude in the vertical direction due to several

pedestrians could be determined by multiplying the mean vibration amplitude due to

one pedestrian by the square root of the number of pedestrians np on the structure at

any time. This coefficient of magnification is always called a crowd magnification

factor Cmf

pmf nC = (2.4)

Some simplifications in this equation for calculating the crowd magnification factor

are evident. On the one hand, it may be conservative to consider that all pedestrians

would be walking at the same pacing rate. But on the other hand, it may not be

conservative to consider the phase differences between the loads of each pedestrian

to be entirely random, and also for all the pedestrians to have the same weight. Some

results from Mouring and Elligwood [1994] showed that considering the pacing rate

and weight as constants did not affect the response of the system to the level of

accuracy required for the serviceability evaluations investigated. The same

conclusion regarding the insignificant difference between common or random pacing

rates was also obtained previously by Wheeler [1982] from simulations on

footbridges.

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A series of investigations were carried out to model the force induced by group of

people. It is believed that a person will never generate exactly the same force-time

history during repeated experiments and every day’s activities [Saul et al. 1985].

Based on this fact, the probabilistic approach to walking force model is proposed to

get a reliable estimate of force from group of people by combining the force from

individuals. For a single person, the force can be still assumed to be periodic and the

randomness can be taken into account by probability distributions of person’s

weight, pacing rate and so on. For several persons, the probability distribution of

time delay between people can be included. Tuan and Saul [1985] carried out a large

amount of measurements of the forces from different human activities, particularly

jumping, mainly typical for grandstands to describe the reliable statistical property of

human forces. Ebrahimpour [1987] continued this work and used a specially

constructed force platform to conduct the measurements of different types of forces.

A single jump and continuous jumping with controlled frequency at 2, 3 and 4 Hz

were investigated. The first three harmonics and force repeating periodically were

chosen for a statistical description of continuous jumping force-time histories from

individuals. In order to use the time delay distribution together with statistically

described time histories to estimate the resulting forces from any number of people,

Ebrahimpour identified the time delay distribution between two people who were

trying to perform synchronised jumping. Unfortunately, the procedure was

experimentally verified for only four persons. Further computer simulation revealed

that the force peak amplitude per person decreased with increase in the number of

people. However, this model was hardly applicable in practice as only the peak

amplitude is not enough to describe the force. In order to improve the model and

make it applicable, subsequent research work was carried out by Ebrahimpour and

Sack [1989] as well as Ebrahimpour et al. [1989]. They [1992] finally presented a

much more practical design suggestion by proposing the design curves for the first

three harmonics of jumping load as a function of the group size.

Ebrahimpour et al. [1996] measured the vertical dynamic load generated by a group

of moving pedestrians while walking and a linear regression model was used to

statistically characterize the footstep load-time history. A design proposal for only

the DLF of first harmonic was given as a function of a number of pedestrians and the

walking frequency (Figure 2.11). Higher harmonics were not considered as it was

- 24 -

probably believed that the first harmonic was most important for the force induced

by group of people. However, the fact that pedestrian in large crowds sometimes

adjust their step according to the movement of others was not taken into account

even though the number of pedestrians they considered was up to 100.

Figure 2.11 DLF for the first harmonic of walking force as a function of a number of people and walking frequency [Ebrahimpour et al. 1996]

Synchronisation always occurs when uncoordinated pedestrians crossing a vibrating

structure with natural frequencies close to the pacing rate [Grundmann et al. 1993].

According to previous test results, this would contribute to an increase in the crowd

magnification factor since the pedestrians would trend towards coordinated motion.

Based on previous work in which the probability of synchronisation was

investigated, Grundmann et al. [1993] proposed the crowd magnification factor Cmf

as following:

)( maxaPnkC splmf = (2.4)

Here, np is the number of pedestrians, Ps(amax) is the probability of synchronisation

as function of the acceleration amplitude amax, and kl is a factor to account for the

spatial distribution of the load along the structure, which depends on the mode shape

associated with the natural frequency under consideration. For a single span

footbridge, kl is assumed to be 0.6. It should be mentioned, this equation could only

be normally applied if the frequency of vibration is within the range of pacing rate. A

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typical value for the Ps was suggested as 0.225 corresponding to acceleration

amplitude of 0.7 m/s2 at 2 Hz.

It should be addressed that the mechanism of synchronization is quite complicated

and the load effect of group or crowd people on slender structures is affected by

many factors such as the group size, the structural natural frequency, particularly the

interaction of pedestrian-structure. These will be discussed later in this chapter.

2.3 Dynamic performance of footbridges

As mentioned before, modern footbridges are designed and constructed to be much

slender and flexible. Such slender structures with small stiffness and mass always

have low natural frequencies and are susceptible to vibration. If the natural

frequencies of a footbridge are close to the range of pacing rate of pedestrians, the

bridge structure would suffer excessive vibration induced by the pedestrians.

Unfortunately, most footbridges have the natural frequencies close to the range of

normal pacing rate and many vibration serviceability problems arise, as a result.

2.3.1 Vibration properties of footbridge structures

Footbridge structures may take different structural forms, such as arch, simple beam,

truss, cable-stayed or suspension etc, and they may also be constructed with different

materials such as timber, concrete, steel or composite. However, their function is the

same, that is, to carry pedestrians and cyclists. Therefore they all are subjected the

dynamic loads induced by pedestrians and same dynamic problems could arise,

depending on the vibration properties.

Since the typical pacing frequency of human activities can vary from 1.3 Hz to 3.4

Hz (Table 2.1), it is suggested that natural frequency of footbridges in this range

should be avoided. Further more, the frequency from 3.5 Hz to 4.5 Hz of footbridges

with low damping ratio is also suggested to be avoided as the vibration modes in this

frequency range could be excited by the second harmonic of human-induced loads

[Bachmmann, 2002]. Unfortunately the fundamental vertical natural frequencies of

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most footbridges are within these frequency ranges. Figure 2.12 shows the

distribution of vertical fundamental frequencies of 67 beam-type footbridges as the

function of the main span. Bachmann et al. [1995] recommended the fundamental

frequency following a general formula for prediction purpose:

73.01 6.33 −= Lf (2.5)

where L is the span length in meters. It can be seen that when the span length is

greater than 20 m, most of the fundamental frequencies in vertical direction are

smaller than 5.0 Hz. And from the prediction formula, the natural frequency is less

than 2.0 Hz when the span length is greater than 50 m.

Figure 2.13 [Dallard et al. 2001a; 2001b] shows the lateral natural frequencies of

some footbridges. It can be seen that all the lateral frequencies are less than 2.5 Hz

when the span length is greater than 50 m.

Pirner and Ficher [1998] also gave an envelop formula of the lowest natural

frequencies for some kinds of concrete stress ribbon footbridges.

112

~ 217

925.0431.1 LLf = (2.6)

From this formula, the fundamental frequency (Hz) is less than 3.0 Hz when the span

length (m) is greater than 50 m.

It seems that most footbridges have the fundamental natural frequency below 5.0 Hz

in vertical direction and 2.5 Hz in the lateral direction. When the span length

increases, the footbridges would become more slender and flexible, therefore the

natural frequencies could be even smaller and fall into the frequency range of human

activities. When they are subjected to the human-induced dynamic loads, these

frequencies are easy excited by the first or second harmonics and the footbridges

have a great possibility to experience excessive vibration at or near resonance.

Another important vibration property is damping. Damping presents energy

dissipation in a vibrating structure and reduces the structural response to dynamic

- 27 -

excitations. When a structure vibrates at or near resonant condition, damping is the

governing factor which can be used to control dynamic response.

Figure 2.12 Footbridge fundamental frequencies as a function of span [Bachmann et al. 1995]

Figure 2.13 Lateral natural frequencies of footbridges [Dallard et al. 2001b]

In a structure, there are several dissipation mechanisms and each mechanism has its

own contribution for the total energy dissipation. In practice, “effective damping” is

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used to describe the overall damping which comprises of all of the mechanisms in

the structure. And it is this “effective damping” which is actually measured as modal

damping in practice and adopted in dynamic analysis. Although there are several

damping models proposed to describe the damping properly, the most often used is

the viscous one because of its simplicity. In this damping model, the damping is

expressed as damping ratios ζn defined for each vibration mode separately.

Table 2.3 Common value of damping ratio for beam-type footbridge [Bachmann 2002]

.

For footbridge structures, the damping may be fairly low, especially in the case of

steel or steel-concrete composite footbridges. Table 2.3 shows the common values of

equivalent damping ratio of beam-type footbridges measured when one pedestrian

was walking at the bridge’s fundamental frequency. Although other footbridges such

as cable-stayed bridges, suspension bridges or arch footbridges may have different

damping ratio, it is true that modern construction technologies have brought a

reduction of damping in structures because of a significant decrease in the amount of

friction which was present in old structures. In mid-1940s, the minimum damping

ratio was considered as 1.6%, and until 1960 it was believed damping ratio would

not be below 0.8%, whereas nowadays, modern steel bridges regularly exhibit

damping ratio of 0.5% or less [Wyatt 1977].

Since it is very hard to predict, damping is always determined by experimental

testing. In general, the measurement of damping is carried out along with those of

natural frequencies and vibration modes. As all these dynamic properties are

important to describe the dynamic characteristics of a structure, a large amount of

research has been conducted to develop the test technology and experimental

methods, and several measurements have been carried out on highway bridges as

well as footbridges.

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Leonard and Eyre [1974; 1975] investigated some highway bridges and footbridges

with steel box girder and concrete deck and they reported that the supports and end

conditions had great influence on the damping. They also found that the damping

increased with the increasing of vibration amplitude and suggested that damping was

amplitude dependant. Eyre and Tilly [1977] carried out measurements on 23 steel

and composite bridges, many of them being footbridges. The bridges they

investigated were steel box girder or steel plate girder with different number of span

as well as different span lengths. They found that single span bridges had higher

damping and than multi-span ones. Higher vibration modes generally had higher

damping. They concluded that damping was dependent on the number of spans and

vibration mode. They also confirmed that damping is dependent on the vibrating

amplitude. Tilly et al. [1984] also concluded that it was incorrect to generalise that

damping increases in higher vibration modes or it is dependent on the stiffness and

span length. They also suggested that it is always necessary to quote the measured

damping with the level of response amplitude because of the amplitude dependence.

In order to get better results of the dynamic properties, many techniques were

introduced to experimental testing, and many testing methods have been developed

for full-scale structural experiments. Since different theoretical assumptions are

introduced for these methods, they can produce slightly different results. Among

these methods, time-domain free decay method, frequency-domain half-power

bandwidth method, time-domain-based random decrement method and frequency

response function curve fitting method are often used. The principles of these

methods can be found in standard textbooks dealing with modal identification [Ewins

2000; Maia et al. 1997]. Using these techniques and methods, numerous

measurements were carried out on large bridge structures [Rainer and Van Selst

1976; Abdel-Ghaffar 1978; Brownjhon et al. 1987; Rainer and Pernica 1979;

Cantieni and Pietrzko 1993; Pavic and Reynold 2002a; Pavic et al. 2002a; Hamm

2002; Caetano and Cunha 2002].

Table 2.4 shows some key results related to damping measurements of footbridges.

These results were published in literature and reviewed comprehensively by

Zivanovic et al. [2005]. In this table, damping ratio ζv, ζh and ζl for the first two

vertical, lateral and torsional modes ware given whenever the data were available.

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Table 2.4 Measured damping ratios (for vertical ζv, horizontal ζh and torsional ζt) for some footbridges [Zivanovic et al. 2005]

It can be seen that most of the measurements were carried out for the damping of

vertical vibration modes. It is also shown that for the same footbridge structure, the

value of damping ratio is different if the measurement was conducted by different

methods. Zivanovic et al. [2005] gave detailed and comprehensive review on the

published data and the damping estimation methods in this table. Finally they

concluded that it is not possible to define unique value(s) for footbridge damping. To

overcome this, Bachmann et al. [1995] suggested using the common value of

damping ratios in Table 2.3 for design guidance, and these published results show

those recommendations made by Bachmann et al. still look very reasonable.

2.3.2 Vibration serviceability of footbridges

As the main purpose of a footbridge is to carry pedestrians, the reaction of

pedestrians to the vibration of footbridge governs the vibration serviceability.

However, the reaction of human beings to vibration is a very complex issue, different

people react differently to the same vibration condition, and even the same person

would likely react differently on different days [Lippert 1947]. Since the human

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sensitivity is high and the reaction of human beings to vibration is important for the

design criteria and vibration serviceability, a series research has been carried out in

this area.

Early research work was mainly conducted on the human reaction to the vibration of

buildings and some of the important research was carried out by Reiher and Meister

[Write and Green 1959], Goldman [1948] and Dieckmann [1958]. Reiher and

Meister investigated the effect of harmonic vibration on people having different

posture such as lying, sitting and standing on a test platform driven by different

amplitudes, frequency and direction of vibration. They finally classified the human

perception into six categories and as a function of vibration amplitude and frequency.

Goldman [1948] studied and summarized all known work to that time and he defined

three categories of human reaction to vibrations: perception, discomfort and

maximum tolerable levels. In his study, the perception value was only 0.25%g, and

the minimum discomfort level was about 4.6%g which occurred around the resonant

frequency of the human body (5.0 Hz). Dieckmann [1958] considered the vibrations

below 4.0 Hz in different directions and found that human sensitivity for horizontal

vibrations was higher than that for vertical vibrations.

Regarding the human perception of vibration on bridge structures, Wright and Green

[1959] observed that the real vibrations on bridge structures are much more complex

than the harmonic vibrations on a platform in laboratory. They realized the

applicability of the results obtained in laboratory conditions to footbridges was

questionable and many other parameters specific to bridge vibrations should be taken

into account. For example, a pedestrian on bridge is not stationary but moving, the

duration of exposure to vibration is limited and so on. All these factors will influence

the human perception of vibration on bridge structures. They [1963] measured the

peak oscillations on many highway bridges under normal traffic and concluded that

the scale based on long-time vibration might not be appropriate for bridge vibrations

where peak vibrations usually lasted only for a short period of time. The duration of

vibration was considered as the most important factor to human perception and it

depended to some extent on the bridge damping.

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To study the human perception related to walking and standing people under

vibration with limited duration, Leonard [1966] carried out a laboratory experiment

on a 10.7 m long beam which was driven by sinusoidal excitation at different

amplitudes (up to 0.2” i.e. 5.08 mm) and frequencies (1.0 – 4.0 Hz). About forty

walking and standing persons participated in these tests and the vibration amplitude

was kept at a constant level for one minute. Results clearly indicated that a standing

person is more sensitive to vibration than a walking one, and that the Reiher and

Meister scale was fairly inappropriate for application to bridges. He also suggested

using the curve applicable to standing people for the vibration serviceability of

crowd as well as that in lateral horizontal direction. Smith [1969] conducted an

experiment using just a single pedestrian walking on a flexible aluminium alloy

plank, and twenty-six persons participated in the experiment to classify the vibration

level into three groups: acceptable, unpleasant and intolerable. But finally he decided

only to define the regions of acceptable and unacceptable vibration (Figure 2.14) and

his threshold curve was much higher that that given by Leonard [1966]. And he

explained the possible reason was Leonard tried to draw a lower limit curve rather

than a mean curve.

Figure 2.14 Different scales of human perceptions [Smith 1969]

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Some experiments similar to Leonard’s test were carried out by Kobori and

Kajikawa [1974; 1977] and eleven subjects participated on a vertically vibrating

shaking platform driven by sinusoidal excitation in the frequency range of 1 – 10 Hz.

It was found that the vibration velocity was the main parameter which influenced the

human perception. They also formulated analytically the relationship between

vibration perception and the vibration velocity.

Blanchard et al. [1977] proposed a simple design value for the vibration

serviceability of footbridges. In this proposal, the mean value of Leonard’s and

Smith’s research results were used to define the level of acceptable acceleration a1imit

(m/s2):

fa 5.0limit = (2.7)

where f (Hz) is the footbridge fundamental frequency. This value is still adopted in

the current British standard BD 37/01 [2002] for assessing the vibration

serviceability of footbridge.

From the collected data on human response to vibration from different sources based

on laboratory and tests on full-scale structures, Irwin [1978] constructed either the

perception or maximum allowable magnitude curves for different types of structures

and different types of vibrations, including the limits for root-mean-square (RMS)

accelerations for bridges. The limits were given separately for everyday usage and

storm conditions (Figure 2.15). The maximum sensitivity for everyday curve was

0.07 m/s2 between 1.0 Hz to 2.0 Hz when expressed as an equivalent harmonic peak

value and the curve for storm conditions was obtained by multiplying the base

everyday curve by the factor of 6. Horizontal motion was considered only for the

storm conditions and other curves for different purpose can be obtained by

multiplying the base everyday curve by some factors. Unlike the other research, the

perceptibility curves were expressed by the RMS acceleration:

12

22

1

)(

tt

dttxRMS

t

t

−=∫ &&

(2.8)

where )(tx&& is the acceleration time history, and t1, t2 are the beginning and end of the

time interval considered.

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Figure 2.15 Acceptability of vibration on footbridge after different scales [Smith 1988]

One of the recommendations for acceptable footbridge vibrations is given in the ISO

10137 guideline for serviceability in buildings [ISO 1992] and it is based on RMS

acceleration limits. The recommendation suggests using the base curve for vibration

in both vertical and horizontal directions given in ISO 2631-2 [1989] multiplied by

the factor of 60.

A comparison of the vibration limits from standards and design codes was given by

Pimentel [1997] and the results were presented as in Figure 2.16 based on the peak

acceleration. Here the curve for ISO was obtained by converting the RMS

acceleration to peak value. From this figure, it can be seen that the BS 5400 allows

the highest level of vibrations over the typical range of footbridge response

frequencies. Since different people react differently, Obata et al. [1995] suggested

curves of 25%, 40%, 50%, 60% and 75% probability for each of four defined

perception levels. To simple this issue, on other hand, Bachmann et al. [1995]

proposed a constant acceleration acceptance level of 0.5 m/s2 for vertical vibration of

footbridges.

The only guideline which recommends a horizontal vibration limit for footbridge is

ISO 10137 [1992]. Figure 2.17 presents the perception curve. The highest sensitivity

is in the frequency range up to 2.0 Hz and is at about 3.1%g peak acceleration.

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Figure 2.16 Acceptability of vertical vibration in footbridges after different scales [Pimentel 1997]

Figure 2.17 Acceptability of horizontal vibration this base curve should be multiplied by the factor 60 for footbridge [ISO 1992]

Some research findings related to human perception of horizontal vibrations in

buildings are meaningful to evaluate the lateral vibration serviceability of

footbridges. For example, Chen and Robertson [1972] studied the human perception

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threshold to horizontal sinusoidal vibration with frequencies between 0.067 and 0.20

Hz. The most important factors typical for this issue were identified as the frequency

of vibration, body movement, expectancy of motion and body posture. They found

that the perception threshold of walking people is higher that that of stationary

person. And the vibration perception threshold is lower but the tolerance level is

higher if one expects the movement, regardless the vibration direction. Nakata et al.

[1993] also investigated the perception threshold to horizontal vibration. Forty sitting

people were exposed to horizontal sinusoidal vibration at the frequency range 1.0 to

6.0 Hz. It was found that the fore-aft perception threshold was higher than the side-

to-side threshold in the frequency range 1.0 to 3.0 Hz, while in the range 3.0 to 6.0

Hz, the opposite was true.

Wheeler [1982] noticed that human perception of vibration in a walking crowd on

footbridges was different from that for a single person. The same conclusion was

obtained by Ellis and Ji [2002] from an experiment with a jumping crowd where the

jumpers didn’t concern though the measured acceleration was 0.55g. Nakamura

[2003] carried out some field test on a full-scale suspension footbridge and studied

the tolerance level to the lateral vibration induced by crowd loading. He found that

the amplitude of deck displacement of 45 mm (corresponding to an acceleration of

1.35 m/s2) is a reasonable serviceability limit. He also reported that deck

displacement amplitude of 10 mm (0.3 m/s2) was tolerable by most pedestrians,

while a displacement of 70 mm (2.1 m/s2) would make people to feel unsafe and

prevent them from walking.

2.3.3 Dynamic performance of footbridges under human-induced loads

Under human-induced dynamic loads, each footbridge might behave differently. The

dynamic performance of a footbridge depends not only on the dynamic excitations,

but also on the vibration properties of the bridge and damping system. However,

modern footbridges are more slender and flexible than ever due to the increasing

span length and application of light weight and high strength materials. Most of

them have natural frequencies within the frequency range of human activities. As a

consequence, they are more prone to human-induced vibration and some kind of

human-structure interaction almost inevitably occurs when they are subjected to

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human-induced loads. In general, there are two aspects regarding the dynamic

performance of footbridges under human-induced loads and the human-structure

interactions. The first considers the changes of dynamic properties of footbridge

structures due to the presence of pedestrians. The second aspect concerns the

dynamic response of footbridge structures and behaviour of pedestrians.

2.3.3.1 Dynamic properties of footbridges under moving people

Human body is a complex system which has mass, stiffness and damping. When a

person occupies a structure, the dynamic properties of the structure will be affected

by the person as the human body can behaviour like a damped dynamic system

attached to the structure, and the structure and human body form a new dynamic

system. Though this effect is small for one person and depends on the occupied

structure, it is greater if more people are present [Ellis and Ji 1997; Sachse 2002]. In

order to identify the effect and investigate the human-structure interaction, single

DOF system [Ji 2000; Zheng and Brownjhon 2001] and multi DOF system [Williams

et al. 1999] are proposed to model the human body. In the single DOF system model

proposed by Zheng and Brownjhon, the human body had a damping ratio of 39% and

a natural frequency of 5.24 Hz. However, the simplified single DOF system model

has been shown to be frequency-dependent and cannot be always represented by the

same set of mass, stiffness and damping parameters [Sachse 2002; Sachse et al.

2002].

It should be mentioned that most of the research on the effect of human body on the

dynamic properties of structures was carried on floor structures. Some of the results

may not be applicable to slender footbridge structures. For example, Ellis and Ji

[1994] reported that a person running and jumping on the spot cannot change

dynamic characteristics of the structure and therefore should be treated only as load.

This conclusion was made based on their investigation carried out on a simply

supported beam which had high fundamental vertical frequency of 18.68 Hz. A

similar conclusion was obtained by the same researchers on the effects of a moving

crowd on grandstands [Ellis and Ji 1997].

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Regarding the dynamic properties of footbridge structures, Willford [2002]

mentioned that it has been observed that the damping in vertical modes of structures

increases with the presence of stationary people, but there is little data in the

literature regarding the effect of walking people. Investigation on the London

Millennium Bridge with 250 walking people suggested that the effective vertical

damping was about 3% of critical, higher than the measured damping of unloaded

bridge (0.7% of critical). The author also mentioned that the effect of people walking

can be equivalent to negative damping for the lateral modes. Brownjhon [2004]

conducted a comprehensive study on a pedestrian bridge which was a steel framed

elevated walkway with fundamental vertical frequency of about 5 Hz and low

damping ratio of 0.85%. After investigating the effect of size of crowd on the

damping and vibration properties, he reported that the slop of the curve for additional

damping is 0.26% per person and for reduction in frequency is -0.26% per person.

Although there is little information in the literature regarding the effect of

pedestrians on the natural frequency of footbridges, published data show that the

mass of modern footbridges decreases due to new technology and application of light

weight and high strength materials. When these light footbridges are subjected to

congested crowd, the change of structural mass as well as natural frequency will be

significant. For example, the structural mass per unit area (bridge mass divided by

the deck width) for Millennium Bridge in London is about 500 kg/m2, for T-Bridge

(cable-stayed) in Japan is about 800 kg/m2 and for M-Bridge (suspension) in Japan is

about 400 kg/m2 [Nakamura 2003]. It was reported that during the opening day of the

Millennium Bridge, the maximum crowd density was about 1.3 to 1.5 people/m2.

And the maximum crowd density of T-bridge under congested condition was about

1.0 to 1.5 person/m2. If the maximum density of 1.5 persons/m2 is considered and

the average weight is assumed to be 700 N/person, then about more than 100 kg/m2

will be added to the structure. This increased mass is about 25% for M-Bridge, 20%

for T-Bridge and 12.5% for Millennium Bridge.

Another effect of pedestrians on slender footbridges is the dynamic force. Although

there is no direct evidence showing how the dynamic force was affected by

pedestrians, some research carried on footbridges and slender floors as well as other

structures showed that the force induced by human activities measured on slender

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and flexible structures is quite different from that measured on rigid surface. Ohlsson

[1982] investigated the force spectrum induced by human activities on flexible

timber floor and reported that the spectrum was different from that measured on rigid

surface. Pimentel [1997] conducted some experimental test on one composite

footbridge and one stress ribbon footbridge, and he found that the dynamic load

factors (DLFs) were much lower on real and moving footbridges in comparison with

those measured on rigid surface. Yao et al. [2002] also reported that jumping forces

are lower on flexible structure. Furthermore, Pavic et al. [2002c] carried some

research to investigate lateral component of jumping forces. After comparing the

horizontal jumping forces directly measured on a force plate and indirectly measured

on a concrete beam, they found that the forces on the structure was two times lower

than the one on the force plate.

It is clear that for slender footbridge structures, some of human-structure interaction

occurs inevitably and it changes the dynamic properties of footbridges and affects the

dynamic performance.

2.3.3.2 Synchronization between walking people in groups

It is believed that the human body is the most complex system and has the ability to

automatically adjust the frequency and phase of its movement in different activities.

This ability makes its behaviour more complicated and flexible than any other

dynamic structure.

As mentioned before, the presence of groups of pedestrians on a footbridge is a

common situation and early studies on the effects of crowd pedestrians showed that it

was a rather conservative approach to evaluate the total load as the load of a single

person multiplied by the number of people involved.

Matsumoto et al. [1972] gave a simple proposal to consider the effect of group

people. According to this proposal, the total response of group people can be

obtained by multiplying the response of a single person by a crowd magnification

factor defined in Equation 2.4. This proposal was believed to be appropriate at least

for footbridges within natural frequencies in the range of pacing rate (1.8 – 2.2 Hz).

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While for footbridges with frequencies in the range of 1.6 – 1.8 Hz and 2.2 – 2.4 Hz,

a linear reduction of the magnification factor was suggested with its minimum value

of 2 at the ends of these intervals when more than four people are present on the

footbridge at the same time [Bachmann and Ammann 1987]. Mouring [1993] also

simulated a vertical force from walking group people and concluded that the effect of

group of people should be considered even in case of the footbridges with

fundamental frequency outside the walking frequency range (1.8 – 2.2 Hz).

However, this proposal didn’t consider the possibility of synchronization between

people walking in group or in a dense crowd.

Synchronization is believed to be a common phenomenon in group and crowd

movements. Ebrahimpour and Fitts [1996] reported that the optical sense plays an

important role in the synchronization of people’s movement. In the case where the

jumping frequency was controlled by an audio signal, two jumping persons who

could see each other synchronized their movements better than when they were

looking in opposite direction. Errikson [1994] also claimed that the first harmonic of

the walking load could be almost synchronized perfectly for highly correlated people

in a group, while the higher harmonics should be treated as completely uncorrelated.

It was not surprising that only the first harmonic was treated in Ebrahimpour et al.’s

force model for group of people [Ebrahimpour et al. 1996].

For bridge structures, the load density sometimes could be extremely dense.

Wolmuth and Surtees [2003] studied the crowd-related failure of bridge structures

and mentioned that for some case the crowding could vary from 0.3 – 15.4

persons/m2 of span. For example, on the opening day of the First Bosporus Bridge,

Istanbul, Turkey in 1973, the crowd density was reported as high as 5.0 persons/m2.

An estimated 60,000 to 100,000 people surged onto the 1074 m long main

suspension span, and about another 2,000 people per minute poured onto the

structure at the each end. The latter flow was stoped when the bridge began to sway.

However, Bachmann and Ammann [1987] reported that the maximum possible

density for walking crowd can normally be 1.6 – 1.8 persons/m2 and it was found

that normal walking becomes difficult at crowd density above about 1.7 persons/m2

[Dallard et al. 2001d]. Some “lively” footbridges were reported being subjected to

such dense loads. For example, the maximum density of T-Bridge in Japan was

- 41 -

reported to be about 1.0 – 1.5 person/m2 in a congested condition, and the maximum

density of Millennium Bridge in London was reported to be around 1.3 – 1.5

persons/m2 during the opening day. It is widely now accepted that people working in

a group or a crowd would subconsciously adjust and synchronise their foot steps due

to the limited deck space and the possibility that they can see each other and are

aware of others’ movements. Schlaich [2002] presented the relationship of bridge

capacity and crowd density (Figure 2.18). It was noticed that in any case, the crowd

density influences the bridge capacity, the walking speed, the degree of

synchronization between people and the intensity of human-induced force.

Figure 2.18 Relationship between the bridge capacity, pedestrian density and their velocity [Schlaich 2002]

To describe the different behaviour of walking people and take in account of the size

of group, Grundmann et al. [1993] proposed three models corresponding to different

pedestrian configurations on footbridges.

Model 1: When people walk in a small group, they will be probably walking with the

same speed, slightly different pacing rate and step length. In such cases, some

synchronization is expected between these people when the bridge frequency is

within the range of normal pacing rate.

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Model 2: When people walk freely on the bridge deck and their pacing rates are

randomly distributed and no synchronization is expected. The upper limited density

of this kind of unconstrained free walking is assumed to be 0.3 persons/m2.

Model 3: When pedestrian density increases to 0.6 – 1.0 persons/m2, unconstrained

free walking becomes impossible and pedestrians are forced to adjust their step

length and speed to match the motions of other pedestrians. In such cases,

synchronization occurs inevitably.

These models indicated that synchronization often occurs when people walk in group

or in crowd. In general, group of walking people always refers to several people

walking with the same pacing rate, step length and speed, while crowd of walking

people refers to densely packed walking people and some of these people have to

adjust the pacing rates, step speed and length to suit the space available.

2.3.3.3 Synchronization excitation and dynamic performance of footbridges

Synchronization occurs not only between people, but also between the walking

people and the vibrating footbridge structures. When crossing a bridge which is

vibrating at a frequency within the frequency range of human activities such as

walking and running, pedestrians trend to change their pacing rates to move in

harmony with the bridge vibration. This mechanism leads to large amplitude

synchronous vibration, and this kind of synchronization will affect the dynamic

performance of footbridges and incurs excessive vibration and vibration

serviceability problems. Table 2.5 shows some case reports of excessive vibrations

due to human-induced loads in footbridges [Pirmental 2002].

Considering effect of human-structural interaction on the dynamic response,

Grundmann et al. [1993] tried to quantify the probability of synchronization in

vertical direction by defining the probability of synchronization as a function of

acceleration amplitude of the vibrating structures. Then they proposed that the

dynamic response to group people on the structure can be obtained by multiplying

the response of single person by crowd magnification factor Cmf which is expressed

in Equation 2.4. To simplify this issue, they finally suggested that for group of up to

- 43 -

10 people, the magnification factor can be taken as the value from the chart shown in

Figure 2.19, with that maximum value of 3.0 for the vertical natural frequencies from

1.0 Hz to 2.5 Hz. Also the same factor was proposed for the lateral direction but

corresponding to two times lower natural frequencies, although they believed that the

probability of synchronization in the lateral direction was much lower than in the

vertical direction and this proposal was a conservation solution.

Table 2.5 Case reports of excessive vibrations in footbridges [Pirmental 2002]

Tanaka and Kato [1993] investigated eighty cases for the trial design of simple

pedestrian bridges in which resonance was easily caused by human walking, and

developed a simplified formula which was able to calculate accurately the maximum

vertical response amplitude for bridge design. They studied and calculated the

natural frequencies of each designed pedestrian bridge and the response amplitude

when a person walked on the bridges with the same pace as its natural frequency.

They also discussed the vibration serviceability of the pedestrian bridges at

resonance. Their research results showed that the pedestrian bridge of which dead

load was more than 2.0 tons/m had enough serviceability even at resonance.

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Figure 2.19 Magnification factor for groups of up to 10 peoples [Grundmann et al. 1993]

Dallard et al. [2001c] suggested using random vibration theory to predict the

dynamic response due to crowd. They gave a formula to calculate the mean square

acceleration response E(a2) induced by N pedestrians with normal distributed pacing

rates:

−≈M

Fp

c

NaE nnn ω

σµω

σωπ

16)( 2 (2.8)

where ωn , M and c are natural frequency, modal mass and critical damping ratio,

while Fωn and p are the amplitude of harmonic human-induced force and probability

density function for normally distributed pacing rates with the mean value µ and

standard deviation σ. However, this formula was reported to be conservative and also

the distribution of pacing rates within a crowd is unknown.

Extensive research was also conducted to estimate effect of synchronization on the

dynamic response due to crowd people. For example, Mouring [1993] identified

importance of the degree of correlation between people in a crowd to the effect of

synchronization; McRobie and Morgenthal [2002; 2003] suggested using the theory

of wind engineering to assess the liveliness of footbridge in the vertical direction

under crowd load. Brownjhon et al. [2004] proposed a mathematical model for the

spectral density of continuous vertical forces on pedestrian structure due to walking.

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As pedestrians are more sensitive to low frequency motion of vibrating structures in

the lateral than in the vertical direction, synchronization phenomenon occurs more

likely on slender footbridges. Although Grundmann et al. [1993] believed the phase

range and consequently the phase spread among the pedestrians was higher for

vibration in the lateral direction, and they considered that the possibility of

synchronization was lower in the lateral direction than in the vertical direction,

Bachmann and Ammann [1987] suggested that the synchronization was more

pronounced in the lateral direction. They explained that when a pedestrian noticed

the lateral vibration, he would attempt to re-establish his balance by moving his body

in the opposite direction; as a consequence, the load he exerts on the deck would be

directed so as to enhance the structural dynamic response and result in vibration

instability. This synchronization phenomenon between pedestrians and bridge

structure in lateral direction is now widely known as lock-in effect [Dallard et al.

2001c; Bachmann 2002].

Although it has been known for a long time that the lateral synchronization

phenomenon exists in footbridges structures and many footbridges have experienced

excessive lateral vibrations due to human-induced loads, it was only after the

opening of Millennium Bridge in London that the lateral synchronization

phenomenon has attracted the attentions of researchers and bridge engineers and a

series of research has been carried out to investigate this issue and improve the

vibration serviceability of slender footbridge structures. As one of the consequences,

the first international conference on footbridges (Footbridge 2002) was held in 2002

to address the issue of design and dynamic behaviour of footbridges.

Although many footbridges have been reported having experienced excessive

vibrations due to lateral synchronous excitation, only a few cases have been

investigated intensively.

Bachmann and Ammann [1987] reported a steel box girder footbridge with the main

span of 110 m suspended from an angular arch having experienced strong lateral

movements during the opening ceremony. It was reported that natural frequency of

the lowest lateral mode was about 1.1 Hz, very close to half the mean pacing rate at

2.0 Hz, producing an almost resonating vibration. Improvement was achieved by

- 46 -

installing tuned vibration absorbers to be effective in the lateral direction. Bachmann

[1992] also reported another reinforced concrete footbridge with fundamental lateral

frequency of 1.0 Hz which experienced considerable lateral vibrations. Many other

bridges with different structural forms were also reported having experienced large

lateral vibrations induced by pedestrians [Dallard et al. 2001a; 2001b; 2001c]. These

bridges include Grove Suspension Bridge, Chester, UK; Link Bridge, Birmingham,

UK; Auckland Harbour Road Bridge, New Zealand; Pont du Solferino Arch

footbridge, Paris, France; Alexandra Bridge, Ottawa, Canada et al. However, no

further published information on the details can be found.

One of the footbridges which has been investigated intensively is the T-Bridge (Toda

Park Bridge) in Japan (Figure 2.20). The footbridge structure is a cable-stayed type

with two-span continuous steel box girder and concrete tower. The total span length

is almost 179 m (134 m+45 m) with the deck width of 5.25 m and tower is 61.4 m

high. The lowest lateral frequency is about 0.93 Hz with the damping ratio of about

0.008 [Nakamura 2004]. Field measurements were conducted several times to

establish the nature and mechanism of vibration. Fujino et al. [1993] reported that

under typical congested conditions, more than 20,000 people sometimes passed over

the bridge within 20 mins or so, and as many as 2,000 pedestrians walked

simultaneously on the bridge. Under such situation, not only vertical vibration but

also noticeable lateral vibration in the girder was often observed. The lateral

amplitude of the girder occasionally exceeded 1 cm which was much bigger than the

vertical amplitude, and the horizontal amplitude of some cables was found to be of

the order of 30 cm. By video recording and observing the movements of people’s

heads in crowd, Fujino et al. [1993] concluded that 20% of the pedestrians on the

main span of the bridge were perfectly synchronised to the girder vibration, and the

force produced by the rest of pedestrians cancelled each other and had no effect on

the dynamic response. They suggested that 35 N was a reasonable lateral force

generated by one pedestrian during lateral vibration. They also explained the

synchronization phenomenon as a self-excited nature. When a small lateral motion

was induced by the random lateral human walking forces, some pedestrians

synchronised the walking. Then resonant force acted on the girder, and the girder

motion was increased consequently. Finally walking of more pedestrians were

synchronised and this mechanism resulted in excessive lateral vibration. In order to

- 47 -

suppress the excessive vibration, a large number of small tuned liquid dampers

(TLD) were installed inside the box girder [Fujino et al. 1992a; 1992b; 1992c]. Ten

years late, Nakamura and Fujino [2002] investigated this bridge again and they found

that the girder vibration was not seen as often as it was used to be ten years ago even

though many of the TLDs seemed not to be working properly due to the evaporation

of water and lack of maintenance. Furthermore, using image processing technique for

tracing people’s movements on the bridge, Yoshida et al. [2002] estimated the

overall lateral force generated by the crowd of 1,500 pedestrians at about 5016 N,

which gave an average of only 3.34 N per pedestrian. They also found that the bridge

vibrated with the maximum lateral amplitude of 9.0 mm at the frequency of 0.88 Hz.

Further research [Nakamura 2004] on this footbridge also found that the smaller

bridge damping and the smaller bridge mass produced the larger girder response, and

higher pedestrian density also increased the girder response unless they were too

crowded to walk normally.

Figure 2.20 T-bridge in Japan: (a) – layout; (b) – girder section [Nakamura 2004]

The other footbridge which has been investigated intensively is the Millennium

Bridge in London. The Millennium Bridge is a three-span shallow suspension

footbridge [Dallard et al. 2001a; 2001b; 2001c] with the deck width of 4 m, main

span of 144 m and two side span of 108 m and 80 m. In the main span, the cable sag

is 2.3 m in vertical direction, about six times shallower than a more conventional

suspension bridge structure. During the opening day, it was estimated that about

80,000 to 100,000 people crossed the bridge and about 2,000 people walked on

bridge deck at any one time which resulted in a maximum density of between 1.3 to

1.5 people per square meter. Unexpected excessive lateral vibration occurred at about

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0.5 Hz on the main span, and 0.8 Hz and 1.0 Hz on the side spans. The maximum

lateral acceleration of 0.2g – 0.25g was recorded corresponding to lateral

displacement amplitude of up to 7 cm [Dallard et al. 2001c]. Two days later, the

bridge was closed and a series of research had been carried out to investigate the

cause of excessive vibration and for retrofit procedure.

