Magnificent Hyperbolic Paraboloid
Those who met me on beBee know that I am always "pushing" the contents about interesting engineering stuff. I even created the Interesting Engineering, Technology and Discoveries hive, besides the already existing hives, to activate the beBee construction community to be more engaged in sharing and commenting. On this not such an easy task, my fellow bee and ‘dirt doctor’ Ken Boddie has been of great help.
After my post about geometry of the Platonic solids, which has been very well received on several social networks thanks to huge help in promoting it by dear Milos Djukic (widely known for his love of fractals), I decided to write about one of my favourite geometric forms, hyperbolic paraboloid, or shorten - hypar. The term 'hypar' was first introduced by the architect Heinrich Engel.
Shortly, a hyperbolic paraboloid is a doubly curved surface that resembles the shape of a saddle. It has a convex form along one axis and a concave form along the other. This unique design gives it a soaring shape.
Its name – hyperbolic paraboloid, arises from the fact that the horizontal cross sections are hyperbolas, while vertical cross-sections (parallel to the other two coordinate planes: xz and yz plane) are parabolas. The traces, as shown in the image below, are simply the intersections of a surface with planes parallel to the coordinate planes. When a hyperbolic paraboloid being sliced by horizontal planes we get hyperbolic traces (explained in a short video). The horizontal cross-sections are also called level curves.
Hypar is both a translational and a ruled surface
Hypars can also be formed by two sets of straight lines called generatrices, or rulings. This means that despite being a curved surface, a hypar can be thus constructed with straight lines, just like in this video showing a hypar model made of skewers.
A surface that contains two sets of straight lines, or rulings, is called a doubly ruled surface. Through every one of its points (T) there are two distinct lines that lie on the surface, as shown in the image above.
An important reason for the use of hypars in construction is that they can be formed using straight lines. Consequently, the hyperbolic paraboloid is an ideal shell form, allowing the use of straight boards for formwork.
Particularly interested is how hypars can be joined together, at their edges, to make beautiful structures.
Strength Through Curvature - Hypar Thin Shell Concrete Roofs
Fernand Aimond (1902-1984) can be considered as the father of the thin hypar concrete shells, who also formulated the membrane theory of the hypar. He designed and constructed several hypar roofs in France in the 1930s for aircraft hangars and workshops.
But a master of hypars and joining hypars was Spanish architect, engineer, structural artist and builder Felix Candela (1910-1997), who popularized the use of hypars for reinforced concrete construction. Candela was influenced by Aimond and gave credit to his work. He also followed the fundamentals of the Eduardo Torroja's work, a famous Spanish structural engineer renowned for his development of reinforced concrete and thin concrete shell structures.
I must say.... that although an architect by training, in practice I am a constructor and building contractor.
He designed and built many innovative thin shell concrete roof structures. Most of these structures were built in Mexico in the 1950s and 60s. That was the “golden era” for thin shell construction. Candela constructed over 300 shells during these decades. He was the greatest practitioner of shell design whose name has become synonymous with the hyperbolic paraboloid.
His thin concrete vaulted shells were made of reinforced concrete mixed and poured in situ, over wooden formwork. The thin shell roofs allowed the minimum thickness of concrete while having great strength and stiffness at the same time.
The strength comes from form, not mass. Hypar roofs gain its strength from double curvature that allows hypar structures the great resistance to bending. The doubly curved shape strikes a balance between tension and compression forces, allowing hypar structure to remain thin, yet surprisingly strong.
Candela constructed his first hypar shell for the University of Mexico in 1951. The Cosmic Rays Laboratory main structure is a shell made of two adjacent hypar saddles (12 meters in total length and 11 meters wide) supported by a platform and stiffened by three arches. It is one of the thinnest cast-in-place (in situ) concrete structures ever built. At its thickest point, the concrete reaches only 15 mm that the cosmic radiation could be measured through it. The shell remains in a good condition for more than 60 years.
The second image clearly shows the straight-line form boards following the ruling lines (double ruling) of the geometric form.
This early hypar structure was an overture to Candela's outstanding career as a shell builder.
In 1952 Candela started experimenting with the umbrella structures of four identical quadrants, called tympans. The shell geometry was created by joining four straight-edged hypars whose sides rose upwards away from the central column (as if an umbrella blown upward). Umbrella structure is a type of inverted pendulum. Precisely, it is a cantilevered structure. The basic structural scheme of an umbrella consists of a foundation, only one column (with an embedded pipe to catch the rainwater) and a roof.
Inspired by Aimond’s sketches, Candela created the solution of umbrella shell, without the edge beams. Candela's concept also included the ingenious design of an inverted umbrella structure as footings to distribute the weight of the structure onto the poor quality Mexican subsoil. The footing was simply an inverted form of the umbrella shell.
