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Funicularity and Equilibrium for High-Performance Conceptual Structural Design

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Curved structures are characterized by the critical relationship between their geometry and structural behaviour, and selecting an appropriate shape in the conceptual design of such structures is important for achieving material efficiency. However, the set of bending-free geometries are limited and, often, non-structural design criteria (e.g., usability, architectural needs, aesthetics) prohibit the selection of purely funicular or antifunicular shapes. In response to this issue, this thesis studies the possibility of achieving an axial-only behaviour even if the geometry departs from the ideally bending-free shape. This dissertation presents a new design approach, based on graphic statics that shows how bending moments in a two-dimensional geometry can be eliminated by adding forces through an external post-tensioning system. his results in bending-free structures that provide innovative answers to combined demands on versatility and material optimization. The graphical procedure has been implemented in a free-downloadable design-driven software (EXOEQUILIBRIUM) where structural performance evaluations and geometric variation are embedded within an interactive and parametric working environment. This provides greater versatility in finding new efficient structural configurations during the first design stages, bridging the gap between architectural shaping and structural analysis. The thesis includes the application of the developed graphical procedure to shapes with random curvature and distribution of loads. Furthermore, the effect of different design criteria on the internal force distribution has been analyzed. Finally, the construction of reduced- and large-scale models provides further physical validation of the method and insights about the structural behaviour of these structures. In summary, this work strongly expands the range of possible forms that exhibit a bending-free behaviour and, de facto, opens up new possibilities for designs that combine high-performing solutions with architectural freedom. Free download at: http://oa.upm.es/39733/1/Leonardo_Todisco.pdf
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... Before the analytical formalization of the idea of funicularity, occurred between 15 th and 17 th century, many funicular structures had already been constructed thanks to experience and static considerations of the designers and constructors [2]. In order to find some written essays concerning this topic, one needs to look at the 13 th century. ...
... Starting from the 15 th century, the first studies on arches and cables appear. Leon Battista Alberti (1404-1472), Andrea Palladio (1508-1580), Leonardo da Vinci (1452-1519) and Simon Stevin (1548-1620) are some of the most celebrated scientists to give a fundamental contribution to the formulation of the behaviour of curved structure and of the arch equilibrium [2]. Galileo Galilei (1564Galilei ( -1642 was the first one who attempted to give a mathematical description of a cable; in his writing "Dialogues Concerning Two New Sciences" (1638), mistakenly using an erroneous analogy with the parabolic motion of projectiles, he confused catenary and parabola [6]. ...
... Joachim Jungius (1587-1657) rejected Galilei's statement, demonstrating the difference between the two, which increases when the span to sag ratio decreases. Jungius's writing "Geometria Empyrica" was published in 1669 after his death [2]. The correct equation of a cable's geometry was written in 1691 by Gottfried Wilhelm von Leibniz (1646-1716), Christiaan Huygens (1629-1695) and Johann Bernoulli (1667-1748) [7]. ...
... During the Roman period, builders seem to have some awareness of funicularity expressed in attempts to change load distributions to achieve better structural stability (e.g. use of filler materials or use of concrete with graded density) [6]. In order to find some written essays concerning this topic, one needs to look at the 13th century. ...
... Theoretical definitions attempt to justify and formalized what was experimentally evident. Leon Battista Alberti (1404-1472), Andrea Palladio (1508-1580), Leonardo da Vinci (1452-1519) and Simon Stevin (1548-1620) are some of the most celebrated scientists to give a fundamental contribution to the formulation of the behavior of curved structure and of the arch equilibrium [6]. Galileo Galilei (1564Galilei ( -1642 was the first one who attempted to give a mathematical description of a cable; in his writing "Dialogues Concerning Two New Sciences" (1638), mistakenly using an erroneous analogy with the parabolic motion of projectiles, he confused catenary and parabola [10]. ...
... Joachim Jungius (1587-1657) rejected Galilei's statement, demonstrating the difference between the two, which increases when the sag to span ratio decreases. Jungius's writing "Geometria Empyrica" was published in 1669 after his death [6]. The correct equation of a cable's geometry was written in 1691 by Gottfried Wilhelm von Leibniz (1646-1716), Christiaan Huygens (1629-1695) and Johann Bernoulli (1667-1748) [11]. ...
