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On the behaviour of bending-active plate structures

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By distinguishing bending-active structures based on the geometric dimension of their elements, 1-d and 2-d systems can be identified. Rods and gridshells typically belong to the first category, plates to the latter. Plate-based bending-active structures present limited formability compared to rods due to the higher out-of-plane rotational inertia of the plates’ cross-section. Nonetheless, by following simple generative rules it is possible to substantially expand the formal possibilities of bending-active plates. This approach was tested and employed for the construction of a series of full-scale prototypes in order to demonstrate the wide range of shapes that can be achieved in this way. The research conducted so far was primarily focused on the main geometric and mechanical aspects associated to the form-finding process. The current paper looks instead into the global structural behaviour of bending-active plate shells. The process of converting an arbitrary freeform surface into a buildable plate shell requires the introduction of voids in the surface to allow the bending process to take place. The effect of this operation on the global mechanical behaviour of the shell will be analysed and discussed. Considerations on the scalability and buckling of bending-active plate structures will be also presented to highlight the potential of this approach to be employed for larger structural systems.
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Proceedings of the IASS Annual Symposium 2017
“Interfaces: architecture . engineering . science”
September 25 - 28th, 2017, Hamburg, Germany
Annette Bögle, Manfred Grohmann (eds.)
Copyright © 2017 by Riccardo LA MAGNA and Jan KNIPPERS
Published by the International Association for Shell and Spatial Structures (IASS) with permission.
On the behaviour of bending-active plate structures
Riccardo LA MAGNA*, Jan KNIPPERSa
* Institut für Tragkonstruktionen und Konstruktives Entwerfen (ITKE), Keplerstr. 11 70174, Stuttgart, Germany
ric.lamagna@gmail.com
a Institut für Tragkonstruktionen und Konstruktives Entwerfen (ITKE)
Abstract
By distinguishing bending-active structures based on the geometric dimension of their elements, 1-d
and 2-d systems can be identified. Rods and gridshells typically belong to the first category, plates to
the latter. Plate-based bending-active structures present limited formability compared to rods due to the
higher out-of-plane rotational inertia of the plates’ cross-section. Nonetheless, by following simple
generative rules it is possible to substantially expand the formal possibilities of bending-active plates.
This approach was tested and employed for the construction of a series of full-scale prototypes in order
to demonstrate the wide range of shapes that can be achieved in this way.
The research conducted so far was primarily focused on the main geometric and mechanical aspects
associated to the form-finding process. The current paper looks instead into the global structural
behaviour of bending-active plate shells. The process of converting an arbitrary freeform surface into a
buildable plate shell requires the introduction of voids in the surface to allow the bending process to
take place. The effect of this operation on the global mechanical behaviour of the shell will be analysed
and discussed. Considerations on the scalability and buckling of bending-active plate structures will be
also presented to highlight the potential of this approach to be employed for larger structural systems.
Keywords: Bending-active, plates, form-finding, shells
1. Introduction
A good example for the new possibilities emerging from a physically informed digital design process is
the research on bending-active structures. Bending-active describes a wide range of structural systems
that employ large deformations as a form giving and self-stabilizing strategy (Lienhard [1]). These
systems use the elastic deformation of planar, off-the-shelf building materials to generate structures
emerging from the combination of curved elements (Knippers et al. [2]). While the traditional maxim
in engineering is to limit the amount of bending in structures, this typology actually harnesses bending
for the creation of complex and extremely lightweight designs.
Many aspects qualify a structure as bending-active, the large elastic deformation of its constituent
elements being the most prominent and the main constraint for a proper design. To maintain the strain
within the elastic limits of the material, the building elements must be necessarily thin and slender,
making this type of structures extremely lightweight. The requirement of thin cross-sections is opposed
by the necessity of withstanding external loads to be considered structurally efficient. This inherent
contradiction is at the heart of bending-active systems, which need to counterbalance the drastic
reduction of available material by employing alternative measures. The answer to this lies in the
geometric configuration of the structure, as different assemblies have a major influence on the global
behaviour of the system. Geometry is therefore the fundamental aspect for the correct design of bending-
active structures. In the current paper the mechanical behaviour of elastically formed plate-based
systems will be studied. The research presented here focuses in particular on the trade-offs between
geometry, fabrication, assembly, and structural capacity of plate-based bending-active structures.
