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Funicular Shell Design: Geometric approaches to form finding and fabrication of discrete funicular structures
Abstract
Addressing both architects and engineers, this dissertation presents a new framework for the form finding and design of fabrication geometry of discrete, funicular structures in the early design phase. Motivated by ongoing debates about digital architecture and funicular shell form finding, it introduces a new methodology for structurally-informed design of curved surface architecture through the use of geometrical rather than analytical or numerical representations of the relation between form, forces and fabrication. Based on Thrust Network Analysis (TNA), new algorithms are presented that enable an interactive exploration of novel funicular shapes, enriching the known formal vocabulary of shell architecture. Using TNA, the framework adopts the same advantages of techniques like graphic statics, providing an intuitive and educational approach to structural design that ranges from simple explorations to geometry-based optimisation techniques. Complementary to this structurally-informed design process, the work reflects on the latest building technologies while also revisiting historic construction techniques for stereotomic stone masonry and prefabricated concrete shells to develop efficient fabrication design strategies for discrete funicular structures. Based on architectural, structural and fabrication requirements, several tessellation approaches for given thrust surfaces are developed for the design of informed discretisation layouts of any funicular shape. The flexibility and feasibility of the form-finding framework is demonstrated in several case studies employing the new structural design tool RhinoVAULT, which implements the developed form-finding methods. The use of fabrication design strategies is discussed in a comprehensive case study that shows project-specific tessellation design variations and first fabrication results for a complex stone masonry shell.
... Regarding continuum optimization, well-known algorithms include the SIMP, 1 density-based method (Bendsøe and Kikuchi 1988;Rozvany et al. 1995;Sigmund 2001;Andreassen et al. 2011), and the level-set method (Allaire et al. 2002;Wang et al. 2003;van Dijk et al. 2013), among others. For the particular case of shell structures, various contributions and proposals for optimization algorithms for aesthetically appealing surfaces are available in literature (Block 2009;Rippmann 2016;Adriaenssens et al. 2014), and in the even more specific context of slabs (Ma et al. 2023), to name a few. As for discrete optimization, the first "true" optimal layout optimization work was carried out by Michell (1904) using an infinitely dense network of truss (axial-only) elements, which paved the way for further research such as Dorn et al. (1964), Sokól (2011), Sokół (2014), Zegard and Paulino (2014). ...
... ability to handle non-perpendicular loads on the shell surface (Ma et al. 2023;Cascone et al. 2021); 2. shell structures can have a combination of compression and tension zones, as well as out-of-plane bending, i.e., Nervi (Huxtable 1960), b Tennis Center by H. Isler (Adriaenssens et al. 2014). (Color figure online) the method is not limited to funicular (tension-only) or statically determined (compression-only) dome shells (Rippmann 2016;Oval et al. 2018); and 3. the reinforcing elements layout does not need to conform to the shell discretization (Li et al. 2017), i.e., the discrete element discretization vertices can disagree with those of the continuum shell. ...
This work presents a mechanics-informed technique to layout and design reinforcing ribs in a (quasi-)optimal manner for a thin-shell structure. The element layout follows the principal stress lines of the thin-shell continuum. The mechanics problem is solved using finite element analysis, and the result is then post-processed to determine the principal stress lines using techniques based on “streamlines.” In principal stress lines, unlike fluid mechanic streamlines, multiple directions exist at every node. This is a problem of “incomplete information,” where possible directions exist only at the nodes and the direction selection and interpolation are not trivial. Four algorithms of varying complexity and nature are proposed to address this problem, including gradient-based optimization formulations which provide an acceptable balance between computational complexity and quality of the results. The resulting principal stress lines do not necessarily coincide with the finite element nodes, which is another key novelty of this work. The (discrete) reinforcing members contribute to the overall shell stiffness by distributing (or spreading) their stiffness onto the shell, using an energy conservation approach. Finally, the laid out reinforcing members are optimally sized using a gradient-based optimizer therefore constituting a two-step procedure. The capabilities, shortcomings, and constraints of the proposed method are illustrated by means of three examples. These examples suggest that the proposed method outperforms structured and orthogonal layouts of the shell reinforcing elements.
... Later, it discusses numerous scholars' clarifications and identifications of Parametricism, finalizing by interpretation of styles and Parametricism and ending with heuristics in Parametricism. Figure 5 reflects one of the initial efforts to use parametric equations originated in Robert Hook's hanging chain model, which Antonio Gaudí then established in his ideas [46]. It is problematic to distinguish if Gaudí was constrained by the diverse investigators and mathematicians who had previously applied parametric formulas to illustrate geometry [47]. Mark Burry, the contemporary authorized architect of Gaudí's Sagrada Familia, utters that there is for all aims and objectives none composed by Gaudí himself about the motivations, conjectures, and repetition that pushed him to spread the limitations [48]. ...
