June 10, 2000 the Millennium Pedestrian Bridge was open to the public. Built to mark the new millennium, the celebrated footbridge spanned 333m over the River Thames, linking the City of London. On opening day, an estimated upwards of 100,000 people gathered to cross with a maximum of 2000 on the bridge at a given time. As soon as pedestrians began to walk across the bridge there was immediate and excessive lateral movement. The movement became so large that people found it hard to maintain there balance. To remedy the situation an occupancy limit was enforced in an attempt to reduce the swaying. Ultimately, fearing public safety concerns, the bridge was closed for investigation on June 12, 2000, only two days after opening (Dallard et al. 2001b; Newland 2003).

Awarded to and conceived by Arup engineers, Foster & Partners architects, and sculptor Sir Anthony Caro, the Millennium Bridge conveyed a minimal, slender design. The bridge was composed of three separate spans; north 81m, center 144m, south 108m, connected by shallow cables fixed at each support by steel brackets on slender concrete piers.

Superstructure
The bridge is a shallow suspension bridge with two group of four 120mm cables anchored side by side anchored to the opposite abutments. The low cable design was to allow for unobstructed views of the city. Transverse steel box sections spanned the cable groups under the deck every 8m. This supported the 4m wide deck comprised of aluminum box sections bridging between two edge tubes. At the river supports the cables are saddled through a steel V bracket connected to a tapering concrete pier via high strength pretensioned steel. It is important to note that the bridge deck is supported solely by the steel box sections carried by the cables, the deck does not touch the piers (Dallard et al. 2001a; Dallard et al. 2001b).

Substructure
The two river supports were each comprised of tapered concrete piers supported large 6m concrete caissons cast into the river bed. The concrete abutments at both the North and South ends were placed on cast-in-place concrete piles underneath a reinforced concrete pilecap (Dallard et al 2001a). Dynamic analysis was performed on the structure to calculate the dynamic properties of the structure under prescribed loading effects. This included all gravity load analysis as well as lateral design from wind loading and pedestrian traffic. All dynamic properties including the natural frequencies coincided with similar structures.

Fig. 1: SDOF EOM Formulation (Courtesy: FEMA)

Structural Dynamics

The dynamic response of structures is broken into two separate categories, single degree of freedom (SDOF) and multi-degree of freedom (MDOF) systems. These are defined by the minimum number of variables used to describe the movement of a structure. Consider the SDOF shear frame in Figure 1, where the mass is excited by horizontal force F(t). The movement of the mass is defined in one direction as lateral displacement to the right.

Using Newton’s second law of motion and principles of static equilibrium, the equation of motion (EOM) for the frame can be expressed as the differential equation m(a) + c(v) + k(u) = F(t). Mass(m), damping(c), and stiffness(k), are defined properties of a structure while the corresponding differential motion parameters are acceleration(a), velocity(v), and displacement(u). This expression can also by expressed by the time varying function Fi(t) + Fd(t) + Fs(t) = F(t), where Fi, Fd, and Fs represent the mass inertial force, damping force, and spring force respectively. Altering the system properties will inherently change the response of the system.

The natural frequency (wn) at which a system vibrates is given by wn= sqrt(k/m). Therefore, as stiffness is increased while mass remains unchanged, the structure will vibrate at a quicker pace. Damping is the process in which vibration is steadily decreased. The energy of the vibrating structure is diminished through various mechanisms in the structure itself or by outside influences such as dampers. The damping property of a structure is shown by the damping coefficient Z=c/ccr where c is the damping provided and ccr is the critical damping needed to prevent vibration.

These basic principles can be applied to MDOF systems as well. In the case of the Millennium bridge, the motion of the bridge has many degrees of freedom represented by vertical, horizontal, and torsional movement. Rather than one single response, or natural shape, the structure has a number of shapes called modes shapes. Depending on the forcing function and initial conditions, the structure can vibrate in any single or combination of these shapes. With each shape comes a corresponding frequency needed to produce that shape. This concept is shown below with a three story shear frame subjected to harmonic excitation. The lateral movement of each floor can be idealized and compared to the lateral movement of each bridge span. The separate scalar equations of motion for each floor is represented as a combined EOM in matrix form, [m]{a} + [c]{v} + [k]{u} = p*sin(t). Deriving the response then involves a modal eigenvalue analysis to compute the natural frequencies and mode shapes of the structure. For a more in depth discussion on structural dynamics as it relates to MDOF and modal analysis please refer to Dynamics of Structures: Theory and Applications to Earthquake Engineering (Chopra 2012).

