Tom Van Mele

On the equilibrium of funicular polyhedral frames and convex polyhedral force diagrams

This paper presents a three-dimensional extension of graphic statics using polyhedral form and force diagrams for the design of compression-only and tension-only spatial structures with externally applied loads. It explains the concept of 3D structural reciprocity based on Rankines original proposition for the equilibrium of spatial frames. It provides a definition for polyhedral reciprocal form and force diagrams that allows including external forces and discusses their geometrical and topological characteristics. This paper furthermore provides a geometrical procedure for constructing a pair of reciprocal polyhedral diagrams from a given polyhedron representing either the form or force diagram of a structural system. Using this method, this paper furthermore suggests a design strategy for finding complex funicular spatial forms in pure compression (or tension), based on the construction of force diagrams through the aggregation of convex polyhedral cells. Finally, it discusses the effect of changes in the geometry of the force diagram on the geometry of the form diagram and the distribution of forces in it. Three-dimensional extension of graphic statics using polyhedral form and force diagrams.Defining the topological and geometrical relationships of 3D reciprocal diagrams.Design of compression and tension-only spatial structures with externally applied loads.Designing complex funicular spatial forms by aggregating convex force polyhedral cells.CAD implementation to manipulate the geometry of the force and explore its effects on the forms.

Shaping Tension Structures with Actively Bent Linear Elements

To achieve sufficient anticlastic (negative) curvature, membrane structures are tensioned between high and low anchor points, attached to the ground, buildings or poles. By integrating flexible bending elements in the membrane surface, an internal support and shape-defining system is created that provides more freedom in design and allows reducing the amount of external supports compared to traditional membrane structures. This paper presents a computational framework for form finding of tension structures with integrated, elastically bent, linear elements, based on three-dimensional bending moment vectors and a mixed force density formulation. With an implementation of this framework in CAD modelling software, users can control form and forces by prescribing any combination of force densities, forces, stiffness or lengths to the spline and cable-net elements. Sparse matrix operations are used to compute the resulting equilibrium shapes. The shape-defining possibilities of integrating ‘bending-active’ elements in tension structures are demonstrated through a series of design studies with various boundary conditions and spline configurations. The presented framework and implementation provide a straightforward method for the design of this hybrid structural system, and, therefore, facilitate its further exploration.

Scissor-hinged retractable membrane structures

This paper introduces scissor-hinged retractable membrane structures, a system for retractable membrane roofs that require a fully retractable supporting structure and multiple stable roof-configurations. A vaulted, foldable, supporting structure is developed consisting of two scissor-hinged frames that can retract towards opposite sides of the space below. Structural membranes are spanned in these frames in a ridge-and-valley configuration to form the roofs outer surface. Actuators are integrated to control the tension in the membrane surface in different roof-configurations. Transformation from one configuration to another is controlled by cables running through the supporting structure over series of pulleys. The characteristics of these components are discussed, and their implementation illustrated with a design for a retractable roof over a tennis arena. The structural behaviour of this roof is analysed under representative load conditions. A procedure for such analyses using conventional software tools for the design and analysis of tensile surface structures is presented.

Automated Generation of Knit Patterns for Non-developable Surfaces

Knitting offers the possibility of creating 3D geometries, including non-developable surfaces, within a single piece of fabric without the necessity of tailoring or stitching. To create a CNC-knitted fabric, a knitting pattern is needed in the form of 2D line-by-line instructions. Currently, these knitting patterns are designed directly in 2D based on developed surfaces, primitives or rationalised schemes for non-developable geometries. Creating such patterns is time-consuming and very difficult for geometries not based on known primitives. This paper presents an approach for the automated generation of knitting patterns for a given 3D geometry. Starting from a 3D mesh, the user defines a knitting direction and the desired loop parameters corresponding to a given machine. The mesh geometry is contoured and subsequently sampled using the defined loop height. Based on the sampling of the contours the corresponding courses are generated and the so-called short-rows are included. The courses are then sampled with the defined loop width for creating the final topology. This is turned into a 2D knitting pattern in the form of squares representing loops course by course. The paper shows two examples of the approach applied to non-developable surfaces: a quarter sphere and a four-valent node.

