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Proceedings of the IASS Symposium 2018

Creativity in Structural Design

July 16-20, 2018, MIT, Boston, USA

Caitlin Mueller, Sigrid Adriaenssens (eds.)

Copyright © 2018 by Stijn Brancart, Olga Popovic Larsen, Lars De Laet and Niels De Temmerman.

Published by the International Association for Shell and Spatial Structures (IASS) with permission.

Bending-active reciprocal structures: geometric parameters and

their stiffening effect

Stijn BRANCART*, Olga POPOVIC LARSENa, Lars DE LAETb, Niels DE TEMMERMANb

*Researcher at Vrije Universiteit Brussel (VUB), Department of Architectural Engineering (ARCH)

Pleinlaan 2, 1050 Brussels, BELGIUM

Stijn.Brancart@vub.be

a Professor at Royal Danish Academy of Fine Arts (KADK), School of Architecture

b Professor at Vrije Universiteit Brussel (VUB), department of Architectural Engineering (ARCH)

Abstract

Bending-active reciprocal structures consist of elastically bent beams in a mutually supporting weave

pattern in which each connection joins only two beams. Aside from the advantages for fabrication and

assembly, this system has an extensive geometric potential, allowing reuse and reconfiguration.

However, the load-bearing behaviour of these structures is limited by the required flexibility of the

components. This paper discusses the impact of geometric parameters on the stiffness of bending-active

reciprocal geometries for two cases: single-layered domes and double-layered components. It concludes

by illustrating the use of the double-layered component for the development of a kit-of-parts structure:

the ReciPlyDome. Thanks to the elastic capacity of the components and the very simple connections,

the ReciPlyDome can be very easily and rapidly assembled and manufactured. This in contrast with the

high technical complexity of many other kit-of-parts or grid structures. The analyses underline the

importance of the form-force relationship when developing bending-active reciprocal structures and

show the potential of active bending for the development of pre-assembled beam components. This

research is framed within the wider scope of developing lightweight, transformable and temporary

structures.

Keywords: Active bending, reciprocal frame structures, lightweight structures, transformable structures, parameter study,

design for stiffness, prototyping, geometry

1. Introduction

Through their mutually supporting beam configurations, reciprocal structures have the advantage of

connecting no more than two beams at each node (Popovic Larsen [6]). Although this does simplify the

joint design, the development of reciprocal structures is complicated by the complex, non-hierarchical

relation between geometric parameters, such as the component eccentricity that results from their cross-

sectional height (Parigi and Kirkegaard [8]). Bending-active components however can employ their

reversible and adaptable curvature to develop structural height. As such they can be used to develop

very thin, curved reciprocal geometries. Several geometric explorations by CODA [2], Alison Grace

Martin [4] and Hiroshi Murata [5], as well as contributions to the field of rotegrity [3] show the potential

of this technique to develop lightweight and rapidly assembled structures. Like most bending-active

structures however, the challenge lies in developing adequate stiffness with flexible components. As the

thickness of the components is limited by the elastic limit of the material, other design parameters will

need to be controlled to influence the load-bearing behaviour. This paper discusses these design

parameters and analyses their effect on the stiffness of the structure, focussing on single-layered

polyhedral dome geometries. Previously, the authors of this paper introduced the concept of a pre-bent,

double-layered component (Brancart et al. [1]). Where this component offers advantages for the

fabrication and assembly of bending-active reciprocal structures, it can also be employed to stiffen the

Proceedings of the IASS Symposium 2018

Creativity in Structural Design

2

structure. This paper presents a brief analysis of this stiffening effect, considering the impact of the

coupling of the layers as well as the effect of the residual pre-stress in the pre-bent components. Finally,

it shows how this component was used for the development of a kit-of-parts structure: the ReciPlyDome.

This case study shows the potential of bending-active reciprocal structures in developing low-tech

systems for rapidly assembled structures.

