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STRUCTURES OF WARPED SURFACES

COMBINATIONS 01“ UNITS OF HYPERBOLIC PARABOLOIDS

THE STUDENT PUBLICATION OF THESCHOOL OF DESIGN, RALEIGH, N. C.

VOLUME 10, NUMBER 1

Copyright MCMLX, The Student Publication of the School of Design

INTRODUCTIONThe Hyperbolic Paraboloid, a double-curved surface gen-

erated by the displacement of a straight line, and commonlydescribed as a saddle shape, has become the unit—theme ofmany structures built all over the world during the lastdecade. The first known structural development based uponsuch units was introduced in France by Bernard Laf’faille,who in 1933, built at Dreux the two-sided cantilevered struc—ture shown in Figure 1. As the result of such experimentalwork, he published two years later the “Memoirs sm- L’EtudeGenerale des Surfaces Gauches” in the journals of the SecondCongress of the International Association of Bridges andStructural Engineering.

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Figure 1In 1936, the French engineer F. Aimond published the most

complete study ever made on the subject, in the fourthvolume of the above mentioned journal, called “Treatise onthe Statics of Hyperbolic Paraboloid Shells not Stifi in Bend-ing.” This study covers a structural analysis of these warpedsurfaces, as well as both simple and elaborate geometricalcombinations of Hyperbolic Paraboloid units to enclosevaried spaces. During the same year, L. Issenmann Pilarski,in his book Calcule des Voiles Minces en Beton Arme, pub-lished by Dunod in Paris, France, included part of the studiesmade by Lafi'aille and Aimond, thus completing the originalbibliography on the subject.Although the Hyperbolic Paraboloid had been well known

as a geometric surface, it was not used until 1933 as a struc—ture. Only Antonio Gaudi, the Spanish architect, saw thearchitectural and structural possibilities of such surfaces.before Laffaille and Aimond. In the basement of La Sagrada

Familia, Gaudi’s unfinished church in Barcelona, Spain, thereare two plaster models of structures formed by three rhom-boidal units of Hyperbolic Paraboloids, combined in a hexa—gonal plan. They are advanced for the period in which theywere conceived, and constitute perhaps the best examplesof Gaudi’s structural ideas.

After the first structures were built by Lafl’aille, theItalian engineer Giorgio Baroni built several industrial roofsusing units of Hyperbolic Paraboloids for the Alfa Romeofactory in Italy. The impact of these constructions was re-duced, unfortunately, due to the outbreak of the SecondWorld War, which completely paralyzed all civil construc-tion in Europe.The rebirth of the Hyperbolic Paraboloid came in 1950,

when it was used as a saddle-shape in the Cosmic Ray Pavi—lion of the University City of Mexico. This constructionconstitutes the beginning of uninterrupted structural devel-opments by the Spanish architect, Felix Candela. With thepersistence present in the best builders’ tradition, and withhis constant exploration of the structural and visual richnessof the Hyperbolic Paraboloid, he has made a lasting contri—bution to the art of construction.

The material published here is based upon studies madeby the author since 1952 as a part of his courses in archi-tectural design at the School of Design in Raleigh, NorthCarolina, and later at the School of Architecture and Plan-ning at the Massachusetts Institute of Technology in Cam-bridge. Some of these studies were developed by students intheir attempt to understand the geometric, structural, andarchitectural characteristics that result from the combinationof Hyperbolic Paraboloid units, through the construction ofmodels in varied materials, techniques, and scales.The accompanying plates attempt to convey to the reader

how combinations of these nonplanar, four—sided surfaces ofgreat structural efficiency can create almost endless archi—tectural spatial relationships.We hope that those who may be interested in these forms

for architectural use do not blindly translate them into build-ings. Here, they purposely have been represented solely asthree dimensional organizations of the four-sided units, inde-pendent of the lengths of their sides; of the angles formed bytheir sides; of their curvatures; materials and surface treat-ment; and independent of their scale in relation to anyelement of reference or to any given environment. Misinter-pretation of these variables undoubtedly will transform apotentially valid visual event into an actual visual offense.

UNITS OF HYPERBOLIC PARABOLOIDS—THEIRPOSITION IN SPACE

The structures described in the accompanying plates arebased on the combination of four-sided warped surfaces,which are portions of the boundless surface of double curva-ture, called the Hyperbolic Paraboloid. This surface eithercan be generated by the translation of a given parabola,parallel to a xz- plane, along any given parabola containedin a yz- plane, or by the displacement of a straight line

Figure 2: First Position

called a generatrix. Such a generatrix is displaced parallelto a plane director, along two nonplanar straight lines, calleddirectrices. Thus, the Hyperbolic Paraboloid is a ruled sur-face of compound curvature. This curvature usually isdescribed as a negative curvature, because the focus of eachcurve is placed at different sides of the double-curved surface.The small portions of the Hyperbolic Paraboloid previously

mentioned can have any position in space, depending uponthe position in space of the particular Hyperbolic Paraboloidto which they belong. There are three typical positions which

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a Hyperbolic Paraboloid may have, related to orthogonalplanes of reference.

