Geometry, mechanical behavior, and the construction of cylindrical shells

In essence, laminar structures are those whose geometry corresponds to a thin and continuous surface, where two of its dimensions predominate over the dimension of thickness; their behavior is based on the fact that all internal stresses are normal, be they traction or compression, and tangential. For this to occur, the load that acts on the structure must preferably be “uniformly distributed and should not present sudden variations or punctual loads” (^{Morales, 2009}, p. 940), in such a way that its mechanical resistance is considered a virtue of its form.

In general terms, laminar structures (also called active surface structures) can be classified into three groups (^{Engel, 1970}): without curvature (folded sheets, for example), with simple curvature (like the cylindrical shells studied here), and with double curvature (as in the case of hyperbolic paraboloids). In the case of singly curved shells, their structural behavior can be explained in light of the membrane theory, which assumes that all stresses are tangent to the shell and are evenly distributed in its thickness (^{Candela, 1951}), in such a way that only two-dimensional stresses are developed along its surface (^{Salvadori & Heller, 1966}). Flexural rigidity and shear stiffness are negligible, as is resistance to compression (Figure 1).

Source: Laura Henao, 2018, CC BY.

Figure 1 Two-dimensional stresses considered in membrane-type structures 

In the particular case of simple cylindrical shells, with horizontal and symmetrical axes, and according to specialized literature (^{Billington, 1965}), they can be differentiated into short and long according to their support conditions and structural behavior. Thus, the so-called short ones are those that rest on their longitudinal edges (or generatrices) and have a ratio r/L>0.6 (where r is the interior radius of the directrix and L is the length of the membrane), while the long ones, at a ratio of r/L<0.6 (ASCE, 1952), rest on their ends (or on the lines of the generatrices) and exhibit a similar behavior to a beam (Figure 2a).

Source: Laura Henao, 2018, CC BY.

Figure 2 Geometry and behavior of a cylindrical shell 

In any case, cylindrical shells should not be confused with vaults, despite similarities in their shape, since both have a different mechanical behavior (^{Torroja, 1957}). The behavior of vaults is based on the interconnection of parallel arcs that transmit the same stress on continuous supports under the generatrices; they are usually constructed with materials resistant to compression and their thickness can sometimes be considerable. On the other hand, cylindrical shells behave in a similar way to longitudinal prismatic beams, although characterized with a curved profile and reduced thickness, and are capable of covering large spans without continuous lateral supports and with a predominance of bending stresses.

Thus, when a cylindrical shell comes under load, even under the action of its own weight, the generatrices get deformed, curving in such a way that the area located in the lower part of the shell works with tension and the upper part works with compression (Figure 2b). If the thickness of the shell is small in relation to its length, this flexion is reduced and has little influence on its general behavior.

From the point of view of construction, cylindrical shells can be made of reinforced concrete or reinforced ceramics. In both cases, the manufacturing process follows the same sequence: formwork (from straight molds since it is a developable surface), reinforcing (continuous or reticular depending on whether it is in concrete or ceramics, respectively), concreting (complete or on the small beams in the same cases), and uncasing. Depending on the number of shells and their dimensions, it can be a repetitive or simultaneous process, where it is possible to reuse the molds, which always implies savings in the quantity of constituent materials.

Genesis of cylindrical shells

The origin of cylindrical shells in the first decades of the 20^th century is closely related to the advances made by the German company Dyckerhoff & Widmann AG (Dywidag), under the guidance of two of its most prominent engineers: Franz Dischinger and Ulrich Finsterwalder (^{Stegmann, 2009}), responsible for calculating the first thin roofs in reinforced concrete: the Zeiss Planetarium in Jena² (1922), which had a circular plan with a 25 m radius and a spherical shell thickness of 6 cm (Figure 3); the market in Frankfurt am Main (1927), where each cylindrical vault covered a surface area of 50 x 17 m, and the market in Leipzig (1929), that had two twin domes on octagonal layouts with a diameter of 65.8 m, among others (^{May, 2012}).

