Figure 3-2 - uploaded by Leonardo Todisco
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sketch of the Golden Horn Bridge designed by Leonardo da Vinci in 1502 (Biblioteque Institute Paris ).
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... Also, scientific works have been developed with load-adapted arches such as those in Refs. [8] or [9]. ...
In the specific case of curved cable-stayed bridges, the horizontal component of the load introduced by the stay cables on the deck is variable, concentric and dependent on the connection configuration between the tower and the cables, becoming a challenge in the design of these type of bridges. Hitherto, designers have dealt with this challenge in different ways, either by optimizing the position of the tower and its geometric characteristics, or by modifying the morphology of the stay cable system. This paper proposes the use of funicular and anti-funicular curves of the horizontal concentric load, introduced by the stay cables, to design the curved deck directrix, reducing lateral forces on the deck under the self-weight hypothesis. For the design of the deck directrix, two different formulations are considered: one discrete by means of summations and the other continuous by means of non-linear differential equations. Both formulations study the two possible signs of the axial force which will govern the design (funicular and anti-funicular curves). A least squares approximation is developed to facilitate the implementation of these formulations. The paper introduces a method to liberate the deck from its most important lateral loads, i.e., the concentric loads introduced by the stay cables. This way, it develops a deck dominated by axial forces instead of lateral ones (Bending moment with vertical axis, Mz, and lateral shear force, Vy), which can be critical for its design and decrease the stay-cable system efficiency. It explains, by different methods, how this directrices vary with different design decisions, so that the designer can develop the directrix that suits his design. Finally, two examples of directrices are given as a conclusion.
... This progress has focused primarily on pedestrian [7] use, where the freedom of design is greater and where engineers such as Schlaich have developed very successful systems with suspension and arch bridges, and scientific works have been achieved with load-adapted arches such as those in Refs. [8] or [9]. ...
In the specific case of curved cable-stayed bridges, the horizontal component of the load introduced by the stay cables on the deck is variable, concentric and dependent on the connection system between the tower and the cables. The problem has hitherto been addressed in different ways, either by optimizing the position of the tower and its geometric characteristics, or by modifying the morphology of the stay system. This paper proposes the use of funicular and anti-funicular curves of this concentric load to reduce lateral forces on the deck under the self-weight hypothesis. It will show the principles of the method to develop pedestrian and road bridges. For the design of the deck directrix, two different formulations are considered: one discrete by means of summations and the other continuous by means of non-linear differential equations. Both formulations study the two possible signs of the axial force which will govern the design. A least squares approximation is developed to facilitate the implementation of these formulations. Two examples are given as a conclusion.
... En los últimos años, gracias a las nuevas tecnologías (3) (4), se ha podido ver una cierta evolución en diferentes tipologías de puentes curvos, como puentes atirantados, colgantes o en arco. Este progreso se ha centrado fundamentalmente en el uso peatonal, donde ingenieros como Schlaich han desarrollado sistemas muy acertados con puentes colgantes y arco (5) o trabajos científicos con arcos adaptados a las cargas como los realizados por Lorenz Lachauer y Toni Kotnik (6) o Leonardo Todisco (7). ...
The aim of the following study is to optimize the layout of a curved bridge deck. The guideline proposed is intended to
ensure that the entire board is free of vertical axis bending moments. Two mathematical approaches are developed for
the directive, a discrete and a continuous. For both, positive and negative sign for the axial load in the deck is studied.
To make the possible implementation easier, a least square approximation is made for both signs. Two examples will be
developed to end the article.
... By using this method, inner forces are calculated in the continue structure, no discrete network of elements is employed, and material properties have to be implemented leading to time-consuming procedures, without guaranteeing a stable convergence [66]. This approach has been adopted by the author for finding the antifunicular shape of arches supporting a curved deck [107]. ...
... An application of a design-oriented tool was implemented by the author to find an antifunicular arch configuration for a deck with any geometry in the 3dimensional space. Methodology, tool and results have been presented in his thesis submitted for the master degree in Engineering of Structures, Materials and Foundations [107]. ...
