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NESTED CATENARIES

Author: Defne Sunguroglu Hensel, Research Fellow AHO

Words: 1372

In October 2010, a series of events at AHO Oslo School of Architecture and Design focused on the

topic of structural masonry exemplified by the innovative works of Eladio Dieste. 1 As part of these

events the brick construction experiments workshop commenced with a line of inquiry based on the

basic principles of the catenary arch derived through a form-finding method first utilized three-

dimensionally by Antoni Gaudi, the hanging chain model.

The catenary arch and the theory of the thrust line have a long history and tradition in structural

masonry, architecture and engineering. The optimum tension form can be derived by hanging a

flexible chain from its two ends, which results in the specific catenary curve. Inverting this catenary

results in an optimum compression form. 2 Johann and Jacob Bernoulli describes the hanging chain

as “a mechanical system consisting of very many small, rigid parts, its links. Hence the equilibrium

state should be characterized by the lowest position of the center of gravity. Once the chain has

reached a configuration in which it cannot lower any link without in turn raising another, it will be in

equilibrium.”3 Antoni Gaudi instrumentalized this method of form finding three-dimensionally for the

design of the Colonia Guell Chapel of which only the crypt was built during the period between 1908-

1914. Gaudi’s model was constructed with chains suspended from a base plate in order to extract

catenary arch geometries under different loading as polygonal funicular arches. The model facilitated

the empirical data that underlie the structural form and the complex spatial organization of a network

of masonry vaults and domes supported by inclined columns and walls that follow the catenary

curve.4 Similarly, the catenary is present in Eladio Dieste’s Free-Standing and Gaussian vaults.

The brick construction experiments aimed at exploring some of the heretofore unexplored potentials

of a catenary arch arrangements, focusing on the spatially organized network of interacting

Catenaries. The aim was to accomplish an undulating wall made from nested catenaries. This

research challenge formed the basis of a ten days intense brick construction experiments workshop

run together with the master mason Øyvind Buset and nineteen master level students from the AHO

Auxiliary Architectures Studio. Methodologically the research employed a combination of physical

1 Eladio Dieste Advancing Architecture Through Material Systems Innovation Exhibition and Symposium was organized by

Michael Hensel, Defne Sunguroğlu Hensel and Birger Sevaldson and sponsored by Byggutengrenser.no, Wienerberger, Weber

and Einar Stange at the Oslo School of Architecture and Design, AHO during 08-22 October 2010. 2 Although the exact history of the Catenary cannot be traced, it was Robert Hooke who defined the theory of the thrust line as

early as 1675 published under the title of Heliscopes and some other instruments. The exact form of a thin Catenary arch was later described mathematically by Jakob Bernoulli using the new differential and integral calculus in 1704. 3 This quote is taken from the book The Parsimonious Universe written by Stefan Hildebrandt and Anthony Tromba published

by Springer-Verlag New York in 1996, page 135. 4 Antoni Gaudi produced his famous hanging chain model during the years between 1898 and 1908, which was destroyed

during the Spanish civil war. The in-depth study and the reconstruction of this model based on the remaining documents was carried out by the team led by Frei Otto at the Institute of Lightweight structures, commissioned in 1982. The Gaudi group at TH Delft, directed by Jan Molema was closely involved in the project.

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form-finding experiments with hanging chains, digital parametric and associative modelling, and 1:1

scale tests. Three teams focused at the tasks at hand.

The form-finding team developed complex arrangements of hanging chain models in an iterative

manner investigating the various configurations that the chain arrangements developed under their

own self-weight. The experiments were carried out across different scales ranging from 1:1, 1:4 to

1:10 studying analogously the underlying geometric principles, the parametric definition and

behaviour starting from a single chain towards more complex arrangement of chains that are point

loaded as a result of a nested assembly. The latter introduced incremental loading to the larger

system of chains and thus affected shape change over interacting regions of the arrangements in

relation to the applied force and the resultant force vector. When a catenary is point loaded its

geometry changes from a single catenary curve to two catenary curves that meet at a single point of

connection. These sets of experiments studied both the planar and spatial arrangement of catenaries

and the resulting global form by varying the distribution of hanging points from a straight line to a

sinusoidal curve.

The computation team worked on several tasks simultaneously. One area of investigation focused on

the digital registration of the results gained from the physical form-finding experiments, using a

mechanical Digitizer and employing photometric readings to extract empirical information. Two

methods for computing nested chain behaviour were developed and investigated in parallel, which

constitute [i] the development of an associative parametric set-up [ii] the implementation of the

Kangoroo physics engine to the Rhino Grasshopper set-up.5 The findings from both the chain models

and the 1:1 partial physical tests informed the development of the digital models with the aim to

develop the design and construction drawings for the construction of the full-scale prototype. These

included drawings for the production of the formwork and implementing brick arrangements in the

digital models with varying mortar gaps for the different arches. Once the three-dimensional

orientation of the start and the end bricks was defined, the spatial rotation of bricks around the axis of

the thrust line could be interpolated.