In an attempt to describe these typical positions, each ofthe four accompanying figures presents the vertical, hori-zontal, and third projections of a Hyperbolic Paraboloid con-taining a given unit. For clarity, in the figures the boundlesssurface of the Hyperbolic Paraboloid has been limited by acircular dotted line. In each figure, the vertical projection atupper left shows the front view of a Hyperbolic Paraboloid

Figure 3——Second Position

with a unit limited by four straight line generatm’ces. Thehorizontal projection, at lower left, shows only the plan ofthe unit with its two sets of generatm'ces. The third projec-tion, at the right, shows the lateral View of the HyperbolicParaboloid containing the same unit.FIRST POSITION: Figure 2. The Hyperbolic Paraboloid

has its Z axis parallel to the planes xz and yz. Under thiscondition each set of generatrices is horizontally projected asparallel lines.

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SECOND POSITION: Figure 3. The Hyperbolic Paraboloidhas its Z axis parallel only to one plane, either xz or yz.Under this condition each set of generatrices is horizontallyprojected as non-parallel lines.THIRD POSITION: Figures 4 and 5. The Hyperbolic Para-

boloid has its Z axis non-parallel to either plane xz or yz.Variation A: Figure 4. When the Hyperbolic Paraboloid isinclined along a plane parallel to one of either set of genera-tm'ces (Plane Director), only such a set of generatrices ishorizontally projected as parallel lines. Variation B: Figure

Figure 4—Third Position—A.

5. When the Hyperbolic Paraboloid is inclined in any posi—tion other than the examples previously described, both setsof generatm'ces are horizontally projected as non-parallellines.

Generalizing then, it can be said that the degree of paral—lelism shown by the horizontal projection of the generatm’cesindicates the relative position in space of the mentioned units.The closer the parallelism among the generatm'ces of each set,the more vertical is the Z axis of the Hyperbolic Paraboloidthat contains the axis.

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This degree of verticality of the Z axis has paramountimportance in the determination of the values of the internalstresses developed within each unit. Structurally speaking,a Hyperbolic Paraboloid performs more efficiently with itsZ axis parallel to the forces impinging on it, which arefundamentally vertical ones.

The reader, by observing most of the plans of the followingplates, can easily determine, through the varied degrees ofparallelism among the generatrices of each set, how inclinedthe Hyperbolic Paraboloid to which the unit belongs, is.

Figure 5—Third Position—B.

Through awareness of such inclination, we can determinehow close the unit is, to the ideal position in space, describedin the First Position.

Structures designed by combining units belonging to Hyper-bolic Paraboloids with Second or Third Positions in spacehave not been frequently used, perhaps because a more com—plex stress analysis than the one required for the FirstPosition is involved; and also because Hyperbolic Paraboloidshave not yet been studied long enough to discover all theirstructural, three-dimensional, and architectural possibilities.

The Author ofSTRUCTURES OF WARPED SURFACESEDUARDO F. CATALANO was born in Buenos Aires,

Argentina in 1917. He was graduated in architecture fromthe University of Buenos Aires and then received Master ofArchitecture degrees from the University of Pennsylvania in1944 and from Harvard University in 1945.He has been in private practice in Argentina and the United

States since 1941. During 1950—51 he taught at the Archi-tectural Association School in London. In 1951 he acceptedan appointment in the School of Design at North CarolinaState College where he subsequently served as professor andhead of the Department of Architecture. In September 1956he left North Carolina to become Professor of Architecture inthe School of Architecture and Planning at the MassachusettsInstitute of Technology.

ACKNOWLEDGEMENTSSpecial acknowledgement is given to the following students of theMassachusetts Institute of Technology for their contribution of the modelsand photographs of the following plates: THOMAS BLOOD—Plate 3:LEONARD SAULNIER and K1 SUH PARK—Plate 10; RALPH MOR—RILL——-Plate 11; ROBERT CHESTER and ELIZABETH ROSS#Plate 12:ROGER MARSHALL—Plate 13; RICHARD PAINTER and ALANCOOPERiPlate 16: RAYMOND DeLUCIA and DOROTHY GINSBERG#Plate 8'0.ILLUSTRATIONS:GLORIA CATALANORALPH KNOWLES—Fig. 2; RICHARD LEAMANflPlate 1, top; Plate 29,bottom; PAUL SHIMAMOTO—Plate 2, bottom; Plate 15, bottom; Plate 21;Plate 25, bottom; ROGER JACKSON—Plate 2, top; Plate 26; Plate 27.COVER:Model and Photograph by ROGER MARSHALL.

