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Structural Reciprocity: Critical Overview and Promising Research/Design Issues


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Reciprocity is a structural principle that has fascinated designers and builders throughout the world since ancient times. Despite the topic's having been studied by various academics, designers and researchers, a critical overview of the references is still missing, as is an outline and discussion of the current and future promising research/design issues. Further, no single text provides an exhaustive definition of the principle of structural reciprocity and it must be critically recon-structed from several different sources. This paper aims to fill in these gaps, pro-viding a complete and annotated list of references, in which historical examples, as well as patents, research articles and terminological issues are discussed. A con-sistent definition of structural reciprocity is also proposed, and the promising developments of such a principle are outlined in order to guide designers and researchers in the future.
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Structural Reciprocity: Critical Overview
and Promising Research/Design Issues
Alberto Pugnale Mario Sassone
Published online: 4 March 2014
ÓKim Williams Books, Turin 2014
Abstract Reciprocity is a structural principle that has fascinated designers and
builders throughout the world since ancient times. Despite the topic’s having been
studied by various academics, designers and researchers, a critical overview of the
references is still missing, as is an outline and discussion of the current and future
promising research/design issues. Further, no single text provides an exhaustive
definition of the principle of structural reciprocity and it must be critically recon-
structed from several different sources. This paper aims to fill in these gaps, pro-
viding a complete and annotated list of references, in which historical examples, as
well as patents, research articles and terminological issues are discussed. A con-
sistent definition of structural reciprocity is also proposed, and the promising
developments of such a principle are outlined in order to guide designers and
researchers in the future.
Keywords Structural reciprocity Reciprocal frames (RFs) Nexorades
Mutually supported elements (MSE) Leonardo da Vinci Leonardo
grids Spatial structures Timber constructions Form-finding
Morphogenesis History of construction
A. Pugnale (&)
Faculty of Architecture, Building and Planning, University of Melbourne,
757 Swanston St., Melbourne, VIC 3010, Australia
M. Sassone
Department of Architecture and Design (DAD), Politecnico di Torino,
Castello del Valentino, Viale Pier Andrea Mattioli, 39, 10125 Torino, TO, Italy
Nexus Netw J (2014) 16:9–35
DOI 10.1007/s00004-014-0174-z
The principle of reciprocity is based on the use of load-bearing elements which,
supporting one another along their spans and never at the extremities, compose a
spatial configuration with no clear structural hierarchy.
An illustrative example will help the reader better understand the concept. Let us
consider three glasses, arranged on a table at the vertices of a hypothetical
equilateral triangle, and then imagine covering that area using just three kitchen
knives. Considering that the glasses are the only supports, the knife handles should
first be placed over the glass openings, and the blades should be made to overlap one
another, like a fan. The resulting configuration is the simplest reciprocal structure
made of three elements (Fig. 1).
Such a system has been used throughout the world since ancient times.
However, a comprehensive, annotated list of references regarding them has
not been developed yet. Furthermore, since designers and researchers works
frequently in independent ways, the current studies still need to be grouped
and compared systematically in order to outline the future promising design/
research activities. It is an explicit aim of this essay to respond to these needs
by proposing a critical overview of the topic supported by: (1) an annotated
list of references in both Western and Eastern cultures; (2) a complete lists of
patents on reciprocity; (3) a complete list of publications grouped by topic;
(4) a complete and annotated list of scientific terms coined and used by
different authors; (5) a consistent definition of structural reciprocity; (6)
several future research and design directions based on the current state of the
Differences and Similarities of a Fragmented History
The first applications of structural reciprocity date back to Neolithic pit
dwellings, Eskimo tents and Indian tepees, as documented by Popovic Larsen
(2008). However, it was also present during the Roman Empire, when Julius
Caesar used it for the construction of a bridge over the Rhine: the structure was
made of interlocked timber elements, with the main purpose of simplifying the
joints. The project was described in Caesar’s ‘‘Commentaries on Gallic and
Civil Wars’’ and was later reconstructed by Palladio (Gros and Beltramini
2003). Apart from these ancient applications, independent from one another,
reciprocity has been independently studied and used in both Western and
Eastern cultures for centuries. The needs and purposes have been different, but
the structural outcomes have been similar. During the last few decades,
reciprocity has also become a research topic for academics, who have started to
study the mechanical, geometrical and construction aspects of reciprocal
10 A. Pugnale, M. Sassone
Reciprocity in Western Culture: Timber and Short-Beams
In Europe, structural reciprocity has mainly been used, at least until the twentieth
century, to span distances longer than the length of the available timber beams.
Several architects, builders, mathematicians and scientists all reasoned separately
about this problem, without evidence of being in contact with one another.
Between 1225 and 1250, Villard de Honnecourt drew some short-beam arrange-
ments in his sketchbook (Villard de Honnecourt 1959) for the construction of
reciprocal floors. Between 1220 and 1235, Alexander of Lincoln designed and built
Lincoln Cathedral using reciprocal supports, as reported by Hewett (1974).
Leonardo da Vinci explored at least six different spatial configurations based on the
principle of reciprocity, as can be seen from his sketches in the Codex Atlanticus,
folio 899v (Leonardo da Vinci 2000) (Fig. 2). To read the transcription of this page,
refer to Williams (2008).
After Leonardo, Sebastiano Serlio discussed the construction of planar floors
with short-beams in his first book on architecture, but unfortunately proposed an
unbuildable structure (Serlio 1566; Yeomans 1997). John Wallis wrote about
different types of floors made of reciprocal elements in his ‘‘Opera Mathematica’
(Wallis 1695), and this was the first scientific text to be supported with structural
calculations, as it has been described in detail by Houlsby (2014).
Amongst the treatises on carpentry, E
´my, who was professor of Fortification at
the Royal Military School in Saint-Cyr, included some reciprocal examples in his
´de l’art de la charpenterie’’ (E
´my 1837). Reciprocity was also present in the
Fig. 1 An illustrative example of a simple reciprocal structure. Image:authors
This problem was also solved using an alternative construction technique, based on the overlapping of
layers of short load-bearing elements. This technique can be found, for instance, in the roof structure of
the Square Hall building in the Old Nisa, Mesopotamia, as described by Pizzetti and Zorgno (1980). It
was also proposed by Guarini for his Chapel of the Holy Shroud in Turin, Italy.
A more detailed description of historical works on reciprocity can be found in Popovic Larsen (2008).
Instead, a complete list of historical treatises, dealing with reciprocal structure, has been provided by Vito
Bertin on his website:
Structural Reciprocity 11
so-called ‘flat-vaults’ described by Fre
´zier (1737), but which were invented by
Joseph Abeille and Se
´bastien Truchet. It was finally described by Rondelet (1810)
and Tredgold (1837).
As we mentioned before, reciprocal structures were used in Europe for
technological and construction reasons. Recent surveys and renovations of old
British buildings seem to confirm this statement. The layout of the sub-floor
structure of the Wollaton Hall (Fig. 3) shows an irregular pattern of timber short-
(b) (c)
Fig. 2 a Leonardo da Vinci, Codex Atlanticus, fol. 899v, with different concepts identified and
numbered. The reciprocal systems represented by Leonardo are redrawn in bh.Continuous lines are
used to represent the working parts of the arrangements, which are clearly drawn and reproducible with
physical models. Unclear potions of Leonardo’s sketches, as well as construction lines are reproduced
with hidden lines. Hatches are then used to identify parts of the structures which were meant to be clad.
