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1. Bio-Based Composite Spatial Shell Structures

   Akbari, Mostafa and (July 2023, Proceedings of the {IASS} Annual Symposium 2023)

   The authors of this research investigate the possibility of fabricating shell-based cellular structures using knitting techniques. Shellular Funicular Structures are two-manifold single-layer structures that can be designed in the context of graphic statics. These are efficient compression/tension-only structures that have been designed for a certain boundary condition. Although the shellular funicular structures are efficient geometries in transferring the forces, the fabrication process is challenging due to the geometric complexity of the structure. Since Shellular structures comprise a single surface, they are suitable candidates to be fabricated using knitting technique, a method by which yarn is manipulated to create a textile or fabric. Using knitting approach, one can fabricate shellular structures with minimum production waste in which the knit can work as a formwork for actual structure or act as a composite structure combined with bio-based resin. This research proposes a workflow to fabricate shellular structures using knitting that can be scaled up for industrial purposes. In this process, the designed shellular structures are divided into multiple sections that can be unrolled into planar geometries. These geometries are optimized based on the elastic forces in the knitted network and knitted and sewn to make a topologically complex geometry of the shellular systems. After assembling the knitted parts and applying external forces at the boundaries, the final configuration of the structural form in tension is achieved. Then this form is impregnated with custom bio-resin blends from chitosan, sodium alginate, and silk fibroin to stiffen the soft knit structures into a compressed system. Although this method is an efficient fabrication technique for constructing shellular structures, it needs to be translated into an optimized method of cutting, knitting, and sewing with respect to the complexity of the shellular geometry. As a proof of concept of the proposed workflow, a mesoscale shellular structure is fabricated. Keywords: Biocomposite Structures, Shellular Funicular Structures, Knitting, Graphic statics. 

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2. From design to the fabrication of shellular funicular structures

   Akbari, M.; Lu, Y.; Akbarzadeh, M. (September 2021, Proceedings of the Association for Computer-Aided Design in Architecture (ACADIA))

   Shellular Funicular Structures (SFSs) are single-layer, two-manifold structures with anticlastic curvature, designed in the context of graphic statics. They are considered as efficient structures applicable to many functions on different scales. Due to their complex geometry, design and fabrication of SFSs are quite challenging, limiting their application in large scales. Furthermore, designing these structures for a predefined boundary condition, control, and manipulation of their geometry are not easy tasks. Moreover, fabricating these geometries is mostly possible using additive manufacturing techniques, requiring a lot of support in the printing process. Cellular funicular structures (CFSs) as strut-based spatial structures can be easily designed and manipulated in the context of graphic statics. This paper introduces a computational algorithm for translating a Cellular Funicular Structure (CFS) to a Shellular Funicular Structure (SFS). Furthermore, it explains a fabrication method to build the structure out of a flat sheet of material using the origami/ kirigami technique as an ideal choice because of its accessibility, processibility, low cost, and applicability to large scales. The paper concludes by displaying a design and fabricated structure using this technique. 

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3. The form-finding and fabrication of an ultra-thin funicular bridge prototype made up of hollow glass units

   LU, Yao; CREGAN, Matthew; CHHADEH, Philipp A; SEYEDAHMADIAN, Alireza; BOLHASSANI, Mohammad; Schneider, Jens; YOST, Joseph R; AKBARZADEH, Masoud (October 2021, International Association of Shell and Spatial Structures)

   The recent development of three-dimensional graphic statics (3DGS) has greatly increased the ease of designing complex and efficient spatial funicular structural forms [1]. The reciprocal diagram based 3DGS approaches not only generate highly efficient funicular structures [2], but also result in planarity constraints due to the polyhedron nature of the reciprocal diagrams [3]. Our previous research has shown the feasibility of leveraging this planarity by using planar glass sheets to materialize the 10m-span, double-layer glass bridge [3]. This paper is framed as a proof of concept for the 10m bridge and explores the form-finding, detail configuration, fabrication constraints, and assembly logic by designing and constructing a small-scale bridge prototype with a span of 2.5m. The prototype is designed in a modular approach, where each polyhedral cell of the form is materialized using a hollow glass unit (HGU) (Figure 1a), which can be prefabricated and preassembled, and therefore, greatly simplifies the assembly of the whole bridge. The compression-only form of the prototype is generated using the PolyFrame beta [4] plug-in for Rhinoceros [5]. The form-finding is carried out with a comprehensive consideration of a variety of parameters, including fabrication constraints, assembly ease, construction cost, and practicality. To start the form-finding process, a group of closed convex force polyhedrons is aggregated, controlling the topology of the form diagram and the orientations of the form elements. By manipulating the face tilting angles of the force diagram, the supported edges at the end of the bridge are all made horizontal, reducing the difficulty of the support design. Then, vertex locations and edge lengths of the form diagram are constrained, determining the final dimensions of both the bridge and the cells. After getting the geometry of the bridge, the detail developments are streamlined. Each of the 13 HGUs consists of two flat deck plates and a series of side plates (Figure 1b). To interlock the adjacent cells and prevent possible sliding, a male-female connection mechanism is introduced to the conjoint side plates of the HGUs (Figure 1b). Additionally, to eliminate the direct contact of the glass parts and prevent the stress concentration, two softer transparent materials are involved for connecting purposes. Within each HGU, silicon-based binding agent is used to hold the glass parts together; between the neighboring HGUs, plastic sheets are placed as interface materials (Figure 1b). Figure 1. a) The 2.5m-span small-scale prototype dome, b) Exploded view showing deck plates, side plates, male-female connection, and interface material For the fabrication of the glass parts, 5-axis Waterjet cutting techniques are applied. While the glass sheets for the deck plates can be purchased from the market, the irregular side plates with male-female connections need to be made from kiln-cast glass. In terms of the Waterjet cutting constraints, there is a max cutting angle of 60 degrees from vertical. With respect to this, all the glass parts are examined during the design process to ensure they all satisfy the cutting angle requirements. Aiming to achieve a fast and precise assembly, several assistant techniques are developed. On the local HGU level, assembly connectors are designed and 3D-printed to help locate the glass parts. On the global prototype level, the assembly sequence of the HGUs are simulated to avoid interference. Besides, a labeling system is also established to organize the fabricated parts and guide the entire assembly process. The design and construction of this small-scale prototype provide important information for the future development of the full-scale bridge regarding the interlocking detail design, the fabrication constraints, and assembly logic. The actual structural performance of the prototype awaits further investigation through-loading experiments. 

