JOURNAL OF THE INTERNATIONAL ASSOCIATION FOR SHELL AND SPATIAL STRUCTURES: J. IASS
COMPUTATIONAL DESIGN OF BISTABLE DEPLOYABLE
SCISSOR STRUCTURES: TRENDS AND CHALLENGES
Liesbeth I.W. ARNOUTS1,2, Thierry J. MASSART1, Niels DE TEMMERMAN2 and Péter Z. BERKE1
1 BATir
Department, Université libre de Bruxelles (ULB), Avenue Franklin Roosevelt 50, 1050 Brussels, Belgium,
larnouts@ulb.ac.be, thmassar@ulb.ac.be, pberke@ulb.ac.be
2 Department of Architectural Engineering, Vrije Universiteit Brussel (VUB), Pleinlaan 2, 1050 Brussels, Belgium,
Niels.De.Temmerman@vub.be
Editor’s Note: This space reserved for the Editor to give such information as date of receipt of manuscript, date of receipt of
revisions (if any), and date of acceptance of paper. In addition, a statement about possible written discussion is appended.
DOI: Digital Object Identifier to be provided by Editor when assigned upon publication
ABSTRACT
Mobile deployable scissor structures are transportable and can be transformed rapidly from a compact folded
state offering a huge volume expansion. Intended geometrical incompatibilities during transformation can be
introduced as a design strategy to obtain bistability, which allows instantaneously achieving some structural
stability in the deployed state. In such bistable deployable structures, these incompatibilities result in the elastic
bending of some specific members that are under compression with a controlled snap-through behaviour. Attempts to optimally design deployable bistable structures remain scarce, since the underlying structuralmechanical concepts are complex. Furthermore, the requirement of flexibility during deployment while ensuring
some structural stability in the deployed state prevents the use of simple design methodologies relying on the
structural behaviour under service loads only. In this contribution, the trends and challenges of using computational tools in the structural analysis and design process of deployable bistable structures are discussed.
Computational tools are crucial for the geometrical and structural design, for the definition of a rigorous design
methodology and for a deeper understanding of the complex transformation behaviour of these structures.
Keywords: deployable structures, transformable structures, scissor structures, bistability, snap-through, computational tools, parametric design, nonlinear computational mechanics, structural analysis, finite elements
1. INTRODUCTION
Nature itself is constantly evolving through transformations. Populations adapt through evolution
and individuals transform during their lifetime to
adapt to their environment. In contrast to biological
systems, buildings are often designed to fulfil a
single set of functions throughout their lifetime. By
consequence, it is a challenge for them to adapt to
changing circumstances. In our dynamic and rapidly evolving society there is a need for flexible
spaces. Convertible, temporary and lightweight
structures can adapt their shape or function to meet
changing needs and circumstances on either the
short or on the long term. They are sustainable because of their modular design, reusability and easy
adaptivity [1]. The design of transformable structures is inherently a design for change, since it
includes explicitly the concept of time and different
stable structural states.
Transformable structures can be either kit-of-parts
systems, having reversible connections allowing for
disassembly, or deployable structures, having the
ability to transform from a compact state to a deployed configuration [2]. This contribution focuses
on the latter having the advantage of fast and easy
transformation
without
requiring
on-site
(dis)assembly operations. Applications of deployable structures within the field of architecture and
civil engineering include mobile or temporary structures for disaster relief, for construction and
maintenance sites, for military operations or for
recreational purposes. Examples can be found for
bridges [3], towers [4] and shelters [5]. Alternatively, they form transformable extensions to static
constructions, such as retractable roof structures [6]
for halls or sports stadiums [7] or adaptable façades
[8]. It is worth mentioning that deployable systems
are also heavily used in the space industry for technologies such as deployable antennas [9], masts and
satellites [10], however produced with very differ-
Copyright © 2019 by Liesbeth I. W. Arnouts, Thierry J. Massart, Niels De Temmerman and Péter Z. Berke.
Published by the International Association for Shell and Spatial Structures (IASS) with permission.
1
Vol. 60 (2019) No. 1 March n. 199
ent manufacturing tolerances and cost compared to
civil engineering structures, and intended for use in
a zero-gravity environment, with often a one-time
deployment. Transformable structures also have
everyday applications for smaller devices, such as
umbrellas and foldable chairs, as medical devices
[11] and in the toy industry [12].
The main common characteristics of all deployable
structures is that they are both mechanisms and
structural systems. As mechanisms, they transform
between multiple predetermined configurations. As
structural systems, they can bear loads in their stable configuration(s). Research on deployable
structures is therefore located at the intersection of
different disciplines. Besides the geometrical and
kinematical aspects (the kinematic design of the
transformation is usually based on rigid wireframe
models), there are also the structural and mechanical aspects (taking the applied loads and structural
compliance into account), which makes the design
problem complex. Furthermore, because deployable
structures are mechanisms, they require tolerances
that are stricter than for usual civil engineering constructions [13]. In the past, most design approaches
were heavily simplified for the ease of application.
Today, the continuously improving computer technology and advanced numerical methods allow for
dealing with highly complex problems [14], making
the use of computational tools efficient and indispensable as part of the design process.
The aim of this contribution is to describe the trends
and challenges in the use of computational tools as
part of the design of deployable structures with a
special focus on bistable structures. Also, an illustration is given of the use of computational tools in
the design of a large bistable deployable dome.
2. DESIGN OF DEPLOYABLE STRUCTURES
2.1.
Classification of deployable structures
Several authors such as A. Hanaor [15] and C.
Gengnagel and N. Burford [16] attempted to classify deployable structures based on their
morphological and kinematical properties, the interested reader can refer to G.E. Fenci and N.G.R.
Currie [1]. In most cases the distinction is made
between skeletal structures and continuous structures. Skeletal structures are e.g. pantographic
structures, bars with articulated joints, ruled surfaces, reciprocal grids, tensegrity or bending-active
structures. Continuous structures are for instance
rigid or curved origami structures, pneumatic struc-
2
tures, tensioned membranes or sliding or retractable
plates.
This contribution focuses on pantographic structures, also called scissor structures. They consist of
scissor-like elements (SLE’s). An SLE is the assembly of two beams connected at a point along
their length by a revolute joint (Fig. 1). This joint
allows large rotations around its hinge axis being
perpendicular to the common plane of the connected beams. By altering the position of this scissor
hinge, three basic unit types are obtained: translational, polar and angulated units. For translational
units (Fig. 1a), unit lines that connect the upper and
lower nodes of one SLE remain parallel during
deployment. For polar (Fig. 1b) and angulated units
(Fig. 1c), they intersect at a single point. For angulated units, the angle between the unit lines is
constant while for polar units, the angle varies during deployment. Units are linked to each other by
the beam extremities. The assembly of different
types of units in the modular deployable structure
results in different unfolded shapes and transformation kinematics.
(a) translational
(b) polar
(c) angulated
Figure 1: A translational, polar and angulated unit.