... Later, it discusses numerous scholars' clarifications and identifications of Parametricism, finalizing by interpretation of styles and Parametricism and ending with heuristics in Parametricism. Figure 5 reflects one of the initial efforts to use parametric equations originated in Robert Hook's hanging chain model, which Antonio Gaudí then established in his ideas [46]. It is problematic to distinguish if Gaudí was constrained by the diverse investigators and mathematicians who had previously applied parametric formulas to illustrate geometry [47]. Mark Burry, the contemporary authorized architect of Gaudí's Sagrada Familia, u ers that there is for all aims and objectives none composed by Gaudí himself about the motivations, conjectures, and repetition that pushed him to spread the limitations [48]. ...
Architectural styles have evolved, with distinct periods marked by unique styles that reflect broader artistic movements. Despite extensive research on traditional and contemporary architectural styles, there remains a gap in understanding the distinctiveness and contributions of Parametricism as a 21st-century architectural style. This study aims to critically examine the emergence and development of Parametricism, aiming to highlight whether it represents a fundamental evolution in architectural thought and practice or primarily a methodological innovation. Through a comprehensive investigation of its formal characteristics, historical roots, and relationships with other architectural movements, this study seeks to discuss Parametricism’s unique contributions to modern architecture. This research is crucial as it addresses the underexplored area of parametric design’s impact on architectural theory and practice, offering a critical analysis that situates Parametricism within the broader context of 21st-century architectural styles. By clarifying the definition and position of Parametricism, this study aims to enrich both the state of the art and the state of practice in the field, providing valuable insights for scholars, practitioners, and students of architecture. Through this critical assessment, the paper contributes to the ongoing discourse on the evolution of architectural styles, highlighting the originality and significance of Parametricism in contemporary architecture.
... The viability of this form-finding approach has been demonstrated through historical precedents, notably in Nervi's innovative floor systems where he successfully implemented stress-line-based patterns in projects like the Gatti Wool Factory and Palace of Labor, achieving both structural efficiency and architectural expressiveness through the alignment of material with natural force flows (Halpern et al., 2013). Vector-based approaches like TNA and VFIFE offer alternative methodologies: TNA employs graphic statics to establish relationships between form and forces (Rippmann, 2016), while VFIFE generates forms through physical models and validates membrane stress distribution (Li et al., 2017). For the compatibility with Karamba optimization, this study adopts the PSL method to determine the distribution of ribs. ...
Green roofs (GRs) play an important role in urban sustainability initiatives, offering reduced building energy consumption and enhanced ecological performance through additional growing medium and vegetation layers. Nevertheless, these functional components introduce challenges such as extra structural load, greater slab thickness, and increased construction complexity. This research proposes a design-fabrication method for a thin-vaulted green roof prototype, featuring a compression-only surface and upstand ribs along unevenly distributed stress lines to achieve both lightweight properties and material efficiency compared to a conventional flat roof with GR systems. To realize such a non-standard and multifunctional structure, a hybrid formwork system combining 3D clay printed (3DCP) molds and a reusable, doubly curved foam base is proposed to handle geometric complexity and functional integration by incorporating stay-in-place formwork as the growing medium. An empirical construction experiment, conducted at a 1:5 scale, validates the proposed method with optimized fabrication procedures for a proposed application scenario in a community renovation project.
... The history of graphical statics begins with a study published by Varignon in 1725. However, it was not until the 1860s that Maxwell and Cremona developed the method of graphical statics [1]. A significant milestone in graphical statics is Die Graphische Statik by Culmann [2,3]. ...
... However, the final product tends to be non-homogeneous since it is built layer by layer. Subtractive methods, like wire and blade cutting and robotic carving and milling [11], offer an alternative that results in homogeneous blocks. ...
This paper establishes a connection between reciprocal form and force diagrams and edge cable-supported membrane structures under isotropic stress. While various form-finding techniques exist for tensile membrane structures, reciprocal diagrams provide an intuitive graphical approach that is particularly effective during early design stages. The study introduces the foundational concepts of reciprocal diagrams, outlines their construction for membrane and cable elements, and explores their application to isotropically stressed membranes. The relationship between form and force diagrams is analyzed, highlighting the similarities between different graphical conventions. A method for three-dimensional form-finding is presented, starting with planar equilibrium informed by force densities and iteratively updating these densities to achieve isotropic stress. Several case studies are discussed, demonstrating the method’s capability to accurately calculate geometry and stress distributions compared to reference solutions. This work contributes to the field by offering a clear, visual approach to form-finding in membrane structures, combining graphical methods with computational techniques to simplify design and analysis processes.