Below is the famous photo of his second experimental umbrella in 1953, in which he tried to find the optimum rise and area of an umbrella. It shows load testing with twenty-five workers who climbed the shell.
Below is shown an umbrella structure under construction, with scaffolding and formwork, and carefully placed reinforcement.
Candela experimented with umbrella forms during the 1950s. The umbrellas became asymmetrical and with various inclinations. Over the years, his common umbrella form progressed to the folded hypar. By placing several umbrellas, side by side, Candela designed large roof surfaces, mostly for low-cost industrial construction. These structures were built in a very short time and with reusable formwork.
Later, he developed a new form of the rectangular umbrella by arranging folded hypars. First such umbrella structure was constructed for the Candelaria Subway Station in Mexico City, built in 1968.
Based on this form, he also created some interesting structures that appeared to defy gravity, such as the band shell in Santa Fe, Mexico City, completed in 1956.
The ground-breaking design of the hypar umbrella was inspired by natural trees complemented by Candela's imagination and the implementation of technical innovations.
The folded hypars, an adaptation of the common umbrella design, Candela used to create dramatic roofs of large spaces, such as the roof for the Church of the Miraculous Medal (Milagrosa), built in 1955 in Mexico.
The church is a remarkable building. It is somewhat reminiscent of the Gaudi's forms. The structure is a combination of super-warped hypar surfaces with a thickness of 4 cm or less. Even the bell tower is made of hypars. It seems as if Candela wanted to see how much can be done with this type of structure.
As clearly explained in the image below, the form for the roof structure consists of the hypars derived from asymmetrical umbrella forms that were tilted and then pleated to create triangular openings.
The roof shells and interior columns merge together in an extraordinary geometry. Candela shows in Milagrosa his virtuous performance in the use of hyperbolic paraboloids.
More about this design on the following video
The Milagrosa is also one of the early examples of the brutalist-style building with the sculptural exposed concrete.
Outside and inside view of this amazing roof with visible imprints of the form boards
Here Candela used a similar construction procedure as on his previous structures, including the use of inverted umbrella form for the foundations. The formwork for Milagrosa differed from that of his other shells because the warping is too large. To provide adequate curvature, Candela used the wedge-shaped thick boards.
The casting of the roof with no mechanical equipment - concrete was placed manually using buckets
One of his major works is the Lomas de Cuernavaca Chapel (“hills of Cuernavaca”), Mexico, built in 1958. Candela used a single hypar surface to get an open-air chapel with a structural form of pure aesthetics. The Cuernavaca Chapel shell is self-supporting with a span of 30 m and dramatic curvature. The open end of the chapel rise to a height of 24 meters and has visible thinness of 4 cm that represents the frontiers of the form.
As seen in the image below, the straight timber members (framework) were used to develop curve form, because a double curved hypar can be generated by straight lines. The form boards were then laid on the top in the other direction. Imprints of these form boards are visible on the inner side of the concrete shell.
(1) The formwork clearly shows the hyperbolic paraboloid as a ruled surface
(2) Scaffoldings for the concrete shell
(1)The concrete shell immediately after the removal of the scaffolding
(2)Visible imprints of the form boards
In 1958, Candela completed his most significant work, the Los Manantiales Restaurant in Xochimilco, Mexico City.
The Xochimilco shell is composed of four intersecting hypars with free curved edges. The soaring shape at the edges resulted in a simple and gracefully thin shell structure.
Most shells prior to Candela have edge ribs to prevent crack from the edges inward. Candela found the free-edge innovative solution which was elegant and efficient at the same time. His solution has focused on the groins rather that the edges. Each of Los Manantiales’ eight groins is thickened into a V-shaped beam. These beams enhance the groins’ stiffness, reducing bending and deflection, and thereby eliminating normal edge forces. At the supports, the V-beams anchor into umbrella footings.
In these original drawings, Candela clearly shows the form of Manantiales Restaurant shell. It is an eight-sided groin vault composed of four hyperbolic paraboloid saddles that intersect at the centre point. The form was original and unexplored at that time. The innovative use of V-shaped beams allows edges free of stiffening beams and produced the radical shell thickness of only 4 cm.
The plan is radially symmetric with a maximum diameter of ca. 42 meters. Groins spanning around 32 meters between supports. At the centre the shell has a height of 5.8 m and the edges rise to 9.9 m.
Narrow boards, used as formwork, followed the straight-line rulings that form the hypar surface. A layer of cement grout, laid under concrete, created a smooth inner surface. The concrete was placed by hand, one bucket at a time.
The V-shaped beams were reinforced with steel, while the shell has only nominal reinforcement to address temperature effects that can cause cracking.
The almost 60-year-old shell is still structurally intact and generally in fine condition. It represents a testimony of Candela as a structural artist of great merit.