Source: ^{Cassinello, Schlaich & Torroja (2010)}.

Figure 3 Dome in Jena (1922), by Dyckerhoff & Widmann AG 

Thanks to the first of these projects, Dischinger obtained-jointly with his partner in this work, Walther Bauersfeld-the patent that ended up being known as the “Zeiss-Dywidag system of thin roof shells” (^{Billington, 2013}) or “Z-D system.” In this case, steel bars were arranged following tension trajectories, with specially located reinforcements in the corners and other singular regions subjected to stresses. The diffusion of this system in the following years was due to a well-planned commercial strategy that led Dywidag to build laminar structures in several European countries, and even in Latin America: subsidiaries of this German company based in Uruguay and Argentina built at least seven concrete membranes in the period between 1929 and 1933, either through direct contracting or through technical consulting during the construction process. In any case, between 1923 and 1939, the systems patented by Dywidag were used in at least 35 countries, and reached more than 280,000 m² of covered area (^{May, 2015}).

In the specific field of the structural analysis of cylindrical shells, the works of Knud W. ^{Johansen (1944)} and Hegel ^{Lundgren (1949)} were added to those developed by Dywidag engineers. Lundgren made use of classical equations in the analysis of beam bending in order to determine longitudinal tensions, and combined them with an analysis of the arc-shaped part of the structure to find the value of transverse bending moments. This same principle was used by Félix ^{Candela in the design of his first long cylindrical shell for the roof of the Almacén Pisa in 1951} (^{Martínez & Echeverría, 2017}), as well as by the Uruguayan engineer Eladio Dieste in calculating several of his membranes built in reinforced ceramics since 1959 (^{Cabeza & Almodóvar, 1996}). Similarly, this calculation system was still used in the structural design made by August Komendant for the Kimbell Art Museum, designed by Louis Kahn in 1972 (^{Buonopane & Osanov, 2015}).

In Colombia, interest in laminar structures was manifested early and especially in the academic field at the Universidad Nacional, as evidenced by articles published since 1939 in the journal Ingeniería y Arquitectura, printed publication of the university’s Faculty of Engineering, where both professors and students, as well as international guests participated with articles. Thus, the first note on the subject published in the journal was written by the Dominican engineer Rafael Bonnelly, who aimed to find “practical formulas that facilitate the rapid design of membranes, to overcome thus the limitations imposed by patents” (Bonnelly, 1942, p. 18), which then restricted knowledge about the calculation of cylindrical shells.

His explanation was developed based on the case of a semi-cylindrical vault in reinforced concrete of 84 m long, 40 m wide, and 7.5 cm thick, for which values of inertia moment, stresses, and the section of steel bars were determined through a simplified, but mathematical process. Three years later, Bonnelly would submit to the journal Ingeniería y Arquitectura a detailed and illustrated review of a laminar vault built by himself in Santo Domingo, which verified in practice his theoretical appreciations (^{Bonnelly, 1945}).

Another sign of academic interest in the calculation of laminar structures in Colombia is manifested through a series of studies translated into Spanish by Professor Julio Carrizosa and published in Ingeniería y Arquitectura (^{Pucher, 1943a}, ^1943b). These articles divulged in the country the methods developed by the German engineer Adolf Pucher, who, in addition to having worked for Dywidag, would be well known in Latin America for his textbook on reinforced concrete (^{Pucher, 1958}), used in the training of civil engineers.

For his part, in 1944, Henry Cornelissen, engineer and professor at the Universidad Nacional, described in the same journal new modes of analysis for cylindrical shells, based on two outstanding methods for the solution: those of Dr. U. Finsterwalder and Dr. F. Dischinger, respectively, “given that today the calculation of these structures is too complicated and laborious for engineers not specialized in this area” (^{Cornelissen, 1944, p. 5}). As examples, the author mentions the roofs of the Budapest Market Hall (1931), with membranes of 41.5 m span length, 12 m width and 6 cm thickness, designed by Finsterwalder, and the Hershey Arena, designed by the engineer Anton Tedesko (1936) in the United States. The article’s bibliography included authors such as Dischinger, Finsterwalder, Moltke and Kalinka, Schorer, Tedesko and Timoshenko, all of great influence in the field of calculation of reinforced concrete membranes.