Scissor structures can be further classified on the
basis of their structural behaviour in foldable, incompatible and bistable structures. Foldable
structures are geometrically compatible throughout
all stages of deployment, which means that their
beams remain straight throughout the transformation process. They therefore behave like pure
mechanisms (no internal stresses are generated during transformation). Once deployed, external
manipulation is required (e.g. bracing) for them to
bear (even small) loads. Incompatible structures are
geometrically incompatible in the folded or deployed configuration, i.e. internal stresses are
always present. These stresses are undesired especially in the deployed configuration because they
make the structure more susceptible to buckling, but
they also can be a desired feature in order to obtain
JOURNAL OF THE INTERNATIONAL ASSOCIATION FOR SHELL AND SPATIAL STRUCTURES: J. IASS
self-deploying structures. Finally, bistable structures are geometrically compatible in the folded and
deployed configuration while they are incompatible
throughout transformation (Fig. 2). The transformation of bistable structures generates internal
stresses that increase and then drop to zero when
going from one stable configuration to the other. It
takes an extra effort to overcome this incompatibility, after which the structure snaps into the next
compatible configuration. When structures are geometrically
incompatible
throughout
the
transformation phase, they are self-locking [17],
since they will attempt to fix themselves into one of
their stable configurations. Other examples of bistable structures are bistable wings [18], morphing
structures [19], and tape springs [20].
Figure 2: Top view of a bistable scissor module made by
K. Roovers [21] going from the compact to the deployed
state. Some elements bend due to angular distortions.
Figure 3: Deployment of a bistable scissor structure made
by K. Roovers [21]. Some modules on the perimeter deform
due to peripheral incompatibilities.
The geometric incompatibilities in bistable scissor
structures during deployment originate from angular distortions or peripheral incompatibilities [21].
Angular distortions occur when the angles between
SLE planes at a central node do not add up to 2π
during transformation (Fig. 2). Peripheral incompatibilities occur during transformation when the
lengths of the beams at the perimeter are insufficient to encircle the grid that lies within (Fig. 3).
During deployment, the surface area of a grid might
increase relatively to the available length of the
perimeter. The modules on the perimeter deform in
order to cope with this relative decrease in perimeter length, leading to an incompatible grid for
certain deployment stages. The size of the snapthrough effect can be altered by changing the ge-
ometry or stiffness of the beams subject to buckling, by adding or removing members or by using
flexible joints or telescopic beams [21]. The complex behaviour of bistable deployable structures
during deployment makes their design complicated,
which is a drawback that is balanced by the benefits
of compact storage, transportability and easy
(un)folding.
2.2.
Trends in the design of scissor structures
The invention of concepts for movable and rapidly
deployable structures dates back several millennia,
with ancient applications in umbrellas, Mongolian
yurts and the Egyptian hunter’s chair [22]. In medieval times, scissor linkages were encountered in
Leonardo da Vinci’s drawings [23]. Throughout
history, particularly the simple plane translational
scissor linkage, called a lazy-tong mechanism, has
been well known and applied.
In the early 1960’s, E.P. Piñero was the first to
build scissor structures with more complex configurations and of larger scales [24]. He designed
mobile theatres, pavilions and exhibition buildings.
Since Piñero’s structures behaved as mechanisms,
stability was achieved by adding cables. Later, F.
Escrig presented the foldability conditions for scissor structures; demonstrated how structures could
be obtained by placing pantographs in a grid; and
showed how curvature could be introduced [25].
Together with J.P. Valcárcel [26] and J. Sánchez
[27], he proposed several geometric models. Some
of his grids exhibited a ‘snap-through’ effect due to
geometric incompatibilities. According to Escrig,
this was a problem because energy input is required
for transformation. T.R. Zeigler was the first to
theoretically identify the ‘snap-through’ phenomenon. He proposed a self-supported spherical dome
which does not require additional members for stabilization [28]. The design aiming for a limited
range of geometric shapes resulted in the existence
of bent elements in the deployed configuration
which make the structure more prone to buckling.
R. Clarke further developed this idea and explained
the origin of the geometric incompatibilities in Zeigler’s dome [29].
W.P. Zalewski and S. Krishnapillai improved Zeigler’s work and found flat and spherical polygonal
modules that were geometrically compatible in the
folded and deployed configuration [30], but bending some of the elements during transformation.
Krishnapillai constructed simple models following
a trial-and-error procedure, verifying the feasibility
3
Vol. 60 (2019) No. 1 March n. 199
of his ideas. Based on this work, R.D. Logcher and
Y. Rosenfeld carried out trial and error approaches
with drawings and physical models [31]. In parallel,
C.J. Gantes, together with J.J. Connor, developed a
geometric design approach and studied the structural response of structures that consist of assembled
polygonal modules, using finite element models
[32]. Meanwhile, C.H. Hernandez used the scissor
principle to construct cylindrical barrel vaults which
behave as mechanisms [33]. Next to the employment of scissor systems in civil engineering
applications, A. Kwan used scissor systems in space
technologies and developed masts and satellite panels [10]. Active cables zig-zagged over small
pulleys, located at the top and bottom nodes of his
structures.
More recently, Y. Akgün and K. Korkmaz, amongst
others, developed a new modified SLE [44]. With
the use of actuators, these scissor elements can be
adapted with changes of the global shape of the
scissor sytem. The polygonal hyperboloids of M. Al
Khayer and H. Lalvani [45] are other remarkable
examples of innovative designs of deployable scissor structures. Other individual researchers are
worth mentioning, among who J. Patel and G.K.
Ananthasuresh [46], who have explained the geometry and kinematics of scissor systems based on
angulated elements and plates, as well as A. Kaveh
and A. Daravan [47], B.P. Nagaraj et al. [48] and J.S. Zhao and J.-Y. Fengh [49], who made an in
depth geometric and kinematic analysis of planar
and spatial scissor structures.
Using purely geometric approaches, A. Zanardo
[34] and T. Langbecker [35] extended the foldability condition of Escrig to determine the foldability
of scissor structures and to analyse their kinematics.
Also based on the previous studies of Escrig, W.
Chen et al. [36] and E. Gutierrez and J. Valcarcel
[37] developed scissor systems in order to obtain
planar or dome-like structures. I. Raskin and J.
Roorda focused numerically on the stiffness and
stability characteristics of linear structures that behave as mechanisms [38].
Japan has formed a major source of research on
deployable scissor grids with contributions by e.g.
K. Kawaguchi, K. Atake, T. Kokawa and M. Saito.
One of the innovative proposals is the design of a
scissor arch which uses zigzag cables with pulleys
to (un)fold [50]. The research in Japan has led to a
remarkable amount of prototypes, e.g. a deployable
bistable geodesic full sphere [51]. In Belgium, N.