Image elaboration: authors. bTranscription of drawings A1 and A2. cTranscription of drawing B1.
dTranscription of drawing B2. eTranscription of drawing B3. fTranscription of drawing C2.
gTranscription of drawing E1. hTranscription of drawing E2
12 A. Pugnale, M. Sassone
beams which are arranged in a reciprocal way. They were clearly conceived not to
be seen as the photograph of the finished ceiling does not refer anyhow to its
reciprocal nature (Fig. 4). The same happens at the home of William Morris,
Kelmscott Manor, where the presence of reciprocal beams in the sub-floor structure
was only revealed during the most recent works of restoration (Insall 1972,2008)
(e) (f)
Fig. 2 continued
Structural Reciprocity 13
(Fig. 5). A unique example that contradicts such logic can be found at Palazzo
Piccolomini in Pienza. The ceiling of its ‘‘Music room’’ presents a reciprocal
arrangement of timber beams which appears to be purely decorative: such a room is
one of the smallest of the Palazzo, but it is the only one using short-beams to span a
relatively short distance (Fig. 6).
Reciprocity in the East: Interwoven Structures and Symbolism
In Eastern culture, interest in reciprocal structures derives from two separate
concepts. On the one hand, especially in China, the use of interwoven strips of
bamboo for the realization of baskets is an old tradition that has been transferred to
objects of larger scale; the so-called ‘‘Rainbow Bridge’’ in Shandong is the main
example (Baverel 2000; Rizzuto et al. 2002; Di Carlo 2008). On the other hand, the
religious concept of Mandala, with its symbolic ‘magic circle’ shape (Gombrich
1979), has inspired the construction of circular reciprocal roofs in many Buddhist
temples all over Asia. Unfortunately, written documents concerning this practice are
only found after the year 1275, i.e., when Marco Polo arrived in China at the end of
the Song Dynasty.
A few contemporary Asian architects, such as Kazuhiro Ishii (Fig. 7), Yasufumi
Kijima and Yoichi Kan, still take advantage of structural reciprocity for their
constructions (Gutdeutsch 1996; Popovic Larsen 2008,2009).
The Basis of a Scientific Approach
Even though structural reciprocity has not had a linear history, and the evidence of
its use seems to be fragmented, a main point in common in both Western and
Eastern cultures is the use of timber as a construction material. In Europe, this
occurred for functional reasons, with the aim of developing flat configurations. In
(g) (h)
Fig. 2 continued
14 A. Pugnale, M. Sassone
Asia, designers were mainly guided by spiritual considerations in the construction of
3D structures, which recalled symbolic shapes.
In this historical framework, Leonardo da Vinci was the only one in the West to
study the potential of elaborating reciprocal structures as complex 3D geometries
Fig. 3 Wollaton Hall, sub-floor structure survey. This layout shows an irregular pattern of timber short-
beams which was clearly conceived not to be seen. Image: Ed Morton, The Morton Partnership LTD,
London, UK, reproduced by permission
Fig. 4 Wollaton Hall, ceiling. The reciprocal arrangement of sub-floor timber beams is not visible as it
was clearly conceived to solve a construction matter. Photo: Peter Langley, reproduced by permission
Structural Reciprocity 15
Fig. 5 Sub-floor structure of the William Morris house (Kelmscott Manor) during the most recent
renovation works. This reciprocal arrangement of timber beams was clearly due to structural and
construction reasons, as the carpentry was not designed to be exposed. Photo: Donald Insall, reproduced
by permission
Fig. 6 Palazzo Piccolomini, Ceiling of the ‘‘Music room’’. The reciprocal arrangement of timber beams
appears to be purely decorative and designed to be seen. Such a room is one of the smallest of the
Palazzo. However, it is the only one using short-beams to span a relatively short distance. Photo: Il
Cenacolo srl, Rome, reproduced by permission. Image elaboration:authors
16 A. Pugnale, M. Sassone
(Leonardo da Vinci 2000). It was John Wallis who first approached structural
reciprocity scientifically, like a modern research topic (Wallis 1695; Houlsby 2014).
The Years of Patents and Scientific Research
Over the last century, the principle of reciprocity has continued to stimulate the
interest of designers and, for the first time, has become a topic of interest in the field
of scientific research. As shown in ‘‘Appendix 1’, several patents were registered by
different authors between 1924 and 2012. The first was the lamella construction
system by Zollinger with reciprocal joints, which has been described in detail by
Popovic Larsen (2014). A foldable structure by Emilio Perez Pin
˜ero, which
proposed a kinematic roof structure with nodes based on the superimposition of
bars, was then registered in the USA and Canada in 1965 (Figs. 8,9). A detailed
description of this project can be found in the monograph about Pin
˜ero by Escrig
The German engineers Erwin Walle and Sigurd Prinz patented a reinforced
concrete ceiling made of prefabricated units hinged together, which curiously
recalls the reciprocal slabs designed by Kahn for the Mill Creek public housing
project in Philadelphia, dated 1952–1953. The patent was filed in 1971 and then
published in three different parts. The building element for the construction of
interlocking grids patented by Gat is also worth mentioning. This was actually the
first work to be also published as an article in a scientific journal (Gat 1978).
However, it was the 3D closed circuit of sticks patented by Graham Brown that
Fig. 7 Seiwa Bunraku-Kan, Puppet theatre complex designed by Kazuhiro Ishii. Photo: Kentaro
Tsukuba, reproduced by permission
Even though Gat’s system was the first patent also published in a scientific journal, it was not the first
article ever to discuss reciprocity. A paper by Donald Dean about a ‘new’ structural system called
‘Lamella Grid’ had in fact already been published in 1964 (Dean 1964).
Structural Reciprocity 17
definitively inspired and kick-started academic research on structural reciprocity.
Brown established contact with the University of Nottingham, with the intention of
increasing his understanding of the mechanical behaviour of such a system, which
he called ‘‘Reciprocal Frame’’. This led John Chilton’s group to make several
publications during the 1990s (Chilton and Choo 1992,1994; Choo et al. 1994;
Chilton et al. Chilton 1995a,b), including the Ph.D. thesis by (Popovic 1996), which
has recently been revised and printed as a book (Popovic Larsen 2008). A complete
list of patents involving structural reciprocity is given in ‘‘Appendix 1’’.
Fig. 8 Emilio Perez Pin
patent. Detail of the node
Images: United States Patent and
Trademark Office, reproduced
by permission
Fig. 9 Emilio Perez Pin
˜ero’s patent. Superimposition of bars allows and controls the kinematic
movement of the assembly Images: United States Patent and Trademark Office, reproduced by permission
18 A. Pugnale, M. Sassone
Even though John Chilton’s group was the first to use the term ‘Reciprocal
Frame’ in a scientific paper (it is here that a definition of the concept of closed
circuits of elements is found), other names by other academics and research groups
also exist (see ‘Appendix 2’ for further details). Such terminological heterogeneity
creates a certain level of confusion that has been avoided, in this text, by adopting
the most logical and generic term ‘structural reciprocity’, or simply ‘reciprocity’.
