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The principle of structural reciprocity: history, properties and design issues

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Abstract

This paper deals with the principle of structural reciprocity, considering its origins in both Occidental and Orient culture and aiming to highlight the definition, main characteristics and interesting aspects of such concept referring to its application to the world of construction. Issues spanning from history, form-finding and morphology, structural behaviour and construction techniques are discussed in the paper, which should be considered as a starting point to stimulate future research and design directions/approaches.
The principle of structural reciprocity: history, properties and design issues
Alberto PUGNALE
Assistant Professor, PhD
Aalborg University
Aalborg, Denmark
apu@civil.aau.dk
Dario PARIGI
Assistant Professor, PhD
Aalborg University
Aalborg, Denmark
dp@civil.aau.dk
Poul H. KIRKEGAARD
Professor, PhD
Aalborg University
Aalborg, Denmark
p
hk@civil.aau.d
k
Mario SASSONE
Assistant Professor, PhD
Politecnico di Torino
Torino, Italy
mario.sassone@polito.it
Summary
This paper deals with the principle of structural reciprocity, considering its origins in both
Occidental and Orient culture and aiming to highlight the definition, main peculiarities and
interesting aspects of such concept referring to its application to the world of construction. Issues
spanning from history, form-finding and morphology, structural behaviour and construction
techniques are discussed in the paper, which should be considered as a starting point to stimulate
future research and design directions/approaches.
Keywords: reciprocity, spatial structures, timber constructions, form-finding, morphogenesis,
history of construction.
1. Introduction
The principle of reciprocity in structural design and construction, i.e. the use of load bearing
elements to compose a spatial configuration wherein they are mutually supported one another, has
been known since the antiquity. Evidence of its application can be initially found in Neolithic pit
dwellings, Eskimo tents and Indian tepees, as documented by Popovic Larsen [1][2], as well as
during the Roman Empire, when Julius Caesar used it for the construction of a bridge on the Rhine,
which was made of interlocked timber elements (described in his Commentaries on Gallic and Civil
Wars) with the purpose of simplifying the joints among elements, as also reconstructed and drawn
by Palladio [3]. However, apart from these ancient applications, independent one another and
developed worldwide, the principle of reciprocity has been then autonomously studied and used, on
the basis of different needs and purposes, in both Occidental and Oriental culture for centuries, until
it became a research topic for academics. During the last decades, they started a new set of works
related to structural, geometrical and constructive issues of 'reciprocal' structures.
In this framework, the present paper aims to present a historical-critical overview on the principle
of reciprocity in the world of construction, highlighting its origins in different cultures with
differences and similarities, involved research fields and architectural design issues.
2. Different roots and curious similarities of a fragmented history
2.1 Reciprocity in the Occidental culture: timber and short-beams
In Europe, the concept of spanning distances longer than the length of the available timber beams
was the main reason for the use and development of the principle of reciprocity until the last
century. In spite of the fact that there was other techniques to deal with large span issues and short-
beams, such as that used in the roof structure of the 'Square Hall' building in Old Nisa in
Mesopotamia, dated around 2000BC and described by Pizzetti and Zorgno [4] (after also used by
Guarini for the design of his domes), different architects, scientists, mathematicians and
constructors reasoned about this problem without relations and documented continuity on one
another. Villard de Honnecourt drew in his sketchbooks some grillage assemblies for the
construction of floors inspired to the principle of reciprocity between 1225 and 1250 [5], then
Alexander designed and built the Lincoln cathedral using reciprocal supports between 1220 and
1235, as reported by Hewett [6]. Leonardo Da Vinci explored at least five different spatial
configurations based on the principle of reciprocity, as shown his sketches inside his "Codex
Atlanticus" [7] [8], practically spanning through the main regular and non-regular 2D and 3D
geometrical configurations. Furthermore, Sebastiano Serlio discussed the construction of planar
floors with short beams in his first book on architecture (in Italian), dated 1566 [9] [10], and John
Wallis wrote in his "Opera Mathematica" [11] about different kind of planar configurations of
structures with reciprocally supported elements, also providing examples of calculation worth to be
further investigated. Thus, Émy, who was professor of Fortification at the Royal Military School
Saint-Cyr, dedicated his efforts in developing a book about Carpentry, the "Traité de l'art de la
charpenterie" [12], in which some examples of reciprocal structures can be found, also for the
construction of the so-called 'flat-vaults' invented by Joseph Abeille and Sébastien Truchet in 1699
[13], and finally Rondelet [14] dealt with the principle of 'short beam' for the construction of floors
in his treatise dated 1810, in a similar way to what was presented in 1837 by Thomas Tredgold in
his "Elementary Principles of Carpentry" [15].