Measurements on the bridge showed that the excessive lateral vibration took place at

the natural frequencies of each span and the damping for each mode was around

0.6% to 0.8%. Dallard et al. [2001c] found that the excessive lateral movement was

clearly caused by a substantial lateral loading effect which had not been anticipated

during design and this loading effect was due to the synchronization of lateral

footfall forces within a large crowd of pedestrians on the bridge. They explained that

this arose because it was more comfortable for pedestrians to walk in

synchronization with the natural swaying of bridge, even if the degree of swaying

was initially very small. A series of publications and discussions [Dallard et al. 2000;

2001a; 2001b; 2001c; 2001d; Newland 2003a; 2003b; Henderson 2001; Perera 2001;

Pavic et al. 2002a; 2002b; Willford 2002; and et al.] regarding the problem and

solution of the Millennium Bridge were published to make bridge engineers as well

as researchers aware of this synchronization phenomenon, and address the

inadequacy of guidelines in design standards. They further concluded that the same

problem can occur on any footbridge, independent of structural forms, with a lateral

natural frequency below around 1.3 Hz and with a sufficiently large crowd of

pedestrians crossing the bridge structure.

Based on the testing results and their observations, Dallard et al. [2001c] proposed

that the dynamic force F(t) induced by pedestrians, after synchronising their

footsteps with the vibration of bridge structure, was proportional to the deck lateral

velocity v(t):

)()( tkvtF = (2.9)

where k is a lateral force coefficient. They also defined a limiting number of people

NL to avoid bridge vibration instability as:

k

cfMNL

π8= (2.10)

- 49 -

where c, f and M are modal damping ratio, lateral frequency and corresponding

modal mass. For the Millennium Bridge, k was found to be 300 Ns/m in the lateral

frequency range 0.5 – 1.0 Hz.

In order to investigate the dependence between the probability of synchronization

between people and the amplitude of bridge vibration, Willford [2002] carried out

some measurements with single pedestrians on a 7.2 m long walkway subjected to

lateral vibration in a laboratory. The tests indicated that at 1 Hz amplitudes of the

motion as low as 5 mm caused a 40% probability of synchronization, and the lateral

dynamic load factor increased from 0.05 to 0.10 as the amplitude of the deck

increased from zero to 30 mm. The results showed that the relationships were

nonlinear and dependent on the frequencies of the bridge movements.

Other research was also carried out to investigate the synchronization and excessive

vibration of footbridges. Stoyanoff et al. [2002] noticed the similarity between the

synchronization of human-induced vibration due to lock-in effect and vortex-

shedding caused by wind, and they suggested using wind engineering theory to

quantify the vibration due to crowds. A correlation factor cR(N) was proposed to treat

the crowd loading problems in a moderate crowd of N people when the density is

below 1.0 person/m2.

NR eNc γ−=)( (2.11)

where the decay coefficient γ was defined from the assumed N and cR(N) at

saturation (cR=0.2 for the maximum congested footbridge). Yoneda [2002] proposed

a simple method to evaluate the vertical and lateral maximum dynamic velocity

responses due to synchronized walking of many people. In the simple method, it was

assumed that the lateral dynamic forces increased in proportion to the increase of the

number of pedestrians. Barker [2002] proposed a different theory to explain the

mechanism of excessive lateral vibration of footbridges based on the observations on

Millennium Bridge. He presented a simple non-linear model of pedestrian loading

from which correlated forces were derived using completely unsynchronised input

and even from input at wrong frequency and he claimed that the response to crowd

movement may increase without any synchronization between people. Further,

Dinmore [2002] suggested treating the human-induced force as a wave which

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propagated through the structures and recommended that using different materials to

vary the dynamic stiffness and provide energy loss due to wave reflection and

refraction on their contact. Blekherman [2005] considered the nonlinear

(autoparametric) resonance in footbridges as a reason for excessive lateral vibration

induced by walking pedestrians, and proposed a physical model (an elastic

pendulum) to describe this phenomenon. He reported that under special frequency

conditions (the ratio between frequencies of vertical and lateral beam modes is about

2, or 2:1), nonlinear resonance became possible if the vertical excited mode was near

a primary resonance and a load parameter (a static displacement caused by

pedestrians) was equal or more than its critical value.

Nakamura [2003] conducted some field measurements of lateral vibration on a lively

suspension footbridge, M-bridge, in Japan. It was reported that the M-bridge suffered

from lateral vibration since it opened in 1999. As the girder vibration was fairly

large, some pedestrians felt unsafe. Measurements showed that the footbridge

vibrated in the third asymmetric mode with a natural frequency of 0.88 Hz or the

fourth symmetric mode with a natural frequency of 1.02 Hz, depending on the

distribution of pedestrians on the bridge. It was also found that the synchronization

was unlikely to occur at the girder natural frequency under 0.60 Hz. Research results

showed that when a pedestrian walked on the vibrating bridge, the person

synchronised to the girder frequency with a phase shift between 120o and 160o ahead

the girder. It was also confirmed that smaller bridge mass and damping produces

largest girder response.

2.4 Slender cable supported footbridges and vibration control

2.4.1 Slender cable supported footbridges Cable supported bridge is one of the main structural forms of modern bridges to

cross large span [Irvine 1992; Gimsing 1998]. With the development of new

materials, the cable supported bridge has increased in popularity to become the main

type of bridge structure for large and long span bridges and it has the ability to

overcome large spans from 200 m to 2000 m (and beyond). Table 2.6 shows the

leading ten long-span bridges worldwide by the year 2005, and all these are

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suspension bridges. Furthermore, the longest span bridge currently being planned is

the Strait of Messina Bridge, Italy. The plan calls for a single-span suspension bridge

with a central span of 3,300 m (about 2 miles). This would be more than 60% larger

than the Akashi-Kaikyo Bridge (1,991 m) in Japan, currently the largest suspension

bridge in the world. It is obvious that cable supported bridges are more versatile than

other forms of bridge structures to cross large span (Figure 2.21).

Table 2.6 The Leading 10 long-span bridges worldwide by the year 2005 (All of these are suspension bridges)

No Bridge Span Location Year

1 Akashi-Kaikyo 1991 m Kobe-Naruto, Japan 1998

2 Great Belt East 1624 m Korsor, Denmark 1998

3 Runyang South 1490 m Zhenjiang-Yangzhou, China 2005

4 Humber 1410 m Hull, Britain 1981

5 Jiangyin 1385 m Jiangsu, China 1999

6 Tsing Ma 1377 m Hong Kong, China 1997

7 Verrazano-Narrows 1298 m New York, NY, USA 1964

8 Golden Gate 1280 m San Francisco, CA, USA 1937

9 Höga Kusten 1210 m Kramfors, Sweden 1997

10 Mackinac 1158 m Mackinaw City, MI, USA 1957

Similar to long span cable supported highway bridges, cable supported footbridges

also have the ability to cross longer spans than other types of pedestrian bridge

structures and can be constructed to be more pleasing and in different structural

configurations. However, compared with long-span and large cable supported

highway bridge structures, the design loads for footbridges are relatively smaller as

pedestrian bridge structures are mainly designed for pedestrians and cyclists. As a

consequence, the girder of cable supported footbridges is often weak and therefore

the structural stiffness is mainly provided by the suspending cables or stayed cables.

Due to the application of high strength and light weight materials and new

technology, modern cable supported footbridges become more slender and flexible

than ever. Slender and flexible footbridges with low stiffness, low mass and low

damping are prone to vibration induced by human activities.

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Figure 2.21 The Maximum span length of bridges [Ito 1996]

Although modern cable supported footbridges can take different configurations, the

main types of bridge structures are suspension bridges, cable-stayed bridges and

ribbon bridges.

In general suspension footbridges, similar to the long span suspension bridges,

comprise four components: towers, supporting cable systems, girder and deck, and

anchor blocks. Since the loads are relatively lighter and the spans are much smaller

than suspension highway bridges, the towers are usually not very tall and the girders

are relatively weak. Though the bridge structures can have three spans, most modern

suspension footbridges are single-span structures, and the supporting cables are

gravity anchored at the anchor blocks. In order to improve the ability to resist the

lateral forces such as wind, the girder systems always take the form of plane braces

[Brownjohn 1994], stiffening trusses, or other measures may be taken by inclining

the supporting cables and hangers, adding lateral resistant systems such as wind

ropes. For example, the M-Bridge (Maple Valley Great Suspension Bridge) in Japan

(Figure 2.22) [Nakamura 2003] is a suspension footbridge built in 1999. It has a

main span of 320 m and two back spans of 60 m each. The tower is 26.2 m high and

made of steel pipe. A cable consists of seven spiral strand ropes with each diameter

of 46 mm. The girder consists of two steel H-beams, not having stiffening trusses or

box girders, and is therefore very flexible in the vertical direction. In order to resist

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the wind forces, the girder is stiffened by sway bracing consisting of steel H-beams

and is also supported by wind ropes on a near-horizontal plane (4o below the pure

horizontal plane). The cables sags are 23.4 m for the supporting cable and 13.3 m for

the wind rope.

Figure 2.22 the M-bridge in Japan [Nakamura 2003]

Macintosh Island Park Suspension Bridge (Figure 2.23), Gold Coast, Australia is the

largest timber suspension footbridge In Australia. This timber suspension bridge is

about 100 m long with the main span of 60 m and two side spans of 20 m. The

timber tower is 5 m high and the walkway width is 1.6 m. Three rectangular beams

sit on the cross rectangular timber beams which are hung by the hangers, and timber

trusses along the two sides act as handrails. The cross beams are connected by weak

brace. Since this timber suspension bridge is very slender and lively, notice boards

with the words “natural movement underfoot will be expected as you cross the

bridge” and “enjoy the sensation as you cross” are placed on each end of the bridge

to remind pedestrians that the bridge vibration is “live”. In order to improve the

dynamic behaviour, two cables are added to underneath the deck and another two

ropes are added on each side to increase the lateral resistance (Figure 2.23(b)).

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However, it seems the timber suspension bridge still has large vibration when

pedestrians cross.

(a)

(b)

Figure 2.23 Macintosh Island Park Suspension Bridge, Gold Coast, Australia (a) – layout; (b) – additional cables

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Cable-stayed footbridges have become more popular in modern bridge industry, and

a large number of pedestrian cable-stayed bridges have been constructed worldwide.

In order to satisfy the aesthetic requirements, most of the cable-stayed footbridges

have asymmetric spans and the pylons can be designed in deferent shapes. The stay

cables can also be arranged in many styles to suit the footbridge structure. The girder

systems can have different structural forms such as truss, stiffening girder, or box

girder. For example, the lively T-bridge (Figure 2.20) in Japan is a cable-stayed

footbridge with two-span continuous steel box girder. The bridge has a main span of

134 m, a side span of 45 m and two cable planes with 11 stays per plane. The tower

is 61.4 m high and made of reinforced concrete [Nakamura and Fujino 2002].

Ribbon bridge structure is another kind of slender cable supported bridges. Since it

takes the form of suspension with small cable sag and its cable profile is very flat, it

can be considered as a shallow suspension bridge. In general, long-span suspension

bridges have large cable sags varying from 1/6 to 1/10 of their spans, and they

achieve their load and deformation resistance and stability under vibration,

oscillation and galloping effects using tension cables with significantly large sags

hanging from tall towers interacting with stiffening girders or trusses. Ribbon

bridges, however, have small cable sags varying from 1/40 to 1/60 or even smaller of

their spans, and they do not need tall towers and can cross large spans.

Stress ribbon bridge structure [Morrow et al. 1983; Del Arco et al. 2001] is the main

type of ribbon bridges, which was developed in the 1960s. The stress ribbon concept

borrows from the suspension bridge principle but develops it further by using high

strength materials and modern engineering technology, especially precasting and pre-

stressing methods. It may be thought of as degenerate suspension bridge in the sense

that the deck and suspension cables have the same profile in elevation. In a

prestressed concrete stress ribbon bridge, high strength steel cables pass through a

series of precast concrete components, the deck assembly of which can be tensioned

from stiff abutments. Whereas in a suspension bridge the main load carrying

component is the cable with deck acting as a stiffening element, in a stress ribbon

bridge both the cable and the deck can be independently tensioned, thus adding

considerable rigidity to the structures. This kind of bridges are mainly designed and

constructed for roadway, pedestrian bridges. Since they are slender, pleasing in

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appearance and easy and practical to be constructed, a number of such bridges have

been constructed in various countries such as Japan, Switzerland, United Kingdom,

former Czechoslovakia, United States, and other places. The work by Strasky and co-

workers is outstanding because of their contributions to both the practical and the

theoretical aspects [Strasky 1987; 1995; 2002; 2005; Redfield et al. 1992]. One of

the most remarkable prestressed concrete footbridges built is the one over the

Sacramento River in California [Redfield et al. 1992] with a 130m span for a depth

of only 38cm.

Figure 2.24 Stress ribbon bridge, Prague-Troja, Czechoslovakia [Strasky 1987]

A stress ribbon bridge generally refers to the cable supported pre-stressed concrete

bridge. However this kind of structure can also be designed with a steel deck [Wheen

and Wilson 1977] or a deck of other light materials [Block and Schlaich 2002]. For

example, Tanaka et al. [2002] carried out a study to develop two types of hybrid

stress-ribbon pedestrian bridges called as “stress-ribbon cable-stayed suspension”

and “stress-ribbon suspension” footbridge with a very light full-steel or a light

concrete-steel girder.

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The Millennium footbridge in London (Figure 2.25) is a three-span tension ribbon

bridge or shallow suspension bridge which was designed by the consulting

engineering company Ove-Arup and opened to pedestrians in 2000. The

superstructure has three spans of cable supported structure with maximum 2.3 m

cable sag and the bridge is 144 m long in the middle span. Transverse arms span

between cables along the two sides and the deck structure comprises two steel edge

tubes which span onto the transverse arms and the extruded box section aluminium

deck span on the tubes. In the bridge superstructure, the cables have been designed to

be as much as possible below the level of the bridge deck to allow uninterrupted

views from the deck. The supporting cables are two groups of four 120 mm diameter

locked coil cables spaning from bank to bank over two piers, and the structural

stiffness is mainly provided by there supporting cables.

Figure 2.25 the Millennium Footbridge in London, U.K. (Photo by Ian Britton)

It can be seen that though cable supported footbridges can have different types of

superstructures, the decks often span on the supporting cables or on cross frames

which are hung on the cables. Since the girder is always weak, the whole structural

stiffness is mainly provided by the supporting cables. Slender cable supported bridge

structures often have complex behaviour and some important characteristics, such as

coupled load performance in lateral and vertical directions as well as coupled or

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multi-modal vibration, have been often ignored by bridge engineers. For example, Ji

et al. [2003] once pointed out that for cable suspended bridges, if the cables were

inclined, the deck would experience both horizontal and rotational movements when

it was subjected to an asymmetrically applied vertical load. Cable suspended bridges

are more likely to be sensitive to horizontal movements induced by the vertical load

than other kinds of structures, and it is pertinent to examine the horizontal movement

induced by both vertical and horizontal loads. However, there is little research

considering this kind of complex behaviour in dynamic performance of slender cable

supported footbridges.

2.4.2 Measures against excessive vibration of slender footbridges

There are substantial references in literature regarding to vibration control of

different structures. According to the theory of dynamics, the dynamic response of a

structure depends on many factors such as structural mass and stiffness, natural

frequencies and vibration modes, structural damping, applied loads and the load

frequencies. The dynamic response could be changed if any of the factors changed.

Based on the theory of dynamics, many measures and techniques have been

developed to reduce the excessive vibration and most of them can be applied into

bridge structures. The main measures against excessive vibration of footbridges

include: frequency tunning; detailed vibration response assessment and amplitude

limitation; introducing special measures and increasing damping [Bachmann 2002].

Frequency tunning is a measure related to the structural natural frequencies. The

natural frequencies, especially the fundamental frequency, can be “tuned” by

changing the stiffness and mass system to avoid the resonant vibration. According to

Bachmann [2002], the vertical natural frequency of footbridge structures, in the

range 1.6 – 2.4 Hz should be avoided, moreover, the vertical frequency in the range

of 3.5 – 4.5 Hz also should be avoided in the case of footbridges with a small

damping ratio (mainly steel and composites bridges). Further more, if footbridges are

often crossed by running persons, the natural frequencies in the range of 2.1 – 2.9 Hz

should also be avoided. In the lateral direction, the frequency around 1.0 Hz (i.e. 0.7

to 1.3 Hz) and in the case of very light and lowly damped bridges perhaps around 2.0

Hz or even 3.0 Hz should be avoided. In any case a “high tuning” to greater than 3.4

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Hz is a safe solution. However, lower frequency modes can also be excited. For

example, the lowest lateral mode corresponding to the frequency of about 0.5 Hz of

Millennium Bridge in London was excited during the opening day. Pimentel [1998]

also mentioned that the handrail of a footbridge was identified as a potential way to

increase the structural stiffness and thus the natural frequencies. A selective

distribution was the best strategy of frequency tuning.

Detailed vibration response assessment was always carried out to make sure the

dynamic amplitudes are within limitation and satisfy the vibration serviceability

requirements in the contemporary design procedures. However, the results and their

reliability are often questionable [Zivanovic et al. 2005] because of many uncertainty

of modelling assumptions.

Some special measures may be adopted to improve the dynamic behaviour of slender

footbridges. These special measures include changing the structural form to increase

the stiffness; adding damping materials and improving the connections to increase

the damping, especially for structures with low damping.

Increasing damping is considered as the most convenient and effective measure to

control excessive vibrations of footbridges. The damping can be increased by

installing extra damping devices to the primary bridge structure. Although many

different kinds of damping devices have been developed nowadays, the popularly

used devices in bridge structures are tuned mass dampers (TMDs), tuned liquid

dampers (TLDs) and viscous dampers. A tuned mass damper [Hatanaka and Kwon

2002] is a spring-mass-damper system and its natural frequency and damping ratio

are tuned to the relevant properties of the primary structural system resulting in an

optimum frequency and damping ratio to reduce the dynamic response of primary

bridge structure. However, a tuned mass damper is only efficient in a narrow

frequency range and when exactly tuned to a certain natural frequency of the primary

system, and it doesn’t work efficiently if the primary system exhibits several

narrowly spaced natural frequencies, such as a flexural bending and a torsional

fundamental frequency. A tuned liquid damper (TLD) is similar to TMD and uses a

large mass of liquid. This is only effective when the forcing frequency is near the

natural frequency of the primary system. In addition it has advantages, such as low

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cost, almost zero trigger level, easy adjustment of natural frequency, easy

installation, and low maintenance [Noji 1990; 1991]. Although different kinds of

viscous dampers have been developed, a viscous damper is mainly a vibration

absorber which has the ability to convert the kinetic energy into heat and increase

damping to the primary system and therefore reduce the dynamic response.

In order to reduce excessive vibrations of bridges, one or more different measures

may be employed. Fujino et al. [1992a; 1992b; 1992c] reported that 600 plastic tanks

with 34 mm of water were used as tuned liquid damper to suppress the lateral

excessive vibration of the T-bridge in Japan. All these TLDs were placed inside the

box girder and the mass ratio was only about 0.007. To reduce the in-plane vertical

oscillation of the stay cables of the same footbridge, secondary wires were also

employed to connect stay cables. In another lively footbridge, Millennium Bridge in

London, 37 fluid viscous dampers were installed on the bridge structures to suppress

mostly the excessive lateral vibration. As a consequence, the total structural damping

ratio increased from 0.5% to 20% and the maximum near-resonant accelerations

were reduced by about 40 times.

2.5 Some existing footbridge design codes on human-induced

vibration

Bridge codes provide the guidelines for bridge structure design. However, for slender

and flexible bridge structures, these guidelines are not adequate for the bridge

engineers as the flexible bridge structures are prone to enhanced levels of vibration.

When the Millennium Bridge in the UK suffered severe unexpected lateral

movements as pedestrians crossed the bridge, Dallard et al. [2000] confirmed the

inadequacy of existing codes to address the synchronous excitation on footbridges

and the complexity of multi-modal vibration experienced by the bridge.

Bridge Codes approach the vibration problem induced by pedestrians in two ways.

The simplest way is to determine the natural frequencies of the footbridge structure

and ensure the frequencies lie out of the range of pedestrian pacing rate. Another way

is to model a pedestrian walking or running across a footbridge or jumping on the

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bridge, determine the maximum dynamic response of the bridge structure and ensure

that this is within the acceptable limits.

The British code, BS5400: Part 2 [1978] defined a procedure to check the vertical

vibration serviceability due to a single pedestrian for the footbridges having

fundamental vertical natural frequency up to 5 Hz. According to this procedure, the

maximum acceleration is calculated by modelling the load produced by one

pedestrian as a dynamic pulsating load F(t) which moves along the footbridge. Such

a load has a frequency (pacing rate) coinciding with the first natural frequency f0 of

the unloaded footbridge in the vertical direction:

)2sin(180)( 0tftF π= (N) (2.11)

where t is the time (second), and the pedestrian is assumed to have a weight of 700N

and move with a speed v=0.9 f0 (m/s). The term 180N is the amplitude of the first

harmonic of the walking load, and the maximum acceleration must be smaller than

0.5 0f (m/s2). If the first frequency of the unloaded footbridge exceeds 5.0 Hz, no

further checks are required on the vibration serviceability. For value of the first

natural frequency greater than 4.0 Hz, the calculated maximum acceleration is to be

reduced by an amount varying linearly from 0.0 at 4.0 Hz to 70% at 5.0 Hz.

An updated version of BS 5400: BD 37/01 [2002] requires a check on the vibration

serviceability in the lateral direction. A detailed dynamic analysis is required for all

footbridges having fundamental lateral natural frequency below 1.5 Hz. However,

the checking procedure is not given in the code. Although the BS 5400: part 2 and

BD 37/01 addressed dynamic excitation of footbridge by pedestrians, they did not

cover synchronous vertical and lateral excitation arising from groups or crowds of

pedestrians under normal usage. Such loadings can be significantly greater than

normal code provision.

Similar to the BS 5400 [1978], the Ontario Highway Bridge Design Code [OHBDC

1991] contains the same provisions for the assessment of vibration serviceability of

footbridges. A dynamic analysis is required to check the dynamic response of a

bridge due to a footfall force, simulated by a moving sinusoidal force with amplitude

of 180 N and a frequency equal to the fundamental vertical natural frequency or 4.0

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Hz, which is ever lower. However, the limit acceleration defined graphically is lower

than that in BS 5400. The Code also required that the lateral and longitudinal

frequencies of the superstructures should not be less than the smaller of 4.0 Hz and

1.5 times of the fundamental vertical frequency. There is no clause regarding the

assessment of lateral vibration induced by pedestrians. The Canadian Highway

Bridge Design Code [CSA 2000] provides the same provisions for the vibration

serviceability of footbridge structures.

Although the American Guide specification [AASHTO 1997] kept silent on the

vibration serviceability due to pedestrians, it proposed to avoid fundamental vertical

frequencies of footbridges below 3 Hz. In the case of low stiffness, damping, mass,

and if running and jumping were expected, all frequencies below 5 Hz should be

avoided. However, the lower limits for those dynamic properties were not

mentioned.

Eourocode 5 [1997] contained some guidelines for timber bridge design related to

the loadings induced by pedestrians. It required the calculation of acceleration

response of a bridge due to small groups or streams of pedestrians in both vertical

and lateral directions, with the proposed frequency-independent acceleration limits of

0.7 and 0.2 m/s2 in these two directions respectively. These limits of acceleration

should be checked for bridges with vertical natural frequency below 5 Hz and lateral

frequency below 2.5 Hz. And a procedure to calculate the vertical acceleration for

bridges with one, two or three spans was provided based on the response due to a

single pedestrian.

The new Australia standard for bridge design AS 5100.2-2004 (Part 2: bridge loads)

[SAI 2004] provided a clause for the vibration serviceability of footbridges. It

required that the vibration of superstructures of pedestrian bridges with resonant

frequencies for vertical vibration in the range 1.5 Hz to 3.5 Hz should be investigated

as a serviceability limit state. To assess the dynamic response, one pedestrian was

assumed to have a weight of 700 N and cross the superstructure at an average

walking speed from 1.75 to 2.5 footfalls per second. Unlike the BS 5400 using the

maximum acceleration as limit of vibration serviceability, it required the maximum

dynamic amplitude not to be greater than the graphically frequency-dependent

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dynamic amplitude limit (Figure 2.26). For the pedestrian bridges with the

fundamental frequency of horizontal vibration below 1.5 Hz, The Code required that

special consideration should be given to the possibility of excitation by pedestrians

of lateral movements of unacceptable magnitude. However, the limit of lateral

deflection is not given. The Code also gave a note to mention that bridges with low

mass and damping and expected to be used by crowds of people, are particular

susceptible to such vibrations and specialist literature should be referred to.

Figure 2.26 Dynamic amplitude limits for pedestrian bridges [SAI 2004]

2.6 Summary

Due to new technology and application of light weight and hight strength materials,

modern pedestrian bridges are designed and constructed slender and flexible.

However, with low stiffness, structural mass and damping ratio, slender bridge

structures are prone to vibrations induced by human activities such as walking,

running and jumping. If the natural frequencies of the bridge structures are within the

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frequency range of human-induced dynamic loads, they are easy to be subjected to

synchronous excitations and serious vibration serviceability problems may arise.

In this chapter, a comprehensive review has been carried out on the human-induced

dynamic loads, vibration serviceability and dynamic performance of footbridges

under such loads. The main aspects reviewed include: the measurements and

modelling human-induced loads; vibration properties and dynamic performance of

footbridges; and design guidelines related to the vibration serviceability of

footbridges. The review shows that the whole issue is very complex and further

investigations are required to appropriately assess dynamic response of modern

slender footbridges under human-induced loads and ensure enhanced levels of safety

and reliability. Some conclusions from the literature reviewed are summarised

below:

• The force induced by human activities normally is frequency-dependent and

has three components in vertical, lateral and longitudinal directions. Most

measurements and modellings in the past were focused on the vertical

component. The lateral and longitudinal components were seldom investigated

and further investigation including measurement and modelling are required.

• The effect of groups or crowd of people is very complex and is significantly

affected by the synchronization between people and the synchronization of

human-structure interaction. Furthermore, the force measured on slender

structures is reported to be smaller than that measured on rigid surface and is

affected by the human-structure interaction. However the probability of

synchronization is not clear. Investigation is required to appropriately model

group and crowd loads and the forces on slender structures.

• Human perception of vibration is a complex issue and affected by many

factors such as vibration frequency, body movement, expectancy of motion and

body posture. It is clear that pedestrians are sensitive to low frequency lateral

motion of the vibrating structure they are crossing.

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• Modern footbridges are slender and flexible with low stiffness, low mass and

low damping. They are more prone to vibrations induced by human activities

than ever. When the spans increase, most footbridges have vertical and lateral

fundamental frequencies below 5 Hz and within the frequency range of human

activities.

• Synchronization phenomenon in lateral vibration can happen on any

footbridge, independent of structural form, with a lateral natural frequency

below 1.3 Hz and with a sufficiently large crowd of pedestrians crossing the

bridge. Lateral vibration mode with frequency lower than the frequency of

normal walk can also be excited, and so do the higher vibration modes

depending on the load distribution. However, it is still not clear how the

synchronization is developed and what kind of relationship exists among the

probability of synchronization, dynamic response, vibration modes and human-

structure interaction.

• Many bridge design codes provided guidelines to assess the vibration

serviceability of footbridges in vertical direction by considering one pedestrian

crossing the footbridge at the natural frequency. Although dynamic analysis is

required by some design codes, no detail procedure is given. Slender

footbridges continue to be “live” and cause vibration serviceability problem.

This indicates that further research should be carried out to provide reliable

guidance regarding to the human-induced lateral vibration of footbridges.

Some knowledge gaps on the vibration serviceability of slender footbridges under

human-induced loads are listed as follows:

• There is no frequency-dependent force function available for modelling the

lateral and longitudinal forces induced by walking and other human activities,

and there is no force model proposed to simulate synchronous excitation and to

study lateral vibration.

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• Pedestrians are sensitive to low frequency lateral vibration, but the limit of

vibration serviceability in the lateral direction of slender footbridges is not

clear. The relationships among the probability of synchronization, dynamic

response, vibration modes and human-structure interaction are also not clear. .

• Slender footbridges with low mass and low stiffness are prone to human-

induced vibration. It is known that pedestrians have some influence on the

vibration properties and dynamic performance of such structures. These effects

are presently not understood as they have not been comprehensively

investigated.

• Modern slender suspension footbridges can be designed and constructed as

ribbon bridges with small sag ratio (cable sag to span) less than 1/40, but they

have complex vibration behaviour which is not fully understood at present.

These aesthetically pleasing bridges are becoming increasingly popular, but

have experienced vibration problems due to inadequate design guidance.

The present thesis addresses some of the knowledge gaps identified above and in

particular aims to generate new information on the dynamic characteristics of slender

suspension bridges, with a view of providing design guidance.

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Suspension footbridge model with pre-tensioned reverse profiled cables

3.1 Introduction

When pedestrians walk cross a bridge structure, the pacing rates can vary from 1.4

Hz to 2.4 Hz, and even lower under congested condition. If the footbridge structure

has natural frequencies within this range, it is susceptible to dynamic force induced

by pedestrians. On the other hand, pedestrians trend to change their pacing rates to

move in harmony with the bridge vibration and the bridge structure vibrates at near

resonant frequency under such synchronous excitation. This mechanism could lead to

large amplitude vibration and cause serious vibration serviceability problems in

footbridges. It is therefore important to study the load performance, vibration

properties and dynamic behaviour of footbridges with natural frequencies within the

frequency range of human-induced dynamic loads.

In this conceptual study, a slender suspension footbridge model with reverse profiled

pre-tensioned cables in the vertical plane and pre-tensioned side cables in the

horizontal plane is proposed to investigate the structural behaviour and dynamic

characteristics of slender footbridges under human-induced dynamic loads. In this

bridge model, the transverse bridge frames with top, bottom and side legs hang from

the top suspending cables and further restrained by the reverse profiled pre-tensioned

bottom cables and pre-tensioned side cables. The deck units span across beams

which are simply supported on the bridge frames. The pre-tensioned suspension

footbridge model is chosen and designed due to the ease of modifying the dynamic

properties by varying the cable profiles and cable forces, and hence obtaining a range

of low natural frequencies. When pre-tensions are introduced into the reverse

profiled bottom and/or side cables, the structural stiffness in vertical and lateral

planes can be improved and the natural frequencies can be altered. This feature will

be useful in investigating the load deformation performance and dynamic behaviour

at different natural frequencies of such footbridges under human induced loads.

3

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In this bridge model, the structural stiffness is entirely provided by the cable systems.

When the structure is subjected to applied loads, all the loads can be balanced by the

tension forces in the cables with deformed cable profiles since these forces can

provide components in different directions. As the main concern of this conceptual

study is the load deformation performance and dynamic behaviour, the connection

details and anchorages of cables are not considered for the analysis, although they

are very important in the design and construction of real footbridges.

3.2 Description of the proposed suspension footbridge model

This pre-tensioned suspension footbridge model is shown in Figure 3.1. In this

bridge model, the cable systems are composed of three groups of cables which may

have same or different profiles: top supporting (or suspending) cables, bottom pre-

tensioned cables (Figure 3.1(a)) and side pre-tensioned cables (Figure 3.1(b)). The

top cables are two parallel supporting cables which have catenary profiles and

provide tension forces to support the whole structural gravity, applied loads and

internal forces induced by the pre-tensioned bottom cables. Two parallel bottom

cables are designed to have reverse profiles in the vertical plane and their function is

to introduce pre-tension forces and provide internal vertical forces to transverse

bridge frames and the top supporting cables. The side cables are a pair of bi-concave

cables which have the same cable profiles in the horizontal plane, and their main

function is to provide internal horizontal forces and horizontal stiffness. When the

pre-tensioned bottom and/or side cables are slack, they could carry small tension

forces only to support their own gravity and cannot resist any external loads. In this

case, they couldn’t contribute stiffness and tension forces to the structure. However,

these small tensions can provide sufficient restraining forces to prevent the transverse

frames from swaying in the longitudinal direction.

Transverse bridge frames have been designed to support the deck and hold the

cables. These frames (Figure 3.1(c)) comprise cross members (for the support beams

and deck), top and bottom vertical legs as well as horizontal side legs and they form

a set of spreaders for the cables to create the required profiles. They have in plane

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stiffness to protect against collapse under in plane forces and contribute very little in

the way of longitudinal, lateral and rotational stiffness for the entire system. The

transverse bridge frames are hung from the top cables, and further restrained by the

lower reversed profile cables as well as the side cables. Two support beams of

rectangular section are simply supported on cross members of the adjacent bridge

frames, and the deck units are simply supported at the ends on these beams.

(a)

(b)

40004003000

30

0030

00

400 3000

1F1F

F2

F2

F3F3

Top Cable

Bottom Cable

Side Cable

(c)

Figure 3.1 Pre-tensioned cable supported bridge model: (a) – elevation; (b) – top view; (c) – middle transverse bridge frame

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It is noticed that similar reverse cable systems have been proposed and used in long

span suspension bridges and footbridges. For example, in a long span suspension

bridge model proposed for the Straits of Messina Bridge, Italy, stabilising reverse

cables were designed to improve the torsional and lateral stiffness as well as the

aerodynamic behaviour [Borri et al. 1993], and in the M-bridge, Japan, wind ropes

with reverse profiles in nearly horizontal plane were designed to increase the lateral

resistance against wind loads [Nakamura 2003]. However, there is little information

regarding the structural behaviour of such bridges and influence of parameters, and

further information is needed to understand their performance under loads.

In order to simplify the problem, all the transverse bridge frames are assumed to have

the same size, and hence the weight of frame and deck acting on the cables can be

considered as equal concentrated loads.

3.3 Cable profiles and initial distortions

A typical symmetric cable profile with equal concentrated loads is shown in Figure

3.2. In the bridge model, there are three groups of cables and they can be designed

with different cable profile. In the following description, different cables and cable

profiles are defined by the subscript j, where j =1, 2, 3 represents the top, bottom and

side cables as well as their profiles respectively.

Figure 3.2 A typical cable profile

For a cable supported bridge model with N uniform segments in the horizontal

direction, the forces from the N-1 transverse bridge frames can be modelled as N-1

equal concentrated loads acting on the cables. Assuming the horizontal distance

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between two adjacent transverse bridge frames (or loads) to be a., the span length

will be defined as:

NaL = (3.1)

For the jth symmetric cable with the cable sag Fj , the sag Fj is located at the middle

segment or the middle node K. Choosing local x – y coordinates as shown in Figure

3.2, the coordinates for the node K can be obtained as:

Kax jK = jjK Fy = )2/int(NK = (3.2)

Where int( ) is an integer function. For a symmetric cable subjected to equal

concentrated loads, it is easy to obtain the vertical and horizontal reactions by using

static equilibrium equations. Using these reactions, equal concentrated loads and

cable sag, the cable profile and the tension forces as well as the tensile deformation

in the segments can be also calculated. The coordinate of the i th node, j th cable can

be expressed by

iax ji = jiji Fy α= Ni ,,2,1,0 ⋅⋅⋅= (3.3)

Here the coefficient αi can be calculated by the following equation.

)](/[)( KNKiNii −−=α (3.4)

The tension force Tji and tensile deformation ∆Lji of the i th segment, jth cable can be

obtained by

WT jiji β= )/( jijijiji AEWaL γ=∆ 3,2,1 ;,,2,1,0 =⋅⋅⋅= jNi (3.5)

Eji and Aji are Young’s modulus and area of cross section of the i th cable segment, jth

cable. W is the applied equal concentrated load. The coefficients βji and γji are shown

to be as follows

[ ] [ ]22 12)/)((21 +−+−= iNFaKNK jjiβ (3.6)

221 )/()(1 aFjiijiji −−+= ααβγ (3.7)

In the bridge model, all the cables are stretched to keep the designed cable sags or

cable profiles and then the decks can be kept in a horizontal plane before the bridge

structure is subjected to the applied loads. This can be done by introducing initial

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distortions to the cables according to their cable sags, cross sectional areas, material

properties, loads such as the weight of bridge frame and decks as well as cables, and

extra internal forces produced by pre-tensioned reverse profiled cables or horizontal

side cables.

Figure 3.3 Extra internal forces in cables

Assuming the bottom cables have a diameter D2, Young’s modulus E2, and cable sag

F2, if the internal vertical force Wint at each bridge frame is induced (Figure 3.3), the

initial distortion ∆L2i introduced to the i th cable segment of one bottom cable can be

determined to be:

)/(2 222int22 DEaWL ii πγ−=∆ (3.8)

The side cables are a pair of bi-concave cables in the horizontal plane which have

opposite cable profile to each other. When they are pre-tensioned, only internal

horizontal forces can be introduced to the bridge frames. If the side cables have

diameter D3, Young’s modulus E3, and cable sag F3 (in horizontal plane), and

internal horizontal force Qint at each bridge frame is induced by the pair of side

cables (Figure 3.3), the initial distortion ∆L3i introduced to the i th cable segment of

one side cable is determined as:

)/(4 233int33 DEaQL ii πγ−=∆ (3.9)

When the internal vertical force Wint is induced at each bridge frame by pre-tensioned

bottom cables, the top supporting cables are subjected to the weight (gravity) of the

whole structure and the extra internal vertical forces. If the top supporting cables

have diameter D1, Young’s modulus E1, and cable sag F1, and the total weight of one

bridge frame, the cables and decks between adjacent frames is G , the following

initial distortion ∆L1i in the i th cable segment of one top cable should be introduced:

- 73 -

)/()(2 211int11 DEaWGL ii πγ +−=∆ (3.10)

After the initial distortions are introduced to the cable systems, the cable profiles can

have the designed cable sags and the bridge deck will be kept in the horizontal plane

before it is subjected to the applied loads.

3.4 Suspension footbridge models for numerical analysis

3.4.1 Structural analysis softwares

The structural analysis softwares Microstran [Engineering Systems 2002] and

SAP2000 [CSI 2004] are used for the numerical (Finite Element) analyses in this

conceptual study. This numerical analysis is carried out in two phases. The first

phase consists of static and free vibration analyses and includes choosing bridge

models for the research project, study of the structural performance under static

applied loads, and dynamic properties of the proposed bridge models as well as

effect of structural parameters. The second phase is essentially dynamic analyses and

includes modelling of crowd walking dynamic loads, study of dynamic

characteristics of slender suspension footbridge models and their non-linear time

history behaviour under crowd walking dynamic loads. In the numerical analysis, all

members except the cables will be modelled as frame or beam elements. Microstran

and SAP2000 have different methods to model the cable members. Microstran has

catenary cable element included in the software package and it is easy to model the

cable supported structures. SAP2000 has a powerful capacity for dynamic analysis

including non-linear time history analysis. However, it uses only frame/beam

elements, with variable section properties, to simulate the behaviour of slender

cables. When a bridge model has slack cables, it will take long time for the software

to check the cable’s status, whether it is in tension or buckled. Therefore, in the first

phase, Microstran is adopted to choose and design the slender suspension footbridge

models, and to investigate the load performance and dynamic properties. In the

seconde phase, SAP2000 is used to investigate the non-linear time history dynamic

response, and in most of the footbridge models, the pre-tensions are introduced into

the reverse profiled cables to obtain the required natural frequencies.

