4th Int’l Specialty Conference on The Conceptual Approach to Structural Design: 28 – 29 June 2007, Venice, Italy
CONCEPTUAL DESIGN OF “UEHARA MINIMUM KIOSK”, AN ULTRA-LIGHTWEIGHT STRUCTURE THAT BORROWED GENIUS FROM NATURE AND PRODIGIOUS MINDS Humberto Uehara*, Uehara Group, Engineers & Architects, Mexico.
Abstract Through million of years nature has come up with a great variety of shapes and geometries some of which exhibit a magnificent order, an exceptional beauty and high efficiency, which are requisites to arrive at outstanding designs such as honeycombs built by bees and immersion bells made by aquatic spiders, among others. On the other hand, regarding some interesting animal locomotion patterns, the author has been fascinated by the run of the basilisk lizard on water surface without sinking itself; a spirally-shaped flight of the peregrine falcon before impacting to its prey as well as the whirlwind´s motion of a shoal when attacked by sea predators. Taking into account nature´s engineering genius, the author has designed a so-called “Uehara Minimum Kiosk”, using a combination of a steel-made single layer polyhedral truss and a catenoid-shaped membrane. The whole structure is supported by three slender columns that are anchored onto the ground by means of screw-type anchors. The shape of this spatial truss was based on the geometry of a truncated icosahedron, a polyhedron discovered by Archimedes of Syracuse (287-212 B.C.). It is known that a hexagon encloses the maximum surface utilizing the least amount of material, a mathematical keynote well understood by bees that are good at economizing wax. In order to minimize deformations under wind loads, the hexagonal arrays of this truss were stabilized by prestressed steel cables, just like the rigid bones in our body are held by flexible tendons and ligaments, giving an intimate example of a tensegrity structure. With regard to the enclosing space, I proposed a fabric-made catenoid, the only curved minimal surface of revolution discovered by Euler in 1744, whose tension characteristics resembles those of an immersion bell built by an aquatic spider, that weaves a silk-made shell and fills up its inner space with an air bubble that brings down from the water surface carrying it on its abdomen. In my long pursuit of arriving at a flexible structure having minimum weight an maximum rigidity, the author undertook a “Multicriteria Optimization” and he idealizes the motion of the structure under wind action as a “music of osculating conics” which lies on a scale that goes from melodious sounds or consonances, (elliptical paths) to semitones (parabolic paths) and dissonances (hyperbolic-paths). The author is challenging to project these harmonic oscillations on some black screens for different wind patterns as in the case of string musical instruments. A variety of these tunes can be displayed nicely and some of them correspond to Lissajous´s figures. On the other hand, the author is planning to undertake an interferometric analysis of deformations to study the effect of wind turbulence. The so-called “music of osculating conics” in our light- weight structure is also associated to the precessional rotation of our kiosk´s
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structure, which can be taken to safety limits by tensioning the strings up to their optimum tunes. Finally, the author designed three kiosks, a 5 meter diameter for a “green house”, a 10 meter one for a “dinner party hall” and a 20 meter one for a coffee shop of “Alfalfares Amusement Park”, actually under construction in the city of Queretaro, Mexico. With the purpose of predicting the expected self-weight and member diameters when scaling up geometrically similar structures, the author compared the above data with that obtained by applying Galileo Galilei´s (1564-1642) similitude principle, finding a good agreement. Keywords: efficiency of nature structures; music of osculating conics; interferometric analysis of deformations, precessional rotation of kiosk´s structure. 1. FOUR NATURE´S OUTSTANDING DESIGNS THAT INSPIRED ME TO DESIGN “UEHARA MINIMUM KIOSK” I am convinced that engineers and architects have a lot to learn about nature engineering. However, I will describe four achievements that for the “survival of the fittest” were undertaken by nature engineering genius, thus capturing my attention during the conceptual design of “Uehara Minimum Kiosk”. 1.1
Efficient geometry that economizes wax
We know that one of the most beautiful hexagonal arrays is the honeycomb constructed by bees. The French physicist R.A.F. de Réaumur (1683 – 1757) thought that the hexagonal structure of the bee´s honeycomb should follow from a minimum principle: the bee would build its cells with the greatest economy in order to use as little wax as possible. (Fig.1) The question whether the bees have hit upon the optimal honeycomb was investigated by the Hungarian mathematician Fejes Tóth in 1964, who found out a cell whose savings is less than 0.35% of the area of an opening. This prominent mind challenged the minimum principles of bee´s engineering (Fig.2).