In addition to academic publications, the physical presence of the Spanish engineer Eduardo Torroja in the Colombian context was very important, who stayed in Bogotá for five days in 1952. On that occasion, he gave several conferences, among which one was entitled “Doubly and simply curved laminar roofs” (^{Escobar & Cárdenas, 2006}), in which he explained the mechanical principles of cylindrical vaults, among other things.

By then, many of the buildings designed by Torroja were already recognized by the international community, among which his project of a recessed dome with a curvature radius of 44.1 m and a thickness of 9 cm for the Algeciras market (1934) stands out. However, his most famous project would be the roof for the La Zarzuela Racecourse (1935) in Madrid, equipped with double curvature shells, without ribs, that extends 12.7 m from the support towards the track, with 5.3 m width in each of its modules and a thickness of 5 cm at the end of the cantilever. Additionally, in the field of cylindrical vaults, his design for the roof of the Frontón Recoletos stands out, also in Madrid, designed with two semi-cylindrical lobes of simple curvature, in reinforced concrete (^{García, 2017}).

In this last project, which unfortunately was partially destroyed during the Spanish Civil War, Torroja had deliberately eliminated the possibility of constructing a thick prismatic beam under the line where the two laminar vaults were joined (Figure 4). In his words, “if there exists a longitudinal beam, it is constituted by the same shell, which, leaving behind the narrow mechanical frame of prismatic pieces, develops a resistance process that is much more complex and complete, but no less defined” (^{Torroja, 1936, p. 4}).

Source: ^{Cassinello, Schlaich & Torroja (2010)}.

Figure 4 Frontón Recoletos (1935), Madrid, by Eduardo Torroja 

After the Second World War, cylindrical shells were extended throughout the United States and Europe as a roofing solution for industrial buildings, warehouses, vehicle exhibition halls and sports venues, among others, with a special care in their geometric form, the way of their support, and the construction process, giving rise to a singular typological variety (^{García & Osuna, 2009}). This phenomenon coincides with an interest aroused by this design in engineers and architects in Colombia, who already had the mathematical apparatus necessary for its structural calculation, and who aimed to develop an experimental work in the construction field in order to achieve practical and versatile structures at a low cost.

It is no coincidence that the interest of Colombian professionals in the design and construction of light roofs with large spans came at a time when the country was beginning an industrialization process in its most important cities. Such situation made it necessary to design new spaces with good lighting and ventilation conditions, but without intermediate supports that would hinder the equipment, and especially at a low cost so that its construction would be viable.

Case studies in Colombian architecture

The interest in laminar structures has already been demonstrated in the academic context of the Universidad Nacional de Colombia, which was manifested in articles written by foreign professors and collaborators who worked at the university; however, in the field of practice, the first constructions of this type were slow and progressive.

Thus, it is known that between 1942 and 1946 the architect José María Montoya Valenzuela and the engineer Fausto Galante built vaults in reinforced concrete to cover the chapel of the Minor Seminary of Bogotá, but its influence in the local and national context was limited. It was the German architect-engineer Leopoldo Rother who would made the first attempt to take advantage of the benefits of flat or slightly curved laminar structures, as evidenced by two separate projects for the Universidad Nacional de Colombia: the Alfonso López Stadium (1937) and the headquarters of the Printing House (1945), today Museum of Architecture (^{Niño, 2003}; ^{Pinilla, 2017}).

