The force as a function: Towards analytical graphic statics for spatial structures

TL;DR

This paper develops a dimension-independent framework for representing forces as functions in graphic statics, extending classical planar concepts to spatial structures using Grassmann algebra.

Contribution

It introduces a novel, dimension-independent method for treating forces as functions, unifying planar and spatial graphic statics through Grassmann algebra.

Findings

Provides a unified analytical approach for 2D and 3D force analysis.

Recovers known stress functions for planar and spatial structures.

Lays groundwork for advanced spatial graphic statics methods.

Abstract

One of the most influential early works in graphic statics is one of Maxwell, where he introduced the idea of a discontinuous stress function and the use of a 3D projective polarity for a planar problem. A recent work gave an analytical description with the goal to provide explaining power, treating planar forces as linear functionals (moment functionals) forming a 3D vector space, so force and form diagrams of planar problems can be interpreted in a 3-dimensional way. The linear combination of these moment functions can naturally be used as a discontinuous stress function since the Airy stress function is known to correspond to moments of planar forces. The main contribution of this work is to present a dimension independent way of treating forces as functions, that returns the known stress-functions of planar and spatial graphic statics. This is done by relying on the multi-linear function definition of Grassmann-algebra.