Philippe Block

Designing unreinforced masonry models

We present a complete design pipeline that allows non-expert users to design and analyze masonry structures without any structural knowledge. We optimize the force layouts both geometrically and topologically, finding a self-supported structure that is as close as possible to a given target surface. The generated structures are tessellated into hexagonal blocks with a pattern that prevents sliding failure. The models can be used in physically plausible virtual environments or 3D printed and assembled without reinforcements.

Interactive Vault Design

This paper presents a new computational framework based on Thrust Network Analysis (TNA) for the design of funicular structures. Fast and robust solving algorithms enable the interactive exploration of these constrained structural systems. By giving explicit, bidirectional control over the internal force distribution and overall geometry to the designer, free exploration of these statically highly indeterminate systems is made possible. The equilibrium of funicular compression networks is represented by reciprocal diagrams, which visually express the force dependencies between different parts of the structure. By modifying these diagrams in real-time, the designer is able to explore novel and expressive vaulted geometries that are blurring the difference between shapes associated to typical compression-only forms, obtained e.g. with hanging networks, and freeform surface structures. The power of this framework for design is demonstrated by a user-friendly software implementation, which has been used to design and build a freeform, thin-tile masonry vault.

Assembling self-supporting structures

Self-supporting structures are prominent in historical and contemporary architecture due to advantageous structural properties and efficient use of material. Computer graphics research has recently contributed new design tools that allow creating and interactively exploring self-supporting freeform designs. However, the physical construction of such freeform structures remains challenging, even on small scales. Current construction processes require extensive formwork during assembly, which quickly leads to prohibitively high construction costs for realizations on a building scale. This greatly limits the practical impact of the existing freeform design tools. We propose to replace the commonly used dense formwork with a sparse set of temporary chains. Our method enables gradual construction of the masonry model in stable sections and drastically reduces the material requirements and construction costs. We analyze the input using a variational method to find stable sections, and devise a computationally tractable divide-and-conquer strategy for the combinatorial problem of finding an optimal construction sequence. We validate our method on 3D printed models, demonstrate an application to the restoration of historical models, and create designs of recreational, collaborative self-supporting puzzles.

Three-Dimensional (3D) Equilibrium Analysis of Gothic Masonry Vaults

This study presents a practical method for three-dimensional static equilibrium analysis for masonry vaults using funicular networks. The method, a nonlinear extension of Thrust Network Analysis, is explained, and through three exemplary case studies, the potential of this new research is demonstrated. These examples discuss different assumptions on the “flow of forces” in Gothic quadripartite vaults; visualize the flat-vault equilibrium of rose windows under wind loading; and provide a stability analysis of the intricate nave vaults of Sherborne Abbey, Dorset, England. The presented approach provides insights in structural redundancy of unreinforced masonry structures by quantifying lower bounds on the geometric safety factors. The method for efficient funicular analysis of complex vault geometries furthermore provides the foundation for a fully three-dimensional funicular analysis implementation, extending thrust line analysis to three-dimensional thrust networks, for historic masonry.

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.

Algebraic graph statics

Abstract This paper presents a general, non-procedural, algebraic approach to graphical analysis of structures. Using graph theoretical properties of reciprocal graphs, the geometrical relation between the form and force diagrams used in graphic statics is written algebraically. These formulations have been found to be equivalent to the equilibrium equations used in matrix analysis of planar, self-stressed structural systems. The significance and uses of this general approach are demonstrated through several examples and it is shown that it provides a robust back-end for a real-time, interactive and flexible computational implementation of traditional graphic statics.

Parametric self-supporting surfaces via direct computation of airy stress functions

This paper presents a method that employs parametric surfaces as surface geometry representations at any stage of a computational process to compute self-supporting surfaces. This approach can be differentiated from existing relevant methods because such methods represent surfaces by a triangulated mesh surface or a network consisting of lines. The proposed method is based on the theory of Airy stress functions. Although some existing methods are also based on this theory, they apply its discrete version to discrete geometries. The proposed method simultaneously applies the theory to parametric surfaces directly and the discrete theory to the edges of parametric patches. The discontinuous boundary between continuous patches naturally corresponds to ribs seen in traditional vault masonry buildings. We use nonuniform rational B-spline surfaces in this study; however, the basic idea can be applied to other parametric surfaces. A variety of self-supporting surfaces obtained by the proposed computational scheme is presented.

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.

Interactive Equilibrium Modelling

This paper presents a novel method for computer-aided equilibrium modelling of structures in early design stages. Based on the force density method, an iterative procedure is developed that enables the generation of spatial kinematic pin-jointed structures that are in equilibrium close to a given input geometry, while satisfying additional constraints on both geometry and forces. This method forms the core of an interactive form-finding process that consists of alternating steps of modelling and computational optimization. In each modelling step, the user is able to modify geometry, topology, external forces and constraints of the structure. In each optimization step, equilibrium is re-established while respecting the user-defined constraints. A prototype has been implemented within an existing CAD software package, and three examples illustrate the use of the presented method, ranging from a playful exploration of surprising shapes to the rationalization of structural geometry. The method allows to intuitively explore the formal freedom of spatial equilibrium shapes with mixed compression and tension forces, within hard, user-defined constraints. In conclusion, it is claimed that by providing interactive equilibrium modelling methods, the design of new, surprising spatial forms with efficient structural behaviour is facilitated.

New Design and Fabrication Methods for Freeform Stone Vaults Based on Ruled Surfaces

Thin concrete and steel grid shells show elegantly how shell design is used for contemporary freeform architecture. Their natural beauty is coupled to an inherent efficiency due to minimal bending, result from their good structural form. Thanks to digital form finding tools, streamlined planning processes and automated fabrication, the technical and economic difficulties to design and build those structures, especially grid shells, decreased significantly [1].