ArticlePDF Available

Analytical Investigations on Elementary Nexorades


Abstract and Figures

Nexorades or reciprocal frames can be seen as a practical way to reduce the complexity of connections in spatial structures by connecting reciprocally the members by pairs. This reduction of the technological complexity of the connections is however replaced by a geometrical complexity due to numerous compatibility constraints. The purpose of this article is to make explicit these constraints for elementary structures and to solve analytically the resulting system of equations. Applications to regular polyhedrons are presented and a practical realization (a 3 m high dodeca-icosahedron) is shown. In the brief conclusion, perspectives for complementary analytical developments for spatial structures are drawn forth.
Content may be subject to copyright.
Engineers studying spatial structures often face the
problem of linking numerous structural elements to
each other. A current solution is to design nodes
specifically dedicated to the structure. Such
connectors are often complex, most of the time
expensive and strongly depending on the geometry of
the members to be linked and on the overall geometry
of the structure, especially when free form design is
concerned [1]. Moreover, concentrating members in a
node makes the stress higher at these points, which
rises specific dimensioning needs not to result in a
structural weakness. Starting from these observations,
some authors began investigating the problem of
complex connections by replacing them by multiple
connections and studying the old concept of multi-
reciprocal frames. This term indicates structures in
which members support each others, reciprocally.
This construction principle has a long tradition all
over the world as one can see for example in the
book by Popovic which provides an up-to-date state-
of-art of the architectural knowledge on reciprocal
Analytical Investigations on
Elementary Nexorades
B. Sénéchal
, C. Douthe
, O. Baverel
Université Paris-Est, Laboratoire Navier, ENPC, 6 & 8 av. B. Pascal, Champs-sur-Marne, 77455 MLV Cedex 2
Université Paris-Est, IFSTTAR, IFSTTAR, DSOA/EMS, 58 bd Lefebvre, 75732 Paris Cedex 15
AE&CC Ecole Nationale Supérieure d’Architecture de Grenoble, 60, av. de Constantine BP 2636 Grenoble cedex 2
(Received 04/02/2011, Revised version 21/06/2011, Acceptation 06/10/2011)
Abstract: Nexorades or reciprocal frames can be seen as a practical
way to reduce the complexity of connections in spatial structures by
connecting reciprocally the members by pairs. This reduction of the
technological complexity of the connections is however replaced by a
geometrical complexity due to numerous compatibility constraints. The
purpose of this article is to make explicit these constraints for elementary
structures and to solve analytically the resulting system of equations.
Applications to regular polyhedrons are presented and a practical realization
(a 3 m high dodeca-icosahedron) is shown. In the brief conclusion,
perspectives for complementary analytical developments for spatial
structures are drawn forth.
Key Words: Reciprocal frame, nexorade, regular polyhedron, analytic
International Journal of Space Structures Vol. 26 No. 4 2011 313
systems, with a detailed history of this construction
process and many build examples [2]. This structural
principle offers a wide variety of forms and patterns
which inspired architects like Shigeru Ban [3] or
artists like Roelof [4]. Their approaches remain
generally empirical and few studies proposed an
investigation of the structural behaviour of reciprocal
frames or of their form. Among these authors, one
can mention the numerical and experimental work of
Rizzuto on dodecahedra [5] or the numerical study of
Sanchez et al [6] on grids inspired by drawings by
Leonardo da Vinci.
A generalization of the multi-reciprocal grids
concept was proposed by Baverel et al and called
“nexorade” according to a neologism of Nooshin [7].
The general idea of this generalization is that, contrary
to historical reciprocal systems, the connections
between the members must not necessary be in
compression. One can thus vary the respective
positions of the members and build grids with an even
wider variety of forms, from flat grids to double
curved shapes (see [8] and [9]). In nexorades or
*Corresponding author e-mail:
reciprocal systems, the members are not converging to
the connections points, but are resting on each others
or below each others by pairs. This process induces a
lot of geometrical constraints. Hence with nexorades,
the problem of connecting numerous members shifts
from technological complexity to geometrical
complexity. To investigate the family of structures that
can be built with so many geometrical constraints,
form-finding methods were developed. Baverel et al
[10] proposed a numerical method based on a genetic
algorithm, while Douthe et al [11] used the dynamic
relaxation method and a fictitious mechanical
behaviour to define the suitable geometrical
For a designer, these two methods have an
important inconvenient: they are not fully
deterministic. Like for most structures requiring form-
finding, it is not possible to impose a desired form: the
final form of the structure is the result of an entire
process which includes geometrical and mechanical
constraints. Nevertheless, it is possible to do some
analytical calculations on the geometry of elementary
nexorades with standard dispositions and hence to
predict exactly the set of parameters that allows the
construction of the structure. This paper is aimed at
presenting those analytical investigations. Section 2
will present the necessary vocabulary of nexorades
and detail the principle of the transformation that will
be studied. Then section 3 develops analytical
calculations of the nexorades geometry and ends with
a set of three characteristic non-linear equations.