It is also worth mentioning Rother’s design for the Laverde Building (1943) in Bogotá, whose roof finish was developed using an undulating cover constructed from membranes that were most likely made of reinforced concrete (^{Devia, 2006}). In addition, in his project for the National Building of Barranquilla (1945), it is clear the use of short cylindrical shells as a roofing solution (^{Arango, 1989}), calculated by the engineer Guillermo González Zuleta. These two professionals were responsible for the first building covered with cylindrical shells in the country: the Girardot market.

The Girardot market

Designed by Leopoldo Rother starting from 1946, its construction was completed in 1954. It counted with the collaboration of José Antonio Parra, calculating engineer, and Guillermo González Zuleta, supervising engineer, who graduated from the Universidad Nacional de Colombia in 1941. It is a building conceived as a large horizontal plane that provides shade for market vendors protecting them from solar radiation, almost without the presence of enclosing walls, which allows the air to circulate freely through the building (Figure 5).

Source: Laura Henao, 2018, CC BY.

Figure 5 Current view of the Girardot market 

The structural elements are clearly inscribed in a mesh with orthogonal axes separated by 7 m in one direction and 2.5 m in the other; these measurements correspond to the dimensions of the roof shells and wooden formwork elaborated without auxiliary equipment.

In effect, the roof plan is formed by 198 cylindrical shells that are 7 m long, 2.5 m wide, 5 cm thick with a 60 cm depth, called here “membranes” by their designers (^{Rother, 1967}). In their longest direction, each of the 22 rows of membranes aligns 9 shells in such a way that, for the observer, they acquire a continuous length of 63 m. This is achieved through a simple but effective resource: a design called “invisible beam” by Rother himself, which is nothing else than a prismatic concrete piece from where each cylindrical shell hang, and which embraces them on the upper surface of their ends.

In turn, each of the 10 invisible beams reaches a length of almost 55 m and a height of 70 cm counted from the highest point (or “crest”) of the membranes. The upper edge of these beams is horizontal, while the lower is wavy and it extends to the valley formed between the shells, where it reaches a total height of 1.3 m. The width of the beams is always 40 cm, and in the transverse direction they are connected by section braces of 25 x 50 cm, separated by 7.5 m from each other, without getting into contact with the membranes. At the ends, the beams adopt a curved profile, which prevents them from poking over the edge of the roof plane (Figure 6).

Source: Laura Henao, 2018, CC BY.

Figure 6 Digital reconstruction of the roof, top view 

Each beam is supported on a total of 8 inclined columns arranged in a V-shape, which accentuates the transparency of the building and allows the viewer to understand the light character of the structure.

Although the traditional historiography describes them as concrete structures, ^{Rother (1984)}-in the book about his father’s work-explains how, faced with difficulties inherent in the calculation of this type of structures, it was necessary to build scale models³ that were subjected to load, opting for hollow bricks as a lightening system to achieve membranes with a thickness of 5 cm.

Finally, the drainage system designed for the roof deserves a special mention. Given that rainwater got accumulated in the valleys between the membranes, it was necessary to design a system of tubular downpipes attached to the V-shaped columns, after giving to the exterior finish of said valleys a slope to conduct the water towards the scuppers. Since not all the valleys coincided with the upper end of a downpipe, horizontal channels crossed the lower part of the membranes over some of the transverse axes, and altered the formal cleaning of the lower face of the roof (Figure 7).

Source: Original plan, dated July 27, 1949. General Archive of the Nation, Bogotá. Fund INVÍAS (Mapoteca 1, Planoteca 03, Bandeja 09, Ref. 171).

Figure 7 Partial view of the detail of the rainwater evacuation system 

^{Rother himself (1967)} mentioned that the water retained in some of the valleys between the membranes could freely evaporate as a mechanism to preserve a fresh environment inside the market place; he was so concerned that he designed on the crest of the edge membranes a cord of 15 cm high, in order to prevent retained water from overflowing towards the outside wall of the building.

Over the years, deficiencies of the rainwater evacuation system have contributed to the deterioration of the building, and caused pathologies that affect its structural integrity in the long term.