2.2 Reciprocity in Orient: interwoven structures and symbolism
In the Oriental culture, the interest in the use of reciprocal structures derived from different
inspiration concepts. On the one hand, especially in China, the use of interwoven strips of bamboo
for the construction of baskets is an old tradition that has been then transferred to objects of larger
scale, such as bridges of which the first example is the so-called 'Rainbow Bridge' in Shandong
[16][17][18]. On the other hand, the religious concept of Mandala and the symbolic shape of the
'magic circle' [19] has been used as a direct reference for the development and construction of
simple circular 'fans' made of reciprocally supported elements, used in many Buddhist temples all
around Asia. Unfortunately, written documents concerning that period can only be found for the
period after 1275, in which year Marco Polo arrived in China and therefore at the end of the Song
Dynasty. However, in the last decades different Asian architects, such as Kazuhiro Ishii, Yasufumi
Kijima and Yoichi Kan, took advantage of this principle for their constructions, as described by
Popovic Larsen [2][20] and Gutdeutsch [21].
2.3 The bases of a scientific approach
It should be underlined that Leonardo Da Vinci was the only one in Occident who studied
reciprocal structures in their potential of being elaborated as complex geometrical configurations,
also in the three-dimensional space [7]. John Wallis is also worth to be cited as the first author who
approached the principle of reciprocity with a modern scientific approach [11]. He considered a
precise and well-defined problem related to covering a long span with a flat grillage of reciprocal
short-beams, and he proposed a rational and logical calculation of the forces acting on each element
of the proposed configurations. Therefore, he dealt with the problem in mathematical and structural
terms, but also giving inspiration and focusing on practical aspects of construction of timber frames.
In summary, the principle of reciprocity has not had a linear history and the evidences of its
knowledge and use around the world seems to be largely unrelated to one another. However, a
common point in the development and application of the principle of reciprocity is the strict
relationship with the use of timber as constructive material, in both Occidental and Oriental culture,
but it was more a practical and constructive issue in Europe, aimed to the development of flat
configurations, while it was also a spiritual reason in Asia, with the realization of three-dimensional
structures recalling magical and religious symbolic shapes.
2.4 The years of patents and the development of research
In the last century, the principle of reciprocity continued to stimulate the interest of designers, and
for the first time it became a topic of interest in the field of scientific research. Ten different patents
based on the principle of MSE was registered between 1965 and 1992, starting with the foldable
structure by Piñero (filed in 1961 in USA, published in USA and Canada in 1965) presenting a
MSE spatial configuration of bars [22][23][24]. Thus, the reinforced concrete ceiling made of
prefabricated units hinged together by Walle and Prinz (filed in 1971 and published in 1975) [25];
the MSE geodesic construction proposed by Bijnen in 1977 [26]; the building element for
construction of interlocking grids by Daniel Gat in 1981 [27], also described in a scientific article
previously appeared in 1978 in "Architectural Science Review" [28]; the interwoven construction
developed by Melvin Crooks in 1978 (and patented in 1980) made of flexible so-called 'A'-shaped
elements, intended to be used for three-dimensional structures such as domes [29]; the three-
dimensional structure of reciprocally supported timber beams connected with notched joints
registered by Graham Brown in UK, Canada and as international patent between 1989 and 1993
[30][31][32], and finally the Rotegrity system, conceived as a interwoven structure made of flexible
elements based on the principle of reciprocity, developed and patented in 1992 by the Simple
Science Toys based in Eugene, Oregon, USA.