Section 4 concerns the application of these equations
to the investigation of the transformation of regular
polyhedrons into nexorades. It includes also
illustrations of practical realizations. A brief
conclusion summarizes the article and suggests some
tracks for further research.
2.1. Definition and vocabulary
A “nexorade” indicates hence a space structure in
which members support each others reciprocally with
connection which might be in tension or in
compression (see figure 1). The members forming
nexorades are called “nexors” and their nodes may be
called “fans” because of their fan aspect (see figure 2).
One can easily imagine that the number of nexors
involved in a fan cannot be less than three, whereas it
has no theoretical upper limit. Beside the section of the
nexors can be of various shapes and even non-uniform
314 International Journal of Space Structures Vol. 26 No. 4 2011
Analytical Investigations on Elementary Nexorades
along their axis [13]. However in this document, the
analyses focus on circular-section nexors.
In order to make a thorough analysis of some
elementary geometrical properties of nexorades, some
geometrical parameters must be set, namely the nexors
lengths (L), their engagement lengths (
) and
eccentricities (e). The engagement length measures the
length of the nexor which is engaged in the fan (which
therefore may also be called “engagement window”).
It is the distance between the two points of a nexor
which are connected to other nexors involved in the
same fan (see figure 2). The eccentricity is the distance
between the axes of two connected nexors at
connection points. Since this paper focuses on
cylindrical nexors with identical radius, the
eccentricity at connection points is also the diameter of
the nexors.
2.2. The rotation method
Generally, it is more comfortable for a designer to
reason on a structure with members converging at
nodes and to ignore the geometrical constraints of
nexorades. A major point of nexorades design is thus
to make explicit the way to change from an ideal
punctual connection (like the vertices of the cube on
figure 3) to a fan connection (figure 4). The
transformation method which will be developed in the
following section consists in the composition of a
rotation and a lengthening of each member (and that is
why it is called “method of rotation”).
Figure 1. Nexorade geodesic dome.
Engagement window
Engagement length
A fan
Figure 2. An elementary fan with three members.
B. Sénéchal, C. Douthe, O. Baverel
International Journal of Space Structures Vol. 26 No. 4 2011 315
three nexors and to nexors zzwith different
geometrical properties. Doing so, one will increase the
number of parameters and, as the relations between
those parameters are non-linear, one will, in general,
rely on numerical solutions.
3.1. Definition of the members geometry
In the following, the nexors will be considered as
straight members with identical circular cross sections.
They will be modelled as segments representing their
centroïd and will be linked to each other by small
segments corresponding to the eccentricity between
their axes. The transformation from a “standard
converging structure” (namely a typical three
members connection as one can find on the cube in
figure 3) into a nexorade is illustrated in figure 5. In
figure 5a, the nexors (numbered from 1 to 3) are
converging at the point I, where the three ends of the
nexors meet. On figure 5b, a fan has been created and
there are three connections instead of one. Each nexor
is connected to two other nexors, one at the
engagement length (point D
) and one at the upper
extremity (point B
). The free extremity of the nexors
are named C
. The points C
are unchanged during the
For convenience reasons, it will be considered that
the points C
are located at the middle of the nexor and
not at the end. Indeed, in the specific case of
polyhedrons, the symmetry of the structure is such
that, if points C
are located in the middle of the
polyhedron’s edges, it is possible to deduce the
transformation of the whole structure by simple
rotations of the elementary transformation of figure 5.
The length L of segment C
will therefore represent
the half of the nexor total length.
3.2. Definition of the transformation
It was demonstrated in [8] that, in this transformation,
the axis of rotation of a nexor passes through point C
Figure 3. A cube with standard vertices.
Figure 4. A cube after transformation into a nexorade.
(a) (b)
Figure 5. Elementary transformation of a fan with three identical nexors.
It is focused here on fans with three identical nexors,
for simplicity and clarity reasons. Nevertheless, the
method can be easily extended to fans with more than
and through the centre C of the sphere which contains
the C
and is tangent to the three nexors (C
I) before
the transformation (see figure 6). So let us call α the
angle made by the nexor and the plane containing the
points C
in the initial converging configuration
(figure 6 left), L
the initial length of the nexor,
rotation angle around CC
which characterizes the
transformation and k the extension factor (k = L / L
These parameters (
, k and L
) are not independent
and the relations between them which had been found
numerically in [8], are formulated analytically in an
explicit manner in the coming paragraphs.