Proceedings of the IASS Annual Symposium 2017
Interfaces: architecture.engineering.science
2
Figure 1: The form conversion principle. By strategically removing material from the surrounding, the radial
strips of the plate are free to bend in multiple directions
2. The form conversion approach
Bending-active structures can be classified based on their geometric properties. By categorising
bending-active systems depending on the geometrical dimension of their constituent elements, it is
possible to distinguish between 1D and 2D systems. Rods and elastic gridshells typically belong to the
first category, whilst plate structures to the latter. Structural systems that achieve form and stability
through the elastic deformation of thin plates fall into the category of bending-active plate structures.
The most prominent example can be found in Buckminster-Fuller’s 1957 Plydome (Fuller [3]) and more
examples have been emerging recently. Plate-dominant bending-active structures have not yet received
much attention and are generally considered difficult to design (Schleicher and La Magna [4]). One
reason is that plates have a limited formability since they deform mainly along the axis of weakest inertia
and thus cannot be easily forced into complex geometries. Although less investigated and put into
practice, undergoing research and a series of recent prototypes have been exploring the structural and
form inducing potential of bending-active plate structures.
The principal limit to the formal potential of bending-active structures lies in the restrictions on the
material formability. The only deformations that can be achieved within stress limits are the ones that
minimise the stretching of the material fibres. For strips and plate-like elements, these reduce to the
canonical developable surfaces: cylinders and cones. Attempting to bend a sheet of material in two
directions will either result in irreversible, plastic deformations or ultimately failure. Such a strict
requirement severely limits the range of structural and architectural potential for plate-based bending-
active systems. To expand the range of achievable shapes, it is therefore necessary to develop
workarounds for the induction of Gaussian curvature. To overcome such limitations, multidirectional
bending can be induced by strategically removing material and freeing the strips from the stiffening
constraint of the surrounding. A similar approach is presented by Xing et al. [5] and referred to as band
decomposition. The key principle is illustrated in Figure 1. Here a full disc is bent by applying a support
displacement towards the centre. The von Mises stress plots show the stress concentrations that arise in
the material because of the non-isometric deformation that the disc undergoes. If material is strategically
removed from the disc as shown in Figure 1b, the radial strips are now free from the constraint of the
neighbouring material and can be easily bent into shape. Moreover, the stress levels are radically reduced
as the elements are only subjected to non-extensional deformation. Only at the intersection between the
petals a small area of stress concentration arises. The approach presented here has been termed form
conversion (La Magna et al. [6]) as it establishes a one-to-one correspondence between the base
geometry and the bending-active plate system that emerges from it.
Proceedings of the IASS Annual Symposium 2017
Interfaces: architecture.engineering.science
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Figure 2: Local curvature values a. Positive Gaussian curvature (synclastic surface), b. Zero Gaussian curvature
(cylindrical surface), c. Negative curvature (anticlastic surface), d. Monkey saddle
With the form conversion method, it is possible to span multiple directions with varying degrees of local
curvature, therefore faithfully reproducing the global curvature of the source geometry. Taking as an
example the 4-branched element in Figure 2, different combinations of bending directions will
reproduce discrete versions of different curvature points. For instance:
1. A positive rotation around the local y axis of each branch will produce an area of positive
Gaussian curvature (Figure 2a).
2. A positive (or negative) rotation around the local y axis of two branches will produce a
cylindrical area of zero Gaussian curvature (Figure 2b).
3. Positive rotation around the local y axis of two branches and a negative rotation of the remaining
ones will produce an area of negative Gaussian curvature (Figure 2c).
4. A negative rotation around the local y axis of three branches and a positive rotation of the fourth
one will have the effect of building a local monkey saddle (Figure 2d).
The previous remark demonstrates the universal character of the method. Most shapes can be reproduced
following the form conversion process. The only constraints are imposed by the limits of the material
and the achievable bending radii. For architectural and structural applications this method is very well
suited, as the radii of curvature are normally not extremely pronounced.
