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Graphic statics, graphical kinematics, and the geometry of equilibrium

Graphic statics provides a geometric picture of the forces in a 2D structure in terms of a diagram dual to the original structure. This picture allows an engineer to design the forces to be present in the structure by manipulating the “design degrees of freedom” of the dual, prior to any member sizing or analysis. These design degrees of freedom depend on the geometry and topology of the structure in a way that is not always intuitive.

The goal of this talk is to provide a method to exactly describe these design degrees of freedom by using a separate but related duality: the duality between statics and kinematics. This duality is linear-algebraic, rather than geometric: it relies on the relation between the four fundamental matrix subspaces disclosed by the Fundamental Theorem of Linear Algebra.

We will first review the Fundamental Theorem and discuss how it applies to structures, then show numerous examples of how the Fundamental Theorem relates graphic statics to graphic kinematics of structures. This relation allow us to use the kinematics of the dual diagram to exactly define the design degrees of freedom of the structure. We will then use this insight to develop a new result that gives a geometric picture of the design degrees of freedom for 3D structures.

This result happens to coincide precisely with a group of techniques for geometric design and manipulation of single-layer meshes that are already well-developed in the computer graphics / CAGD field. This connection allows these techniques to be used directly in the engineering design of cable net and grid shell structures. We conclude with examples of applications and an interesting theoretical connection between 3D graphic statics and the stress function for shell structures.

Author(s):

Toby Mitchell    
Skidmore, Owings & Merrill LLC
United States