Fig.1 Bee´s honeycomb cell
Fig.2 Tóth´s optimal cell
Analogy applied to Uehara Minimum Kiosk: As a main geometry for the dome skeleton frame, I proposed half of a truncated icosahedron, a polyhedron discovered by Archimedes of Syracuse (287-212 B.C.), which can be considered an efficient polyhedron as demonstrated below. The apex of our structure is shaped by a single pentagon surrounded by hexagonal arrays, a geometry that coincides with that of bees. However, because of the need of stabilizing our hexagonal arrays to minimize deformations due to external loads, the author triangulated the truncated icosahedron at the outer and inner faces by joining the extreme nodes of each hexagon with radial cables that are prestressed during structure´s erection stage and they are separated by some compression spreaders. Fig.3.
S V I V 1 3 for 23 different polyhedron in terms of a geometrical invariant expressed by the equation I S V , where S It is worthwhile mentioning that it is possible to define surface to volume ratios
is the measured surface area and V the measured volume of polyhedron. Since “I” is dimensionless, its value for a given type of polyhedron is independent of its size. A small value of “I” means a figure with the least area and the largest possible volume.
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Volume 1. 2. 3. 4. 5. 6.
“I”
Sphere Icosahedron Truncated icosahedron Truncated octahedron Regular octahedron Cube
4.83598 5.14836 5.00296 5.31474 5.71911 6.00000
Fig.3 General geometry and components of Uehara Minimum Kiosk 1.2 A clever diver that builds an underwater dual-function shell “Argyroneta” is a spider that lives underwater and constructs a silk bell-shaped dome, whose inner space is filled out with an air bubble which this clever diver brings down from water surface, carrying it on its abdomen. This beautiful bell may be considered a wrinkle-free surface, that follows the principle of constant surface stress as in the case of soap-films. This membrane is known in mathematical language as a stable minimal surface.
Fig.4 Aquatic spider´s immersion bell
Fig.5 Optimization of a bell to preserve high tonal quality. Prof. E. Ramm. Stuttgart University
Analogy applied to Uehara Minimum Kiosk: “Argyroneta” spider engineering genius inspired me to propose a catenoid-shaped membrane for the kiosk´s enclosure space, a geometry known as the only curved minimal surface of revolution discovered by Euler in 1744. This membrane is attached to the dome skeleton frame by means of an upper tension ring and a lower compression ring respectively. As we know, doubly curved surfaces have higher strength per unit of invested resources than planar systems built of an equivalent amount of the same material.
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Fig.6 Catenoid geometry
1.3 Aerodynamics of a lizard that runs above water surface “Basilisc” (Fig.7) is a light-weight lizard that runs over water surface at a mean speed of 12 km/hr without sinking, thanks to the aerodynamic characteristics of its body that reduces air resistance. As a matter of fact, biologists J. Glasheen and T. Mcmahon determined the ratio between this lizard´s mass (M) to the dynamic force (F) it has to develop, to walk on the water surface smoothly without sinking. According to some research, the optimized intermittent motion that corresponds to this lizard’s run 0,17 -0.17 it is characterized by a velocity proportional to M and a stride proportional to M . (Fig.8).
Fig.7 A basilisk during its feat
Fig.8 Void triangles correspond to runner lizards
Analogy applied to Uehara Minimum Kiosk: Although it may sound showy, this lizard blessed by a well-proportioned aerodynamic body was my source of inspiration to design the frame skeleton for Uehara Minimum Kiosk. The prestressing cables (tendons) do not only play an important role in resisting the tensile forces that may change drastically during a strong windstorm, but additionally serve to stabilize the hexagonal assemblies (bones) which are in danger of buckling. The following equation [19] was obtained by great professor Yasuhiko Hangai (passed away) to whom I met in Kuala Lumpur in 1997 , and professor Yoshikatsu Tsuboi (Structural Designer of Tokyo Olympics Swimming Pool Facility,1964) to calculate the member buckling loads of reticulated single-layer space frames.