Focusing on scientific publications in the last decades, an increasing number of papers started to
appear in conference proceedings and journals. As already mentioned, the first was written by
Daniel Gat referring to his SIGMA system [28]. Thus, different researches based on the principle of
reciprocity have been developed and all of them are referring to the initial investigation by Chilton
et al., published in 1992, 1994 and 1995 [33][34][35][36][37]. These authors were the first using the
term 'Reciprocal Frame' in a scientific paper (a first introduction to terminological heterogeneity in
the scientific research on reciprocity is provided by Popovic Larsen [2]) and they also provided a
precise definition and description of the principle restricted to closed circuits of elements. In fact, in
that period John Chilton started a collaboration with Graham Brown at Nottingham University, who
recently patented a reciprocal three-dimensional structure, previously mentioned, and decided to
increase his level of understanding of its structural behaviour. At the same university, the first PhD
thesis on that topic was published. It was written by Olga Popovic and completed in 1996 [1].
Chilton was the advisor of the research involving historical references and architectures based on
the RF principle, also focusing on geometrical, structural and construction considerations.
The following papers deal with different aspects of structures designed on the principle of
reciprocity, and can be roughly classified in: (1) related to morphology and geometry of spatial
structures based on reciprocity [38][39], also in their polyhedrical configurations [40][41][42][43];
(2) dealing with form-finding and morphogenesis of such structures with computational tools
[44][45][46], recently also combined with the rapid prototyping of the timber elements[47]; (3)
concerning joints and connections of reciprocal structures [48][49]; (4) focusing on the structural
behaviour [50][51][46]; (5) investigating the kinematic potential in order to generate adaptive
structures/architectures [52][53][54][55][56] mainly inspired from Leonardo Da Vinci's work [57]
and on the Piñero's patents [22][23]; (6) working on material and sections for their realization [58],
(7) or presenting and discussing architectures, constructions and prototypes [1][2][18][59].
2.5 Reciprocity in contemporary art, architecture and industrial design
Considering realized architectures in the last century, some famous and unrelated examples
implementing the principle of reciprocity have been built in order to covering long spans, such as in
the Mill Creek Public Housing project by Kahn designed in 1952-53, in the Berlin Philarmonie by
Scharoun, built in 1960-63, and in a salt storage building in Lausanne by Atelier Gamme
Architecture, realized in 1989. Some structures have been also realized with the aim of giving shape
to spiritual philosophies, therefore involving symbolic meaning, such as in the Casa Negre by Josep
Maria Jojul (1915-26) and in all Graham Brown and previously cited Japanese constructions.
Also some experimental pavilions have been constructed, such as the Forest Park structure by
Shigeru Ban and ARUP AGU, or the two interesting experimentations uniquely designed by Cecil
Balmond and ARUP AGU - the H-edge pavilion realized together with the students at Penn
University and the Serpentine Gallery 2005, designed in collaboration with Siza. In both examples,
the principle of reciprocity has been explored in its architectural potential of being used with
different materials, element sections, joints and planar or three-dimensional configurations,
providing a fundamental evidence of much this pseudo-typology should be still explored by
architectural designers.
In the field of industrial design, Pino Pizzigoni realized during the Fifties timber and marble chairs
and tables made of reciprocally supported elements, called by the architect as structures with
'interlocked members' emphasizing the attention on the joints (See Pizzigoni Archive: PIZ N, 1948)
[60]. Finally, the sculptors George Hart and Rinus Roefols [61] have been interested in the principle
of reciprocity, using it for the conception of some of their geodesic domes and complex three
dimensional sculptures.
3. Peculiarities and interesting aspects of reciprocity
The term reciprocity derives from the Latin 'reciprocus' which is composed by the two parts 'recus',
meaning backwards, and 'procus', translated as forwards. The etymological significance of
reciprocity is therefore 'moving backwards and forwards', implying the practice of exchanging
things with others for mutual benefits. Such definition stresses the obliged, forced return of a certain
action, slightly differing from 'mutuality', which refers to a voluntary exchange, or 'alternative',
which focuses more on the time aspect of acting at regular intervals.
In the world of construction, the application of the principle of reciprocity requires:
- the presence of at least two elements allowing the generation of a certain forced interaction;
- that each element of the composition must support and be supported by another one;
- that every supported element must meet its support along the span and never in the vertices (in
order to avoid the generation of a space grid with pin-joints).
A structure or spatial configuration respecting such conditions has some peculiar characteristics and
interesting aspects which have to be taken into account during design but also stimulate the
exploration of their expressive potential in architectural terms.