The transformation will be described in the
coordinate system (O, x, y, z) of figure 6 which is
defined by O, its origin obtained by projection of I on
the plane containing the C
and the three coordinate
axes, (Ox) which points toward C
, (Oz) which points
toward I and (Oy) which is deduced from the two
others so that (O, x, y, z) is direct. After some
calculations, the matrix R
of the rotation of nexor 3
along the axis CC
can be defined as:
So, if I
is the image of I by the rotation R
and B
the image of I
by the extension k, the general
transformation for the nexor is defined by:
33 3
33 33
33 3
i II
−+ cos (cos ) cos sin sin cos (cos
11 1
αθ αθ ααθ
cos sin cos sin sin
sin cos (cos ) sin
αθ θ αθ
αα θ
ααθ α θ θ
sin cos ( cos ) cos
316 International Journal of Space Structures Vol. 26 No. 4 2011
Analytical Investigations on Elementary Nexorades
One then remarks that, as the nexors are identical,
the coordinates of the points of the two other nexors
can be deduced by simple rotation of 120° around the
(Oz)-axis (see figure 6) which is given by the
following matrix:
The coordinate of the extremity of nexor 1 which is
connected to nexor 3 is thus given by:
One has now found the relations between the initial
geometrical parameters (initial length L
| and
) and the transformation parameters (angles
and extension k).
3.3. Definition of the geometrical
As the nexors are identical (same length, same
engagement length and same diameter) and as the
initial configuration is unchanged when rotated by a
120° angle around (Oz), one deduces that the problem
can be limited to the study of one connection, for
example between nexor 3 and nexor 1. To find the
four characteristic equations of the model, one will
thus write the geometrical constraints of this
3.3.1. Perpendicularity of eccentricity and nexors
By definition, the eccentricity is the minimal distance
between the axes of nexor 1 and 3. It is thus
perpendicular to the axes of nexor 1 and nexor 3. The
first constraint equation is thus obtained by writing the
perpendicularity of the eccentricity and nexor 1. In
figure 7, the eccentricity is represented by the segment
and the nexor 1 by the segment C
. The first
geometrical constraint is thus given by:
In the same manner, C
represents the nexor 3 so
that the perpendicularity of the eccentricity and nexor
3 is given by:
33 3 1
R11 33
= i
12 3 2 0
32 12 0
(Nexor 3)
Figure 6. Transformation parameters.
3.3.2. Consistency of the engagement length and
After transformation into a fan, the points B
and D
shall be such that the distance between them equals the
engagement length
(see figure 8). So considering
that the extension is defined by:
One deduces the geometrical constraint of the
engagement length:
The last geometrical constraint is obtained writing
that the distance between B
and D
is the eccentricity e:
3.4. Characteristic equations of a fan
In the preceding section, the geometrical constraints of
the transformation were established (equations 5, 6, 8
and 9). By combining these equations with the
parameters of the transformation given in equation 1 to
4, one will obtain the characteristic equations of the
fan. The resolution of the system requires some
algebraic calculations that are not detailed here.
Nevertheless one can remark that equation 7 gives
explicitly the relation between the extension k, the
BD e
B. Sénéchal, C. Douthe, O. Baverel
International Journal of Space Structures Vol. 26 No. 4 2011 317
Figure 8. Characteristic length of the fan.
Figure 7. Perpendicularity of the eccentricity and nexor 1
and 3.
final length L and the initial length L
. Only four
independent parameters thus remain: the rotation angle
, the eccentricity e, the final length L and the
engagement length
for three independent equations.
One parameter can thus be chosen independently from
the others. The simplest equations are obtained with
the angle
as initial parameter:
It would be more convenient to have the
eccentricity or the final length as initial parameters
because they are the ones at the designer disposition
but an explicit relation between those parameters
could not be found. To inverse expression 10 to 12,
numerical solutions have thus to be used as will be
illustrated hereafter.
4.1. General properties of regular convex
A regular polyhedron is a polyhedron whose faces are
congruent (all alike) regular polygons which are
assembled in the same way around each vertex. The
regular convex polyhedron are often called “platonic
solid”. They are five in number: the tetrahedron,
the cube (or hexahedron), the octahedron,
the dodecahedron and the icosahedron. These
polyhedrons are associated in pairs called duals, so that
to the vertices of the one correspond the faces of the
other. The dual of the dual is the original polyhedron. A
way of finding one polyhedron’s dual is to define a
vertex of the future dual at the center of each face of the
polyhedron, then to join each of these vertices to all his
nearest neighbouring vertices. The cube and the
octahedron are duals, the dodecahedron and the
icosahedron also and the tetrahedron is self-dual.
