Emergence of the London Millennium Bridge instability without synchronisation.

The pedestrian-induced instability of the London Millennium Bridge is a widely used example of Kuramoto synchronisation. Yet, reviewing observational, experimental, and modelling evidence, we argue that increased coherence of pedestrians' foot placement is a consequence of, not a cause of the instability. Instead, uncorrelated pedestrians produce positive feedback, through negative damping on average, that can initiate significant lateral bridge vibration over a wide range of natural frequencies. We present a simple general formula that quantifies this effect, and illustrate it through simulation of three mathematical models, including one with strong propensity for synchronisation. Despite subtle effects of gait strategies in determining precise instability thresholds, our results show that average negative damping is always the trigger. More broadly, we describe an alternative to Kuramoto theory for emergence of coherent oscillations in nature; collective contributions from incoherent agents need not cancel, but can provide positive feedback on average, leading to global limit-cycle motion.

Stability of trusses by graphic statics.

This paper presents a graphical method for determining the linearized stiffness and stability of prestressed trusses consisting of rigid bars connected at pinned joints and which possess kinematic freedoms. Key to the construction are the rectangular areas which combine the reciprocal form and force diagrams in the unified Maxwell-Minkowski diagram. The area of each such rectangle is the product of the bar tension and the bar length, and this corresponds to the rotational stiffness of the bar that arises due to the axial force that it carries. The prestress stability of any kinematic freedom may then be assessed using a weighted sum of these areas. The method is generalized to describe the out-of-plane stability of two-dimensional trusses, and to describe three-dimensional trusses in general. The paper also gives a graphical representation of the 'product forces' that were introduced by Pellegrino and Calladine to describe the prestress stability of trusses.

Graphic kinematics, visual virtual work and elastographics.

In this paper, recent progress in graphic statics is combined with Williot displacement diagrams to create a graphical description of both statics and kinematics for two- and three-dimensional pin-jointed trusses. We begin with reciprocal form and force diagrams. The force diagram is dissected into its component cells which are then translated relative to each other. This defines a displacement diagram which is topologically equivalent to the form diagram (the structure). The various contributions to the overall Virtual Work appear as parallelograms (for two-dimensional trusses) or parallelopipeds (for three-dimensional trusses) that separate the force and the displacement pieces. Structural mechanisms can be identified by translating the force cells such that their shared faces slide across each other without separating. Elastic solutions can be obtained by choosing parallelograms or parallelopipeds of the appropriate aspect ratio. Finally, a new type of 'elastographic' diagram-termed a deformed Maxwell-Williot diagram (two-dimensional) or a deformed Rankine-Williot diagram (three-dimensional)-is presented which combines the deflected structure with the forces carried by its members.

Correction to 'The geometry of structural equilibrium'.

[This corrects the article DOI: 10.1098/rsos.160759.].

The geometry of structural equilibrium.

Building on a long tradition from Maxwell, Rankine, Klein and others, this paper puts forward a geometrical description of structural equilibrium which contains a procedure for the graphic analysis of stress resultants within general three-dimensional frames. The method is a natural generalization of Rankine's reciprocal diagrams for three-dimensional trusses. The vertices and edges of dual abstract 4-polytopes are embedded within dual four-dimensional vector spaces, wherein the oriented area of generalized polygons give all six components (axial and shear forces with torsion and bending moments) of the stress resultants. The relevant quantities may be readily calculated using four-dimensional Clifford algebra. As well as giving access to frame analysis and design, the description resolves a number of long-standing problems with the incompleteness of Rankine's description of three-dimensional trusses. Examples are given of how the procedure may be applied to structures of engineering interest, including an outline of a two-stage procedure for addressing the equilibrium of loaded gridshell rooves.

Parallels between wind and crowd loading of bridges.

Parallels between the dynamic response of flexible bridges under the action of wind and under the forces induced by crowds allow each field to inform the other. Wind-induced behaviour has been traditionally classified into categories such as flutter, galloping, vortex-induced vibration and buffeting. However, computational advances such as the vortex particle method have led to a more general picture where effects may occur simultaneously and interact, such that the simple semantic demarcations break down. Similarly, the modelling of individual pedestrians has progressed the understanding of human-structure interaction, particularly for large-amplitude lateral oscillations under crowd loading. In this paper, guided by the interaction of flutter and vortex-induced vibration in wind engineering, a framework is presented, which allows various human-structure interaction effects to coexist and interact, thereby providing a possible synthesis of previously disparate experimental and theoretical results.

Modeling walker synchronization on the Millennium Bridge.

On its opening day the London Millennium footbridge experienced unexpected large amplitude wobbling subsequent to the migration of pedestrians onto the bridge. Modeling the stepping of the pedestrians on the bridge as phase oscillators, we obtain a model for the combined dynamics of people and the bridge that is analytically tractable. It provides predictions for the phase dynamics of individual walkers and for the critical number of people for the onset of oscillations. Numerical simulations and analytical estimates reproduce the linear relation between pedestrian force and bridge velocity as observed in experiments. They allow prediction of the amplitude of bridge motion, the rate of relaxation to the synchronized state and the magnitude of the fluctuations due to a finite number of people.

Theoretical mechanics: crowd synchrony on the Millennium Bridge.

Soon after the crowd streamed on to London's Millennium Bridge on the day it opened, the bridge started to sway from side to side: many pedestrians fell spontaneously into step with the bridge's vibrations, inadvertently amplifying them. Here we model this unexpected and now notorious phenomenon--which was not due to the bridge's innovative design as was first thought--by adapting ideas originally developed to describe the collective synchronization of biological oscillators such as neurons and fireflies. Our approach should help engineers to estimate the damping needed to stabilize other exceptionally crowded footbridges against synchronous lateral excitation by pedestrians.

The Big Flood: North Sea storm surge.

In the 50 years since the catastrophic southern North Sea storm surge of 31 January-1 February 1953, there have been technological advances in the engineering of flood protection, increased understanding of physical processes in shallow seas and estuaries, and developments in the mathematical statistics of extreme events. This introductory paper reviews how the scientific understanding of surge events, their impacts and the human responses to them is evolving on many fronts, often across disciplinary boundaries. The question of how the long-term nature of the problem itself will be influenced by possible climate, land use and policy changes is addressed, along with their associated uncertainties.

Diversity and adaptation of shelters in transitional settlements for IDPs in Afghanistan.

The diversity of shelters used in transitional settlements for internally displaced persons (IDPs) in Herat, Afghanistan is described. The information is based on a field survey undertaken in March 2002 and highlights the adaptation techniques, which IDPs undertake to improve any provided shelter. Potential areas for improvement are indicated; for example, the possibility for using insulated, demountable liners to prevent cold-related deaths without sacrificing shelter flexibility along with the likely need for better agency coordination of the shelter responses they provide. The wider context in which the technical recommendations would be implemented must also be considered. Such issues include agency resources, political impediments to providing the desired option, and the preference of many IDPs that the best shelter would be their home.