3.1 Reciprocity vs. hierarchy
According to the second requirement of a structure developed on the principle of reciprocity, each
element of the configuration must work, at the same time, as support and supporter of other
elements, generating, with the repetition of such scheme, a pattern wherein all the components of
the system play the same role, without differences in terms of structural hierarchy. This situation is
clearly opposite to that of other structures wherein the elements have distinct functions depending
on their respective positions and relations. For instance, a frame is 'read' as girders and columns, in
a truss the upper and lower chords transfer bending while the web elements transfer shear, even the
well-known theory of shells shaped as revolution surfaces refers to parallels and meridians to
interpret the tie effect of the formers on the latter, giving the possibility to the designer of 'reading'
the structural hierarchy of the composition, and thereby distinguishing between main and secondary
elements, vertical and lateral systems, bracing, slabs, etc. In structures based on the principle of
reciprocity the possibility of an intuitive understanding of the structural behaviour is intrinsically
lost. It was already clear in the past, when Wallis was studying reciprocal grillages [11], that
following the load paths inside the structure is a good starting point to evaluate the internal forces
acting in each element. However, it was equally clear that the path choice, i.e. the sequence of
elements to use for computing the internal reactions is totally arbitrary. Recently such issue has
been also investigated by Kohlhammer and Kotnik [51], confirming previous studies. Reciprocal
structures seem to have an intrinsic 'generative' property, being understandable more as a
reproduction of elementary/minimum patterns (fans) rather than instantiations of higher level
abstract schemes.
The prevalence of reciprocity on hierarchy has consequences both on the architectural conception
and structural design levels. The absence of a hierarchical logic in the composition of structural
elements is relatively far from the Occidental culture, in which from gothic cathedrals to
Calatrava’s bridges the discovery of load paths and the compression of the load bearing
mechanisms are an integral part of the architectural experience, and many aspects of the mechanical
behaviour, such as redundancy and robustness are related to this aspect, as explained in the paper by
Balfroid et al. [62].
3.2 Form-finding/morphogenesis techniques and design approaches
The definition of simple two- and three-dimensional configurations of reciprocal structures, as well
as of those based on regular polyhedra, can be found by means of different geometrical techniques,
as clearly described by Popovic Larsen [2], Baverel [44] and Stacchetti [63]. However, the use of
physical models is probably the most diffused tool to 'find' the form of reciprocal configuration due
to its more direct interaction with the designer who can better see the assembly of elements in space.
However, the adaptation of the principle of reciprocity to more general reference shapes requires
the use of some computational aid in order to predict the geometrical configuration of the frame.
Furthermore, another degree of difficulty in the description of its final geometry could be given by
the use of non-regular frames, characterized by different unit cells in shape and dimensions, or non-
regular frames with non-regular topology, that also implies a variation in the dimensions of bars and
position of joints.
At present, three main form-finding techniques has been studied in order to deal with these
problems. The first is known as Iterative Additions and is based on the addition of set of bars to the
frame, starting from the basic configurations of three or four elements [47]; the second is related to
the use of optimization strategies, such as Genetic Algorithms [44] or Relaxation [45], which
iteratively find the spatial definition of the frame (the solution) by reducing a measure of
geometrical errors (fitness function) from a non-compatible configurations (tentative individuals,
population); the third technique implies the modelling of the behaviour of kinematically
undetermined configurations in order to define their final shape.
It should be underlined that the definition of reciprocal spatial configurations on the basis of
physical models is always based on taking advantage of the main characteristics of reciprocity,
allowing the designer to follow a so-called 'bottom-up' approach bringing to spatial complexity
from the composition and modification/deformation of elementary/minimum configurations. On the
contrary, the use of computational techniques can only start from an over-imposed reference
geometry/shape, in a forced 'top-down' approach in which the final configuration is not a result of
the peculiar properties of reciprocity and could then, from the ontological point of view, by another
structural typology such as grid-shell. However, such discussion about design approaches should be
developed in a more general way, comparing for instance the design of reciprocal configurations
with that of more consolidated typologies such as grid-shells with planar quadrilateral elements.
Also in that case, the use of geometric rules of translation and scaling of curves, defined by Schaich
and Shober, could be considered as a bottom-up approach, while the 'a priori' definition of a
reference surface, and then the application of an optimization procedure, is always a top-down
design process. In addition to this geometrical problems of form-finding of reciprocal structures, a
new set of questions and potential experimentation fields are arising from the development of
digital fabrication techniques and CNC machines, which require a completely different approach to
the design of the frame elements since the new way of component production [47].
