Algebraic 3D graphic statics: Reciprocal constructions

Abstract

The recently developed 3D graphic statics (3DGS) lacks a rigorous mathematical definition relating the geometrical and topological properties of the reciprocal polyhedral diagrams as well as a precise method for the geometric construction of these diagrams. This paper provides a fundamental algebraic formulation for 3DGS by developing equilibrium equations around the edges of the primal diagram and satisfying the equations by the closeness of the polygons constructed by the edges of the corresponding faces in the dual/reciprocal diagram. The research provides multiple numerical methods for solving the equilibrium equations and explains the advantage of using each technique. The approach of this paper can be used for compression-and-tension combined form-finding and analysis as it allows constructing both the form and force diagram based on the interpretation of the input diagram. Besides, the paper expands on the geometric/static degrees of (in)determinacies of the diagrams using the algebraic formulation and shows how these properties can be used for the constrained manipulation of the polyhedrons in an interactive environment without breaking the reciprocity between the two. Keywords: Algebraic three-dimensional graphic statics, polyhedral reciprocal diagrams, geometric degrees of freedom, static degrees of indeterminacies, tension and compression combined polyhedra, constraint manipulation of polyhedral diagrams.

Geometry-based structural form-finding to design architected cellular solids

“…This paper showed that 3DGS enables us to design infinite efficient light-weight cellular materials for different boundary conditions with various internal topology and force distribution. In this system, the resulted geometry depends on the external force polyhedron, force diagram's internal subdivision, and geometrical degree of freedom of the pin-jointed system [Hablicsek et al 2018]). This method lets us understand the relationship between the internal flow of force, geometry, and external force-flow with their geometrical relationships.…”

Spherical harmonic shape descriptors of nodal force demands for quantifying spatial truss connection complexity

“…Nodal equilibrium is represented as a closed polygon (2D) or polyhedron (3D). Although initially used as a procedural method of analysis and design, recent algebraic formalizations of graphic statics principals have allowed computational graphic statics to resurge as a modern tool in design and analysis [1,19,25,26,37].…”

Spherical harmonic shape descriptors of nodal force demands for quantifying spatial truss connection complexity

“…Nodal equilibrium is represented as a closed polygon (2D) or polyhedron (3D). Although initially used as a procedural method of analysis and design, recent algebraic formalizations of graphic statics principals have allowed computational graphic statics to resurge as a modern tool in design and analysis (Van Mele and Block, 2014;Akbarzadeh, 2016;Hablicsek et al, 2019;Nejur and Akbarzadeh, 2021;Mozaffari et al, 2021).…”