... The author implemented a tool named SOFIA (Shaping Optimal Forms with an Interactive Approach), based on incremental analysis, that starting from a generic arch geometry allows to obtain its three-dimensional antifunicular shape for a curved deck [107]. ...
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
... Shaping Optimal Forms with an Interactive Approach), che rende più agevole la definizione geometrica di un ponte ad arco anti-funicolare per un impalcato di qualsiasi geometria [6]. ...
I concetti di funicolarità ed equilibrio sono stati utilizzati da sempre, in maniera più o meno intuitiva, per la progettazione di strutture resistenti per forma. L'esatta definizione geometrica di una struttura funicolare o anti-funicolare rappresenta una fase fondamentale nella progettazione di tali strutture. In questo articolo dopo una breve presentazione dello stato dell'arte e di una classificazione delle metodologie esistenti per definire una geometria funicolare, si applica il concetto di funicolarità a due problemi progettuali: il primo riguarda la definizione geometrica di un ponte ad arco con impalcato curvo, mentre il secondo studia la possibilità di trasformare qualsiasi struttura in una anti-funicolare mediante l'applicazione di forze esterne. In entrambi i casi sono stati sviluppati dei software, implementati in un ambiente di lavoro interattivo e parametrico, che possono riabilitare metodologie oggigiorno poco utilizzate, come ad esempio la statica grafica. In definitiva questo articolo incentiva l'innovazione nella progettazione strutturale combinando consolidati concetti strutturali, esistenti metodologie di form-finding e recenti design-oriented software per il conceptual design di strutture efficienti e versatili.
... This paper applies the concept of funicularity [1][2][3][4] to curves that are originally not funicular under dead loads, in order to take advantage of the structural efficiency due to a funicular behavior while also allowing for architectural and geometric flexibility. ...
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
... The concept of funicularity has been well-documented in the literature for two-dimensional and three-dimensional geometries (Allen and Zalewski 2009;Block 2005Block , 2009Block et al. 2006a;Huerta 2008;Todisco 2014); this section provides a brief overview. ...
Funicular geometries, which follow the idealized shapes of hanging chains under a given loading, are recognized as materially efficient structural solutions because they exhibit no bending under design loading, usually self-weight. However, there are circumstances in which nonstructural conditions make a funicular geometry difficult or impossible. This paper presents a new design philosophy, based on graphic statics, that shows how bending moments in a nonfunicular two-dimensional curved geometry can be eliminated by adding forces through an external posttensioning system. An interactive parametric tool is introduced for finding the layout of a posttensioning tendon for any structural geometry. The effectiveness of this approach is shown with several new design proposals.
... This paper applies the concept of funicularity [1][2][3][4] to curves that are originally not funicular under dead loads, in order to take advantage of the structural efficiency due to a funicular behavior while also allowing for architectural and geometric flexibility. A classical design approach for non-funicular structures is based on the use of bending solutions, in which axial forces and bending coexist, or trusses. ...
... This paper applies the concept of funicularity [1][2][3][4] to curves that are originally not funicular under dead loads, in order to take advantage of the structural efficiency due to a funicular behavior while also allowing for architectural and geometric flexibility. ...
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.
p>Bridge designs usually exhibit significant geometric variations between different structural solutions, which implies a low degree of reuse of the models in similar projects. To overcome this limitation, a parametric approach is proposed as an answer. Generative design enhances the bridge design process, increasing efficiency by reducing time and effort. The proposed methodology is based on the creation of a flexible geometric model through the introduction of parameters and numerical relationships between them. Therefore, from a generic generative development, different geometric and structural solutions of composite bridges could be created by modifying the parameter values in a bridge model. The objective of the present work is to define the workflow for a multi-girder composite bridge project based on parametric design and optimization in Grasshopper/Rhino to model the bridge Karamba3D, for structural analysis, and Tekla Structures, for 3D representation. This article describes the methodology implemented, starting with the design of the script into a visual programming interface that runs inside Rhino. Thanks to Grasshopper-Tekla live link, the 3D model is generated by using a set of Grasshopper components that can create and interact with objects in Tekla Structures. Afterwards, the algorithm for FEM analysis is created with Karamba3D. Finally, an optimization process is defined to reduce material waste and achieve an efficient design.</p