2. Bending-active reciprocal structures: geometric parameters

Reciprocal structures are grid structures that derive their strength and stability from a mutually

supporting beam organisation in which no more than two beams are connected at each node. As such,

they can span large spaces with short, discretised elements without additional joints. Yet, their design

and modelling are complex due to the intricate geometric relationship between the shape-defining

parameters: the eccentricity or distance between the beam axes, caused by the cross-sectional heights;

the engagement ratios of the beams, resulting from the location of the connections along the beam

lengths; and the positioning of the beams, sitting either on top or at the bottom of each other (Parigi and

Kirkegaard [8]). As the relation between these parameters is non-hierarchical, changing one of them will

invariably lead to changing the others. Bending-active reciprocal structures consist of elastically curved

beam members. Since these flexible members are usually very thin compared to those in conventional,

rigid reciprocal structures, their curvature will largely define the structural height of the bent beam

member. Figure 1 shows how this allows developing the same geometry with much thinner beam

profiles. Where rigid components often require complex cut-outs to compensate for the different

inclinations of the beams at the connections, the flexibility of bending-active components allows

developing tangential, in-plane connections (for example simple bolted connections).

Figure 1: Using (elastically) curved beam members allows developing structural height and thus eccentricity

with low member thickness. As the curvature is reversible and adaptable, the components can be used for

multiple configurations.

As the curvature of bending-active components is elastic and reversible, their eccentricity is adaptable.

This offers benefits for the design and development of reconfigurable and kit-of-parts structures, in

which the same components might be used for different geometric configurations. The authors of this

paper developed a methodology for the design of bending-active reciprocal structures based on

equilateral polyhedral dome geometries ((Brancart et al. [1]). Figure 2 shows four reciprocal geometries,

all consisting of the same components. The transformation from one configuration to the other is made

by rotating the components, thus moving from an icosahedral base shape to its dual, the dodecahedron.

Practically, this conforms with shifting the intermediate connections.

Proceedings of the IASS Symposium 2018

Creativity in Structural Design

3

Figure 2: Rotating the beam members generates an array of potential geometries. Practically, this means shifting

the intermediate connections along the beam, i.e. changing the engagement length. As such, each geometry has a

dual configuration, achieved by switching the intermediate connections (in this case a and d, and b and c).

Combining the bolt holes creates a “universal component” that can generate all four configurations (e).

3. Single-layered, bending-active reciprocal domes: a parameter study

Although for bending-active reciprocal structures the geometry does not depend on the cross-section of

the components, the elasticity constraint does limit the component thickness and consequently the load-

bearing behaviour. This section discusses three other (geometric) parameters that allow improving this

behaviour: the engagement ratio by rotating the components, the topology and the width of the

components. Although it is possible to develop a wide variety of curved surface or shell structures, this

paper focusses on the aforementioned polyhedral domes. All domes have the same radius and thus span

and are subjected to two load cases: a self-weight or dead load, and a combination of point loads at the

five vertices of the top pentagon. Due to the high complexity of the structural modelling process, these

first analyses do not consider the residual pre-stress in the structures. The effect of the pre-stress, the so-

called stress-stiffening effect, will be discusses in the next section. Figure 3 shows the effect of moving

the intermediate connections closer towards or farther away from the nodes. A critique on reciprocal

structures is that they induce bending by connecting the members intermediately. However, it is clear

that these bending-active structures benefit from a more equal division of the beam member (for example

in thirds). Moreover, this allows distributing loads, for example due to a cladding system, more evenly

over the structural surface, avoiding unfavourable concentrations. This effect is of course related to the

topology of the structure. Figure 4 shows the behaviour of three domes based on different polyhedra:

the icosahedron, the rhombic triacontahedron and the truncated icosahedron. Each of them consists of a

different number of components of different length. While the icosahedral dome uses less material than

the other ones, it performs just as bad under self-weight as point loading. Although using roughly the

same amount of material, the truncated icosahedron performs a lot better than the rhombic

triacontahedron, indicating a clear impact of the topological organisation of the beams and potential

preference for shorter beam lengths. Figure 5 finally shows the impact of the beam width. While the

thickness of the beam members is limited by the elastic limit stress of the material, it is possible to

increase the width without (considerably) increasing the stress in the structure. The results show how

this is a very efficient approach for increasing the stiffness, especially under point loads. Since the dead

load increases with increasing width, the effect is slightly less in this case.

Proceedings of the IASS Symposium 2018

Creativity in Structural Design

4

Figure 3: Moving the intermediate connections along the beam’s length results in different configurations,

concentrating the nodes more or less at the beam ends. A more equal division of the beams, e.g. in three identical

segments, increases the stiffness and allows distributing nodal loads more evenly over the surface of the

structure.