3.3 The use of timber: connections and joints
A structural system where all the elements work as beams subjected to shear and bending is
naturally related with timber constructions and technologies, which have been always preferred, in
the Occidental tradition, for the application in works where the functionality is the prevailing aspect,
such as simple bridges, cranes and machinery, slabs and roofs, instead of masonry, which has been
mainly used, on the contrary as reference material for significant buildings and infrastructures. In
this context, the principle of reciprocity has been mainly conceived and used to solve practical
construction problems, and by means of that, for instance, simple bridges can be rapidly built
without the need of iron keys or complex joints. In all cases, the superposition joint, with or without
a friction component, is an essential part of the timber system, and its design can be refined, as
when a small over-thickness is suggested to correct slab deflections with a suitable camber [11].
The strict relationship between timber elements and reciprocal joints deserves to be compared with
current glulam construction, where joints are frequently realized with steel plates, clamps, or pins,
connected to timber elements by bolts or needles. Such kind of constructions, efficient form the
point of view of constructability and structural behaviour, keep the joints separated from the base
material, suggesting that it could be replaced by other materials, as for instance the architect
Shigeru Ban does by using cardboard pipes. In reciprocal structures, connections are parts of the
members themselves and the design and detailing cannot be separated by the whole conception.
However, the mechanical behaviour of joints, in terms of degrees of freedom, presence of friction,
local deformability, displacement capability, etc, directly contribute to define the global behaviour
of the structure [62].
3.4 Static and kinematical determination
According to the previously stated third requirement of reciprocity, in this kind of structures
supported elements cannot meet their supports in the vertices. Such characteristic allows to define
the constraint between supporting and supported members in a wider way, considering sliding
hinges and prismatic joints as possible substitutes of rotation hinges, and offering to configurations
obtained with such a modelling of constraints unique kinematic properties which are difficult to
intuitively predict. A preliminary research, carried through the analysis of the kinematic matrix,
showed how different constraint patterns can lead to unexpected kinematic behaviour, which is
worth of interest in the context of this kind of structures [52]. More recent developments [54][55]
propose a kinetic system based on the concept of reciprocity called Kinetic Reciprocal System
(KRS), i.e. a specific morphogenetic process, aimed to the generation of suitable KRFs with
assigned overall behaviour starting from sets of kinematic parameters, that involve the creation of
complex geometries of intersecting curves and lines which cannot be predicted directly from the
input parameters.
4. Conclusions and further developments
The principle of reciprocity has been studied and applied worldwide, starting from different
inspiration concepts and intended to achieve distinct purposes. However, some relevant historical
references still need to be further investigated for their scientific approach to the topic, such as the
"Opera Mathematica" by Wallis [11], and a set of architectural issues characterizing this principle
can offer new ideas for future research projects, such as the question of connections and joints in
timber reciprocal configurations, which stimulate the use of the material as a 'whole' in the
construction, in terms of defining elements and connections, and the possibility of studying
reciprocal structures as assemblies of elements with a high interest kinematic potential.
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... This principle of reciprocity in the design and construction of structures has been known since antiquity [32,35]. It is included in the main construction treaties. ...
... Volume terzo" (1878). Since the beginning of the 20th century, until the 1950s, reciprocal structures have been mentioned in all Italian building treatises and manuals [35,36]. ...
... Pugnale took a historical and critical look at the principle of reciprocity in construction [35]. They reviewed the historical background of reciprocal structures in Western and Eastern culture. ...
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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.
... The limited length of trunks is relevant in the consideration of reciprocal structures [7] (Figs. 7 and 8), also called nexorades [8]. These structures with members mutually supporting one another are of high architectural value and great antiquity. ...
... Each beam supports neighboring beams and it itself is supported by neighboring beams [11], which creates non-hierarchical organization of beams that complicates the design of reciprocal structures. In other words, the position of each beam determines, and is at the same time determined not only by the position of the immediately adjacent beams, but also by all the beams of the structure [7]. The result of this specific organization is that the shape of a reciprocal structure is neither easy to control or predict [12]. ...