For the restaurant design, Candela re-elaborated the shell structure built in 1958 in Xochimilco (Mexico) composed of eight radially symmetrical lobes, each lobe as part of a hypar. The shell is only 6 cm thick that gradually increases up to 22.5 cm at the intersection of the ribs.
The geometry of the concrete shell structure for the entrance building was based on the hand-drawn sketches by Candela. This shell consists of the intersection of three hyperbolic paraboloids.
The geometry of the restaurant and entrance building
The underwater restaurant and the entrance building
The oceanographic park is integrated into the complex known as the City of Arts and Sciences, a cultural and architectural complex designed by Spanish architect Santiago Calatrava in partnership with Candela. The park was opened in 2003, while the whole complex was finished in 2010.
The shells are made of steel fibre reinforced concrete (SFRC) with reinforcement netting on the mid-surface of the shell. The concrete was applied with a shotcrete process onto wooden formwork that was supported by shoring towers. The process also included superficial finishing. The use of SFRC improved the shell’s tensile stress capabilities.
(1) (2) Restaurant’s roof under construction – 3D space framework up to a height of 13.70 m
(3) The shotcrete process on the entrance building roof
Concrete shells immediately after the removal of framework
Both buildings were designed to open out from the ponds like water lilies
Candela’s work continues to inspire contemporary designers such as Santiago Calatrava, another great Spanish architect who is also PhD in civil engineering (structural engineer). Candela, who was teacher of Calatrava, highly influenced the Calatrava’s works.
Geometry Meets Music
The Philips Pavilion was a temporary structure designed for Expo '58 in Brussels. The design was done by Le Corbusier in collaboration with the Greek composer Iannis Xenakis, an architect and engineer, who carried out most of the designing.
The goal was to build the pavilion to demonstrate the sound and light possibilities of Philips' technologies in those days. The structure supposed to be self-supporting and column-free, a hollow structure with the acoustic qualities. Xenakis found a solution for the shape of the building in a series of conjoined curved planes forming a tent-like enclosure.
The chosen design was a cluster of nine hyperbolic paraboloids with a stomach-shaped floor plan (shown on the model below). Over 500 spectators might have been accommodated inside the pavilion for an eight-minute spectacle of light and sound called Poème électronique, by composer Edgard Varèse. People were sitting and lying in the dark room and watched visual effects along the walls of the pavilion.
Following video is a short presentation of the used design and its geometry.
The final structure was 22 meters high and certainly pushed the limits of engineering of that time.
Coloured photographs of the Philips Pavilion at the Brussels World's Fair in 1958
To build such a complex form Xenakis proposed a structural system of prefabricated concrete panels hung in tension on the 7 mm wire cables, which were laid on both side of the shell. This ingenious solution also included about 2000 precast panels 5 cm thick panels, built in a hangar shed from simple sand moulds that matched the curvature of the pavilion. The size of the precast slabs was about 1 square meter with a light reinforcement mesh to prevent breakage during transportation to the site.
Moulded sand hill for casting panels
The steel cables were positioned inside and outside of the shell, following the ruling lines of hypar geometry, and anchored to the concrete ribs and foundation beam that formed the base of the pavilion. The "Strabed " system used for mounting prestressing cables. After erecting the panels, the entire structure was post-tensioned. Once the panels were erected, the entire structure had been post-tensioned, making a monolithic structure. The outside of the shells including the prestressing wires were covered with aluminium paint and the interior plastered with a sound-absorbent layer of asbestos felting.
(1) The precast slabs had been placed in the position and filled with mortar
(2) After erecting the panels, the prestressing wires were tensioned with a special jack
At the edges of each hyperbolic paraboloid shell (lines of intersection) were cylindrical concrete ribs, 40 cm in diameter, as the structural elements cast in situ. The scaffolding wooden beams followed the ruling lines of the hyperbolic paraboloid shells.
The second image below shows the numerous prestressing wires of high-tensile steel, which were applied on both surfaces of the shell and left visible.
The original pavilion was demolished in 1959. In recent years, there have been attempts to rebuild the pavilion in the hometown of the Philips Company, Eindhoven, Holland.This work of art represented a phenomenon through its synthesis of architecture, visual media and music, and left its mark on history.
In this post, I presented only a part of many astonishing structures based on the geometry of the hyperbolic paraboloid. Besides the form and shape of these structures, I also tried to present the construction process which is quite remarkable. The used methods of construction were both material and labour intensive. At that time, almost 60 years ago, those methods were the only possible.
The thin concrete shells are not as popular as they once were, in the golden era of concrete shell structures prior to 1968. Recent advances in concrete technology should affect the cost of construction and may help revive the interest in this structural form, particularly in the magnificent hyperbolic paraboloid form.
6. The Princeton University Art Museum