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.

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.

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.

This paper extends the overview (Mitchell et al. [11]) relating graphic statics and reciprocal diagrams to linear algebra-based matrix structural analysis. Focus is placed on infinitesimal mechanisms, both in-plane (linkage) and out-of-plane (polyhedral Airy stress functions). Each self-stress in the original diagram corresponds to an out-of-plane polyhedral mechanism. Decomposition into sub-polyhedra leads to a basis set of reciprocal figures which may then be linearly combined. This leads to an intuitively-appealing approach to the identification of states of self-stress for use in structural design, and to a natural "structural algebra" for use in structural optimisation. A 90° rotation of the sub-reciprocal generated by any sub-polyhedron leads to the displacement diagram of an in-plane mechanism. Any self-stress in the original thus corresponds to an in-plane mechanism of the reciprocal, summarised by the equation s = M* (where s is the number of states of self-stress in one figure, and M* is the number of in-plane mechanisms, including rigid body rotation, in the other). Since states of self-stress correspond to out-of-plane polyhedral mechanisms, this leads to a form of "conservation of mechanisms" under reciprocity. It is also shown how external forces may be treated via a triple-layer Airy stress function, consisting of a structural layer, a load layer, and a layer formed by coordinate vectors of the structural perimeter.

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We examine the relationship between infrastructure provision and poverty alleviation by analysing 500 interviews conducted in serviced and non-serviced slums in India. Using a mixed-method approach of qualitative analysis and regression modelling, we find that infrastructure was associated with a 66% increase in education among females. Service provision increased literacy by 62%, enhanced income by 36%, and reduced health costs by 26%. Evidence suggests that a gender-sensitive consideration of infrastructure is necessary and that a 'one size fits all' approach will not suffice. We provide evidence that infrastructure investment is critical for well-being of slum dwellers and women in particular.

The fundamental theorem of linear algebra establishes a duality between the statics of a pin-jointed truss structure and its kinematics. Graphic statics visualizes the forces in a truss as a reciprocal diagram that is dual to the truss geometry. In this article, we combine these two dualities to provide insights not available from a graphical or algebraic approach alone. We begin by observing that the force diagram of a statically indeterminate truss, although itself typically a kinematically loose structure, must support a self-stress state of its own. Such an "extra" self-stress state is described by the fundamental theorem of linear algebra. We show that the self-stress states of a truss are in a one-to-one correspondence with linkage-mechanism displacements of its reciprocal, and the relative centers of rotation of these mechanism displacements correspond to centers of perspective of a projection of a plane-faced three-dimensional polyhedral mesh. We prove that this polyhedral mesh is exactly the continuum Airy stress function, restricted to describe equilibrium of a truss structure. We use the Airy function to prove James Clerk Maxwell's conjecture that a two-dimensional truss structure of arbitrary topology has a self-stress state if and only if its geometry is given by the projection of a three-dimensional plane-faced polyhedron. Although a very limited number of engineers have been aware of the relationship between trusses and a polyhedral Airy function, the authors believe that this is the first truly rigorous elucidation. We summarize the properties of this "dual duality," which has the Airy function at its core, and conclude by showing applications to design of tensegrities, planar panelization of architectural surfaces, and optimization of trusses.

A method is presented for computing and displaying the Maxwell (two-dimensional) or Rankine (three-dimensional) diagram reciprocal to a truss under load. Reciprocals are constructed via the two-dimensional Airy stress function or its three-dimensional analogue. A linear combination of the coordinates of each original node and the coordinates of its reciprocal object leads to the finished diagram. This is related to an underlying Minkowski sum of the polyhedral stress functions of the original and reciprocal diagrams. Some regions of the resulting combined diagram have units of length L, and other regions use units of force F. These regions are connected in a consistent manner, in two dimensions by rectangular areas and in three dimensions by prismatic volumes, each having work units of dimension FL, and which give a visual representation of the F L ⋅ terms in Maxwell's load path theorem. A linear combination weighted in favour of the original leads to a diagram where original bars are augmented by a thickness (in two-dimensional) or a polygonal crosssectional area (in three-dimensional). The diagrams thus give an appealing visual representation of the material required for a constant stress design. Although the method gives a complete description of loaded trusses in two dimensions, in three dimensions, it is currently only applicable to those trusses which possess a Rankine reciprocal.