De Temmerman proposed various architecturally
and structurally viable concepts for mobile applications in architecture [52]. T. Van Mele, combined
scissor structures with membrane structures to design covering systems [7]. L. Alegria Mira
developed methods for a structural assessment and
optimization of scissor grids in conceptual design
stages [5]. She developed the concept of the Universal Scissor Component, which can be
reconfigured and reused to generate a wide variety
of scissor grids [53]. A. Koumar designed a deployable scissor arch that enables quick sheltering
solutions for areas struck by disaster [54]. K. Roovers enlarged the spectrum and application
possibilities of scissor structures by revealing the
mathematical principles required to convert surface
geometries into scissor systems [21].
A different type of scissor structures was developed
and commercialized by C. Hoberman [12]. His
designs use angulated units, i.e. units with ‘kinked’
beams, which allow forming closed-loop linkages.
For structural analyses, Hoberman uses a ‘snapshot’
method, where the structure is analysed in a particular state of transformation. The various innovative
sculptures and architectural projects of Hoberman
Associates brought scissor grids to the attention of a
broad public. S. Pellegrino investigated pantographic structures as deployable antennas, together with
Z. You, and made further progress on the angulated
unit by investigating multi-angulated elements [39].
W. Gan and Pellegrino explained the geometry of
structural mechanisms in analytical and numerical
ways [40]. The concept of the multi-angulated element was further developed by P.E. Kassabian by
providing this type of structure with cover elements
which provide, both in the open and closed positions, a weatherproof surface [41]. F.V. Jensen
found that it is possible to remove the angulated
elements and connect the plates directly to each
other [42]. S.D. Guest and S. Pellegrino developed
foldable plate cylinders which fold down to flat
polygons [43].
4
2.3. Computational tools in the design of scissor structures
Developing and analysing scissor structures is a
contemporary subject in engineering research. Often an experimental approach is used on reduced
scale models in which, in the case of bistable scissor structures, researchers try to combine an easy
deployment process and adequate stiffness in the
deployed state. This is achieved by starting from an
initial guess and iteratively fine-tuning it to get a
JOURNAL OF THE INTERNATIONAL ASSOCIATION FOR SHELL AND SPATIAL STRUCTURES: J. IASS
good balance between having a self-locking effect
for stability with a limited force needed for deployment. Trial and error experiments on reduced
scale models [21,31,51] give a valuable insight in
the structural behaviour, but they are inefficient as a
design approach for large civil engineering structures.
Computational tools can contribute significantly to
a deeper understanding of the complex transformation behaviour of bistable deployable structures
and to the development of a rigorous design methodology
[17].
Designing
the
complete
transformation behaviour in terms of kinematics
and applied forces is a prerequisite for conceiving
practical applications, which are currently rare in
civil engineering. The design of bistable deployable
structures relies on different computational tools:
computer-aided design and nonlinear structural
analysis.
Computer-aided drawing and modelling can be used
to produce drawings and renderings. Moreover, it
simplifies the complicated geometric design process
and it allows finding the geometry parametrically.
The geometric design approach for scissor structures is based on the requirement for geometric
compatibility in the deployed as well as in the folded configuration, which means that the beams are
located theoretically on one line in the folded configuration. This can be translated into the
deployability constraint [25]:
(1)
efficient and intuitive design approach can be provided by integrating graphical design methods in
digital design environments in the form of parametric models [21]. The locus of all valid intermediate
hinges that comply with the deployability constraint
is an ellipse, with the common end nodes of both
units as its focal points [52]. This graphic representation of the deployability constraint makes it
possible to draw the geometry of scissor structures
using ellipses or circles, depending on the desired
shape (Fig. 7).
Computational structural analysis (linear for the
deployed state and non-linear for the transformation) allows assessing the structural performance
of a given design. Since scissor structures usually
have a complicated 3D geometry and are subject to
large geometric and structural transformations, no
efficient analytical alternative exists to numerical
modelling. Finite element models can be used to do
a linear analysis of scissor structures in the deployed state subjected to service loads (e.g. wind,
snow) [5,54]. A nonlinear analysis of bistable or
incompatible scissor structures is required during
deployment, subjected to large displacements and
geometric incompatibilities [32]. Although some of
the researchers in the past used the finite element
method to perform structural analyses in the deployed state, modelling the complete transformation
cycle of scissor structures is very scarce, since the
underlying phenomena and modelling tools and
concepts are complex. Nevertheless, simulation of
the deployment process is a crucial part of the analysis,
requiring sophisticated
computational
modelling techniques.
2.4. Challenges in the design of bistable deployable structures
Figure 4: Deployability constraint.
This constraint (Fig. 4) ensures that all units in the
assembly simultaneously reach their most compact
state. Using this constraint, together with other geometric constraints [17,21,52], a rigid wireframe
model (i.e. neglecting the compliance of the structural elements) can be obtained that is compatible in
the folded and deployed configuration. Verifying
mathematical expressions indicates if a given assembly is compatible. It is however impractical as a
tool to search for new design possibilities or to
study a variety of shapes, due to the amount of parameters and equations that arise using irregular
units or when creating larger assemblies. A more
Analytical or numerical work that investigates the
behaviour of bistable scissor structures is inherently
complex. Often a trial and error approach is used
which is laborious and does not allow to fully understand their behaviour [21,31,51]. Additionally,
such approaches are not a general design methodology and limit the number of design alternatives.
Other approaches were based on the modelling of
structures without discrete joints or imperfections,
based on a large number of simplifications.
The difference in the behaviour of bistable scissor
structures during deployment and in the deployed
configuration prevents the formulation of a straightforward, simple design methodology (Fig. 5). To
obtain feasible designs, a bistable deployable struc-
5
Vol. 60 (2019) No. 1 March n. 199
ture should be lightweight, flexible during deployment and should provide sufficient stiffness in the
deployed state. These desired requirements are often contradicting, and the challenge is thus to find
an acceptable compromise in the computational
design. A low mass and a compact structure in the
folded state are important for transportation. Flexibility of specific members during deployment is
important to ensure a limited force needed for deployment and additionally an elastic material
behaviour must be ensured. Not taking the transformation stage into account could lead to
inefficient, expensive and potentially unfeasible
designs. Sufficient stiffness in the deployed state is
needed to bear self-weight and possibly service
loads without needing external bracing.
Figure 5: Conflicting requirement during transformation
and in the deployed configuration, as presented by C.J.
Gantes [55]. The magnitude of the geometric incompatibilities is given in function of the utilisation ratio, which is the
actual performance value divided by the maximum allowable performance value.
Optimisation methods have been developed for
structures with contradicting objectives, subject to
several constraints. These methods can solve a wide
range of problems, but their drawback is that each
iteration requires a structural analysis with a considerable computational effort. A number of
remedies have been suggested by Gantes to reduce
the number of required exact analyses to optimise
bistable scissor structures [17]: breaking down the
problem into a number of smaller sub-problems,
using approximations and simplified models. In the
above design approach however, there are several
iterations which have to be performed manually,
making it difficult to automatize.