The selection of well-conditioned sub-matrices is a critical concern in problems across multiple disciplines, particularly those demanding robust numerical stability. This research introduces an innovative, AI-assisted approach to sub-matrix selection, aimed at enhancing the form-finding of reticulated shell structures under the xy-constrained Force Density Method (also known as Thrust Network Analysis), using independent edge sets. The goal is to select a well-conditioned sub-matrix within a larger matrix with an inherent graph interpretation where each column represents an edge in the corresponding graph. The selection of ill-conditioned edges poses a significant challenge because it can render large segments of the parameter space numerically unstable, leading to numerical sensitivities that may impede design exploration and optimisation. By improving the selection of edges, the research assists in computing a pseudo-inverse for a critical sub-problem in structural form-finding, thereby enhancing numerical stability. Central to the selection strategy is a novel combination of deep reinforcement learning based on Deep Q-Networks and geometric deep learning based on CW Network. The proposed framework, which generalises across a trans-topological design space encompassing patterns of varying sizes and connectivity, offers a robust strategy that effectively identifies better-conditioned independent edges leading to improved optimisation routines with the potential to be extended for sub-matrix selection problems with graph interpretations in other domains.
Mutallib-khan mosque is an abandoned and derelic building in the city of Khoy, north-west of Iran. It was built in 1839 with intentions to Serve as a mosque, in which muslims say their prayers in congregation, and also as a tomb for the proprietor and patron of it, Mutallib-khan himself. But upon his unfortunate demise, the building remained incomplete, and the architect's work remained undone. without a dome or any kind of roof or closure for about two century, it has been susceptible to any and every kind of damage, ranging from natural predicament to mankind misuses and has always been desolated. The buildings interior space is completely open to the Element and therefore it is fair to say that it does not have an interior space with a quality that house an interior function. On the other hand the very exposure of the building has always been the abode to sun, and daylight has always been present, and this has brought a quality to the space for the past two centuries that has become the quintessence of the building: it's character. This article studies ways in which it is possible to rehabilitate the building through redefining an interior space. The framework of this studies is divided into four main sections. In the first section the building itself is studied from an architectural and historical point of view. Second section is dedicated to a comparison between two extremes in how to treat a classic building in rebuilding its dome. In the third and fourth section methods and approaches to designing a dome-like shell is studied, in the context of iranian domes and contemproray shells. The outcome of these four sections is a generic proposal of a two layer transluscent dome with an interior geometry of a catenary arch, and an exterior shape of a Chamaneh arch that could be 3D printed directly on the building.
SiDMACIB (Structurally informed Design of Masonry Assemblages Composed of Interlocking Blocks) is the first numerical model capable of extending the equilibrium problem of limit analysis to interlocking assemblies. Adopting the concave formulation, this model can compute the stress state at the corrugated faces with orthotropic behaviour, such as their combined torsion-shear capacity. Generally speaking, finding the plastic torsion-shear capacity of planar faces shared between conventional blocks is still a fresh topic, while investigating this capacity for interlocking interfaces is particularly rather unexplored. Upon the authors’ previous works that focused on interlocking blocks with a single lock, in this paper, an extension to blocks composed of several locks (multi-lock interfaces) is presented and the SiDMACIB model is upgraded accordingly. For this purpose, the shear-torsion results obtained from the original SiDMACIB formulation are validated and subsequently compared with those derived from distinct element analysis conducted using the 3DEC 7.0 software. Based on this comparison, revisions to the SiDMACIB model are proposed, involving a reduction in the number of locks affecting torsion-shear capacity.
This research investigates the assembly of funicular shell structures using a single layer of flat ceramic tiles. The objective is to synthesize recent advances in structural prediction software with existing means and methods of on-site assembly. The primary area of investigation is at the scale of the tectonic unit-most specifically how introduction of geometric intelligence at the scale of the unit can simplify the assembly of forms that are difficult to realize in the context of modern construction. The project simulates an industrial production scenario in which components for a given shell structure can be fabricated using a wire cutter-equipped 6-axis robotic arm. It aims to increase the adaptability and applicability of ceramic shell structures.