Fig. 3: Lateral Mode Shapes (Courtesy: FEMA)

Fig. 2: MDOF EOM Formulation (Courtesy: FEMA)

Investigation

Fig. 4: Resonance (Courtesy: FEMA)

Based on eyewitness and video evidence it was clear that the lateral vibration of the bridge was caused by lateral pedestrian loading. The theory was that the lateral frequency produced by crowd movement coincided with several lateral and torsional modal frequencies of the bridge causing large oscillations. This phenomenon is known as resonance and occurs when the ratio of forcing to natural frequencies (BETA) equals 1. While vertical and lateral crowd motion was considered, lateral crowd syncrony, or in tune walking, was not a a loading effect that was anticipated in design (Dallard et al. 2001b).

To prove this theory, modal response testing was performed on the bridge to validate the dynamic properties calculated by the original designers and to predict the forces exerted in the bridge by pedestrians. This was done by way of Frequency Response Curve (FRC) generation. Essentially, an FRC represents the steady state response

Fig. 5: Crowd Results (Courtesy: Strogatz et al. 2005)

response of a structure at different harmonic forcing frequencies (Pavic et al. 2002). The curve allows for accurate estimation of the natural frequencies, damping, and mode shapes of the structure. Ultimately, the combined response and parameters will help govern the proper retrofit strategies.

Fugro Structural Monitoring was hired by Arup to develop and install a horizontal shaker and preform data acquisition. The shaker developed consisted of moving a 1000kg mass at forces high enough to excite the bridge at a wide range of frequencies. Resonant frequencies were found by increasing the movement of the shaker mass until the oscillation of the bridge mirrored it (Dallard et al. 2001b).

Arup also commissioned testing using actual crowds to measure displacement to compare with video evidence. As the adjacent figure shows, when crowds of around 100 crossed the bridge, a slight wobble would occur as a result of clashing loading and bridge modal frequencies. Consequently, this wobble feedback would cause larger crowd to step generally in unison producing large oscillations. This effect is also illustrated in the phase coherence, or resonance, among pedestrians quantified by the order parameter. Smaller crowds produced an unsynchronized or out of phase order parameter shown by a fluctuating order parameter. However, critical crowd sizes generated a more in-phase feedback. This represents crowds walking in sync (Stogatz et al. 2005).

Both testing methods confirmed the original modal properties of the bridge identified susceptible modal frequencies. Crucial lateral modes were identified in the range of 0.5 to 1.0Hz. While Arup considered lateral crowd vibration of approximately 2.0Hz, they did not calculate for large crowd syncrony which lowers the forcing function to near 1.3Hz, close enough to resonate with several modal frequencies (Taylor). To combat this problem, all crucial modal frequencies would need to raised above the resonant frequency of 1.3Hz (Dallard et al. 2001a).

Retrofit

Based on the fundamental principles of structural dynamics the lateral displacements of the bridge could be reduced by limiting the number of people on the bridge at a given time, increasing the lateral stiffness, and/or adding damping. Based on feedback from the city, limiting the bridge capacity was not an option.

Stiffness
This measure would entail adding bracing or additional piers and essentially shift the frequencies so they are in a range no longer problematic. Investigation of the pedestrian induced lateral vibration concluded that resonance could occur at any frequency below 1.3Hz. In order to reduce possibility of excessive vibrations, all frequencies corresponding to mode shapes with large lateral movements would need to be greater. It was found that the first lateral mode at the center span had the lowest natural frequency at approximately 0.5Hz, roughly 3 times less than the operating frequency. Idealizing that the mass of the bridge will not change, the stiffness would need to be nine times greater to eliminate movement. Increasing stiffness nine-fold would require extensive and complex structural additions, essentially ruining the unique architecture of the bridge. Thus the stiffening option was not ideal (Fitzpatrick and Ridsdill Smith 2001).