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4. Developing a polyhedral graphic statics formulation for tetrahedral truss systems

   MOZAFFARI, Salma; HABLICSEK, Márton; AKBARZADEH, Masoud; VOGEL, Thomas (October 2021, International Association of Shell and Spatial Structures)

   This paper presents procedures to generate truss topologies as an input form for polyhedral graphic statics and develops an algebraic formulation to construct their force diagrams. The study's ultimate goal is to extend the authors' previous research in 2D [1] to generate 3D strut-and-tie models and stress fields for reinforced concrete design. The recent algebraic formulation constructs reciprocal polyhedral diagrams of 3D graphic statics with either form or force as input [2]. However, the input is usually a set of polyhedrons or self-stressed networks [3]. Another implementation of polyhedral graphic statics [4] includes general truss topologies. But the starting geometry is usually the global force diagram, and based on its modification or subdivision, a form diagram is constructed. Therefore, currently, there exists no formulation to analyze a spatial truss using polyhedral graphic statics. This paper develops an algorithm to build upon the algebraic 3D graphic statics formulation and notation [2, 5] to construct the force diagram for input geometries that do not include all closed cells. The article also shows how the proper definition of the external spaces between the applied loads and reaction forces and the tetrahedral subdivision of the truss makes it possible to construct the reciprocal force diagram. This technique can be further explored to generate various truss topologies for a given volume and identify an optimized solution as the strut-and-tie model for reinforced concrete. Figure 1 illustrates an example of a spatial truss with two vertical applied loads and four vertical supports, the subdivision of the inner and outer space, the constructed force diagram, and the Minkowski sum of the dual diagrams (i.e., the geometrical summation of the form and scaled force diagram). 

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5. Web-based Interactive Polyhedral Graphics Statics Platform

   CHAI, Hua; AKBARZADEH, Masoud (October 2021, International Association of Shell and Spatial Structures)

   This paper introduces a web-based interactive educational platform for 3D/polyhedral graphic statics (PGS) [1]. The Block Research Group (BRG) at ETH Zürich developed a dynamic learning and teaching platform for structural design. This tool is based on traditional graphic statics. It uses interactive 2D drawings to help designers and engineers with all skill levels to understand and utilize the methods [2]. However, polyhedral graphic statics is not easy to learn because of its characteristics in three-dimensional. All the existing computational design tools are heavily dependent on the modeling software such as Rhino or the Python-based computational framework like Compass [3]. In this research, we start with the procedural approach, developing libraries using JavaScript, Three.js, and WebGL to facilitate the construction by making it independent from any software. This framework is developed based on the mathematical and computational algorithms deriving the global equilibrium of the structure, optimizing the balanced relationship between the external magnitudes and the internal forces, visualizing the dynamic reciprocal polyhedral diagrams with corresponding topological data. This instant open-source application and the visualization interface provide a more operative platform for students, educators, practicers, and designers in an interactive manner, allowing them to learn not only the topological relationship but also to deepen their knowledge and understanding of structures in the steps for the construction of the form and force diagrams and analyze it. In the simplified single-node example, the multi-step geometric procedures intuitively illustrate 3D structural reciprocity concepts. With the intuitive control panel, the user can move the constraint point’s location through the inserted gumball function, the force direction of the form diagram will be dynamically changed from compression-only to tension and compression combined. Users can also explore and design innovative, efficient spatial structures with changeable boundary conditions and constraints through real-time manipulating both force distribution and geometric form, such as adding the number of supports or subdividing the global equilibrium in the force diagram. Eventually, there is an option to export the satisfying geometry as a suitable format to share with other fabrication tools. As the online educational environment with different types of geometric examples, it is valuable to use graphical approaches to teach the structural form in an exploratory manner. 

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