1.4 Two spiral-shaped spectacles that symbolizes whirlwinds Peregrine falcons are the fastest birds on earth colliding towards their preys at speeds of up to three hundred kilometers per hour. Results of a wind-tunnel experimental research conducted by biologist Vans A. Tucker of Duke University in North Carolina, 2000, show that falcons keep their head straight and taking advantage of their razor-sharp vision follow a logarithmic spiral (Fig. 9) that maximizes speed. Another worthwhile sight spectacle is the “spiral dance” of a shoal when attacked by sea predators (Fig.10), and a spider web (Fig.11).
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Fig.9 Peregrine path that maximizes speed
Fig.10 A beautiful shoal´s spectacle
Fig.11 A spider web
Analogy applied to Uehara Minimum Kiosk Since we are dealing with a light-weight structure, its behavior should be investigated with the aid of reduced-scale models that are exposed to turbulent flow when blown by several fans. This turbulence effect produces drag forces that are not only random with respect to time “t” but are random with respect to the height of the structure “H”. Benoit Mandelbrot [10,146-147] wonders up to what extend the no-intermittent wind turbulence developed in wind tunnels can be representative of the same physical phenomenon for “intermittent natural turbulence”, that occurs in the atmosphere.
2 2 2 2 ytg x a x y x ytg x a y 6 6 y 2 2 2 2 xa tg 6 y x y y xtg 6 x a y
Fig.12 Turbulence geometry patterns in clouds (left) and water flow (right) look similar. The above equation may explain the transitional whirls linked by a cascade as proposed by Lewis Fry Richardson (1926).
ENVIRONMENTAL ASPECTS: With regard to environmental aspects considered in the design of our kiosks, the uppermost part of them is furnished with a circular opening whose function is to remove excess of heat through a stack effect, thus maintaining an inner ventilated space comfortable to people. Fig.14. SUN
LOW PRESSURE CONE
DIRECT SUNLIGHT
M AR W R AI CO MN LU
D AR T PW IF U R F D O AL E M S R BA HE T
REFLECTED SUNLIGHT
COLD AIR
Fig. 14 Deformed configuration of our structure under a strong wind and the thermal stack effect that provides a comfortable inner ventilation
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2.- Why japanese technology of pagodas became another source of inspiration to develop my kiosk ? During my nearly 10 year-stay in Japan, I became particularly interested in the morphology of Japanese old buildings like the 34 m height five-storied Pagoda of Horyuji in Nara which was built in the seventh century. I admire the courage and genius of Japanese ancient builders who built all those deep canopies by cantilevered beams resting on the top of columns that are counterbalanced by the weight of the upper stories. Theses monuments to flexibility have survived many centuries of big storms and earthquakes. It is not astonishing that those ancient builders avoided overturning of large canopies by introducing a kind of three-dimensional truss action at the top roof of these buildings ?. How they could induce an effect that after a jolt the building will sway sinuously like a snake ?. The secret may be hidden in the fact that each story was designed to rock independently around a central anchoring pillar. Worldwide structural engineers became interested in this construction system and they hope to apply it to the construction of modern buildings. Structural engineer Toshihiko Kimura [11] carried out a dynamic analysis for these kind of buildings, arriving at the following interesting conclusions: 1. The natural period of a pagoda is very large (between 1 and 1.5 seconds) as compared to other traditional structures, and in general it is much larger than the subsoil period. 2. Pagodas have enough strength to withstand considerable lateral forces. 3. Pagodas can undergo large deformations before failure occurs. 4. Pagodas posses a big structural damping.