Structural reciprocity is a very extensive research topic, and has been approached
from very different points of view. Looking at the current publications, at least
fourteen different sub-topics can be defined: (1) morphology and geometry of
reciprocal spatial structures, including (2) polyhedrical configurations; (3) form-
finding and morphogenesis of such structures with computational tools, recently
also combined with (4) the rapid prototyping of timber elements; (5) the design of
joints and connections; (6) analysis of the structural behaviour; (7) the study of the
kinematic potential; (8) investigations on materials and sections for construction; (9)
the use of planar panels; (10) discussion/criticisms of projects and prototypes; (11)
teaching structural reciprocity; (12) history of reciprocal structures; (13) art and
sculpture, (14) reciprocal structures based on Leonardo’s grids. ‘‘Appendix 3’ lists
the main research works published in each category.
Reciprocity in Contemporary Architecture, Art and Industrial Design
Considering the works of architecture realized over the last century, we find several
unrelated examples that have implemented structural reciprocity. Long spaces were
spanned with short beams in the Mill Creek Public Housing project, designed by
Louis Kahn in 1952–1953, but also in the Berlin Philarmonie by Hans Scharoun,
which was built in 1960–1963, and in a salt storage building in Lausanne, built by
Atelier Gamme Architecture in 1989 (Natterer et al. 1991). In order to introduce
spiritual philosophies into shapes, reciprocity was used in the Casa Negre by Josep
Maria Jujol in 1915–1926, (Fig. 10) (Ligtelijn and Saariste 1996), as well as in all
the structures by Graham Brown and the Japanese constructions mentioned earlier.
Some experimental pavilions, with the Arup Group’s Advanced Geometry Unit
group as the engineering consultant, have also been constructed: the Forest Park by
Shigeru Ban (McQuaid 2003), the H-edge pavilion by Cecil Balmond, realized with
the students at Penn University (Balmond 2007), and the Serpentine Gallery 2005
by A
´lvaro Siza (Sakamoto et al. 2008). These examples clearly illustrate that the
architectural potential of structural reciprocity still needs to be explored—the use of
different materials, as well as the definition of element sections and joints, are some
of the possible research/design issues of the next few years.
In the field of industrial design, Pino Pizzigoni realized reciprocal chairs and
tables made of timber and marble elements. Two examples can be seen in Figs. 11
and 12 (Pizzigoni Archive: PIZ N, 1948; Pizzigoni 1982; Deregibus and Pugnale
2010). Shifting to art, the sculptor George Hart used reciprocity to create simple
geodesic domes. Rinus Roelofs, instead, focused on complex three-dimensional
sculptures based on Leonardo’s grids (Roelofs 2005,2008). Some of his projects are
actually impracticable and are therefore just rendered as drawings or prototyped
though 3D printing techniques.
Structural Reciprocity 19
Characteristics and Interesting Aspects
The term ‘reciprocity’ comes from the Latin reciprocus, a composite of recus,
backwards, and procus, forwards. The etymological significance of reciprocity is
therefore that of ‘back and forth’, evoking an exchange for mutual benefit. In the
world of construction, situations in which structural systems imply an exchange of
actions is frequent—the voussoirs of an arch, for instance, achieve equilibrium
through mutual action/reaction. However, reciprocity differs from mutuality
because a transitive relation between at least two elements occurs only if such a
relation is perfectly symmetric.
The concept of structural reciprocity therefore refers to a specific subset of
structures, which are characterized by two main properties.
First, in each and every element the functions of supporting and being supported
by other elements must be separated instead of being overlapped. For instance, in
simply supported beams constraints are placed at the extremities and loads act at the
midpoints—the functions correspond to different position on the element and no
inversion is possible. In contrast, constraints and loads both occur at the ends of
truss bars—so the functions overlap here and the definition of supporting and
supported ends becomes merely conventional. This first property also implies that
only beams and two-dimensional elements can form a reciprocal structure. The
forces are transferred through bending and shear with beams, both in-plane or out-
of-plane for 2D components.
Second, each and every element must be supported by the other one it supports.
As stated above, a perfectly symmetric relationship is needed to distinguish a
reciprocal structure from a simply mutual one.
Fig. 10 Casa Negre, by Josep Maria Jujol. One of the few European examples of a reciprocal structure
conceived for aesthetical purposes. Photo: Jaime Segura, reproduced by permission
20 A. Pugnale, M. Sassone
An example will be helpful here. Let us consider the beam arrangements shown in
Fig. 13. In Fig. 13a, each beam B is simultaneously supported by and supporting other
beams, while beam A directly takes the load and the beams C rest on the ground. This
system is not reciprocal because the supporters and those being supported are not the
same: beams B are supported by beams C, while they sustain, instead, beam A. In
Fig. 11 Chair for a Pedrini
collection, designed by Pino
Pizzigoni, 1948. Photo:
Pizzigoni Archive, PIZ N, card
3. Scanned by Carlo Deregibus
and Alberto Pugnale,
reproduced by permission
Fig. 12 Table prototype
designed by Pino Pizzigoni,
1948. Photo: Pizzigoni Archive,
PIZ N, card 4. Scanned by Carlo
Deregibus and Alberto Pugnale,
reproduced by permission
Structural Reciprocity 21
Fig. 13b, beam B is supported by beam A and supports beam C, while beam A is
supported by beam D, which is also supporting beam C. Such circularity allows us to
state that beam B is, at the same time, supported by and supporting beam A. This is also
visible from the actual arrangement, which does not organize the elements in a
sequence, but rather it forms a ‘loop’. In sequential systems, the positions and roles of
Fig. 13 The relationship between elements in mutual and reciprocal structures: aleft antisymmetric,
which can be found in the Sindone chapel by Guarini; bright symmetric, which corresponds to the
arrangement of the Puppet Theatre by Ishii (see Fig. 7). The drawings are schematic and not to scale.
Images: authors
Fig. 14 Circularity of force flows in a reciprocal beam. Image: authors
22 A. Pugnale, M. Sassone
the members are related, while in loops the concepts of beginning and end do not exist,
and the positions of the elements are totally interchangeable.
Force Flows
Loops are not only related to the geometrical configurations, but are also
intrinsically present in the equilibrated force flows. This can demonstrated in the
simplest reciprocal system, i.e., the beam made of two aligned elements as shown in
Fig. 14. Let us assume that the structure is subject to a load condition of two forces
directed downward. A circular flow can be verified by following how shear (S) and
axial (N) forces act in the different parts of the system.
The same circularity occurs in the configuration shown in Fig. 15 (or in
Fig. 13b). Let us assume a generic load condition applied downward. The four
vertical links only transfer compression, while the inner parts of the elements
transmit shear with the same sign. This produces a circular force flow, or loop.
Neither force values nor the application points affect the presence of a force loop
in any of the examples mentioned above. However, the distribution of external loads
is closely related to it. The scheme shown in Fig. 16 demonstrates that at least one
exception exists, and that exception implies an antisymmetric load condition.