Because of the progress in computational efficiency
over the recent years, several researchers in the
6
field of pantographic structures did an effort to use
optimisation methods to solve a variety of problems. Foldable scissor structures were optimised by
Z. You to minimise the weight or maximise the
stiffness [56]. An initial effort optimising bistable
scissor structures was done by C. Gantes [57]. Genetic algorithms were used to minimise the weight
by solving a non-linear problem that simulates the
deployment and a linear analysis. A. Kaveh minimised the weight of scissor structures which behave
as mechanisms [58]. For movable bridges composed of linkages, A.P. Thrall employed shape and
size optimisation to minimise the self-weight and
the power required for operation. She also used size
and shape optimisation for pantographic structures
to minimise the weight [59]. L. Alegria Mira did a
topology, shape and size optimisation of the Universal Scissor Component to minimise its weight
[53]. A. Koumar performed for the first time a multi-objective optimisation of a barrel vault, leading to
a pareto front with several optimal solutions. The
mass and the compactness were minimised [54].
Structural optimisation can guide the designer in
choosing a feasible design or can suggest improvements, without requiring a trial and error approach.
One of the main challenges in the field of bistable
deployable structures today is to efficiently perform
a multi-objective size and shape optimisation to
obtain lightweight bistable scissor structures that
provide stiffness in the deployed state and flexibility during deployment by choosing the best
combinations of the geometry, the cross-sections of
the beams and the material properties.
3. EXAMPLE OF THE USE OF COMPUTATIONAL TOOLS IN THE DESIGN OF A
BISTABLE SCISSOR STRUCTURE
To illustrate how computational tools are used in
the particular case of bistable deployable structures,
a large structural example is presented in the following paragraphs. A bistable deployable dome
with a radius of 5 m is designed which consists of a
triangulated double layer grid of polar units with
concurrent unit lines (final design shown on Fig.
12). The structure is bistable due to angular incompatibilities. By consequence, the structure is a selflocking doubly curved dome with a triangulated
grid that can be reduced to a compact bundle of
bars.
JOURNAL OF THE INTERNATIONAL ASSOCIATION FOR SHELL AND SPATIAL STRUCTURES: J. IASS
Figure 6: A flowchart of the design process.
This structure could potentially be used as a mobile
and temporary structure for disaster relief, military
operations or for exhibition or recreational purposes.
The complete design process is given in Fig. 6. The
first part is the geometric design, in which the rigid
wireframe design, i.e. the beam lengths and the
position of the hinges, is defined. The second part is
the structural design, in which the bistable scissor
structure is structurally analysed during transformation and in the service state using non-linear
finite element modelling.
3.1.
Geometry and kinematics
Digital parametric design environments present a
powerful tool to handle the high degree of complexity of the design process of scissor structures in an
interactive and efficient manner. General design
methods for the rigid wireframe design of certain
types of scissor units can be translated into algorithms, which translate user-defined input
parameters into a deployable scissor grid in a stepwise and interactive manner. It can provide direct
visual and numerical feedback on the solution
throughout the process. By changing the input parameters, a myriad of solutions can quickly be
generated and explored. The various alternatives
can efficiently be evaluated and compared, leading
to better informed designs. These tools can also be
extended e.g. to simulate the deployment or to per-
form linear elastic analyses [5].
The computational tool used in this contribution is
the 3D modelling software Rhinoceros® and its
parametric design plug-in Grasshopper® [60].
Grasshopper’s visual interface, relying on a visual
programming approach, knows various plug-ins
such as the live-physics plug-in Kangaroo which
can be used for form finding, for the optimisation of
meshes and scissor grids, and for deployment simulations that are not based on finite elements [61]. By
using Kangaroo, the transformation of scissor structures can be approximated by allowing rotations
between the beams, however the structural stiffness
and the deformability of the elements is not properly considered at this stage. Rigorous finite element
modelling is required to capture the correct transformation behaviour.
In the geometric design, two stable states are considered i.e. the folded and the deployed state. The
movement between those two configurations (the
kinematics) is of course implicitly present and is
properly incorporated only in the non-linear finite
element analysis in the structural design stage.
For three-dimensional scissor grids, the deployability constraint is translated into networks of tangent
rotational ellipsoids placed along a surface. However, in some specific cases, such as for flat structures
or the designed dome, this graphical representation
can be simplified to a circle packing (Fig. 8a),
Figure 7: A geodesic pattern covered by a circle packing (blue) and folding stages of the resulting bistable deployable
dome. The red members bend during deployment due to angular incompatibilities.
7
Vol. 60 (2019) No. 1 March n. 199
which is a configuration of circles with specified
patterns of tangency [21]. Each pair of tangent circles (ci;cj) holds an SLE with unit lines normal to
the circles and going through the respective centres
(Ci;Cj) of the circles. The intermediate hinge (revolute joint) Ii of the SLE is located at the point where
the circles touch (Fig. 8b). In the case of this dome,
all of the intermediate hinge points will lie on the
surface of the sphere and all of the unit lines intersect at the centre O of the sphere.
the bistable deployable dome in the deployed configuration subject to service loads as well as during
deployment. In general, the larger this angle, the
higher the stiffness of the dome in the deployed
state and the higher the force required for deployment.
Figure 9: Bistable scissor grids based on the same circle
packing with a different angle between their beams in the
deployed configuration.
(a) circle packing
Spheres can be covered by an infinite variety of
circle packings and therefore give rise to several
scissor grids (Fig. 10). First, an initial grid (i.e. a
network of lines) is designed on the spherical surface, which is optimised to hold a circle packing
and then populated by scissor units (Fig. 7). The
design freedom therefore largely lies in shaping the
initial grid. It can result from e.g. an arbitrary point
cloud, projecting a planar grid onto a sphere or a
geodesic pattern. For this example of a bistable
deployable dome, the scissor grid is based on a
geodesic pattern (Fig. 7).
(b) tangent circles c0 to c2; triangle formed by points T0
to T2 tangent to the base sphere at points I0 to I2
Figure 8: Populating a spherical circle packing with polar
scissor units, based on the work of K. Roovers [21].
Figure 10: Bistable scissor grids based on different initial
grids.
For a given spherical circle packing for a bistable
deployable dome, there is one free parameter to
choose, which is the angle φ between the two
beams of an SLE in the deployed configuration
(Fig. 8b). The design satisfies the deployability
constraint (Eq. 1) because this angle is the same for
all of the SLE’s in the structure, as the intermediate
hinges are located at the points where the circles
touch and as the unit lines go through the centres of
the circles as well as through the centre of the
sphere. Changing the angle φ changes outer and
inner dimensions of the dome (Fig. 9). Varying this
parameter also changes the structural behaviour of
In general, the denser the circle packing (i.e. the
shorter the beams of the scissor structure), the higher are the stiffness of the dome in the deployed state
and the force required for deployment. Increasing
the density of a circle packing does not change the
global motion because the same geometry is enveloped when the structure is deployed. Nevertheless,
the number of elements and joints are increased by
using a denser circle packing, which makes the
structure locally more complex. Moreover, since
finite elements are used to analyse the structure
during transformation, the number of the degree of
freedom of the system increases when the number
8
JOURNAL OF THE INTERNATIONAL ASSOCIATION FOR SHELL AND SPATIAL STRUCTURES: J. IASS
of joints and elements increases. The parameters
that can be fine-tuned to obtain a feasible design
with a given external shape and dimensions of the
structure, are the angle φ between the two beams of
an SLE in the deployed configuration (Fig. 9) and
the initial grid to hold a circle packing (Fig. 10).
chosen in the design procedure, which corresponds
to realistic beam cross-sections for scissor structures [5,17,52]. The mass of the structure is 360.5
kg.
