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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)