Fig. 6: Frictionless Hermetic Fluid Damper (Courtesy: Taylor Devices Inc.)

Damping
Based primarily on feasibility and maintaining the original design of the bridge, damping was chosen to dissipate bridge movement. While every structure has a designed level of damping through its inherent material properties, damping can be added through the use damping devices. These devices function to mitigate vibrations of a structure through passive or active systems. Active systems are essentially computer controlled force generators that actively push on or counteract a movement in a structure. Passive dampers dissipate energy created from relative movements and often require no computer control. Using the frequency of mode shape one as it relates to the critical damping, 0.49Hz requires damping of over 20% critical damping, compared to roughly 0.5% designed for the structure (Fitzpatrick and Ridsdill Smith 2001). In order to reach this level of critical damping several damper type were examined, including passive tuned mass dampers, passive fluid viscous dampers, and active control. Fluid dampers were eventually chosen for their small weights, fatigue and maintenance lifespans, and easy installation. Passive fluid dampers were also chosen for their ability to achieve damping at a wide range of frequencies and small amplitudes. The Millennium Bridge required control of frequencies ranging from 0 to 5Hz and small amplitudes measuring 0.025 to 25mm. A total of 37 viscous dampers, specially designed by Taylor Devices Inc., were installed to decrease vibration amplitudes in the crucial lateral and torsional modes. Dampers were placed primarily under the bridge deck and on top of the transverse box sections to reduce lateral deck and pier movement as well as vertical deck to ground movement. After several crowd tests, it was concluded that the retrofit was successful. The damping solution reduced the dynamic response by a factor of 40 and no resonance or crowd syncrony was reported. The Millennium Bridge officially re-opened to the public on February 22, 2002. Estimates predict the bridge to be used by approximately 4 million people per year (Taylor).

Photo 3: Horizontal Chevron Damper (Courtesy: Taylor Devices Inc.)

Photo 2: Vertical Dampers (Courtesy: Taylor Devices Inc.)

Conclusions

While the design engineers followed every protocol for the structural and dynamic design of the Millennium bridge, they did not foresee the formation and effects of the phenomenon that is crowd step syncrony. However, the study of this loading effect along with the modal testing and damping techniques have proven useful in the design considerations of similar pedestrian bridges and structures. In today's design and construction processes, buildings, bridges, and other structures are becoming lighter and more slender. This is a result of a combination of factors including improved composite materials, a call for more sustainable structures, and an the overall sophistication of modern structural design. As illustrated by the Millennium bridge, serviceability performance such as deflection and specifically vibration are now governing considerations. While the bridge was not in danger of immediate structural collapse, the deflection was excessive and considered a failure. Consequentially, new research in crowd loading and damping techniques have adapted to these serviceability issues. Tuned mass and fluid viscous dampers are now being studied and installed in structures around the world to decrease vibration issues.

Chopra, Anil K. (2012). Dynamics of Structures: Theory and Applications to Earthquake Engineering. 4th ed. Englewood Cliffs, N.J.: Prentice Hall

Textbook: Structural Dynamics and Earthquake Engineering

Dallard, P., Fitzpatrick, T., Flint, A., Low, A., Ridsdill Smith, R., Willford, M., and Roche, M. (2001). "London Millennium Bridge: Pedestrian-Induced Lateral Vibration." J. Bridge Eng., 6, 412-417. (October 3, 2015). http://ascelibrary.org/doi/pdf/10.1061/(ASCE)1084-0702(2001)6:6(412)

Journal: Highlights the response of pedestrian-induced lateral vibration on footbridges. Discusses retrofit designs.