- 74 -

As mentioned before, the cables in a suspension footbridge model are stretched to

keep the cables at the designed cable sags and deck in horizontal plane before the

bridge is subjected to the applied loads. This can be done by introducing different

initial distortion in the cables depending on their initial tension forces. When

Microstran is adopted for the structural analysis, the initial distortions can be applied

directly as initial distortion loads. While when SAP2000 is used, the initial

distortions can be introduced by temperature loads. If the thermal factor of the cable

materials is assumed to be α, then the initial distortions ∆L1i , ∆L2i and ∆L3i defined

by equations (3.10), (3.8) and (3.9) for the i th segment of top, bottom and side cables

can be represented approximately by the temperature loads ∆T1i , ∆T2i and ∆T3i

respectively for the cable with small cable sag:

)/()(2 211int11 DEWGT ii παγ +−=∆ (3.11)

)/(2 222int22 DEWT ii παγ−=∆ (3.12)

)/(4 233int33 DEQT ii παγ−=∆ (3.13)

3.4.2 Bridge models in Microstran and SAP2000

In the proposed suspension bridge model, stainless steel (Young’s modulus 2.0×1011

Pa and density 7850 kg/m3) is chosen for the transverse bridge frames and support

beams, and Aluminium (Young’s modulus 6.5×1010 Pa and density 2700 kg/m3) is

chosen for the deck units to reduce the total structural weight. Stainless steel cables

are chosen for all the cable systems and the material properties are the same as those

of bridge frames. The thermal coefficient of cables is assumed to be 0.117×10-5. In

the numerical analysis, the horizontal distance between the adjacent bridge frames is

set to be 4 m and the width of the deck for applied loads is set to be 4 m. The cable

sags, cable sectional areas (diameters) and span length are important structural

parameters and can be changed to suit the research aim.

In numerical analysis and parameter study, two set of sections are chosen for the

members of bridge frames, support beams and deck units: solid rectangular sections

and hollow rectangular sections. A footbridge with the members of solid rectangular

- 75 -

sections is noted as Solid Section Bridge (SSB) while the footbridge model with the

members of hollow section noted as Hollow Section Bridge (HSB).

SSB: In a solid section footbridge model, it is assumed that all members of the

transverse bridge frames have uniform rectangular cross section dimensions of

250×300 mm and the support beams have uniform rectangular cross section

dimensions of 200×250 mm. 8 deck units have the same dimensions of

4000×500×50 mm.

(a) (b) (c)

Figure 3.4 Hollow sections of bridge members (HSB): (a) – member of bridge frame; (b) – supporting beams; (c) – deck units

HSB: In a hollow section footbridge model, hollow rectangular sections (Figure 3.4)

are chosen for all bridge frames and support beams and extruded section is used for

the aluminium deck units to reduce the entire structural weight. The hollow

rectangular section dimensions for all the member of bridge frames are 250×300×20

mm, here the 20 mm is the thickness and 250 mm and 300 mm are the outer width

and height. The section dimensions for support beams are 200×250×20 mm. The net

area of the section of a deck unit is about 100×125 mm2, i.e. 10 small sections of

10×125 mm2 within the width of 500 mm.

It should be noted that the main concern of this conceptual study is the load

performance and dynamic structural behaviour as well as the effects of important

parameters. For the proposed pre-tensioned suspension footbridge model, the

sections of bridge frames, supporting beams and deck units, as well as the connection

details of members and anchorages of cables are not important for the overall bridge

structural behaviour, although they are very important in the design and construction

of real footbridges. All the bridge members, no matter with solid section or hollow

- 76 -

section, are designed to have enough stiffness to prevent them and bridge frames

from collapse. And the main difference between the members with solid or hollow

sections is that they can have different gravity, structural mass as well as maximum

tension forces. For such footbridge structures the important structural parameters are

span length, cable profiles, cable section and pre-tensions.

Cable configuration is not a parameter, but it shows the layout of footbridge structure

and cable systems. Therefore, different footbridge models can be represented by their

cable configurations. As the proposed footbridge model consists of three groups of

cables: top supporting cables, pre-tensioned bottom cables and pre-tensioned side

cables, and different cable configuration can be obtained by choosing the

combination of different cable systems.

Figure 3.5 Footbridge model C123 in SAP2000

As described before, different cables and their cable profiles are defined by a

subscript j, when j equals to 1, 2 or 3, the number represents the top supporting

cables, pre-tensioned bottom cables or pre-tensioned side cables respectively. The

cable configuration can also be defined by the combination of cable numbers. By this

definition, the following cable configurations or bridge models will be mentioned in

future analysis and discussion: C123, C120, C103 and C100. Here the C refers to the

cable configuration, and the order of number refers to top, bottom and side cables,

and “0” indicates the corresponding cables being removed from the footbridge

structure or having no contribution to the structural stiffness.

- 77 -

C123 – footbridge model with top, bottom and side cables;

C120 – footbridge model with top and bottom cables but without side cables;

C103 – footbridge model with top and side cables but without bottom cables;

C100 – footbridge model with top cables as well as slack bottom and side cables.



- 78 -

In the bridge model C123 (Figure 3.5), the reverse profiled bottom and side cables

are pre-tensioned and the structural stiffness is provided by all the cable systems.

When the footbridge model has the cable configuration C120 (Figure 3.6), the

structural stiffness is provided by the top supporting cables and pre-tensioned bottom

cables. The side cables as well as side legs are always removed to reduce the total

structural weight and maximum tension forces. For the footbridge model C103

(Figure 3.7), the stiffness is provide by the top supporting cables and pre-tensioned

side cables. The bottom cables are removed from the structure together with the

bottom legs. While in the bridge model C100 (which has the same cable

configuration as bridge model C123), the structural stiffness is provide only from the

top supporting cables. The bottom and side cables are let to be slack and have no

contribution to the structural stiffness.

Figure 3.8 Footbridge model in Microstran

In the following chapters, bridge models with different cable configurations will be

studied for different research purposes. In chapter 4, the main concern is the

characteristics of load deformation and vibration properties. The bridge model

(Figure 3.8) with solid section members (SSB) and all reverse profiled cables will be

adopted in the numerical analysis. Though this bridge model has the cable

configuration C123, it will behaviour like bridge model C100, C120 or C103 when

the reverse profiled bottom and/or side cables are let to be slack. The main reason is

to keep the structural weight same in all the footbridge models in order to investigate

the effect of some structural parameters such as cable sections. As all the bridge

models have the same cable configuration C123, these models will be mentioned as

- 79 -

“pre-tensioned bridge” model or “un-pre-tensioned bridge” model. In chapter 5, a

scaled physical suspension bridge model will be designed and constructed to have

similar structural features to those of the proposed footbridge model. The physical

model with configuration C123 and C120 will also be studied for calibration

purpose. While in chapter 6, the main concern is the dynamic behaviour of slender

suspension footbridge structures and the bridge model with hollow section members

(HSB) will be adopt to reduce the total structural weight and maximum tension

forces in the cable systems. Therefore, the cables without contribution to the

structural stiffness are removed and the bridge models will be referred by their cable

configurations such as bridge model C120, bridge model 103 or bridge model C123.

3.4.3 Finite element modelling of bridge models

In order to study the structural behaviour and dynamic performance of slender

suspension footbridges, extensive numerical analyses are carried out on the proposed

bridge model by using Finite Element method. As mentioned before, Microstran has

catenary cable element included in the software package, while SAP2000 uses

frame/beam elements to simulate the behaviour of slender cables. Therefore similar

finite element modellings of all the bridge members except the cables are used for

the proposed footbridge models in Microstran and in SAP2000 though the members

sections are different.

The proposed footbridge model is modelled as a space frame structure with three-

dimensional prismatic beam (cable) elements. The deflection of the structural model

is governed by the displacements of the joints. Every joint of the structural model

have up to six displacement components: three translations along the local axes and

three rotations about its axes (Figure 3.9). Different supports and boundary

conditions are simulated by applying corresponding joint restraints. A beam element

(Figure 3.10) has two end nodes connected to two different joints and each end node

has six displacement degrees of freedom. Different connections among structural

members can be modelled by using end releases and member offsets. Figure 3.5 to

Figure 3.7 show the numerical suspension footbridge model with different cable

configurations in SAP2000, while Figure 3.8 shows the computer model in

Microstrian. In these computer models, the bridge deck units are assumed to be

- 80 -

simply supported by the supporting beams, they are modelled as 3D beam elements

with released ends: the two ends are supposed to have the same pin connections to

the supporting beams and can carry torques and axial forces in order to keep the

structure symmetric about the bridge centre line. The supporting beams are also

modelled as 3D beam elements with released ends (pin connections), but one end can

not carry torque and axial force. Therefore the axial force and torque in a supporting

beam are only caused by the loads on the supporting beam. For the bridge frame, the

members are modelled as 3D beam elements rigidly connected together at the

intersection points. All the joints on the bridge frames at the two ends of the bridge

model are assumed to have fixed joint restraints (i.e. zero translations and rotations),

and therefore these frames have almost no effect on the structural performance and

vibration properties.

Figure 3.9 Displacement degrees of freedom in the local coordinate system [CSI 2004]

Figure 3.10 Beam/frame element and corresponding coordinate systems [CSI 2004]

halla


halla


- 81 -

When Microstran is adopted for the numerical analysis, all the cables are modelled as

catenary cable elements and the end releases are not permitted. These cable members

have axial tension only – no other member force components exist, and the catenary

formulation of the cable element permits the accurate computation of the equilibrium

position of each cable under load. As the initial tension force in a cable element is

affected by the unstrained cable length, the chord length is taken as the unstrained

cable length in the modelling, and pre-tension forces are applied by initial distortion

loads.

When SAP2000 is used to model cable supported structures, the cables can only be

modelled as frame or beam elements with non-linear properties added. These

elements are supposed to be tension only members (or easily buckled members) and

have large deflections. In order to simulate the flexible behaviour of cables by using

beam/frame elements, two measures should be taken: modifying the section

properties and modelling a cable member by using enough numbers of segments or

elements. Since the behaviour of cable structures is quite different from that of frame

structures, the bending and torsional stiffness of a cable member are very small, and

modification factors should be applied to the torsional constant and moments of

inertia of the beam element when it is used to model a cable member. It should be

mentioned that a cable in real bridge structures can carry small bending moments and

torques, and a cable which can carry only axial tension force is just an ideal member

and it hardly exists in the real world. This modification of section properties ensures

the element naturally buckle if the element goes into compression. However, no

detailed information is provided on how to select the modification factors as different

types of cables have different section properties. As there is no further information

on how to modify the section properties for a cable member, the modification factors

of moments of inertia of section and torsional constant are set to be 0.01 when

SAP2000 is adopted for the following numerical analysis. To make the cable

member flexible and take into account of the effect of axial force and large deflection

on structural stiffness, a cable member should also be divided into many small

segments (elements) as a cable member always has large deflection under load. The

default number of segments in SAP2000 is 10 for a cable member [CSI 2004]. While

in the modelling of the proposed suspension footbridge, this number is taken as 20 to

- 82 -

ensure the elements to be small enough and the relative rotations within each element

to be small. Further discussion on the number of segments will be given in chapter 5.

In the finite element model, the connections of the cables and bridge frames are

modelled as rigid connections, and end releases (pin connections) are only applied at

the connections between the cables and end bridge frames.

- 83 -

Load deformation performance and vibration properties of the proposed slender suspension footbridges

4.1 Introduction

Suspension bridges are slender tension structures, and hence the structural stiffness

and vibration properties such as natural frequencies and vibration modes depend

mainly on the tension forces in the cable systems. When the bridge structures are

subjected to external applied loads, the load resistances in different directions are

mainly provided by the components of the tension forces. Further more the vibration

properties can be altered by changing the tension forces. In general, traditional

suspension bridges have only supporting cables in near vertical planes and tension

forces depend on the cable profiles, structural weight and applied loads. As a

consequence, the bridges are weak in the lateral direction and unable to offer

sufficient resistance to lateral vibration.

In the proposed suspension footbridge model, there are more design structural factors

to improve the structural behaviour, and to control the vibration properties to some

extent. When the reverse profiled bottom and/or side cables are pre-tensioned, extra

tension forces can be introduced into the cable systems and provide more tension

forces to resist the external applied loads and improve the structural stiffness. The

study of load deformation performance and free vibration analysis are important and

helpful to understand the structural behaviour of such slender suspension footbridges

and the effect of structural parameters.

The structural analysis software package Microstran is adopted to investigate the

load deformation performance and vibration properties of the proposed suspension

footbridge and the footbridge models analysed in this chapter are bridge models with

members of solid sections (SSB), that is, all the members of bridge frames,

supporting beams and deck units have solid rectangular sections. In most of the

numerical analysis, the span length is set to be 80 m. The main structural parameters

4

- 84 -

studied include cable sags, cable sections, pre-tensions in the reverse profiled bottom

and side cables. Here the pre-tensions introduced into the bottom and side cables are

represented by extra internal vertical force (Wint) and extra internal horizontal forces

(Qint) respectively.

4.2 Load deformation performance under quasi-static loads

4.2.1 Applied quasi-static loads

Since footbridges are mainly designed to carry pedestrians, pedestrian load and wind

force are the main design loads, though other loads such as seismic load also should

be considered if the bridge is constructed in a seismic area. Normally, the weight of

pedestrian can be modelled as quasi-static vertical load acting on the bridge deck,

and the wind force can be modelled as quasi-static lateral horizontal force acting on

the projected area in bridge elevation. In order to study the load deformation

performance under different loading cases, symmetric vertical load and asymmetric

vertical load (or eccentric vertical load) are considered. The symmetric load is

modelled as a uniformly distributed load acting on the deck (Figure 4.1(a)) and the

asymmetric load (eccentric vertical load) as uniform load distributed along the half

width on bridge deck (Figure (4.1b)). The horizontal static load is simply modelled

as uniform load acting on the deck in the transverse direction (Figure 4.1(c)).

According to the Austroads Bridge Design Code [1992], the pedestrian load intensity

for footbridge with loaded area greater than 100 m2 is 4 kPa and the ultimate limit

state load factor for design traffic loadings (pedestrian loading) is 2.0. Therefore the

load density for this footbridge structure has been chosen as 8 kPa for the vertical

symmetric and asymmetric loads. Since the lateral loads are usually caused by

walking pedestrians and wind, and maximum lateral force generated from walking

varies from 3 to 10 percent of the pedestrian’s weight [Willford 2002]. In the quasi-

static analysis, the lateral load density is set to be one tenth of the vertical loads (i.e.

0.8 kPa). In order to study the effects of structural parameters, the cable sag will vary

from 1.2 m to 2.4 m, and cable diameter for the all the cables from 120 mm to 339

mm, and the extra internal vertical forces (Wint) and extra internal horizontal forces

(Qint) induced by the pre-tension cables are selected according to the load cases.

- 85 -

In order to compare the results and describe the structural behaviour effectively, two

types of cable supported bridge models are treated as discussed in the following

sections. Pre-tensioned bridge refers to a suspension footbridge model with pre-

tensioned bottom and/or side cables (bridge models C123, C120 and C103). Un-pre-

tensioned bridge, on the other hand, refers to a cable supported bridge model with

slack bottom and side cables (Bridge model C100) which have no contribution to the

structural stiffness but carry small tension forces to support their own gravity loads

and prevent the transverse bridge frames from swaying freely in the longitudinal

direction. To make a cable slack, a small initial distortion (extension 0.01m) is

introduced to these cables before the loads are applied.

(a)

(b)

(c)

Figure 4.1 Applied loads: (a) – symmetric vertical loads; (b) – eccentric vertical loads; (c) – lateral horizontal loads

- 86 -

4.2.2 Un-pre-tensioned footbridges under symmetric vertical load: effect of

cable sags

In a suspension bridge structure, the supporting cables are the most important

structural members which provide tension forces to support the entire structure and

the applied loads, for the bridge structure with or without pre-tensioned cables. In

order to show the effects of cable sag and cross sectional area of the top supporting

cables, un-pre-tensioned bridge model (C100) has been studied and this bridge model

can be used to simulate the behaviour of traditional suspension footbridges. For an

un-pre-tensioned cable supported bridge structure, the whole structural stiffness is

provided by the top supporting cables, as the bottom as well as the side cables are

slack and have no contribution the structural stiffness.

Numerical results show that, compared with the deformation of cables, the transverse

bridge frames deform slightly and they can be considered as rigid members under

gravity and applied loads. The position of the maximum vertical deflection is at the

mid point of the cross member of the middle frame and maximum tension force in

the cables occurs at the two end segments. In order to show the deflection of the

cables and bridge frame, in the following analysis, the maximum deflections

represent the deflections at intersection point of legs and cross member of the central

bridge frame (almost at the same place as the maximum cable sag) (Figure 4.2), and

the maximum tension force represents the maximum tension force in the end

segment of a top, bottom or side cable, when they are mentioned in text or shown in

the figures and tables.

U

Uv

l

A

A'

Figure 4.2 Deflections and deformed bridge frame

- 87 -

Figures 4.3 and 4.4 show the effect of cable sags on maximum vertical deflection and

maximum tension force in one of the top supporting cables under the applied loads

when the top supporting cables have a diameter of 240 mm (D1=240 mm) but

different cable sags (F1). It can be seen that when the cable sag increases, the vertical

structural stiffness increases and both the deflection and maximum tension force

decrease.

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.0 0.8 1.6 2.4 3.2 4.0 4.8 5.6 6.4 7.2 8.0

Applied Load (kN/m2)

Def

lect

ion

(m)

F1=1.2 m

F1=1.8 m

F1=2.4 m

Figure 4.3 Maximum vertical deflections under symmetric applied vertical loads with different cable sags

0.0E+00

5.0E+06

1.0E+07

1.5E+07

2.0E+07

2.5E+07

3.0E+07

0.0 0.8 1.6 2.4 3.2 4.0 4.8 5.6 6.4 7.2 8.0


Ten

sion

For

ces

(N)

F1=1.2 m

F1=1.8 m

F1=2.4 m

Figure 4.4 Maximum tension forces in top cables under applied vertical loads with different cable sags

- 88 -

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.0 0.8 1.6 2.4 3.2 4.0 4.8 5.6 6.4 7.2 8.0


Def

lect

ion

(m)

D1=120 mm

D1=169 mm

D1=240 mm

D1=339 mm

Figure 4.5 Maximum vertical deflections under applied vertical loads with different top cable cross sectional area (diameter)

0.0E+00

5.0E+06

1.0E+07

1.5E+07

2.0E+07

2.5E+07

0.0 0.8 1.6 2.4 3.2 4.0 4.8 5.6 6.4 7.2 8.0


Ten

sion

For

ce (

N)

D1=120 mm

D1=169 mm

D1=240 mm

D1=339 mm

Figure 4.6 Maximum tension forces in top cables under applied vertical loads with different top cable cross sections

Figure 4.5 and Figure 4.6 show the variations of the maximum vertical deflection and

maximum tension force in one top cable with cross sectional area (or diameter) of the

top cables when the cable sag is set to 1.8 m (F1=1.8 m). In order to show the effect

of the cable’s cross sectional area, the total weight of the whole bridge structure is

kept the same by changing the diameters of the slack side cables. Results show that

- 89 -

at the same initial cable sag, the structural stiffness increases (as expected), while the

maximum tension force (which mainly depends on the sag of deformed cable)

increases slightly when the cross sectional area increases.

4.2.3 Effect of Pre-tension Forces in the Bottom Cables (Internal Vertical

Forces)

When the reverse profiled bottom cables are pre-tensioned, extra internal vertical

forces are induced to the bridge frames and catenary supporting cables. In order to

investigate the effects of pre-tension in the bottom cables as well as the structural

parameters of the pre-tensioned bottom cables such as cable sectional area and cable

sag, the side cables are assumed to be slack (Qint=0), and the top supporting cables

are stretched to keep the deck in horizontal plane before it is subjected to the applied

vertical loads.

The effect of pre-tension has been investigated by changing the internal vertical force

(Wint), while the top and bottom cables, as well as the slack side cables, are supposed

to have the same cable sag of 1.8 m and diameter of 240 mm (F1= F2= F3=1.8 m, D1=

D2= D3=240 mm). To illustrate the variation of structural stiffness and the effects of

cross sectional area and pre-tension, results are compared with those of an un-pre-

tensioned bridge model, in this section as well as in the others. In the un-pre-

tensioned bridge model (UPTB) (F1=1.8 m, D1=339 mm), it is assumed that all the

cable profiles are the same as those of the pre-tensioned bridge model, but the

sectional area of the top supporting cables is equal to the sum of sectional areas of

the top and bottom cables in the pre-tensioned bridge model, and the diameters of the

slack bottom and side cables are set to be 190 mm (F2= F3=1.8 m, D2= D3=190 mm)

to keep the same gravity loads as that of the pre-tensioned bridge model.

Figure 4.7 shows the variation of maximum vertical deflection under applied

symmetric vertical load. Figure 4.8 shows the maximum tension force in one of the

top cables and Figure 4.9 the maximum tension force in a bottom cable. From these

figures, it can be seen that for a pre-tensioned suspension footbridge, the structural

behaviour depends not only on the top supporting cables, but also on the pre-

tensioned bottom cables as well as the pre-tensioned forces in the bottom cables (or

- 90 -

extra internal vertical forces), and the performance can be described in two phases. In

the first phase when the bottom cables are pre-tensioned and provide vertical forces

to the top supporting cables, the pre-tension forces in the bottom cables decrease

while the tension forces in top cables increase with the applied vertical load. The

structural stiffness in this phase is almost the same as that of the un-pre-tensioned

bridge model (UPTB). This feature demonstrates that in a pre-tensioned footbridge,

the structural stiffness depends on the total cross sectional areas of the top and

bottom cables, irrespective of their profiles, i.e. catenary or reverse profile. In the

second phase, the pre-tension forces have been released, the bottom cables gradually

become slack, and they have no ability to provide extra internal vertical forces to the

top supporting cables and can only carry the tension forces to support their own

gravity. In this case, the bottom cables do not contribute to the structural stiffness

again and the bridge structure behaves as an un-pre-tensioned one, since the

structural stiffness depends only on the top cables.

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.0 0.8 1.6 2.4 3.2 4.0 4.8 5.6 6.4 7.2 8.0


Def

lect

ion

(m)

Wint= 0 kNWint= 10 kNWint= 20 kNWint= 30 kNWint= 40 kNUPTB

Figure 4.7 Maximum deflection under applied vertical loads with different internal vertical forces

Figure 4.10 shows the total horizontal tension force in a bridge section. It can be seen

that in a pre-tensioned suspension footbridge structure, the total horizontal force

remains almost constant with increase in applied load except when the pre-tensioned

bottom cables slack, for which case the total horizontal force increases with applied

load. These are interesting features of load transfer and balance in this type of

- 91 -

structure. The reason is that the top cables and pre-tensioned bottom cables form a

self-balancing system when extra internal vertical forces exist. When the internal

vertical forces have been released, the self-balancing system disappears and the

applied loads are resisted only by the top supporting cables.

1.0E+07

1.2E+07

1.4E+07

1.6E+07

1.8E+07

2.0E+07

2.2E+07

0.0 0.8 1.6 2.4 3.2 4.0 4.8 5.6 6.4 7.2 8.0


Ten

sion

For

ce (

N)


Figure 4.8 Maximum tension force in top cables under applied vertical load with different internal vertical forces

0.0E+00

5.0E+05

1.0E+06

1.5E+06

2.0E+06

2.5E+06

0.0 0.8 1.6 2.4 3.2 4.0 4.8 5.6 6.4 7.2 8.0


Ten

sion

For

ces

(N)


Figure 4.9 Maximum tension force in bottom cables under applied vertical load with different internal vertical Forces

- 92 -

2.0E+07

2.5E+07

3.0E+07

3.5E+07

4.0E+07

4.5E+07

0.0 0.8 1.6 2.4 3.2 4.0 4.8 5.6 6.4 7.2 8.0


Tot

al H

oriz

onta

l For

ces

(N)


Figure 4.10 Sum of total horizontal forces of top and bottom cables under applied vertical load with different internal vertical forces

4.2.4 The Effects of cable sag and cross sectional area of bottom cables

The sectional area and cable sag of the pre-tensioned bottom cables can affect the

structural performance to some extent. These effects have been shown in Figure 4.11

to Figure 4.14. In these figures, the bottom cables are pre-tensioned to provide 30

kN extra internal vertical force (Wint=30 kN, Qint=0) to the top cables at each bridge

frame before the symmetric vertical load is applied and the cable sag and diameter of

the top cables are assumed to be 1.8 m and 240 mm respectively (F1=1.8 mm,

D1=240 mm).

Figure 4.11 and Figure 4.12 show the maximum deflection and the maximum tension

force in the bottom cables when the bottom cables have diameter of 240 mm with

different cable sags. It can be seen that when the cable sag of the pre-tensioned

bottom cables is greater than that of the top cables, the structural stiffness can be

greater than that of an un-pre-tensioned bridge (UPTB, D1=339 mm, D2= D3=190

mm). The bottom cables are easier to slack when they have greater cable sag. Figure

4.13 and Figure 4.14 show the maximum deflection and maximum tension force in

the bottom cables respectively when the bottom cables have different diameters.

- 93 -

Here the total weight of the bridge structures is kept the same. When the diameter of

the pre-tensioned bottom cables is larger, the structural stiffness is larger and the

bottom cables are easier to slack.

0.00

0.05

0.10

0.15

0.20

0.25

0.0 0.8 1.6 2.4 3.2 4.0 4.8 5.6 6.4 7.2 8.0


Def

lect

ion

(m)

F2=1.2 m

F2=1.8 m

F2=2.4 m

UPTB

Figure 4.11 Maximum deflection under applied load with different bottom cable sags

0.0E+00

4.0E+05

8.0E+05

1.2E+06

1.6E+06

2.0E+06

0.0 0.8 1.6 2.4 3.2 4.0 4.8 5.6 6.4 7.2 8.0


Ten

sion

For

ces

(N)

F2=1.2 mF2=1.8 mF2=2.4 mUPTB

Figure 4.12 Tension forces in bottom cables under applied load with different bottom cable sags

- 94 -

From these figures, it can be concluded that when the pre-tensioned bottom cables

have small sectional area or small cable sag, they are slender and not easy to slack,

and the extra internal forces induced by the pre-tensioned bottom cables are released

very slowly.

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.0 0.8 1.6 2.4 3.2 4.0 4.8 5.6 6.4 7.2 8.0


Def

lect

ion

(m)

D2=120 mmD2=169 mmD2=240 mmD2=339 mm

Figure 4.13 Maximum deflection under applied load with different bottom cable sections

0.0E+00

4.0E+05

8.0E+05

1.2E+06

1.6E+06

2.0E+06

0.0 0.8 1.6 2.4 3.2 4.0 4.8 5.6 6.4 7.2 8.0


Ten

sion

For

ce (

N)

D2=120 mmD2=169 mmD2=240 mmD2=339 mm

Figure 4.14 Tension forces in bottom cables under applied load with different bottom cable sections

- 95 -

4.2.5 Effect of pre-tension in the side cables (internal lateral forces)

When the horizontal side cables are pre-tensioned, internal horizontal forces can be

provided to the bridge frames and the horizontal stiffness can be improved

significantly. However, the vertical structural stiffness increases only slightly, since

the horizontal cables are flexible in the vertical direction and provide small vertical

force when they deform. Figure 4.15 shows the maximum vertical deflection with

pre-tensioned bottom and side cables when the symmetrical vertical load is applied.

Here it is assumed that all the top, bottom and side cables have the same cable sag of

1.8 m and diameter of 240 mm (F1=F2=F3=1.8 m, D1=D2=D3=240 mm). It can be

seen that the vertical stiffness mainly depends on the top and bottom cables and the

effect of pre-tensioned side cables on the vertical stiffness is much smaller than that

of the pre-tensioned bottom ones.

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.0 0.8 1.6 2.4 3.2 4.0 4.8 5.6 6.4 7.2 8.0


Def

lect

ion

(m)

Wint= 0 kN Qint= 0 kNWint= 0 kN Qint=10 kNWint= 0 kN Qint=20 kNWint=30 kN Qint= 0 kNWint=30 kN Qint=10 kNWint=30 kN Qint=20 kN

Figure 4.15 Maximum vertical deflection with pre-tensioned bottom and side cables

4.2.6 Performance under lateral horizontal loads and eccentric vertical loads

Bridge structures are always subject to lateral horizontal loads (such as wind) and

eccentric vertical loads. To illustrate the performance of the pre-tensioned bridge

model under such loadings, bridge models with different pre-tensions are studied.

Here it is supposed that all the bridge models have the same cable profiles and all the

cables have the same cable sag of 1.8 m and diameter of 240 mm (F1=F2=F3=1.8 m,

- 96 -

D1=D2=D3=240 mm). The lateral horizontal load is modelled as a distributed uniform

load with density of 800 N/m2 acting on the bridge deck (Figure 4.1(c)), and

eccentric vertical load is modelled as distributed uniform load (with density of 8

kN/m2) acting on the half width of the deck (Figure 4.1(b)).

0.00

0.02

0.04

0.06

0.08

0.10

0.00 0.08 0.16 0.24 0.32 0.40 0.48 0.56 0.64 0.72 0.80


Def

lect

ion

(m)

Wint= 0 kN Qint= 0 kNWint=20 kN Qint= 0 kNWint=40 kN Qint= 0 kNWint= 0 kN Qint=10 kNWint= 0 kN Qint=20 kNWint=20 kN Qint=10 kNUPTB (D1=339 mm)

Figure 4.16 Horizontal deflection under lateral horizontal applied loads

Figure 4.16 shows the maximum horizontal deflection at the end of the cross member

in the middle transverse bridge frame (Figure 4.2) under horizontally applied load.

Results show that the horizontal stiffness is much smaller than the vertical stiffness

for bridge structures without pre-tensioned bottom and side cables (UPTB), even if

the sectional area of the top supporting cables are increased. The reason is that the

top cables are in the vertical plane, and their tension forces have only small

components in the lateral horizontal direction to resist the lateral loads, after they

deform in the lateral direction. When the bottom cables are pre-tensioned, the lateral

horizontal stiffness can be improved since the tension forces in the deformed top and

bottom cables can provide more force components in the lateral horizontal direction.

However, the most effective measure to improve the lateral stiffness is to introduce

the pre-tensioned side cables. This can be seen from the Figure 4.16 that, after the

side cables have been pre-tensioned, the cables in the vertical plane have only slight

effect on the lateral structural stiffness.

- 97 -

0.00

0.01

0.02

0.03

0.04

0 2 4 6 8 10 12 14 16 18 20

Number of Frame

Def

lect

ion

(m)

Wint= 0 kN Qint= 0 kN Wint= 0 kN Qint=10 kNWint=20 kN Qint= 0 kN Wint= 0 kN Qint=20 kNWint=40 kN Qint= 0 kN Wint=20 kN Qint=10 kNUPTB (D1=339 mm)

Figure 4.17 Lateral horizontal deflection along bridge under eccentric vertical loads

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0 2 4 6 8 10 12 14 16 18 20

Number of Frame

Def

lect

ion

(m)

Wint= 0 kN Qint= 0 kN Wint= 0 kN Qint=10 kNWint=20 kN Qint= 0 kN Wint= 0 kN Qint=20 kNWint=40 kN Qint= 0 kN Wint=20 kN Qint=10 kNUPTB (D1=339 mm)

Figure 4.18 Vertical deflection along bridge under eccentric vertical loads

Under eccentric loads, lateral horizontal deflection is produced accompanying the

vertical deflection, since the structural stiffness is mainly provided by the cable

systems which are often weak in the lateral direction, the torsion can change the

direction of the vertical loads and produce small lateral horizontal component, and

large lateral deflection can be induced. Figure 4.17 and Figure 4.18 show the lateral

- 98 -

horizontal and vertical deflections of the ends of cross members at the side of applied

eccentric loads along the bridge longitudinal direction. It also can be seen that

although the pre-tensioned bottom cables can improve the vertical structural

stiffness, the best measure to suppress the lateral horizontal deflection is to introduce

the pre-tensioned side cables.

4.3 Vibration properties of slender suspension footbridges with

pre-tensioned cables

Free vibration analysis is an important part towards understanding the dynamic

characteristics of a structure. Natural frequencies and vibrational modal shapes are

the basic vibration properties for structures, and they influence the response under

dynamic loads. In most bridge design codes, provision for dynamic effects is made

via the dynamic allowance factors for quasi-static structural analysis based on the

fundamental flexural frequency. However, for slender cable supported bridge

structures, the lateral and coupled modes are more significant, and the situation is

more complex for slender shallow suspension bridges as some features could not be

revealed by traditional 2D analysis. Since the pre-tensioned cables have been added

to the proposed slender suspension footbridge, the natural frequencies can be “tuned”

when different pre-tensions are introduced. This feature is useful to investigate the

dynamic behaviour of such footbridges under human-induced dynamic loads, and it

provides a greater chance to improve the structural behaviour.

4.3.1 Vibration Mode Shapes

In general, a bridge structure can have three main kinds of vibration modes: lateral

modes, torsional modes and vertical modes. However cable supported bridge

structures, especially suspension bridge structures, have four main kinds of vibration

modes [Xu et al. 1997]: lateral modes, vertical modes, torsional modes and

longitudinal modes. With the lateral vibration modes, the entire bridge structure

rather than bridge individual components vibrate in the horizontal plane, sometimes

in conjunction with very small vertical, torsional and longitudinal vibration at the

lowest natural frequency. When a bridge vibrates under the vertical vibration modes,

- 99 -

the entire bridge structure will vibrate in the vertical plane, and under the torsional

vibration modes, the deck will twist. The longitudinal modes can be classified as

pure longitudinal modes and modes associated with other vibration types. The pure

longitudinal modes mean that they have distinct frequencies and modal

configurations, while the other modes participate in lateral, torsional, and vertical

modes as well as their combinations.

For the slender suspension footbridges (pre-tensioned or un-pre-tensioned) with

shallow cable profiles, there are also four main kinds of vibration modes: lateral

modes, torsional modes, vertical modes and longitudinal modes. However, it is

found that the lateral modes and torsional modes do not always appear as pure lateral

or torsional vibration modes. Most of the time, they are combined together and form

two kinds of coupled vibration modes. Here, the coupled modes are noted as coupled

lateral-torsional vibration modes (LmTn) and coupled torsional-lateral vibration

modes (TmLn), where L and T represent Lateral and Torsional modes respectively,

m and n are the number of half-waves. In order to illustrate the typical vibration

modal shapes, a un-pre-tensioned solid section bridge model (SSB) with the cable

sag of 1.8 m and cable diameter of 240 mm for all the cables (F1=F2=F3=1.8 m,

D1=D2=D3=240 mm, Wint=0, Qint=0) is adopted for the free vibration analysis.

4.3.1.1 Coupled lateral-torsional vibration modes

When the footbridge structure vibrates with the coupled lateral-torsional vibration

modes, one of the top supporting cables has the lateral and downward movement and

the other has lateral and upward movement. Then the movement of the deck appears

as if it has lateral movement and sways about a point above the deck. To some

extent, the coupled lateral-torsional vibration modes can be called as lateral sway

vibration modes. Figure 4.19 shows the first three coupled lateral-torsional vibration

modes of the deck.

Results show that these vibration modes are dominated by the lateral modes in

conjunction with torsional vibration. For the first coupled lateral-torsional vibration

mode (L1T1), the component of torsional movement will decrease and the mode can

reduce to pure lateral mode if the cable sag of the top supporting cables increases or

- 100 -

a large pre-tension force has been introduced for pre-tensioned bridge models. For

the other coupled modes, the lateral vibration modes are always combined with

torsional vibration.

(a)

(c)

(b)

L1T1 (0.37312 Hz)

(a)

(c)

(b)

L2T2 (0.53434 Hz)

(a)

(c)

(b)

L3T3 (0.79064 Hz)

Figure 4.19 Coupled lateral-torsional vibration modes (a) -- elevation; (b)-- top view; (c)-- side view

- 101 -

(a)

(c)

(b)

T1L1 (0.63597 Hz)

(a)

(c)

(b)

T2L2 (1.0035 Hz)

(a)

(c)

(b)

T3L3 (1.5111 Hz)

Figure 4.20 Coupled torsional-lateral vibration modes (a) -- elevation; (b)-- top view; (c)-- side view

- 102 -

4.3.1.2 Coupled torsional-lateral vibration modes

The coupled torsional-lateral vibration modes can be considered as the reverse of

coupled lateral-torsional vibration modes (Figure 4.20). One of the top supporting

cables also has the lateral and downward movement and the other has the lateral and

upward movement. However, the deck has lateral movement like the cables and

sways about a point underneath the bridge deck. This type of vibration mode can be

called reverse lateral swaying vibration mode. The dominant modes are torsional

vibration modes and the first mode (T1L1) will reduce to torsional one when high

pre-tensions are introduced.

V1 (0.66259 Hz)

V2 (0.82865 Hz)

V3 (1.24940 Hz)

Figure 4.21 Vertical vibration modes

4.3.1.3 Vertical vibration modes

When the footbridge structure vibrates with the vertical vibration modes, the cables

and deck have same upwards and downwards movement. These are common

- 103 -

vibration modes for bridge structures. In general, most vertical vibration modes

appear as pure vertical modes, without corresponding lateral or torsional ones. Figure

4.21 shows the elevation of the first three pure vertical vibration modes. Here, V

represents the vertical modes, and the number represents the number of half-wave.

However, for cable supported bridge structures, coupled vertical vibration modes

may exist. The most common coupled mode is the one half-wave symmetric vertical

mode (V1) coupled with the three half-waves symmetric one (V3). Coupled

asymmetrical vertical modes were not obtained in this research project.

LSW1 (0.63597 Hz)

LSW2 (1.3214 Hz)

LSW3 (1.9700 Hz)

Figure 4.22 Longitudinal swaying vibration modes

4.3.1.4 Longitudinal vibration modes

Longitudinal vibration modes exit in most cable supported bridge structures with

weak connections between the bridge frames. When the footbridge structure vibrates

in this type of modes, the bridge frames sway in the longitudinal direction. This kind

of vibration modes can hence be called as longitudinal swaying (LSW) vibration

modes. As mentioned before, the longitudinal modes can be classified as pure

longitudinal modes and modes associated with other vibrations. The pure

longitudinal modes have distinct frequencies and modal configurations, while the

other modes participate in lateral, torsional, and vertical modes only. Figure 4.22

- 104 -

shows the first three longitudinal vibration modes. The first mode (LSW1) and the

third mode (LSW3) are almost pure longitudinal mode, while the second mode is

coupled with the first symmetric vertical vibration mode. It should be mentioned that

the longitudinal vibration modes are significantly affected by the connection between

the bridge frames. When the connection is weak, these modes will correspond to low

frequencies. Otherwise, they may correspond to high frequencies or disappear. For

example, for a pre-tensioned cable supported bridge model, these modes will

disappear when the pre-tension force in the bottom cables increase to a high level.

4.3.2 Natural frequencies

Natural frequencies and their corresponding vibration mode shapes are important

vibration properties for dynamics of structures and they are affected by many factors

such as structural stiffness, mass and structural geometry. Cable supported bridge

structures are always flexible and slender with low natural frequencies, since the

structural stiffness is mainly provided by the cable systems. In this section, the

effects of important structural parameters on the natural frequencies and their

corresponding vibration modal shapes have been investigated. These structural

parameters include cable sag, cable cross sectional area (diameter), applied mass,

pre-tensions (extra internal vertical and horizontal forces) in the pre-tensioned

bottom and side cables, and etc. In numerical analysis, the major parameter study is

carried out on pre-tensioned bridge models with the span length of 80 m (L=80 m),

and when the effect of span length is investigated, a parameter study will be

conducted again. When the effect of cable cross sectional area is investigated, the

total gravity load of the entire bridge structure is kept the same by changing the

dimeters of the side cables.

In order to present useful information and to avoid superfluous data, only the

frequencies corresponding to the first six coupled lateral-torsional modes, the first

four coupled torsional-lateral modes and the first five vertical modes are shown in

most of the tables. These frequencies are often within the first twenty frequencies

and more important in dynamic analysis than the other frequencies corresponding

high order vibration modes. The first three longitudinal swaying modes are shown

only when the un-pre-tensioned bridge models are studied, as these vibration modes

- 105 -

are not important in practical design and are sensitive to the connections between the

bridge frames. As mentioned earlier, when the pre-tensions are introduced in the

reverse profiled cables, these vibration modes disappear from the first twenty natural

frequencies.