Reciprocity vs. Hierarchy
According to the second property mentioned above, each and every element of a
reciprocal structure must be simultaneously supported by and supporting other
components. It must generate, with the repetition of such a scheme, a pattern in
which all the members of the system play the same role, without differences in
terms of structural behaviour. This situation is clearly in contrast to that of other
structures in which the elements have distinct functions depending on their
respective positions and relationships. The difference between reciprocity and
hierarchy is exemplified by the schemes shown in Fig. 13.
In reciprocal structures, an intuitive understanding of the structural behaviour is
intrinsically lost. This was already clear in the seventeenth century, when, for the first
time, John Wallis attempted to calculate reciprocal grillages by following the load
paths within the structure in order to evaluate the internal forces acting in each element
(Wallis 1695; Houlsby 2014). However, the choice of a specific path (i.e., the sequence
of elements to use to compute the internal reactions) was totally arbitrary. Recent
research by (Kohlhammer and Kotnik 2011) also seems to confirm this.
Reciprocal structures have an intrinsic generative property, which can be
considered more as a reproduction of elementary/minimum patterns, or fans, rather
than instantiations of higher level abstract schemes. The absence of hierarchy
affects the architectural as well as the structural design of such systems. In the
Vito Bertin proposes an alternative description of how structural reciprocity works mechanically. He
states that a reciprocal configuration should present three characteristics: (1) a set of levers that form a
static system, (2) a set of elements that mutually support each other and (3) self-connected components,
i.e. which behave at the same time as elements, nodes and connectors. Visit Vito Bertin’s website for
further details:
Structural Reciprocity 23
Western culture in particular, the presence of a hierarchical logic in composition has
always been an integral part of the building experience. Furthermore, many aspects
of the mechanical behaviour of a structure, such as redundancy and robustness, are
also related to hierarchy.
Form-Finding/Morphogenesis Techniques and Design Approaches
The form-finding of 2D and 3D configurations of reciprocal structures, as well as of
those based on regular polyhedra, can be performed through different computational
techniques. Illustrative case studies have been provided by Baverel (2004), Popovic
Larsen (2008) and Stacchetti (2005).
At present, three main families of such techniques can be identified. The first is
known as ‘iterative additions’. Starting from the basic configurations of three or four
elements, these techniques work by adding sets of new bars to the elementary frame
(Proll et al. 2010). The second is related to the use of optimization strategies, such as
genetic algorithms (Baverel et al. 2004) or relaxation (Douthe and Baverel 2009).
Such techniques find the frame geometry (the solution) by iteratively reducing a
measured value of the geometrical errors (the fitness function) from a set of non-
compatible configurations (the tentative solutions). The third models the behaviour
of kinematically undetermined configurations in order to define their final shapes.
However, in architecture, the use of physical models is probably still the most
diffused way of exploring reciprocal configurations, as designers can intrinsically
S>0 S>0
Fig. 15 Circularity of force
flows in a reciprocal grillage.
Image: authors
Fig. 16 Two directional force flows in a reciprocal beam under an antisymmetric load condition. Image:
24 A. Pugnale, M. Sassone
take advantage of the characteristics of reciprocity. With the aid of computation,
instead, a design process can only start from an over-arching reference geometry/
shape, a forced ‘top-down’ approach in which reciprocity does not inspire the
project but is just adapted to it.
In addition to these problems of structural and geometrical form-finding, a new set of
questions and fields of experimentation are emerging from the development of digital
fabrication techniques, mostly because the new way in which the components are
produced automatically requires a different approach to their design (Proll et al. 2010).
The Use of Timber: Members and Joints
In reciprocal systems made of elongated members, all the elements are beams
generally joined by simple overlapping. This naturally relates them to timber
construction, in which high flexural strength members are usually available, while
the realization of joints is a complex problem. Furthermore, working with raw
timbers without the need of iron keys or complex connections made reciprocal
construction fast and reliable, as shown by the application to military bridges, such
as the one described by Caesar (Gros and Beltramini 2003).
The idea of joining members by overlapping has also a conceptual relevance: the
overlaps do not require any specific device, such as bolts and pins which are made
of timber or steel, but they influence the shape and the organization of the whole
structure. The curvature of a reciprocal bridge made of raw timbers is ruled by the
size of the timbers themselves and by the position of overlaps. Hence, there is an
intrinsic relationship between members and joints (joints are in fact parts of the
members) and the design and detailing is an integral part of the whole conception. In
terms of mechanical behaviour, such as for the number of degrees of freedom, the
effect of friction and of local deformability, and displacement capability, joints
directly contribute to define the global behaviour of the structure.
Let us compare reciprocal structures in wood with the current industrial detailing
of timber structures, such as glulam. Glulam structures generally present connec-
tions based on the use of additional steel devices, plates, clamps or pins, which are
bolted to timber. Even though they are efficient in terms of constructability and
structural behaviour, steel joints do not appear to belong to the base material (see
Fig. 17). The conceptual and tectonic distance between the two materials is even
more emphasized when the timber members are replaced by composite or even
cardboard pipes, such as in the projects by Shigeru Ban (McQuaid 2003), and there
are no significant changes in the steel joint system (see Fig. 18).
Static and Kinematical Determination
According to the first property of reciprocity mentioned above, the supported
elements of reciprocal systems cannot come into contact with their supports at the
vertices. This allows the constraint between the supporting and supported members
to be defined in an extensive manner: sliding hinges and prismatic joints become
possible substitutes of rotation hinges, and this offers unique kinematic properties to
the obtained configurations, which are difficult to predict intuitively.
Structural Reciprocity 25
Preliminary research, carried out through the analysis of the kinematic matrix,
has shown how different constraint patterns can lead to unexpected kinematic
behaviour (Parigi et al. 2009; Parigi 2011; Parigi and Sassone 2011a). Recent
developments have proposed a kinetic system based on the concept of reciprocity,
which is called kinetic reciprocal system (KRS). Through a morphogenetic
procedure, suitable Kinetic Reciprocal Frames are generated with assigned overall
behaviour; the process starts from sets of kinematic parameters that create complex
Fig. 17 Scottish Parliament
designed by EMBT, detail of the
chamber roof structure. Photo:
Yeh-Lun Chou, reproduced by
Fig. 18 Paper tower at the London Design Festival 09, designed by Shigeru Ban. Photo: Tom Parkin,
reproduced by permission
26 A. Pugnale, M. Sassone
geometries of intersecting curves and lines, and which cannot be predicted directly
from the input data (Parigi 2011; Parigi and Sassone 2011b). The latest results of
such a research activity led to the development of a form-finding tool which is
called the ‘Reciprocalizer’ (Parigi and Kirkegaard 2014; Parigi et al. 2012,2014).
Other conceptually similar research is found in Goto et al. (2011) and Kidokoro and
Goto (2011).
Use of Planar Elements
When a reciprocal system is designed with elongated elements (i.e., with a set of
components that behave like beams) design efforts are mainly focused on three
aspects: (1) the definition of the elementary fans that have to be assembled, (2) the
study of their composition possibilities and (3) the selection of the jointing systems.