In the previous line models, the joints were represented by points. Joints connect several SLE’s at
their end points in a three-dimensional configuration and are of finite size. To represent realistic
hubs, joint lines are added to the line model that
create space between the different rotation axes of
the beams. A relatively simple method is chosen to
include in the parametric design algorithm and that
maintains the concurrency of the unit lines throughout transformation i.e. the unit lines intersect in a
single point during the whole transformation phase
[21]. One parameter can be chosen, which is the
angle λ (Fig. 11). A disadvantage of this method is
the resulting amount of different hubs that is needed
to build the structure because the lower hubs are
smaller than the upper hubs.
(a) top view
(b) side view
(c) 3D view
Figure 12: Final geometric design of the dome with hinges
and openings.
Figure 11: Joints in the line model such that the unit lines
and the central axes of the joints are concurrent. Based on
the work of K. Roovers [21].
The design of the proposed closed dome is realistic
for a roof type of application. For a temporary tentlike structure however, as aimed for here, openings
have to be created. The desirable amount of SLE’s
on the bottom of the dome can be removed to open
space for entrance on the dome surface. In this example, five SLE’s were removed, creating openings
with a height of 2.5 m. The top view, side view and
3D view of the resulting structure is shown in Fig.
12. The dome has an outer diameter of 12.85 m and
a total height of 6.05 m. The inner diameter of the
dome is 9.6 m and the inner height is 4.9 m.
The beams of the structure are made of aluminium
with a Young’s modulus of 70 GPa, a poisson’s
ratio of 0.35, a density of 2700 kg/m³ and a yield
strength of 160 MPa. Hollow rectangular aluminium beam cross-sections of 25x50x5 mm were
3.2.
Structural analysis
The same software used for the kinematic design
(Rhinoceros) has a plugin Karamba3D that can be
applied for linear structural analysis using the finite
element method [62]. The advantage of doing the
geometric/kinematic and preliminary linear structural design in the same software is that there is
immediately some structural feedback in the early
design stages [5]. As it is still impossible to analyse
snap-through problems in Rhinoceros and Grasshopper, another finite element software is needed.
In this example, Abaqus is used [63] for the detailed
analysis of the non-linear transformation phase as
well as for the linear analysis of the deployed configuration under service loads. Rhinoceros and
Grasshopper were used to design the structure geometrically, to create the input file for the finite
element model in Abaqus, to run the simulation in
Abaqus and to read the results of the analysis. The
9
Vol. 60 (2019) No. 1 March n. 199
coupling of Rhinoceros/Grasshopper with Abaqus
is done by using in house written Python scripts.
For all of the structural members, 2-noded geometrically nonlinear Timoshenko beam finite elements
in Lagrangian formulation are used. Four beam
finite elements are used to model the semi-length of
each beam, which was verified to be a converged.
The joints, which allow the beams to rotate around
a rotation axis, are simulated with the Abaqus connector type ‘hinge’. The rotation axis of the joints
between the two beams in an SLE is defined perpendicular to the common plane of the two
connected beams and is updated during structural
transformation. This hinge axis co-rotates with the
beams and remains perpendicular to the common
plane of both beams during transformation. 3D
hubs, which connect several beams of different
SLE’s, are modelled here as a stiff grid composed
of short beam elements (Fig. 13) with the Abaqus
connector type ‘hinge’ at their extremities.
2. The maximum deflection is L/100 and the maximum horizontal displacement at any point of
the structure is H/100 with L the length and H
the height of the structure. This constraint is
less strict than for traditional structures [54].
3. An analytical verification is implemented for the
local buckling of each member by taking into
account the normal force and the bending moments around the two axes following Eurocode
9, based on [54].
4. The calculation of the global buckling is based
on an eigenvalue-based buckling analysis of the
model.
The results from the linear analysis of the bistable
scissor structure in the deployed configuration show
that, as expected, the maximum displacement is the
vertical displacement in the lower centre node,
which is 3.6 cm (Fig. 14). This is lower than H/100
since the height of the structure is 6.05 m. The maximum von Mises stress in the beams is 13.7 MPa
(Fig. 15), which is lower than the yield strength of
Aluminium and local buckling has been checked for
all of the structural members.
Figure 13: Dome with finite joint dimensions (left) and
detail of the stiff grid of beams that represents the hubs
(right). The rotation axes of the hinges are given by the
small lines perpendicular to the beams.
The lower corner points of the structure are fixed in
the vertical direction as if the structure was standing
on a surface. To prevent rigid body modes, the lower centre point is allowed to move only in the
vertical direction while rotations around the vertical
axis are prohibited. In the deployed configuration,
the only applied force is gravity. This corresponds
to a bistable scissor structure just after deployment,
before the structure is fixed to the ground. In real
life applications wind-bracing could be added before the structure is subjected to other service loads
such as wind or snow. Therefore, these are disregarded in this simplified simulation.
Figure 14: The displacement of the beams on the deformed
configuration of the dome subjected to gravity.
To obtain a feasible structure in the deployed configuration, the following constraints are considered:
1. The maximum stress is less than the yield stress
of the material.
10
Figure 15: The von Mises stress in the beams of the dome
subjected to gravity.
JOURNAL OF THE INTERNATIONAL ASSOCIATION FOR SHELL AND SPATIAL STRUCTURES: J. IASS
Buckling load magnitudes and associated buckling
modes are calculated for global buckling of the
structure. The obtained buckling load factor is the
number by which the applied load must be multiplied to obtain the buckling load magnitude and the
corresponding buckling mode presents the shape of
the structure when it buckles. The first buckling
mode (Fig. 16) which has the smallest buckling
load magnitude has a buckling load magnitude of
8.84, which means that the structure will buckle
when the magnitude of the applied loads is the
magnitude of the gravity loads multiplied by 8.84.
Figure 16: The first buckling mode of the structure subjected to gravity.
The results of the analysis in the deployed configuration show that the structure will be a feasible
structure just after deployment since it bears gravity
loads without needing further bracing. To be able to
use the dome in the deployed configuration as a
structure, additional computations should be performed in which realistic fixations to the ground,
wind-bracing and a cover (e.g. a membrane attached to the structure) are be considered. Doing so
is technically straightforward, requiring the modification of the FE input files only, however it relies
on making complex design choices (e.g. membrane
attachment) and is therefore out of the scope of this
contribution.