The form conversion approach for bending-active plate structures was tested on two full-scale
prototypes (Figure 3). Each case study presented different characteristic in terms of geometry, topology,
and shape (La Magna et al. [6]). Nonetheless, the core generative strategy remained the same for both
cases. The prototypes were built out of 3.5mm plywood sheets. The main difference between the
Berkeley Weave (left) and the Bend9 (right) pavilions derives from the topology of the base mesh,
quadrangular in the first case and triangular in the second case.
Figure 3: View of the Berkeley Weave installation (left), View of the Bend9 pavilion (right)
Proceedings of the IASS Annual Symposium 2017
Interfaces: architecture.engineering.science
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4. Structural behaviour of plate-based form conversion structures
Figure 4: Wrapping process of the form conversion hemisphere
In this section, the effect of prestress on the load-bearing behaviour of bending-active structures derived
through the form conversion process is studied. To do so we concentrate on a hemispherical shell
structure which represents the base test case for the analysis (La Magna [7]). The half sphere is compared
to the form conversion counterpart derived from a triangular base mesh as well as a quad based mesh
dual. As we are interested in assessing the influence of prestress over the mechanical behaviour of the
shells, the structural systems must be chosen as similar as possible to be able to extrapolate comparable
results. The spheres are chosen as single layered shells made of an ideal isotropic material with a
Young’s modulus of 15000 N/mm2. A value for the shear modulus was omitted to focus exclusively on
the bending action and avoid parasitic stress arising from transversal secondary strain effects.
To compare the continuous hemisphere with the form conversion geometries, a full form-finding of the
shells had to be performed. By doing so, the bent hemispheres are the outcome of the form-finding
process and possess all the forces and stress information that might have an influence over the structural
behaviour of the shells. In the first place the discrete shell geometry is completely unrolled on the plane.
As the initial generated surface from the form conversion process is everywhere either flat or at most
single curved, we know that the unrolling of the surface can occur without stretching or shrinking. In
other terms, the planar, unrolled surface is completely isometric to the wrapped shape.
The bending process of the hemispheres is shown in Figure 4. The unrolled pattern is completely bent
into shape by using successive form-finding steps based on the ultra-elastic cable approach (Lienhard et
al. [8]). A series of elastic cables are attached to the nodes of the mesh that need to be connected together
and gradually shrunk until the length of each cable element vanishes. The form-finding process of the
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Interfaces: architecture.engineering.science
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spheres had to be broken down into several steps, as attempting to perform the simulation at once by
first attaching and then shrinking all the cables through prestress inevitably led to high instabilities in
the system. Successive series of ultra-elastic cables pull the flat elements together until the whole
structural system is restored as seen in Figure 4. The fact that the full bending process of the hemisphere
could be fully simulated, is also a strong guarantee that the shape deriving from the form conversion
process is geometrically compliant with the limits of the material.
For each set of ultra-elastic cables, after the shrinking process is completed nodes that were distant from
one another in the starting mesh are now coincident. The ultra-elastic shrinking cables method restores
the C0 continuity of the mesh, meaning that continuity is guaranteed only in terms of position. The
tangent between now coincident elements is not smooth, so that in most cases a kink develops along the
contact line. This could be easily solved by adding a rotation of the nodes to restore the continuity of
the tangent, but the computational effort was not deemed necessary for the evaluation of the global
result. From Figure 4 it can be seen how the stress level in the quad elements are very low. Here the
colour gradient shows the von Mises stress averaged over each element. Almost everywhere the mesh
is unstrained (central areas) or axially bent (bridging elements), meaning that the stress derives
exclusively from bending. Only in few, localised zones a rise in stress can be seen. As expected, these
coincide with the intersection areas between multiple elements and at the sharp corner transitions
between branches spanning different directions. Despite of these localised stress concentrations, the
bending stresses are generally way below the maximum acceptable levels of the material.
The bent hemispheres are analysed in two different flavours: once in their bent configuration, meaning
that the prestress in the plate elements arising from the form-finding process is taken into consideration.
The second analysis is performed on the dead geometry (Gengnagel et al. [9]). This means that the
geometry on which the simulation is performed is the same as the one deriving from the form-finding,
but the stress state is completely disregarded. The base geometry is stress free in this case. In this way,
the effect of the prestress on the stiffness and the mechanical behaviour of the shell can be better
recognised.
