Analytical graphic statics

Abstract

Graphic statics is undergoing a renaissance, with computerized visual representation becoming both easier and more spectacular as time passes. While methods of the past are revived, little emphasis has been placed on studying the mathematics behind these methods. Due to the considerable advances of our mathematical understanding since the birth of graphic statics, we can learn a lot by examining these old methods from a more modern viewpoint. As such, this work shows the mathematical fabric joining different aspects of graphic statics, like dualities, reciprocal diagrams, and discontinuous stress functions. This is done by introducing a new, three dimensional force diagram (containing the old two dimensional force diagram) depicting the three dimensional equilibrium of planar force systems. A corresponding three dimensional “form diagram” (dual diagram) is introduced, in which forces are treated as linear functionals (dual vectors). It is shown that the polyhedral stress function introduced by Maxwell is in fact a linear combination of these functionals; and the projective dualities connecting these three dimensional diagrams are also explained.