Fig.15 Deformed shape of a japanese pagoda under earthquake excitation
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3. Structural concept of R. Buckminster Fuller´s geodesic structures Fuller played with spheres in the evenings trying to find out the most compact packing of spheres around a central one. He has shown that the number of spheres on each successive layer is given by 2 10n + 2, where n is the number of the layer. Besides, Fuller has pointed out that if the center sphere of a 13-sphere cuboctahedral array is allowed to “shrink” in diameter, the twelve outer spheres shift into a new contracted position to form the twelve vertices of the icosahedron (Fig.16). This constitutes the most symmetrical distribution of spheres packed around a common center point. Fuller invented and, in 1954 patented the geodesic dome as it is known today. He knew the structure of a radiolarian. Fuller combined the advantage of the sphere (resistance to distributed loads) with that of the tetrahedral/octahedral space frame (resistance to local loads) by wrapping the double layer space frame around the sphere. With these principles in mind, Fuller designed the United States Pavilion, Expo 67, Montreal.
Fig.16 Fuller´s studies of sphere-packing show how he arrived at the tetrahedral, the simplest undeformable body we can build with the least number of bars. 2. Structural concept of Uehara Minimum Kiosk As far as my structural concept is concerned, astonishingly, a Soddy´s hexlet fits into the hexagonal arrays of our kiosk structure, by letting the center of each sphere coincides with the vertex of the hexagons. Professor Frederick Soddy (1877-1956) of Oxford, winner of the 1921 Nobel Prize for his work on radioactive substances and isotopes, discovered this configuration (necklace of spheres) in 1936, and it was derived independently in Japan over a century before Soddy, by Japanese samurai-class mathematicians as shown by Japanese temple geometry tablets from 1822 in the Kanagawa prefecture of Japan. This beautiful and amazing result states that the radii ri of this collar of six spheres are related by:
1 1 1 1 1 1 , regardless of the position of the first sphere. r1 r4 r2 r5 r3 r6 Soddy´s bowl of integers contains an infinite number of nested hexlets. According to this theorem which can be proved with the aid of geometric inversion, the center of the variable sphere is a hyperbola with foci at the centers of two fixed spheres. It follows that if we now remove the restriction that the center of the variable sphere must lie in a plane, its locus become a hyperboloid of revolution with the same foci as before. Thus, regardless of the third fixed sphere, all spheres of the hexlet must have their centers on this hyperboloid. In addition, they lie in a plane. But the plane section of a hyperboloid of revolution is a conic. Therefore, in Soddy´s hexlet this conic is an ellipse, in the limiting case it is a parabola, and in the curvature reversing case it is a hyperbola, which means dissonance.
In our kiosk structure, these conics will osculate during wind action, evoking what the author has called a “music of osculating conics”. According to Kepler´s research there exist a mystic connection between the physical original experience of the melodious sound and certain numerical relationships. With regard to our structure´s motion we should taut their strings to induce melodious sounds or consonances, namely the octave 1:2, the fifth 2:3, the fourth 3:4, the big third 4:5, the little third 5:6, the big sixth 3:5 and the little sixth 5:8. These harmonic ratios can be seen by an harmonic graph in the
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form of Lissajous´s figures (Fig.22, right).Jules Lissajous, a french mathematician discovered in the middle of century XIX that two strings of equal tautness and quality produces a melodious sound, if their lengths were kept proportional to certain numerical relationships. My japanese ancestors had surely this fact in mind during the earthquake and strong wind resisting design of their pagodas that move beautifully like eels.
Fig.17 Soddy´s necklace of spheres that fits into hexagonal arrays of “Uehara Minumum Kiosk”.
Fig.18 “Music of osculating conics”
Fig.19 A catenoid fulfils its function as an enclosure space (left) and helical anchors (rights) are responsible for a safe foundation. In nature, bat´s wings (center) and gecko´s feet (right) are examples of transitional shapes that fulfil a variety of functions. 3. The meaning of Harmonic Motion When we say harmony we think first of all about music. If we induce a small lateral displacement to a slender vertical wire of uniform circular cross section built in at its lower end A and carrying at its upper end a luminous ball of weight W, inside a dark room, we are able to watch very nicely an outline of an elliptical path as shown in Fig.20 (left). Moreover, if the cross section of the wire is very nearly square so that the two spring constants k and k1 corresponding to the two principal planes of bending of the wire are almost, but not quite, equal, we obtain a very interesting motion of the luminous ball as depicted in Fig. 20 (right). This device is called Wheatstone´s kaleidophon and it was invented by Charles Wheatstone in 1827.