However, as suggested by the rare natural example of the cocolith, planar
components of different shapes can also be used to transfer forces in a reciprocal
way. The cocolith features circular tiles, but also squares, triangles and more
articulated, or irregular, geometries could also be considered.
In order to guide future morphological research activities, we have distinguished
five main categories of reciprocal configurations based on planar components. The
first type uses planar elements as ‘thick’ elongated elements. This category includes
all those reciprocal structures in which the planar elements are approached in the
same way as elongated ones. For instance, Werner Blaser designed tables and chairs
made of fans, in which planar timber panels were reciprocally interlocked. A few
buildings have also been designed with thick linear elements; the Serpentine Gallery
2005 by A
´lvaro Siza and Cecil Balmond is probably the most relevant of all. A
second category considers planar elements as groups of elongated elements. This
includes all those reciprocal structures in which the planar elements can be
substituted by fans made of sticks. A simple example is that of triangular tiles that
replace fans of three linear elements each. The tiles can have three different
engagement lengths, whereas only two are possible for elongated elements. A third
family includes all those configurations in which the planar elements are part of a
truss. A forth category groups the reciprocal systems in which the members can
transmit bending moment via the notches, but are assembled differently than in the
previous three categories. The fifth and last category collects all the possible
remaining configurations (Baverel and Pugnale 2012,2013).
The research and experiments on reciprocal structures currently underway are the
heirs of an ancient tradition in both the East and West. We hope that the attempt at
synthesis of the state of the art (definition, terminology, methodologies of form-
finding and analysis) will allow those engaged in such research to view and present
their efforts in terms of a broad context, overcoming the fragmentation that has
prevented reciprocal structures from receiving the attention they deserve, and
fostering a fruitful exchange of information, resources and results.
Structural Reciprocity 27
Appendix 1
See Table 1.
Table 1 List of patents involving the principle of structural reciprocity, arranged according to the release
Author Title Patent No. Date issued Country
Friedrich Zollinger Means of manufacturing space-
covering structures such as
roofs, frame walls, and the
US 1483037 A 5 February 1924
(filed in 1922)
Emilio Pe
Folding three-dimensional
reticular structure
CA 707069 6 April 1965
(no filing date)
Emilio Pe
Three dimensional reticular
3185164 25 May 1965
(filed in 1961)
Guy Monteilhet Nouvelle structure pour la
couverture des ba
FR 1560454 21 March 1969
(filed in 1968)
Erwin Walle,
Sigurd Prinz
Vorrichtung und Verfahren zur
Herstellung ebener
¨chentragwerke aus
vorgefertigten Einzelteilen
DE 2152580 A 3 May 1973
(filed in 1971)
Wilhelm Johannes
Tragwerk fu
¨user, Hallen
und a
¨hnliche Bauwerke
LU 72572 A1 9 May 1975
(filed in 1975)
Wilhelm Johannes
Draagconstructie voor huizen,
hallen, en dergelijke
NL 7507054 A 13 June 1975
(filed in 1975)
Erwin Walle,
Sigurd Prinz
Reinforced-concrete ceiling
carrier-grid beams as
prefabricated units hinged
together with zones covered
by prefabricated slabs
DE 2152580 B2 17 July 1975
(filed in 1971)
Wilhelm Johannes
Charpente pour maisons, halles
et e
´difices analogues
BE 835087 A1 16 February 1976
(filed in 1975)
Wilhelm Johannes
Perfeccionamientos relativos a
estructuras de soporte para
casas, cobertizos y edificios
ES 438831 A1 17 September
1976 (filed in
Wilhelm Johannes
Tragwerk fu
¨user, Hallen
und a
¨hnliche Bauwerke
DE 2514036 A1 30 September
1976 (filed in
Wilhelm Johannes
Charpente pour maisons, halles
et e
´difices analogues
FR 2306309 A1 29 October 1976
(filed in 1975)
Erwin Walle,
Sigurd Prinz
Suhlbetondecke mit als
Plattenbalken wirkenden
DE 2152580 C3 27 June 1977
(filed in 1971)
Johannus Maria
Een Geodetiese
NL 7603046 A 27 September
1977 (filed in
Wilhelm Johannes
Tragwerk fu
¨r Bauwerke,
insbesondere Ha
¨user oder
CH 594114 A5 30 December
1977 (filed in
28 A. Pugnale, M. Sassone
Appendix 2
See Table 2.
Table 1 continued
Author Title Patent No. Date issued Country
Melvin Crooks Building construction of
A-shaped elements
4182086 8 January 1980
(filed in 1978)
Daniel Gat
Research and
foundation Ltd)
A building Element for
construction of interlocking
IL 55404 A 27 February 1981
(filed in 1978)
Graham Brown Three-dimensional structures WO 89/08172 8 September
Graham Brown Three-dimensional structures GB 2235479 A 6 March 1991
(filed 1989)
The United
Graham Brown Three-dimensional structures CA 1320812 C 3 August 1993
(filed in 1989)
Christopher Allen,
Roderick Robbie
Stadium building US 5351449 A 4 October 1994
(filed in 1993)
Richard Dick
Geodesic structure US 7389612 B1 24 June 2008
(filed in 2001)
Timothy William
Gerald Baddeley
Improvements relating to roof
truss structures (A bracket for
connecting roof joists in a
reciprocal frame)
GB 2483263 A 7
March 2012
(filed in 2010)
The United
Table 2 List of terms used in scientific literature to indicate structural reciprocity or those somehow
related to such a principle, sorted according to publication date (historical books, treatises, projects and
patents are not included)
Term and related terms Variants and related
First use in
Lamella beams 1D circular arrays of
lamella beams,
lamella beams in a
circular pattern
Dean et al.
In 1997, Dean (1997)
published an article on
lamella beams arranged
in circular patterns
explicitly referring to the
RF studied in
Nottingham by Chilton
et al.
Selfsupporting interlocking grids
for multiple applications
interlocking grids
Gat at (1978) Coined by Daniel Gat at
the suggestion of A. Ben-
Arroyo, his colleague at
Technion, Israel Institute
of Technology
Structural Reciprocity 29
Table 2 continued
Term and related terms Variants and related
First use in
Mandala-Dach Mandala roof Chilton and
Probably coined by
Graham Brown, together
with a German
colleague, and first cited
by Chilton and Choo in
1992. Later cited by
Olga Popovic Larsen and
Roland Schumacher
Reciprocal frame (RF) Reciprocal structure,
reciprocal system
Chilton and
Coined by Graham Brown,
even though the term is
not used in the
descriptive text of his
Serlio floor Serlio’s device,
Serlio’s prototype
Used only by David
Yeomans in his article
Multi-reciprocal grid (MRG) Saidani et al.
Proposed by John Chilton
to Olivier Baverel when
he showed him his early
research on reciprocal
Nexorade Nexor, fan Baverel
et al. (2000)
Coined by Hoshyar
Nooshin to identify the
structures studied by
Olivier Baverel after he
moved to the University
of Surrey
Leverworks Mutually supported
stick structures,
lever beam
Bertin and
Coined by Vito Bertin to
explain his research on
reciprocal systems. Such
a name emphasizes the
lever principle involved
in structural reciprocity
Multi-reciprocal element (MRE) Rizzuto et al.