4.3.2.1 Effects of cable sag and cross sectional area of the top supporting cables

(un-pre-tensioned bridge model)

Cable sag and cross sectional area (diameter) have great effects on the structural

stiffness. They also have evident effects on the natural frequencies and the

corresponding mode shapes, particularly on the vertical vibration modes. As

mentioned before, when an un-pre-tensioned suspension footbridge model is treated,

the bottom and side cables are allowed to be slack and they can carry only small

tension forces to support their own gravity loads and have no contribution to the

structural stiffness. Numerical results show that these small tension forces will

mainly affect the natural frequencies in the longitudinal direction and have only

slight effect on the other vibration modes.

Table 4.1 shows the natural frequencies and their corresponding vibration mode

shapes when the cable sags of the top supporting cables (as well as the slack bottom

and side cables) are set to 1.2 m, 1.8 m and 2.4 m. It can be seen that when the cable

sag increases, the maximum tension force in the top supporting cables (T1) decreases.

All the frequencies decrease, except the first coupled lateral-torsional vibration mode

which changes slightly and the frequency corresponding to the one half-wave

symmetric vertical mode increases. This illustrates that the cable sag has a significant

effect on the natural frequencies, since it changes the tension force in the top

supporting cables and thereby influences the stiffness. It can also be seen that the

cable sag will change the order of vibration modes and the mode shapes. When the

cable sag is set to 1.2 m and 1.8 m, the one half-wave symmetric vertical mode (V1)

is the lowest vertical mode, but when the cable sag increases to 2.4 m, the two half-

waves asymmetric vertical mode (V2) has become the first vertical mode. This

unusual feature occurs under certain circumstances and has been explained in detail

by Irvine [1992] and Gimsing [1997]. Also the first coupled torsional-lateral mode

- 106 -

(T1L1) has reduced to a pure torsional vibration mode, while the other coupled

modes retain the same mode shapes.

Table 4.1 Natural frequencies and corresponding modes with the cable sag

Bridge parameter SSB: L=80 m; D1=D2=D3=240 mm

Wint (kN) 0 0 0 Internal force

Qint (kN) 0 0 0

F1 (mm) 1200 1800 2400

F2 (mm) 1200 1800 2400 Cable sag

F3 (mm) 1200 1800 2400

T1 (N) 2.23E+07 1.49E+07 1.12E+07

T2 (N) 5.77E+04 5.80E+04 5.83E+04 Cable tension

T3 (N) 5.74E+04 5.76E+04 5.78E+04

L1T1 0.36066 0.37312 0.35474

L2T2 0.59589 0.53434 0.50658

L3T3 0.87993 0.79064 0.75538

L4T4 1.15080 1.03740 0.97815

L5T5 1.41100 1.27330 1.20520

Coupled lateral-torsional

L6T6 1.65400 1.49970 1.42160

T1L1 0.66666 0.59431 0.64282*

T2L2 1.29070 1.00350 0.82500

T3L3 1.92240 1.51110 1.25300 Coupled torsional-lateral

T4L4 2.52830 1.99430 1.64500

V1 0.61541 0.66259 0.75684

V2 1.01160 0.82865 0.72073

V3 1.51560 1.24940 1.10360

V4 1.99930 1.64020 1.42380

Vertical

V5 2.48120 2.03840 1.77460

LSW1 0.68305 0.63597 0.48824

LSW2 1.37390 1.32140 1.15930 Longitudinal swaying

LSW3 2.04790 1.97000 1.68340

Notes: * pure torsional mode

Table 4.2 shows the natural frequencies and their corresponding vibration mode

shapes when the cables have different cross sectional area (diameter). Here the cable

sag is set to 1.8 m for all the cables. From this table, it can be seen that when the

sectional area (diameter) of the supporting cables increases, all frequencies for the

coupled lateral-torsional modes increase. The effect on the coupled torsional-lateral

modes and vertical modes is interesting. The frequency of the first coupled torsional-

lateral mode (T1L1) and vertical mode (V1) increase, however, frequencies for the

higher coupled torsional-lateral and vertical modes change slightly. All the

- 107 -

frequencies are arranged in the table according to number of half-waves of their

vibration modes.

Table 4.2 Natural frequencies and corresponding modes with the cross sectional area (diameter)

Bridge parameter SSB: L=80 m; F1=F2=F3=1.8 m

Wint (kN) 0 0 0 0 Internal force

Qint (kN) 0 0 0 0

D1 (mm) 120 169.7 240 339.4

D2 (mm) 240 240 240 190 Cable diameter

D3 (mm) 317 294 240 190

T1 (N) 1.49E+07 1.49E+07 1.49E+07 1.53E+07

T2 (N) 5.68E+04 5.68E+04 5.68E+04 3.57E+04 Cable tension

T3 (N) 1.01E+05 8.64E+04 5.76E+04 3.62E+04

L1T1 0.31304 0.34081 0.37312 0.39713

L2T2 0.51605 0.52191 0.53434 0.55892

L3T3 0.76104 0.77030 0.79065 0.83762

L4T4 0.99456 1.00180 1.03740 1.08530

L5T5 1.22800 1.24250 1.27330 1.33700


L6T6 1.44610 1.46320 1.49970 1.57540

T1L1 0.51212 0.53557 0.59431 0.70211

T2L2 1.01830 1.01980 1.00350 0.97555

T3L3 1.51800 1.51550 1.51110 1.47130 Coupled torsional-lateral

T4L4 2.00420 2.00080 1.99430 1.93670

V1 0.49129 0.55534 0.66259 0.81798

V2 0.82990 0.82949 0.82865 0.82610

V3 1.23990 1.24290 1.24940 1.27150

V4 1.63970 1.64000 1.64020 1.63380

Vertical

V5 2.03330 2.03520 2.03840 2.03780

LSW1 0.66574 0.65630 0.63597 0.53799

LSW2 1.37850 1.36030 1.32140 1.12020 Longitudinal

LSW3 2.04280 2.01700 1.95960 1.66990

4.3.2.2 Effects of pre-tensions in the reverse profiled bottom and side cables

(extra internal vertical and horizontal Forces)

In pre-tensioned cable supported bridge structures, if the pre-tension forces are

introduced to the reverse profiled bottom cables, the tension forces in the top

suspending cables will increase due to the extra internal vertical forces induced by

the pre-tensioned bottom cables (Figure 3.3). If the side cables have been pre-

tensioned, the horizontal structural stiffness will be improved greatly. Both the extra

- 108 -

internal vertical forces and horizontal forces will enhance the connection between the

transverse bridge frames, and therefore, the frequencies corresponding to the

longitudinal vibration modes increase or the longitudinal modes disappear from the

first twenty frequencies. Frequencies corresponding to all the other vibration modes

will increase when the tension forces in the cable systems increase.

Table 4.3 Internal vertical forces and the natural frequencies and their corresponding modes

Bridge parameter SSB: L=80 m; F1=F2=F3=1.8 m; D1=D2=D3=240 mm


Qint (kN) 0 0 0 0

T1 (N) 1.49E+07 1.56E+07 1.60E+07 1.66E+07


T3 (N) 5.76E+04 5.76E+04 5.76E+04 5.76E+04

L1T1 0.37312 0.40813 0.42359 0.43836

L2T2 0.53434 0.62016 0.65486 0.69166

L3T3 0.79064 0.88525 0.94023 0.99736

L4T4 1.03740 1.14920 1.22120 1.29790

L5T5 1.27330 1.41530 1.50510 1.59620


L6T6 1.49970 1.66950 1.77760 1.89150

T1L1 0.59431 0.68580 0.71579 0.72983

T2L2 1.00350 1.04940 1.06910 1.09060


T4L4 1.99430 2.04850 2.08690 2.13010

V1 0.66259 0.79349 0.83294 0.84947

V2 0.82865 0.86172 0.88590 0.91274

V3 1.24940 1.31590 1.35540 1.39500

V4 1.64020 1.70890 1.75670 1.80970

Vertical

V5 2.03840 2.12510 2.18470 2.25030

Table 4.3 shows the natural frequencies and their corresponding vibration modes

with different extra internal vertical forces induced by the pre-tensioned bottom

cables. Table 4.4 shows the dynamic properties with different extra internal

horizontal forces when the side cables have been pre-tensioned. Here in the bridge

model, the cable sags of all the cable systems are set to 1.8 m and the diameters of all

the cables are 240 mm (L=80 m, F1=F2=F3=1.8 m, D1=D2=D3=240 mm). It can be

seen that all the frequencies increase when the extra internal forces increase. Since

- 109 -

the main function of the pre-tensioned bottom cables is to improve the vertical

structural stiffness, as mentioned before, the frequencies corresponding to the

vertical vibration modes increase rapidly when the extra internal vertical forces

increase, although the other frequencies also increase at the same time. After the side

cables have been pre-tensioned, the frequencies corresponding to the coupled lateral-

torsional modes as well as the coupled torsional-lateral modes, increase rapidly when

the pre-tension forces in the side cables increase.

Table 4.5 shows the effect of the cross sectional area (diameter) of the pre-tensioned

bottom cables. It can be seen that the frequencies corresponding to the coupled

lateral-torsional modes change slightly when the cross sectional area increases. The

frequencies corresponding to the first coupled torsional-lateral mode (T1L1) and the

first vertical mode (V1) increase while all the others remain almost the same or

change slightly.

Table 4.4 Internal horizontal forces and the natural frequencies and their corresponding modes



Qint (kN) 0 10 20 10

T1 (N) 1.49E+07 1.49E+07 1.49E+07 1.60E+07


T3 (N) 5.76E+04 1.17E+06 2.23E+06 1.17E+06

L1T1 0.37312 0.55257 0.57335 0.64353

L2T2 0.53434 0.60216 0.65811 0.75219

L3T3 0.79064 0.90682 0.99061 1.04990

L4T4 1.03740 1.17160 1.28450 1.34350

L5T5 1.27330 1.44680 1.58800 1.66080


L6T6 1.49970 1.70540 1.87400 1.95320

T1L1 0.59431 0.68081 0.70218 0.74982

T2L2 1.00350 1.07680 1.10820 1.12710


T4L4 1.99430 2.06480 2.12910 2.15650

V1 0.66259 0.67261 0.68202 0.84195

V2 0.82865 0.85713 0.88501 0.91408

V3 1.24940 1.29640 1.33770 1.39640

V4 1.64020 1.70150 1.75750 1.81420

Vertical

V5 2.03840 2.11540 2.18560 2.25730

- 110 -

Table 4.5 Cross sectional area (diameter) of the pre-tensioned bottom cables and the vibration properties

Bridge parameter SSB: L=80 m; F1=F2=F3=1.8 m


Qint (kN) 0 0 0 0

D1 (mm) 240 240 240 240

D2 (mm) 120 169 240 339 Cable diameter

D3 (mm) 317 294 240 0

T1 (N) 1.65E+07 1.65E+07 1.66E+07 1.67E+07


T3 (N) 1.01E+05 8.64E+04 5.76E+04 0.00E+00

L1T1 0.43196 0.43487 0.43836 0.44246

L2T2 0.69280 0.69324 0.69166 0.69217

L3T3 0.99903 0.99791 0.99736 1.00680

L4T4 1.30550 1.30240 1.29790 1.30010

L5T5 1.61320 1.61090 1.59620 1.60320


L6T6 1.90210 1.89790 1.89150 1.89390

T1L1 0.63003 0.66424 0.72983 0.83620

T2L2 1.05130 1.06400 1.09060 1.15570


T4L4 2.05760 2.07990 2.13010 2.25720

V1 0.73632 0.77783 0.84947 0.94295

V2 0.91242 0.91227 0.91274 0.91820

V3 1.37980 1.38430 1.39500 1.42100

V4 1.80540 1.80650 1.80970 1.82340

Vertical

V5 2.23970 2.24290 2.25030 2.27190

4.3.2.3 The effects of structural weight and applied loads

In general, natural frequencies will decrease when the structural weight (or mass)

increases, but for cable supported bridge structures, effect of structural weight could

be a little more complicated. Table 4.6 and Table 4.7 show the effect of additional

structural weight (or mass) on the natural frequencies of un-pre-tensioned and pre-

tensioned footbridge models. Here the additional weight is applied to the bridge

structure by increasing additional mass which is assumed to be uniformly distributed

on the deck and is modelled as lumped masses on the supporting beams. After the

additional mass is applied, the top suspending cables are stretched to the required

cable sag and to keep the deck in horizontal plane. In this case, the total structural

weight consists of the original structural weight and weight from the applied mass.

Table 4.6 shows the effect of extra structural weight on a un-pre-tensioned bridge

(Wint=0, Qint=0), while Table 4.7 shows the effect on the pre-tensioned one (Wint=40

- 111 -

kN, Qint=0). For the un-pre-tensioned bridge model, when the structural weight

increases, the frequencies of the one half-wave coupled lateral-torsional mode

(L1T1) and one half-wave vertical mode (V1) decreases slightly and the frequency of

the one half wave coupled torsional-lateral mode (T1L1) keep almost the same.

While the frequencies of higher coupled modes and vertical modes increase slightly.

For the pre-tensioned bridge model, on the other hand, the frequencies of the first

coupled modes (L1T1 and T1L1) decrease while frequencies of higher coupled

lateral-torsional modes change only slightly. The frequency of the first vertical mode

(V1) decreases more rapidly that those of higher vertical modes. It can be seen that

though the structural weight has more significant effect on the natural frequencies of

pre-tensioned footbridges than those of un-pre-tensioned ones, all these effects are

slight. The reason is that the natural frequencies depend mainly on the tension force

and structural mass, and the tension force increases with the increase of structural

weight. The gain in stiffness and mass more or less minimises changes in natural

frequency.

Table 4.6 Effects of additional weight on the natural frequencies of un-pre-tensioned suspension footbridges


Wint (kN) 0 0 0 0 0 Internal force

Qint (kN) 0 0 0 0 0

Additional mass (weight) m (kg/m2) 0 200 400 600 1000

T1 (N) 1.49E+07 1.67E+07 1.84E+07 2.02E+07 2.37E+07

T2 (N) 5.80E+04 5.80E+04 5.80E+04 5.80E+04 5.80E+04 Cable tension

T3 (N) 5.76E+04 5.76E+04 5.76E+04 5.76E+04 5.76E+04

L1T1 0.37312 0.37160 0.3702 0.36893 0.36667

L2T2 0.53434 0.54400 0.5521 0.55896 0.56987

L3T3 0.79064 0.80458 0.81626 0.82616 0.84181

L4T4 1.03740 1.05700 1.0734 1.08720 1.10890

L5T5 1.27330 1.28760 1.3168 1.33290 1.35720


L6T6 1.49970 1.52690 1.5522 1.57140 1.59980

T1L1 0.59431 0.59420 0.59404 0.59387 0.59351

T2L2 1.00350 1.01330 1.0221 1.03020 1.04420

T3L3 1.51110 1.52860 1.5412 1.55420 1.57650 Coupled torsional-lateral

T4L4 1.99430 2.01560 2.0344 2.05090 2.07830

V1 0.66259 0.64169 0.62409 0.60904 0.58467

V2 0.82865 0.82855 0.82846 0.82838 0.82818

V3 1.24940 1.24760 1.2616 1.25330 1.25020

V4 1.64020 1.64400 1.6468 1.64880 1.65090

Vertical

V5 2.03840 2.04610 2.0518 2.05600 2.06100

- 112 -

Table 4.7 Effects of additional weight on the vibration properties of pre-tensioned suspension footbridges



Qint (kN) 0 0 0 0 0

Additional mass (weight) m (kg/m2) 0 200 400 600 1000

T1 (N) 1.71E+07 1.89E+07 2.06E+07 2.24E+07 2.59E+07


T3 (N) 5.76E+04 5.76E+04 5.76E+04 5.76E+04 5.76E+04

L1T1 0.453 0.44577 0.43999 0.43499 0.42675

L2T2 0.727 0.72241 0.71794 0.71378 0.70642

L3T3 1.052 1.0469 1.0419 1.03700 1.028

L4T4 1.371 1.3675 1.3629 1.35790 1.3479

L5T5 1.697 1.6924 1.6863 1.67970 1.6648


L6T6 1.999 1.9969 1.9922 1.98590 1.9707

T1L1 0.740 0.73327 0.72725 0.72167 0.71168

T2L2 1.112 1.1143 1.1171 1.12010 1.1263


T4L4 2.174 2.1781 2.1831 2.18820 2.1981

V1 0.860 0.82642 0.79759 0.77273 0.73196

V2 0.939 0.92868 0.91978 0.91228 0.90035

V3 1.434 1.4153 1.4001 1.38720 1.3666

V4 1.862 1.845 1.8301 1.81720 1.7957

vertical

V5 2.315 2.297 2.2809 2.26650 2.2418

The effect of applied load is different from that of additional weight. When a

footbridge is subjected to pedestrian load, the bridge structure deforms and the

pedestrian load also contributes to the increase in structural mass. Therefore, the

natural frequencies are subjected to some change caused by the pedestrians and

bridge deformation.

Table 4.8 and Table 4.9 show the effects of applied load on the vibration properties

of un-pre-tensioned and pre-tensioned footbridges respectively. Here the applied load

is also added by increasing additional uniformly distributed mass. For un-pre-

tensioned bridge model, the effect is slight, all the frequencies decrease slightly

except that of the one half-wave coupled torsional-lateral mode (T1L1). However,

for pre-tensioned bridge model, the effect is significant. All frequencies decrease

significantly when the applied load (mass) increases. The reason is that for a pre-

tensioned bridge model, the total horizontal tension force in a bridge cross section

- 113 -

changes only slightly while the structural mass increases when the bridge structure is

subjected to applied load.

Table 4.8 Effects of applied load on the vibration properties of un-pre-tensioned suspension footbridges


Wint (kN) 0 0 0 0 0 Internal force Qint (kN) 0 0 0 0 0

Applied mass (load) m (kg/m2) 0 200 400 600 1000

T1 (N) 1.49E+07 1.60E+07 1.70E+07 1.80E+07 1.99E+07

T2 (N) 5.80E+04 5.65E+04 5.53E+04 5.41E+04 5.21E+04 Cable tensions

T3 (N) 5.76E+04 5.76E+04 5.75E+04 5.75E+04 5.75E+04

L1T1 0.37312 0.36810 0.36332 0.35881 0.35061

L2T2 0.53434 0.53287 0.53095 0.52870 0.52363

L3T3 0.79064 0.78883 0.78623 0.78308 0.77575

L4T4 1.03740 1.03590 1.03340 1.03000 1.02180

L5T5 1.27330 1.27190 1.26880 1.26450 1.25340


L6T6 1.49970 1.49930 1.49670 1.49240 1.48040

T1L1 0.59431 0.59667 0.59866 0.60027 0.60242

T2L2 1.00350 0.99221 0.98236 0.97358 0.95832


T4L4 1.99430 1.97450 1.95690 1.94090 1.91230

V1 0.66259 0.65147 0.64152 0.63257 0.61697

V2 0.82865 0.81124 0.79612 0.78276 0.75999

V3 1.24940 1.22390 1.20970 1.18760 1.15380

V4 1.64020 1.61020 1.58350 1.55930 1.51680

vertical

V5 2.03840 2.00440 1.97360 1.94520 1.89470

4.3.2.4 Effect of span length

Span length is an important structural parameter and has great effect on the vibration

properties. Two pre-tensioned cable supported bridge models with the span lengths

of 40 m and 120 m have been studied and similar numerical analysis to the above

footbridge models have been carried out. Here, all the cable sags are assumed to be

1.8 m. For the model with span length of 40 m, all the cable diameters are set to be

120 mm in order to avoid over rigid stiffness and low stress in the top supporting

cables, while the diameters for the bridge model with span length of 120 m are set to

be 240 mm. Table 4.10 shows the natural frequencies and their corresponding

vibration modes with different internal forces for the bridge model with span length

of 40 m, and Table 4.11 for the bridge model with span length of 120 m.

- 114 -

Table 4.9 Effects of applied load on the vibration properties of pre-tensioned suspension footbridges



Qint (kN) 0 0 0 0 0

Applied mass (load) m (kg/m2) 0 200 400 600 1000

T1 (N) 1.71E+07 1.77E+07 1.84E+07 1.91E+07 2.08E+07


T3 (N) 5.76E+04 5.76E+04 5.77E+04 5.78E+04 5.80E+04

L1T1 0.453 0.42646 0.40393 0.38339 0.35975

L2T2 0.727 0.67872 0.63499 0.59881 0.56150

L3T3 1.052 0.97999 0.91413 0.85931 0.81339

L4T4 1.371 1.27750 1.19180 1.12220 1.06630

L5T5 1.697 1.57500 1.47050 1.38210 1.30910


L6T6 1.999 1.86470 1.73820 1.63280 1.54610

T1L1 0.740 0.71863 0.69497 0.65515 2.37360

T2L2 1.112 1.07800 1.04970 1.02640 0.60831


T4L4 2.174 2.10570 2.05060 2.00760 1.48820

V1 0.860 0.81188 0.76463 0.69937 0.61871

V2 0.939 0.89019 0.84929 0.81699 0.78057

V3 1.434 1.35940 1.29650 1.24330 1.18360

V4 1.862 1.76880 1.69030 1.62800 1.55780

vertical

V5 2.315 2.20270 2.10750 2.03130 1.94560

For the 40 m suspension footbridge model, the structural stiffness is relatively rigid

and the vibration modes are affected significantly by the short span length. It can be

seen from Table 4.10 that the vibration modes become more complicated. The one

half-wave vertical mode (V1) is not always the fundamental vertical mode as its

corresponding frequency is greater that that of the two half-waves asymmetric

vertical mode (V2). And it would even disappear when the reverse profiled bottom

and side cables are pre-tensioned (Wint=20 kN, Qint=10 kN). This phenomenon also

occurs in the coupled vibration modes. The two half-waves couped torsional-lateral

mode (T2L2) becomes fundamental in most situations. Even the one half-wave

symmetric coupled lateral-torsional mode (L1T1) will disappear or become the

second mode. However, for a longer span suspension footbridge model (L=120 m), it

can be seen from the Table 4.11 that all the vibration modes are almost arranged

according to the number of half-waves. These results demonstrate that the span

length and pre-tension forces have important effects on vibration modes and their

natural frequencies.

- 115 -

Table 4.10 Effect of internal forces on the vibration properties of footbridge with span length of 40 m



Qint (kN) 0 0 10 10

T1 (N) 2.88E+06 3.16E+06 2.88E+06 3.16E+06


T3 (N) 1.46E+04 1.46E+04 2.95E+05 2.95E+05

L1T1 0.40968 0.45197 --

1.24360

L2T2 0.55224 0.77242 0.63642 1.02520

L3T3 0.79279 0.97987 0.86653 1.03150

L4T4 1.04330 1.28870 1.20960 1.46120

L5T5 1.26230 1.54100 1.24410 1.75100


L6T6 1.44960 1.77230 1.48870 1.98600

T1L1 1.03590 1.33390 1.05310 1.46870

T2L2 1.02000 1.17910 1.24430 1.31720


T4L4 1.97980 2.10360 2.08960 2.21180

V1 1.07690 1.23490 1.10930 --

V2 0.83193 0.89780 0.86071 0.93306

V3 1.41530 1.75150 1.45500 1.27710

V4 1.60440 1.80360 1.68030 1.82130

Vertical

V5 1.97310 2.18570 2.06460 1.82640

Table 4.11 Effect of internal forces on the vibration properties of footbridges with span length of 120 m

Model SSB: L=120 m; F1=F2=F3=1.8 m; D1=D2=D3=240 mm

Wint (kN) 0 20 0 20 Extra internal forces

Qint (kN) 0 0 10 10

T1 (N) 3.34E+07 3.59E+07 3.34E+07 3.59E+07

T2 (N) 5.77E+04 2.51E+06 5.77E+04 2.51E+06 Cable tensions

T3 (N) 5.74E+04 5.74E+04 2.50E+06 2.50E+06

L1T1 0.31272 0.37174 0.37864 0.43572

L2T2 0.52884 0.63252 0.59646 0.70210

L3T3 0.78187 0.92271 0.88605 1.01630

L4T4 1.03340 1.21550 1.16700 1.33400

L5T5 1.27180 1.50480 1.44480 1.65440


L6T6 1.50570 1.78700 1.71390 1.96190

T1L1 0.50347 0.54611 0.54543 0.58053

T2L2 1.00050 1.05740 1.05110 1.09690


T4L4 2.00530 2.09200 2.06870 2.15870

V1 0.47483 0.55102 0.48778 0.56227

V2 0.82632 0.88409 0.85555 0.91189

V3 1.23960 1.32860 1.28410 1.37060

V4 1.64460 1.75970 1.70470 1.81660

Vertical

V5 2.05270 2.19180 2.12460 2.26420

- 116 -

4.4 Summary

Suspension footbridges with shallow cable profiles are slender and flexible

structures, their load deformation performance and vibration properties are affected

by many structural parameters. From the above analysis, some features of the

structural behaviour and effects of important structural parameters can be

summarized in the following, and this information will be used to study the dynamic

characteristics of slender suspension footbridges with coupled vibration modes under

human-induced dynamic loads:

• The structural stiffness of the proposed suspension footbridge depends not only

on the cable sag and cross sectional area of the top supporting cables, but also

on cable cross sections and pre-tensions in the reverse profiled pre-tensioned

cables. Pre-tensioned reverse profiled bottom cables can improve the vertical

structural stiffness. Before the pre-tensioned bottom cables slack, the vertical

stiffness depends on top and bottom cables’ cross sectional area and cable sag,

irrespective of cable profiles, catenary or reverse profiled. After the pre-

tensioned bottom cables slack, the vertical stiffness is provided only by the top

supporting cables.

• The stiffness in lateral direction is much smaller than that in the vertical one.

Eccentric vertical loads can not only produce vertical deflection and torsion,

but also induce lateral horizontal deflection. This can contribute to large

horizontal sways and consequent problems.

• Introducing pre-tensioned side cables in horizontal plane can significantly

improve the lateral stiffness, and greatly suppress the lateral deflection induced

by lateral applied loads as well as eccentric vertical loads.

• Suspension footbridges with shallow cable sags always have four kinds of

vibration modes: lateral, torsional, vertical and longitudinal modes. The lateral

and torsional modes are often combined together and become two new coupled

modes: (1) coupled lateral-torsional modes, (2) coupled torsional-lateral modes.

- 117 -

When the cable sags increase or the pre-tensions in the bottom and/or side

cables to some extent, the one half-wave symmetric coupled lateral-torsional

mode or coupled torsional-lateral mode can reduce to pure lateral and torsional

modes.

• Modes with lowest frequencies are often coupled lateral-torsional or pure

lateral modes, and not flexural vertical vibration modes. For different kinds of

vibration modes, the one half-wave symmetric mode corresponds generally to

the lowest natural frequency. However, when the structural stiffness increases,

the frequency corresponding to the two half-wave mode will become the

fundamental and the one half-wave mode will become the higher mode or even

disappear.

• Vibration properties are affected by many structural parameters such as cable

profiles, cable sections, span length and pre-tensions in reverse profiled cables.

Structural weight has slight effect on the natural frequencies and vibration

modes. For pre-tensioned suspension footbridge models, the effect of applied

load is significant, particularly for slender and light weight bridge structures.

• For pre-tensioned suspension footbridges, the vibration properties can be

“tuned” by introducing different pre-tensions into the reverse profiled bottom

and/or side cables. This feature is useful to improve the structural behaviour of

such kind of cable structures and minimise excessive lateral vibration.

- 118 -

- 119 -

Experimental testing and calibration of physical bridge model

5.1 Introduction

Based on the research information generated in chapter 4, a physical cable supported

bridge model was designed and constructed for experimental testing and calibration

of computer models. This physical bridge model has features similar to those of the

(proposed) analytical pre-tensioned suspension footbridge model and has variable

vibration properties covering a range of low natural frequencies. Experimental

testing will be carried out in the laboratory to measure the natural frequencies and the

deflections under static loads for the purpose of calibration.

The structural analysis software packages Microstran (V8) and SAP2000 (V9) are

used to model and analyse this bridge structure. Numerical results will be compared

with those from experimental testing to calibrate the computer models. The two

programs have difference in modelling the cable supported bridge as mentioned in

chapter 3. Here, Microstran is mainly used in the initial design of the physical bridge

model, and SAP2000 will be used in the next chapter to carry out extensive non-

linear time history numerical analyses and to study the dynamic performance of

slender suspension footbridges under human-induced dynamic loads.

5.2 Physical bridge model and experiment system

5.2.1 Design of physical bridge model and experimental system

In order to carry out experimental testing on the designed physical bridge model, the

whole experimental system was considered to consist of two parts (Figure 5.1):

physical bridge model and support system. The physical bridge model is the main

part and its load performance and vibration properties are the main concern of

experimental testing. The support system provides anchorage to the physical bridge

5

- 120 -

model. Load cells are installed to measure the tension forces in the cable system and

also different pre-tensions can be introduced to the supporting and reverse profiled

cables through this support system to obtain different structural stiffness and

vibration properties.

Figure 5.1 The experimental bridge model and support system

Figure 5.2 Details of transverse bridge frame

- 121 -

Physical bridge model

The physical bridge model is designed initially to have a span of 4.5 m, and it is

possible to change the span to 3.0 m or 1.5 m for further research. The bridge model

comprises of cable system, transverse bridge frames and deck units. The cable

system includes three groups of cables: top supporting cables, pre-tensioned reverse

profiled (bottom) cables in vertical plane and pre-tensioned bi-concave side cables in

the horizontal plane. Nine transverse bridge frames are supported by the cables and

enable the cable system to retain the designed cable profiles. The horizontal distance

of two adjacent bridge frames is set to be 450 mm. The deck units are simply

supported on the transverse bridge frames. The details of the transverse bridge frame

are shown in Figure 5.2. In the bridge frame, a total of six cable plugs are designed to

hold the cables with required cable profiles. Four cable plugs are arranged at the two

sides of the bridge frame for the catenary and reverse profiled cables. These cable

plugs can move to different locations along the sockets in the frame’s side members

by tuning the thread rods, and the maximum moving distance of these cable plugs is

280 mm. Another two cable plugs are arranged underneath the cross member of the

bridge frame to hold the side cables in the horizontal plane. These two cable plugs

have maximum moving distance up to 100 mm.

Support system

The support system is composed of headstocks, cable clamps, load cells, turnbuckles,

end supports and basements. Two headstocks are designed to provide different

supports to the cables for different cable configurations. The elevation detail of the

headstock is shown in Figure 5.3. The cable clamps (Figure 5.4) are designed to be

attached on the headstocks and their main functions are: (1) to lock the cables and to

reduce slip of cables; (2) to change the directions of the cables to align with the load

cells. Load cells are Aluminium specimens with strain gauges and are used to

measure the tension forces in the cables. The turnbuckles are used to provide

different cable tension forces by changing the cable lengths. The end supports are

designed to provide supports to the load cells and transfer the forces to the basement.

The basement is composed of two Parallel Flange Channel steel beams (150PFC)

- 122 -

and is designed to keep the bridge model with the required span length by bolting the

headstocks at different locations.

Figure 5.3 Elevation of headstock

The connection of the support system at one end of the physical bridge structure is

illustrated in Figure 5.5. When the cables go through the headstocks and cable

clamps, they are connected to the load cells and turnbuckles, and then fixed on the

- 123 -

end supports. The tension forces can be adjusted by tuning the turnbuckles and

measured by the load cells at the same time. After the required tension forces have

been obtained, the cables are locked by the cable clamps to minimize the slip of the

cables.

Figure 5.4 Details of cable clamp

Figure 5.5 Connection of support system

End Support Turnbuckl Load cell Cable clamp Headstock

Basement Basement

Adjusting tension force

Fixing Cable

Strain gauge

Measuring tension force

Cable Bridge

structure

- 124 -

5.2.2 Construction of physical bridge model

The physical bridge model was constructed in the Concrete Lab at Queensland

University of Technology for the experimental testing. The whole bridge system is

mainly made of Aluminium and stainless steel materials to prevent the structure from

rusting and reduce the total weight of bridge model. For example, the bridge deck

units and main parts of the transverse bridge frames are made of Aluminium, while

stainless steel thread rods are used to adjust the location of cable plugs for different

cable profiles, and stainless steel wires are chosen for the cable systems. For the

supporting system, the headstocks and end supports are made of Aluminium and the

basement is composed of two Parallel Flange Channel steel beams (150PFC). Figure

5.6 shows the physical bridge model constructed in the laboratory.

Figure 5.6 The physical bridge model constructed in the laboratory

Figure 5.7 shows the support system at one end of the bridge model and the strain

indicator as well as strain gauge connection boxes which connected to load cells.

Before the physical bridge model is constructed, 12 Aluminium load cells with strain

gauges have been tested to establish the relationships of tension force and strain for

each load cell, and these relationships are used in constructing the physical bridge

- 125 -

model and to measure and control the cable tension forces. Different tension forces

in the cable systems are introduced by tuning the turnbuckles according to the type of

bridge model. In each model type, when all the tension forces have been adjusted to

the designed values, the cables are locked by the cable clamps to reduce the effect of

cables as well as load cells, turnbuckles and shackles in the support systems on the

behaviour of bridge model during testing.

Figure 5.7 The support system of physical bridge model

Table 5.1 Material properties of stainless steel wires

Stainless steel wire (7x19)

Nominal diameter

Density

Modulus

Breaking strength

(mm) (kg/m3) (MPa) (N)

1/8 inches 3.0 7582 58460 6597.6

1/16 inches 1.5 6315 14530 1714.7

For calibration purpose, all cable plugs in the bridge frames were adjusted for the

bridge model to have cable sag of 100 mm for the supporting top cables and reverse

profiled bottom cables in vertical plane, cable sag of 75 mm for the reverse profiled

side cables in horizontal plane. Here, the 1/8 inches 7x19 stainless steel wire is used

for the supporting top cables, while the 1/16 inches 7x19 stainless steel wire is

- 126 -

chosen for the bottom and side cables. Their material properties measured in

laboratory are listed in Table 5.1. It is found these wires have nearly linear force-

strain relationships. The Young’s modulus in the table is calculated according to the

nominal cable diameter and average ratio of tension force to the strain. The mass

density of Aluminium measured in laboratory is 2780 kg/m3 on average. The weight

of a bridge frame is about 20.80 N (or 2.12 kg), and the total weight of a frame with

cables, deck units between two adjacent bridge frames as well as other accessories

such as screws etc. is about 48.03 N (or 4.90 kg) on average. In computer

simulations, the Aluminium is supposed to have standard Young’s modulus of

6.5×1010 MPa and Poisson’s ratio of 0.33, while the stainless steel of the thread rods

is assumed to have the Young’s modulus of 2.0×1011 MPa and Poisson’s ratio of

0.25 with the mass density of 7850 kg/m3.

5.3 Experimental testing and calibration of bridge model

5.3.1 Bridge model cases and testing procedure

For the purpose of calibration, free vibration and load deformation performance

under static vertical loads of the bridge model are first analysed by two structural

analysis softwares (Microstran and SAP2000) respectively, and the structural

parameters such as cable sags and tension forces in the computer simulations are then

applied in the construction of physical bridge model and experimental testing. Figure

5.8 and Figure 5.9 show the computer bridge model in Microstran and SAP2000

respectively. Four bridge model cases are considered to investigate the effect of

reverse profiled cable on the structural behaviour. In case 1 and case 2, the bridge

model is designed to have a cable configuration with only supporting top cables and

pre-tensioned reverse profiled bottom cables, and the internal vertical force

introduced at each bridge frame is supposed to be 10 N and 30 N respectively; while

in case 3 and case 4, the bridge model is designed to have all the pre-tensioned

reverse profiled bottom and side cables, the internal vertical force at each bridge

frame is let to be 30 N and the internal lateral force introduced by the pre-tensioned

side cables is let to be 5 N and 15 N respectively.

- 127 -

Figure 5.8 The experimental bridge model in Microstran

Figure 5.9 The experimental bridge model in SAP2000

In the experimental testing of free vibration, 8 accelerometers are installed at 5

different locations to measure the natural frequencies and the distribution of

accelerometers is illustrated in Figure 5.10. These locations are the intersection

points of cross members and side members (see Figure 5.2) of the middle bridge

frame and the second frame to the bridge end. At the points 1, 2, 3, two

accelerometers are installed in lateral and vertical directions respectively to get the

signals of the lateral and vertical accelerations, while at the point 4 and 5,

accelerometers are installed to pick up only the signals of vertical acceleration. Here

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lateral and vertical accelerations are noted as AL and AV respectively for different

points. For instance, AL1 and AV1 represent the lateral and vertical accelerations of

point 1 respectively. Figure 5.11 shows the accelerometers installed at the point 1

(middle bridge frame). This arrangement of accelerometers is helpful to recognize

the natural frequencies and their corresponding vibration modes. During the

experimental testing, the acceleration signals are recorded automatically by the

automatic data acquisition system Somat TCE eDAQ (version 3.8.6)

(http://www.somat.com) and then analysed by the data analysis software Somat

InField (version 1.5.1) (http://www.somat.com). These data acquisition system is

shown in Figure 5.12.

Figure 5.10 Distribution of accelerometers on physical bridge model

Figure 5.11 Accelerometers installed on the middle bridge frame

- 129 -

Figure 5.12 Data acquisition and analysis system

Figure 5.13 Applied static vertical loading system

- 130 -

The static vertical load is applied by adding a series of standard 1 kg weights. Four

Aluminium bars form the loading system and transfer the vertical load to the deck

units, and therefore the deck units at the two middle segments are subjected to line

uniform vertical loads at their own half span lengths. Figure 5.13 shows the loading

system and applied standard weights. The deflections of the cross member of the

middle bridge frame are measured at the locations near the two ends. The weight of

four Aluminium bars is 28.45 N (2.9 kg). For calibration purpose, five standard 1.0

kg weights are applied gradually on the loading system.

5.3.2 Free vibration and natural frequencies

In order to excite different vibration modes, three kinds of initial excitations are used

in the experimental testing of free vibration: (1) initial lateral excitation – one bridge

frame is pulled out or pushed away laterally for a small distance and then released;

(2) initial torsional excitation – one bridge frame is twisted and then released; (3)

initial vertical excitation – one bridge frame is pushed down and released. To obtain

more signal samples for all low frequency vibration modes, these initial excitations

were applied at different locations on different bridge frames.

The bridge model vibrates differently when subjected to different initial excitations.

Figure 5.14 to Figure 5.16 show the typical accelerations of point 1 at the middle

bridge frame in bridge model case 1 when different initial excitations are applied. It

can be seen that when the initial lateral excitation is applied, the bridge model

vibrates mainly in the lateral direction and the amplitude of vertical acceleration

(AV1) is relatively smaller than that of the lateral one (AL1) (Figure 5.14); while

when the initial torsional or vertical excitation is been applied, the lateral

acceleration is smaller than the vertical one (Figure 5.15 and Figure 5.16), and this

indicates that point 1 moves mainly up and down. The difference between torsional

vibration and vertical one can be seen by comparing the vertical accelerations at

point 1 and point 4 which locate at different sides of the bridge deck. Figure 5.17

shows the accelerations when initial torsional excitation is applied. It can be seen that

these two points have different phases and the bridge vibrates mainly in torsional

modes, i.e., one point moves up and the other moves down. Figure 5.18 shows the

vertical accelerations of point 1 and point 4 when initial vertical excitation is applied.