Article
The study of new design methods targeted to minimize the use of materials is a theme of great relevance nowadays; structural designers pursue structural solutions characterized by efficiency, sustainability and optimization. Funicular systems adopt the "right" shape in accordance with the applied load and are ideally able to act without introducing bending. In this work an effective and easy-to-read method to study and quantify the funicularity is presented and applied to structural shells obtained using form finding, and analyzed under different static loads. In order to formulate the new method, the classical funicularity concept has been extended and the definition of Relaxed Funicularity (R-Funicularity) introduced. The parameter used to define the funi-cularity is the eccentricity and a structural shell is called R-Funicular when the eccentricity is included into an admissibility interval.
... The graphical construction to convert a non-funicular shape into a bending-free one is described in detail in other authors' papers [11][12][13][14]. Here, it is shortened in two main steps, and applied to a circular arch ( Figure 2). ...
... Yet, the problem is indeterminate: an infinite number of solutions exist to make the starting geometry bending-free. This indeterminacy, which is in-depth studied in the first author's thesis [11], conceptually corresponds to the fact that as infinite axial-only geometries match to a single loading distribution, an infinite sets of loads correspond to a single geometry. Axial forces in the arch and the post-tensioning force applied to the tendon depend on the chosen solution: the smaller the distance between the compressive curve and the post-tensioning system, the higher the forces are in both components. ...
... Furthermore, several applications of the graphical methodology are described in the other research of the authors [11]. Results point out that: ...
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Non-structural design criteria (e.g., usability, architectural needs, esthetics) may prohibit the selection of purely funicular or anti-funicular shapes. In response to this issue, this paper illustrates the possibility of achieving an axial-only behavior, even if the geometry departs from the ideally bending-free shape. This is achieved by adding forces through an external post-tensioning system, with a layout defined through graphic statics. The paper briefly illustrates examples of this approach and its implementation within a design-driven software where structural performance and geometric variation are embedded within an interactive and parametric working environment.
... Another common constraint is related to the elimination of the horizontal reaction at the supports. This constraint was already explored by Todisco [11]. It implies that the parameter related to the posttensioning horizontal reaction remains fixed, compensating the reaction caused by the compression chord. ...
... As observed by Todisco [11], imposing that the horizontal reaction remains null, implies that the posttensioning system has to be placed entirely below the compression chord. With a similar reasoning based on constrained graphic statics (Fivet and Zastavni [5]), it can be shown that the domain of positions for the pole of the post-tensioning system that allows placing it above the compression chord is limited by the vectors in the diagram of forces corresponding to the first and last elements of the compression chord. ...
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External post-tensioning allows designing efficient structures in bending. While using very light elements in tension, the forces generated by the post-tensioning can compensate permanent loads acting in the structure. Additionally, efficiency under live loads is also improved by increasing the structural depth between the tension and compression chords. This efficiency depends on multiple interrelated factors, such as the geometric and mechanical characteristics of the compressed chord and the post-tensioning system, inclination of the struts in-between, initial prestressing force, boundary conditions, and the distribution of the live load itself. This paper expands the design space of planar externally post-tensioned structures using graphic statics. On this extended design space, constraints are applied to find solutions with particular structural interest. Finally, an assessment of the efficiency of the structures under permanent and live loads is proposed, allowing to compare different solutions and produce optimal configurations.
... The procedure developed by authors to define the layout of a posttensioning system to convert any twodimensional curve into an axial-only solution under permanent loads is based on graphic statics, 7,8 and for reader's convenience can be found in other recent works. [9][10][11][12] It is out of the scope of this paper to describe the graphical construction and only the main concepts are explained here. ...
... The material-efficiency of these bending-free structures under permanent loads has been studied in other works, 9,11,12 but their behavior for other load distributions has not been analyzed yet. ...
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Bending moments due to permanent loads can be strongly reduced in a generally shaped structure by introducing a set of additional forces through an external posttensioning system. The response of these structures under live loads has not been studied yet; therefore, this work investigates the behavior of these structural systems under load distributions that differ from the permanent ones. First, a definition of a moment‐based efficiency is proposed. Next, a parametric analysis is carried out to evaluate the performance of these structural systems by varying the two main design variables and other relevant parameters (strut connection and orientation, discretization, efficiency definition, etc.). Finally, the methodology is applied to a case of study where several posttensioned alternatives are compared to the standard one (i.e., without posttensioning) to prove the benefits of adopting external posttensioning in terms of material‐saving.
... Starting from 15 th century, the first studies on arches and cables appear. Leonardo da Vinci (1452-1519) and Simon Stevin (1548-1620) are two of the most celebrated scientists to give a fundamental contribution to the formulation of the arch equilibrium [13]. ...