Dallard, P., Fitzpatrick, T., Flint, A., Le Bourva, S., Low, A., Ridsdill Smith, R., and Willford, M. (2001). “The London Millennium Footbidge.” The Structural Engineer, 2001, 79, no.22, 17-33. (October 3, 2015). https://researchcourse.pbworks.com/f/structural+engineering.pdf

Journal: Description of the Millennium Bridge as well as retrofit ideas to control vibrations (fluid viscous dampers, tuned mass dampers)

Pavic, A., Reynolds, P., Wright, J., and Armitage, T. (2002). "Methodology for Modal Testing of the Millennium Bridge, London." Proceedings of the ICE - Structures and Buildings, May 2002, 111-21. (October 3, 2015). http://www.icevirtuallibrary.com/doi/pdf/10.1680/stbu.2002.152.2.111

Journal: Modal testing was used on the Millennium Bridge to create a Frequency response curve. This method as opposed to the original vibration response analysis on the bridge is crucial in determining the modal contributions to the lateral displacement.

Website: Manufacturer and installer of the dampers.

Dallard, P., Fitzpatrick, T., Flint, A., Low, A., Ridsdill Smith, R., Willford, M., and Roche, M. (2001). "London Millennium Bridge: Pedestrian-Induced Lateral Vibration." J. Bridge Eng., 6, 412-417. (October 3, 2015). Journal: Highlights the response of pedestrian-induced lateral vibration on footbridges. Discusses retrofit designs. http://ascelibrary.org/doi/pdf/10.1061/(ASCE)1084-0702(2001)6:6(412) Dallard, P., Fitzpatrick, T., Flint, A., Le Bourva, S., Low, A., Ridsdill Smith, R., and Willford, M. (2001). “The London Millennium Footbidge.” The Structural Engineer, 2001, 79, no.22, 17-33. (October 3, 2015). Journal: Description of the Millennium Bridge as well as retrofit ideas to control vibrations (fluid viscous dampers, tuned mass dampers) https://researchcourse.pbworks.com/f/structural+engineering.pdf

Daniel, Y., Lavan, O., and Levy, R. (2012) "Multi-Modal Control of Pedestrian Bridges Using Tuned-Mass-Dampers." Structures Congress 2012, 471-482. (October 3, 2015). Journal: Discussion of tuned mass dampers http://ascelibrary.org/doi/pdf/10.1061/9780784412367.042 Fitzpatrick, T., and Ridsdill Smith, R. (2001) "Stabilizing the London Millennium Bridge." Ingenia, 18-22. (October 3, 2015). Article: ARUP’s solution to damping the vibration http://www.ingenia.org.uk/ingenia/articles.aspx?Index=123 Newland, David E. (2003). "Vibration of the London Millennium Bridge: Cause and Cure." International Journal of Acoustics and Vibration 8, no. 1, 9-14. (October 3, 2015). Journal: Vibration discussion of the bridge. http://www.iiav.org/ijav/content/volumes/8_2003_710551272264886/vol_1/385_firstpage_196081286806609.pdf Pavic, A., Reynolds, P., Wright, J., and Armitage, T. (2002). "Methodology for Modal Testing of the Millennium Bridge, London." Proceedings of the ICE - Structures and Buildings, May 2002, 111-21. (October 3, 2015). Journal: Modal testing was used on the Millennium Bridge to create a Frequency response curve. This method as opposed to the original vibration response analysis on the bridge is crucial in determining the modal contributions to the lateral displacement. http://www.icevirtuallibrary.com/doi/pdf/10.1680/stbu.2002.152.2.111 Ricciardelli, F., and Pizzimenti, D. (2007). "Lateral Walking-Induced Forces on Footbridges." J. Bridge Eng., 6, 677-88. (October 3, 2015). Journal: Research used to modal lateral vibration due to walking-induced forces. Date used to develop design criteria for footbridges. http://ascelibrary.org/doi/pdf/10.1061/(ASCE)1084-0702(2007)12:6(677)

Chris S. Ulrich, M.Eng. Civil Engineering, Penn State, 2015## Introduction

## Table of Contents

## Background/Design

Awarded to and conceived by Arup engineers, Foster & Partners architects, and sculptor Sir Anthony Caro, the Millennium Bridge conveyed a minimal, slender design. The bridge was composed of three separate spans; north 81m, center 144m, south 108m, connected by shallow cables fixed at each support by steel brackets on slender concrete piers.SuperstructureThe bridge is a shallow suspension bridge with two group of four 120mm cables anchored side by side anchored to the opposite abutments. The low cable design was to allow for unobstructed views of the city. Transverse steel box sections spanned the cable groups under the deck every 8m. This supported the 4m wide deck comprised of aluminum box sections bridging between two edge tubes. At the river supports the cables are saddled through a steel V bracket connected to a tapering concrete pier via high strength pretensioned steel. It is important to note that the bridge deck is supported solely by the steel box sections carried by the cables, the deck does not touch the piers (Dallard et al. 2001a; Dallard et al. 2001b).