31 1
cos cos
cos ) cos )
ccos ) 3cos cos )
13 1
cos cos )
cos [cos ( cos )[ cos cos
sin ]
αθ α θα
cos cos sin ]
cos ( cos ) cos (
θα θ
θα θ
31 21
ccos )
4.2. Transforming regular polyhedron
into nexorades
In this part, the validity and interest of the
mathematical expressions (10 to 12) obtained by the
“method of rotation” are illustrated by examples for
which numerical solutions had already been presented
and detailed in [8] or [12]. Practically, for each
polyhedron, the method of rotation is applied with the
-parameter for all the vertices so that, every
nexor, rotating around its middle, sees the same
transformation for both its vertices. The example of
compatible parameters which allow for the
transformation of a dodecahedron into an icosahedron
is shown in figures 9 and 10.
Figure 9 represents some typical three-dimensional
views of this transformation. The dodecahedron has
three edges and three pentagonal faces converging to
each vertex. When transformed into a nexorade, it
gives birth to triangular engagement windows defined
by an engagement length λ and an eccentricity e.
However these engagement windows can also be seen
as the faces of a polyhedron transformed into a
nexorade, the former faces becoming hence the new
engagement windows. This polyhedron has five edges
and five triangular faces converging to each vertex; it
can thus be seen as the transformation of an
icosahedron into a nexorade defined by an engagement
length 2L–
and an eccentricity e. The transformation
into a nexorade can be thus seen as a continuous way to
go from a polyhedron to its dual.
Figure 10 represents then the variations of the total
length of the nexors (2L), their engagement length (
and the eccentricity (e) (which is here taken equal to
the nexors’ diameter) with the angle of rotation
radians) for the dodeca-icosahedron (Note that the
scale is so that the distance between the centre of the
318 International Journal of Space Structures Vol. 26 No. 4 2011
Analytical Investigations on Elementary Nexorades
polyhedrons and its vertices equals 10 units). These
values obtained from expressions (10 to 12) fit very
well with the numerical values shown in [8] or [12].
One notices that the associated values for the
icosahedron can be deduced easily from that of
figure 10, considering a rotation of
= π
and an
engagement length of
= 2L
and keeping the total
length and the eccentricity unchanged. Beside, it is
worth remarking in figure 10 that the engagement
length is always larger than the eccentricity
(considering either the hexahedron or the tetrahedron).
The practical consequence of this is that the
construction of dodeca-icosahedron is much easier
than that of a tetrahedron where the space for installing
the connections is much reduced.
4.3. Construction of an dodeca-
icosahedron nexorade
On July 2006 it was decided to test the feasibility of such
regular polyhedra on an icosahedron. The available
Figure 9. Views of dodeca-icosahedron for
= 5°, 10°, 20°, 30°, 45°, 60°, 70°, 80°, 85° (cf. [8]).
0 0.2 0.4 0.6
0.8 1 1.2 1.4
Figure 10. Characteristic parameters of the dodeca-
material in the laboratory was for the time-being
composite tubes for the nexors and swivel scaffolding
elements for the connections between the nexors. The
size of these scaffolding elements was about 75 mm
from tube axis to tube axis (the tube diameter being of
42 mm) and corresponds to the given eccentricity of the
structure to be built. The length of the available tubes
was 2000 mm, out of which 50 mm were reserve at each
extremity for the positioning of the connectors, so that
finally 1900 mm was taken for the length of the nexors.
Then, by referring to figure 10 and equations (10), (11)
and (12), one could deduce from the nexors’ length and
the eccentricity, the suitable engagement length:
115 mm.
Once the characteristic parameters of the dodeca-
icosahedron had been set, the construction began. No
major difficulty was reported, even for the setting of
the last members of this highly hyperstatic structure.
The whole nexorade was assembled by two people in
about 4 hours. The final structure can be seen on
figure 11 and its characteristic pentagonal fan on
figure 12.
B. Sénéchal, C. Douthe, O. Baverel
International Journal of Space Structures Vol. 26 No. 4 2011 319
4.4. Extension to other polyhedrons
Considering the method and examples introduced in
this paper, some obvious extensions to semi-regular
and truncated polyhedrons seem possible, for
instance using non-constant engagement lengths.