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It can be seen that when the bridge vibrates mainly in vertical modes, these vertical

accelerations almost have the same phase.

Figure 5.14 Case 1: accelerations at point 1 (initial lateral excitation)

Figure 5.15 Case 1: accelerations at point 1 (initial torsional excitation)

Figure 5.16 Case 1: accelerations at point 1 (initial vertical excitation)

- 132 -

Figure 5.17 Case 1: vertical accelerations at point 1 and point 4 (initial torsional excitation)

Figure 5.18 Case 1: vertical accelerations at point 1 and point 4 (initial vertical excitation)

For different bridge model cases, the natural frequencies can be obtained by carrying

out a series of spectrum analysis on the accelerations recorded when different initial

excitations are applied on the bridge model. The experimental results for the natural

frequencies of the bridge model for the four cases are presented in Table 5.2 together

with the main structural parameters of these cases. Here the mass density is the total

mass of one bridge frame, the deck units of one segment as well as other accessories

divided by the width of deck (400 mm) and the length of one segment (450 mm). The

measured tension forces of different cables are average values. For example, there

are two top supporting cables and all the tension forces at the end of bridge model

are supposed to be the same. However, these forces are controlled by different

turnbuckles and can not be adjusted to be exactly the same in the physical bridge

model, and therefore the average tension force of the two cables at the four ends is

listed in these tables. The natural frequencies corresponding to different vibration

modes are obtained from accelerations when the bridge model is subjected to

- 133 -

different initial excitations: the frequencies of lateral vibration modes are obtained

based on the acceleration signals recorded when the initial lateral excitation has been

applied on the bridge model, while the frequencies of torsional and vertical modes

are obtained based on the acceleration signals when the bridge model is subjected to

initial torsional and vertical excitations respectively.

Table 5.2 Natural frequencies from experimental testing

Model case Case 1 Case 2 Case 3 Case 4

Mass density (kg/m2) 27.15 27.15 27.2 27.2

Wint (N) 10 30 30 30 Internal force

Qint (N) 0 0 5 15

T1 (N) 1637.5 2235.4 2232.9 2232.9

T2 (N) 284 882.4 871.6 871.6 Tension force

T3 (N) -- -- 381.4 1182.7

L1T1 2.005 2.536 2.787 2.924

L2T2 3.984 4.968 5.48 5.89 Lateral

L3T3 5.852 7.470 8.22 8.602

T1L1 4.337 5.214 5.404 5.568 Torsional

T2L2 6.204 8.857 9.559 9.811

V1 2.875 3.355 3.551 3.77

V2 4.187 5.286 5.725 6.19 Vertical

V3 6.187 7.819 8.514 9.841

Figure 5.19 Case 1: spectra of accelerations at point 2 (initial lateral excitation)

- 134 -

Figure 5.20 Case 1: spectra of accelerations at point 2 (initial torsional excitation)

Figure 5.21 Case 1: spectra of accelerations at point 2 (initial vertical excitation)

- 135 -

Figure 5.19 to Figure 5.21 show the spectra of the lateral and vertical accelerations at

point 2 in bridge model case 1 when different initial excitations have been applied. It

can be seen from Figure 5.19 that when the bridge model vibrates mainly in the

lateral direction, 3 natural frequencies can be easily recognized in the spectra of

lateral acceleration, and the same natural frequencies appear in the spectra of vertical

acceleration. This phenomenon indicates that these vibration modes have both lateral

and vertical components and appear as coupled lateral-torsional modes. The natural

frequencies corresponding to torsional and vertical modes are recognized by

comparing the spectra of vertical accelerations at different locations (points). When

the bridge model is subjected to initial torsional excitation, it is found different

vibration modes besides torsional ones have been activated and the spectra are

complicated with the different frequencies (Figure 5.20). Two torsional modes have

been recognized and it is found that the torsional mode (4.337 Hz) has apparent

lateral component while the other tosional mode (6.204 Hz) does not. This implies

that some torsional modes also exist as coupled torsional-lateral modes. When the

bridge model is subjected to initial vertical excitation, it vibrates mainly in vertical

direction and the frequencies of vertical modes be easily recognized from the spectra

of vertical acceleration (Figure 5.21). However, it is found that the natural frequency

(2.875 Hz) corresponding with first vertical mode also has a small component in the

lateral acceleration, but the spectra density of this component is much smaller than

the vertical one. This result is different from those from computer simulations, and it

is probably caused by the errors of tension force in the cable systems.

From these figures, it is also found that the natural frequencies are affected slightly

by the dynamic performance. For example, the frequency corresponding to the

coupled lateral-torsional mode L1T1 is 2.005 Hz when the bridge model vibrates

mainly in the lateral direction. However, it changes to 1.994 Hz and 2.031 Hz when

the bridge model vibrates mainly in torsional modes and vertical modes respectively.

The frequency corresponding to the vertical mode V1 also appears as different

values, 2.885 Hz, 2.849 Hz and 2.875 Hz, when the bridge model vibrates mainly in

lateral, torsional and vertical modes respectively. This phenomenon illustrates that

suspension bridges are non-linear slender structures and the vibration properties are

affected by the tension forces and the structural performance. When the bridge

- 136 -

vibrates in different ways, the tension forces in cable systems change, and so do the

natural frequencies.



- 137 -


Figure 5.22 to Figure 5.24 show the spectra of accelerations in bridge model case 2

when the internal vertical force increases from 10 N to 30 N on each bridge frame.

From Figure 5.22, it can be seen that three frequencies corresponding to lateral

vibration modes are precisely shown in the spectra of lateral acceleration, but these

frequencies have not been found in the spectra of vertical acceleration. The

frequencies shown in the spectra of vertical acceleration are those corresponding to

vertical vibration modes. From Figure 5.23, it is clearly shown that two torsional

modes have components in both lateral and vertical accelerations and this result

indicates that these modes are coupled torsional-lateral modes. In Figure 5.24, the

frequency corresponding to the second vertical mode is found in both the spectra of

lateral and vertical accelerations; while for the first vertical mode, it is found that

there is a frequency (3.419 Hz) in the spectra of lateral acceleration which does not

corresponding to any lateral or torsional vibration mode. Since it is close to the

frequency of 3.355 Hz, it is probably a variation of the frequency of the first vertical

mode. This phenomenon indicates that the vertical modes have small components in

lateral direction.

- 138 -

Figure 5.25 to Figure 5.27 show the spectra of accelerations at point 2 for case 3,

while Figure 5.28 to Figure 5.30 show those in bridge model case 4. From these

figures, it is found that the first torsional mode is always coupled vibration mode

(Figure 5.26 and Figure 5.29). For the first lateral mode, a vertical component is

found with a frequency close to that of this mode (Figure 5.25 and Figure 5.28). As

this frequency is much less than that of the first vertical mode, it is probably a

variation of the vertical component of the first lateral mode. This phenomenon is

also found in the first vertical mode (Figure 5.30, as well as Figure 5.24). It should

be noted that this variation is easy to be found for the first lateral, torsional and

vertical modes, but difficult for the higher vibration modes.

From the above spectra analysis, it can be seen that the dynamic behaviour of the

physical bridge model is complex. All the natural frequencies will change slightly

when the bridge model vibrates in different ways. Furthermore, coupled vibration

modes may have components in lateral and vertical directions but with slightly

different frequencies. The results of experimental testing confirm that the natural

frequencies of a suspension bridge structure can be controlled to some extent by

introducing reverse profiled cable systems and different pre-tensions.


- 139 -



- 140 -



- 141 -


Table 5.3 Static vertical deflections measured from experimental testing

Model case Case 1 Case 2 Case 3 Case 4

Applied load Deflection Deflection Deflection Deflection

(kg) (N) (mm) (mm) (mm) (mm)

0 0 0 0 0 0

2.9 28.45 -5 -3.43 -3.12 -2.46

3.9 38.26 -6.58 -4.57 -4.26 -3.39

4.9 48.07 -8.2 -5.75 -5.29 -4.26

5.9 57.88 -9.77 -6.95 -6.33 -5.22

6.9 67.69 -11.42 -8.16 -7.42 -6.17

7.9 77.5 -13.07 -9.32 -8.48 -7.15

5.3.3 Load performance under static vertical load

Table 5.3 shows the vertical deflections at the end of cross member of the middle

bridge frame with the applied loads in different bridge model cases. It is found the

vertical deflections at the two ends of the cross member change slightly when the

standard weights are applied carefully at the centre of the loading system. Here

shown in the table are the average value of those at the two ends of the cross

- 142 -

member. It can be seen that the vertical deflection can be reduced when different pre-

tensions have been introduced in the reverse profiled bottom and/or side cables.

Table 5.4 Natural frequencies of physical bridge model in case 1 and case 2

Model case Case 1 Case 2

Method Microstran SAP2000 Experiment Microstran SAP2000 Experiment

Mass density (kg/m2) 27.15 27.15 27.15 27.15 27.15 27.15

Wint (N) 10.0 10.0 10.0 30.0 30.0 30.0 Internal force

Qint (N) 0 0 0 0 0 0

T1 (N) 1638.2 1638.2 1637.5 2227.9 2227.9 2235.4

T2 (N) 285.2 285.2 284.0 874.9 874.6 882.4 Tension force

T3 (N) -- -- -- -- -- --

L1T1 2.0514 2.0371 2.005 2.6225 2.6009 2.536

L2T2 3.9530 3.8476 3.984 5.1183 4.9618 4.968

L3T3 5.7524 5.4822 5.852 7.4442 7.0134 7.470 Lateral

L4T4 7.3865 6.9026 -- 9.5271 8.6953 --

T1L1 3.6428 3.7360 4.337 4.3338 4.4672 5.214

T2L2 6.1147 6.1489 6.204 7.6683 7.7520 8.857

T3L3 9.1113 9.0505 -- 11.4620 11.3870 -- Torsional

T4L4 11.5350 11.5840 -- 14.4620 14.5430 --

V1 2.6146 2.6265 2.875 3.0814 3.0957 3.355

V2 4.0951 4.1227 4.187 5.1981 5.2316 5.286

V3 6.0473 6.0745 6.187 7.6626 7.6916 7.819 Vertical

V4 7.7952 7.8347 -- 9.8853 9.9377 --

5.4 Comparison of results and discussion

5.4.1 Results from computer simulations and experimental testing

Table 5.4 and Table 5.5 present the results of natural frequencies from experimental

testing and from computer simulations using Microstran and SAP2000. Since

SAP2000 uses beam elements to simulate the cable members, the modification

factors for the moments of inertias and torsional constant of the cable sections is set

to be 0.01 and number of elements for one cable member between two adjacent

bridge frames is set to be 20 in the numerical analysis ( as discussed earlier). From

these tables, it is found that the differences of the results from different structural

analysis softwares are quite small. Compared with the results from SAP2000, the

- 143 -

natural frequencies of lateral modes from Microstran are slightly greater, while the

frequencies of torsional and vertical modes are slightly smaller. The numerical

results from computer simulations using Microstran and SAP2000 show that the

(physical) bridge model has coupled lateral-torsional modes and coupled torsional-

lateral modes which are dominated by lateral modes and torsional modes

respectively, while the vertical modes appear as pure modes.

Table 5.5 Natural frequencies of physical bridge model in case 3 and case 4


Method Microstran SAP2000 Experiment Microstran SAP2000 Experiment

Mass density (kg/m2) 27.20 27.21 27.20 27.20 27.21 27.20

Wint (N) 30.0 30.0 30.0 30.0 30.0 30.0 Internal force

Qint (N) 5.0 5.0 5.0 15.0 15.0 15.0

T1 (N) 2230.6 2230.9 2232.9 2230.4 2230.3 2232.9

T2 (N) 874.9 874.9 871.6 874.9 874.3 871.6 Tension force

T3 (N) 381.1 381.2 381.4 1178.2 1180 1182.7

L1T1 2.8066 2.7874 2.787 3.1138 3.1011 2.924

L2T2 5.4743 5.3283 5.480 6.1016 5.9803 5.890

L3T3 7.9687 7.5681 8.220 8.9007 8.5635 8.602 Lateral

L4T4 10.2090 9.4355 -- 11.4280 10.7770 --

T1L1 4.3693 4.5208 5.404 4.4428 4.6282 5.568

T2L2 7.7624 7.8670 9.559 7.9465 8.0733 9.811

T3L3 11.3210 11.5100 -- 11.6220 11.6900 -- Torsional

T4L4 14.6650 14.7360 -- 15.5240 15.0900 --

V1 3.2167 3.2314 3.551 3.4842 3.5008 3.770

V2 5.5137 5.5448 5.725 6.1103 6.1415 6.190

V3 8.1197 8.1512 8.514 8.9979 9.0234 9.841 Vertical

V4 10.4790 10.5240 -- 11.6190 11.6490 --

It is found that the natural frequencies of the lateral modes from experimental testing

and computer simulations agree well with the maximum error less than 8% for the

first three lateral modes, though the experimental results are slightly smaller. The

natural frequencies of the vertical modes from experimental testing are found to be

slightly greater (about 8% for the first vertical mode) than those from computer

simulations. However, the natural frequencies of torsional modes are much greater

than those from computer simulations. This result is mainly caused by the method of

modelling the bridge deck units and it will be discussed later.

- 144 -

Table 5.6 and Table 5.7 show the vertical deflections at the end of cross member of

the middle bridge frame from experimental testing and computer simulations when

the physical bridge model is subjected to static vertical load which is applied

carefully at the centre of bridge. Here in the tables, the experimental results are the

average deflection of those at the two ends of the cross member. It is found that the

experimental results are slightly smaller than those from computer simulations with

maximum error less than 5%. It also can be seen from these tables that the vertical

deflections from Microstran and SAP2000 agree very well.

Table 5.6 Static vertical deflection of physical bridge model in case 1 and case 2 (in mm)


Applied load

(kg) (N) Microstran SAP2000 Experiment Microstran SAP2000 Experiment

0 0 0 0 0 0 0 0

2.9 28.45 -5.0969 -5.1014 -5.00 -3.5889 -3.5943 -3.43

3.9 38.26 -6.8053 -6.8114 -6.58 -4.8090 -4.8162 -4.57

4.9 48.07 -8.4894 -8.4971 -8.20 -6.0203 -6.0293 -5.75

5.9 57.88 -10.1499 -10.1592 -9.77 -7.2228 -7.2336 -6.95

6.9 67.69 -11.7872 -11.7981 -11.42 -8.4166 -8.4292 -8.16

7.9 77.50 -13.4019 -13.4145 -13.07 -9.6018 -9.6162 -9.32

Table 5.7 Static vertical deflection of physical bridge model in case 3 and case 4 (in mm)


Applied load

(kg) (N) Microstran SAP2000 Experiment Microstran SAP2000 Experiment

0 0 0 0 0 0 0 0

2.9 28.45 -3.2729 -3.2776 -3.12 -2.7695 -2.7743 -2.46

3.9 38.26 -4.3882 -4.3945 -4.26 -3.7164 -3.7228 -3.39

4.9 48.07 -5.4967 -5.5046 -5.29 -4.6591 -4.6672 -4.26

5.9 57.88 -6.5985 -6.6079 -6.33 -5.5977 -5.6074 -5.22

6.9 67.69 -7.6935 -7.7045 -7.42 -6.5322 -6.5434 -6.17

7.9 77.50 -8.7819 -8.7945 -8.48 -7.4625 -7.4754 -7.15

5.4.2 Variation in results and discussion

As shown in the above analyses, the experimental results confirm that the physical

bridge model has coupled vibration modes. However, lateral components have been

- 145 -

also found with the vertical vibration modes, though their spectra densities are very

small. Theoretically, the vertical modes always appear as pure vibration modes as the

bridge model is a symmetric structure. In the construction of the physical bridge

model, errors can be introduced in the tension forces. For example, the tension forces

in the two supporting cables are controlled by 4 turnbuckles with load cells. It is

difficult to adjust the turnbuckles to keep all the tension forces as exactly the same

value at each end of the two supporting cables. For the entire physical bridge model,

there are 12 turnbuckles, and it is very difficult to keep 12 tension forces at the 3

specified values. Therefore the bridge model can not be a real symmetric structure

and small lateral components could possibly appear in the vertical modes due to this

error. For coupled vibration modes, the ratios of spectra densities of the non-

dominant components to the dominant ones are higher than those caused by error

introduced in the construction of bridge model.

The differences in natural frequencies (Table 5.4 and Table 5.5) and vertical

deflections (Table 5.6 and Table 5.7) from experimental testing and computer

simulations could be a result of loss of tension forces in the cable system. When a

cable goes through a headstock and a cable clamp and then is connected to the load

cell as well as the turnbuckle, some friction exists between the cable and headstock

as well as cable and cable clamp even when the contacted faces are lubricated very

well. This friction makes the tension force of the cable in bridge model less than that

measured by the load cell. As a consequence, the cable sag is slightly larger than the

one in computer simulations, and this phenomenon has been observed during the

experimental testing. Therefore, the lateral stiffness decreases while the vertical

stiffness increases, and hence the natural frequencies of lateral modes go down but

those of vertical modes go up, and the vertical deflections are smaller than expected.

Another reason for the differences in the natural frequencies is the structural stiffness

which is used to calculate the natural frequencies. Cable supported bridges are

slender and non-linear structures, their structural stiffness depend significantly on the

tension forces in the cables and the deformed cable profiles (geometry). As shown in

the spectra of accelerations, the natural frequencies are affected by the dynamic

performance, and a vibration mode has different frequency values when the bridge

model vibrates in different ways. However, in the computer simulations, the natural

- 146 -

frequencies are calculated based on the structural stiffness when the bridge model is

static. Therefore, different natural frequencies can be obtained even without the loss

of tension forces.

As shown in Table 5.4 and Table 5.5, the natural frequencies of torsional modes from

experimental testing are greater than those from computer simulations. As mentioned

before, this could be caused by the modelling of bridge deck units. In computer

simulations, the deck units are modelled as beam elements which have line

distributed mass on the axial lines. As the distribution of mass in the width direction

has been neglected, it will introduce error to the moments of inertia of mass about an

axis parallel to the member axis. For lateral and vertical modes, this kind of

modelling would not cause error as the distributions of mass have no effect on these

modes. However, for the torsional modes, the natural frequencies depend on the

moment of inertia and the distributions of mass have significant effect on the

moment of inertia. When axial lines of the deck units are in the bridge longitudinal

direction, this error will have significant effect on the moment of inertia of mass

about the axis in bridge longitudinal direction and hence on the torsional modes.

Furthermore, in the physical bridge model, the mass of one bridge frame is 2.12 kg,

while the mass of 8 deck units between two adjacent bridge frames is about 2.67 kg.

It means that the deck units contribute more than half the structural mass and the

error of modelling in the moments of inertia would incur some difference between

the results from computer simulation and from experiment testing.

Despite the variations discussed above, the results for natural frequencies and

deflections under static loads obtained from experimental testing and computer

simulations compare reasonably well, across the 4 bridge model cases. This provides

adequate confidence in the accuracy of the computer model which will be used in the

further investigations.

5.5 Modification factor of moment of inertia of cable section properties

In SAP2000, the frame/beam element is used to model cable members and the

moment of inertia of the cable section properties should be modified to simulate the

flexible behaviour of cables. It is recommended by the SAP2000 manual [CSI 2004]

- 147 -

that small modification factors should be taken for the moment of inertia of cable

section properties as cables can carry small bending moments and torque in real

cable supported structures. However, there is no further information about what

kinds of factors should be adopted as the flexibility is quite different for different

types of cables and the materials. For example, the type of 7x19 stainless steel wires

is more flexible than the type of 7x7 wires, while type of 1x7 or 1x19 wires is much

rigider than the type of 7x7 wires.

To illustrate the effect of modification factors for moment of inertia of cable section

properties, Table 5.8 shows the natural frequencies of the physical bridge model in

case 4 when different modification factors have been chosen in the computer

simulation. It can be seen that when the modification factors for the moments of

inertia and torsional constant change from 0.005 to 0.5, the tension forces are almost

the same and the natural frequencies change very slightly. These results imply that in

the computer simulation, these cables carry only very small bending moments and

torque, and they behaviour as axial rods even if they are modelled as beam elements.

One reason for these results might be that the cable sections are so small compared

with the other structural members that they have very small contribution to the whole

structure.

When modelling the cable members, one cable member is divided into several

segments in SAP2000 to make the cable flexible. The default number is 10, and this

number has been modified as 20 in the above computer simulation.

In the next chapter, SAP2000 will be adopted for the extensive numerical analysis to

investigate the dynamic performance of slender suspension footbridges under

human-induced dynamic loads. When modelling the slender footbridge structure, the

number of segments for a cable member between two adjacent bridge frames is set to

be 20 and the modification factors for moments of inertia and torsional constant are

set to be 0.01 to make sure that the cables behaviour flexibly. Since all the deck units

have axial lines perpendicular to the bridge longitudinal direction, modelling the

deck units as beam elements will not incur significant error to the moment of inertia

of mass about an axis in bridge longitudinal direction and hence to the coupled

torsional-lateral vibration modes and their corresponding natural frequencies.

- 148 -

Table 5.8 Effect of modification factors of cable section properties

Model case Case 4

Method SAP2000 Microstran Experiment

Modification factors 0.005 0.01 0.05 0.1 0.5 -- --

Wint (N) 30.0 30.0 30.0 30.0 30.0 30.0 30.0 Internal force

Qint (N) 15.0 15.0 15.0 15.0 15.0 15.0 15.0

T1 (N) 2230.3 2230.3 2230.3 2230.3 2230.3 2230.4 2232.9

T2 (N) 874.4 874.3 874.4 874.4 874.4 874.9 871.6 Tension force

T3 (N) 1180.0 1180.0 1180.0 1180.0 1180.0 1178.2 1182.7

L1T1 3.1009 3.1011 3.1023 3.1036 3.1095 3.1138 2.924

L2T2 5.9800 5.9803 5.9822 5.9842 5.9939 6.1016 5.89

L3T3 8.5631 8.5635 8.5661 8.5689 8.5828 8.9007 8.602 Lateral

L4T4 10.7760 10.7770 10.7800 10.7830 10.8010 11.4280 --

T1L1 4.6280 4.6282 4.6300 4.6319 4.6416 4.4428 5.568

T2L2 8.0727 8.0733 8.0783 8.0835 8.1108 7.9465 9.811

T3L3 11.6890 11.6900 11.6980 11.7060 11.7500 11.6220 -- Torsional

T4L4 15.0890 15.0900 15.1010 15.1120 15.1740 15.5240 --

V1 3.5006 3.5008 3.5023 3.5039 3.5107 3.4842 3.77

V2 6.1411 6.1415 6.1441 6.1467 6.1589 6.1103 6.19

V3 9.0230 9.0234 9.0262 9.0290 9.0432 8.9979 9.841 Vertical

V4 11.6490 11.6490 11.6530 11.6570 11.6790 11.6190 --

- 149 -

Dynamic response of slender suspension footbridges under crowd walking dynamic loads

6.1 Introduction

It is known that when crossing a bridge which is vibrating at a frequency within the

range of walking rates, pedestrians trend to change their pacing rates to move in

harmony with the bridge vibration, and consequently the bridge structure vibrates at

or near a resonant frequency. This mechanism can lead to large amplitude

synchronous vibration which may cause serious vibration serviceability problems in

footbridges. Synchronization occurs more likely in the lateral direction for slender

footbridge structures with weak girder, as pedestrians are more sensitive to vibration

in this direction than that in vertical direction, and they try to change their pacing

rates to keep the body in balance. Investigation of Millennium Bridge in London

showed that lateral synchronous excitation can be caused by the combination of high

pedestrian density and the presence of lateral modes of vibration below 1.3 Hz,

independent of the bridge structural form [Dallard et al. 2001a].

Since a footbridge structure vibrates at or near a resonant natural frequency when

synchronization occurs, the dynamic response of footbridge structure can be

simulated by resonant vibration. In other words, one of the vibration modes will be

excited by the synchronised pedestrians if the corresponding natural frequency is

within the range of human activities such as walking or running, and the vibration

mode excited depends on the distribution of pedestrian load.

In order to study the dynamic characteristics of slender suspension footbridges under

synchronous excitation, non-linear time history analyses are carried out on the

resonant vibration induced by crowd walking pedestrians with different pacing rates.

For this purpose, the hollow section bridge model (HSB) is employed, as it is a light

bridge model with hollow section bridge frames and extruded aluminium deck units

and the maximum tension forces can be kept at a reasonable range for different

6

- 150 -

footbridge model covering a range of natural frequencies. In the numerical analysis,

the structural analysis software package SAP2000 (V9) is adopted to study the

vibration properties and dynamic response under walking dynamic loads, and Hilber-

Hughes-Taylor method is used for the non-linear direct-integration time history

analysis, and small time step (about 0.01 second, which is less than one fiftieth of the

period) is used to ensure the accuracy. It is assumed that the walking dynamic loads

are distributed uniformly on the bridge deck and different force-time functions are

employed based on the pacing rate. The lateral resonant vibration is the main

concern though other vibration modes will also be considered.

Since damping in real structures is very complex and structure dependant, it is not

possible to define unique value(s) for the damping as the damping ratio is different if

the measurement was conducted by different methods [Zivanovic et al. 2005].

Common value of damping ratio in Table 2.3 has been suggested for design guidance

[Bachmann et al. 1995]. In SAP2000, only the proportional damping has been used

for direct-integration time-history analysis. In this proportional damping model, the

damping matrix is calculated as a linear combination of the stiffness matrix and mass

matrix with two defined coefficients. In this research, these two coefficients are

selected according to the periods (or natural frequencies) and it is assumed that both

the first and second vibration modes in the same type (i.e. L1T1 and L2T2, or T1L1

and T2L2, or V1 and V2) have the same damping ratio.

6.2 Crowd walking dynamic loads

As mentioned before, synchronous excitation can be caused by the combination of

high density of pedestrians and low natural frequencies within the frequency range of

pacing rate. Fujino et al. [1993] reported that when synchronization occurred on the

T-bridge in Japan, almost 20 percent of pedestrians completely synchronized their

footfalls to the bridge vibration. However, there is no other information regarding the

degree of synchronization on other slender “lively” footbridges. Willford [2002] also

mentioned that the lateral dynamic load factor (DLF) of the first harmonic of walking

force can vary in the range of 0.03 to 0.1 (about 23 N to 70 N per person with the

weight of 700 N). It seems there are many uncertainties in the description of lateral

- 151 -

force induced by walking pedestrians and further research is required to model the

lateral force accurately.

To simulate the synchronous excitation induced by walking pedestrians, a model of

crowd walking dynamic loads is proposed to investigate the resonant vibration of

slender suspension footbridges under synchronous excitations in this conceptual

study. In this proposed model, the following assumptions are adopted:

a) A portion of groups or crowd of walking pedestrians participate fully in the

synchronization process and generate vertical and lateral dynamic forces, and

the remaining pedestrians generate only vertical static force on the bridge deck

as they walk with random pacing rates and phases.

b) The lateral dynamic force induced by a footfall has the same force-time

function as its vertical component, but the magnitude is only a small portion

(4%) of the vertical one. The force-time function of vertical dynamic force is

frequency-dependant and follows the functions provided by Wheeler [1982].

c) The pedestrian loads are uniformly distributed on the bridge deck and load

density is taken as 1.5 persons per square meter and the average weight per

person is assumed to be 700 N.

0.0

0.4

0.8

1.2

1.6

2.0

0.0 0.2 0.4 0.6 0.8

Time (s)

Dyn

amic

for

ce /

stat

ic w

eigh

t slow walk (1.67 Hz)brisk walk (2.37 Hz)normal walk (2.00 Hz)fast walk (3.13 Hz)

Figure 6.1 Vertical force-time functions of a footfall (after Wheeler [1982])

- 152 -

Figure 6.1 shows the typical vertical force-time functions [Wheeler 1982] from slow

walk to fast walk. It can be seen that the peak value and shape of a force function

vary according to pacing rate. However, it should be mentioned that in the reference,

the pacing rate of normal walk is greater than that of brisk walk, but its curve is

much flatter. In order to model the walking dynamic loads, the walking activities are

classified into four types according to their pacing rates and each type of activity

covers a range of frequency:

slow walk – smaller than 1.8Hz; normal walk – 1.8 Hz ~ 2.2Hz;

brisk walk – 2.2 Hz ~ 2.7 Hz; fast walk – greater than 2.7 Hz

Time

Dyn

amic

for

ce /

stat

ic w

eigh

t

0 T ncT n

F n [t ]

left foot right foot

Figure 6.2 Vertical force-time function of normal walk

The crowd walking dynamic loads can be modelled according to the pacing rate.

Taking the normal walk for example, if the vertical force-time function of one

footfall is defined as Fn[t], and the period and foot contact time are Tn and Tnc (Figure

6.2) respectively, then this function has the following feature:

nc

nc

nn Tt

Ttt

tFtF

≤≤><

=0

or 0

][

0][ (6.1)

The continuous vertical force function Fnv(t) (Figure 6.3) and lateral force function

Fnl(t) (Figure 6.4) can therefore be expressed according the pacing rate fp or load

period Tp (Tp=1/ fp).

- 153 -

∑∞

=

−=0

)]([)(k

pnnv kTtFtF η (6.2)

)]})12(([)]2([{)(0

pk

npnnl TktFkTtFtF +−−−=∑∞

=

ηη (6.3)

pn TT /=η or np ff /=η ( Hz 2.2 Hz 8.1 <≤ pf ) (6.4)

Where η is a time factor, fn and Tn are the pacing rate and period (fn=1/ Tn) shown in

Figure 6.2 for normal walk (i.e. fn =2.0 Hz).

0.0

0.5

1.0

1.5

2.0

2.5

0.0 0.5 1.0 1.5 2.0 2.5

Time (s)

Dyn

amic

for

ce /

stat

ic w

eigh

t left footright footvertical force

Figure 6.3 A typical continuous vertical force-time function

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

2.5

0.0 0.5 1.0 1.5 2.0 2.5

Time (s)

Dyn

amic

for

ce /

stat

ic w

eigh

t left footright footlateral force

Figure 6.4 A typical continuous lateral force-time function

- 154 -

In numerical analysis, the static load is modelled as ramp load in order to reduce the

fluctuation of dynamic response at the beginning of time history analysis. If the

degree of synchronization (portion of pedestrians involved in synchronization) is

assumed to be ϕ, then the walking loads for normal walk (1.8 Hz ≤ fp < 2.2 Hz)

including vertical dynamic force (VDF) qnv(t), lateral dynamic force (LDF) qnl(t) and

vertical static (ramped) force (VSF) qsv(t) can be modelled as:

)(1050)( tFtq nvnv ϕ= )(N/m2 (6.5a)

)(42)( tFtq nlnl ϕ= )(N/m2 (6.5b)

−−

= )1(1050

)10/()1(1050)(

ϕηϕ t

tqsv )(N/m

)(N/m2

2

η

η10

100

≥<<

t

t (6.5c)

In general, the synchronization is affected by the interaction between pedestrians and

vibrating structure, and the degree of synchronization should be a function of

dynamic response. However, the basis of synchronization cannot be established and

further research is required to establish the relationship of pedestrians and structural

dynamic response. In this conceptual research, the degree of synchronization ϕ is

assumed to be 0.2 for crowd walking pedestrians. For a small group of pedestrians,

all pedestrians can be expected to walk at the same pacing rate and phase, and the

degree of synchronization can be set to 1.0. As the numerical analysis considers

crowd loading, the degree of synchronization is set to be a constant value (0.2),

though it may change in real life situations of synchronous excitation due to the

interaction of human-structure, and depend on the dynamic response of the vibrating

bridge structure.

The crowd walking dynamic loads with other pacing rates can also be defined and

modelled similarly by following the same procedure.

6.3 Dynamic performance and resonant vibration under crowd

walking dynamic loads

For slender footbridges, the fundamental lateral natural frequencies are always low

and these low frequencies are easy to be excited by crowd of walking pedestrians.

- 155 -

For example, the lowest lateral frequency of Millennium Bridge in London is about

0.48 Hz and for M-bridge in Japan, it is about 0.27 Hz. The excited frequencies of

lateral vibration modes are 0.48 Hz, 0.8 Hz and 1.0 Hz for Millennium Bridge (by

crowd pedestrians) [Dallard et al. 2001c], 0.88 Hz and 1.02 Hz for M-bridge (by

groups of pedestrians or single person) [nakamura 2003]. It seems low frequency

vibration is more important to slender footbridge structures.

In order to illustrate the dynamic behaviour of slender footbridges with coupled

vibration modes under walking dynamic loads, the hollow section bridge models

(HSB) with the frequency of 0.75 Hz of first coupled lateral-torsional mode have

been chosen for the dynamic analysis. The bridge models are supposed to have span

length of 80 m, cable sags of 1.8 m and cable diameters of 120 mm (L=80 m;

F1=F2=F3=1.8 m; D1=D2=D3=120 mm). Three bridge models with different cable

configuration are considered. These bridge models are noted by their cable

configurations as mentioned before. The bridge model C123 (Figure 3.5) is a

footbridge model with all cable systems: top supporting cables, pre-tensioned bottom

and side cables; bridge model C120 (Figure 3.6) is a footbridge model with top

supporting cables and pre-tensioned bottom cables but without side cables and side

legs in the bridge frames; while bridge model C103 (Figure 3.7) is one with top

supporting cables and pre-tensioned side cables but without bottom cables and legs.

The vibration properties of the different bridge models are shown in Table 6.1. It can

be seen that for bridge models with different cable configurations, the fundamental

lateral natural frequency can be the same by tuning the pre-tension forces in the pre-

tensioned cables. For the bridge model C120, the lateral stiffness is provided by the

top and bottom cable in vertical plane, and the tension forces in the cables are the

highest to get the same frequency of the first coupled lateral-torsional mode L1T1.

For the model C123, all the tension forces reduce since the side cables provide

tension forces in lateral direction and improve the lateral stiffness. While for the

bridge model C103, the tension force in the top supporting cables can not be changed

by the side cables but the side cables provide most of the lateral stiffness. For this

bridge model, the frequencies of the first coupled lateral-torsional (L1T1), first

coupled torsional-lateral (T1L1) and first vertical modes are very close to each other

and this could lead to vibration problems to the bridge structure.

- 156 -

Table 6.1 Vibration properties of different Bridge models

Bridge parameter HSB; L=80 m; F1=F2=F3=1.8 m; D1=D2=D3=120 mm

Bridge model C120 C103 C123

Mass density M (kg/m2) 363.80 363.45 465.84

T1 (N) 6987428 3265192 5536132

T2 (N) 3722268 -- 1356765 Cable tension

T3 (N) -- 1439100 1110712

L1T1 0.7500 0.7500 0.7500

L2T2 1.4585 0.9354 1.0980

L3T3 2.1634 1.4012 1.5602

L4T4 2.8656 1.8382 2.0340

L5T5 3.5654 2.2926 2.5246


L6T6 4.2572 2.7388 3.0111

T1L1 1.1949 0.7658 0.8982

T2L2 1.8718 1.2535 1.4158

T3L3 2.7238 1.8053 2.0593 Coupled torsional-lateral

T4L4 3.5793 2.3738 2.7023

V1 1.0943 0.7531 0.9062

V2 1.5151 1.0100 1.1633

V3 2.2866 1.5255 1.7597

V4 3.0239 2.0162 2.3203

Vertical

V5 3.7785 2.5208 2.8998

6.3.1 Resonant vibration at the frequencies of first vibration modes

In general, the first coupled lateral-torsional mode (L1T1) and first vertical mode

(V1) are one half-wave symmetric modes and they are easier to be excited by crowd

walking dynamic loads than other higher vibration modes when the entire bridge

deck is full of walking pedestrians. While the first coupled torsional-lateral mode

(T1L1) or pure torsional mode (T1) is not easy to be activated as the crowd loads are

supposed to be distributed uniformly and symmetrically on the deck and the load

effect on this kind of mode theoretically is almost zero, although it is also one half-

wave vibration mode.

In order to compare the dynamic responses of different bridge models, they are

assumed to be subjected to the crowd walking dynamic loads distributed on the entire

bridge deck with the pacing rate equal to their own natural frequency, and the

damping ratio for all bridge models and all types of vibration modes is assumed to be

0.01 (ζ=0.01). The dynamic responses (deflections or accelerations) are picked up

- 157 -

from same point at the intersections of the legs and cross member at the middle

bridge frame (Figure 4.2).

6.3.1.1 Bridge model C120

First coupled lateral-torsional vibration mode (L1T1, fp=1.50 Hz)

When pedestrians walk at pacing rate of 1.5 Hz (slow walk), the first coupled lateral-

torsional mode (L1T1) is excited at the frequency of 0.75 Hz (half of pacing rate).

Figure 6.5 and Figure 6.6 show the resonant lateral and vertical dynamic deflections

respectively. From these figures, it can be seen that the footbridge structure resonates

in the lateral direction at the corresponding natural frequency (0.75 Hz); while in the

vertical direction, the bridge structure vibrates at the load frequency (1.5 Hz) with

normal dynamic response, as resonant vibration does not occur in this direction. The

amplitude of the lateral deflection increases to the maximum value then fluctuates

and finally becomes constant, i.e., the vibration trends to be steady after several

fluctuations. The vertical vibration amplitude is much smaller than the lateral one,

though the vibration also trends to be steady after several fluctuations. The vertical

vibration is actually contributed by three parts: vertical static force, vertical dynamic

force and the lateral sway of bridge frame under lateral dynamic force, with the static

force being most dominant. It is evident that the footbridge does not resonate in the

vertical direction when subjected to the vertical dynamic force and the maximum

vertical dynamic deflection is mainly produced by the resonant lateral sway. The

details of dynamic deflections in steady vibration can be seen in the Figure 6.7.

Numerical results show that the curves of the lateral and vertical accelerations

(Figure 6.8 and Figure 6.9) have the same dynamic features as their corresponding

deflections.

The statistics of dynamic deflections and accelerations during the full and steady

vibration are shown in Table 6.2 and Table 6.3. Here and in the following tables, the

maximum and minimum steady deflections and accelerations are the maximum and

minimum peak values of the steady vibrations within a period of fifteen seconds after

the vibrations become steady. The dynamic amplitudes and mean value are

- 158 -

calculated based on their maximum and minimum values. Taking the steady

deflection for example, the amplitude and mean value are obtained by:

2/)( minmax stdstdustd UUA −= (6.6)

2/)( minmax stdstdustd UUM += (6.7)

The dynamic amplification factor (DAF) is calculated as:

staticustdustd UADAF /= (6.8)

-0.06

-0.04

-0.02

0.00

0.02

0.04

0.06

0 25 50 75 100 125 150 175

Time (s)

Lat

eral

def

lect

ion

(m)

Figure 6.5 Bridge model C120: lateral dynamic deflection at pacing rate of 1.5 Hz (ζ=0.01)

-0.07

-0.06

-0.05

-0.04

-0.03

-0.02

-0.01

0.00

0 25 50 75 100 125 150 175

Time (s)

Ver

tica

l def

lect

ion

(m)

Figure 6.6 Bridge model C120: vertical dynamic deflection at pacing rate of 1.5 Hz (ζ=0.01)

- 159 -

Here the lateral static deflection is the one caused by the quasi-static lateral force

corresponding to the lateral dynamic force (LDF) (Equation 6.5(b)) with the load

density equal to the amplitude of dynamic force (i.e. 42ϕ). The vertical static

deflection is the one generated by the quasi-static vertical force having the load

density of 210ϕ as in the vertical dynamic force (VDF) (Equation 6.5(a)). The

negative sign of vertical deflection in the figures and tables indicates downward

deflection.

The acceleration factor AFastd of steady vibration is obtained by

gAAF astdastd /= (6.9)

here g is the gravity acceleration (9.81 m/s2). A similar procedure is applied to the

acceleration of full vibration.

