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SHAJAY BHOOSHAN STUDIO PHILIPP SIEDLER FEDERICO BORELLO BEGUM AYDINOGLU

Graphic statics is a method for structural form finding that originates in the pre-digital era, but continues to be used and developed even today. In graphic statics, the geometry and equilibrium of forces of a structural system are represented by two reciprocal diagrams: the form and the force diagram. Since the geometrical relationship between these diagrams provides explicit control over both form and forces of a structure simultaneously, graphic statics is considered as an intuitive technique for structural design, relevant to architects, engineers, researchers, and students.

Despite its clear strengths and advantages, existing method of graphic statics has some limitations. The most important one is that a designer can only design three dimensional structures by reducing them to a combination of two dimensional sub-systems, for example, by using projections. Seminal projects have been explored as examples of application of the graphic static method in 19th century to solve vault structures like the St John cathedral in New York (Image 8) and St. Francis de Sales in Philadelphia (Image 9) both designed by Rafael Gustavino, as well as the Salginatobel Bridge by Robert Maillart (Image 10).

1Masoud Akbarzadeh, Tom Van Mele, Philippe Block, On the equilibrium of funicular polyhedral frames and convex polyhedral force diagrams, Computer-Aided Design 63 (2015) 118–128. 2Luigi Cremona, Le figure reciproche nella statica grafica, 1872.

Image 8. Use of graphic static method to design the vault structure of the St John cathedral in New York by Rafael Gustavino 1888.

Image 9. Tile dome of St. Francis de Sales in Philadelphia designed by Rafael Gustavino 1907.

The Salginatobel Bridge is Robert Maillart’s design for a competition held in summer 1928 to link the villages of Schuders and Schiers in the Swiss canton of Graubünden. Since its completion in 1930, the 90m-span bridge has received considerable praise: while some have celebrated its striking new art form, others have emphasized its brilliant economical and structural efficiency. It does indeed represent a great example of optimal and elegant design which did not have to undergo any repair work in its first 45 years, despite the poor quality of the concrete. There is therefore interest in the design methods employed, especially since Maillart only did handwritten workings and looked for straightforward methods to define the structure. Built in 1929, this masterpiece has since received extensive recognition both for its structural elegance and its efficiency. However investigations into the design process enabling this degree of perfection remain incomplete. Studying Maillart’s original working drawings, this paper reviews the earliest stages chronology of the Salginatobel Bridge design process. It focuses on three methods: the use of graphically controlled parabolas, the minimization of bending moments within the bridge and the geometrical definition of the foundation block. These graphical methods reveal how Maillart simultaneously dealt with geometry and the flow of forces throughout using a straightforward, handy and efficient form-finding process which is still relevant today. Maillart used parabolas throughout the entire design process in order to define any non-straight geometry. Circles and catenaries are almost non- existent in the first working drawings. A parabola is a regular curve frequently used for sketching arches to approximate catenary or funicular configurations under distributed loads. However, surprisingly, they are not meant to be used predominantly in that way here, but rather as a graphical convenience that is accurately reproducible and easy to handle with control points. This feature becomes essential since the geometry will be readjusted many times due to successive refining steps. The specific orientation of most parabolas and traces of their construction attest to their use as a non-structural, but purely graphical tool.

Image 12. Reactions of the equilibrated half bridge in red; funicular polygon and its construction in blue; thrust line and its construction in orange.

Image 13. Line of centroids in red, thrust line in orange, eccentricities in red hatching.

Equilibrium of funicular polyhedral frames and convex polyhedral force diagrams Graphic statics is a method for structural form finding that originates in the pre-digital era, but continues to be used and developed even today. In graphic statics, the geometry and equilibrium of forces of a structural system are represented by two reciprocal diagrams: the form and the force diagram. Since the geometrical relationship between these diagrams provides explicit control over both form and forces of a structure simultaneously, graphic statics is considered as an intuitive technique for structural design, relevant to architects, engineers, researchers, and students. Despite its clear strengths and advantages, existing method of graphic statics has some limitations. The most important one is that a designer can only design three dimensional structures by reducing them to a combination of two dimensional sub-systems, for example, by using projections. This paper therefore presents a three-dimensional version of graphic statics using reciprocal polyhedral form and force diagrams for the design and analysis of spatial frames with externally applied loads.

1Masoud Akbarzadeh, Tom Van Mele, Philippe Block, Three-dimensional Compression Form Finding through Subdivision, Proceedings of the International Association for Shell and Spatial Structures (IASS) Symposium 2015, Amsterdam.

Image 3. A complex 3D branching structure designed using 3D form and force diagrams.

The computational research consisted in the exploration of different network topologies via algorithmic modeling. The load condition is expressed through vectors applied perpendincularly to each corresponding face of the polyhedra (force diagram). As previously mentioned the algorithm consisted of three main operations: 1. Definition of the force diagram via convex polyhedra aggregation sharing faces, corresponding edges and vertices. 2. Each polyehdra was scaled from the geometrical center to achieve the nodal center. 3. Each of the faces were projected and extruded to the corresponding polyhedra faces to generate the dual graph of each polyhedra. 4. The algorithm iteratively align the polyhedra edges with the corresponding faces resulting in a form diagram with edges always perpendicular to their corresponding faces. The result of the algorithm application given the specific load condition resulted into network in a compression only state without bending moments leading to a reduction of the material need in the fabrication process. Multiple iterations have been tested differentiating the polyhedra agreggation, scale and orientation.

Topological studies exploring the aggregation of different convex polyhedra (force diagram) and the corresponding resulting compression networks (form diagram).

The algorithm explored provides another relevant advantage thanks to the geometrical costruction process of the network topology itself. All the faces of the networks mesh are planar because the perpendicularity to the corrisponding polyhedra face which makes theam suitable for multiple fabrication processes and material. Pair of faces of the mesh have one edge aligned on the same plane that makes them continuous and developable into strips with single curvature. This method allows potentially to have a structure nodes free which relies only on the face’s edges connectivity as for example a finger joints strategy. The behaviour of the membrane and the cable net have been simulated through dynamic relaxation via particle-springs system with the aim of synchronising the digital model with the real material behaviour, observed in parallel through small scale prototypes. This allowed to extend the topological research digitally without relying on physical prototyping.

Developable mesh strips of a compression network structure resulting from the aggregation of one cube and four pyramids.

Iterative subdivision of the mesh faces into continuous planar strips.

Dynamically relaxed model to simulate steel cables winding and membrane wrapping.

Digital simulation of the membrane behavior and the cable net through dynamic relaxation via particle-springs system.

Digital simulations with the aim to forecast the material behaviour and the global geometric output of the threading and wrapping strategy.