To simulate the transformation phase, the deployed
configuration is used as the initial configuration,
because in the geometrically ideal folded configuration all nodes would lie on a straight line and
because scissor structures are designed geometrically in the deployed configuration. The structure is
folded to its most compact configuration in force
driven simulations using the modified Riks solution
strategy to capture the snap-through. Around 60
loading steps are used to compute the load magnitude and its nonlinear evolution during
transformation. The finite element model has 1500
elements, 4732 nodes and 10752 variables. The
computational time of a single folding simulation is
around 1 minute.
Figure 17: Applied loads to fold the bistable deployable
dome.
The lower corner points of the structure are fixed in
the vertical direction. Several sets of forces can be
used to transform scissor structures. In this example, horizontal radial forces are applied to the upper
corner points of the structure and an additional vertical force is needed on the lower centre point of the
structure (Fig 17). The horizontal forces have the
same magnitude. The vertical force is 200 times
larger than the horizontal forces to be able to fold
the structure. Gravity is also taken into account
during deployment.
(a) Total applied load vs. displacement during transformation.
(b) Deformed configurations during folding.
Figure 18: Transformation behaviour.
11
Vol. 60 (2019) No. 1 March n. 199
The required total folding force (i.e. the sum of the
magnitudes of the forces) is given as a function of
the displacement in Fig. 18a and the structural configurations corresponding to points A, B, C, D and
E are shown in Fig. 18b. Point A corresponds to the
initial deployed configuration with gravity applied.
Due to the geometric incompatibilities, some of the
beams bend, requiring an increasing force to fold
the structure (A-B). The maximum folding load is
reached at point B, followed by a snap-through
from B to D. In point C is an unstable configuration
in which no externally applied force is required.
The required load to keep on folding becomes negative (i.e. changes direction) from C to E. This
implies that without this restraining force, the structure would ‘snap’ to point E in a dynamic fashion.
Between points D and E, the magnitude of the vertical restraining force decreases because the bent
beams become straight again and internal stresses
relax. Point E can be considered as the final folded
configuration i.e. a transformed configuration in
equilibrium under gravity load.
To obtain a feasible mechanism during deployment, the following constraints are considered:
1. The maximum stress during deployment is lower
than the yield stress of the material.
2. The maximum negative load during the folding
process should be lower than zero, so the structure does not unfold under gravity from the
folded configuration.
Figure 19:Maximum von Mises stress as a function of the
displacement of the upper center node.
The results of the analysis during transformation
show that the structure is realizable. The maximum
von Mises stress during transformation is 139 MPa.
(Fig. 19), which is below the yield strength of Aluminium (depending on the alloy). Figure 19 shows
graphically how internal stresses build up and relax
during transformation. In this case, the deployment
is more critical than the deployed configuration in
12
terms of stress levels reached. The maximum negative load during the folding process (point D in Fig.
18a) is well below zero.
For this example, the mass of the structure is 360.5
kg and the required folding force is 1.52 kN. This
means that the transformation cannot be controlled
by human power but that machines such as cranes
or a set of cables connected to winches will be
needed for transportation and transformation. To
decrease the mass and the required folding force,
the structure can be optimised by changing the geometry, the material or the cross-sections of the
beams or by changing the way the loads are applied
e.g. by using a cable running over pulleys to transform the structure. Another option is to use two
groups of elements with different cross-sections and
material properties e.g. aluminium for beams that
do not bend during transformation and a material
that can easily bend elastically for the beams that
bend during transformation. Optimising a bistable
scissor structure requires several iterations with the
geometry, cross-sections, materials and applied
loads. Using computational optimisation algorithms
would be an ideal solution for the design of bistable
scissor structures in terms of mass, stiffness in the
deployed state and flexibility during transformation
and it is part of future work.
4. CONCLUSIONS AND OUTLOOK
The aim of this contribution is to investigate trends
and challenges in the use of computational tools for
the design of bistable deployable structures.
In the past most design aspects had to be heavily
simplified in order to be understood and used. Today, the continuously improving computer
technology and advanced numerical methods allow
for dealing with highly complex problems, making
the use of computational tools crucial and efficient
as part of the design process. The structural assessment of bistable deployable structures relies on
different computational tools: computer-aided design and nonlinear structural analysis. An example
was given of the use of such computational tools for
the design of a large, real life scale bistable deployable dome. Parametric models were used to
geometrically design a dome, which was afterwards
structurally analysed in the deployed configuration
as well as during transformation using finite element models. The importance to take the nonlinear
structural response during transformation into account makes the design of bistable structures a nontrivial problem.
JOURNAL OF THE INTERNATIONAL ASSOCIATION FOR SHELL AND SPATIAL STRUCTURES: J. IASS
Generally, as for the example of the bistable deployable dome, the geometry, kinematics and
structural analysis are developed and investigated as
separate entities with different tools or software
packages. However, the desired requirements to
obtain a lightweight structure which is flexible during deployment while offering sufficient stiffness in
the deployed state are often contradicting and the
challenge is to find an acceptable compromise between the geometry, cross-sections and materials of
the beams. To do this, the geometric design and
structural analysis in the deployed state as well as
during transformation have to be considered. As
future development, an optimisation approach to
such structures is envisioned which links the geometry to structural and transformation analyses.
Next to the use of computational tools in the design
of bistable scissor structures, a realizable design
should include a geometric design and a cover
adapted to its function, detailing of the beams, connections and hubs, and the use of physical scale
models and prototypes.
Journal, vol. 6, pp. 50-64, 2012.
DOI: 10.2174/1874155X01206010050
[5]
L. Alegria Mira, A. P. Thrall and N. De
Temmerman, "Deployable scissor arch for
transitional
shelters,"
Automation
in
Construction, vol. 43, pp. 123-131, 2014.
DOI: 10.1016/j.autcon.2014.03.014
[6]
J. Cai, J. Feng and C. Jiang, "Development
and analysis of a long-span retractable roof
structure," Journal of Constructional Steel
Research, vol. 92, pp. 175-182, 2014.
DOI: 10.1016/j.jcsr.2013.09.006
[7]
T. Van Mele, N. De Temmerman, L. De Laet
and M. Mollaert, "Scissor-hinged retractable
membrane structures," International Journal
of Structural Engineering, vol. 1, no. 3/4, pp.
374-396, 2010.
DOI: 10.1504/IJStructE.2010.033489
[8]
A. Vergauwen and N. De Temmerman,
"Analysing the applicability of deployable
scissor structures in responsive building
skins," WIT Transactions on the Built
Environment, vol. 124, pp. 493-504, 2012.
DOI: 10.2495/HPSM120441
[9]
A. Kwan, "A Parabolic Pantographic
Deployable Antenna," International Journal
of Space Structures, vol. 10, no. 4, pp. 195203, 1995.
