The cúpulas of double curvature

The structures laminares, or shells, are thin surfaces, curves, of little thickness, compared with the global dimensions of the structure. The surfaces distinguish by his generation in surfaces of revolution, of traslación and ruled. His alabeo is variation of curvature. His pandeo is the flexión of a straight element. In the plate the efforts of traction and compressesion are maximum and the ones of invalid cut. The feeblest zones will be the ones of lower curvature and the ones of contact with the elements of anchorage, where the tensions are greater. The membrane is a leaf of material so thin, that can not develop compressesions, that is to say, does not have rigidity. We tend it between elements compressesed, topmasts and rings.

The biological designs of the Nature with his shells and shells show us how attain envolventes of suitable form for covers with little weight. They resist by his form the loads of own weight and the external loads by means of normal efforts of compressesion and/or traction, and tangential. They are membranes tended between points of anchorage. His efficiency is due to his structure and to the alabeo of the planes. Membranes of double curvature, subjected to tension, without any compressesion, can resist perpendicular loads to his surface. It remembers us the cloth of an umbrella. The textile architecture employs material tightened, fabrics, plates, the correct term is Tensile architecture, in English.

In the antiquity the cúpulas built of stone or mampostería, happening afterwards by the brick, and even, the wood. The Romans used them a lot, the greater is the one of the Panteón of Rome (years 120-124), with 44 m of diameter, done of concrete in mass. The employment of cloths in constructions is very ancient: the travelling circus with linen, or canvas of hemp, with diameters of until 50 m. This coverage fulfils always two conditions: double curvature and is pretensada. The ancient covers were of membrane tightened, no pretensada. It is interesting to see the evolution from the ancient conical shop, supported in a vertical support, until the covers of membranes pretensadas, that demand double curvature, very little deformables in front of the action of the loads.

In the 20th century abandon the cúpulas traditional, and with armed concrete cover spaces of big lights, with thin surfaces of small thickness. The 1ª thin shell is the cúpula of the Zeiss Planetarium (Jena 1925) of 6 cm of thickness, 25 m of diameter and 12,5 m of height, work of Walter Bauersfeld. The armour is formed by a mesh triangulada of steel, on which pours the concrete. Today day the majority of the stadiums, terminals of airport and big shopping centres are covered by membranes tightened, with upper surfaces to the 100.000 m².

Require minimum elements of support of rigid structure, and allow a level of acceptable diurnal natural sunlight. The membrane is able to bear the loads that require in the Technical Code of Edificación. In cold countries the reinforcement of the membrane allows to face the incidence of a snowfall. The covers tightened employ topmasts, tensors and cables, to tighten the plate by his extremes, in steerings and opposite senses, even out of plane. The advantages on the conventional ancient cover are a minimum weight. His coefficient of transmission of light allows us dispense of the glass, with his problems of weight and rigidity. The commonly used membranes transmit the 13-16% of the solar light incident.

The membrane receives in roll, that it is necessary to cut according to the design. The patterns cut have to join to form the cover. There are unions sewed, soldered by high frequency, ultrasound, hit, etc. Afterwards is necessary to join the membrane to the topmasts perimetrales, to transfer the normal efforts, or tangential, of the membrane to the system of edge. They exist flexible edges curvados and rigid edges. Finally they are the intermediate topmasts or perimetrales (pivotantes, or articulated), necessary to attain the suitable tension.

The EFTE has transformed the architecture of the 3 last decades. EFTE Is the acronym of Etileno Tetra Fluoro Etileno. The resin is the copolymer of this molecule, a transparent plastic of extraordinary durability. It possesses a high chemical resistance and mechanics to the cut and to the abrasion, and bears until 170 °C. The resin is procesable by extrusion and moldeo by injection. His more remarkable quality is his high resistance to the ultraviolet rays; thus it does not be yellowish during his exhibition to the solar light. It is the alternative to the glass in the construction. The EFTE weighs 100 times less than the glass. In Spain have EFTE in the cúpula of the Millenium, in the square of the Millenium, Valladolid.

Is simple (curvature desarrollable) when the curvature in a point given is of the same sign in all the steerings, except in one of them: the straight generatriz, that costs 0. For example: cylinders and cones. The values of pretensado are very inferior to the maximum resistance of the membrane, but are sufficient to compensate the losses by fatigue of the material along the time. The solution in front of the loads are the triangular networks, by his capacity to resist any system of strengths of traction and compressesion.

There are two classes of double curvature, no desarrollable:

(To) Sinclásticas: The curvature in a point given is of the same sign in all the steerings, the centre of curvature is in the same side of the membrane: examples, the sphere, the elliptical paraboloid, etc.

(B) Anticlásticas: Two centres of curvature to both sides of the membrane. The curvature in a point is positive in some steerings, and negative in others: the paraboloid hjiperbólico, the chair to mount. Geometrically the surfaces anticlásticas result to move a curve on another, of reverse curvature (the paraboloid), or by the rotation of a curve around an axis (the catenoide). The double curvature contributes a double action of arch.