Used by Joe Rizzuto to
differentiate his research
from that on MRG
published by Messaoud
Saidani and colleagues
Mutually supported element
supported elements
Rizzuto and
Used by Joe Rizzuto in an
attempt to generalize the
principle of reciprocity
Leonardo’s lattice Leonardo grid Roelofs
First coined by Rinus
Roelofs. Also Used by
Duvernoy (2008) and
other authors in the
special issue of the
Nexus Network Journal
on Leonardo’s reciprocal
Selfsupporting framework Proll et al.
Only used by Mischa Proll
and Andreas Gu
¨nther in
their thesis and related
30 A. Pugnale, M. Sassone
Appendix 3
See Table 3.
Table 3 The main publications on reciprocity, classified according to the topic
Topic Publication
Morphology and geometry Baverel and Saidani (1999), Baverel et al. (2000,2006), Baverel
and Popovic (2011), Se
´chal et al. (2011)
Polyhedrical reciprocal structures Rizzuto et al. (2001b), Baverel (2007), Rizzuto and Hulse (2007),
Di Carlo (2008), Rizzuto (2009)
Form-finding and morphogenesis
with computational tools
Baverel et al. (2004), Douthe and Baverel (2009), Brocato and
Morandini (2010), Goto et al. (2011), Parigi and Kirkegaard
(2012), Parigi et al. (2012), Parigi et al. (2014), Peng et al.
Rapid prototyping Parigi and Kirkegaard (2014), Gu
¨nther and Proll (2013), Song-
Ching Tai (2012)
Joints and connections Rizzuto et al. (2001a), Rizzuto and Popovic Larsen (2010)
Structural behaviour Brocato (2011), Douthe and Baverel (2014), Gelez et al. (2011),
Kohlhammer and Kotnik (2011), Saidani and Rizzuto (2004),
´nchez and Escrig (2011)
Kinematic/adaptive reciprocal
Parigi et al. (2009), Sassone and Parigi (2010), Goto et al. (2011),
Kidokoro and Goto (2011), Parigi (2011), Parigi and Sassone
(2011a,b), Parigi and Kirkegaard (2012), Parigi et al. (2012)
Materials Pizzigoni (2009), Brocato and Morandini (2010)
Use of planar panels Blaser (1992), Baverel and Pugnale (2012,2013)
Table 2 continued
Term and related terms Variants and related
First use in
Lamella flock Tamke and
Used as the title of a
research project by
CITA, based in
Copenhagen, in which
free-form lamella
structures, inspired by
Zollinger’s work, have
been studied
Interlocking frame Song-Ching
Tai (2012)
Used by Alan Song-Ching
Tai to identify short
members that span
across large surfaces and
lock to each other with
simple notches.
Reciprocal Frames are
considered as a structural
sub-family of IFs, which
are not necessarily
Structural Reciprocity 31
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Goto (2011), Tho
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Pugnale and Parigi (2012)
History Popovic (1996), Di Carlo (2008), Duvernoy (2008), Popovic
Larsen (2008), Williams (2008), Pugnale et al. (2011), Houlsby
Art and sculpture Roelofs (2008)
Leonardo’s grids Popovic (1996), Sa
´nchez and Escrig (2006), Duvernoy (2008), Di
Carlo (2008), Popovic Larsen (2008), Roelofs (2005,2008),
Williams (2008), Sa
´nchez and Escrig (2011)
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Alberto Pugnale is a lecturer in Architectural design at the University of Melbourne, Australia. In 2007,
he won the fifth edition of the IASS HANGAI Prize, related to the study of complex architectural/
structural bodies. He has been a assistant professor at Aalborg University, Denmark (2010–2012), and an
invited lecturer in France and Italy. At present, he is member of the International Association for Shell
and Spatial Structures (IASS) and is a licensed architect in Europe. His research interests are in the
computational morphogenesis of free-form structures, reciprocal structures and history of construction.
Mario Sassone Ph.D. and Assistant Professor at Politecnico di Torino, is member of the 209 Committee
of American Concrete Institute and of the International Association of Spatial and Shell Structures
(IASS). His research field mainly concerns the computational analysis of time dependent effects
behaviour in concrete and steel concrete composite structures, and the structural and architectural
optimization of RC shells by means of artificial intelligence techniques. He is currently working on
bridging the gap between architectural issues and computational approaches to structural and physical
problems. Several members of his research group have been awarded with the IASS HANGAI prize.
Structural Reciprocity 35
... A regular pattern is formed with these short beams (Karakoç, 2017). The construction principle of reciprocal structures is that none of the load-bearing beams support each other from the end point and the load is transferred in a hierarchical order (Pugnale and Sassone, 2014). Reciprocal structures, which are formed by overlapping many structural elements, have several advantages over other structural systems used to pass large-spans (Başoğul, 2021). ...
... There are many examples of the historical development of reciprocal systems from ancient times to the Middle Ages and from the Middle Ages to the present. In Leonardo da Vinci's (1452-1519) "Codex Atlanticus," there are sketches for clarity with regular and irregular geometric configurations (Pugnale and Sassone, 2014). In addition to these, thinkers such as Villard de Honnecourt and Sebastiano Serlio have studied the planar elements of these systems (Pizzigoni, 2010). ...
... At least 2 elements are required in the formation of the system. It is important that each supported structural member never comes into contact with the corner points along the span (Pugnale and Sassone, 2014). In this system, which is created by transferring loads to each other and overlapping each other, interlocking systems can also be constructed by opening notches on the beams to create a planar pattern. ...
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Interior Architecture education is an educational discipline in which the experience-based learning that takes place in studios is at the center of the design concept. In this paper, it is planned to experience the basic design principles and elements that are required to be understood in the basic design course within the interior architecture education process, and to contribute to awakening and stimulating the minds of the students for various inventions thorugh raising awareness of nature-based learning. As a result of the pilot study designed for this purpose, an application-based flowchart is presented as a model for designing new patterns for basic design courses. In the creation of the model, the concept of pattern is researched in different disciplines within a broad framework in line with the principles of consistency and inclusiveness. The stages of creating the model include the stages of analyzing the pattern systematics based on abstraction in interior architecture basic education and reconstructing the systematic in a way that will increase the capacity to be used in the creation of new patterns. In line with these stages, it is aimed to define basic design elements such as point, line, plane, volume and basic design principles such as contrast, emphasis, repetition, balance, and rhythm in the formation principles of natural elements, and to systematize the same principles strategically to be used in the shaping of design patterns.
... Due to low strength and low stiffness of members, the span of the structure is small. The members of single-layer reciprocal frame are mainly subjected to bending moment, but the bending stiffness of the member is low, which is the main reason for low stiffness of the single-layer reciprocal frame [16][17][18]. As shown in Figure 6, by transforming the members to lattice-type members, the transformation of the single-layer reciprocal frame to the double-layer reciprocal frame is realized, and the stiffness and strength of the structure have been improved. ...
... As shown in Figure 6, by transforming the members to lattice-type members, the transformation of the single-layer reciprocal frame to the double-layer reciprocal frame is realized, and the stiffness and strength of the structure have been improved. The double-layer reciprocal frame retains the features and advantages of the reciprocal frame and the strength and stiffness have been improved [17][18][19]. ...