Each case is analysed by progressively applying a point load in the negative z direction at the top of the
hemisphere. The point load has an intensity of 0.1 kN, and at each successive iteration the load is
increased by a factor λ. By comparing the load factor λ with the deflection in z direction measured in
mm, we obtain the load-deflection graphs as shown in Figure 5 (Charpentier et al. [10]). For the form
conversion hemispheres, the continuous lines of the graph represent the bent geometries, whilst the
dashed lines are for the dead geometries. The continuous red line is representative of the full hemisphere.
Focusing on the results, it can be seen that all test cases initially behave pretty much linearly. The starting
branches of each graph are consistent with the hypothesis made so far: a reduction of material implies a
proportional reduction of stiffness. As expected, the full hemisphere is stiffer than the form conversion
counterparts, as more material is involved in the transfer of forces. It is interesting to notice that the
triangular version and the quad version also behave very similar to each other, as their respective load-
deflection curves almost perfectly overlap. This suggests a minor influence of the type of pattern on the
mechanical behaviour of the shells. What is rather more worthy noticing, is that the difference between
the bent geometries and the dead geometries is minimal. The curves in the graph only show a slight
deviation between the two variations, which means that the prestressing effect emerging from the form-
finding does not seem to have a major influence on the load-bearing behaviour of the discontinuous
shells. Furthermore, a beneficial effect deriving from the prestressing would always be dependent on
the direction of loading. A material strip bent in one direction will behave radically different whether
loaded from the top or the bottom, as the stress distribution deriving from bending will be working
against the deformation in the first case, but will be adding stress in the latter.
Proceedings of the IASS Annual Symposium 2017
Interfaces: architecture.engineering.science
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Figure 5: Load-deflection curves for the concentrated point load applied on the prototypical hemispheres
It is therefore not possible to conclude in advance whether the prestressing effect of bending does have
a beneficial or detrimental effect on a global scale. Moreover, materials that are suitable for bending-
active structures also tend to be extremely susceptible to creeping effects, which overtime tend to nullify
the effect of prestressing as it is lost during the relaxation process. In the light of these arguments, it is
therefore safe to disregard effects deriving from the prestressing of the plates. This can dramatically
simplify the task of planning and designing bending-active plate structures, as only the geometry of the
system is required to preliminary evaluate the internal stress state of the elements after form-finding.
Proceedings of the IASS Annual Symposium 2017
Interfaces: architecture.engineering.science
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Figure 6: Load-deflection curves for the self-weight load applied on the prototypical hemispheres
This is also confirmed in the following graph (Figure 6), where the five test cases are also analysed
under self-weight. The additional self-weight study is meant to highlight aspects otherwise lost in the
previous point load study, as under concentrated loads shells tend to be more susceptible to local
imperfections. The x-axis of the diagram shows the maximum vertical displacement of the shell. In this
case, the bent spheres behave worse than their dead counterparts showing a less stiff response. Besides,
failure generally occurs at a lower load factor. Comparing Figure 5 with Figure 6 it is therefore not
possible to decide in absolute terms whether the prestressing effect will be beneficial or not to the global
behaviour of the shell.
From Figure 6 it can be noted that the full sphere is considerably stiffer than the form conversion spheres.
On the other hand, failure will occur rather suddenly due to local buckling in the continuous shell, as
opposed to a larger plastic deformation zone which characterises the discontinuous cases. It is also
interesting to notice the difference between the triangular pattern and the quadrangular pattern. The quad
sphere shows a noticeably stiffer response curve. A possible explanation may lie in the geometry of the
voids. By better approximating a triangular field, the quad pattern introduces increased in-plane
stiffness, just like a triangulated gridshell is considerably stiffer than one with square fields.