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Fig. 20
4. The work of Johannes Kepler (1571-1630). Kepler formulated a model in which polyhedra envelop other polyhedra. That model is Kepler´s planetary system (Fig.21).
Fig.21 5. Pythagoras stated that Nature adjusts itself to laws expressed in mathematical form. It is attributed to Pythagoras that a pleasant sensation for musical harmony it is produced when the proportion of frequencies can be expressed with simple numbers. Pythagoras passed in front of a blacksmith´s and stopped himself. The different sounds coming from the strike of hammers at the smith´s anvil sounded him like common harmonies. Pythagoras came in to prove whether the weight of those hammers was responsible for the sounds of different notes. He was impressed about the close relation that exists between music and mathematics (Fig.22). Lissajous´s figures (right).
Fig.22
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5. Interferometric analysis of deformations An experimental approach actually investigated by the author is to read and interpret the “nodal motions” for critical structural joints through an interferometric analysis of deformations, whose “geometrical tendencies” may be compared with those given by Lissajous´s figures. (Fig. 22, right). Discord signals are synonymous with structural instabilities. The technique of terrestrial interferometers was invented by the American physician Albert Michelson (1852-1931), who was awarded the Physics Nobel Prize in 1907. Is not the purpose of this paper to deepen into this subject. Rather, by looking at the pictures below we intend to transmit the following abbreviated message: With the aid of new light technologies we can superpose two light waves to produce a third one. By passing through retarding lines, those waves are received by a computer and then processed till creating an image with a high resolution. With this technique it is possible to carry out an analysis of deformations.
Fig.22
Fig.23 6. Precessional rotation concept of Uehara Minimum Kiosk
Since our kiosks posses a set of radial cables that contribute to stabilize its hexagonal arrays, these cables are furnished with an optimum initial tension. We recommend to verify their prestressing force once a year with the aid of a transducer devise, in order to assure a good performance during strong wind storms. Since this structure undergoes large deformations, the author proposes a “critical precession angle Ө” (Fig. 24, left), as the maximum allowable angle that our structure may withstand without developing a collapse mechanism. Fig.24 Trace of an “Herpolody” on the floor by the vertex of an “imaginary cone” from which it is possible to have an insight about the “precessional rotation” of our structure.
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7. Similitude principle by Galileo Galilei The square-cube law is a principle, drawn from the mathematics of proportion, that is applied in engineering and biomechanics. It was first demonstrated in 1638 in Galileo´s Two New Sciences. It states: When an object undergoes a proportional increase in size, its new volume is proportional to the cube of the multiplier and its new surface area is proportional to the square of the multiplier. 3
l v2 v1 2 l1 Where v1 is the original volume, v2 is the new volume, l1 is the original length and l2 is the new length. 2
l A2 A1 2 l1 Where A1 is the original surface area and A2 is the new surface area.
Fig.25. Galileo noticed that the weight a bone carries is proportional to the animal volume (length3) whereas the strength of this same bone is proportional to its cross section area (lenght²). Although this principle may be applicable to our kiosks of 5 m, 10 m and 20 m diameter, the refined formulas proposed by Steven M. Stanley showed a good agreement with the element diameters for our various structures. Similarity Formulas Characteristic Proportions Bone diameters
General Formula
1 d x
dM
1 x 2
Geometric
X 1 dM
Elastic
X 2 13
dM
3 38
Constant Stress
X1
2
d M25
Fig.26. Steven M. Stanley refined Galileo´s statement and proposed the above relations. According to the principle of elastic similarity, bones “should be scaled in such a way as to deform under gravity in a geometrically similar fashion”. In other words, the ratio of bone-deformation to bone-length will remain constant with increasing size.