Figure 4: The same dome shapes can be made through different topological configurations, consisting of less but

longer components, or more and shorter. Although they use only slightly more material, the dome types with

more components clearly perform better.

Proceedings of the IASS Symposium 2018

Creativity in Structural Design

5

Figure 5: While the thickness of the bent beams is limited by the elastic deformation range, the width of the

components does not increase the residual pre-stress considerably. Therefore it is an efficient approach to

increase the stiffness, even despite the increase in self-weight.

4. Double-layered components: stiffening

As the load-bearing behaviour of single-layered domes is limited by the low component thickness,

double layering can be an efficient way of increasing the stiffness without increasing the residual

prestress too much. This section discusses the behaviour of a double-layered component that was

developed in the scope of the ReciPlyDome project, which will be discussed in the final sections. The

component consists of two interconnected laths of different lengths, curved and restrained by their

geometric incompatibility. Although bending-active, this component functions as a static, pre-curved

beam and does not transfer bending moments to its adjoining members. Figure 6 shows the results of a

basic structural analysis of a reciprocal module consisting of three bending-active beams: one with single

layer (a) and one with double-layered components (b). Each is subjected to a set of point loads in the

three nodes, pushing downwards first and pulling upwards in a second load case. The results include

load-displacement curves and stress progressions for the pre-bent configuration with residual stress as

well as for the reference geometry. These reference geometries does not include the residual pre-stress

(compared to the other models, which are elastically bent prior to the load simulation). As such they

allow analysing the stress-stiffening effect. It is clear that the double-layered module performs

considerably better in maintaining its structural form, both for downward and upward loading, without

very much increasing the residual stress in the structure. Although the effect is considerable, it can be

increased by better connecting the two layers, for example by adding spacers or developing a truss-like

system with interconnecting cables. While this falls out of the scope of this publication, a more detailed

analysis of this component will be the topic of future work.

Apart from a slight benefit under small upward loading, the stress-stiffening effect seems to influence

the load-bearing behaviour negatively overall. For the single-layered module, this corresponds with

earlier research discussing the destabilising effects of pre-compression forces in elastically pre-bent

structures (Lienhard [7]). Since pre-tension is generally accepted to have a beneficial influence on the

stiffness of structures, think of pre-tensioned concrete reinforcement or tensile surface structures, the

double-layered component fails to employ the pre-tensioning in its bottom layer for the analysed load

cases.

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Creativity in Structural Design

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Figure 6: Using pre-bent, double-layered components is an efficient way of developing stiffness without

increasing the pre-stress too much. Although the double-layered component does contain pre-tensioning forces in

its bottom layer, the stiffness does not benefit from it in these two load cases.

Where the analysis in Figure 6 starts from a statically indeterminate module with all fixed connections,

Figure 7 shows what happens when pulling up a module that is allowed to slide on two of its legs (and

is thus connected statically determinate). It is clear that this load case results in a considerable stress-

stiffening effect. The compressive action on the components, which is clear in the reference geometry

case, is counteracted by the tension in the bottom layer. Although the overall stiffness of the systems in

decreased by changing the boundary conditions, it does activate the stress-stiffening effect and increase

the interaction between the top and (pre-tensioned) bottom layer.

Although this seems like a theoretical or hypothetical case, this analysis underlines the effect of loading

directions and boundary conditions on the stress-stiffening and overall load-bearing behaviour. In the

fully assembled dome geometries, the beam elements will be loaded in different directions depending

on their location in the structure. From these simulations we can moreover derive that the stress-

stiffening effect will mainly be employed for high displacements, which are evidently more common in

bending-active structures.

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Creativity in Structural Design

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Figure 7: Allowing two of the module’s ends to slide activates the stress-stiffening effect thanks to the tension

forces in the bottom layer of the pre-bent components.

5. Prototype: the ReciPlyDome

The main benefit of active bending lies in its simplicity to create curved members or structures from

initially planar or linear components. Additionally, the reciprocal configuration allows for simple nodes.