Bus terminal in Bogotá

This building, designed in 1950 by the architects Álvaro Ortega and Gabriel Solano, with structural calculations by Guillermo González Zuleta, was intended to serve as terminal station for the municipal buses of the tram company, located in the city of Bogotá. The same team of professionals had already worked together on the design of the “11 de Noviembre” Baseball Stadium (1947) in Cartagena de Indias, together with the architect Jorge Gaitán Cortés, which displays the first large double-curvature membrane vault in Colombia (^{Vargas & Galindo, 2015}).

The project of this new building demanded several independent buildings (Figure 8) organized around a large maneuvering yard: [a] a two-story office building located on the main facade, [b] a covered area dedicated to fuel supply, [c] washing and greasing area, [d] machinery workshops; and [e] bathrooms and warehouses.

Source: Laura Henao, 2018, CC BY

Figure 8 Digital reconstruction of the compound 

All buildings were constructed using cylindrical roof shells based on a module with a square plan of 4.8 m on each side. Building [a] had two stories and a rectangular plan covered with a single membrane 4.8 m wide and 24 m long, with an 80 cm deflection, built in reinforced ceramics and supported, on its longer sides, on the load walls that constituted the facades. Buildings [b] and [c] were double-height structures with a system of enclosures independent from the roof system, characterized by a set of membranes in series (4 in the first building, 6 in the second) 4.8 m wide and 21.2 m long (corresponding to 19.2 m plus two end cantilevers of 1 m long each).

Although in this project the membranes were organized in a similar way to those of the Girardot market, here the cylindrical shells were not placed below invisible beams, but were confined between two of them, which were parallel and separated from each other by a distance of 9.6 m between axes. In turn, transverse joists (2 in the first building, and 3 in the second), spaced by 4.8 m, braced both beams without contact with the membranes in order to leave the valleys free, where rainwater was collected through downspouts attached to circular pillars over square pedestals (Figure 9). In cases when the valley did not coincide with a column where the drainpipe could be attached, a scupper was installed from which a hidden fiber-cement pipe was hanging behind the inner face of the invisible beam, which was joined to the downpipe on the next structural axis.

Source: Laura Henao, 2018, CC BY.

Figure 9 Digital reconstruction of the roof system in building [c] 

For its part, the largest roof was the one that covered the workshop area, or building [d], which was formed by 3 aligned vaults 24 m long and 16 m wide each, supported by cable-stayed arches, which, at the same time, were supported on circular columns with a diameter of 30 cm. The three vaults, identical and continuous, had a deflection of 2 m in height. Next to them, but in a perpendicular sense, there was a set of 14 membranes of lower height that covered building [e], 4.8 m wide and 8 m long, confined at their end walls by edge beams supported on columns (Figure 10). In all cases, waterproofing was achieved, according to the authors of the project, with a lime-based plaster on the upper face of the vaults and direct water discharge on the pavement.

Source: Laura Henao, 2018, CC BY

Figure 10 Digital reconstruction of the roof system of building [d], in the foreground, and of building [e] on the right 

In this project, transversal spans were modest and they were all perfectly modulated, which allowed using mobile wooden frameworks (^{Villazón, 2001}) that were light and cheap, and of simple and economic handling. The membranes were constructed in such a way that in the joints (10 and 12 cm thick), between the bricks (of 38 x 23 cm), the armor formed by 3/8” steel bars was arranged in such a way they could work when the concrete had completely hardened. In the valleys between the membranes, thrusts were neutralized by reinforcing them with four ¼” bars embedded in concrete.

The building not only came to be included in the sample presented by ^{Hitchcock for the Museum of Modern Art in New York in 1955} (Hitchcock, 1955), but also served as an example to illustrate articles such as the one published in the journal Architectural Forum (^{Shell Concrete Today, 1954}), which reviewed important projects with laminar structures from around the world, as a summary of the academic event organized on the subject by the Massachusetts Institute of Technology (MIT), in Boston, in June of 1954.