DOI: 10.1177/026635119501000402
ACKNOWLEDGMENTS
This work was supported by a Research Fellow
(ASP – Aspirant) fellowship of the Fund for Scientific Research – FNRS (F.R.S.-FNRS) (grant
number FC 23469).
REFERENCES
[1]
[2]
G. E. Fenci and N. G. R. Currie, "Deployable
structures
classification:
A
review,"
International Journal of Space Structures,
vol. 32, no. 2, pp. 1-19, 2017.
DOI: 10.1177/0266351117711290
N. De Temmerman, L. Alegria Mira and A.
Vergauwen, "Transformable structures in
architectural engineering," WIT Transactions
on The Built Environment, vol. 124, pp. 457468, 2012.
DOI: 10.2495/HPSM120411
[3]
I. Ario, M. Nakazawa, Y. Tanaka, I. Tanikura
and S. Ono, "Development of a prototype
deployable bridge based on origami skill,"
Automation in Construction, vol. 32, pp. 104111, 2013.
DOI: 10.1016/j.autcon.2013.01.012
[4]
Z. Cai, H.-C. Zhao, J.-S. Zhao and F.-L. Chu,
"Stiffness of a Foldable Tower for Wind
Turbine," Open Mechanical Engineering
[10] A. Kwan, Z. You and S. Pellegrino, "Active
and
Passive
Cable
Elements
in
Deployable/Retractable Masts," International
Journal of Space Structures, vol. 8, no. 1&2,
pp. 29-40, 1993.
DOI: 10.1177/0266351193008001-204
[11] K. Kuribayashi, K. Tsuchiya, Z. You, D.
Tomus, M. Umemoto, T. Ito and M. Sasaki,
"Self-deployable origami stent grafts as a
biomedical application of Ni-rich TiNi shape
memory alloy foil," Material Science and
Engineering: A, vol. 419, pp. 131-137, 2006.
DOI: 10.1016/j.msea.2005.12.016
[12] C. Hoberman, "Radial expansion/retraction
truss structures". United States Patent
5,024,031, 18 June 1991.
13
Vol. 60 (2019) No. 1 March n. 199
[13] L. I. W. Arnouts, T. J. Massart, N. De
Temmerman
and
P.
Z.
Berke,
“Computational
modelling
of
the
transformation of bistable scissor structures
with
geometrical
imperfections”,
Engineering Structures, vol. 177, pp. 409420, 2018.
DOI: 10.1016/j.engstruct.2018.08.108
[14] D. Lee, O. Larsen and S. Kim,
"Computational Tools for the Design of a
Deployable Dome Structure: Beyond their
Limits," Structures and Architecture, pp.
267-274, 2016.
DOI: 10.1201/b20891-34
[15] A. Hanaor and R. Levy, “Evaluations of
Deployable Structures for Space Enclosures,”
International Journal of Space Structures,
vol. 16, no. 4, pp. 211-229, 2001.
DOI: 10.1260/026635101760832172
[16] C.
Gengnagel
and
N.
Burford,
“Transformable Structures for Mobile
Shelters,” in Adaptables2006, International
Conference on Adaptive Building Structures,
Eindhoven: TU/e, 2006, pp. 10-280 – 10284.
[17] C. J. Gantes, Deployable Structures: Analysis
and Design, Southampton: WIT Press, 2001.
[18] O. Bilgen, A. F. Arrieta, M. Friswell and P.
Hagedorn, "Dynamic control of a bistable
wing under aerodynamic loading," Smart
Materials and Structures, vol. 22, no. 2, p.
025020, 2013.
DOI: 10.1088/0964-1726/22/2/025020
[19] Q. Guo, H. Zheng, W. Chen and Z. Chen,
"Modeling Bistable behaviors in Morphing
Structures
through
Finite
Element
Simulations," Bio-Medical Materials and
Engineering, vol. 24, pp. 557-562, 2014.
DOI: 10.3233/BME-130842
[20] A. Brinkmeyer, S. Pellegrino and P. M.
Weaver, "Effects of Long-Term Stowage on
the Deployment of Bistable Tape Springs,"
Journal of Applied Mechanics, vol. 83, no. 1,
p. 011008, 2016.
DOI: 10.1115/1.4031618
[21] K. Roovers, Deployable Scissor Grids -
14
Geometry and Kinematics, PhD thesis: Vrije
Universiteit Brussel, 2017.
[22] F. Escrig, "Transformable Architecture,"
Journal of the International Association for
Shell and Spatial Structures, vol. 41, no. 132,
pp. 3-22, 2000.
[23] J. P. Valcárcel and F. Escrig, "Pioneering in
Expandable Structures: the "Madrid I"
Notebook by Leonardo Da Vinci," Bulletin of
the International Association for Shell and
Spatial Structures, vol. 35, no. 114, pp. 3345, 1994.
[24] E. P. Piñero, "Project for a mobile theatre,"
Architectural Design, vol. 12, pp. 154-155,
1961.
[25] F. Escrig, "Expandable Space Structures,"
International Journal of Space Structures,
vol. 1, no. 2, pp. 79-91, 1985.
DOI: 10.1177/026635118500100203
[26] F. Escrig and J. P. Valcárcel, "Geometry of
Expandable Space Structures," International
Journal of Space Structures, vol. 8, no. 1&2,
pp. 71-84, 1993.
DOI: 10.1177/0266351193008001-208
[27] F. Escrig and J. Sanchez, "New designs and
geometries of deployable scissor structures,"
in Adaptables2006, International Conference
on Adaptive Building Structures, Eindhoven:
TU/e, 2006, pp. 18-22.
[28] T. R. Zeigler, "Collapsible self-supporting
structure". United States Patent 3,968,808,
1976.
[29] R. Clarke, "The kinematics of a novel
deployable space structure system," in
Proceedings of the 3rd International
Conference on Space Structures, 1984.
[30] A. Krishnapillai, "Deployable structures".
United States Patent 5,167,100, 1992.
[31] Y. Rosenfeld, Y. Ben-Ami and R. D.
Logcher, "A Prototype "Clicking" ScissorLink Deployable Structure," International
Journal of Space Structures, vol. 8, no. 1-2,
pp. 85-95, 1993.
DOI: 10.1177/0266351193008001-209
JOURNAL OF THE INTERNATIONAL ASSOCIATION FOR SHELL AND SPATIAL STRUCTURES: J. IASS
[32] C. J. Gantes, J. J. Connor, R. D. Logcher and
Y. Rosenfeld, "Structural Analysis and
Design of Deployable Structures," Computers
and Structures, vol. 32, no. 3/4, pp. 661-669,
1989.
DOI: 10.1016/0045-7949(89)90354-4
[33] C. H. Hernandez, "New ideas on deployable
structures," in Mobile and Rapidly Assembled
Structures II, vol. 21, F. Escrig and C.
Brebbia, Eds., Southampton, Computational
Mechanics Publications, 1996, pp. 64-72.
[34] A. Zanardo, "Two-dimensional articulated
systems developable on a single or double
curvature surface," Meccanica, vol. 21, no. 2,
pp. 106-111, 1986.