There may also be other possible extensions to more
complicated spatial structure, still with the same
basic method of rotation.
Nexorades or reciprocal frames can be seen as a
practical way to reduce the complexity of
connections in spatial structures by connecting
reciprocally the elements by pairs. This
technological simplification induces however
numerous geometrical constraints, so that the
problem of construction becomes mainly geometric.
Form-finding methods were thus developed to
explore the form universe of these structures, but it
is possible in some specific cases to find analytical
solutions and to link with exact mathematical
formulas the different parameters between them. The
aim of this paper was hence to present a general
analytical method to express the geometrical
constraints induced by the transformation of a
standard structural frame into a reciprocal frame.
Applications to regular polyhedra were shown
(achieving hence the numerical work started in [8])
and illustrated by the realization of a large scale
Furthermore, the principal of the presented method
is quite general and it seems to have good potential
for future developments of constructive approaches
of reciprocal frames, especially for double layer
spatial grids in which regular polyhedra play an
important role.
1. Schlaich J.; Schober H. and Kürschner K., New Trade
Fair in Milan – Grid Topology and Structural
Behaviour of a Free-Formed Glass-Covered Surface,
International Journal of Space Structures, 2005, 20
(1), 1–14.
2. Popovic Larsen O, Reciprocal Frame Architecture,
Architectural Press, London, 2007.
3. McQuaid M, The Japanese pavilion, in Shigeru Ban, ed
Phaedon, 2006, pp. 60–67.
4. Roelofs R., Three-Dimensional and Dynamic
Constructions Based on Leonardo Grids, International
Journal of Space Structure, Sept 2007, 22 (3),
5. Rizzuto JP., Hulse R., Dodecahedric Mutually Supported
Element Space Structure: Experimental Investigation,
International Journal of Space Structure, June 2007, 22
(2), 107–121.
Figure 11. The dodeca-icosahedron.
Figure 12. A characteristic pentagonal fan of the dodeca-
6. Sanchez J, Escrig F, Adaptable Leonardo, in
Scheublin F. & Pronk A. eds, Proc of int. Conf. on
adaptable building structures, Eindhoven 2006, vol. 1
pp 28–32.
7. Baverel O, Nooshin H, Kuroiwa Y, Parke GAR,
Nexorades, International Journal of Space Structure
2000, Vol. 15 No. 2.
8. Baverel O., Nexorades: a family of interwoven space
structures, PhD Thesis, University of Surrey, décembre
9. Baverel O, Saidani M, The Multi-Reciprocal Grid
System, in Obrebski J.B. ed, Proc. Int. Conf. on
Lightweight Structures in Civil Engineering, 1998,
Warsaw, Poland, pp 66–71.
320 International Journal of Space Structures Vol. 26 No. 4 2011
Analytical Investigations on Elementary Nexorades
10. Baverel O, Nooshin H, Kuroiwa Y, Configuration
processing of nexorades using genetic algorithms,
Journal of the I.A.S.S. 2004, 45 no 2, p. 99–108.
11. Douthe C., Baverel O., Design of nexorades or reciprocal
frame systems with the dynamic relaxation method,
Computer and structures, 87 (2009) 1296–1307.
12. Baverel O, Nooshin H, Nexorades Based on Regular
Polyhedra, Nexus Network Journal October 2007,
Vo lume 9, Number 2 pp. 281–299 Birkhäuser.
13. Rizzuto JP, Saidani M, Chilton JC, Polyhedric Space
Structures using reciprocally supported element of
various cross-sections, Journal of the International
Association for Shell and Spatial Structures, 2001, vol
42, no 3, pp. 149–159.
... First, the form-finding of geometrical configurations of nexorades requires efficient non-linear solvers, which do not provide any certainty about their output [1]. Besides elementary cases, like regular polyhedra [6,10] cylindrical polyhedra or regular planar tilings, no theoretical result has been derived on the nature of the space of feasible nexorades. Second, nexorades are not as efficient as other structural systems mapping free-forms, like gridshells, because of their low node valence and low structural redundancy [11]. ...
... Dynamic relaxation is a non-linear method that can converge to local minima, but do not systematically converge towards the global minimum. Few papers deal with the analytical solution of the form-finding problem for nexorades: Senechal et al. studied the transformation of regular polyhedra by rotations [6], while Baverel studied analytically transformations by translations of regular polyhedra [12]. Finally, some papers approach form-finding of nexorades by setting the eccentricities as soft constraints and perform iterative least-square optimisation [14]. ...