... Joachim Jungius (1587-1657) rejected Galilei's statement, demonstrating the difference between the two, which increases when the sag to span ratio decreases. Jungius's writing "Geometria Empyrica" was published in 1669 after his death [13]. The correct equation of a cable's geometry was written in 1691 by Gottfried Whilelm von Leibniz (1646-1716), Christiaan Huygens (1629-1695) and Johann Bernoulli (1667-1748) [4]. ...
Conference Paper
The study of new design methods targeted to minimize the use of materials is a theme of great relevance nowadays; structural designers pursue structural solutions characterized by efficiency, sustainability and optimization. Funicular systems adopt the "right" shape in accordance with the applied load and are ideally able to act without introducing bending. In this work an effective and easy-to-read method to study and quantify the funicularity is presented and applied to structural shapes obtained using form finding and analyzed under different static loads. Moreover, a modal analysis of these structures is performed and the same method is applied to the modal stress distribution obtained in order to study the intrinsic funicularity of the structures according to their dynamic behavior. A number of different numerical examples are presented. This paper is of interest to practitioners and educators interested in the domain of analysis of thin and thick shell structures under seismic loading.
... Since ancient times, prestressing has been used to fasten joints between elements, to improve stability and material efficiency in structural systems, or to tune their vibratory behavior. From vessels of ancient Egypt [1] or masonry in ancient Rome buildings or in gothic cathedrals [2,3], to the modern use of prestressed concrete [4], cable nets or pneumatic membranes [5], the potential of prestress in structural engineering has been demonstrated throughout history. In addition, the prestress becomes widespread in Nature. ...
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In this paper we propose a constructive procedure to determine the radial pretension in an axially-symmetric orb-web, on the basis of one eigenfrequency and its corresponding vibration mode. The method can be applied both to axisymmetric modes, corresponding to the solution of a regular Sturm–Liouville problem, and to non-axisymmetric modes, corresponding to the solution of a singular Sturm–Liouville problem. In the absence of measurement errors of the eigenfunction, the identification of the pretension field is almost perfect. On the contrary, if the measurement presents some noise, then identification is imprecise. In this case, a regularization technique applied to the measurement of the eigenfunction, or to its derivative, is proposed, as well as a filtering process, which significantly improves the reconstruction of the pretension field. The fundamental mode of vibration, sampled with a suitable number of points, is the one that provides the lowest identification error.
... Todisco et al. [1,2] developed a methodology using graphic statics that permits attaining bending-free behavior for curved structures in two dimensions whose geometry departs from the ideal funicular shape. The desired state of equilibrium is obtained through the introduction of an additional set of loads into the structure that is deduced using graphic statics from the structure's reciprocal polygon of forces. ...
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Bending-free structures are of enormous interest due to their high material efficiency. However, structural concerns are commonly conditioned by other design criteria, such as function or aesthetics. In order to overcome these limitations, Todisco et al. developed a methodology that allowed achieving bending-free behavior in two-dimensional structures whose geometry departed from the ideal funicular shape. An external system of post-tensioned cables was defined using graphic statics to introduce new loads into the structure, which permitted minimizing bending. The aim of this paper is to present the basis of a new procedure to extend this methodology to three-dimensional structures using the new developments of 3D graphic statics. Simple spatial structures under different loading conditions have been studied, exploring the wide range of solutions allowed by the indeterminacies of the problem. In addition, reduced-scale models have been constructed for several configurations, as first conceptual proof of the method. This procedure allows exploiting the potentialities of 3D graphic statics for the design of bending-free spatial structures. This is applicable to explore a new and wide range of high-efficient geometries whose shape is imposed a priori.
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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
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Funicular structures, which follow the shapes of hanging chains, work in pure tension (cables) or pure compression (arches), and offer a materially efficient solution compared to structures that work through bending action. However, the set of geometries that are funicular under common loading conditions is limited. Non-structural design criteria, such as function, program, and aesthetics, often prohibit the selection of purely funicular shapes, resulting in large bending moments and excess material usage. In response to this issue, this paper explores the use of a new design approach that converts non-funicular planar curves into funicular shapes without changing the geometry; instead, funicularity is achieved through the introduction of new loads using external post-tensioning. The methodology is based on graphic statics, and is generalized for any two-dimensional shape. The problem is indeterminate, meaning that a large range of allowable solutions is possible for one initial geometry. Each solution within this range results in different internal force distributions and horizontal reactions. The method has been implemented in an interactive parametric design environment, empowering fast exploration of diverse axial-only solutions. In addition to presenting the approach and tool, this paper provides a series of case studies and numerical comparisons between new post-tensioned structures and classical bending solutions, demonstrating that significant material can be saved without compromising on geometrical requirements.
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