Algebraic 3D Graphic Statics: Constrained Areas

Abstract

This research is a continuation of the Algebraic 3D Graphic Statics Methods that addressed the reciprocal constructions in an earlier publication (Hablicsek et al. 2019). It provides algorithms and (numerical) methods to geometrically control the magnitude of the internal and external forces in the reciprocal diagrams of 3D/Polyhedral Graphic statics. 3D graphic statics (3DGS) is a recently rediscovered method of structural form-finding based on a 150-year old proposition by Rankine and Maxwell in Philosophical Magazine. In 3DGS, the form of the structure and its equilibrium of forces are represented by two polyhedral diagrams that are geometrically and topologically related. The areas of the faces of the force diagram represent the magnitude of the internal and external forces in the members of the form diagram. The proposed method allows the user to control and constrain the areas and edge lengths of the faces of general polyhedrons that can be convex, self-intersecting, or concave in a group of aggregated polyhedral cells. In this method, a quadratic formulation is introduced to compute the area of a face based on its edge lengths only. This quadratic function is then turned into a linear formulation to facilitate the non-trivial computation of reciprocal polyhedral diagrams. The approach is applied to force diagrams, including a group of polyhedral cells, to manipulating the face geometry with a predefined area and the edge lengths. The method is implemented as a multi-step algorithm where each step includes computing the geometry of a single face with a target area and updating the polyhedral geometry. One of the remarkable results of this framework is to control the construction of the zero-area faces as proposed by McRobie (2017b). The zero-area faces represent a member with zero force in the form diagram. This research shows how self-intersecting faces, including the zero-area faces, can be constructed with additional edge constraints in a group of polyhedral cells without breaking the reciprocity of the form and force diagrams. Thus, it provides more hints on the generalization of the principle of the equilibrium of polyhedral frames. It also suggests a design approach where the boundary conditions and internal forces of compression-only systems can be manipulated to the design systems with both compression and tensile forces with no change in the geometry or the faces’ planarity of the form diagram.