EDITORS: ARTHUR J. HAMMILL—JOSEPH V. MOROGBUSINESS MANAGER: JAMES M. STEVENSONCIRCULATION MANAGER: WERNER F. HAUSLERFACULTY ADVISOR: HORACIO CAMINOS

THIS BI-ANNUAL PUBLICATION WAS ORIGINATED BY THESTUDENTS OF THE SCHOOL OF DESIGN IN 1951 AND IS MAIN-TAINED AS AN INDEPENDENT STUDENT PROJECT. ALL MATERIALIN THIS ISSUE IS COPYRIGHTED BY THE STUDENT PUBLICATIONOF THE SCHOOL OF DESIGN AND NO PORTION MAY BE RE—PRODUCED WITHOUT WRITTEN PERMISSION.

SUBSCRIPTION RATES: PATRON, $10 PER YEAR; REGULAR,$2.50 PER YEAR; INDIVIDUAL COPIES, $1.50: INDIVIDUALCOPIES THIS ISSUE $2.00. ALL PRICES ARE NET AND PRO-VISIONAL.

ADDRESS ALL ENQUIRIES: STUDENT PUBLICATION OF THESCHOOL OF DESIGN, BOX 5273, RALEIGH, NORTH CAROLINA.ALSO AVAILABLE THROUGH: WITTENBORN AND CO., 1018MADISON AVENUE, NEW YORK 21, N. Y.; ALEC TIRANTI, LTD.,72 CHARLOTTE STREET, LONDON, W. |.. ENGLAND.

PATRONSEdgar J. Kaufmann, Jr., New York, N. Y.; Harrison andAbramovitz, New York, N. Y.; William W. Atkin, New York,N. Y.; Richard L. Aeck, Atlanta, Ga.,' Bill J. Addison, Lynch-bura, Va.; Anthony Lord, Asheoille, N. C.; Ballard, Todd andSnibbe, New York, N. Y.; Pietro Belluschi, Cambridge, Mass.;Richard M. Bennett, Chicago, Ill.: George Bireline, Raleigh,N. 0.; Joseph N. Boaz, Raleigh, N. 0.; Harold Boericke, Jr.,Washington, D. 0.; Leslie Boney, Wilmington, N. C.; Richard R.Bradshaw, Van Nuys, Calif.; Clark and Enerson, Lincoln, Neb.;Joseph H. Cox, Raleigh, N. 0.; Curtis and Davis, New Orleans,La.; Gilleland and Strutt, Ottawa, Ontario; Charles M. Goodman,Washington, D. 0.; Irving Grossman, Toronto, Canada; RoyGussow, Raleigh, N. C..‘ Douglas Haskell, New York, N. Y.;John Hertzman, Raleigh, N. C..' W. N. Hicks, Raleigh, N. 0.;Robert L. Humber, Greenville, N. 0.; Henry L. Kamphoefner,Raleigh, N. C.; Katz-Waisman—Blumenkranz-Stein-Weber Archi-tects Associated, New York, N. Y.; Edgar Kaufmann CharitableTrust, Pittsburgh, Pa.; Ketchem and Konkel, New York, N. Y.;Ketchum, Gina and Sharp, New York, N. Y.; Leavitt Associates,Norfolk, Va.; Jeffrey Lindsay, Beverly Hills, Calif.; Lyles,Bissett. Carlisle and Wolff, Columbia, S. C.; Sibyl Moholy—Nagy,New York, N. Y.; A. G. Odell, Jr., Charlotte, N. C.; Frei Otto,Berlin; Oliver and Smith, Norfolk, Va.; Richard Proctor Swal-low, Austin, Tex.; Sch. of Arch., Princeton Univ., Princeton,N. J.; G. Milton Small, Raleigh, N. 0.; Harry J. Spies, Clark,N. J.; Duncan R. Stuart, Raleigh, N. C.; E. Wayne Taylor,Raleigh, N. C.,' Edwin G. Thurlow, Raleigh, N. C.; B. 0.Vannort, Charlotte, N. C.,' William H. Wainwright, Ca/mbridge,Mass. Especial appreciation is expressed to Mr. Edgar J. Kauf-mann, Jr. and to Mr. Wallace K. Harrison for their interestand support.

