Algebraic 3D Graphic Statics: Constrained Areas

Abstract: This research 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 (3DGS). In 3DGS, the form of the structure and its equilibrium of forces is 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 system. For the first time, the methods of this research allow 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. As a result, a designer can explicitly control the force magnitudes in the force diagram and explore the equilibrium of a variety of compression and tension-combined funicular structural forms. In this method, a quadratic formulation is used to compute the area of a single face based on its edge lengths. The approach is applied to manipulating the face geometry with a predefined area and the edge lengths. Subsequently, the geometry of the polyhedron is updated with newly changed faces. This approach is a multi-step algorithm where each step includes computing the geometry of a single face and updating the polyhedral geometry. One of the unique results of this framework is the construction of the zero-area, self-intersecting faces, where the sum of the signed areas of a self-intersecting face is zero, representing a member with zero force in the form diagram. The methodology of this research can clarify the equilibrium of some systems that could not be previously justified using reciprocal polyhedral diagrams. Therefore, it generalizes the principle of the equilibrium of polyhedral frames and opens a completely new horizon in the design of highly-sophisticated funicular polyhedral structures beyond compression-only systems.