A Physical Costa Surface 1/3 : History of The Surface

In the spring of 2016, Ruy Marcelo Pauletti, associate professor at University of São Paulo, taught the graduate class “Non-linear analysis of prestressed structural systems” at Princeton University. For the final project students applied techniques of design, numerical analysis and fabrication to a tensile sculpture that is on display in Princeton.

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The student’s animated rendering of a modified Costa surface with six holes

Tension structures embody purity of forms. There is little hidden in a pre-stressed membrane that serves both  as a form definer and a structure. Membrane structures are structurally efficient, span large distances and can express elegance. They are also very relevant for sculptures since they are lightweight and compelling to engineers, architects,  mathematicians and visual artists. The students in Prof. Pauletti’s course use dynamic relaxation and natural density as form finding methods for the design of the tensile sculpture and use patterning methods for its fabrication.

For their sculptural qualities, minimal surfaces (embodied by the form of soap films) are the focus of this design process. Those surfaces are characterized by the minimization of total area for a given boundary. As such, the student’s project is the design and fabrication of a Costa surface (see rendering above). The history behind this surface is remarkable. In 1982 Celso Costa, then a mathematics graduate student in Rio de Janeiro, disproved a longstanding conjecture. This conjecture said that the plane, the catenoid and the helicoid were the only finite topological types of complete embedded minimal surfaces in R3 (surfaces with no boundary and no intersection with themselves). He intuited that a three-time punctured torus with two catenoidal extreme ends and a planar middle end also satisfies the criteria of such a surface. For the mathematically inclined reader, his original work can be found here.

The first rendering of the surface was done by David Hoffman at University of Massachusetts at Amherst with help of programmer James Hoffman in 1985. You can watch David Hoffman talk about minimal surfaces and the history of the Costa surface in the video below.

Only at a few occasions has the Costa surface been materialized at an architectural form before. It is relatively simple to create the form from the mathematical equations but to build a tensile Costa surface sculpture is more complex. The challenges are producing the pattern layout, sewing and working with a flexible material.

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Happy (rammed) earth day!

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(image credit: Pat Dumas, flickr)

In celebration of earth day, we want to show you the explorations of earth construction at the Form Finding Lab, in particular our focus on rammed earth. Rammed earth buildings are constructed by pouring soil into a form work, similar to the one used to make concrete elements. This soil is then compacted in successive layers, either by hand or using tampers to create highly compacted dirt walls like those in the Tuscon Mountain house.

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Rammed earth walls at the Tucson Mountain House designed by Architect Rick Joy (photo credit: designmilk – flickr)

For his senior thesis, Aaron Katz (’16) studied rammed earth buildings and the earthquake loading capacity of rammed earth walls.  His research project evaluated the application of limit state analysis developed for masonry to assess the overturning of rammed earth walls during earthquakes.

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Japanese Lineages In Structural Design

Japan has a wealth of traditional craftsmanship. The traditional Japanese craftsman takes pride in the quality of his work and in the lineage of masters of which he is the living descendant. In traditional Japan, a young man entered a craft by becoming the disciple of a master and learning the practice as an apprentice. As the student’ skills and knowledge increased, he progressed through the stages of journeyman, craftsman and eventually master. This process was slow and took an average 8 years. When the student was finally approved by the master as a fully pledged craftsman, he would receive the lineage of the master’s name which he would inscribe in bold letters above his new shop door and on his tools.

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The structural designer’s new toolbox

It is said that “an engineer is a (wo)man who can do for a dime what any fool can do for a dollar.” While it is customary for an engineer to be responsible for achieving a specific technological need for the lowest economic cost, this saying is crippling to both the engineer’s creativity and the design’s potential. With the available new digital and numerical tools, the role of the bridge designer needs to be emphasized.

Traditionally, designers use tools such as back of- the-envelope hand calculations, physical experiments and tests, design charts, and 2D sketches to develop and advance the preliminary design process. The digital and numerical computational techniques are new tools in this existing toolbox; they allow for the rapid generation of a large set of design alternatives that fulfill specific requirements. These alternatives might be unusual and surprising to the traditional bridge designer, who is grounded in intuition and accumulated knowledge; yet they present unexplored feasible domains in the design space. I would like to demonstrate this statement with the design development of one of my favorite footbridges. The Knokke-Heist footbridge in Belgium (designed in 2008 by Ney and Partners), was designed unlike traditional structures, where the loads in the longitudinal and transverse directions were decoupled. The bridge features on the cover of our book “Laurent Ney: Shaping Forces” and was a key feature in the exhibition with the same name held at the Palais des Beaux Arts, Brussels.

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Cover of the book “Laurent Ney: Shaping Forces”
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Exhibition at Palais des Beaux Arts: Laurent Ney (third from the right), Sigrid Adriaenssens (right)

From a topological point of view, the bridge is based on a cutout, curved shape that efficiently carries external loads through membrane action and satisfies the site requirements. The shape of the steel bridge was numerically form-found, like a network of connected springs supported at the abutments and the mast heads. The grid is allowed to relax or “fall” under gravity loads applied at the spring connections. The idea behind the curved structural form is to carry all loads within the steel surface shell without needing additional structural elements.

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Knokke-Heist footbridge in Belgium

Once the overall shape has been found, the geometry is further refined to comply with the CNC manufacturing constraint of single-curvature steel sheet bending, and then it is numerically optimized to maximize the overall stiffness of the bridge. The latter task presents a typical topology optimization problem that consists of distributing a given amount of material in a design domain subject to load and support conditions, such that the stiffness of the structure is maximized. The figure below shows the optimal thickness distribution in the shell for different values of the thickness ρmean, which is a measure of the total material volume constraint. Topology optimization provided a powerful tool for the preliminary design of this thin-shelled bridge. By combining topology optimization with form finding and CNC manufacturing constraints, a 3D typology that might not have been conceivable in a purely analytical or intuitive fashion was generated. You can find out more about the topology optimization study we carried out on this bridge here.

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Optimal material distribution in the thickness of the shell

Author: Sigrid Adriaenssens