Between the surfaces anticlásticas interest particularly those that are ruled, this is: they can obtain by the trip or rotation of a straight on another family of curves, resulting a surface alabeada. As his characteristic is the existence of two opposite curvatures, will develop always a double action of arch and wire, in addition to the action of cut resulted of the alabeo. The double curvature is more efficient that the simple curvature. So that a surface do not develop efforts of flexión, the necessary condition is that it can not deform, and this only occurs, if the curvature is double. But it is sensitive to the pandeo, consequence of his minimum thickness.

The ruled surfaces more employees are the conoide, the helicoid, the hyperboloid and the hyperbolic paraboloid. The surfaces of the techumbre can generate by rotation or traslación. In the first case the curve turns around an axis called of rotation, whereas in the second moves simultaneously to himself same. They are surfaces of revolution: the sphere, the paraboloid, the hyperboloid and the cylinder.

If the surface can describe by the movement of a straight line, calls ruled surface, like the cylinder and the cone.

An ideal membrane is a leaf of material, so thin in comparison with his lateral dimensions that only can develop tractions that will be two-dimensional along his surface. When being of small thickness the rigidity to flexión and the sharp is despicable. Equally the resistance to compressesion, since of the contrary, with a small compressesion pandearía.

The stability of the membranes is due to his geometry and to the tensions that develops under the load. The theory of the membrane takes like base for the design of surfaces curves. But it does not reflect necessarily the true distribution of tensions, since they assume conditions estáticamente determinate. This approximation is valid to understand the structure, since the real analysis is complex.

The real surfaces are the sufficiently thin as to consider despicable the tensions of flexión, but the sufficiently thick as for no pandear low small tensions of compressesion. The superficial loads bear no only through superficial tractions, but also by means of superficial compressesions.

The most ideal material is the armed concrete or pretensado, but also uses the wood, the metals, the plastics reinforced. In general the tensions of membrane are used to to be small, by what many times the thickness of the structures laminares comes determined by the moments induced by the elements of support. In spite of this, the strengths of membrane have to evaluate by several reasons:

• Determine where will develop tensions to have the armour, or suitable reinforcement, if the material works fundamentally to compressesion.
• Determine which is the tension of greater compressesion, with the end to check the pandeo.
• Determine the travel of the extremes and the tensions of flexión that produce when limiting total or partially these travel.

The membranes are not adapted to resist punctual loads, by what have to avoid whenever it was possible, as it can produce a collapse local.

Are surfaces of clear-cut revolution by the rotation of a flat curve around a vertical axis. They allow to cover big lights, even a diameter of 100 m, and shut the greater quantity of space with the minimum surface. Thus they result very apt to cover rail stations, fields of sport, swimming pools, palaces of congresses, rooms of exhibitions, etc. in where it needs that the inner spaces are diáfanos.

If we think in a sphere subjected to inner pressesure, the two families of main steerings are the parallel and meridian working to tension. If the pressesure is external, is very intuitive to see that the meridians work to compressesion, like arches, and the parallel tie them, so that those no pandean. A very recessed socket work all he compressesed.

To do an idea of the level of mechanical efficiency of the cúpula, suffice to remember that they are sufficient thicknesses in the order of 1/400, twenty lower times that the ones of the beams of equivalent light. As all cascarón the cúpula will have a limitation by pandeo, that will require to increase his thickness. The triangulation of the sphere applied first to the recuadros formed by the network of meridians and parallel, 19th century.

Another solution for the sphere indeformable consisted in the cúpula geodesic, that results to project an icosahedron on the spherical surface. The application of the tightened is typical in the ring of edge, to counter the efforts of traction, the parallel of support.

The majority of the cúpulas current are almost hemispherical, to avoid push them lateral, since when they are more recessed need rings of traction to resist them. The cúpula spherical resists the load with a system of internal strengths situated in his surface. Usually it is used to to have a main strength of compressesion along the meridians, vertically, and a horizontal strength minor, generally of traction, acting around her, in the parallel.

So that the theory of the membrane can apply to the cúpula has to be creame uniformly along his edges. On the other hand the trip of the cúpula has to be compatible with the trip of the structure sustentante, fact that does not be used to to occur in the reality. This incompatibility produces flexiones, although they are used to to remain limited to the next zones to the edges.

The usual thickness of the cúpulas of miscellaneous armed concrete of 7,5 cm to 11,5 cm for lights of 30 to 60 m, with an increase of thickness of a 50 to 75% in the periphery. If the lights are greater will increase his thickness, rigidizará with nerves, or, will form a double layer, to avoid the pandeo to compressesion.