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To decrease the impact of the construction sector on climate change, sufficiently economic processes must urgently be developed to encourage the use of minorly transformed timber, such as tree trunks. However, the question remains how to inexpensively incorporate tree trunks with irregular morphology into structural systems from design to fabrication. It is only within recent years that robotic tools and parametric design software have made it possible to both quickly and affordably build with elements of geometry as variable as tree trunks. The objective of this article is to describe an automatic process, from parametric design to robotic fabrication, that makes it possible to build a full-scale structure from reshaped tree trunks. This research brings to light the scientific and technical challenges that must be undertaken before industrialization can be realized, therefore the geometric parametrization required of trunk-to-trunk connections is detailed herein. Finally, the authors describe a case study from design to fabrication of a full-scale 10 m × 5 m reciprocal structure prototype made from fourteen tree trunk elements fabricated as described above using robotic arms. It is seen that reciprocal structures can span large distances with logs of limited length, but their structural complexity brings additional challenges to the design and construction processes. This study highlights not only the need for future development to accelerate the processes of automatic scanning and cutting of timber logs, but also for mechanical characterization of logs using non-destructive methods and better prediction of transmission of stresses in wood-wood assemblies.
... The geometry of reciprocal frames mainly depends on the eccentricity at the connections, generally governed by the thickness and superposition of the beams, and the grid topology [13]. While this does provide a variety of design parameters to control the structural grid geometry, the strong interdependency between these control parameters complicates the design and modelling process [14]. ...
... Moreover, the flexibility and curvature of the components allows tangential alignment at the connections. This reduces the complexity at the nodes and facilitates the development of very lowtech structural systems, like CODA's Panikkar, structures by Hiroshi Murata and Alison Grace Martin, or other developments based on the concept of rotegrity [13]. ...
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While transformable structures allow rapid assembly, reuse and reconfiguration, their technical complexity and related production or cost issues often hinder architectural application. Yet, vernacular or nomadic structures, like teepees or yurts, show how reversibility and transportability can be achieved with low-tech structural systems. The project presented in this paper is part of a broader research that investigates how material understanding and manipulation through elastic deformation can contribute to the development of low-tech and rapidly assembled kit-of-parts structures. More specifically, the ReciPlyDome project combines the concepts of bending-active and reciprocal structures. The parametric design and modelling of the system are based on regular polyhedra to guarantee uniformity and allow reconfiguration of the components. While this works from a design perspective, the structural performance is limited by the flexibility of the components. Therefore, we developed a double-layered component to increase the structural height and improve the stiffness of the structure. Preliminary FEM analyses provide assessment of this layering and stress stiffening, i.e. the stiffening effect of the residual stress. A full-scale prototype illustrates the rapid and low-tech manufacturing and assembly process. Through design, analysis and finally construction, this project illustrates the potential of active-bending in developing more low-tech and rapidly assembled structures.
... In order to better contextualise RFs among discrete grid member structures, and according to many authors, the hallmarks of a RF structure are: 1) it is formed by expanding and adding single RF units to the perimeter of the single unit to form a grid structure (Popovic Larsen 2007); 2) each element must work, simultaneously, both as support and supporter of other ones, without any clear structural hierarchy (Pugnale et al. 2011); 3) its elements support one another along their span and never at the extremities (Baverel and Pugnale 2014); 4) the length of each element is shorter than the distance to be spanned by the whole structure (Pugnale et al. 2011); 5) its elements are generally joined using friction, notching, nailing or tying (Song et al. 2013); sometimes, in larger structures, mechanical joints such as scaffolding swivel clamps are used (Sénéchal et al. 2011). ...
... In order to better contextualise RFs among discrete grid member structures, and according to many authors, the hallmarks of a RF structure are: 1) it is formed by expanding and adding single RF units to the perimeter of the single unit to form a grid structure (Popovic Larsen 2007); 2) each element must work, simultaneously, both as support and supporter of other ones, without any clear structural hierarchy (Pugnale et al. 2011); 3) its elements support one another along their span and never at the extremities (Baverel and Pugnale 2014); 4) the length of each element is shorter than the distance to be spanned by the whole structure (Pugnale et al. 2011); 5) its elements are generally joined using friction, notching, nailing or tying (Song et al. 2013); sometimes, in larger structures, mechanical joints such as scaffolding swivel clamps are used (Sénéchal et al. 2011). ...
