A0. Intro to GS, TNA, RV2
Learning Goal:
understand Thust Network Analysis
understand concepts: pattern, FormDiagram, ForceDiagram, ThrustDiagram
understand horizontal and vertical equilibrium
Graphic Statics
Graphic statics is a design and analysis method for two-dimensional discrete structures that rely on geometrical rather than analytical or numerical representations of the relation between a structure's geometry and the equilibrium of its internal forces. The reciprocal form and force diagrams provide valuable insights for designers to understand a structure's behavior through a visual medium.
The magnitude of the internal force in a member of a structure and the static equilibrium is the length of the corresponding edge in the force diagram. which are parallel to one another. 2D graphic statics is used by engineers of the 19th century to design and analyze complex truss structures intuitively and visually.
For a given form diagram as shown in Fig. 2a, the topology of the force diagram can be taken by constructing a dual diagram, connecting each of the areas, or subspaces, of the form diagram and crossing its edges (Fig. 2b). Once the connection between the two diagrams is established, i.e. each edge of the form diagram corresponds to an edge in the force diagram, the equilibrium will be achieved if there's a configuration of the force diagram for which the reciprocal edges are parallel and have their length proportional to the force applied (Fig. 2c).
2.5 D Thrust Network Analysis & Form Finding
Graphic Statics combining computational form-finding is a fundamental theory and principle behind thrust network analysis (TNA). TNA is a form-finding methodology for generating compression-only vaulted surfaces and networks, which computes a network of compressive forces from given loads (self-weight and external vertical forces) and boundary conditions.
In TNA, the horizontal projection of the vaulting shell structure is used as a form diagram. ForceDiagram of this projected form diagram represents the horizontal equilibrium of the shell structure. The thrust network or the thrust diagram is a network that represents the shell structure which is discretized into line segments. Using projective geometry, duality theory, and linear optimization, TNA provides a graphical and intuitive method, adopting the same advantages of techniques such as graphic statics, but offering a viable extension to fully three-dimensional problems.
RhinoVAULT 2
RV2 is a Rhino plugin that uses the TNA approach to intuitively create and explore compression-only structures. The workflow of RV2 is based on the theoretical framework of TNA, which can be broken down into 7 main steps.
Diagrams
FormDiagram: horizontal projection of the Thrust Network
ForceDiagram: topological dual of the Form Diagram
ThrustDiagram: final geometry (identical to thrust network)
FormDiagram (Pattern)
The form-finding process always with the FormDiagram. Its topology is called a Pattern, the discretized version of the shell to be designed and all the lines carry axial compression forces. Once the Pattern, the topology of the final shell, is defined, the boundary conditions as support vertices, unsupported boundaries, and potential openings can be added. Then the FormDiagram will be generated from the Pattern. Based on boundary conditions, mesh faces and edges in the FormDiagram will be modified compared with the input Pattern, in order to compute the correct dual ForceDiagram.
ForceDiagram
The ForceDiagram represents the equilibrium of the nodes in the FormDiagram.
The ForceDiagram is the dual diagram of the ForceDiagram. It is created as the "centroidal dual" of the FormDiagram. This means that the dual graph is defined by placing its vertices at the centroids of their corresponding faces in the FormDiagram. Two diagrams have the same number of edges and each edge in the FormDiagram corresponds to an edge in the ForceDiagram. A node in the FormDiagram is a closed polygon in the ForceDiagram, and vice versa. The two diagrams are not yet reciprocal, meaning that the corresponding edges in the diagrams are not perpendicular to each other. So the ForceDiagram cannot represent the equilibirum of FormDiagram yet.
Equilibrium
horizontal equilibrium: orthogonalize the edges in Form and Force Diagram.
vertical equilibrium: generates the Thrust Network.
Horizontal equilibrium
Horizontal equilibrium modifies the edges so that they are perpendicular to the corresponding. Both diagrams can be modified based on a mediation variable. When the perpendicularity criteria (within desired angle tolerance) are met, two diagrams are considered reciprocal. Two diagrams are reciprocal if they are dual, and if their corresponding edges are at a constant angle with each other. Now the ForceDiagram can represent the equilibrium in the FormDiagram and the horizontal equilibrium of the final ThrustDigram. Once the horizontal equilibrium has been established, the distribution of horizontal forces is fixed.
Vertical Equilibrium
ThrustDiagram is initiated as a copy of the FormDiagram. Given a desired target height of the eventual ThrustDiagram, vertical equilibrium solver iteratively re-scales the ThrustDiagram in the z-axis, until the highest vertex of the ThrustDigram lies at the desired target height. Then the Z-coordinates of the ThrustDiagram are updated.
Vertical equilibrium calculates a scale factor of the ForceDiagram, or the magnitude of the horizontal forces, which determines the depth of the final ThrustDiagram. A higher scale factor results in higher horizontal forces and therefore a shallower three-dimensional shape. Vice versa, a lower scale factor results in lower horizontal forces and thus we obtain a deeper solution.
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