In Spain an example of this type of structure is the cúpula spherical of the Market of Algeciras (1934-1935, of Manuel Sánchez Arks and Eduardo Torroja) with a diameter of 47,8 m and 10 cm of thickness. The cúpula covers a space octogonal, of 18,2 m sideways, and rests on 8 pillars, in each one of the corners. The tension of horizontal traction absorbs by means of a ring octogonal, that joins the upper extremes of the supports.

Is a ruled surface of double negative total curvature, as the centres of the radii of curvature of each game of curves are situated in opposite sides of the surface. That is to say, can generate by two families of parabolas of opposite curvature, or by two families of straight to 45º with the planes of the parabolas. It is a surface without alabeo. The membrane, if has 4 supports, are 4 supports that do not find in the same plane. The double curvature increases the rigidity of the shell. The straight elements of edge result traccionados or compressesed and free of flexiones.

The chair to mount is generated by two families of generatrices straight, that allows to be hormigonada on a simple formwork realizar to base of wooden tables, that follow his generatrices straight. In the dimension 2 find the conical curves: elipse, parabola and hyperbole, and in the dimension 3 have the quadric surfaces. These surfaces have to see with the curves that appear like sections with planes: in the hyperbolic paraboloid these sections are parabolas and hyperbolas.

Gaudí was one of which employed it, but the one who more has worked it has been Félix Candela. His interest was motivated by the fact that the hyperbolic paraboloid, even being a surface curvada, can build with straight slats. The only that has to do is to go varying the angle of inclination of a straight that moves on of another curve, is the ruled surface. Gaudí gave the precise instructions to the capataz and to his workers, when these built a hyperbolic paraboloid in the ceiling of the Holy Family, of Barcelona, initiated in the year 1883.

The process of construction of the paraboloid follows these steps: Dice 4 points in the space, that are not in a same plane, there is an only hyperbolic paraboloid that happens precisely by these 4 points. This is the same property that says that two points determine an only straight. So much Gaudí like Candela took advantage of the mathematically clear-cut surfaces, with some perfectly determinate equations and a way to build them totally established. This indicates the lack of freedom of these architects in the design of the form wished in his maquetas. Only they could use a family of surfaces, depending on some parameters.

The behaviour of membrane with axial and tangential strengths constants along the surface bases in an ideal situation under uniform load, and without additional tensions been due to the supports, situation that does not coincide exactly with the reality.

The fact that the load was not uniform, that was not articulated in the outline, that was not despicable the rigidity of the elements of edge in the plane of the surface, drives to the existence of efforts of flexión and sharp efforts, additional to the next zones to the edges. It can consider adapted the approximation to the membrane, if the quotient light/height is in the order of 9, and the slope of the beams of edge, for rectangular pieces, miscellaneous of 1/5 to 1/3.

Eduardo Torroja in 1935 built the hippodrome of the Zarzuela, Madrid, with a consistent roof in a battery of semi-detached cylindrical plates in a big voladizo of very pose thickness. But the visual and architectural wealth of the hyperbolic paraboloid has been exploded by the Mexican Félix Candela, after the second world-wide war. Some of his works are the pavilion of the Cosmic Rays (Autonomous University of Mexico, 1951) formed by two hyperbolic paraboloids parallel, of 1,5 cm of thickness. Another work of Félix Candela is the underwater Restaurant of the Oceanografic, of Valencia (2002).

The fabrics and the networks of cables are the two more developed materials, to work in traction and admit double curvature. The metallic wire is the one who has more provision, given his resistance of break and high elastic limit, little deformabilidad. It can incorporate protections like the galvanised, or enfundarse with plastic materials. The fundamental element of the rope is the anchorage, that can be a simple thread in the wire macizo. Also they are possible anchorages curvados in macizos of concrete. All they are weak points.

The textile membranes are of materials composed in which the armour can be of fibre of glass, or polyester, and the coating of PVC, teflon or silicone. The fibre of glass recubierta of teflon is much more permanent. With fibre of glass-teflon reach transparencies of 16%; with silicone of 80%. It is necessary to remember that they are material steeringal, because of the orientation and crosslinking of fibres.

Southern Cross Station, Melbourne, Australia.

Has been designed by Grimshaw Jackson Architects, and built in three years: 2004-2007, with a surface of 60.000 m², surface covered 36.700 m². The customer is Spencer Street Station Authority, Victorian Government. Cost 439 million euros. With cúpulas of double curvature.

King's Cross Station, LONDRES.UK.

Designed by John McAslan + Partners Architects. Customer Network Rail. Delivered the Work in 2013. Surface 8.000 m². Cost 600 million euros. Cover of double curvature.

Queen Alia International Airport. Amman, Jordan.

Designed by Foster + Partners Architects. Customer The Hashemite Kingdom of Jordan Ministry of Transport. Airport International Group. Beginning of the works in 2007. Finished the first Terminal in 2013, continue the Works in the 2º Terminal, will finalise in 2014. Surface 116.000 m². 99 cúpulas of spherical section on square plant.

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