As a proof-of-principle, we therefore applied the concept of bending-active reciprocal structures for the

development of a kit-of-parts system for rapidly assembled and temporary structures: the ReciPlyDome

(Figure 8). Based on the rhombic triacontahedron polyhedral shape, the dome consists of only two

different beam types (40 identical beams + 5 shortened beams to create the bottom edge). This allowed

very fast fabrication of the pre-assembled components in barely more than one day. As the components

could be pre-bent in the workshop, the erection of the dome required no on-site bending. To align the

components at the connections however, they need to be slightly torqued. As such, the dome can be set

up in only a couple of hours. The project shows how apparent geometric complexity can be achieved

with a very simple system. This is also the case for the textile cladding, which was added to a second

version of the prototype (Figure 8, right). Based on the polyhedral base shape, the textile consists of a

minimum of different cutting patterns. Further research will analyse the effect of textile or other cladding

on the load-bearing behaviour of bending-active reciprocal structures.

Figure 8: The ReciPlyDome consists of only two different beam types. Thanks to the pre-assembly of the

bending-active components, the simplicity of the connections and the uniformity of the system the dome can be

set-up in only a couple of hours.

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Creativity in Structural Design

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6. Conclusions

This paper presented a study of geometric parameters and their influence on the load-bearing behaviour

of bending-active reciprocal structures. While it is clear that active bending of the beams increases the

geometric potential for reciprocal structures and facilitates the development of rapidly assembled and

reconfigurable structures, the stiffness is often limited by the flexibility of the components. Although

careful consideration of geometric parameters such as the topology, location of the intermediate

connections and width of the beams can considerably improve the behaviour, the real potential lies in

developing structural thickness by layering. The double-layered component that was analysed in this

paper is only a first step in exploring this approach for the stiffening of bending-active reciprocal (and

in extension all bending-active) structures. Moreover, the results show how a stress-stiffening effect can

occur depending on the loading direction and level of pre-tensioning in the components. Further research

will explore this effect in more detail, study potential stiffening of the double-layered component by

increasing the connectivity between the different layers and investigate the effect of cladding and hybrid

(textile, cables) components on the load-bearing behaviour of bending-active reciprocal domes. The

ReciPlyDome project shows how the utilisation of elastic deformation can contribute to the simplicity

of the system. It allows developing much simpler connections so the structure can be assembled more

easily.

Acknowledgements

This research was funded by the Flemish Institute for Innovation through Science and Technology (IWT,

now VLAIO) and the Vrije Universiteit Brussel. The collaboration between the Vrije Universiteit

Brussel (VUB) and Royal Danish Academy of Fine Arts (KADK) was made possible thanks to the

funding of the VELUX Foundation’s visiting professorship programme and the European COST Action

TU1303 on Novel Structural Skins. The ReciPlyDome was funded by the KADK and constructed with

the help of students Mikkel Asbjørn Andersen, Niklas Munk-Anderson and Christian Jespersen from

Danish Technical University (DTU), and Veronika Petrova from the KADK.

References

[1] Brancart, S., Popovic Larsen, O., De Laet, L., De Temmerman, N., Bending-active reciprocal

structures based on equilateral polyhedral geometries, in IASS 2017: Interfaces, Proceedings of the

International Conference of the International Association for Shell and Spatial Structures,

Hamburg, 2017.

[2] CODA, Panikkar, http://coda-office.com/work/Panikkar, consulted on January 30, 2017.

[3] Deviant Art, Explore Rotegrity, http://www.deviantart.com/tag/rotegrity, consulted on January 30,

2017.

[4] Flickr, Alison Grace Martin, https://www.flickr.com/photos/109333486N07/, consulted on

January 30, 2017.

[5] Hiroshi Murata, The Da Vinci Dome, http://www.hiroshi-murata.com/home/the-da-vinci-dome,

consulted on January 30, 2017.

[6] Larsen, O.P., Reciprocal Frame Architecture, Architectural Press/Elsevier, London, 2008.

[7] Lienhard, J., Bending-Active Structures: using elastic deformation in static and kinematic systems

and the structural potentials therein, PhD dissertation, University of Stuttgart, 2014.

[8] Parigi, D., Kirkegaard, P.H., The Reciprocalizer: an Agile Design Tool for Reciprocal Structures,

in Nexus Network Journal, 2014.