-0.08

-0.06

-0.04

-0.02

0.00

0.02

0.04

0.06

0.08

150 151 152 153 154 155 156 157 158 159 160

Time (s)

Def

lect

ion

(m)

lateral deflection

vertical deflection

Figure 6.7 Bridge model C120: steady dynamic deflections in details at pacing rate of 1.5 Hz (ζ=0.01)

It should be noted here that the steady vibration is more meaningful in practice and

analysis, as the effect of initial conditions vanishes. Although the maximum and

minimum peak values as well as the amplitudes and mean values of the dynamic

responses during the entire vibration have been listed and assessed by the equations

similar to Equation (6.6), (6.7) and (6.8) which are applied to the steady vibration

analysis, these dynamic quantities are less meaningful and will not be discussed, as

- 160 -

they are affected significantly by the initial conditions and some other factors such as

assessment method. For example, when the bridge model C120 vibrates in the mode

L1T1 under crowd walking dynamic loads, the maximum and minimum vertical

deflections during the full vibration is 0.00025 m (upward) and -0.06980 m

(downward) from Table 6.2 and Figure 6.6, and these values results in a fake

amplitude and mean value of 0.03502 m and -0.03477 m respectively for the full

vibration.

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

0 25 50 75 100 125 150 175

Time (s)

Lat

eral

acc

eler

atio

n (m

/s2 )

Figure 6.8 Bridge model C120: lateral dynamic acceleration at pacing rate of 1.5 Hz (ζ=0.01)

-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

0 25 50 75 100 125 150 175Time (s)

Ver

tica

l acc

eler

atio

n (m

/s2 )

Figure 6.9 Bridge model C120: vertical dynamic acceleration

at pacing rate of 1.5 Hz (ζ=0.01)

- 161 -

Table 6.2 Bridge model C120: Dynamic deflections of the first vibration modes excited by pedestrians

Bridge parameter HSB: L=80 m; F1=F2=1.8 m; D1=D2=120 mm


Vibration mode excited L1T1 V1 T1L1

Pacing rate fp (Hz) 1.5000 1.0943 1.1949

Damping ratio z 0.010 0.010 0.010

Displacement U Ul Uv Ul Uv Ul Uv

Static displacement Ustatic (m) 0.00126 -0.01452 0.00126 -0.01452 0.00126 -0.01452

Umax (m) 0.05510 0.00025 0.00724 0.16211 0.00878 0.00689

Umin (m) -0.05221 -0.06980 -0.00393 -0.28027 -0.00619 -0.08667

Aumax (m) 0.05365 0.03502 0.00558 0.22119 0.00749 0.04678

Mumax (m) 0.00144 -0.03477 0.00166 -0.05908 0.00129 -0.03989

Full vibration

DAFu 42.4 2.4 4.4 15.2 5.9 3.2

Ustdmax (m) 0.04145 -0.05057 0.00555 0.16211 0.00614 -0.03821

Ustdmin (m) -0.03853 -0.06728 -0.00207 -0.28027 -0.00378 -0.07890

Austd (m) 0.03999 0.00836 0.00381 0.22119 0.00496 0.02035

Mustd (m) 0.00146 -0.05892 0.00174 -0.05908 0.00118 -0.05856

Steady vibration

DAFustd 31.6 0.6 3.0 15.2 3.9 1.4

Table 6.3 Bridge model C120: Dynamic accelerations of the first vibration modes


Bridge models C120 C120 C120


Pacing rate fp (Hz) 1.5000 1.0943 1.1949

Damping ratio z 0.010 0.010 0.010

Acceleration A Al Av Al Av Al Av

Amax (m/s2) 1.18794 0.70175 0.11733 10.74397 0.14846 1.97922

Amin (m/s2) -1.18117 -0.78584 -0.11481 -10.29827 -0.15090 -1.80830

Aamax (m/s2) 1.18456 0.74380 0.11607 10.52112 0.14968 1.89376

Mamax (m) 0.00338 -0.04205 0.00126 0.22285 -0.00122 0.08546

Full vibration

AFa (g) 0.121 0.076 0.012 1.072 0.015 0.193

Astdmax (m/s2) 0.90320 0.60751 0.11261 10.74397 0.09732 1.27703

Astdmin (m/s2) -0.90389 -0.56496 -0.10494 -10.29827 -0.09614 -1.15111

Aastd (m/s2) 0.90355 0.58624 0.10878 10.52112 0.09673 1.21407

Mastd (m) -0.00034 0.02128 0.00384 0.22285 0.00059 0.06296

Steady vibration

AFastd (g) 0.092 0.060 0.011 1.072 0.010 0.124

- 162 -

-0.004

-0.002

0.000

0.002

0.004

0.006

0.008

0 25 50 75 100 125 150 175 200Time (s)

Lat

eral

def

lect

ion

(m)

Figure 6.10 Bridge model C120: dynamic lateral deflection at pacing rate of 1.0943 Hz (ζ=0.01)

-0.30

-0.25

-0.20

-0.15

-0.10

-0.05

0.00

0.05

0.10

0.15

0.20

0 25 50 75 100 125 150 175 200

Time (s)

Ver

tica

l def

lect

ion

(m)

Figure 6.11 Bridge model C120: dynamic vertical deflection at pacing rate of 1.0943 Hz (ζ=0.01)

- 163 -

-0.15

-0.10

-0.05

0.00

0.05

0.10

0.15

0 25 50 75 100 125 150 175 200Time (s)

Lat

eral

acc

eler

atio

n (m

/s2 )

Figure 6.12 Bridge model C120: dynamic lateral acceleration at pacing rate of 1.0943 Hz (ζ=0.01)

-12.0

-8.0

-4.0

0.0

4.0

8.0

12.0

0 25 50 75 100 125 150 175 200Time (s)

Ver

tica

l acc

eler

atio

n (m

/s2 )

Figure 6.13 Bridge model C120: dynamic vertical acceleration at pacing rate of 1.0943 Hz (ζ=0.01)

- 164 -

The first vertical vibration mode (V1, fp=1.0943 Hz)

The first vertical vibration mode (V1) can be excited if the crowd of pedestrians walk

at the pacing rate of 1.0943 Hz. This frequency, as well as the one corresponding to

the first coupled torsional-lateral vibration mode, is much lower than the minimum

pacing rate (1.6 Hz) defined by Bachmann [2002], and it is believed that such low

frequencies would not be excited by normal walking pedestrians. However,

synchronization at low frequency can occur if the load density is high and

pedestrians are forced to walk with the stream of crowd. It is supposed that all the

low natural frequency vibration modes can be excited by crowd of pedestrians in this

conceptual study.

Figure 6.10 and Figure 6.11 show the dynamic lateral and vertical deflections

respectively when pedestrians walk at the pacing rate of 1.0943 Hz, while Figure

6.12 and Figure 6.13 show the corresponding lateral and vertical accelerations. It can

be seen that the lateral deflection and acceleration are quite small, and they are

mainly induced by the lateral force. In the vertical direction, the bridge structure

resonates under the vertical dynamic force, and the amplitude of deflection as well as

acceleration increase gradually to steady values without fluctuation. As the

footbridge structure is light with mass density of about 363.8 kg/m2, the amplitudes

of vertical deflection and acceleration can reach 0.221 m and 1.072g respectively

when 20% pedestrians participate in the synchronization.

First coupled torsional-lateral vibration mode (T1L1, fp=1.1949 Hz)

When pedestrian walk at the pacing rate of 1.1949 Hz, the first coupled torsional-

lateral vibration mode is supposed to be excited. However, it seems this does not

happen under the crowd walking dynamic loads. Figure 6.14 and Figure 6.15 show

the lateral and vertical dynamic deflections. It can be seen that the deflections are

relatively smaller compared to the resonant vibration at other natural frequencies of

first coupled lateral-torsional mode L1T1 and vertical mode V1. The dynamic

amplification factors in lateral and vertical directions are about 3.9 and 1.4

respectively (Table 6.2). It is found that lateral deflection is mainly produced by the

lateral dynamic force and in the vertical direction the whole bridge deck vibrates up

- 165 -

and down without torsion. This indicates that the coupled torsional-lateral mode has

not been excited but the vertical mode, because the coupled torsional-lateral mode is

an asymmetric vibration mode about the central line of the bridge structure and the

vertical dynamic force induced by pedestrians is distributed uniformly and

symmetrically on the footbridge deck.

-0.008

-0.006

-0.004

-0.002

0.000

0.002

0.004

0.006

0.008

0.010

0 25 50 75 100 125 150 175

Time (s)

Lat

eral

def

lect

ion

(m)

Figure 6.14 Bridge model C120: dynamic lateral deflection at pacing rate of 1.1949 Hz (ζ=0.01)

-0.10

-0.08

-0.06

-0.04

-0.02

0.00

0.02

0 25 50 75 100 125 150 175Time (s)

Ver

tica

l def

lect

ion

(m)

Figure 6.15 Bridge model C120: dynamic vertical deflection at pacing rate of 1.1949 Hz (ζ=0.01)

- 166 -

It is noticed that if pedestrians walk at the pacing rate of 2.3898 Hz (1.1949 Hz for

the lateral dynamic force), the dynamic responses are much smaller than those at the

pacing rate of 1.1949 Hz. This indicates that the coupled torsional-lateral mode is

possibly excited by the vertical forces, though this mode has lateral deflection

coupling with dominant torsional mode.


For the bridge model C123, the lateral stiffness has been improved by the side pre-

tensioned cables. Since the side cables can provide tension force and enhance the

structural stiffness, the tension forces in the top and bottom cables can be reduced to

get the same natural frequency of the coupled mode L1T1. Therefore the stiffness in

the vertical direction will be smaller than that of the bridge model C120.

Table 6.4 Bridge model C123: Dynamic deflections of the first vibration modes

Bridge parameter HSB: L=80 m; F1=F2=F3=1.8 m; D1=D2=D3=120 mm



Pacing rate fp (Hz) 1.5000 0.90616 0.8982

Damping ratio z 0.010 0.010 0.010



Umax (m) 0.02807 0.00001 0.00420 0.19189 0.00410 0.13807

Umin (m) -0.02580 -0.07900 -0.00184 -0.32084 -0.00182 -0.26872

Aumax (m) 0.02694 0.03951 0.00302 0.25637 0.00296 0.20339

Mumax (m) 0.00114 -0.03950 0.00118 -0.06447 0.00114 -0.06532

Full vibration

DAFu 28.3 2.4 3.2 15.6 3.1 12.4

Ustdmax (m) 0.02014 -0.05819 0.00337 0.19189 0.00315 0.12733

Ustdmin (m) -0.01800 -0.07482 -0.00125 -0.32084 -0.00149 -0.25871

Austd (m) 0.01907 0.00832 0.00231 0.25637 0.00232 0.19302

Mustd (m) 0.00107 -0.06651 0.00106 -0.06447 0.00083 -0.06569

Steady vibration

DAFustd 20.1 0.5 2.4 15.6 2.4 11.8

It is found that the dynamic responses of all the first vibration modes have the same

dynamic features as those of the bridge model C120. Table 6.4 and Table 6.5 show

the statistics of dynamic deflections and accelerations respectively. From these

tables, it can be seen that compared with the bridge model C120, the model C123 has

- 167 -

smaller static lateral deflection and larger static vertical deflection. When the first

lateral-torsional mode (L1T1) is excited, the steady amplitude and dynamic

amplification factor of the lateral deflection are much smaller at 0.01907 m and 20.1

(compared with 0.03999 m and 31.6). In the resonant vibration of first vertical mode

(V1) at frequency of 0.90616 Hz, the steady amplitude the vertical deflection is

0.25637 m but the DAF just changes slightly. However, the steady amplitude of

vertical acceleration is 0.854g, much smaller than 1.072g in the bridge model C120.

Since the frequency of T1L1 is close to that of the V1, larger vertical deflection is

induced when pedestrians walk at the pacing rate same as the natural frequency of

the mode T1L1.

Table 6.5 Bridge model C123: Dynamic accelerations of the first vibration modes


Bridge models C123 C123 C123


Pacing rate fp (Hz) 1.5000 0.90616 0.8982

Damping ratio x 0.010 0.010 0.010

Acceleration A Al Av Al Av Al Av

Gravity acceleration g (m/s2) 9.81 9.81 9.81 9.81 9.81 9.81

Amax (m/s2) 0.59518 0.62471 0.06354 8.35158 0.06134 6.62635

Amin (m/s2) -0.59535 -0.82923 -0.06473 -8.40571 -0.05950 -6.49296

Aamax (m/s2) 0.59527 0.72697 0.06414 8.37865 0.06042 6.55966

Mamax (m) -0.00009 -0.10226 -0.00059 -0.02706 0.00092 0.06670

Full vibration

AFa (g) 0.061 0.074 0.007 0.854 0.006 0.669

Astdmax (m/s2) 0.43613 0.44666 0.04045 8.35158 0.05095 6.30387

Astdmin (m/s2) -0.43195 -0.65788 -0.05206 -8.40571 -0.04793 -6.11550

Aastd (m/s2) 0.43404 0.55227 0.04626 8.37865 0.04944 6.20969

Mastd (m) 0.00209 -0.10561 -0.00581 -0.02706 0.00151 0.09419

Steady vibration

AFastd (g) 0.044 0.056 0.005 0.854 0.005 0.633


When the footbridge has only top and side pre-tensioned cables, the tension force in

the supporting cables can not be “tuned” and depends mainly on the cable sag and

structural weight. The natural frequencies and vibration modes can be only

“adjusted” by introducing different pre-tensions into the side cables. In this situation,

the distribution of natural frequencies may be not reasonable. For example, when the

frequency of first coupled lateral-torsional mode has been “adjusted” to be 0.75 Hz,

- 168 -

the frequencies corresponding to the first vertical and first coupled torsional-lateral

modes are very close to 0.75 Hz (Table 6.1). This frequency distribution can cause

serious vibration problems and make the vibration unstable.

-0.20

-0.15

-0.10

-0.05

0.00

0.05

0.10

0.15

0.20

0 100 200 300 400 500 600

Time (s)

Lat

eral

def

lect

ion

(m)

Figure 6.16 Bridge model C103: dynamic lateral deflection under crowd walking dynamic loads at pacing rate of 1.5 Hz (ζ=0.01)

-0.18

-0.16

-0.14

-0.12

-0.10

-0.08

-0.06

-0.04

-0.02

0.00

0 100 200 300 400 500 600

Time (s)

Ver

tica

l def

lect

ion

(m)

Figure 6.17 Bridge model C103: dynamic vertical deflection under crowd walking dynamic loads at pacing rate of 1.5 Hz (ζ=0.01)

- 169 -

-0.06

-0.04

-0.02

0.00

0.02

0.04

0.06

0 50 100 150 200 250 300

Time (s)

Lat

eral

def

lect

ion

(m)


-0.30

-0.25

-0.20

-0.15

-0.10

-0.05

0.00

0.05

0.10

0 50 100 150 200 250 300

Time (s)

Ver

tica

l def

lect

ion

(m)


- 170 -

-0.15

-0.10

-0.05

0.00

0.05

0.10

0.15

0 50 100 150 200 250 300 350 400 450 500

Time (s)

Lat

eral

def

lect

ion

(m)


-0.20

-0.15

-0.10

-0.05

0.00

0.05

0 50 100 150 200 250 300 350 400 450 500

Time (s)

Ver

tica

l def

lect

ion

(m)


- 171 -

-0.06

-0.04

-0.02

0.00

0.02

0.04

0.06

0 50 100 150 200 250 300Time (s)

Lat

eral

def

lect

ion

(m)


-0.14

-0.12

-0.10

-0.08

-0.06

-0.04

-0.02

0.00

0 50 100 150 200 250 300Time (s)

Ver

tica

l def

lect

ion

(m)


Figure 6.16 and Figure 6.17 show the dynamic lateral and vertical deflections

respectively when pedestrians walk at the pacing rate of 1.5 Hz (natural frequency of

- 172 -

the mode L1T1) and the bridge structure has damping ratio of 0.01. Figure 6.18 and

Figure 6.19 show the dynamic lateral and vertical deflections respectively when

pedestrians walk at the pacing rate of 0.7531 Hz (natural frequency of the mode V1).

Figure 6.20 and Figure 6.21 show the resonant dynamic responses when pedestrians

walk at the pacing rate of 0.7658 Hz, the frequency of first coupled torsional-lateral

mode. From these figures, it can be seen that for footbridges with the configuration

C103 and low damping ratio, the resonant dynamic performance under crowd

walking dynamic loads are very complicated and unstable. However, this

performance can be improved by increasing the damping ratio. Figure 6.22 and

Figure 6.23 show the dynamic lateral and vertical deflections at the pacing rate of 1.5

Hz when the damping ration increases to 0.02. It can be seen that when the damping

increases, the dynamic behaviour is quite different and the vibration amplitudes

reduce dramatically.

Since the dynamic performance of the footbridge model C103 is much complex,

further detailed research is required to reveal their dynamic characteristics and

discussion will not be made again on such kind of footbridge model in this thesis.

6.3.2 Tension forces in resonant vibrations with first modes

Before a suspension footbridge structure is subjected applied loads, initial tension

forces exist in the cable systems due to the gravity of the structure and the pre-

tensions in pre-tensioned cables. It is these initial tension forces which contribute

most of the stiffness to support external loads. When the bridge structure is subjected

to applied loads, the tension forces will change based on the cable deformations and

structural stiffness. In general, the tension forces in the top and bottom cables are

affected significantly by the vertical loads while those in the side cables are

influenced significantly by the lateral forces. However, the effect of lateral force is

often not significant, even in resonant lateral vibration when the footbridge is

subjected crowd walking dynamic loads. This is because the lateral dynamic force is

very small (only about 4% of the vertical dynamic force) and the stiffness in this

direction is much weaker than that in the vertical direction. On the other hand, the

resonant vertical vibration will have significant effect on the tension forces of top

supporting cables and bottom pre-tensioned cables, and influence not only the

- 173 -

dynamic amplitudes but also the mean values. This can be seen from the Figure 6.24

and Figure 6.25 as well as Table 6.6 and Table 6.7.

0.0E+00

1.0E+06

2.0E+06

3.0E+06

4.0E+06

5.0E+06

6.0E+06

0 25 50 75 100 125

Time (s)

Ten

sion

for

ce (

N)

top supporting cable

bottom pre-tensioned cable

side pre-tensioned cable

Figure 6.24 Tension forces of bridge model C123 in resonant vibration with the mode L1T1 at pacing rate of 1.5 Hz (ζ=0.01)

0.0E+00

1.0E+06

2.0E+06

3.0E+06

4.0E+06

5.0E+06

6.0E+06

7.0E+06

0 25 50 75 100 125Time (s)

Ten

sion

for

ce (N

)




Figure 6.25 Tension forces of bridge model C123 in resonant vibration with the mode V1 at pacing rate of 0.9046 Hz (ζ=0.01)

- 174 -

0.0E+00

1.0E+06

2.0E+06

3.0E+06

4.0E+06

5.0E+06

6.0E+06

7.0E+06

100 101 102 103 104 105 106 107 108 109 110Time (s)

Ten

sion

forc

e (N

)




Figure 6.26 Detail of tension forces in bridge model C123 in resonant vibration with the mode V1 at pacing rate of 0.9046 Hz (ζ=0.01)

Table 6.6 Tension forces in bridge model C120

Bridge Model HSB: L=80 m; F1=F2=1.8 m; D1=D2=120 mm

Model case C120 C120

Vibration mode excited L1T1 V1

Pacing rate fp (Hz) 1.5000 1.0943 Damping ratio z 0.010 0.010

Tension force Ttf T1 T2 T1 T2

Initial tension force 6987428 3722268 6987428 3722268

LDF 6987888 3722069 6987888 3722069

VDF 7037776 3672619 7037776 3672619

VSF 7190781 3697047 7190781 3697047 Quasi-static force

VDSF 7242559 3477968 7242559 3477968

Tstdmax (N) 7219029 3546875 7926829 4220556

Tstdmin (N) 7167993 3504015 6524666 2907937

ATstd (N) 25518 21430 701081 656310 Steady vibration

MTstd (N) 7193511 3525445 7225748 3564246

Figure 6.24 shows the tension forces of the bridge model C123 when the bridge

structure resonates under crowd walking dynamic loads at the pacing rate of 1.5 Hz

with the coupled lateral-torsional mode L1T1. It can be seen that the tension forces in

the top supporting cable and bottom pre-tensioned cables vary slightly and the

tension force of side pre-tensioned cables has small dynamic amplitude. Figure 6.25

shows the tension forces of bridge model C123 when the first vertical mode V1 has

- 175 -

been excited by crowd of pedestrians walking at pacing rate of 0.9046 Hz. It can be

seen that the tension forces in the top and bottom cables fluctuate with large

amplitudes while the amplitude of tension force in the side cable is much smaller

compared with the others. The details of variation in tension forces can be seen in

Figure 6.26. For the bridge model C120, similar results are obtained. The tension

forces in the top and bottom cables vary with very small amplitudes in the lateral

resonant vibration but with large amplitudes in vertical resonant vibration.

Table 6.7 Tension forces in bridge model C123


Bridge model C123 C123

Vibration mode excited L1T1 V1

Pacing rate fp (Hz) 1.5000 0.9062

Damping ratio z 0.010 0.010

Tension force T1 T2 T3 T1 T2 T3

Initial tension force 5536132 1356765 1110712 5536132 1356765 1110712

LDF 5536179 1355926 1114229 5536179 1355926 1114229

VDF 5592927 1300434 1110981 5592927 1300434 1110981

VSF 5766346 1135939 1114869 5766346 1135939 1114869 Quasi-static force

VDSF 5825122 1081867 1117189 5825122 1081867 1117189

Tstdmax (m) 5783793 1154066 1182900 6614646 1929134 1206983

Tstdmin (m) 5752032 1112858 1056117 5017380 456341 1107543

ATstd (m) 15881 20604 63391 798633 736397 49720 Steady vibration

MTstd (m) 5767912 1133462 1119508 5816013 1192738 1157263

Table 6.6 and Table 6.7 show the tension forces in steady resonant lateral and

vertical vibrations of bridge models C120 and C123 respectively under different

quasi-static forces and crowd walking dynamic loads. It can be seen that the tension

forces change slightly under quasi-static lateral dynamic force (LDF), as the load

density in the lateral direction is very small. The vertical force has significant effect

on the tension forces of the top and bottom cables for both bridge models C120 and

C123, but slight effect on the tension force of the side cables (bridge model C123).

In steady resonant vibrations, the mean values of the tension forces in top and bottom

cables are between the tension forces under quasi-static vertical static force (VSF)

and quasi-static total vertical force VDSF (VSF and VDF), as the vertical dynamic

force has some component of static force. While the mean value of tension force of

side cables in the bridge model C123 is greater than the tension force under quasi-

- 176 -

static total vertical force VDSF. This is because lateral dynamic force has slight

effect on dynamic response due to the non-linear geometry. These tables also

illustrate that when the footbridges resonate in the lateral direction, the amplitudes of

tension forces of the top and bottom cables in lateral resonant vibrations are much

smaller than those in vertical resonant vibrations. When the bridge model C123

resonates in the vertical direction, the amplitude of the tension force in the side

cables is much smaller than those in the top and bottom cables. All these results

indicate that side pre-tensioned cables are more slender and flexible in vertical

direction while the top and bottom cables are slender in lateral direction.

6.3.3 Resonant vibration at other modes

Other higher vibration modes can also be excited by crowd walking dynamic loads.

However these higher modes are mainly coupled lateral-torsional modes. When the

bridge deck is full of pedestrians, the vertical forces can remain the same, but the

phase of lateral force at different locations can be changed easily by the pedestrians

when a different mode is excited. Though the phase of lateral dynamic force can be

changed at different locations, the coupled torsional-lateral modes are not easy to be

excited by the crowd walking dynamic loads, as torsional modes are asymmetric

about the centre line and are mainly excited by the vertical forces. The higher vertical

vibration modes, however can be easily excited by group loads distributed differently

according to their mode shapes.

Table 6.8 and Table 6.9 show the statistics of resonant deflections at the second

coupled lateral-torsional mode (L2T2) and second vertical mode (V2). Here the

dynamic response are picked up from the intersection point of the bridge frame at the

quarter span length, as the maximum dynamic deflections of the second asymmetric

vibration modes occur at this location. For the bridge model C120, it is found that

when crowd of pedestrians walk at the pacing rate of 1.5151 Hz (the natural

frequency of second vertical mode) on the entire bridge deck, the second vertical

(V2) mode has not been excited but the first coupled lateral-torsional mode (L1T1)

has. This is because the pacing rate in close to the natural frequency (1.5 Hz) of the

mode L1T1 and the mode V2 is asymmetric in the longitudinal direction. However,

both the modes V2 and L1T1 can be excited if the crowd walking dynamic loads are

- 177 -

distributed only on the half span. In the footbridge model C123, the lateral and

vertical deflections are relatively quite small when bridge deck is full of the crowd

walking dynamic loads.

Table 6.8 Bridge model C120: resonant deflections of higher vibration modes



Load distribution Full span Full span Half span

Vibration mode excited L2T2 V2 V2

Pacing rate fp (Hz) 1.4585 1.5151 1.5151

Damping ratio z 0.010 0.010 0.010



Umax (m) 0.01289 0.00000 0.02761 0.00000 0.01488 0.04869

Umin (m) -0.01223 -0.05055 -0.02548 -0.04919 -0.01336 -0.12358

Aumax (m) 0.01256 0.02527 0.02655 0.02460 0.01412 0.08614

Mumax (m) 0.00033 -0.02528 0.00107 -0.02460 0.00076 -0.03744

Entire vibration

DAFu 39.3 2.3 27.9 2.3 22.2 9.3

Ustdmax (m) 0.00825 -0.04115 0.01973 -0.03988 0.01124 0.04869

Ustdmin (m) -0.00758 -0.04806 -0.01752 -0.04736 -0.00977 -0.12358

Austd (m) 0.00791 0.00346 0.01862 0.00374 0.01050 0.08614

Mustd (m) 0.00034 -0.04460 0.00110 -0.04362 0.00074 -0.03744

Steady vibration

DAFustd 24.7 0.3 19.6 0.3 16.5 9.3

Table 6.9 Resonant deflections of higher vibration modes (C123)



Load distribution Full span Full span Half span

Vibration mode excited L2T2 V2 V2

Pacing rate fp (Hz) 1.0980 1.1633 1.1633

Damping ratio z 0.010 0.010 0.010



Umax (m) 0.00737 0.00041 0.00418 0.00004 0.00281 0.07074

Umin (m) -0.00642 -0.05676 -0.00272 -0.05432 -0.00152 -0.15973

Aumax (m) 0.00689 0.02858 0.00345 0.02718 0.00216 0.11524

Mumax (m) 0.00047 -0.02818 0.00073 -0.02714 0.00065 -0.04449

Entire vibration

DAFu 17.5 1.2 4.9 1.1 2.0 5.1

Ustdmax (m) 0.00418 -0.04313 0.00313 -0.04475 0.00210 0.06792

Ustdmin (m) -0.00331 -0.05339 -0.00153 -0.05270 -0.00109 -0.15691

Austd (m) 0.00375 0.00513 0.00233 0.00398 0.00160 0.11241

Mustd (m) 0.00044 -0.04826 0.00080 -0.04872 0.00051 -0.04449

Steady vibration

DAFustd 9.5 0.2 3.3 0.2 1.5 5.0

- 178 -

From these tables, it can be seen that the amplitudes of resonant lateral and vertical

vibrations of higher frequency modes are much smaller than those of fundamental

modes due to the load effect.

6.3.4 Dynamic performance at different pacing rates

It is known that lateral synchronous excitation is often caused by walking pedestrians

if lateral frequency of footbridge structure is less than 1.3 Hz and there are enough

pedestrians crossing the footbridge. Under such dynamic loads, slender footbridges

always resonate and suffer excessive lateral vibration. It should be pointed out that

this kind synchronous excitation includes synchronization between walking people

and synchronization between the vibrating footbridge structure and pedestrians. On

the other hand, if pedestrians walk at pacing rates different to the natural frequencies,

it is believed the synchronization between bridge and pedestrians will not appear as

the this type of synchronization depends on large dynamic responses such as

deflection, velocity or acceleration. However, the synchronization between walking

people is always expected when people are walking in a crowd or groups.

To illustrate the dynamic performance of the slender pre-tensioned suspension

footbridge structures under walking dynamic loads with different pacing rate, it is

assumed that the loading pattern is the same as in resonant vibration but different

pacing rates are applied to the footbridge structures. In other words, it is supposed

that 20% pedestrians walk at the same pacing rate which can vary from 1.0 Hz to 3.5

Hz. In the numerical analysis, the damping ratio is assumed to be 0.01.

The typical dynamic lateral and vertical deflections of bridge model C120 are shown

in Figure 6.27 and Figure 6.28 respectively when 20% pedestrians are assumed to

walk across the footbridge at the pacing rate of 2.0 Hz. It can be seen that the

dynamic amplitudes are quite small. The lateral vibration reaches its maximum and

minimum peak values at the beginning and then gradually becomes steady, while the

vertical deflection increases rapidly under the ramped static vertical force and then

gradually trends to be steady vibration. Since the lateral vibration is small and

couldn’t produce enough vertical deflection due to the lateral sway to affect the

vertical vibration, the vibrations in different directions are mainly excited by the

- 179 -

dynamic forces in those directions. When pedestrians walk at the other pacing rate,

the same dynamic feature can be found in the dynamic responses of bridge model

C123.

The effect of pacing rate can be shown more precisely by the dynamic amplification

factor curves. Figure 6.29 and Figure 6.30 show the DAFs of lateral and vertical

deflections respectively, while Figure 6.31 and Figure 6.32 show their corresponding

values for accelerations in lateral and vertical directions. From these figures, it can

be seen that for the bridge model C120, the DAF of the lateral deflection reaches its

maximum value at pacing rate of 1.5 Hz which corresponds to the natural frequency

(0.75 Hz, half of the pacing rate) of the first coupled lateral-torsional mode L1T1,

and the DAF is quite small when the pacing rate is greater than 2.0 Hz. The

corresponding lateral acceleration also reaches its maximum value at the resonant

pacing rate of 1.5 Hz, however, it attains another peak value at the pacing rate around

3.25 Hz. The DAF of the vertical deflection reaches its maximum at pacing rate

around 1.0 Hz and has another peak value at the pacing rate around 2.0 Hz (about

twice of the frequency of mode V1). The resonant DAF of vertical deflection has not

been shown in Figure 6.30 but it can be found in Table 6.2. For the bridge model

C123, the DAF of the lateral deflection and its corresponding acceleration have two

evident peak values. One peak value occurs at the pacing rate of 1.5 Hz which is the

resonant frequency of the first coupled lateral-torsional mode L1T1, another peak

value occurs at the pacing rate around 3.0 Hz which is close to the natural frequency

of the third coupled mode L3T3. The DAF of the vertical deflection and

corresponding acceleration have three outstanding peak values. The DAF of vertical

resonant deflection is not shown in these figures but as seen in Table 6.3, the first

peak occurs at pacing rate around 1.0 Hz near the fundamental vertical frequency

(0.9062 Hz). The second and third peak values occur at pacing rates around 1.75 Hz

and 3.0 Hz which are supposedly close to the frequencies of the third and fifth

vertical vibration modes (1.7597 Hz and 2.8998 Hz). The second two half-wave

asymmetric modes (L2T2 and V2) can not be shown in these figures as all quantities

are obtained from the dynamic responses at the middle bridge frame and this location

is on the node line of the asymmetric vibration modes.

- 180 -

-0.004

-0.002

0.000

0.002

0.004

0.006

0 25 50 75 100 125

Time (s)

Lat

eral

def

lect

ion

(m)

Figure 6.27 Bridge model C120: lateral deflection at pacing rate of 2.0 Hz

-0.07

-0.06

-0.05

-0.04

-0.03

-0.02

-0.01

0.00

0.01

0 25 50 75 100 125

Time (s)

Ver

tica

l def

lect

ion

(m)

Figure 6.28 Bridge model C120: vertical deflection at pacing rate of 2.0 Hz

- 181 -

0

5

10

15

20

25

30

35

1.0 1.5 2.0 2.5 3.0 3.5

Pacing rate (Hz)

Dyn

amic

am

plif

icat

ion

fact

or Bridge model C120

Bridge model C123

Figure 6.29 DAF of lateral deflection

0.0

0.5

1.0

1.5

2.0

2.5

3.0

1.0 1.5 2.0 2.5 3.0 3.5

Pacing rate (Hz)

Dyn

amic

am

plif

icat

ion

fact

or Bridge model C120

Bridge model C123

Figure 6.30 DAF of vertical deflection

- 182 -

0.00

0.02

0.04

0.06

0.08

0.10

1.0 1.5 2.0 2.5 3.0 3.5

Pacing rate (Hz)

Lat

eral

acc

eler

atio

n ( g

)Bridge model C120

Bridge model C123

Figure 6.31 Lateral acceleration

0.0

0.1

0.2

0.3

0.4

0.5

0.6

1.0 1.5 2.0 2.5 3.0 3.5

Pacing rate (Hz)

Ver

tica

l acc

eler

atio

n ( g

)

Bridge model C120

Bridge model C123

Figure 6.32 Vertical acceleration

- 183 -

6.3.5 Effect of vertical static force on the resonant vibration

As the crowd walking dynamic loads consist of three parts: vertical dynamic force

(VDF), lateral dynamic force (LDF) and vertical static force (VSF), each part has its

own contribution to the vibration. In order to investigate the effect of different forces,

particularly the vertical static force, the bridge models have been analysed under

different load cases when they resonate in the first coupled lateral-torsional mode

and first vertical mode. The load cases considered include pure lateral dynamic load

LD (only LDF) or pure vertical dynamic load VD (only VDF), dynamic loads LVD

(LDF and VDF), vertical loads VDS (VDF and VSF) and crowd loads LVS (LDF,

VDF and VSF).

Table 6.10 Bridge model C120: Resonant deflections of the coupled mode L1T1 under different load cases



Load case LD LVD LVS

Vibration mode excited L1T1 L1T1 L1T1

Pacing rate fp (Hz) 1.5000 1.5000 1.5000

Damping ratio z 0.010 0.010 0.010



Umax (m) 0.06720 0.00717 0.06672 0.01222 0.05510 0.00025

Umin (m) -0.06438 -0.00711 -0.06390 -0.01140 -0.05221 -0.06980

Aumax (m) 0.06579 0.00714 0.06531 0.01181 0.05365 0.03502

Mumax (m) 0.00141 0.00003 0.00141 0.00041 0.00144 -0.03477

Full vibration

DAFu 52.0 0.5 51.7 0.8 42.4 2.4

Ustdmax (m) 0.05553 0.00620 0.05507 0.00882 0.04145 -0.05057

Ustdmin (m) -0.05264 -0.00564 -0.05218 -0.00976 -0.03853 -0.06728

Austd (m) 0.05409 0.00592 0.05362 0.00929 0.03999 0.00836

Mustd (m) 0.00145 0.00028 0.00145 -0.00047 0.00146 -0.05892

Steady vibration

DAFustd 42.8 0.4 42.4 0.6 31.6 0.6

Table 6.10 and Table 6.11 show effects of different load cases on the resonant

dynamic deflections at the modes L1T1 and V1 of the footbridge model C120. It can

be seen that when the footbridge resonates in the L1T1 mode, the vertical dynamic

force has only slight effect on the lateral deflection, but when the vertical static force

- 184 -

is taken into account, the amplitude or DAF of lateral deflection decreases

significantly, while when the vertical mode V1 is excited, the vertical vibration is

mainly caused by the vertical dynamic force. The static vertical force affects mainly

the mean value of the vertical deflection and has only slight effect on the vibration

amplitude. This phenomenon indicates that static vertical force can suppress lateral

vibration of such slender footbridges to some extent. In other words, the lateral

vibration can be reduced by increasing the structural weight.

For the footbridge model C123, the effect of static vertical force is quite different.

Table 6.12 and Table 6.13 show the dynamic deflections of the footbridge model

C123 under different load cases. It is found that the static vertical force has very

slight effect on the amplitude of lateral deflection when the mode L1T1 is excited,

and it mainly affects the mean value of vertical deflection in the resonant vibrations.

It is also found that the DAF and amplitude of lateral deflection are much smaller

than those of the footbridge without pre-tensioned side cables when the coupled

mode L1T1 with the same natural frequency is excited by pedestrians.

Table 6.11 Bridge model C120: resonant deflections of the vertical mode V1 under different load cases



Load case VD VDS LVS

Vibration mode excited V1 V1 V1

Pacing rate fp (Hz) 1.5000 1.5000 1.5000

Damping ratio z 0.010 0.010 0.010



Umax (m) 0.00000 0.21692 0.00000 0.16103 0.00724 0.16211

Umin (m) 0.00000 -0.21907 0.00000 -0.27890 -0.00393 -0.28027

Aumax (m) 0.00000 0.21800 0.00000 0.21996 0.00558 0.22119

Mumax (m) 0.00000 -0.00107 0.00000 -0.05894 0.00166 -0.05908

Full vibration

DAFu 0.0 15.0 0.0 15.1 4.4 15.2

Ustdmax (m) 0.00000 0.21692 0.00000 0.16103 0.00555 0.16211

Ustdmin (m) 0.00000 -0.21907 0.00000 -0.27890 -0.00207 -0.28027

Austd (m) 0.00000 0.21800 0.00000 0.21996 0.00381 0.22119

Mustd (m) 0.00000 -0.00107 0.00000 -0.05894 0.00174 -0.05908

Steady vibration

DAFustd 0.0 15.0 0.0 15.1 3.0 15.2

- 185 -

Table 6.12 Bridge model C123: Resonant deflections of the coupled mode L1T1 under different load cases



Load case LD LVD LVS


Pacing rate fp (Hz) 1.5 1.5 1.5

Damping ratio z 0.010 0.010 0.010



Umax (m) 0.02854 0.00903 0.02853 0.01295 0.02807 0.00001

Umin (m) -0.02629 -0.00906 -0.02632 -0.01380 -0.02580 -0.07900

Aumax (m) 0.02741 0.00905 0.02742 0.01338 0.02694 0.03951

Mumax (m) 0.00112 -0.00001 0.00110 -0.00043 0.00114 -0.03950

Full vibration

DAFu 28.8 0.6 28.8 0.8 28.3 2.4

Ustdmax (m) 0.02050 0.00668 0.02058 0.00779 0.02014 -0.05819

Ustdmin (m) -0.01832 -0.00635 -0.01841 -0.00959 -0.01800 -0.07482

Austd (m) 0.01941 0.00651 0.01949 0.00869 0.01907 0.00832

Mustd (m) 0.00109 0.00016 0.00109 -0.00090 0.00107 -0.06651

Steady vibration

DAFustd 20.4 0.4 20.5 0.5 20.1 0.5

Table 6.13 Bridge model C123: resonant deflections of the vertical mode V1 under different load cases



Load case VD VDS LVS


Pacing rate fp (Hz) 0.9062 0.9062 0.9062

Damping ratio z 0.010 0.010 0.010



Umax (m) -0.00001 0.25792 -0.00001 0.19199 0.00420 0.19189

Umin (m) -0.00001 -0.25481 -0.00001 -0.32083 -0.00184 -0.32084

Aumax (m) 0.00000 0.25636 0.00000 0.25641 0.00302 0.25637

Mumax (m) -0.00001 0.00155 -0.00001 -0.06442 0.00118 -0.06447

Full vibration

DAFu 0.0 15.6 0.0 15.6 3.2 15.6

Ustdmax (m) -0.00001 0.25792 -0.00001 0.19199 0.00337 0.19189

Ustdmin (m) -0.00001 -0.25481 -0.00001 -0.32083 -0.00125 -0.32084

Austd (m) 0.00000 0.25636 0.00000 0.25641 0.00231 0.25637

Mustd (m) -0.00001 0.00155 -0.00001 -0.06442 0.00106 -0.06447

Steady vibration

DAFustd 0.0 15.6 0.0 15.6 2.4 15.6

- 186 -

6.3.6 Effect of damping on the resonant vibration

In general, damping has significant effect on dynamic responses and is the main

control parameter in resonance. However, for the slender shallow suspension

footbridges, the effect of damping on the dynamic responses is quite complex.

