Abstract
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.
Highlights
There has been much recent progress in the field of graphic statics, and in this paper we endeavour to extend that progress to encompass graphic kinematics
This paper has described a new concept which we call ‘elastographics’
This is achieved by combining Williot displacement diagrams with two-dimensional Maxwell or three-dimensional Rankine reciprocal diagrams, thereby providing purely graphical solutions of the elastic structural behaviour of nodally loaded two- and three-dimensional trusses
Summary
There has been much recent progress in the field of graphic statics, and in this paper we endeavour to extend that progress to encompass graphic kinematics. This latter description is capable of describing all six stress resultants (an axial and two shear forces, and a torsional and two bending moments) in rather general three-dimensional frameworks with possibly curved members Another innovation appeared in Zanni & Pennock [8] and McRobie [9] which showed how to combine form and force diagrams for a truss into a unified object, this being called the Maxwell–Minkowski diagram in two dimensions, and the Rankine–Minkowski diagram in three dimensions [9]. For three-dimensional moment-carrying frames, the analogous unified object is the Corsican sum defined by McRobie [7] This progress has, been largely restricted to the notion of statical equilibrium. Throughout we consider two-dimensional trusses first, before showing the natural extension to three-dimensional trusses
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