Having established the minimal influence that prestress has on the structure, we now focus on the
analysis of the continuation of the load-deflection curves beyond the initial linear portion for the point
load test cases. Beyond the linear part, the form conversion shells show a radical departure from the
behaviour of the full sphere. Besides minor differences, both triangular and quad shells display an
extended non-linear structural response. It can be noticed how the shells locally bulge inwards entering
a post-critical phase dominated by the snap-buckling of the spheres’ cap. The graph shows that the cap
of the triangular shell locally buckles at a lower load factor, whilst the quad shell withstands a larger
load intensity, despite becoming softer, until the cap finally snap buckles (the horizontal portion of the
Proceedings of the IASS Annual Symposium 2017
Interfaces: architecture.engineering.science
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diagram). Scaled at the right proportions, the curves still resemble the behaviour of the full sphere,
although the critical states are reached at a lower load. This effect can be explained by looking at a small
portion of the form conversion hemispheres. Locally, the shells are made of developable surfaces
connected together. Despite the topological and geometric arrangement of the elements lock undesirable
deformations that would occur if these were free from the constraining neighbourhood, a certain degree
of flexibility still allows for inextensional deformations of the individual elements to happen. This would
explain why the shells generated through the form conversion method display a softer behaviour and
are more prone to localised buckling. On the other hand, it must be highlighted that these effects appear
at a very late stage and at a high loading factor for the structure. For real life structures at an architectural
scale, the post-buckling effects would definitely not be acceptable as the structure would long have
failed its purpose.
From the point load test cases we have seen that the effects deriving from prestress have a minimal
influence on the load-bearing behaviour and the mechanical response of form conversion bending-active
plate structures. This observation provides a useful argument for the simulation workflow, as it relieves
the analyst from the bulk of integrating the whole form-finding at an early stage of design. Nonetheless,
to capture every detail that might be relevant to understand the structural behaviour, it may still be
necessary to perform a fully-fledged form-finding and analysis. This is also confirmed in the analysis
of the self-weight test cases, where the effects of prestressing are undoubtedly more evident. Although
the analysed prototypes have shown a weak post-critical behaviour, within certain load limits the
structures perform well as expected. For these reasons, the advantages in terms of fabrication and
assembly might overwhelm the relative reduction of structural performance of bending-active plates.
4. Conclusions and future directions
The current paper presented the form conversion approach and its effects on the structural behaviour of
shells built following this strategy. The form conversion approach provides a general method for the
discretisation and construction of bending-active plate based structures. The main ideas guiding this
method showed that the continuity of the surface must be partially sacrificed to be able to build a double
curved surface using exclusively flat strips of material. By manipulating the topology of the base surface,
the form conversion approach guarantees the bendability of the elements and in this way the ability for
two-dimensional elements to span multiple directions and therefore cover the whole space. The validity
of the developed approach was tested on several geometries, and it demonstrated to be applicable to any
surface, as long as the bending and torsion of the elements are kept within the limits of the material. A
series of prototypes were realised to test the form conversion approach at bigger scales. These have
shown to scale well in terms of deformation of the elements, as at architectural scale the curvatures are
generally less pronounced than at model scale. Nonetheless, the inherent contradiction of bending-active
structures, which need to have slender elements to be able to be bent into shape at the expense of the
structural stability, is still not resolved. This can be partially compensated by the coupling of multiple
elements to create multi-layered structures as it was tested in some of the realised prototypes. The
geometries obtained through the form conversion method were tested numerically to assess the structural
potential of the structures created in this way. The simulations showed that the introduction of
discontinuities takes a toll over the capacity of the structure, partially weakening it due to the interruption
of the force flow in the elements. This trade-off is compensated by the ease of construction that the
method ensures and the ability of building double curved shell structures out of flat, off-the-shelf
materials.
Future research will concentrate on the realisation of larger spanning prototypes. The coupling of
multiple layers to achieve higher stiffness is a required step to ensure stability and increased load-bearing
capacity. Strategies for the rigidisation of plate-based bending-active structures is the focus of ongoing
investigations, with the aim of expanding the system’s possibilities in terms of scale and performance.
Proceedings of the IASS Annual Symposium 2017
Interfaces: architecture.engineering.science
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References
[1] Lienhard J., Bending-Active Structures – Form-finding strategies using elastic deformation in
static and kinetic systems and the structural potentials therein, Doctoral thesis, Universität
Stuttgart, 2014.
[2] Knippers J., Cremers J., Gabler M. and Lienhard J., Construction Manual for Polymers +
Membranes: Materials, Semi-Finished Products, Form Finding, Design. Birkhäuser, 2011.