SubstructureThe two river supports were each comprised of tapered concrete piers supported large 6m concrete caissons cast into the river bed. The concrete abutments at both the North and South ends were placed on cast-in-place concrete piles underneath a reinforced concrete pilecap (Dallard et al 2001a). Dynamic analysis was performed on the structure to calculate the dynamic properties of the structure under prescribed loading effects. This included all gravity load analysis as well as lateral design from wind loading and pedestrian traffic. All dynamic properties including the natural frequencies coincided with similar structures.

The dynamic response of structures is broken into two separate categories, single degree of freedom (SDOF) and multi-degree of freedom (MDOF) systems. These are defined by the minimum number of variables used to describe the movement of a structure. Consider the SDOF shear frame in Figure 1, where the mass is excited by horizontal force F(t). The movement of the mass is defined in one direction as lateral displacement to the right.Structural DynamicsUsing Newton’s second law of motion and principles of static equilibrium, the equation of motion (EOM) for the frame can be expressed as the differential equation m(a) + c(v) + k(u) = F(t). Mass(m), damping(c), and stiffness(k), are defined properties of a structure while the corresponding differential motion parameters are acceleration(a), velocity(v), and displacement(u). This expression can also by expressed by the time varying function Fi(t) + Fd(t) + Fs(t) = F(t), where Fi, Fd, and Fs represent the mass inertial force, damping force, and spring force respectively. Altering the system properties will inherently change the response of the system.

The natural frequency (wn) at which a system vibrates is given by wn= sqrt(k/m). Therefore, as stiffness is increased while mass remains unchanged, the structure will vibrate at a quicker pace. Damping is the process in which vibration is steadily decreased. The energy of the vibrating structure is diminished through various mechanisms in the structure itself or by outside influences such as dampers. The damping property of a structure is shown by the damping coefficient Z=c/ccr where c is the damping provided and ccr is the critical damping needed to prevent vibration.

These basic principles can be applied to MDOF systems as well. In the case of the Millennium bridge, the motion of the bridge has many degrees of freedom represented by vertical, horizontal, and torsional movement. Rather than one single response, or natural shape, the structure has a number of shapes called modes shapes. Depending on the forcing function and initial conditions, the structure can vibrate in any single or combination of these shapes. With each shape comes a corresponding frequency needed to produce that shape. This concept is shown below with a three story shear frame subjected to harmonic excitation. The lateral movement of each floor can be idealized and compared to the lateral movement of each bridge span. The separate scalar equations of motion for each floor is represented as a combined EOM in matrix form, [m]{a} + [c]{v} + [k]{u} = p*sin(t). Deriving the response then involves a modal eigenvalue analysis to compute the natural frequencies and mode shapes of the structure. For a more in depth discussion on structural dynamics as it relates to MDOF and modal analysis please refer to

Dynamics of Structures: Theory and Applications to Earthquake Engineering(Chopra 2012).InvestigationBased on eyewitness and video evidence it was clear that the lateral vibration of the bridge was caused by lateral pedestrian loading. The theory was that the lateral frequency produced by crowd movement coincided with several lateral and torsional modal frequencies of the bridge causing large oscillations. This phenomenon is known as resonance and occurs when the ratio of forcing to natural frequencies (BETA) equals 1. While vertical and lateral crowd motion was considered, lateral crowd syncrony, or in tune walking, was not a a loading effect that was anticipated in design (Dallard et al. 2001b).