Three-dimensional graphic statics: Initial explorations with polyhedral form and force diagrams

This article investigates how reciprocal form and force polyhedrons can be used to develop procedures for the design of three-dimensional trusses and funicular structures, analogous to the well-known techniques of graphic statics for two-dimensional structural systems. It demonstrates how global equilibrium of a system of forces can be established by constructing a closed force polyhedron, if the forces can be replaced by a resultant force alone, without a resultant couple. It also describes the three-dimensional equivalent of the “closing string,” which is the basis in graphic statics for the construction of funicular solutions for given loads and support locations. Furthermore, it provides a procedure for constructing a constrained funicular form for a simple, determinate boundary condition. Finally, it discusses some of the difficulties involved with similar constructions and procedures for non-concurrent forces and in particular with those systems of forces that can only be replaced by a resultant force and couple.

Decomposing Three-Dimensional Shapes into Self-supporting, Discrete-Element Assemblies

This study investigates a computational design approach to generate volumetric decompositions of given, arbitrary, three-dimensional shapes into self supporting, discrete-element assemblies. These assemblies are structures formed by individual units that remain in equilibrium solely as a result of compressive and frictional contact forces between the elements. This paper presents a prototypical implementation of a decomposition tool into a CAD software, focusing on user-controlled design to generate such assemblies. The implementation provides an interactive design environment including real time visual feedback, in which the design space of self-supporting block assemblies can be explored and expanded. Some surprising results of such explorations are included and discussed.

Disjointed force polyhedra

Abstract This paper presents a new computational framework for 3D graphic statics based on the concept of disjointed force polyhedra. At the core of this framework are the Extended Gaussian Image and area-pursuit algorithms, which allow more precise control of the face areas of force polyhedra, and consequently of the magnitudes and distributions of the forces within the structure. The explicit control of the polyhedral face areas enables designers to implement more quantitative, force-driven constraints and it expands the range of 3D graphic statics applications beyond just shape explorations. The significance and potential of this new computational approach to 3D graphic statics is demonstrated through numerous examples, which illustrate how the disjointed force polyhedra enable force-driven design explorations of new structural typologies that were simply not realisable with previous implementations of 3D graphic statics.

Equilibrium-Aware Shape Design for Concrete Printing

This paper proposes an Interactive Design Environment and outlines the underlying computational framework to adapt equilibrium modelling techniques from rigid-block masonry to the multi-phase material typically used in large-scale additive manufacture. It focuses on enabling the synthesis of geometries that are structurally and materially feasible vis-a-vis 3D printing with compression-dominant materials. The premise of the proposed research is that the material printed in layers can be viewed as micro-scale bricks that are initially soft, and harden over time—a kind of micro-stereotomy. This insight, and the observation about the necessity of design exploration yields the principal contributions of the paper: A computational framework that allows for geometric reasoning about shape and exploration of associated design-space, and early, proof-of-concept results.

Dieste, González Zuleta and Sánchez del Río: Three approaches to reinforced-brick shell structures

Reinforced-brick shell structures are mainly known through the extraordinary work of the Uruguayan engineer Eladio Dieste. However, other remarkable examples of shells in reinforced masonry were being built during Dieste’s development of the technique or even before. The works by Guillermo González Zuleta in Colombia and Ildefonso Sánchez del Río in Spain deserve a closer look. Analysis of the three engineers’ backgrounds, writings and work reveals different original conceptions of the technique. The conceptual approach to the reinforced brick material and the way they approached challenges and obstacles are very different in the three cases, yet they finally achieved similar structural solutions. This document presents a review of historical reinforced masonry architectural examples by these three master builders. The paper analyses their roles in the development of this technique and presents some of their most influential or striking architectural pieces.

Function Representation for Robotic 3D Printed Concrete

The use of Function Representation (FRep) to synthesise and specify geometries for 3D printing is finding renewed interest. The usefulness and extension of this representation in the synthesis and analysis of geometries for the process of large-scale, layered concrete 3D printing has been previously articulated by the authors. This paper fully extends the implicit representation used previously in shape-design to fabrication-related processing of compressive skeletal structures for realisation by robotic 3D printing of concrete. In particular, we use an initial value formulation of a propagating front to process the nodes and bars of a given funicular spatial structure.