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.

No abstract

No abstract

This article describes the minimum volume layouts for truss structures uniformly loaded and either supported at the ends of a single span (also referred to as "the bridge problem," shown in Figure 1) or supported at equally spaced points as shown in Figure 2 (see also Beghini and Baker 1). The methodology utilized to obtain the layout takes advantage of Maxwell's Theorem of Load Paths and the properties of the reciprocal diagrams in Graphic Statics. This approach has several benefits; in particular, it provides new insights in the solution and therefore it allows to explicitly describe several geometric properties of the optimal layout. The solutions to the problems considered are explored in terms of optimal structures and their duals. A dual structure is associated with the force diagram of the original structure. Such dual truss has the same total load path as the original structure itself, but it is often a completely different geometry corresponding to a different, but related, load condition. Therefore, multiple optimal layouts are available to the practicing engineer and the selection of the best layout depends on the specific design problem as discussed in more detail in section "Dual structure." Optimal truss layouts are investigated for solutions restricted within a semi-infinite plane for the multiplespan problem and in the full two-dimensional (2D) space for the single-span problem. Analytical expressions describing the geometry and the volume of the optimal truss layout and its dual are provided.