DOI: 10.1007/BF01560628
vol. 220, no. 7, pp. 1045-1056, 2006.
DOI: 10.1243/09544062JMES245
[41] P. E. Kassabian, Z. You and S. Pellegrino,
"Retractable
Roof
Structures,"
in
Proceedings of the ICE - Structures and
Buildings, 1999.
[42] F. V. Jensen, Concepts for retractable roof
structures, PhD thesis: University of
Cambridge, 2004.
[43] S. D. Guest and S. Pellegrino, “The folding
of triangulated cylinders, Part I: Geometric
considerations,” ASME Journal of Applied
Mechanics, no. 61, pp. 773-777, 1994.
DOI: 10.1115/1.2901553
[35] T. Langbecker and F. Albermani, "Kinematic
and non-linear analysis of foldable barrel
vaults," Engineering Structures, vol. 23, pp.
158-171, 2001.
DOI: 10.1016/S0141-0296(00)00033-X
[44] Y. Akgün, C. J. Gantes, W. Sobek, K.
Korkmaz and K. Kalochairetis, "A novel
adaptive spatial scissor-hinge structural
mechanism
for
convertible
roofs,"
Engineering Structures, vol. 33, no. 4, pp.
1365-1376, 2010.
DOI: 10.1016/j.engstruct.2011.01.014
[36] W. Chen, G. Fu, J. Gong, Y. He and S. Dong,
"A new design conception for large span
deployable flat grid structures," International
Journal of Space Structures, vol. 17, no. 4,
pp. 293-299, 2002.
DOI: 10.1260/026635102321049556
[45] M. Al Khayer and H. Lalvani, "Scissorsaction deployables based on space-filling
polygonal hyperboloids," in IUTAM-IASS
Symposium on Deployable Structures:
Theory and Applications, 1998.
[37] E. Gutierrez and J. Valcarcel, "Generation of
foldable domes formed by bundle modules
with quadrangular base," Journal of the
International Association for Shell and
Spatial Structures, vol. 43, no. 140, pp. 133140, 2002.
[38] I. Raskin and J. Roorda, "Buckling force for
deployable
pantographic
columns,"
Transactions on the Built Environment, vol.
21, 1996.
[39] Z. You and S. Pellegrino, "Foldable bar
structures," International Journal of Solids
and Structures, vol. 34, no. 15, pp. 18251847, 1997.
DOI: 10.1016/S0020-7683(96)00125-4
[40] W. Gan and S. Pellegrino, "Numerical
approach to the kinematic analysis of
deployable structures forming a closed loop,"
Journal of Mechanical Engineering Science,
[46] Y. Rosenfeld and R. D. Logcher, "New
Concepts
for
Deployable-Collapsable
Structures," International Journal of Space
Structures, vol. 3, no. 1, pp. 20-32, 2007.
DOI: 10.1177/026635118800300103
[47] A. Kaveh and A. Davaran, "Analysis of
pantograph foldable structures," Computers
and Structures, vol. 59, no. 1, pp. 131-140,
1996.
DOI: 10.1016/0045-7949(95)00231-6
[48] B. P. Nagaraj, R. Pandiyan and A. Ghosal,
"Kinematics
of
pantograph
masts,"
Mechanisms and Machine Theory, vol. 44,
no. 4, pp. 822-834, 2009.
DOI:10.1016/j.mechmachtheory.2008.04.004
[49] J.-S. Zhao, J.-Y. Wang, F. Chu, Z.-J. Feng
and J. S. Dai, "Structure synthesis and statics
analysis of a foldable stair," Mechanism and
Machine Theory, vol. 46, no. 7, pp. 9981015, 2011.
15
Vol. 60 (2019) No. 1 March n. 199
DOI:10.1016/j.mechmachtheory.2011.02.001
[50] T. Kokawa, "A Trial of Expandable Arch," in
Spatial structures: heritage, present and
future, vol. 1, G. Giuliani, Ed., Milan, IASS,
1995, pp. 501-510.
[51] K. Kawaguchi and T. Sato, "Development of
Deployable Geodesic Full Sphere with
Scissors Members," in Proceedings of the
IASS 2015 Symposium - Future Visions,
Amsterdam, 2015.
[52] N. De Temmerman, Design and analysis of
deployable bar structures for mobile
architectural applications, PhD thesis: Vrije
Universiteit Brussel, 2007.
[53] L. Alegria Mira, A. P. Thrall and N. De
Temmerman,
"The
universal
scissor
component: Optimization of a reconfigurable
component for deployable scissor structures,"
Engineering Optimization, vol. 48, no. 2, pp.
317-333, 2015.
DOI: 10.1080/0305215X.2015.1011151
[54] A. Koumar, T. Tysmans, R. F. Coelho and N.
De Temmerman, "An Automated Structural
Optimisation Methodology for Scissor
Structures Using a Genetic Algorithm,"
Applied Computational Intelligence and Soft
Computing, 2017.
DOI: 10.1155/2017/6843574
[55] C. J. Gantes, "Overview of scissor-type, bistable deployable structures," in Proceedings
of the IASS Symposium 2018 - Creativity in
Structural Design, Boston, 2018.
[56] Z. You, "Sensitivity Analysis Based on the
Force Method for Deployable CableStiffened
Structures,"
Engineering
Optimization, vol. 29, no. 1-4, pp. 429-441,
1997.
DOI: 10.1080/03052159708941006
[57] C. J. Gantes, P. G. Georgiou and V. K.
Koumousis, "Optimum design of deployable
structures
using genetic algorithms,"
Transactions on the Built Environment, vol.
35, pp. 255-264, 1998.
[58] A. Kaveh and S. Shojaee, "Optimal Design of
Scissor-Link Foldable Structures Using Ant
16
Colony Optimization Algorithm," ComputerAided Civil and Infrastructure Engineering,
vol. 22, no. 1, pp. 56-64, 2007.
DOI: 10.1111/j.1467-8667.2006.00470.x
[59] A. Thrall, M. Zhu, J. K. Guest, I. PayaZaforteza and S. Adriaenssens, "Structural
Optimization of Deploying Structures
Composed of Linkages," Journal of
Computing in Civil Engineering, vol. 28, no.
3, pp. 04014010-1-11, 2014.
DOI:10.1061/(ASCE)CP.1943-5487.0000272
[60] Robert McNeel & Associates, "Rhinoceros 5
[computer software]," 2017. [Online].
Available: http://www.rhino3d.com.
[61] D. Piker, "Kangaroo Physics [computer
software]," 2017. [Online]. Available:
http://kangaroo3d.com.
[62] C. Preisinger, "Linking Structure and
Parametric
Geometry,"
Architectrural
Design, vol. 83, no. 2, pp. 110-113, 2013.
DOI: 10.1002/ad.1564
[63] ABAQUS, ABAQUS Documentation, RI,
USA: Dassault Systèmes, 2011.