... KRF structures, however, have the advantages of requiring less technology to be constructed, with simple connections and linking both in the end and in the middle of the elements. It can be viewed as a practical method to reduce the complexity of connections in spatial structures by reciprocally connecting the elements by pairs (Senechal et al. 2011). This structure can be seen as a construction technique to cover large spaces with short elements assembled together with simple connections (Baverel and Popovic Larsen 2011). ...
... identifying span, height and slope of the structure) of RF structures. The geometric method may help determine the shape of RF frame structures corresponding to their geometric features, such as distances between the components, sectional dimensions, and the number of RF units in the structure [11][12][13][14], according to their geometric relationships. On the foundation of the geometric method, a hybrid approach, in a combination of the global search of the genetic algorithm with the local search of a gradient-based algorithm, was proposed to optimise the geometric forms of RF structures [15]. ...
Reciprocal frame (RF) structures were developed in this work using pultruded glass fibre reinforced polymer (GFRP) members and the formulation of mathematical model for geometric forming and capacity design of such structures were presented. Firstly, governing geometric relationships are identified for the height, span and slope of the RF structure and its overall geometry can be formed accordingly. Once the geometry is determined, the structure can be analysed to calculate the internal forces of each component and connections under given loads. GFRP pultruded members were used to assemble the RF structure due to their high strength and lightweight. Considering the closed shape of the tubular section and the anisotropy of the materials, an innovative connection configuration was developed to facilitate the assembly and minimise the requirement of tools in the construction. Both the conceptual design and modelling analysis are supported with experimental results at connection and structural levels. The load-drift relationships of the connections were experimentally examined. Subsequently, A GFRP reciprocal frame structure was then assembled with a span of 4.5 m and subjected to multi-point bending to verify the proposed evaluation method and to study its mechanical performance. Verified by experimental results, a two-dimensional numerical model of the RF structure was developed and analysed using the finite element (FE) approach where different structural boundary conditions and connection stiffnesses were discussed. Finally, the geometrical nonlinear analysis was also studied for the structure through FE modelling.
... O perímetro gerado entre essas dimensões forma a área chamada de janela de trabalho. Além disso, a configuração espacial gerada pelos leques implica a sobreposição da excentricidade dos nexors, dada como a menor distância entre o eixo de dois elementos conectados(BAVEREL et al., 2000;BAVEREL, 2009;SÉNÉCHAL;BAVEREL, 2011;THONNISSEN, 2014;MESNIL et al., 2018) (Figura 4). ...
Full-text available
Estruturas recíprocas (ERs) são datadas do período neolítico, produzidas originalmente em madeira e compostas por elementos de dimensões reduzidas. Trata-se de um sistema estrutural que apresenta a capacidade de se adaptar a formas livres, complexas e com grande potencial estético, como indicado por diversas pesquisas contemporâneas. No entanto, acredita-se que essas estruturas poderiam fazer melhor uso das tecnologias digitais disponíveis de projeto, análises, fabricação e montagem. Este artigo tem o objetivo de identificar as atuais lacunas em que se pode concentrar esforços de pesquisa no sentido de tornar a tecnologia das ERs mais viável. Foi feita uma revisão sistemática da literatura (RSL) em sete bases de dados diferentes. Encontrou-se um total de 180 artigos, dos quais 49 foram selecionados para análise. Dentre eles, 27 identificaram lacunas existentes, que foram compiladas em cinco categorias: (a) dificuldades na concepção geométrica/estrutural; (b) necessidade de aprimoramento de ferramentas digitais; (c) dificuldade no projeto, análise, fabricação e montagem de conexões; (d) dificuldade de compreensão da real contribuição dos protótipos para aplicações em grande escala; e (e) necessidade de incorporar o processo de montagem no projeto de ERs. Foi possível concluir que o desenvolvimento de novos mecanismos de análise estrutural aliados à criação de novos detalhes construtivos pode contribuir para a expansão do uso das ERs.
... The past decades have showed an increased interest in computational RF design among researchers which resulted into three main approximation methods [5]. First, the analytic approach determines the RF design exact and uses geometrical equations to create an RF that is in most research based on platonic solids [6]. Second, the bottom-up approach uses iterative processes or self-generating geometries to create an RF. ...
Conference Paper
Full-text available
The definition 'Reciprocal Frame' (RF) applies to structures that are feasible by means of circulating shear, compression or tension interactions between their constituent members. This relation indicates that beam depths and connections correlate to an RFs' structural geometry. Till date, a combination of RF form finding that regards both beam depths and structural design of connections has not yet been developed. Although researchers developed computational form finding methods to create geometrical solutions and described the global structural design, computational complexity may have prevented a direct inclusion of detailing in the overall RF design. This research presents a new RF form finding method that includes the structural design of beam dimensions and detailing. A parametric model to be referred to as 'The Timber Reciprocal Frame Designer' (RFD) is developed to design three-and four-member RF assemblies from any arbitrary NURBS surface. The RFD provides a practically verified tool in which designs can be produced by using industry standard machines stimulating structural designers to add RFs to their design pallet.