Table 6.14 Bridge model C120: effect of damping on the coupled mode L1T1




Pacing rate fp (Hz) 1.5000 1.5000 1.5000

Damping ratio z 0.005 0.010 0.050



Umax (m) 0.06758 0.00029 0.05510 0.00025 0.02193 -0.00001

Umin (m) -0.06454 -0.07253 -0.05221 -0.06980 -0.01901 -0.06437

Aumax (m) 0.06606 0.03641 0.05365 0.03502 0.02047 0.03218

Mumax (m) 0.00152 -0.03612 0.00144 -0.03477 0.00146 -0.03219

Full vibration

DAFu 52.3 2.5 42.4 2.4 16.2 2.2

Ustdmax (m) 0.04359 -0.05107 0.04145 -0.05057 0.02163 -0.05081

Ustdmin (m) -0.04066 -0.06763 -0.03853 -0.06728 -0.01872 -0.06425

Austd (m) 0.04213 0.00828 0.03999 0.00836 0.02017 0.00672

Mustd (m) 0.00146 -0.05935 0.00146 -0.05892 0.00145 -0.05753

Steady vibration

DAFustd 33.3 0.6 31.6 0.6 16.0 0.5





Pacing rate fp (Hz) 1.0943 1.0943 1.0943

Damping ratio z 0.005 0.010 0.050



Umax (m) 0.00736 0.35601 0.00724 0.16211 0.00641 0.00607

Umin (m) -0.00407 -0.47180 -0.00393 -0.28027 -0.00332 -0.10775

Aumax (m) 0.00571 0.41391 0.00558 0.22119 0.00486 0.05691

Mumax (m) 0.00165 -0.05790 0.00166 -0.05908 0.00154 -0.05084

Full vibration

DAFu 4.5 28.5 4.4 15.2 3.8 3.9

Ustdmax (m) 0.00555 0.35601 0.00555 0.16211 0.00521 -0.01035

Ustdmin (m) -0.00166 -0.47180 -0.00207 -0.28027 -0.00227 -0.10773

Austd (m) 0.00360 0.41391 0.00381 0.22119 0.00374 0.04869

Mustd (m) 0.00194 -0.05790 0.00174 -0.05908 0.00147 -0.05904

Steady vibration

DAFustd 2.9 28.5 3.0 15.2 3.0 3.4

- 187 -

Table 6.14 and Table 6.15 show the effect of damping on the resonant dynamic

response of the bridge model C120 when the vibration modes L1T1 and V1 are

excited by walking pedestrians respectively. Table 6.16 and Table 6.17 show the

effect of damping on the vibration of the bridge model C123. In numerical analysis,

the damping is applied to the first two natural frequencies or periods according to the

vibration modes. For instance, when the coupled lateral-torsional mode L1T1 is

excited, the damping is applied to the periods (or frequencies) corresponding to the

coupled modes L1T1 and L2T2.

It is found that the damping applied according to the vibration modes has very slight

effect on the vibrations in other modes or directions. When the footbridges resonate

in the coupled mode L1T1 (Table 6.14 and Table 6.16), the dynamic amplitude and

DAF of lateral deflection decrease as the damping ratio increase, however, the

dynamic amplitude and DAF of vertical deflection changes slightly. This

phenomenon also can be seen in the resonant vertical vibrations (Table 6.15 and

Table 6.17). When the damping is applied on the vertical modes and the footbridge

structures resonate in the first vertical mode V1, the effect of damping on the

dynamic lateral deflection is very small.

Table 6.16 Bridge model C123: effect of damping on the coupled mode L1T1




Pacing rate fp (Hz) 1.5000 1.5000 1.5000

Damping ratio z 0.005 0.010 0.050



Umax (m) 0.03340 0.00001 0.02807 0.00001 0.01221 0.00001

Umin (m) -0.03146 -0.08165 -0.02580 -0.07900 -0.00999 -0.07201

Aumax (m) 0.03243 0.04083 0.02694 0.03951 0.01110 0.03601

Mumax (m) 0.00097 -0.04082 0.00114 -0.03950 0.00111 -0.03600

Full vibration

DAFu 34.1 2.5 28.3 2.4 11.7 2.2

Ustdmax (m) 0.02076 -0.05873 0.02006 -0.05820 0.01190 -0.05812

Ustdmin (m) -0.01863 -0.07505 -0.01791 -0.07478 -0.00969 -0.07152

Austd (m) 0.01970 0.00816 0.01899 0.00829 0.01079 0.00670

Mustd (m) 0.00106 -0.06689 0.00107 -0.06649 0.00110 -0.06482

Steady vibration

DAFustd 20.7 0.5 20.0 0.5 11.4 0.4

- 188 -





Pacing rate fp (Hz) 0.9062 0.9062 0.9062

Damping ratio z 0.005 0.010 0.050



Umax (m) 0.00430 0.34975 0.00420 0.19189 0.00386 0.00769

Umin (m) -0.00196 -0.49167 -0.00184 -0.32084 -0.00161 -0.11867

Aumax (m) 0.00313 0.42071 0.00302 0.25637 0.00274 0.06318

Mumax (m) 0.00117 -0.07096 0.00118 -0.06447 0.00113 -0.05549

Full vibration

DAFu 3.3 25.7 3.2 15.6 2.9 3.9

Ustdmax (m) 0.00298 0.32652 0.00337 0.19189 0.00340 -0.00956

Ustdmin (m) -0.00166 -0.46629 -0.00125 -0.32084 -0.00118 -0.11867

Austd (m) 0.00232 0.39640 0.00231 0.25637 0.00229 0.05455

Mustd (m) 0.00066 -0.06988 0.00106 -0.06447 0.00111 -0.06412

Steady vibration

DAFustd 2.4 24.2 2.4 15.6 2.4 3.3

It is also found that the damping has different effects on the vibrations in vertical and

lateral directions. For the footbridge model C120, the DAF of steady resonant lateral

deflection in coupled mode L1T1 reduces from 33.3 to 31.6 while that of the steady

resonant vertical deflection in the mode V1 changes from 28.5 to 15.2 when the

damping ratio increases doubly from 0.005 to 0.01. When the damping ratio

increases ten times to 0.05, the DAF of vertical deflection in mode V1 decreases to

3.4 with almost 88% of vertical deflection suppressed, while the DAF of lateral

deflection in the coupled mode L1T1 drops to 16.0 with only 52% of lateral

deflection suppressed. From Table 6.16 and Table 6.17, similar results are found in

the lateral and vertical resonant vibrations of the footbridge model C123.

6.3.7 Resonant vibration under eccentric walking dynamic loads

When a structural moves horizontally, it is usually considered that the movement is

in response to horizontal forces. However, horizontal movements can also be induced

by vertical loads as structures are three dimensional and movements in the

orthogonal directions are often coupled. Ji et al. [2003] mentioned that horizontal

movements of structures may result from horizontal loading, vertical loading acting

- 189 -

on asymmetric structures, and vertical loading acting asymmetrically on structures.

In particular, they addressed this phenomenon which occurred in railway bridge

structures when high speed trains run on one side of the bridge deck and induced

horizontal vibration. They also pointed out that cable–suspended bridges with

inclined cables will experience both horizontal and rotational movements when they

are subjected to an asymmetric applied vertical load. They explained that due to the

inclination of the cables, the vertical and horizontal movements of the deck are

coupled, and the horizontal movement can be induced by asymmetrically applied

vertical loads because of the geometry of the structural system.

For slender suspension footbridge structures, excessive lateral vibration may be

produced due to this mechanism, as the crowd walking dynamic loads have lateral

dynamic force, vertical dynamic and static forces which can possibly be distributed

asymmetrically. It has been confirmed in chapter 4 that eccentric vertical load can

generates lateral deflection. Moreover, slender suspension footbridge structures often

have coupled vibration modes. It is therefore important to understand the dynamic

behaviour of such slender footbridges under eccentric walking dynamic loads.

Table 6.18 Bridge model C120: Dynamic deflections under eccentric walking dynamic loads




Pacing rate fp (Hz) 1.5 1.0943 1.1949

Damping ratio z 0.010 0.010 0.010



Umax (m) 0.03358 0.00029 0.00816 0.06541 0.04030 0.00434

Umin (m) -0.02559 -0.04558 -0.00107 -0.14367 -0.00252 -0.07600

Aumax (m) 0.02958 0.02293 0.00461 0.10454 0.02141 0.04017

Mumax (m) 0.00400 -0.02265 0.00355 -0.03913 0.01889 -0.03583

Full vibration

DAFu 37.8 2.4 5.9 10.9 27.4 4.2

Ustdmax (m) 0.02700 -0.03338 0.00733 0.06541 0.04030 -0.00370

Ustdmin (m) -0.01895 -0.04441 0.00252 -0.14366 0.02021 -0.07600

Austd (m) 0.02297 0.00552 0.00240 0.10454 0.01004 0.03615

Mustd (m) 0.00403 -0.03889 0.00493 -0.03913 0.03025 -0.03985

Steady vibration

DAFustd 29.4 0.6 3.1 10.9 12.8 3.8

- 190 -

-0.03

-0.02

-0.01

0.00

0.01

0.02

0.03

0.04

0 25 50 75 100 125 150 175 200

Time (s)

Lat

eral

def

lect

ion

(m)

Figure 6.33 Bridge model C120: dynamic lateral deflection under eccentric walking loads at pacing rate of 1.5 Hz

-0.05

-0.04

-0.03

-0.02

-0.01

0.00

0 25 50 75 100 125 150 175 200

Time (s)

Ver

tica

l def

lect

ion

(m)

Figure 6.34 Bridge model C120: dynamic vertical deflection under eccentric walking loads at pacing rate of 1.5 Hz

- 191 -

Table 6.18 shows the statistics of dynamic deflections of the bridge model C120

when pedestrian walk on half the width of deck at different pacing rates. In this table

the static lateral and vertical deflections are produced by the quasi-static dynamic

vertical force (VDF) as this dynamic force is the main excitation when the crowd

walking dynamic loads are distributed on the half width of the bridge deck. Figure

6.33 to Figure 6.38 show the resonant vibrations in lateral and vertical directions.

The coupled mode L1T1 is excited when pedestrians walk at the pacing rate of 1.5

Hz. It is found that the steady amplitude of vertical deflection is quite small and the

bridge structure vibrates mainly with the coupled mode L1T1. Figure 6.33 illustrates

that the dynamic lateral deflection is mainly induced by the lateral dynamic force,

while Figure 6.34 shows the dynamic vertical deflection with small amplitude.

When the vertical mode V1 is excited by pedestrians walking at the pacing rate of

1.0943 Hz, large vertical vibration is induced. Figure 6.35 and Figure 6.36 show the

dynamic lateral and vertical deflections. It can be seen that although the bridge

structure has resonant vibration in the vertical direction, the vertical vibration has

only small contribution to the lateral dynamic deflection, and the bridge structure

vibrates in the lateral direction with the frequency of lateral dynamic force.

Figure 6.37 and Figure 6.38 show the dynamic lateral and vertical deflections when

pedestrians walk on the half width of deck at the pacing rate of 1.1949 Hz, the

natural frequency of the first coupled torsional-lateral mode T1L1. As mentioned

before, this vibration mode is predominately torsional mode and is asymmetric about

the centre line of the bridge deck. It is not easy to be excited by crowd walking

dynamic loads symmetrically distributed on the entire deck, but can be excited by

eccentric loads. When the footbridge structure resonates in this mode, it is found that

both the lateral and vertical deflections have large amplitudes. Figure 6.37 shows that

in the lateral direction, the vibration has large constant amplitude but its mean value

increases almost linearly with time. The lateral deflection is contributed by both the

lateral and vertical dynamic forces. In Table 6.16, the maximum and minimum

steady lateral deflection is chosen from the last periodic vibration cycle. Figure 6.38

shows that the vertical deflection also has increasing mean value although this

increase is very small. In general, the mean value is caused by the static load and/or

- 192 -

the non-linearity of geometry. Under eccentric vertical load, coupled lateral

deflection can be produced simultaneously with the vertical deflection. However,

from these two figures, it is noticed that the mean value of lateral deflection increases

rapidly than that of the vertical deflection.

-0.002

0.000

0.002

0.004

0.006

0.008

0.010

0 25 50 75 100 125 150 175

Time (s)

Lat

eral

def

lect

ion

(m)


-0.15

-0.10

-0.05

0.00

0.05

0.10

0 25 50 75 100 125 150 175

Time (s)

Ver

tica

l def

lect

ion

(m)


- 193 -

-0.01

0.00

0.01

0.02

0.03

0.04

0.05

0 25 50 75 100 125 150 175

Time (s)

Lat

eral

def

lect

ion

(m)


-0.08

-0.07

-0.06

-0.05

-0.04

-0.03

-0.02

-0.01

0.00

0.01

0 25 50 75 100 125 150 175

Time (s)

Ver

tica

l def

lect

ion

(m)


When the pre-tensioned side cables are introduced in the footbridge structure, the

lateral stiffness can be improved, however the vertical stiffness is reduced for the

same natural frequency. Table 6.19 shows the statistics of dynamic deflections when

pedestrians walking along the half width of bridge deck with different pacing rates. It

can be seen from this table that when the coupled mode L1T1 is excited, the

amplitude of vertical deflection is very small and the lateral deflection is mainly

- 194 -

caused by the lateral dynamic force. However, when the vertical mode V1 and

coupled mode T1L1 are excited, the bridge structure also experiences large

vibrations in both lateral and vertical directions. Figure 6.39 and Figure 6.40 show

the lateral deflection under eccentric walking loads with different pacing rates and it

is found that both these lateral deflections have increasing mean values and the

lateral deflections are mainly caused by the vertical dynamic forces. The similarity of

resonant vibration feature under eccentric dynamic loads in the modes V1 and T1L1

is due to their close natural frequencies.

It is obvious that for slender suspension footbridges, large lateral vibration can be

induced by eccentrically distributed walking dynamic loads when coupled vibration

modes are excited. When the footbridge structure resonates in the coupled torsional-

lateral mode L1T1, the lateral deflection often has large dynamic amplitude and

increasing mean value, and this could cause unstable vibration if the eccentric

walking load acts on the bridge deck for long time.

Table 6.19 Bridge model C123: Dynamic deflections under eccentric walking dynamic loads




Pacing rate fp (Hz) 1.5000 0.9062 0.8982

Damping ratio z 0.010 0.010 0.010



Umax (m) 0.01843 0.00001 0.04312 0.09687 0.02371 0.07549

Umin (m) -0.01012 -0.05131 -0.01978 -0.18238 -0.01342 -0.16122

Aumax (m) 0.01427 0.02566 0.03145 0.13962 0.01857 0.11836

Mumax (m) 0.00416 -0.02565 0.01167 -0.04275 0.00514 -0.04287

Full vibration

DAFu 18.2 2.4 40.2 12.9 23.7 11.0

Ustdmax (m) 0.01399 -0.03827 0.04300 0.09294 0.02183 0.07244

Ustdmin (m) -0.00538 -0.04886 -0.01190 -0.17985 -0.00704 -0.15884

Austd (m) 0.00969 0.00530 0.02745 0.13639 0.01444 0.11564

Mustd (m) 0.00430 -0.04357 0.01555 -0.04346 0.00740 -0.04320

Steady vibration

DAFustd 12.4 0.5 35.1 12.6 18.4 10.7

- 195 -

-0.02

-0.01

0.00

0.01

0.02

0.03

0.04

0.05

0 25 50 75 100 125 150 175 200

Time (s)

Lat

eral

def

lect

ion

(m)


-0.015

-0.010

-0.005

0.000

0.005

0.010

0.015

0.020

0.025

0 25 50 75 100 125 150 175

Time (s)

Lat

eral

def

lect

ion

(m)


- 196 -

6.4 Dynamic characteristics of lateral vibration

Pedestrians are sensitive to low frequency lateral vibration of slender footbridge

structures that they are walking across. As the lateral vibration affects the body

balance, pedestrians trend to walk with feet apart to compensate for this lateral

motion and this mechanism results in synchronization and excessive lateral vibration.

Suspension footbridge structures are always slender and have weak stiffness in the

lateral direction. As a consequence, they are prone to the lateral vibration induced by

pedestrians. When the supporting cables have profiles with shallow sags, the

suspension footbridge structures often have coupled vibration modes, and these

coupled vibration modes make the lateral vibration more complex than that in other

forms of bridge structures with pure vibration modes. Furthermore, the dynamic

behaviour of slender suspension footbridges is far more complicated that the quasi-

static load deformation performance and it depends not only on the structural

stiffness, but also on many other factors such as the mass distribution and vibration

properties, damping as well as applied dynamic loads.

6.4.1 Effect of cable section and coupling coefficient

Cable sections have significant effect on the structural stiffness of suspension bridge

structures. In general, increasing cable section can reduce the deflections greatly in

vertical and lateral directions and improve the structural behaviour under different

loadings. However, the effect on the dynamic responses may be different from that

on the load deformation performance under quasi-static loads.

In order to investigate the effect of cable section, three different cable diameters, 90

mm, 120 mm and 180 mm, are considered for the suspension footbridge models

(HSB) which have span length of 80 m and cable sag of 1.8 m. The vibration

properties and relative parameters are shown in Table 6.20. From this table, it can be

seen that for the bridge model C120, the cable section has only slight effect on the

frequencies of coupled lateral-torsional modes and higher vertical modes. The

frequencies of the first coupled torsional-lateral mode T1L1 and fundamental vertical

- 197 -

mode V1 increase while frequencies of higher coupled torsional-lateral modes

decrease when the cable section increases. For the bridge model C123, as the natural

frequency of the first coupled lateral-torsional mode is kept as 0.75 Hz, all other

natural frequencies except the fundamental vertical mode go down significantly

when the cable diameters increase. Only the fundamental vertical frequency

increases with the cable section. It is noted that for the footbridge model C123 with

cable diameter of 180 mm, the fundamental vertical frequency corresponds with the

two half-wave vibration mode, and this frequency as well as that of the first coupled

torsional-lateral mode T1L1 are smaller than the frequency of second coupled lateral-

torsional mode L2T2.

Table 6.20 Vibration properties of footbridges with different cable section

Bridge parameter HSB: L=80 m; F1=F2=F3=1.8 m; D1=D2=D3

Bridge model C120 C120 C120 C123 C123 C123

Cable diameter D1 (mm) 90 120 180 90 120 180

Mass density M (kg/m2) 326.6 363.8 469.3 410.6 465.8 624.6

T1 (N) 6324574 6987428 8896031 5395303 5536132 6122724

T2 (N) 3399147 3722268 4660041 1725983 1356765 484638 Cable tensions

T3 (N) -- -- -- 1724983 1110712 411926

L1T1 0.7500 0.7500 0.7500 0.7500 0.7500 0.7500

L2T2 1.4645 1.4585 1.4485 1.2549 1.0980 0.8178

L3T3 2.1727 2.1634 2.1491 1.8159 1.5602 1.1312

L4T4 2.8775 2.8656 2.8459 2.3827 2.0340 1.4352

L5T5 3.5769 3.5654 3.5460 2.9567 2.5246 1.7783


L6T6 4.2665 4.2572 4.2407 3.5250 3.0111 2.1111

T1L1 1.1259 1.1949 1.3155 0.9205 0.8982 0.8863

T2L2 1.8999 1.8718 1.8199 1.5770 1.4158 1.1963

T3L3 2.7618 2.7238 2.6620 2.2991 2.0593 1.7325 Coupled torsional-lateral

T4L4 3.6301 3.5793 3.4973 3.0226 2.7023 2.2633

V1 0.9853 1.0943 1.2798 0.8569 0.9062 0.9782

V2 1.5175 1.5151 1.5165 1.2945 1.1633 0.9588

V3 2.2831 2.2866 2.3037 1.9476 1.7597 1.4771

V4 3.0263 3.0239 3.0292 2.5798 2.3203 1.9062

Vertical

V5 3.7769 3.7785 3.7926 3.2187 2.8998 2.3827

Table 6.21 shows the statistics of dynamic lateral deflections of different footbridge

models when the first coupled lateral-torsional vibration mode L1T1 is excited by the

crowd pedestrians walk at the pacing rate of 1.5 Hz. The vertical deflections are not

- 198 -

listed here since the footbridge structure does not resonate in vertical direction at this

frequency.

Table 6.21 Resonant lateral deflection of footbridges with different cable section




Vibration mode excited L1T1 L1T1 L1T1 L1T1 L1T1 L1T1

Pacing rate fp (Hz) 1.5 1.5 1.5 1.5 1.5 1.5

Damping ratio z 0.010 0.010 0.010 0.010 0.010 0.010

Displacement U Ul Ul Ul Ul Ul Ul

Static displacement Ustatic (m) 0.00141 0.00126 0.00097 0.00110 0.00095 0.00068

Umax (m) 0.05385 0.05510 0.05317 0.03426 0.02807 0.01259

Umin (m) -0.05054 -0.05221 -0.05095 -0.03162 -0.02580 -0.01106

Aumax (m) 0.05219 0.05365 0.05206 0.03294 0.02694 0.01183

Mumax (m) 0.00166 0.00144 0.00111 0.00132 0.00114 0.00077

Full vibration

DAFu 37.0 42.4 53.6 29.9 28.3 17.5

Ustdmax (m) 0.03840 0.04145 0.04471 0.02400 0.02014 0.00867

Ustdmin (m) -0.03510 -0.03853 -0.04248 -0.02148 -0.01800 -0.00716

Austd (m) 0.03675 0.03999 0.04360 0.02274 0.01907 0.00792

Mustd (m) 0.00165 0.00146 0.00112 0.00126 0.00107 0.00076

Steady vibration

DAFustd 26.0 31.6 44.9 20.7 20.1 11.7

For the bridge model C120, it is found that when the cable diameter increases, the

dynamic amplification factor (DAF) and amplitude of the steady lateral deflection

increase unexpectedly while the mean value decreases. For the bridge model C123,

both the amplitude and mean value of the lateral deflection decrease when the cable

diameter increases. However, the dynamic amplification factor changes slightly

when the cable diameter increases from 90 mm to 120 mm, and drops dramatically as

the cable diameter increases to 180 mm.

In general, the stiffness increases and deflection goes down when the cable cross

sections increase. It is confirmed that the structural stiffness in the lateral direction is

improved by increasing the cable section. Figure 6.41 shows the static lateral

deflection of the footbridges under quasi-static lateral force. It illustrates that the

lateral stiffness increases with increase of cable diameter, and this is also shown in

Table 6.21 by the decrease of the static lateral deflection and the mean value of the

- 199 -

dynamic lateral deflection which depend mainly on the structural stiffness. But

unlike the static load deformation performance, dynamic response depends not only

on the structural stiffness, but also on the vibration properties, damping and applied

dynamic loads. However, as all the footbridge models have the same fundamental

coupled lateral-torsional vibration mode and they are subjected to the same crowd

walking dynamic loads, it seems that the dynamic performances are affected by some

other factors relative to the vibration properties.

0.00

0.01

0.02

0.03

0.04

0 21 42 63 84 105 126 147 168 189 210

Load density (N/m2)

Lat

eral

def

lect

ion

(m)

C120 (D1= 90 mm)C120 (D2=120 mm)C120 (D1=180 mm)C123 (D1= 90 mm)C123 (D1=120 mm)C123 (D1=180 mm)

Figure 6.41 Lateral deflections of footbridges under static lateral force

The vibration properties of a structure are often described by the natural frequencies

and their corresponding vibration modal shapes. For slender suspension bridge

structures, the one half-wave symmetric mode is not always the fundamental one,

and its frequency can be higher than that of the two half-wave asymmetric mode or

others and moreover it even disappear when the structural stiffness is improved.

This has been shown in chapter 4 and explained by many others [Gimisong 1998;

Ivirine 1992]. Another parameter that could be used to describe the frequency is the

increment between two adjacent natural frequencies. However, this increment is

affected by the structural stiffness and mass distribution and there is little in the

literature regarding its effect on the dynamic response. Vibration modal shapes are

- 200 -

important for the dynamic deformation of the entire structure. Normally they appear

as pure modes such as lateral, vertical and torsional modes, and they are independent

from each other. For slender suspension bridges with shallow cable profiles, it is

found that the lateral and torsional modes are often coupled together and form two

kinds of coupled vibration modes: coupled lateral-torsional modes and coupled

torsional-lateral modes. It is obvious that these coupled modes have significant effect

on the dynamic response of slender suspension footbridges.

In order to investigate the effect of coupled modes on the dynamic performance, it is

necessary to introduce a factor called “coupling coefficient” to describe the degree of

coupling in the coupled vibration modes. Here the coupling coefficient y is defined

as the ratio of vertical deflection to lateral one:

lv UU /=ψ (6.10)

The vertical and lateral deflections are picked up from the same point (Figure 4.2)

where the maximum lateral or vertical deflection occurs. For example, the

intersection point of legs and cross member of the middle bridge frame is chosen for

the one half-wave and three half-wave vibration modes and the intersection point of

the bridge frame at the quarter span length is chosen for the two half-waves modes,

as there are anti-nodes with maximum deflections.

Table 6.22 Coupling coefficients of coupled vibration modes of footbridges with different cable sections




L1T1 -0.1374 -0.1092 -0.0776 -0.2829 -0.3372 -0.3935

L2T2 -0.3161 -0.3322 -0.3716 -0.4594 -0.6123 -0.8184 Coupled lateral-torsional

L3T3 -0.3901 -0.4066 -0.4365 -0.5415 -0.6966 -0.9588

T1L1 6.6242 8.1716 11.3563 2.5706 2.1232 1.8875


T3L3 2.3488 2.1244 1.7627 1.4100 1.0752 0.6882

Table 6.22 lists the coupling coefficients of the first three coupled lateral-torsional

modes and first three coupled torsional-lateral modes for footbridges with different

cable sections and cable configurations. Here a negative coupling coefficient shows

- 201 -

that the vibration mode is coupled lateral-torsional one and bridge deck sways about

a point above the deck plane. In the numerical analysis, the upward vertical

deflection and rightward lateral deflection are defined as positive deflections, and

these deflections for the coupling coefficient are picked up from the left intersection

of cross member and legs at the bridge frame shown in Figure 4.2. When the bridge

deck and bridge frames sway about a point above the deck plane, the rightward

lateral deflection is accompanied by downward vertical deflection. Therefore the

coupling coefficient is negative according to Equation (6.10). On the other hand, a

positive coupling coefficient indicates that the vibration mode is coupled torsional-

lateral one and the bridge deck as well as bridge frames sway about a point beneath

the deck plane. The values of coupling coefficients reflect the degree of coupling.

For the coupled lateral-torsional modes, large coupling coefficient indicates high

degree of coupling and large vertical component accompanies the lateral deflection;

while for the coupled torsional-lateral vibration modes, large coupling coefficient

indicates low degree of coupling and the coupled mode trends to be pure torsional

one.

For the bridge model C120, it can be seen that when the cable diameter increases, the

coupling coefficient of the coupled mode L1T1 decreases while the coupling

coefficients of other coupled lateral-torsional modes increase. The coupling

coefficient of mode T1L1 increases greatly but those of other coupled torsional-

lateral modes decrease. These results indicates that the mode L1T1 changes

gradually to become a pure lateral mode and the mode T1L1 changes to be a pure

torsional mode when the structural stiffness is improved by increasing the cable

sections.

However, some different results are found in coupling coefficients of the first

coupled modes L1T1 and T1L1 for the bridge model C123. When the cable diameter

increases, it is found that the coupling coefficient of the coupled mode L1T1

increases and that of the mode T1L1 decreases. This result illustrates that the degree

of coupling increases with the increase of cable section.

Relating the coupling coefficient of the mode L1T1 in Table 6.22 to the steady DAFs

of the lateral deflection in Table 6.21, it is found that for footbridge with the same

- 202 -

cable configuration, the larger the coupling coefficient, the greater the DAF. This

phenomenon illustrates that for slender suspension footbridge structures with coupled

vibration modes, the degree of coupling has significant effect on the dynamic

response, and sometimes this effect is greater than the effect of static structural

stiffness.

Table 6.23 Vibration properties and coefficients with cable sag

Bridge parameter HSB: L=80 m; F1=F2=F3; D1=D2=D3=120 mm


Cable sag F1 (m) 1.2 1.8 2.4 1.2 1.8 2.4

Mass density M (kg/m2) 363.7 363.8 363.9 465.6 465.8 466.0

T1 (N) 8195061 6987428 6511684 8362994 5536132 3414208

T2 (N) 3307207 3722268 4054908 2108728 1356765 269195 Cable tensions

T3 (N) -- -- -- 2104241 1110712 266960

Mode Natural frequencies

L1T1 0.7500 0.7500 0.7500 0.7500 0.7500 0.7500

L2T2 1.4352 1.4585 1.4858 1.3017 1.0980 0.9323

L3T3 2.1323 2.1634 2.2016 1.9203 1.5602 1.1307

L4T4 2.8248 2.8656 2.9121 2.5365 2.0340 1.3419

L5T5 3.5113 3.5654 3.6235 3.1519 2.5246 1.6478


L6T6 4.1889 4.2572 4.3269 3.7596 3.0111 1.9666

T1L1 1.1383 1.1949 1.3201 0.9913 0.8982 0.8537

T2L2 2.0296 1.8718 1.8991 1.7994 1.4158 1.1388


T4L4 3.9633 3.5793 3.3557 3.5136 2.7023 1.8987

V1 0.9483 1.0943 1.2842 0.8663 0.9062 0.9449

V2 1.5694 1.5151 1.5177 1.4526 1.1633 0.8373

V3 2.3601 2.2866 2.2835 2.1835 1.7597 1.3011

V4 3.1346 3.0239 2.9989 2.9000 2.3203 1.6339

Vertical

V5 3.9137 3.7785 3.7501 3.6195 2.8998 2.0475

Mode Coupling coefficients

L1T1 -0.2616 -0.1092 -0.0367 -0.3946 -0.3372 -0.2850

L2T2 -0.4295 -0.3322 -0.1702 -0.4902 -0.6123 -0.7831 Coupled lateral-torsional

L3T3 -0.4648 -0.4066 -0.3112 -0.5321 -0.6966 -1.0816

T1L1 3.2675 8.1716 19.6096 1.6320 2.1232 3.2026


T3L3 1.8607 2.1244 2.4548 1.2718 1.0752 0.7959

6.4.2 Effect of cable sag

It is known that cable sag has considerable effect on the structural stiffness and

vibration properties of suspension bridge structures, as the stiffness and natural

- 203 -

frequencies of such bridges are influenced significantly by the cable tension forces

which depend not only on the bridge gravity or mass, but also on the cable sag. In

general, the stiffness increases while the tension forces decrease with the increase of

cable sag. Most of the natural frequencies except those corresponding with the one

half-wave symmetric vibration modes decrease due to the decrease of tension forces.

Since the stiffness and vibration properties have significant effect on the dynamic

response, it is no doubt that the dynamic behaviour of slender suspension footbridges

will be affected by the cable sag.

Table 6.23 shows the natural frequencies and coupling coefficients of the bridge

models C120 and C123 with different cable sags. Here all the bridge models are

assumed to have span length of 80 m and cable diameter of 120 mm. The cable sag

considered varies from 1.2 m to 2.4 m and the frequencies corresponding to the first

coupled mode L1T1 of all the bridge models are tuned to be 0.75 Hz. From this

table, it can be seen that when the cable sag increase, the mass density changes

slightly. As the tension forces in the top supporting cables caused by the gravity

decrease, more pre-tension forces are required for the bottom and/or side pre-

tensioned cables to get the same fundamental frequency of coupled vibration mode

L1T1. It also can be seen that for the bridge model C120, when the cable sag

increases, the coupling coefficients of the first coupled modes L1T1 and T1L1

change dramatically while the coupling coefficients of other modes change

gradually. These results indicate that when the cable sag increases, the first coupled

modes L1T1 and T1L1 turn to become pure lateral and torsional modes but all the

others remain coupled vibration modes. However, for the bridge model C123, the

coupling coefficients of the coupled modes L1T1 and T1L1 changes slowly, and so

do the other coupled vibration modes when the cable sag increases. These results

illustrate that for the bridge mode C123, the lateral and torsional modes are always

coupled together. It is noticed that for the bridge model C120, the coupling

coefficients of higher coupled lateral-torsional modes decrease and those of higher

coupled torsional-lateral modes increase with the increase of cable sag. However, the

trends are reversed for the bridge model C123. Moreover, when the cable sag

increases to 2.4 m, the vertical component of the coupled lateral-torsional mode

L3T3 is greater than the lateral one, while the vertical components of coupled

torsional-lateral modes T2L2 and T3L3 are smaller than the corresponding lateral

- 204 -

ones. This phenomenon implies that in the higher coupled vibration modes, the

dominant modes will change with the cable sag.

Table 6.24 shows the lateral deflections of the slender suspension footbridge models

with different cable sag when the first coupled lateral-torsional mode L1T1 is excited

by pedestrians walking at pacing rate of 1.5 Hz. It can be seen that for the bridge

model C120, the static lateral deflection changes slightly with the increase of cable

sag, while the dynamic amplification factor and amplitude of lateral deflection in

steady vibration increase significantly but the mean value changes slightly. As the

static deflection depends mainly on the structural stiffness, the slight change of static

lateral deflection indicates that the lateral stiffness is almost the same. However,

dynamic response depends not only on the structural stiffness, but also on the

frequencies of dynamic load and natural frequencies as well as the vibration

properties. The results shown in Table 6.24 illustrate again that the coupling

coefficient has significant effect on the dynamic performance of slender suspension

footbridges with coupled vibration modes.

Table 6.24 Dynamic lateral deflection of footbridge models with cable sag

Bridge parameter HSB: L=80 m; F1=F2=F3; D1=D2=D3=120 mm


Cable sag F1 (m) 1.2 1.8 2.4 1.2 1.8 2.4

Vibration mode excited L1T1 L1T1 L1T1 L1T1 L1T1 L1T1

Pacing rate fp (Hz) 1.5000 1.5000 1.5000 1.5000 1.5000 1.5000

Damping ratio z 0.010 0.010 0.010 0.010 0.010 0.010

Displacement U Ul Ul Ul Ul Ul Ul

Static displacement Ustatic (m) 0.00124 0.00126 0.00126 0.00092 0.00095 0.00094

Umax (m) 0.03356 0.05510 0.08149 0.02361 0.02807 0.01545

Umin (m) -0.03046 -0.05221 -0.07860 -0.02160 -0.02580 -0.01320

Aumax (m) 0.03201 0.05365 0.08004 0.02261 0.02694 0.01433

Mumax (m) 0.00155 0.00144 0.00144 0.00100 0.00114 0.00113

Full vibration

DAFu 25.9 42.4 63.4 24.5 28.3 15.2

Ustdmax (m) 0.02126 0.04145 0.07363 0.01568 0.02014 0.00885

Ustdmin (m) -0.01838 -0.03853 -0.07073 -0.01358 -0.01800 -0.00672

Austd (m) 0.01982 0.03999 0.07218 0.01463 0.01907 0.00779

Mustd (m) 0.00144 0.00146 0.00145 0.00105 0.00107 0.00107

Steady vibration

DAFustd 16.0 31.6 57.1 15.9 20.1 8.2

- 205 -

For the bridge model C123, the effect of cable sag is more complex. When the cable

sag increases from 1.2 m to 1.8 m, the dynamic performance is found to be similar to

that of the bridge model C120. However, when the cable sag increases to 2.4 m, the

dynamic amplification factor and amplitude of the lateral deflection decrease

dramatically although the coupling coefficient of the mode L1T1 decreases with the

increase of cable sag. It seems the dynamic response is influenced by some other

parameters or factors such as higher vibration modes.

6.4.3 Effect of span length

The span length is an important structural parameter and it affects the structural

stiffness and vibration properties such as natural frequency and vibration modes, and

hence affects the dynamic performance of slender suspension footbridges.

In order to study the effect of span length on the dynamic performance, three span

lengths are considered here: 40 m, 80 m and 120 m. Pre-tensioned suspension

footbridge models with span length of 80 m have been studied intensively in the

previous sections, and herein the results will be compared with those of other bridge

models with different span length.

Table 6.25 shows the vibration properties and coupling coefficients of bridge models

with different span length, and all the footbridge models are hollow section bridges

(HSB) and have the same cable sag of 1.8 m. Here the bridge model C123 with span

length of 40 m is not considered as its fundamental frequency of coupled lateral-

torsional modes is greater than 0.75 Hz when the side cables are introduced even

without pre-tension. To avoid over rigid or slender structure, the cable diameter for

the 40 m bridge model C120 is set to be 90 mm and the cable diameter for the bridge

models C120 and C123 with span length of 120 m is set to be 240 mm.

For the bridge model C120, it is found that the span length has only slight effect on

the first four natural frequencies of the coupled lateral-torsional modes but

significant effect on the frequencies of coupled torsional-lateral modes as well as

those of the vertical modes when the fundamental frequency of the coupled mode

- 206 -

L1T1 is tuned to be 0.75 Hz. When the footbridge has short span length (40 m), the

first coupled torsional-lateral mode T1L1 and vertical mode V1 disappear. It is also

found that the span length has great effect on the coupling coefficients of the coupled

vibration modes. Shorter suspension bridge model has smaller coupling coefficient

for the mode L1T1 and greater coupling coefficient for the mode T1L1. These results

indicate that the first coupled modes L1T1 and T1L1 trend to become pure lateral

and torsional modes when the suspension bridge structure becomes shorter in span.

Table 6.25 Natural frequencies and coupling coefficients with span length

Bridge parameter HSB: F1=F2=F3=1.8 m; D1=D2=D3

Bridge model C120 C120 C120 C123 C123

Span length L (m) 40 80 120 80 120

Cable section D1 (mm) 90 120 240 120 240

Mass density M (kg/m2) 319.4 363.8 624.4 465.8 856.4

T1 (N) 1558378 6987428 26913402 5536132 26757930

T2 (N) 819898 3722268 14346491 1356765 9527164 Cable tension

T3 (N) -- -- -- 1110712 9520358


L1T1 0.7500 0.7500 0.7500 0.7500 0.7500

L2T2 1.4645 1.4585 1.4524 1.0980 1.2941

L3T3 2.0927 2.1634 2.1558 1.5602 1.9003

L4T4 2.8323 2.8656 2.8599 2.0340 2.5133

L5T5 3.0153 3.5654 3.5634 2.5246 3.1301


L6T6 3.7452 4.2572 4.2626 3.0111 3.7435

T1L1 -- 1.1949 1.0496 0.8982 0.8973

T2L2 2.0402 1.8718 1.7808 1.4158 1.5624


T4L4 2.9845 3.5793 3.4640 2.7023 3.0462

V1 -- 1.0943 0.9467 0.9062 0.8409

V2 1.5013 1.5151 1.5358 1.1633 1.3860

V3 2.0389 2.2866 2.3064 1.7597 2.0802

V4 2.8249 3.0239 3.0673 2.3203 2.7679

Vertical

V5 3.5752 3.7785 3.8360 2.8998 3.4615


L1T1 -0.0185 -0.1092 -0.1651 -0.3372 -0.3162

L2T2 -0.2034 -0.3322 -0.4192 -0.6123 -0.5488 Coupled lateral-torsional

L3T3 -1.3672 -0.4066 -0.5305 -0.6966 -0.6812

T1L1 -- 8.1716 4.9814 2.1232 2.0729


T3L3 1.3082 2.1244 1.5655 1.0752 0.9964

- 207 -

For the bridge model C123, it is found that the span length has significant effect on

the natural frequencies but small effect on the coupling coefficients. When the span

length increases, the frequency of the first coupled torsional-lateral mode T1L1

changes slightly and the frequency of first vertical mode V1 decreases, while all the

other frequencies increase. This is probably because the tension forces increase more

rapidly than the structural mass does.