To prove this theory, modal response testing was performed on the bridge to validate the dynamic properties calculated by the original designers and to predict the forces exerted in the bridge by pedestrians. This was done by way of Frequency Response Curve (FRC) generation. Essentially, an FRC represents the steady state response

Fugro Structural Monitoring was hired by Arup to develop and install a horizontal shaker and preform data acquisition. The shaker developed consisted of moving a 1000kg mass at forces high enough to excite the bridge at a wide range of frequencies. Resonant frequencies were found by increasing the movement of the shaker mass until the oscillation of the bridge mirrored it (Dallard et al. 2001b).

Arup also commissioned testing using actual crowds to measure displacement to compare with video evidence. As the adjacent figure shows, when crowds of around 100 crossed the bridge, a slight wobble would occur as a result of clashing loading and bridge modal frequencies. Consequently, this wobble feedback would cause larger crowd to step generally in unison producing large oscillations. This effect is also illustrated in the phase coherence, or resonance, among pedestrians quantified by the order parameter. Smaller crowds produced an unsynchronized or out of phase order parameter shown by a fluctuating order parameter. However, critical crowd sizes generated a more in-phase feedback. This represents crowds walking in sync (Stogatz et al. 2005).

Both testing methods confirmed the original modal properties of the bridge identified susceptible modal frequencies. Crucial lateral modes were identified in the range of 0.5 to 1.0Hz. While Arup considered lateral crowd vibration of approximately 2.0Hz, they did not calculate for large crowd syncrony which lowers the forcing function to near 1.3Hz, close enough to resonate with several modal frequencies (Taylor). To combat this problem, all crucial modal frequencies would need to raised above the resonant frequency of 1.3Hz (Dallard et al. 2001a).

## Retrofit

Based on the fundamental principles of structural dynamics the lateral displacements of the bridge could be reduced by limiting the number of people on the bridge at a given time, increasing the lateral stiffness, and/or adding damping. Based on feedback from the city, limiting the bridge capacity was not an option.StiffnessThis measure would entail adding bracing or additional piers and essentially shift the frequencies so they are in a range no longer problematic. Investigation of the pedestrian induced lateral vibration concluded that resonance could occur at any frequency below 1.3Hz. In order to reduce possibility of excessive vibrations, all frequencies corresponding to mode shapes with large lateral movements would need to be greater. It was found that the first lateral mode at the center span had the lowest natural frequency at approximately 0.5Hz, roughly 3 times less than the operating frequency. Idealizing that the mass of the bridge will not change, the stiffness would need to be nine times greater to eliminate movement. Increasing stiffness nine-fold would require extensive and complex structural additions, essentially ruining the unique architecture of the bridge. Thus the stiffening option was not ideal (Fitzpatrick and Ridsdill Smith 2001).

DampingBased primarily on feasibility and maintaining the original design of the bridge, damping was chosen to dissipate bridge movement. While every structure has a designed level of damping through its inherent material properties, damping can be added through the use damping devices. These devices function to mitigate vibrations of a structure through passive or active systems. Active systems are essentially computer controlled force generators that actively push on or counteract a movement in a structure. Passive dampers dissipate energy created from relative movements and often require no computer control. Using the frequency of mode shape one as it relates to the critical damping, 0.49Hz requires damping of over 20% critical damping, compared to roughly 0.5% designed for the structure (Fitzpatrick and Ridsdill Smith 2001). In order to reach this level of critical damping several damper type were examined, including passive tuned mass dampers, passive fluid viscous dampers, and active control. Fluid dampers were eventually chosen for their small weights, fatigue and maintenance lifespans, and easy installation. Passive fluid dampers were also chosen for their ability to achieve damping at a wide range of frequencies and small amplitudes. The Millennium Bridge required control of frequencies ranging from 0 to 5Hz and small amplitudes measuring 0.025 to 25mm. A total of 37 viscous dampers, specially designed by Taylor Devices Inc., were installed to decrease vibration amplitudes in the crucial lateral and torsional modes. Dampers were placed primarily under the bridge deck and on top of the transverse box sections to reduce lateral deck and pier movement as well as vertical deck to ground movement. After several crowd tests, it was concluded that the retrofit was successful. The damping solution reduced the dynamic response by a factor of 40 and no resonance or crowd syncrony was reported. The Millennium Bridge officially re-opened to the public on February 22, 2002. Estimates predict the bridge to be used by approximately 4 million people per year (Taylor).