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PLATE 1 . The central figure shows a portion of a Hyperbolic Paraboloid ABCD divided into four units from which the surroundingtypical four structures are originated. In all these structures each unit preserves its Z axis, vertical, thus belonging to the typical First Positionindicated in figure 2. The structure shown at lower right, is similar to the one of figure 1.

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P L AT E 3 . Structure originated by combining eight units around a central support. It is based upon the introduction of a diagonal divi-sion of each of the four units that generate the structure shown on PLATE 1, upper le t. As in this example, the adjacent edges of the unitsslope toward the support which extends at the center as the common vertical edge of al the units.

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P L AT E 6 . Structure based upon the same combination of units as the one shown on PLATE 5, but designed using different angles anddimensions — for all its components.

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PLATE 8 . Axometric perspective of two structural units similar to the one shown on PLATE 7. Each structural unit is joined to theadiacent ones by means of its horizontal and vertical edges.

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Elevations and plan of a structurePLATE 9.

P LAT E l 0 . Two views of the structure described on PLATE 9. A different visual quality can be obtained by rotating each structuralunit ninety degrees, around its support.

P L AT E 1 l . Structure based upon the combination of four elements, each one containing two units of a Hyperbolic Paraboloid, within asquare plan. Upper: Plan and view of a combination of structural units each one placed at ninety degrees in relation with its two adioiningones. Lower: Plan and views of a combination of the same four structural units where their diagonal edges are converging to a central point.

P L AT E 1 2 . Plans and views of a structure originated by combining eight units resting on four supports. Such supports can be placedrecessed from the horizontal straight edge or at the corners, as shown in both plans.

P L AT E l 3 . Plan and views of a structure originated by combining eight units very similar to the ones adopted for the structure shownon PLATE 12. It only differs from the other on the slope of its adjoining edges which meet at the center of the structure, thus determining acentral high space that changes the visual qualities of the whole.

PLATE 13 in the pothe former structure.PLATE 14. Plan, view, and diagonal section ofsition of the four supports, thus combining each set ofD EJn9M.I I.9 0q. A,m. m)1! q. fi 90w to...muan m.onUr. 1 nu”-mo I1' Wunits within a triangular plan insteaders from the one described onof within a square plan as in

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PLATE 15.square central space can also be designed as an octagonal one.

Plan, view, and transoersal section of a structure originated by the combination of twelve units with four supports recessedfrom the straight horizontal edge. It presents a high space at the center and a low one around the periphery. Plan at bottom shows that the

PLATE 1 6 . Upper: Photograph taken from engraving of the illustration shown on PLATE 15, upper right. Lower: Elevation of astructure obtained by combining twelve units as described on PLATE 17. It is developed within a square plan and with straight horizontalperipheric edges.

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P LATE 1 7 . Plan, view, and diagonal section of a structure originated by the combination of twelve units, very similar to the one de-scribed on PLATES lo and 16. Their difference is based upon the position of the supports and consequently the position of their units.

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P L AT E l 8 . Plan, view, and diagonal section of a structure originated by the combination of twelve units, defining a high space at thecenter and a low one around the periphery. It can be described as a combination of four inclined triangular structural units springing from sin-gle supports. It differs from the structure shown on PLATE 1? in the slope of the edges of the four central units.

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PLATE 20. Section of a structure based upon, the some combination of units as the structure shown on PLATE 19 but with an in-creased vertical dimension for its central space. It shows that a single system can be expressed with different visual qualities by varying thedimension and angles of its component units.

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ight units and resting on four supports.the structure of the bottom are combined as in the typical example indicated on PLATE 1

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y combining five units resting on four supports, within a rectan—r, in the shape of the peripheric units.Plan, view, and longitudinal section of a structure obtained bgular plan. It only differs with the structure described on PLATE 26, lowePLATE 25.

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P L AT E 2 7 . Upper: Plan, view, and elevation of a structure obtained by combining twelve units resting on six supports. Lower: Plan,view, and elevation of a structure obtained by combining seven units. The six peripheric units are originated by dividing the central one witha vertical plane that contains its two supports.

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Structureits central vertex,Upper:f two structures obtained by combining eighteen units within an hexagonal plan.Structure with inclined peripheric edges, resting on twelve supports, and withPlans and views 0with a peripheric horizontal edge. Lower:PLATE 29.

Three views of a structure based upon the use of a single unit resting on two supports. Upper: Diagonal view showingthat the surface is defined by a lattice work of parabolic arches. Center: Underneath view showing that such arches follow the diagonal di-rection determined by the opposite vertexes of the unit. Lower: Side view showing the use of thin conical fillers to complete the surface, andto allow the pass of light.

PLATE 30.