... No.[16][17][18][19][20] Numerical model of rotation angleFig.21. No.[16][17][18][19][20] ...
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A kind of lattice-type steel member is presented and a kind of prefabricated connector suitable for the connection between lattice-type steel members are proposed. The mechanical properties of the connectors are analyzed by using the finite element numerical simulation software ABAQUS. The connectors meet the design goals of the stiffness of connection stronger than members. Parameterized analysis is carried out on the prefabricated connector, and the flexural stiffness expression of the connector is obtained. The suggested values of each component of the prefabricated connectors are given based on the size of connected lattice-type steel members.
... The terms "reciprocal networks"/"reciprocal frames"/"reciprocal structures" are now widely used to designate structures created by arranging short linear elements around a pattern so that they mutually support each other and cover a larger span. The pattern forms a stable geometric configuration in which the elements share and transmit the load to the supports [32][33][34]. ...
... In the 20th century, in 1924, Zollinger [33] filed his patent that adapted the orthogonal mesh (the most frequent solution in documented historical cases) by proposing a rhomboidal and inclined mesh for the construction of steeply pitched gable roofs and for shaping a roof profile in the form of a pointed arch. This introduced a novelty to the resolution of the joint [61], displacing the elements that reach the main one, thereby preventing them from coinciding at the same point, which favors the construction of the structural system (Figure 4e-g). ...
... The constructive definition at this point is so important that, as Pugnale [33] states, in reciprocal frame structures, the joints contribute directly to defining the structural behavior of the mesh. ...
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The application of the systemic approach in architecture aims to promote an integral, holistic view of the architectural design process. The literature reviewed calls for models with systemic behavior, and for these models to be applied in concrete cases. This paper proposes an original approach, using the foundation matrix and the constructive logic matrix. Both matrices are part of a developing model that is being tested on a case study. The work presented here had two objectives: to check this part of the model and gain more knowledge about the model itself. The selected case study, the 2005 Serpentine Gallery Pavilion, is a contemporary ephemeral construction of significant architectural interest. It is a reciprocal frame structure, linked to the construction history. The methodology used was a systemic analysis. In the first phase of the analysis, the reciprocal structures documented historically in the West were reviewed. The other two phases corresponded to the application of the two model matrices. Conceptual diagramming was used in all phases of the process. The results show the importance of the study of historical building solutions. The use of matrices facilitates the identification and understanding of the operations carried out in the design process of the case study. Matrices favor the organization of concepts and relationships from through a systemic approach. Understanding generation operations in an integrated way leads to a type of knowledge (relational knowledge) that allows architecture to be thought about in a holistic way. This makes the systemic view of art and technology as a unit possible, attending to the whole complexity of architectural thinking.
... Indeed, Leonardo da Vinci made several sketches of floors and roofs to cover surfaces and even as bridges to pass over linear obstacles such as rivers, roads, etc. (9), Figure 1. An in-depth review of the scientific literature on reciprocal structures may be found in previous work carried out by other researchers (10,11). Although this is not a very common structural typology, reciprocal structures fell into total disuse after the second half of the 18 th century because the industrial revolution brought about the development of new structural materials and the production of longer elements able to cover long spans. ...
... Nevertheless, in a recipro-cal structure each element in it has to support all of the others to a greater or lesser degree, in a non-intuitive pattern of load transmissions between them. The principle of reciprocity is defined as the use of supporting elements which, resting on each other, form a spatial configuration without any clear structural hierarchy (10). ...
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The elements in conventional structures are perfectly ranked, so that load transmission is logical and follows the usual structural orders. Nevertheless, in reciprocal structures each element has to support all of the others in a less intuitive pattern of load transmission. The purpose of this paper is to understand exactly how load is transmitted between elements, quantifying this analytically by developing a new method which is applicable to a flat structure composed of a basic unit with any number of nexors. It is based on determining the increase in load to which the members in a reciprocal structure are subjected by calculating the coefficient k, or “transference coefficient”. The k coefficient value, and therefore the load transferred between members, falls with the number of nexors, with the proximity of point loads to exterior supports, and with the size of the central space in the structure.
... A planar RF system can consist of multiple RF units with various geometries and patterns. As described by Pugnale and Sassone [7], in an RF system, each element has to be mutually supported by and supporting other elements. This reciprocity shows the main difference with a hierarchical system of a traditional frame structure. ...
... Step 3 Final slide-in (one direction) 7 ...
Conference Paper
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In this paper, we present a preliminary investigation of planar rectangular reciprocal frame (RF) structures considering structural, architectural, environmental, fabrication and assembly aspects. Following a timber-only concept and a low-tech design philosophy, we specifically propose to use salvaged timber and wooden nails only, as materials for fabricating and connecting RF structures, which is in line with an ongoing research project that has initially investigated the characteristics of salvage timber and the structural behavior of wooden nail connection. The present experimental investigation focuses on the single multi-layered element in a basic layout of a four-element planar rectangular RF unit. The single elements are fabricated out of small timber boards to create an opening that serves as a slide-in connection for the ease of assembly. We experimentally investigated the structural behavior of the single element in the proposed unit under bending loads. We also explored the assembly process and the resulting patterns by using the proposed approach to build RF structures. Moreover, to showcase the structural and architectural features of the proposed RF unit, a design case was physically realized.
... Coming from the interactions between the architects and the engineer, the structural concept behind the Serpentine Gallery Pavilion 2005 relied on the reciprocity principle (Sakamoto and Ferré 2008). As explained by Pugnale and Sassone (2014), "the principle of reciprocity is based on the use of load-bearing elements which, supporting one another along their spans and never at the extremities, compose a spatial configuration with no clear structural hierarchy". Led by Cecil Balmond in London, the AGU team applied that principle to the lamella system to achieve the necessary structural stability of the complex form of the pavilion (Figure 4). ...
... With the origin from both Leonardo da Vinci's sketch, Codex Atlanticus, and ancient Chinese bridge construction as depicted in water color painting, Along the River During the Qingming Festival, the structural morphology of RF as part of a free-form surface has been studied widely in the past decades [1]. Larsen et al. documented the design and construction of a full scale timber exhibition hall with beams polar arrayed around an oculus [8]. ...
Conference Paper
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With the advent of polyhedral 3D graphic statics, form-finding processes for multi-layered funicular shells can be attained with considerable ease by leveraging polyhedral reciprocal diagrams. Yet, the resulting space-frame structure consists of multiple struts converging at a single point, and demands a high degree of customization for the connecting node. This poses design, fabrication, and assembly challenges. Reciprocal frame (RF) [1] offers an alternative connection logic with added structural stiffness and aesthetic benefits. We present the design process of a timber frame prototype-a double layered, RF, compression-dominant, funicular shell. The paper, for the first time to our knowledge, applies the polyhedral form-finding method for a compression-only turned compression-dominant shell with RF. The prototype is designed with PolyFrame for Rhinoceros, fabricated by a 3-axis CNC machine and assembled with a 1-axis stacking logic. The optimal performative relationship between structural load, strut geometry, and principle axial stress of multiple wood species are investigated for the final prototype's material specification. The known complex RF geometric vs structural challenges found in free-form RF are addressed in relation to the funicular shell geometry given the varying strut lengths per node. The structural performance of the RF compressional-dominant shell prototype is numerically tested and compared with a conventional space-framed, compression-only shell of the same form. The workflow presented shows potential for application in multi-layered, structurally efficient, spatial structures with simplified fabrication and assembly process with locally sourced timber materials.