Perforated yellow brick ( 22.5*8.5*6.3*) made by Bratsberg was selected for the experiments. This

type of brick made it possible to accomplish the required curvature and provided the required mortar

tolerances. The construction team built different configurations of catenary arrangements, conducted

load tests and investigated different brick laying strategies with focus on the ‘key stones’ at the arch

intersections, and developed practical information for carrying out the construction process. The

construction of the full-scale prototype was conducted in house with low-tech tools available at the

AHO workshop. The final prototype consisted of 950 bricks and covered a floor area of approx. 8000

x 2000 mm reaching 2500 mm high at its highest point. Each catenary was constructed with one layer

of brick.

5 Grasshopper is a parametric modeling plug-in for Rhino, which is a NURBS-based 3-D modeling software. The physics

engine Kangaroo for Grasshopper embeds relaxation script for digitally simulating the physics behind the hanging chain. Currently under development by Daniel Piker.

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The final design was the result of successive decisions made on day-to-day basis, based on findings

from the experiments. Due to the incremental loading of the hanging chain arrangement organised

along a sinusoidal base-curve, the wall inclined up to 300 mm. This inclination and the related impact

to structural behaviour led to constructing a symmetrical second wall with specified areas of contact

intended for mutual support of both inclining arrangements of catenary arches. Upon removal of the

formwork it became apparent that the areas without support were structurally stabile and that the

catenary arch arrangement was not inclined beyond a critical limit. However, a detailed structural

analysis is currently under way. Further areas of inquiry include the development of the mathematical

and geometric description of interacting catenary systems, the related structural behaviour, brick

laying strategies and the detailing of points of intersection, as well as the possibility of utilising

catenary vaults between the arches to stabilise catenary arrangements wherever required or to

produce areas of enclosure wherever architecturally required. At any rate, there is great potential in

the further development of this particular line of research through design and construction.

Project Leaders:

Defne Sunguroğlu Hensel, Research Fellow AHO

Øyvind Buset, Master Mason

Project Team:

Auxiliary Architectures Studio, AHO

Linda Blaasvaer, Mattis Fosse, Marine Giller, Esa Hotanen, Torstein Hågensen-Drønen, Johnbosco Mulwana, Emanuel Ssinabulya, Simen C Lennertzen, Daniela Puga, Joakim Hoen, Rikard Jaucis, Eva Johansson, John Pantzar, Oda Forstrøm, Maximilian Hartinger, Fabian Onneken, Leonard Steidle, Nikolaos Magouliotis, Andre Severin Johansen

Sponsored By:

Byggutengrenser, Wienerberger, Weber and Einar Stange

With Special Thanks To:

Michael Hensel, Birger Sevaldson, Remo Pedreschi, Christoph Gengnagel, Jane Burry, Chris Williams, Daniel Davis, Daniel Piker, AHO and Our Sponsors

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Fig.1 Brick Construction Experiments Workshop, AHO, 2009 focused on the construction of selected structural masonry forms; two of which are the catenary arch (on the right) and the undulating wall (on the left), under the supervision of Defne Sunguroglu Hensel and the master mason Øyvind Buset.

Fig.2 Brick Construction Experiments Workshop, AHO, 2010. The three research teams working simultaneously on the inquiry with physical form-finding and computational methods accompanied with 1:1 scale tests

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Fig.3 Brick Construction Experiments Workshop, AHO, 2010.The construction process of the final piece showing the preparation of the formwork, mortar, special bricks and the brick laying

Fig.4 The empirical studies based on the hanging chain models with dimensional variations, extracting the intermediary stable states of the chain assembly. These studies inform the development of the digital associative parametric set-up focusing on the nested catenary behaviour.

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Fig.5 Digital drawings of the final assembly, showing the ‘thrust line’ that correspond to structural forces in compression (the side and top views) shown as the inversion of the hanging chain

Fig.7 Construction processes of the final wall showing the assembly and formwork procedures

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Fig.8 The freestanding structure shown after the removal of the formwork and the security bars in between the two sides of the

wall

Fig. 9 A small test construction of nested catenaries, organized along a base line developing a planar configuration (on the right) and the undulating wall with nested catenaries organized along a sinusoidal base curve developing a spatially curving arches (on the left)