skip to main content

Title: Developing a polyhedral graphic statics formulation for tetrahedral truss systems
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. more » 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). « less
Authors:
; ; ;
Award ID(s):
1944691
Publication Date:
NSF-PAR ID:
10209910
Journal Name:
International Association of Shell and Spatial Structures
Sponsoring Org:
National Science Foundation
More Like this
  1. 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 butmore »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.« less
  2. 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,more »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.« less
  3. This paper introduces an interactive form-finding technique to design and explore continuous Shellular Funicular Structures in the context of Polyhedral Graphic Statics (PGS). Shellular funicular forms are two-manifold shell-based geometries dividing the space into two interwoven sub-spaces, each of which can be represented by a 3D graph named labyrinth [1]. Both form and force diagrams include labyrinths, and the form finding is achieved by an iterative subdivision of the force diagram across its labyrinths [2]. But this iterative process is computationally very expensive, preventing interactive exploration of various forms for an initial force diagram. The methodology starts with identifying three sets of labyrinth graphs for the initial force diagram and immediately visualizing their form diagrams as smooth and continuous surfaces. Followed by exploring and finalizing the desired form, the force diagram will be subdivided across the desired labyrinth graph to result in a shellular funicular form diagram (Figure 1). The paper concludes by evaluating the mechanical performance of continuous shellular structures compared to their discrete counterparts.
  4. Editor-in-Chief: Joern Ritterbusch, Deputy Editors: (Ed.)
    Owing to the fact that effective properties of low-density cellular solids heavily rely on their underlying architecture, a variety of explicit and implicit techniques exists for designing cellular geometries. However, most of these techniques fail to present a correlation among architecture, internal forces, and effective properties. This paper introduces an alternative design strategy based on the static equilibrium of forces, equilibrium of polyhedral frames, and reciprocity of form and force. This novel approach reveals a geometric relationship among the truss system architecture, topological dual, and equilibrium of forces on the basis of 3D graphic statics. This technique is adapted to devise periodic strut-based cellular architectures under certain boundary conditions and they are manipulated to construct shell-based (shellular) cells with a variety of mechanical properties. By treating the materialized unit cells as representative volume elements (RVE), multiscale homogenization is used to investigate their effective linear elastic properties. Validated by experimental tests on 3D printed funicular materials, it is shown that by manipulating the RVE topology using the proposed methodology, alternative strut materialization schemes, and rational addition of bracing struts, cellular mechanical metamaterials can be systematically architected to demonstrate properties ranging from bending- to stretching-dominated, realize metafluidic behavior, or create novel hybridmore »shellulars.« less
  5. This research investigates the design of structurally performant, lightweight architectural elements produced through concrete 3D printing (C3DP). Traditionally, concrete requires dense and sturdy formwork, whose production adds significantly to the total cost and results in massive and heavy parts after demolding. C3DP offers the unique opportunity to both eliminate the need for formwork and to create lighter parts by introducing internal voids and cavities. The advent of additive manufacturing in a broad range of scales, materials, industries, and applications, led to increased interest and intense research into different types of porous structures, their geometry, and structural performance under various boundary conditions. Precise control over the sparse distribution of material allows not only for parts with similar strength at reduced mass but even for modifications of mechanical properties, like turning brittle materials into elastic or shock-absorbent ones. While with powder-based additive manufacturing processes like metal 3D printing, truss-based lattices have become very popular for the light-weighting of parts or to provide tissue growth scaffolds for medical implants, their geometry – a sparse space frame resulting in numerous individual contour islands and accentuated overhangs – cannot as easily be produced by C3DP, which is based on a continuous material extrusion. Alternative typesmore »of micro-structures, so-called triply periodic minimal surfaces (TPMS), are better suited for this process as they are, as their name suggests, consisting of one continuous surface dividing space into two separate but interwoven subspaces. TPMS are therefore very popular for the efficient design of heat exchangers. We develop and present a continuous and integrated workflow, in which the architectural elements and their structural requirements are designed through transitioning back and forth between the force and the form diagram using 3D graphic statics [1]. The members and their topology from the abstract graph of the conceptual form diagram are seamlessly connected to the volumetric modeling (VM) framework, responsible for the definition of the part geometry [2]. VM represents form assigned distance functions (SDF) and can easily handle complex topologies and flawless Boolean operations of not only the outer shell geometry but also the internal micro-structural infill patterns (Fig. 1, a). In an iterative feedback loop, the infill can be further optimized to leave the material only along certain internal stress trajectories (force flows). This functional grading controlling the relative density is done based on the FE analysis results. The stress distribution is thereby defined as a three-dimensional field (Fig. 1, b). Its values can factor into the SDF equation and be used to modify the wavelength (periodicity) of the TPMS, the local thickness of the surface shell, the solid to void fraction by shifting the threshold iso-value or even the alignment and orientation of the unit cells (Fig. 1, c). They can be arranged in an orthogonal, polar- or even spherical coordinate system to optimally adapt to structural necessities. The TPMS pattern can also gradually transition from one type into another type along the gradient of a spatial function.« less