... The core of the research in this category focuses on computational methods for the form-finding and analysis of reciprocal systems. Which includes form finding process based on surface tessellation and constraint formulation for reciprocal cells coupled with dynamic relaxation [4,7,10,16,19]. Song et al. developed a tool for form finding of reciprocal systems based on two-dimensional pattern creation and conformal mapping on three dimensional forms [17,18]. ...
Conference Paper
Full-text available
ABSTRACT In this research a computational method is developed to study the form-finding process of non-standard reciprocal systems with 2D elements based on the current methods on the morphology of reciprocal systems with 1D elements. The developed form-finding methods will be used in a performance based form exploration process for geometrical and structural performance enhancement. The proposed computational framework will explore new potentials for variations in the assembly design of these systems through the introduction of new geometric parameters both at the component level and the assembly level within the form exploration process. The proposed method integrates parametric assembly design with structural analysis in a stochastic optimization process to explore the design space while minimizing the total weight of the structure. The results of the form exploration process will be stored for the post processing phase in which the solution space is explored to study the variation of the emerging assemblies. In this paper the proposed method is explained and implemented via two case studies towards the further exploration of the concept.
In the face of dwindling resources and to reduce the construction sector’s carbon footprint, it is important to envisage the building’s end of life. It is necessary to consider buildings and their structures as elements that can be dismantled/disassembled for recycling/reuse. To address these issues, this work focuses on non-sequential assemblies that allow reversibility. A full-scale Nexorade prototype has been produced and is used to illustrate the entire article demonstrating the relevance of these assemblies. The first part defines non-sequential assemblies and the multiple possible kinematics for an assembly. The second part focuses on the pavilion’s geometry: the choice of the tessellation and its transposition to a nexorade. The article continues with the mechanical analysis, showing that tilting the beams by 30° reduces the deflection by 35%. The assembly sequence is then chosen to ensure that each node closes non-sequentially. Finally, the pavilion’s construction is detailed, showing the transition from the digital to the physical model and the adjustment needed to switch between these two models. To conclude, the authors discuss the results obtained on non-sequential connections and the inclination of the beams of a nexorade. Possible extensions and improvements for future research are suggested.KeywordsNon-sequential assemblyKinematicNexoradeJoineryDigital fabrication
En este artículo, se aborda la determinación de las condiciones de equilibrio de Superficies de Geometría Variable definidas por Estructuras Recíprocas mediante el estudio de un campo de fuerzas actuantes y del lugar geométrico que describen los elementos de tales estructuras a través de los diferentes estados de transformación. Se establecen las propiedades geométricas y estructurales de un Nodo Recíproco de estas estructuras con tres barras cilíndricas iguales, mediante el desarrollo de una formulación matemático-paramétrica, con el objetivo de deducir el proceso de equilibrio para la obtención analítica de sistemas de estabilización de Superficies de Geometría Variable conformadas por Estructuras Recíprocas Transformables. La finalidad de la investigación presentada aquí es la aplicación de dicho sistema de estabilización mediante un software geométrico-estructural de nueva creación, al diseño y cálculo de distintas tipologías de superficies en la configuración de arquitecturas de geometría transformable.
Reciprocal Frames (RF) are a type of structural system that can be used to build temporary structures with large spans. This is possible due their ease of assembly and the use of relatively short, light-weight elements. However, a challenge is presented when this kind of structure is applied to complex surfaces. In this case, the only way to figure out each connection is through an algorithmic geometric model. Only a few algorithms have already been developed for generating complex RF structures, but they still have some limitations. In this paper we present a new approach represent RF geometry, based on symmetry theory, combining pinwheel patterns with wallpaper symmetry groups. In future works, this will be combined with structural performance analysis and the use of digital fabrication methods for producing the structures’ connections.
Full-text available
The objective of this paper is to discuss the generation of configurations of nexorades using genetic algorithm. The process of genetic algorithm allows one to create a nexorade from an initial simple configuration. An important application of nexorades is for shelters of various sizes and shapes for temporary or permanent purposes.
Full-text available
The objective of this paper is to discuss the characteristics of a family of space structures that are referred to as 'nexorades'. Typically, a nexorade is constructed from scaffolding tubes, connected together with swivel couplers. An important application of nexorades is for shelters of various sizes and shapes for temporary or permanent purposes. In such a shelter, the structural skeleton is provided by a nexorade and the cover is provided by a membrane material.