## Conclusions

While the design engineers followed every protocol for the structural and dynamic design of the Millennium bridge, they did not foresee the formation and effects of the phenomenon that is crowd step syncrony. However, the study of this loading effect along with the modal testing and damping techniques have proven useful in the design considerations of similar pedestrian bridges and structures. In today's design and construction processes, buildings, bridges, and other structures are becoming lighter and more slender. This is a result of a combination of factors including improved composite materials, a call for more sustainable structures, and an the overall sophistication of modern structural design. As illustrated by the Millennium bridge, serviceability performance such as deflection and specifically vibration are now governing considerations. While the bridge was not in danger of immediate structural collapse, the deflection was excessive and considered a failure. Consequentially, new research in crowd loading and damping techniques have adapted to these serviceability issues. Tuned mass and fluid viscous dampers are now being studied and installed in structures around the world to decrease vibration issues.## Related Links

Tacoma Narrows CollapseCable Bridge Failures Overview

Performance of Precast Structures Under Seismic Loading

## Bibliography

Chopra, Anil K. (2012). Dynamics of Structures: Theory and Applications to Earthquake Engineering. 4th ed. Englewood Cliffs, N.J.: Prentice HallDallard, P., Fitzpatrick, T., Flint, A., Low, A., Ridsdill Smith, R., Willford, M., and Roche, M. (2001). "London Millennium Bridge: Pedestrian-Induced Lateral Vibration." J. Bridge Eng., 6, 412-417. (October 3, 2015).http://ascelibrary.org/doi/pdf/10.1061/(ASCE)1084-0702(2001)6:6(412)Dallard, P., Fitzpatrick, T., Flint, A., Le Bourva, S., Low, A., Ridsdill Smith, R., and Willford, M. (2001). “The London Millennium Footbidge.” The Structural Engineer, 2001, 79, no.22, 17-33. (October 3, 2015).https://researchcourse.pbworks.com/f/structural+engineering.pdf

Daniel, Y., Lavan, O., and Levy, R. (2012) "Multi-Modal Control of Pedestrian Bridges Using Tuned-Mass-Dampers." Structures Congress 2012, 471-482. (October 3, 2015).http://ascelibrary.org/doi/pdf/10.1061/9780784412367.042

Fitzpatrick, T., and Ridsdill Smith, R. (2001) "Stabilizing the London Millennium Bridge." Ingenia, 18-22. (October 3, 2015).http://www.ingenia.org.uk/ingenia/articles.aspx?Index=123Newland, David E. (2003). "Vibration of the London Millennium Bridge: Cause and Cure." International Journal of Acoustics and Vibration 8, no. 1, 9-14. (October 3, 2015).http://www.iiav.org/ijav/content/volumes/8_2003_710551272264886/vol_1/385_firstpage_196081286806609.pdfPavic, A., Reynolds, P., Wright, J., and Armitage, T. (2002). "Methodology for Modal Testing of the Millennium Bridge, London." Proceedings of the ICE - Structures and Buildings, May 2002, 111-21. (October 3, 2015).http://www.icevirtuallibrary.com/doi/pdf/10.1680/stbu.2002.152.2.111Ricciardelli, F., and Pizzimenti, D. (2007). "Lateral Walking-Induced Forces on Footbridges." J. Bridge Eng., 6, 677-88. (October 3, 2015).http://ascelibrary.org/doi/pdf/10.1061/(ASCE)1084-0702(2007)12:6(677)Strogatz, S., Abrams, D., McRobie, A., Eckhardt, B., and Ott, E. (2005). "Theoretical Mechanics: Crowd Synchrony on the Millennium Bridge." Nature, 2005, 43-44. (October 3, 2015).http://www.nature.com/nature/journal/v438/n7064/abs/438043a.htmlTaylor, Douglas P. "DAMPER RETROFIT OF THE LONDON MILLENNIUM FOOTBRIDGE – A CASE STUDY IN BIODYNAMIC DESIGN." http://taylordevices.com/Tech-Paper-archives/literature-pdf/66-DamperRetrofit-London.pdf. (October 3, 2015).Dallard,