Article
This paper deals with the use of reciprocal frames in temporary gridshell structures, such as architectural pavilions in expositions and installations. These architectural examples can benefit from the use of short, easy to handle, generally joint-free, and repeatable “modules” in order to create particular self-supporting structures. The lightweight and interwoven grid obtained by connecting short elements according to the reciprocity principle is structurally efficient and, at the same time, aesthetically pleasing, mainly due to the resulting tessellation. The paper firstly investigates the connection between efficiency and aesthetics. The last part of the paper investigates some temporary architectural pavilions from both an aesthetical and parametric point of view. In order to deepen our understanding of these structures, they are re-modelled according to a bottom-up approach by means of a constraint-based parametric CAD modeller. In this way, a reciprocal frame can be explored and modified by the parametric arrangement of its generative elements, which, like a natural organism, grows in self-generating forms.
... A KRF structure in both single and multiple units is a new kinetic structure based on the principle of reciprocity (Pugnale et al. 2011). Despite the fact that KRF structures have piqued the interest of architects and engineers, and extensive research has been conducted into their transformation ability in nonconstant perimeters (Choo et al. 1994;Saidani and Baverel 1998;Parigi and Sassone 2010;Parigi and Kirkegaard 2012;Parigi et al. 2014;Parigi 2021;Pérez-Valcárcel et al. 2021a, 2021b, all proposed examples change their perimeter during deployment and retraction, limiting their application in architectural design and making their use extremely difficult; as a result, they are not yet widely used in construction; in fact, when it comes to the design of kinetic structures, deployability has frequently been associated with a global structural displacement rather than local element displacements. ...
... In the West, on the other hand, history has a theoretical experimental character: architects and mathematicians studied the reciprocal system as a solution to problems of a practical nature. [2] There aren't too many examples of structures made by using the reciprocal method, these are mostly located in Japan and in the United Kingdom, with a small exception in Virginia, in the United States. The examples built present the reciprocal system only in coverage; in fact, the latter is, in all cases, created with a single reciprocal module formed by an always different number of elements. ...
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This study proposes a digital tool able to determinate, using a surface's free form data, a CRFS (complex reciprocal frame structures) made of the aggregation of SRFU (single reciprocal frame units) on a squared grid without changing eccentricity, position and dimensions of the rods and changing only the length. This choice allows to design new structures from synclastic surfaces, even if very complex, but equipped with a fundamental technical feature of being made with equal rods that generate different SRFUs. Using different algorithms already known, the new tool creates spherical cap surface CRFS that approximate to wanted surface. The original contribution of this work in based on this last step which uses dynamic relaxation and brings the spherical CRFS to ease down on the desired non-spherical geometry, without losing the reciprocity's bonds between the rods.
... The principle of structural reciprocity is the use of elements that lean on each other mutually in a three-dimensional structure (Pugnale et al. 2011). A minimum of three elements is needed, and some friction amongst them is required. ...
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Reciprocal structures are constructed by using a wide variety of patterns. These designs are a good source of inspiration when working with laminar constructions (sheets). Using the same formal schemes, laminar constructions feature better bending of their elements along the shape of the model and the application of extra pressure on the connection joints. Many of the geometric constructions made at the University of the Basque Country and presented in this paper show structural, constructive and formal improvements in many reciprocal structures assembled using sheets instead of linear elements.
... Reciprocal spatial structures can also be made out of flexible bent elements. Rotegrity (Fig. 3) is a geodesic reciprocal dome composed of flexible elements [12]. Although originally reciprocal systems were employed for regular surfaces, recent studies have shown that reciprocal structures can approximate any free-form surface by modifying parameters such as thickness and position of basic reciprocal modules [13][14][15][16][17]. [Mr Gat. ...
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Reciprocal structures are composed of mutually supporting rigid elements that are short with respect to the span of the entire structure. Although reciprocal structural systems have received significant interest among architects and engineers, they are not yet commonly employed in construction. The main reason for this nonadoption is the complexity of conceiving a structure from a module to the global scale without adapting the structure's final global shape. As a result, two approaches have emerged for the design of reciprocal structures. The first approach takes the module as primary building block and the final global form emerges as a result of the module's properties. The second approach results from adjusting the module's properties throughout the surface of the structure to fit its predefined global shape. This paper presents a complete design-to-construction workflow for reciprocal frames using a cell-based pattern algorithm. The developed parametric model explores geometry and patterning to adapt any module geometry to any free-form surface by adjusting the eccentricities between the modules. The resulting reciprocal structure is then analyzed and sized using finite elements. Finally, manufacturing layouts are generated and construction processes are discussed. The design-to-construction workflow was validated experimentally with the construction of a 5-meter diameter reciprocal hemispherical dome.