Table 6.26 Resonant lateral deflection with span length

Bridge parameter HSB: F1=F2=F3=1.8 m; D1=D2=D3


Span length L (m) 40 80 120 80 120

Cable section D1 (mm) 90 120 240 120 240

Mass density M (kg/m2) 319.4 363.8 624.4 465.8 856.4

Vibration mode excited L1T1 L1T1 L1T1 L1T1 L1T1

Pacing rate fp (Hz) 1.5000 1.5000 1.5000 1.5000 1.5000

Damping ratio ζ 0.010 0.010 0.010 0.010 0.010

Displacement U Ul Ul Ul Ul Ul

Static displacement Ustatic (m) 0.00141 0.00126 0.00073 0.00095 0.00052

Umax (m) 0.12400 0.05510 0.03044 0.02807 0.01559

Umin (m) -0.12086 -0.05221 -0.02864 -0.02580 -0.01437

Aumax (m) 0.12243 0.05365 0.02954 0.02694 0.01498

Mumax (m) 0.00157 0.00144 0.00090 0.00114 0.00061

Full vibration

DAFu 87.1 42.4 40.4 28.3 28.8

Ustdmax (m) 0.12265 0.04145 0.02270 0.02006 0.01083

Ustdmin (m) -0.11945 -0.03853 -0.02101 -0.01791 -0.00968

Austd (m) 0.12105 0.03999 0.02185 0.01899 0.01025

Mustd (m) 0.00160 0.00146 0.00084 0.00107 0.00057

Steady vibration

DAFustd 86.1 31.6 29.9 20.0 19.7

Table 6.26 shows the resonant lateral deflections of bridge models with different

span lengths and cable configurations when they resonate under crowd walking

dynamic loads at pacing rate of 1.5 Hz (double of the frequency of coupled mode

L1T1). It can be seen that the static and dynamic lateral deflections decrease when

the span length increases. This implies that the longer bridge models are stiffer than

the shorter ones. In general, longer span bridges are slenderer than short ones and

they have smaller fundamental frequency. To get the same fundamental natural

frequency, more pre-tension should be introduced into the cable systems of longer

bridges and therefore their stiffness becomes greater than that of the shorter ones.

However, the effect of span length on the DAF is quite different. For the bridge

- 208 -

model C120; the DAF decreases when the span length increases, while for the bridge

model C123, the span length has only slight effect on the DAF.

Comparing the DAFs with coupling coefficients, it is confirmed again that for

slender suspension footbridge structures with coupled vibration modes, the coupling

coefficient has significant effect on the dynamic amplification factor. The greater the

coupling coefficient is, then the smaller the DAF will be. It also implies that bridge

structures with pure vibration modes have greater DAF than those with coupled

vibration modes.

Table 6.27 Effect of synchronization on the dynamic response of the footbridge with cable configuration C120




Pacing rate fp (Hz) 1.5000 1.5000 1.5000 1.5000 1.5000

Damping ratio ζ 0.010 0.010 0.010 0.010 0.010

Synchronization y 10% 20% 30% 40% 50%



Umax (m) 0.02701 0.05510 0.08470 0.11603 0.15014

Umin (m) -0.02547 -0.05221 -0.08028 -0.11029 -0.14233

Aumax (m) 0.02624 0.05365 0.08249 0.11316 0.14624

Mumax (m) 0.00077 0.00144 0.00221 0.00287 0.00390

Full vibration

DAFu 41.5 42.4 43.5 44.8 46.3

Ustdmax (m) 0.02007 0.04145 0.06458 0.08990 0.11812

Ustdmin (m) -0.01861 -0.03853 -0.06018 -0.08400 -0.11071

Austd (m) 0.01934 0.03999 0.06238 0.08695 0.11442

Mustd (m) 0.00073 0.00146 0.00220 0.00295 0.00370

Steady vibration

DAFustd 30.6 31.6 32.9 34.4 36.2

Displacement U Uv Uv Uv Uv Uv

Static displacement Ustatic (m) -0.00726 -0.01452 -0.02178 -0.02904 -0.03630

Umax (m) 0.00001 0.00025 0.00557 0.01088 0.01620

Umin (m) -0.07125 -0.06980 -0.06836 -0.06680 -0.06503

Aumax (m) 0.03563 0.03502 0.03696 0.03884 0.04062

Mumax (m) -0.03562 -0.03477 -0.03140 -0.02796 -0.02442

Entire vibration

DAFu 4.9 2.4 1.7 1.3 1.1

Ustdmax (m) -0.06168 -0.05057 -0.03912 -0.02730 -0.01495

Ustdmin (m) -0.06994 -0.06728 -0.06454 -0.06176 -0.05894

Austd (m) 0.00413 0.00836 0.01271 0.01723 0.02200

Mustd (m) -0.06581 -0.05892 -0.05183 -0.04453 -0.03694

Steady vibration

DAFustd 0.6 0.6 0.6 0.6 0.6

- 209 -

6.4.4 Effect of synchronization

In general, when people participate in the synchronization with a vibrating

footbridge, the dynamic forces increase and cause large amplitude vibration. On the

other hand, larger amplitude vibration increases the degree of synchronization and

makes more people change their footsteps to keep the body balance and produce

more excessive excitation to the vibrating structure. The degree of synchronization is

an important load factor to the dynamic response of slender footbridge structures as it

changes the portion of different forces generated by walking pedestrians. For slender

suspension footbridge structures, they are often weak in the lateral direction and the

increasing of dynamic forces may affect the dynamic performance. To illustrate the

effect of degree of synchronization, Table 6.27 shows the statistics of dynamic

deflections of the bridge model C120 when the coupled mode L1T1 is excited by

crowd walking dynamic loads with different degree of synchronization. Table 6.28

shows those for the footbridge model C123.

When the degree of synchronization increases, the dynamic lateral and vertical forces

increase while the static vertical force decreases. It is found that when the degree of

synchronization increases, the steady amplitude and DAF of lateral deflection

increase for the bridge model C120, while for the bridge model C123, the dynamic

amplitude increase but the DAF changes slightly. This is because for the bridge

model C120, the DAF of lateral deflection is significantly affected by the vertical

static force. The DAF goes up when the vertical static force decreases, while for the

bridge C123, the effect of vertical static force has only slight effect, and the DAF

changes only slightly even when 50% pedestrians participate in the synchronization.

From these two tables, it is also found that the degree of synchronization has almost

no effect on DAFs of vertical deflections although the dynamic amplitudes of

vertical deflections increase.

6.4.5 Natural frequencies and dynamic amplification factors

An important feature of the suspension footbridge model with reverse profiled pre-

tensioned cables is that the natural frequencies can be altered easily by introducing

- 210 -

different pre-tensions in the bottom and/or side cables. This feature is useful to

investigate the dynamic performance of footbridges under synchronous lateral

excitation at different pacing rates, as synchronous lateral excitation can occur on

any footbridge, independent of structural forms, with a lateral natural frequency

below around 1.3 Hz and with a sufficiently large crowd of pedestrians crossing the

bridge structure.

Table 6.28 Effect of synchronization on the dynamic response of the footbridge with cable configuration C123




Pacing rate fp (Hz) 1.5000 1.5000 1.5000 1.5000 1.5000

Damping ratio ζ 0.010 0.010 0.010 0.010 0.010

Synchronization y 10% 20% 30% 40% 50%



Umax (m) 0.01401 0.02807 0.04221 0.05646 0.07086

Umin (m) -0.01290 -0.02580 -0.03877 -0.05185 -0.06518

Aumax (m) 0.01346 0.02694 0.04049 0.05416 0.06802

Mumax (m) 0.00056 0.00114 0.00172 0.00231 0.00284

Full vibration

DAFu 28.3 28.3 28.4 28.5 28.6

Ustdmax (m) 0.01000 0.02014 0.03013 0.04036 0.05076

Ustdmin (m) -0.00892 -0.01800 -0.02689 -0.03602 -0.04532

Austd (m) 0.00946 0.01907 0.02851 0.03819 0.04804

Mustd (m) 0.00054 0.00107 0.00162 0.00217 0.00272

Steady vibration

DAFustd 19.9 20.1 20.0 20.1 20.2

Displacement U Uv Uv Uv Uv Uv

Static displacement Ustatic (m) -0.00820 -0.01640 -0.02459 -0.03279 -0.04099

Umax (m) 0.00001 0.00001 0.00184 0.00659 0.01151

Umin (m) -0.08057 -0.07900 -0.07746 -0.07592 -0.07440

Aumax (m) 0.04029 0.03951 0.03965 0.04126 0.04295

Mumax (m) -0.04028 -0.03950 -0.03781 -0.03467 -0.03145

Entire vibration

DAFu 4.9 2.4 1.6 1.3 1.0

Ustdmax (m) -0.07010 -0.05819 -0.04601 -0.03372 -0.02123

Ustdmin (m) -0.07837 -0.07482 -0.07098 -0.06718 -0.06333

Austd (m) 0.00414 0.00832 0.01248 0.01673 0.02105

Mustd (m) -0.07423 -0.06651 -0.05849 -0.05045 -0.04228

Steady vibration

DAFustd 0.5 0.5 0.5 0.5 0.5

In order to investigate the dynamic behaviour of suspension footbridges at different

pacing rates, bridge models with different span length and fundamental natural

- 211 -

frequency are cinsidered in this section. In these footbridge models, the cable sags

are set to be 1.8 m and the bridge span varies from 40 m to 120 m. The fundamental

lateral natural frequency considered here for each bridge model varies up to 1.5 Hz

from 0.5 Hz depending on the cable section and span length. The vibration properties

of these footbridge models are shown in Table 6.29 to 6.33.

Table 6.29 Vibration properties and coupling coefficients of bridge model C120 with span length of 40 m

Bridge parameter BMB: L=40 m; F1=F2=1.8 m; D1=D2=90 mm


Mass density m (kg/m2) 319.4 319.4 319.4 319.4 319.4

T1 (N) 893635 1558378 2495647 3704966 5187152 Cable tension

T2 (N) 155064 819898 1757197 2966530 4448705


L1T1 0.5000 0.7500 1.0000 1.2500 1.5000

L2T2 0.9281 1.4645 1.9770 2.4797 2.9767

L3T3 1.2606 2.0927 2.9464 3.6873 4.4166

L4T4 1.6576 2.8323 3.8720 4.8550 5.8011

L5T5 2.0245 3.0153 4.7775 5.9767 7.1183


L6T6 2.3797 3.7452 5.6232 7.0276 8.3518

T1L1 2.5396 -- 2.5021 2.7236 2.9363

T2L2 1.5554 2.0402 2.5675 3.1167 3.6765


T4L4 2.5903 2.9845 4.6390 5.7270 6.8174

V1 -- -- 2.4552 2.6755 2.8397

V2 1.0033 1.5013 1.9995 2.4973 2.9947

V3 1.4148 2.0389 3.2537 3.8800 4.5746

V4 1.9943 2.8249 3.9703 4.9511 5.9250

Vertical

V5 2.3451 3.5752 4.9403 6.1419 7.3353


L1T1 -0.0216 -0.0185 -0.0170 -0.0157 -0.0144


L3T3 -0.9239 -1.3672 -0.0347 -0.1037 -0.0901

T1L1 -0.3665 -- -21.0541 214.5625 81.5062


T3L3 1.6056 1.3082 3.9106 5.8970 8.5391

Table 6.29 shows the natural frequencies and coupling coefficients of the bridge

model C120 with span of 40 m. Here all the cables are assumed to have a diameter of

90 mm. Different pre-tension force in introduced to the bottom cables to obtain

different natural frequencies. The bridge model C123 with span of 40 m is not

considered as the minimum fundamental lateral frequency is greater than 1.0 Hz

- 212 -

when the side cables are added. Table 6.30 and Table 6.31 show natural frequencies

of the bridge models C120 and C123 with span length of 80 m. In these bridge

models, the cable diameters are set to be 180 mm to get reasonable axial stresses in

the top supporting cables for the case of high lateral frequencies. The minimum

lateral frequency of bridge model C123 listed here is 0.75 Hz as the lateral frequency

of 0.5 Hz can not be obtained when the side pre-tensioned cables are introduced to

the bridge model. Table 6.32 and Table 6.33 show the vibration properties of the

bridge models C120 and C123 with span length of 120 m, where the cable diameters

are set to be 240 mm and the maximum fundamental lateral frequency listed in these

two tables is 1.25 Hz. Larger cable diameters are required for higher fundamental

lateral frequency.




Mass density m (kg/m2) 469.3 469.3 469.3 469.3 469.3

T1 (N) 5128303 8896031 14244651 21166903 29663300 Cable tension

T2 (N) 891073 4660041 10009275 16932017 25428943


L1T1 0.5000 0.7500 1.0000 1.2500 1.5000

L2T2 0.8756 1.4485 1.9739 2.4826 2.9834

L3T3 1.2782 2.1491 2.9426 3.7025 4.4432

L4T4 1.6588 2.8459 3.9033 4.9037 5.8664

L5T5 2.0626 3.5460 4.8558 6.0801 7.2377


L6T6 2.4611 4.2407 5.7944 7.2226 8.5387

T1L1 1.0977 1.3155 1.5480 1.8002 2.0652

T2L2 1.3276 1.8199 2.3510 2.8983 3.4519


T4L4 2.5459 3.4973 4.5214 5.5760 6.6390

V1 1.1086 1.2798 1.4502 1.6378 1.8394

V2 1.0230 1.5165 2.0135 2.5117 3.0101

V3 1.5973 2.3037 3.0373 3.7767 4.5169

V4 2.0416 3.0292 4.0192 5.0072 5.9899

Vertical

V5 2.5610 3.7926 5.0245 6.2502 7.4643


L1T1 -0.0931 -0.0776 -0.0640 -0.0528 -0.0435


L3T3 -0.3038 -0.4365 -0.3109 -0.2157 -0.1526

T1L1 -70.0351 11.3563 13.2761 16.1697 19.8023


T3L3 1.1112 1.7627 2.6712 3.9805 5.7543

- 213 -


Bridge parameter BMB: L=80 m; F1=F2=F3=1.8 m; D1=D2=D3=180 mm

Bridge model C123 C123 C123 C123

Mass density m (kg/m2) 624.6 624.6 624.6 624.6

T1 (N) 6122724 10245396 16047212 23805545

T2 (N) 484638 4611580 10417533 18179555 Cable tension

T3 (N) 411926 3891469 10020773 16708540


L1T1 0.7500 1.0000 1.2500 1.5000

L2T2 0.8178 1.5169 2.1429 2.7083

L3T3 1.1312 2.2064 3.1613 4.0141

L4T4 1.4352 2.9030 4.1810 5.3106

L5T5 1.7783 3.6146 5.2038 6.5947


L6T6 2.1111 4.3190 6.2121 7.8481

T1L1 0.8863 1.1630 1.4784 1.7857

T2L2 1.1963 1.7686 2.4178 3.0336


T4L4 2.2633 3.4062 4.6679 5.8564

V1 0.9782 1.1883 1.4066 1.6342

V2 0.9588 1.5469 2.1386 2.6983

V3 1.4771 2.3379 3.2152 4.0460

V4 1.9062 3.0878 4.2614 5.3633

Vertical

V5 2.3827 3.8628 5.3195 6.6794


L1T1 -0.3935 -0.2067 -0.1114 -0.0725

L2T2 -0.8184 -0.4920 -0.2519 -0.1562 Coupled lateral-torsional

L3T3 -0.9588 -0.5699 -0.3106 -0.1960

T1L1 1.8875 3.3103 6.0801 9.3193


T3L3 0.6882 1.2251 2.2403 3.5513

From these tables, it can be seen that lower fundamental frequency always results in

larger coupling coefficient, and the coupling coefficients of the first coupled lateral-

torsional modes of bridge models with cable configuration C120 are much smaller

than those of the bridge models with cable configuration C123. The changes of the

coupling coefficients of the first coupled lateral-torsional modes are graphically

illustrated by the Figure 6.42. For the same bridge model, C120 or C123, when the

fundamental frequency increases, all the coupling coefficients of the coupled lateral-

torsional modes decrease while those of coupled torsional-lateral modes increase.

Among the coupling coefficients, those corresponding to the first coupled modes

- 214 -

L1T1 and T1L1 change more rapidly. These results show that when a suspension

bridge model has high fundamental natural frequency, the degree of coupling effect

is very small and the first couped modes trend to reduce to pure lateral and torsional

vibration modes though other higher lateral and torsional modes are still combined

together and appear as coupled vibration modes.





T1 (N) 15804217 26913402 42772492 63339410 Cable tension

T2 (N) 3235925 14346491 30207025 50774526


L1T1 0.5000 0.7500 1.0000 1.2500

L2T2 0.8930 1.4524 1.9753 2.4839

L3T3 1.3055 2.1558 2.9446 3.7061

L4T4 1.7231 2.8599 3.9086 4.9133

L5T5 2.1427 3.5634 4.8649 6.0989


L6T6 2.5503 4.2626 5.8087 7.2523

T1L1 0.8089 1.0496 1.3151 1.5926

T2L2 1.3083 1.7808 2.2962 2.8301


T4L4 2.5642 3.4640 4.4523 5.4767

V1 0.7583 0.9467 1.1583 1.3832

V2 1.0562 1.5358 2.0274 2.5233

V3 1.5912 2.3064 3.0410 3.7818

V4 2.1083 3.0673 4.0471 5.0319

Vertical

V5 2.6383 3.8360 5.0568 6.2805


L1T1 -0.2298 -0.1651 -0.1188 -0.0875


L3T3 -0.7190 -0.5305 -0.3760 -0.2638

T1L1 3.3989 4.9814 7.0116 9.5692


T3L3 1.0309 1.5655 2.2480 3.2340

For the footbridge models with the same cable configuration, the structural stiffness

varies with their fundamental natural frequencies or vice versa. Bridge models with

high fundamental natural frequencies are stiffer than those with low natural

frequencies. It is difficult to compare the dynamic behaviour of a footbridge model

with that of another model as they have different structural stiffness. To illustrate the

- 215 -

dynamic performance under crowd walking dynamic loads at different pacing rates,

Figure 6.43 and Figure 6.44 show the DAFs of different bridge models in steady

vibration with damping ratio of 0.01 and 0.05 respectively when they resonate at the

fundamental lateral natural frequencies. Here the degree of synchronization of the

crowd walking dynamic loads is assumed to be 20%.


Bridge parameter BMB: L=120 m; F1=F2=F3=1.8 m; D1=D2=D3=240 mm



T1 (N) 17867592 26757930 40686974 59158823

T2 (N) 625855 9527164 23468770 41954705 Cable tension

T3 (N) 620962 9520358 23456616 41945261


L1T1 0.5000 0.7500 1.0000 1.2500

L2T2 0.6730 1.2941 1.8584 2.3920

L3T3 0.9527 1.9003 2.7558 3.5617

L4T4 1.2407 2.5133 3.6553 4.7272

L5T5 1.5310 3.1301 4.5544 5.8846


L6T6 1.8187 3.7435 5.4468 7.0260

T1L1 0.6462 0.8973 1.1873 1.4882

T2L2 1.0959 1.5624 2.0988 2.6565


T4L4 2.1284 3.0462 4.0822 5.1541

V1 0.6432 0.8409 1.0657 1.3027

V2 0.9044 1.3860 1.8991 2.4128

V3 1.3583 2.0802 2.8478 3.6144

V4 1.7956 2.7679 3.7918 4.8085

Vertical

V5 2.2376 3.4615 4.7384 5.9864


L1T1 -0.6137 -0.3162 -0.1680 -0.1019


L3T3 -1.0626 -0.6812 -0.4512 -0.2923

T1L1 0.9861 2.0729 3.9039 6.4342


T3L3 0.9101 0.9964 1.4933 2.2980

From Figure 6.43, it can be seen that for the bridge model C120 with span of 40 m,

the DAF curve fluctuates with the natural frequency and reaches its maximum value

at 0.75 Hz. While for the other bridge models, the DAFs always increase with the

natural frequency. For the bridge models with cable configuration C120, shorter

- 216 -

bridge models have larger DAF; and for the bridge models with cable configuration

C123, the DAFs of longer bridge spans (L=120 m) are larger than those of the

shorter one (L=80 m). These results further confirm that the coupling coefficient has

significant effect on the DAF for bridge model with coupled vibration modes.

-0.70

-0.60

-0.50

-0.40

-0.30

-0.20

-0.10

0.00

0.50 0.75 1.00 1.25 1.50

Lateral natural frequency (Hz)

Cou

plin

g co

effi

cien

t

C120 (L= 40 m; D1= 90 mm)C120 (L= 80 m; D1=180 mm)C120 (L=120 m; D1=240 mm)C123 (L= 80 m; D1=180 mm)C123 (L=120 m; D1=240 mm)

Figure 6.42 Coupling coefficients of the first coupled mode L1T1 with lateral natural

frequency

0

20

40

60

80

100

120

140

0.50 0.75 1.00 1.25 1.50


Dyn

amic

am

plif

icat

ion

fact

or


Figure 6.43 Dynamic amplification factors with the natural frequency (ζ=0.01)

- 217 -

0

4

8

12

16

20

0.50 0.75 1.00 1.25 1.50


Dyn

amic

am

plif

icat

ion

fact

or


Figure 6.44 Dynamic amplification factors with the natural frequency (ζ=0.05)

From Figure 6.44, it can be seen that DAF is also affected by damping. It is found

that damping has greater effect on the DAF at higher natural frequency than that on

the DAF at lower frequency. When the damping ratio increases to 0.05, the curves of

DAFs of the bridge models with cable configuration C120 are much flatter when the

natural frequency is greater than 0.75 Hz and have relatively smaller values at 1.0

Hz; while for the DAFs of bridge model with cable configuration C123 still increase

with the natural frequency.

6.5 Summary

In this chapter, the dynamic performance of slender suspension footbridges under

human-induced synchronous excitation was simulated by resonant vibrations. The

synchronous excitation was modelled as crowd walking dynamic loads which consist

of lateral dynamic force, vertical dynamic force and vertical static force. A series of

numerical analysis was carried out to reveal the dynamic characteristics of slender

suspension footbridges with coupled vibration modes. Studies have been conducted

to investigate the effects of structural parameters such as cable section, cable sag and

- 218 -

span length, etc. on the dynamic behaviour. The main research findings of this

chapter can be summarized as the followings:

• The dynamic behaviour of slender suspension footbridges is quite complex.

When a slender suspension footbridge resonates under crowd walking dynamic

loads, the lateral resonant vibration increases to its maximum value and become

steady vibration after several fluctuations; while the vertical resonant vibration

increases to its maximum peak value directly and becomes steady vibration

without fluctuation.

• When crowd walking dynamic loads are applied on the entire footbridge deck,

the one half-wave coupled lateral-torsional vibration mode and one half-wave

vertical mode are easy to be excited, while the one half-wave coupled torsional-

lateral mode is not. Higher order vertical modes and coupled lateral-torsional

modes can also be excited by groups of walking pedestrians under certain

conditions.

• The amplitude of tension forces in the cable systems is quite small in lateral

resonant vibration but significant in vertical resonant vibration.

• When pedestrians walk cross a slender footbridge at a pacing rate other than

natural frequencies, the vibrations in lateral and vertical directions are very

small. Excessive vibrations are induced when pedestrians walk near/or at a

pacing rate corresponding to one of the natural frequencies.

• For slender suspension footbridges, the effects of damping on the lateral and

vertical resonant vibrations are different. Damping has only a small effect on

the lateral resonant vibration but significant effect on the vertical one.

• For bridge model C120 (no side cables), the vertical static force has significant

effect on the lateral vibration but only slight effect on the vertical one; while for

the bridge model C123 (with side cables), it has only slight effect on both

vertical and lateral vibrations.

- 219 -

• Under eccentric walking dynamic loads (distributed on half deck width),

excessive lateral deflection can be caused by pedestrians walking at different

pacing rates. When the pacing rate corresponds to the frequency of the coupled

mode L1T1, the excessive lateral vibration is induced by the lateral dynamic

force. If pedestrians walk at the pacing rate corresponding to frequency of

vertical mode V1, vertical resonant vibration is induced and large lateral

deflection is caused by the eccentric vertical loads. When the coupled mode

T1L1 is excited, the lateral deflection has large amplitude and increasing mean

value accompanied by large vertical deflection.

• For slender suspension footbridge structures with coupled vibration modes, the

coupling coefficient (defined in this thesis) is an important factor which has

significant effect on lateral dynamic performance, particularly on the dynamic

amplification factors. In general, large dynamic amplification factor for the

lateral vibration is expected when a bridge structure has the fundamental

coupled lateral-torsional mode with a small coupling coefficient.

• Coupling coefficients as well as the lateral vibrations are affected by structural

parameters such as cable section, cable sag and bridge span. The coupling

coefficient decreases with the increase of cable section or cable sag, but it

increases with the increase of bridge span.

• Slender footbridge structures with lower natural frequency always have

coupled lateral-torsional vibration modes with larger coupling coefficients and

smaller dynamic amplification factors when they resonate with coupled

vibration modes.

• Suspension footbridges with side pre-tensioned cables always have

fundamental coupled lateral-torsional modes with large coupling coefficients

and small dynamic amplification factors as well as small lateral deflections.

- 220 -

- 221 -

Conclusions and discussions

7.1 Conclusions

Due to the application of light weight and high strength material as well as the

development of new engineering technology, modern footbridges can cross longer

spans and be constructed lighter and more slender than ever to satisfy the

transportation needs and aesthetical requirements. Such slender footbridge structures

often have low stiffness, low structural mass, low natural frequencies and low

damping ratio and they are prone to vibration induced by human activities. When the

footbridge structures have natural frequencies within the frequency range of human

activities such as walking and running, they can experience synchronous excitations

and suffer unexpected excessive vibrations and serious serviceability problems. This

phenomenon has been shown by the poor dynamic performances of the Millennium

Bridge in London, T-Bridge in Japan and etc. However, bridge design codes

worldwide do not provide sufficient guidelines for such kind of slender bridge

structures to address the human-induced synchronous excitation although this

phenomenon has been known for a long time. This is because slender footbridge

structures often have dynamic behaviours different from those of traditional rigid or

stiffened footbridge structures and there is lack of knowledge on how

synchronization develops on footbridges and other slender infrastructures. This thesis

aimed to address this knowledge gap.

In this thesis, a conceptual study has been carried out to comprehensively investigate

the dynamic characteristics of slender footbridges under human-induced dynamic

loads. A slender suspension footbridge model in full size with pre-tensioned reverse

profiled cables in the vertical plane and side pre-tensioned cables in the horizontal

plane has been proposed for this purpose. Two structural analysis software packages,

Microstran and SAP2000 have been employed in the extensive numerical analysis.

Microstran was used to study the load deformation performance and vibration

7

- 222 -

properties as well as the effects of key structural parameters. SAP2000 was adopted

to investigate the dynamic performance under synchronous excitations when

pedestrians walk at different pacing rates coinciding with the low natural frequencies

of the bridge. A similar scaled physical bridge model was designed and constructed

in the laboratory for experimental testing and calibration to ensure the accuracy of

computer simulations. The synchronous excitation induced by walking has been

modelled as crowd walking dynamic loads which consist of dynamic vertical force,

dynamic lateral force and static vertical force. The dynamic behaviour under

synchronous excitation is simulated by resonant vibration at the pacing rate which

coincides with a natural frequency of the footbridge structure. Based on the extensive

numerical analyses, the main research findings can be summarized as the follows:

• Suspension footbridges with reverse profiled cables have variable structural

stiffness and vibration properties. This feature can be used to improve the static

and dynamic performance of slender structures.

• Slender suspension footbridges always have coupled vibration modes: coupled

lateral-torsional modes and coupled torsional-lateral modes. The mode

corresponding to lowest frequency is often the coupled lateral-torsional mode.

• When the coupled mode L1T1 is excited by walking pedestrians or the coupled

mode T1L1 is excited by walking pedestrians eccentrically distributed on

bridge deck, excessive lateral vibration can be induced.

• Coupling coefficient is an important factor to describe the degree of coupling in

a coupled mode, and it has significant effect on the dynamic amplification

factor of lateral vibration. Large dynamic amplification factor (DAF) is

expected for resonant lateral vibration in a coupled mode with small coupling

coefficient.

• Pre-tensioned reverse profiled side cables can significantly reduce lateral

deflection of slender suspension footbridges, and they can also improve the

- 223 -

coupled vibration modes, and hence improve the lateral dynamic performance

and reduce the dynamic amplification factor (DAF).

7.2 Discussions

Based on the research findings in this thesis, some important aspects on the proposed

suspension footbridge structure and its potential application are discussed.

• Frequency tuning

According to Bachmann [2002], the vertical and lateral natural frequencies of

footbridges in the ranges of 1.6 Hz to 2.4 Hz and 0.7 Hz to 1.3 Hz respectively

should be always avoided. A traditional suspension footbridge usually has only

catenary cables to support the whole bridge structure and resist loads, as the

(structurally) weak deck has only a small contribution to the structural stiffness. As a

consequence, its vibration properties depend mainly on the tension forces in the

catenary cables and bridge mass, and they are very difficult to be altered. On the

other hand, the proposed suspension footbridge consists of pre-tensioned reverse

profiled bottom and/or side cables besides the catenary supporting cables. Therefore

it has more design factors to improve its structural stiffness and dynamic

performance than a traditional suspension footbridge has. Its vibration properties can

be changed easily by introducing different pre-tensions into the reverse profiled

cables without changing the cable profiles. This feature can be used to shift the

natural frequency away from some critical frequency ranges to avoid excessive

vibration.

In general, the natural frequencies go up when pre-tensions are introduced in the

reverse profiled bottom and/or side cables; such frequency tuning might be

reasonable and practical for footbridge structures with natural frequencies around or

greater than 2.0 Hz in vertical direction and/or 1.0 Hz in lateral direction. The natural

frequencies of such structures go up rapidly and the pre-tension forces can be kept at

a reasonable level. For very long span footbridges, their fundamental natural

- 224 -

frequencies are always low and the frequencies of higher modes may be within the

critical frequency ranges of normal walking.

It should be noted that suspension footbridges are tensile structures. Tensile

structures are often slender and flexible, and their behaviour is much complicated

and quite different from those of the footbridges with other structural forms such as

beam or truss. For beam or truss type footbridge structures, the structural stiffness

and vibration properties are usually changed at the same time when the structures are

stiffened. While for suspension footbridges, the structural stiffness depends mainly

on the cable sags and cable sections, but the natural frequencies depend mainly on

the tension forces which are affected by the cable sags. When the cable sags remain

the same, the structural stiffness can be improved significantly by increasing the

cable sections; however, this does not mean the natural frequencies will be

significantly changed at the same time. Furthermore, the structural stiffness can be

improved by increasing the cable sags, but the natural frequencies will go down as

the tension forces decrease (at the same time). The structural stiffness must be

carefully checked if frequency tuning is made by changing the cable sags.

• Lateral vibration and coupling coefficients

Pedestrians are much more sensitive to low frequency lateral vibration when walking

or running than to the vertical vibration. The acceptable amplitudes of acceleration

and deflection in vertical direction are five times of those in the lateral direction

[Bachmann 2002]. On the other hand, suspension footbridges always have much

weaker structural stiffness in the lateral direction than in the vertical direction, and

they are in danger of suffering excessive lateral vibrations. Therefore vibration

control in lateral direction is important for the serviceability of slender suspension

footbridges.

For slender suspension footbridges with natural frequencies in the walking frequency

range, excessive lateral vibration can occur under the following two loading

conditions: pedestrians walking at a pacing rate coinciding with a lateral natural

frequency; or pedestrians walking eccentrically on bridge deck at a pacing rate

coinciding with a vertical or lateral natural frequency. Under the former loading

- 225 -

condition, large amplitude lateral vibration is often expected due to the resonant

vibration. While under the latter loading condition, the lateral vibration is mainly

caused by the slenderness and effect of eccentrically distributed load. It is quite

common on real footbridge structures that pedestrians walk eccentrically along

bridge width on the deck when crossing the footbridges. Since this lateral vibration is

induced by eccentrically distributed vertical load, it is independent of the phases of

footfalls and may make more pedestrians be aware of the lateral vibration and hence

trigger the synchronous lateral excitation. This can happen more easily on slender

footbridges which have nearly integer frequency ratios between vertical and lateral

natural frequencies, as it is probably convenient for pedestrians to adjust their

footfalls to the pacing rates coinciding with the bridge vibrating at its lateral natural

frequency. For example, the Millennium Bridge in London suffered excessive lateral

vibration on the central span at frequencies of just under 0.5 and 1.0 Hz. It was found

that the footbridge has vertical frequencies of 1.15, 1.54, 1.89 and 2.32 Hz and lateral

frequencies of 0.475 and 0.95 Hz at the central span. The vertical frequency of 1.89

Hz is about 4 and 2 times of the lateral frequency of 0.475 and 0.95 Hz respectively

[Blekherman 2005].

For a slender suspension footbridge with coupled vibration modes, it is found that the

resonant lateral vibration is not only affected by the natural frequency, but also

significantly affected by the coupling coefficient of the excited vibration mode. In

general, larger dynamic amplification factor of the lateral vibration is expected when

a bridge structure has the coupled lateral-torsional mode with a smaller coupling

coefficient. This result indicates that lateral dynamic response can be reduced by

increasing the value of coupling coefficient. As coupling coefficients describe the

degrees of coupling of coupled vibration modes, they are influenced by many

parameters such as cable sags, sections and bridge spans. In general, the values of

coupling coefficients go up when the cable sags and/or cable sections decrease, and

footbridges with high fundamental lateral frequencies often have vibration modes

with small coupling coefficients. It seems that for slender suspension footbridges,

there is conflict between improving structural stiffness and reducing dynamic

amplification factor. Increasing cable sections can greatly improve the structural

stiffness, but it will also increase the dynamic amplification factor. This implies that

the dynamic performance of slender suspension footbridges is very complex and

- 226 -

comprehensive consideration should be taken when vibration problems are dealt

with.

• Pre-tensioned reverse profiled cables

Modern suspension footbridges are always slender and encounter more serviceability

problems rather than safety ones due to the application of high strength and light

weight materials. On one hand, the cable materials such as alloy steel, carbon fibre

and aramid fibre have high strengths to guarantee the safety and enable suspension

bridges to cross long spans. On the other hand, the applications of light weight

materials such Aluminium, fibre reinforced polymer (FRP) make the girder and deck

system lighter than ever and hence greatly reduce the tension forces in the supporting

cables. For such slender and light footbridges, the main concern is often the

serviceability rather the strength.

With pre-tensioned reverse profiled cables, suspension footbridges have extra design

parameters which make the vibration properties adjustable to some extent. When the

reverse profiled cables in vertical plane are pre-tensioned, the vertical stiffness is

contributed by both the catenary supporting cables and reverse profiled cables, and

the vertical structural performance can be significantly improved. However, the

effect on the lateral dynamic performance is quite complicated. On the one hand, the

lateral stiffness can be also improved as the total tension forces in the cable systems

increases, and more lateral component can be provided to resist lateral load. On the

other hand, the coupling coefficient of the fundamental coupled lateral-torsional

mode decreases and large dynamic amplification factors can be expected. The lateral

dynamic performance can be improved significantly by introducing pre-tensioned

reverse profiled cables in the horizontal plane. When these reverse profiled cables in

the horizontal plane are pre-tensioned, the lateral stiffness can be noticeably

enhanced and lateral deflection caused by lateral loads or eccentric vertical loads can

be reduced considerably. Furthermore, the coupling coefficient of the fundamental

coupled lateral-torsional mode increases and hence the dynamic amplification factor

is small.

- 227 -

Though the pre-tensioned cables in the horizontal plane can improve the coupled

vibration modes, the pre-tensions in the reverse profiled cables in vertical and

horizontal planes should be comprehensively considered to obtain reasonable

coupling coefficients. Otherwise, some unexpected vibration problems such as

unstable vibration may arise due to the irrational vibration modes and their

combination. It also should be noted that when the pre-tensions are introduced in the

reverse profiled cables, extra horizontal force in the longitudinal direction can be

produced. It is this horizontal force that makes the suspension footbridges to have

variable vibration properties. However, this force will increase the cost of anchorage.

7.3 Contributions to scientific knowledge

The major contributions of this research project to scientific knowledge can be

summarized as:

• A suspension footbridge model with pre-tensioned reverse profiled cables has

been proposed. This innovative structure can have variable structural stiffness

and vibration properties by selecting different structural parameters. These

features have been used in studying the dynamic performance of slender

footbridges under human-induced walking loads which cover a range of low

frequency. These features can also be applied to improve the structural

behaviour of such slender footbridges.

• Extensive numerical analyses have been carried out on the proposed bridge

model to investigate its static and dynamic performance. The research findings

have expanded the knowledge base of dynamics and enhanced our

understanding on the complex dynamic behaviour of slender suspension

footbridges, and the research results are useful to develop design guidance to

ensure the safety and serviceability of such slender structures with social and

economic benefits and to develop technique to retrofit existing “lively” bridge

structures.

• A frequency dependant force model has been proposed to model crowd

walking dynamic loads. As this force model consists of vertical dynamic force,

- 228 -

lateral dynamic force and vertical static force, it has been used in non-linear

time history analysis to simulate human-induced synchronous excitations with

different pacing rates.

• A coupling coefficient has been defined to describe the coupled vibration

modes. It is found that this coupling coefficient has significant effect on the

dynamic amplification factor of lateral resonant vibration and the complex

dynamic behaviour of slender suspension footbridges can be reasonably

explained by this coefficient.

7.4 Suggestions for future work

The dynamic performance of slender suspension footbridges with low stiffness, low

mass, low natural frequencies and low damping ratio is much more complex than

rigid or stiffer footbridges. Further extensive and systematic research is required to

better understand their dynamic behaviour and to ensure the safety and vibration

serviceability. The following topics are suggested for future research in this area:

• Research is required to investigate the effects of pedestrians on the lateral

dynamic performance of slender suspension footbridges. Pedestrians are not

only the source of dynamic excitation, but also cause the changes of structural

mass, vibration properties and structural damping.

• Investigations are needed to establish the relationship between the coupling

coefficient and pre-tensions in the reverse profiled cables in vertical and

horizontal planes, and the detailed relationship between coupling coefficient

and dynamic amplification factor.

• Research is required to study the lateral dynamic performance of slender

footbridges with higher vibration modes excited by groups of pedestrians when

their higher natural frequencies are within the frequency range of normal

human activities such as walking and running.

- 229 -

• Research is needed to investigate the effects of geometrical parameters on the

dynamic performance in vertical and lateral directions. The geometrical

parameters to be considered include the width of deck, locations of cables,

inclined cables, handrails and flexible hangers etc.

- 230 -

- 231 -

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Documents

Dynamic Characteristics of Slender Suspension …characteristics of slender suspension footbridges under human-induced dynamic loads and a footbridge model in full size with pre-tensioned