Full-text available
The objective of this paper is to propose a simple method using dynamic relaxation algorithm to find the form and perform a structural analysis of reciprocal frame systems, also called nexorades. After a brief historical note, details of the parameters that govern the design of nexorades are introduced together with details of the assembly. These parameters induce strong geometrical constrains on this kind of structures so that a form-finding step is required. It is suggested here to introduce a fictitious mechanical behaviour to solve this problem. The dynamic relaxation algorithm is used with a model that takes into account the eccentricity between the elements. Its implementation is explained and its versatility is illustrated through several examples covering various fields of applications going from form-finding problems to non-linear structural analysis of structures. In a last example the structural behaviour of nexorades is compared with more conventional triangulated structures.
Full-text available
The objective of this paper is to discuss the characteristics of nexorades based on regular polyhedra. An important application of nexorades is for shelters of various sizes and shapes for temporary or permanent purposes. In such a shelter, the structural skeleton is provided by a nexorade and the cover is provided by a membrane material.
The purpose of this paper is to consider the construction of polyhedric space structures using reciprocally supported structural elements of circular, rectangular, triangular and trapezoidal cross-section. Platonic and Archimedean polyhedral forms are considered for potential space structure construction. Several examples of these basic forms are investigated using elements that are variously aligned to thepolyhedra surface edges. The impact of element cross-section and orientation alternatives is also considered. A feature of structures constructed using reciprocally supported elements is the way the load bearing elements mutually support one another. The advantages and disadvantages associated with the use of trapezoidal and other element cross-sections suitable for polyhedric space structure construction is reviewed and discussed.
This paper investigates the different shapes that the Multi-Reciprocal Grid could have using regular and semi-regualr grids. The concept is to create forms using grids. The structure is generated by elements mutually supporting each other; it is the addition of elements that creates a 3-dimensional form. The new structure is demountable, transportable, and quick to re-assemble. It could have various types of different geometry, simply or doubly curved, which gives almost an endless possibility of different architecture. As the elements support one another, their connections create a slope between each element, thus 2d grid will not stay flat. They will be curved in one or two direction. It depends on the degree of freedom of the node of the grid. The form of the structure depends on the grid chosen. The new structure may be particularly useful in emergency situations where rapid construction of shelters is needed. The structures could have only one type of element and one type of connection
Leonardo grids is the name I gave to my bar grid construction system with which I was able to construct domes and spheres out of simple elements using one constructing rule. Most of the constructions I made where planar and static. In this paper I want to focus on the nonplanar and dynamic possibilities of the Leonardo grids.
Circuits of mutually supported elements (MSEs) can produce novel 3-dimensional spatial structures. In the creation of a circuit, the basic concept is that the primary members rely on each other for support. These novel structural arrangements may give rise to complexity in the configuration geometry and structural behaviour due to the creation of an eccentricity between elements. An experimental programme was designed to aid understanding of the primary behaviour of structures composed of MSEs. This investigation concentrated on the structural performance, under applied static loading, of a dodecahedric MSE space structure. The main objectives of the experimental investigation were to confirm behaviour as being linear elastic within a predermined appied load range, to understand the distribution of stress and the displacements of the structural elements, later to be compared with that predicted by numerical modelling. Displacements of closed MSE circuits with different spatial orientations were considered. Differences in the recorded strains were also considered. The experiment highlighted the difficulties associated with monitoring MSE circuits, particularly support stiffness and displacements of circuits with arbitrary 3-dimensional spatial orientations. The recorded strains were complex in nature in as far as they included the effects of axial forces, bi-axial bending, shear and torsion. It was found that the maximum applied load did not cause yield of the material according to the von Mises ductile material failure criterion.
The connecting links between the exhibition pavilions of the new trade fair in Milan are mostly covered by innovative steel-glass-structures which are very attractive from an architectural point of view. At the main entrance the 'Logo' arises: A doubly-curved free-formed surface, which ascends to the sky like a volcano. With a height of 37 m, this structure will be identified from afar as the new fair's landmark, Further, a 1,300 m long and 30 m wide free-formed glass roof, the so-called 'Vela', links the individual exhibition halls along the main axis of the trade fair and reflects the nearby Alps in its architectural appearance. Emphasizing the 'Logo', besides the overall architectural concept, this paper mainly deals with the basic principles for the development of the grid topology and the characteristics of the structural system of the arbitrary curved free-formed surfaces.