... (Nan & Wenfeng, 2008) Reciprocal structure is one kind of three-dimensional structure, which is made of three or more sloping beams that support each other along their span. (Pugnale & Sassone, 2014) Reciprocal structure has beautiful mesh patterns which is loved by the architect. This kind of connection between two members avoids the intersection among too many members that intersect to a single point and also make it easy to connect two members together, which reduces the connection complexity. ...
Conference Paper
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Bamboo is a renewable material with abundant resources and short growth cycle. Reciprocal structure is one kind of three-dimensional structure, which is made of three or more sloping beams that support each other along their span. The new structure system, reciprocal bamboo structure with bolt-steel pipe joints, can bring together the advantages of bamboo material and the reciprocal frame. With the new joints that consists of short steel pipes, bolts, nuts, steel strips and steel plates, the connection in the structure system can be assembled in a convenient and efficient way. This project is based on the experiment of bamboo reciprocal structure designed by team of professor Rodolfo Lorenzo. The research object is the structure consists of three bamboo poles and three bolt-steel pipe joints. Based on the test specimen, the finite element model is built to analyze the relationship between the parameters and the displacement. Studying the finite element simulation process of this experiment can contribute to the effective prediction of the deformation characteristics of this new structure.
... The reciprocal frame structure was rediscovered and renamed by Graham Brown in 1987. Various studies have been conducted on such structures over the last three decades (Larsen 1996(Larsen , 2008(Larsen , 2009Douthe and Baverel 2009;Pizzigoni 2010;Larsen and Lee 2013;Pugnale and Sassone 2014;Anastas et al. 2016;Brancart et al. 2017;Gherardini and Leali 2017;Torghabehi 2018;Ramos-Jaime and Sánchez-Sánchez 2020;Pérez-Valcárcel et al. 2021;Hughes et al. 2021), and different terms have been used for this type of structure, such as mutually supporting beams [G. Brown, "Three-dimensional structures," UK Patent No. GB2235479B (1989)], nexorades (Baverel 2000), lever-beam structures (Bertin 2002), and mutually supported element systems (Rizzuto 2007). ...
Consisting of linear elements, reciprocal frame structures are three-dimensional (3D) self-supported structures that can be rapidly assembled. This feature renders them suitable for both temporary and permanent uses. Considered as a type of deployable frame, such structures can be used in architectural and engineering applications. As a subset of reciprocal frame structures, rotegrity structures can be advantageous, owing to their structural form. In this work, a spherical rotegrity structure is constructed on a geodesic sphere through its transformation to a reciprocal structure with mutually supporting elements. Instead of the bolted connections commonly used in practice, a notched connection is proposed to facilitate the construction process of the self-supporting rotegrity structure. The parametric modeling of the rotegrity structure is created in Grasshopper, and two prototypes, having circular and rectangular cross sections, are built. In each prototype, two sets of elements, called Type A and Type B, are used, the linear member lengths and cross sections of which are identical. The cross sections of the members of the two prototypes are different. To assemble the prototypes, first, a number of elements are fabricated through 3D printing and tested in terms of their self-supporting capabilities and connections. Then, stainless steel pipes and wooden bars are used for Prototypes I and II, respectively. It is found that Prototype I, composed of circular hollow section profiles, is unstable, owing to the rotation of elements and because it requires the use of stabilizers. Conversely, Prototype II, which consists of rectangular cross-section elements, becomes stable when the last element is installed in the right place and did not require any additional attachment.
Conference Paper
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The paper discusses some historical precedents for the Reciprocal Frame, which is a patented 3-dimensional beam grillage structural system. Realization of some environmentally-friendly, low-energy timber buildings, constructed in the UK. and using the Reciprocal Frame as the primary roof structure, is described. Small construction projects, such as garden pavilions and gazebos and a single-person dwelling are reported. A timber-framed house, decagonal in plan, built in 1993, a group of three timber-framed buildings, each octagonal in plan, and a decagonal house 13 metres in diameter are also described. Finally, some possibilities for the future development of Reciprocal Frame structures are outlined.
Conference Paper
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This paper is an introduction to the morphology of structures using the Reciprocal Frame (also known as Mandala Dach in Germany) which is a patented three-dimensional beam grillage structural system currently used primarily in roof construction. The principle of the Reciprocal Frame (RF) system is described and some of the varied geometries that are attainable using the principle are explored. Finally, some alternative methods of cladding the basic configuration are proposed and some architectural possibilities are discussed.
Conference Paper
ABSTRACT: This paper describes a structural system, known as the 'Reciprocal Frame' (RF). The planar and 3-dimensional geometry of RFs, which is described in the paper, is related to the physical parameters of the structure (the number of beams, the slope of the beams, the sizes of the inner and outer polygons that define the space filled by the structure, and the vertical spacing of the beams at their intersection points). Graphs are presented to show the inter-relationship between these parameters and the consequence of these variations on architectural design and construction are discussed. The possibility of creating irregular forms of RFs is demonstrated as is the possibility of forming assemblies of RFs constructed from basic units. Methods of covering buildings with RF roofs, such as panels between the beams, conical roof surfaces, hyperbolic paraboloid surfaces or membranes are described. Finally, considering the similarity of the structure, in plan, to the iris of the camera shutter, the possibility of creating kinetic (retractable) RFs outlined.
Conference Paper
A closed circuit of mutually supported beams produces a three-dimensional space structure. This type of structure is known as a Reciprocal Frame (RF). RFs can be connected together to form larger space structures. These larger space structures have recently become known as Multi-Reciprocal Girds or Nexorades. Mutually supported element (MSE) structures describe a family of space structures therefore where the main structural elements support, and are in turn supported by one another. Generally it is only required to connect each of two adjacent elements together in turn to complete a closed RF circuit. The connections between beam elements can be very simple with potentially low overall fabrication costs. This provides potential economic advantages over traditional space structure connection systems. If feasible, it is generally good practice to rationalise the number of section types used in a space structure. The benefits include repetition of details and dimensions, together with design and fabrication detailing effort. Also, the interchangeability of elements used in MSE space structure systems should enable easier production and erection. The aim of this paper is to explore space structure systems composed of mutually supported cylindrical elements and discuss the potential of their future application. Keywords: Reciprocal Frames, Mutually Supported Elements, Space Structures.
Conference Paper
The 'Reciprocal Frame', recently patented, is a three-dimensional beaam grillage structural system used primarily in roof construction. The beams in the grillage both support and are supported reciprocally by each other. The plan view of the beams is similar in appearance to the lines forming the iris of a camera shutter. Its versatility in form and consistency in strength makes it a competitive design for sports arena and stadia, Structural design, drive device, roof operation and covering are discussed.