This paper addresses both the architectural, conceptual motivations and the tools and techniques necessary for the digital production of an architecture of volume. The robotic manufacturing techniques of shaping volumetric materials by hot wire and abrasive wire cutting are discussed through a number of recent projects. A comparative analysis between milling and hotwire cutting is presented and a number of case studies and tool development studies are considered. Finally, the specifics of toolpath generation for robotic wire cutting are introduced.
From Egypt and Classical Greece and Rome through the building booms of the Gothic era and the Renaissance, and from the Industrial Revolution to the present era of digital modelling, Building: 3,000 Years of Design, Engineering, and Construction, charts centuries of innovations in engineering and building construction. This comprehensive and heavily illustrated volume, aimed at students and young professionals as well as general readers, explores the materials, classic texts, instruments, and theories that have propelled modern engineering, and the famous and not-so-famous buildings designed through the ages, from the Parthenon to Chartes Cathedral and the dome of St. Peter's, from eighteenth-century silk mills in England to the Crystal Palace, and on to the first Chicago high-rises, the Sydney Opera House, and the latest "green" skyscrapers.
The book concentrates on developments since the industrial and scientific revolutions of the seventeenth and eighteenth centuries. Incorporated within the continuous narrative are sidebars with short biographies of eminent engineers, excerpts from classic texts, stories of individual projects of major importance, and brief histories of key concepts such as calculus. Also included are extensive reference materials: appendices, a glossary, bibliography, and index.
This book is out of print but now available on line:
https://archive.org/details/building3000year0000addi
This paper describes the development of an unreinforced, freeform vault consisting of 399 discrete limestone blocks with thicknesses ranging from 5 to 12 cm. The vault covers an area of 75 m2 and spans more than 15 m in pure compression, without mortar between the blocks. We discuss how the design of the vault and its individual pieces was entirely driven by constraints related to the fabrication process and to the architectural and structural requirements and timeline of the project. Furthermore, we describe the form-finding process of the shell’s funicular geometry, the discretisation of the thrust surface, the computational modelling and optimisation of the block geometry, and the machining process. Finally, we discuss some of the strategies that were developed for dealing with tolerances during fabrication and construction.
From the 1850s onwards, the civil engineer Robert Henry Bow (1827–1909) became known for his expertise in structural analysis, publishing on the design of bridge and roof trusses, and working with the prolific railway engineer Sir Thomas Bouch (of later Tay Bridge infamy). In the first part of this 1873 publication, Bow describes 337 different truss structures, grouping them into four classes according to their structural characteristics: statically determinate, kinematically determinate, indeterminate, and other. In the second part, he describes a method for graphically analysing truss structures, based on the work of Thomas Maxwell and others, and applies this method to the structures listed in the first part. Perhaps of most interest to the working engineer are the explanations as to which structures are most efficient given typical material constraints, such as girders of uniform cross-section. The work remained a useful resource for practising engineers well into the twentieth century.
Forms are extremely important in concrete construction. They mold the concrete to the required size and shape while controlling its position and alignment (Figure 7.1). Forms are self-supporting structures that are also sufficient to hold the dead load of the reinforcement and fresh concrete and the live load of equipment, workers, and miscellaneous materials (Figure 7.2). In building and designing formwork, three major objectives must be considered: 1. Quality-Forms must be designed and built with sufficient stiffness and accuracy so the size, shape, position, and finish of the cast concrete are attained within the required tolerances.
In einem kurzen Abriß werden in dem nachfolgenden Aufsatz einige der wesentlichen Gesichtspunkte bei der Verwendung pneumatisch gestützter Schalungen angesprochen. bevor näher auf die bisher entwickelten Bauverfahren und die damit gemachten Erfahrungen eingegangen wird.
The objective of this PhD research is to investigate how best to use computational search processes in the early phase of design. Search algorithms are implemented in combination with parametric models of different kinds of geometry, with the purpose of studying various building performances. The early phase of design and the nature of the design problem are stud-ied to get an idea of the kind of search process that would best accompany architects during this phase. Multi-disciplinarity and contrasting objectives are singled out as fundamental required characteristics of such a process. This leads to the proposal of multi-disciplinary studies into architectural shapes, both in the realm of complex curved geometry and in more traditional orthogonal forms. The search process itself is studied in its capacity to generate solutions in a multi-objective setting. The process of selecting and formulating search problems, the parametrization of geometric families for study as well as the selection criteria for outstanding solutions are all topics of discussion. A particular type of algorithm called Genetic Algorithms is implemented and studied in length. Search processes are proposed for 3 architectural design fields: structural, acoustic and energy design