Unfortunately, the building was entirely demolished at the end of the 1970s, but its influence was remarkable: an unsigned article from the journal Proa (No. 50, August 1951) recorded photographs of the finished project and was preceded by an editorial note entitled “The binomial architect-engineer,” where both kinds of professionals were urged to have a greater dialogue about projects.

The Volkswagen Headquarters

Designed by the Italian architect Bruno Violi, the structural calculations of roof membranes were made by Guillermo González Zuleta.⁴ With its main facade on the important 26th street in Bogotá (a highway that quickly connects the center of the city with the airport), the building occupied an area almost equivalent to a block and was dedicated to accommodate several functions for the Volkswagen headquarters. In spite of the property’s irregularities, the structure was organized on a mesh that was close to a module of 6 m, and it was equal to the width of the largest cylindrical roof shells.

The main body [a] (Figure 11), on the 26th Street, was formed by a two-level structure; the first level had double height: 5.35 m, with a mezzanine on the front, while the second was of 3.2 m high, covered by a set of six circumference-segment membranes of 23 m long, 5.93 m wide, and with a depth of 1.05 m.

Source: Laura Henao, 2018, CC BY

Figure 11 Digital reconstruction of the building’s structural and roof system 

Each of these main membranes were supported on three parallel arches of 40 cm wide, supported by a horizontal beam of 15 cm high, in such a way that they formed identical number of axes separated from each other by distances of 6 and 15 m, respectively. At each end, the membranes had a cantilever of 1 m. Its construction was carried out taking into account previous experiences: bricks (5 cm thick and 23 x 38 cm in area) were separated from each other by girders of 10 cm wide that were reinforced with four 3/8” bars. In the areas near the valleys, the entire membrane was made of concrete reinforced with ½” bars in the vicinity of ceramic pieces, and with 5/8” bars at the point of membrane inflection. Reinforcements were uniquely arranged in areas where membranes were located above the supporting arches, which constituted a supplementary reinforcement of a square plan with 3/8” bars spaced 20 cm from each other (Figure 12).

Source: Archive of Bogotá, Plano 101.10.271.13, Fund Guillermo González Zuleta.

Figure 12 Detail of reinforcements of the facade membranes 

The necessity of an optimal drainage system required that membranes extend 50 cm outside the plane of the front and back facades, to allow free transit for the downpipes until the floor level.

Additionally, a reinforced concrete slab was extended towards the back of the property, simultaneously serving as a cover for the areas dedicated to the maintenance of cars and as a parking surface for other cars, which could be reached through a ramp. On the lower side of the plate, supported by circular columns separated from each other by a distance of 12 m, clearly visible prismatic beams were arranged in two directions that formed structural porticos and ended in cantilevers of 3.5 m long on each side.

It is of great interest how two rectangular areas, or building [b], were covered by two sets of membranes. One of them had 9 sinusoidal shells and the other had 8; all of them were 8.8 m long and 4 m wide, with their longitudinal ends confined in an edge beam. While one end of this beam rested on the dividing wall, the other was suspended, by means of concrete columns, from the cantilever of the highest concrete slab, forming a zenith opening between both surfaces through which the air circulated (Figure 13).

Source: Laura Henao, 2018, CC BY.

Figure 13 Digital reconstruction of the suspended membrane system 

In order to further illuminate the space on the first floor, the membranes framed in the moderately suspended structure were constructed with pieces of isolux arranged like structural bricks, following the model proposed by Rother in the central membranes of the Girardot market. The drainage of this series of membranes was achieved through vertical downpipes located in the posterior plane in such a way that water was taken to them through a gentle slope of the surface of the valleys.

Finally, building [c], also destined to car storage, covered its second floor with a series of 8 aligned membranes that ended against an elevated water tank, each 6 m long and 5.93 m wide. As in the previous cases, they were only 5 cm thick.