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Building with raw timber allows to reduce the price of construction and to make it more competitive with respect to concrete or steel construction. For a few years now, the combination of parametric design and robotic tools make possible the fast and precise milling of timber logs for their accurate connection. However, the spans are quickly limited by the logs length. In this context, reciprocal structures are relevant, since they allow to build large spans structures with short beams. Finally, the architectural interest of reciprocal structures is not to prove. However, the choice of the most efficient reciprocal frame, as well as its structural relevance in terms of mass and stiffness is, most of the time, ruled by subjective considerations. This paper focuses on rectangular floors composed of reciprocal moduli and has three objectives: (1) to develop a general mass and stiffness optimization method for reciprocal floors, which is not only necessary to limit the price, but also to reduce their thickness, (2) to define design rules for reciprocal floors, in particular for the choice of the best engagement ratio, and (3) to compare the structural efficiency of reciprocal floors with the one of “traditional” floors with parallel logs. Coming from a dimensionless transformation of the equilibrium equations, the results of this article will thus give the designers keys to better design reciprocal structures, evaluate their structural performances and relevance, and justify their choices.
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In this paper, we present a simple and intuitive approach for designing a new class of space-filling shapes that we call Generalized Abeille Tiles (GATs). GATs are generalizations of Abeille vaults, introduced by the French engineer and architect Joseph Abeille in late 1600s. Our approach is based on two principles. The first principle is the correspondence between structures proposed by Abeille and the symmetries exhibited by woven fabrics. We leverage this correspondence to develop a theoretical framework for GATs beginning with the theory of bi-axial 2-fold woven fabrics. The second principle is the use of Voronoi decomposition with higher dimensional Voronoi sites (curves and surfaces). By configuring these new Voronoi sites based on weave symmetries, we provide a method for constructing GATs. Subsequently, we conduct a comparative structural analysis of GATs as individual shapes as well as tiled assemblies for three different fabric patterns using plain and twill weave patterns. Our analysis reveals interesting relationship between the choice of fabric symmetries and the corresponding distribution of stresses under loads normal to the tiled assemblies.
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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
Full-text available
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
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.
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Conference Paper
A Reciprocal Frame (RF) system is a three-dimensional grillage structure constructed of a closed circuit of mutually supporting beams. A number of RFs connected to each other at the outer end of each radiating beam results in the formation of a Multi-Reciprocal Element (MRE) space structure. The uniqueness of the MRE system is that only two elements contained within module circuits of mutually supporting beams require to be connected to one another. A module cell may theoretically consist of any number of elements however. This simplifies the connections considerably therefore giving this novel system potentially economic advantages over traditional space structure connection systems such as the Mero KK node joint with up to 18 threaded holes. MRE space structures can therefore provide an economic, aesthetic and convenient way of spanning small or very large areas without the need to provide intermediate vertical supports. The aim of this paper is to explore and discuss the historical development of these systems and to consider their present and future application.
Article
Novel architectural forms can be created by connecting reciprocal frame (RF) and mutually supported elements (MSE) circuits together. These networks produce interesting architectural and engineering opportunities and challenges. The opportunities include the creation of roof and standalone structures that have distinctive architectural expression. The challenges include the determination of the often-complex configuration geometry between the elements and their connection system. A key feature of sloping RF and MSE geometry is that at the joint locations the element's centroidal axes generally do not coincide. An eccentricity at these positions has therefore to be incorporated within the connection system. This has a direct impact on element sizing, connection design, fabrication and erection sequences. RF and MSE spatial structure networks give rise to complex structural behaviour. Element-to-element connection eccentricity orientation is a controlling key feature in the determination of how the forces, moments and stresses are distributed between MSEs. The orientation of the eccentricity can be random or aligned to produce a vertical intersection distance as generally used in RF construction. The eccentricity derived from the common perpendicular to the centroidal axes is more commonly used in MSE circuit assembly. This paper considers the various methods used to connect RF and MSE networks and discusses their impact and comparative design advantages and disadvantages.