[3] Fuller R.B., Self-Strutted Geodesic Plydome, U.S. Patent 2,905,113, issued September 22, 1959.
[4] Schleicher S. and La Magna R., Bending-active Plates: Form-finding and Form-conversion, in
ACADIA 2016. Posthuman Frontiers, 2016, 260-269.
[5] Xing Q., Esquivel G., Akleman E. and Chen J., Band Decomposition of 2-Manifold Meshes for
Physical Construction of Large Structures, in Proceedings of the Siggraph Conference, Vancouver,
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[6] La Magna R., Schleicher S. and Knippers K., Bending-Active Plates: Form and Structure, in
Advances in Architectural Geometry 2016, 170-186.
[7] La Magna R., Bending-Active Plates – Strategies for the induction of curvature through the means
of elastic bending of plate-based structures, Doctoral thesis (to appear), Universität Stuttgart, 2017.
[8] Lienhard J., La Magna R. and Knippers J., Form-Finding Bending-Active Structures with
Temporary Ultra-Elastic Contraction Elements, in Mobile and Rapidly Assembled Structures IV,
DOI: 10.2495/mar140091, 2014.
[9] Gengnagel C., Alpermann H. and Lafuente E., Active Bending in Hybrid Structures, in Form-Rule
/ Rule-Form, Innsbruck University Press, 2013.
[10] Charpentier V., Adriaenssens S. and Baverel O., Large Displacements and the Stiffness of a
Flexible Shell, in International Journal of Space Structures, 2015; 30, 3+4; 287-296.
... The proposed material system consists of an initially flat surface, which is tessellated into single components based on a hexagonal grid. Each cell is comprised of a three-layer system, where a pneumatic cushion is placed in the middle of two bending active plates with strategic cut outs [1]. In this way, each cell can be pneumatically actuated and therefore elastically deformed. ...
... This effect appears because the material properties restrict the formability of the plate. To avoid this material failure, material needs to be strategically removed to allow the elastic deformation of an initially planar plate to a double curved surface [1]. Based on this knowledge strategic cut outs are applied which allows the doubled curved deformation of the platebased component ( figure 6, right). ...
... FE simulations of the bending behaviour of a hexagonal plate (left) and a plate with cut outs (right)[1] ...
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Active Bending describes curved beams or surfaces that base their geometry on the elastic deformation of initially straight elements. The bending stress in the bent elements depends on the Young‘s modulus of the material, the height of the cross section and the curvature. To achieve sufficient curvature the height of the cross section and the Young‘s modulus need to be small. Therefore, the stiffness of the lastically-bent elements is small. Additional elements offer the opportunity to create a hybrid structure, whose stiffness is significantly higher than the stiffness of the elastically-bent elements alone. The general influence of Active Bending on stiffness and stress is discussed and numerically analysed. The advantages of and different ways to use elastically bent elements are presented in five case studies. The differences in their structural behaviour compared to curved elements are analysed. It can be shown that, despite the initial stresses and the limitation on the profile’s choice, the structural capacity of a given construction is hardly affected by the elastic bending, which is therefore advantageous over the use of straight elements in terms of fabrication, transport and/or assembling.
Conference Paper
With the design and construction of more and more unusually shaped buildings, the computer graphics community has started to explore new methods to reduce the cost of the physical construction for large shapes. Most of currently suggested methods focus on reduction of the number of differently shaped components to reduce fabrication cost. In this work, we focus on physical construction using developable components such as thin metals or thick papers. In practice, for developable surfaces fabrication is economical even if each component is different. Such developable components can be manufactured fairly inexpensively by cutting large sheets of thin metals or thin paper using laser-cutters, which are now widely available.
Construction Manual for Polymers + Membranes: Materials, Semi-Finished Products, Form Finding
  • J Knippers
  • J Cremers
  • M Gabler
  • J Lienhard
Knippers J., Cremers J., Gabler M. and Lienhard J., Construction Manual for Polymers + Membranes: Materials, Semi-Finished Products, Form Finding, Design. Birkhäuser, 2011.
Self-Strutted Geodesic Plydome
  • R B Fuller
Fuller R.B., Self-Strutted Geodesic Plydome, U.S. Patent 2,905,113, issued September 22, 1959.