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8. Future challenges Over the past nine years the author has designed some domes and pedestrian bridges in Mexico, taking advantage of tension elements like steel cables and rods. For these structures, we usually limit the maximum vertical deflection to a maximum allowable value specified in building codes. I dream with future flexible structures, where we can read from their harmonic graphs displayed on black screens at the inner space, the structural behavior under wind action. I have called this light spectacle Fig.27(bottom-left) the “Music of Osculating Conics” inspired in Kepler´s World Harmony, where he and the author praise a lot the great work of God the Creator. We may soon reach the asymptote given by Robert Le Ricolais´s joke: “zero weight, infinite span”.
TIMPANO FORMADO POR CABLES Ø 3 16 " EN ABANICO
Boulevares dome
δmax= 4.10 cm < L/360 L(span) = 20.0 m
º “La Negreta” pedestrian bridge δmax= 4.30 cm < L/500 L(span) = 23.0 m
“Alfalfares Coffee House” currently under construction 1 0 A1 , A2 , 0 1
structure & roof weight = 15 kg/m² span= 20.0 meters
Fig.27 Structures designed by the author that take advantage of a synergy between tension and compression.
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10. References [1] Humberto Uehara. “Nature Physical Principles and Structural Models as Education Tools for the Conceptual Design of Structures”. First Specialty Conference on the Conceptual Approach to Structural Design. Singapore 2001, (165-173). [2] Kunio Watanabe, Humberto Uehara, “ Genealogy of Three-Dimensional Geometry and Structural Mechanics”, New Architecture in Japan. Magazine (August 1998), 236-241. [3] Humberto Uehara. “Minimum Structures Inspired in Nature Geometry”. 2005, Memories of the National Congress organized by the Institution of Civil Engineers, Queretaro, Mexico. [4] Max Caspar. “Biography of Johannes Kepler (1571-1630)“. 1993, Dover Publications, Inc. New York. [5] S.Timoshenko and D.H. Young. “Engineering Mechanics Dynamics”. 1937, McGraw-Hill Book Company, Inc. New York and London. [6] C. Stanley Ogilvy. “Excursions in Geometry”. 1969, Oxford University Press, New York. [7] Jorge Flores Valdés/Gabriel Anaya Duarte. “Dinámica del Cuerpo Rígido”, 1989, Fondo de Cultura Económica, S.A. de C.V. México. [8] Antonio Fernández Rañada. “Dinámica Clásica”. 2005, Fondo de Cultura Económica, México. [9] R. Buckmister Fuller. “El mago de la cúpula”. 1970, Editorial Diana, México. [10] Benoit Mandelbrot. “La Geometría Fractal de la Naturaleza”. 2003, Metatemas. [11] Architecture of the year 1995, Japan. 42-45. (in Japanese). [12] Tony Rothman/Hidetoshi Fukagawa. “Geometría en los templos del Japón” en el período (16941854). Scientific American, July 1998, 73-79. [13] Israel Katzman. “Cultura, diseño y arquitectura”, Tomo I, 1999, CONACULTA, México. [14] Michel Mayor/Pierre-Yves Frei. “Los Nuevos Mundos del Cosmos, en busca de Exoplanetas”, 2006, Ediciones Akal, S.A., España. [15] John Martineau. “El Libro de las Coincidencias. La Misteriosa Armonía de los Planetas”. 2005, Ediciones Oniro S.A. Printed in Spain. [16] Anthony Ashton. “El Armonógrafo. Las Matemáticas de la Música”. 2005, Ediciones Oniro S.A. Printed in Spain. [17] Peter Pearce. “Structure in Nature is a Strategy for Design”, 1978, MIT Press. [18] W.J.Lewis/T.S. Lewis. “Application of Formian and Dynamic Relaxation to the Form-Finding of Minimal Surfaces”. Department of Civil Engineering, University of Warwick. Journal of the International Association for shell and Spatial Structures: IASS. [19] Masao Saitoh, Yasuhiko Hangai. “Buckling Loads of Reticulated Single-Layer Domes”. Shells, Membranes and Space Frames, Proceedings IASS Symposium, Osaka, 1986, Vol.3.
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