P., Fitzpatrick, T., Flint, A., Low, A., Ridsdill Smith, R., Willford, M., and Roche, M. (2001). "London Millennium Bridge: Pedestrian-Induced Lateral Vibration."J. Bridge Eng.,6, 412-417. (October 3, 2015).Journal: Highlights the response of pedestrian-induced lateral vibration on footbridges. Discusses retrofit designs.

http://ascelibrary.org/doi/pdf/10.1061/(ASCE)1084-0702(2001)6:6(412)

Dallard, P., Fitzpatrick, T., Flint, A., Le Bourva, S., Low, A., Ridsdill Smith, R., and Willford, M. (2001). “The London Millennium Footbidge.”The Structural Engineer,2001, 79, no.22, 17-33. (October 3, 2015).Journal: Description of the Millennium Bridge as well as retrofit ideas to control vibrations (fluid viscous dampers, tuned mass dampers)

https://researchcourse.pbworks.com/f/structural+engineering.pdf

Daniel, Y., Lavan, O., and Levy, R. (2012) "Multi-Modal Control of Pedestrian Bridges Using Tuned-Mass-Dampers."Structures Congress 2012, 471-482. (October 3, 2015).Journal: Discussion of tuned mass dampers

http://ascelibrary.org/doi/pdf/10.1061/9780784412367.042

Fitzpatrick, T., and Ridsdill Smith, R. (2001) "Stabilizing the London Millennium Bridge."Ingenia, 18-22. (October 3, 2015).Article: ARUP’s solution to damping the vibration

http://www.ingenia.org.uk/ingenia/articles.aspx?Index=123

Newland, David E. (2003). "Vibration of the London Millennium Bridge: Cause and Cure."International Journal of Acoustics and Vibration8, no. 1, 9-14. (October 3, 2015).Journal: Vibration discussion of the bridge.

http://www.iiav.org/ijav/content/volumes/8_2003_710551272264886/vol_1/385_firstpage_196081286806609.pdf

Pavic, A., Reynolds, P., Wright, J., and Armitage, T. (2002). "Methodology for Modal Testing of the Millennium Bridge, London."Proceedings of the ICE - Structures and Buildings,May 2002, 111-21. (October 3, 2015).Journal: Modal testing was used on the Millennium Bridge to create a Frequency response curve. This method as opposed to the original vibration response analysis on the bridge is crucial in determining the modal contributions to the lateral displacement.

http://www.icevirtuallibrary.com/doi/pdf/10.1680/stbu.2002.152.2.111

Ricciardelli, F., and Pizzimenti, D. (2007). "Lateral Walking-Induced Forces on Footbridges."J. Bridge Eng., 6,677-88. (October 3, 2015).Journal: Research used to modal lateral vibration due to walking-induced forces. Date used to develop design criteria for footbridges.

http://ascelibrary.org/doi/pdf/10.1061/(ASCE)1084-0702(2007)12:6(677)

Strogatz, S., Abrams, D., McRobie, A., Eckhardt, B., and Ott, E. (2005). "Theoretical Mechanics: Crowd Synchrony on the Millennium Bridge."Nature, 2005, 43-44. (October 3, 2015).Journal: Research used to develop mathematical modeling of lateral excitation by pedestrians on footbridges. Brief discussion on damping.

http://www.nature.com/nature/journal/v438/n7064/abs/438043a.html

Taylor, Douglas P. "DAMPER RETROFIT OF THE LONDON MILLENNIUM FOOTBRIDGE – A CASE STUDY IN BIODYNAMIC DESIGN." http://taylordevices.com/Tech-Paper-archives /literature-pdf/66-DamperRetrofit-London.pdf. (October 3, 2015).Website: Manufacturer and installer of the dampers.

http://taylordevices.com/Tech-Paper-archives/literature-pdf/66-DamperRetrofit-London.pdf