Sam Bouten their Architectural Application Transformable Structures and

Sam Bouten

their Architectural ApplicationTransformable Structures and

Academic year 2014-2015Faculty of Engineering and ArchitectureChairman: Prof. dr. ir. Luc TaerweDepartment of Structural Engineering

Master of Science in de ingenieurswetenschappen: architectuurMaster's dissertation submitted in order to obtain the academic degree of

Counsellor: Ir. Jonas DispersynSupervisor: Prof. dr. ir.-arch. Jan Belis

Transformable Structuresand their

Architectural Application

iii

Acknowledgements

Motivated by Prof. Mónica García Martínez, I attended a 2013 lecture on the work of Spanish

architect Emilio Pérez Piñero, a pioneer in the field of deployable structures. My interest grew by

taking part in a transformable design competition and congress dedicated in his honor.

I decided to further deepen this - by then passionate - interest by writing this master’s thesis on

the subject. I’m grateful to my supervisor, Prof. Jan Belis: he had both a highly motivating outlook

and critical but ever constructive feedback that made me work more driven and precise. His open

approach to the research allowed me to discover widely without losing focus on the important

aspects.

My gratitude also goes to Michiel Van Der Elst and Jonas Van Den Bulcke, fellow students whose

shared interest and curiosity have resonated with mine, and often made me see the topic in new

ways. Jonas’ knowledge of digital fabrication was instrumental in making some of the test models

used throughout the thesis.

Thanks also to Prof. Niels De Temmerman for his expertise-based tips and encouraging words.

Finally and mostly, I’d like to thank Silvia for her continuous support and kind listening.

Permission for use of content

The author gives permission to make this master dissertation available for consultation and to

copy parts of this master thesis for personal use. In the case of any other use, the limitations of

the copyright have to be respected, in particular with regard to the obligation to state expressly

the source when quoting results from this work.

Sam Bouten, May 15 2015

iv

Transformable Structures and

their Architectural Application

By Sam Bouten

Master’s dissertation submitted in order to obtain the academic degree of

Master of Science in de ingenieurswetenschappen: architectuur

Supervisor: Prof. Dr. Ir.-Arch. Jan Belis

Counsellor: Ir. Jonas Dispersyn

Department of Structural Engineering

Chairman: Prof. Dr. Ir. Luc Taerwe

Faculty of Engineering and Architecture

Academic year 2014-2015

Summary

The field of transformable structures is remarkably varied since it transcends the borders of

conventional disciplines and inscribes itself into the modern notion of adaptivity. The main aim of

this thesis is to provide insight in the design of transformable structures on an architectural scale.

In the first part of the thesis, an extensive literature research is done to show the possibilities

that lie in the hands of the designer. Geometrical variations are complemented by examples of

real-life use of each of the addressed categories: scissor-like structures, rigid-foldable origami

and Jitterbug-like mechanisms. Many of them are identified as being variations on

overconstrained linkages.

The second part addresses the kinematic aspects such as the analysis of degrees of freedom and

trajectories. A numerical model for a generalized deployable 4-bar structure is given.

Materialization challenges and pitfalls in the scaling of transformable structures are further

discussed, specifically joint design and actuation.

The third part focuses more deeply on the Sarrus linkage and the different arrays that can be

formed from it. A novel way of introducing a polar angle in the Sarrus linkage by means of a joint

offset is given. Furthermore, a novel array, dubbed the overlap array, is analyzed and its

geometrical aspects discussed. A parametric tool for the design of flat and polar Sarrus arrays is

given. The trade-off between deployability and structural performance of the arrays is discussed

and two case studies finally are used to structurally analyze the different arrays.

Keywords

Deployable structure, scissor-like structure, rigid-foldable origami, Jitterbug-like mechanism,

overconstrained linkage, Sarrus linkage, Sarrus array

v

Contents

Acknowledgements .......................................................................................................... iii

Abstract ...............................................................................................................................iv

List of symbols .................................................................................................................. vii

1. Introduction ............................................................................................................ 1

1.1 Categorization ...................................................................................................... 2

1.2 Basic mechanical concepts ................................................................................ 3

Part I. Review of Literature

2. Scissor-Like Elements ............................................................................................. 7

2.1 Geometrical possibilities ................................................................................... 8

2.1.1 Translational units .................................................................................. 8

2.1.2 Polar units.............................................................................................. 13

2.1.3 Angulated units ..................................................................................... 17

2.2 Architectural application ................................................................................. 20

3. Rigid-Foldable Origami ......................................................................................... 27

3.1 Patterns and tessellations ............................................................................... 28

3.1.1 Miura-ori pattern .................................................................................. 28

3.1.2 Yoshimura pattern ............................................................................... 31

3.1.3 Waterbomb pattern ............................................................................. 34

3.1.4 Resch patterns ...................................................................................... 35

3.2 Flat-foldability .................................................................................................... 37


4. Jitterbug-Like Linkages ........................................................................................ 43

4.1 Geometrical possibilities ................................................................................. 44

4.1.1 Odd-valent vertices .............................................................................. 47

4.1.2 Planar variations .................................................................................. 51


5. Overconstrained Linkages ................................................................................... 55

5.1 Bennett Linkages .............................................................................................. 55

5.2 Goldberg and Myard linkages......................................................................... 59

5.3 Bricard linkages ................................................................................................ 61

5.4 Parallel manipulators ....................................................................................... 62

5.4.1 Modified Wren platforms .................................................................... 62

5.4.2 Sarrus linkages ...................................................................................... 63

vi

Part II. Design Tools

6. Kinematic studies ................................................................................................. 67

6.1 Determining degrees of freedom .................................................................. 74

6.2 Trajectories and envelopes ............................................................................. 75

6.3 Generalized trajectory of 4-bar deployable structures ............................... 84

7. Materialization Challenges .................................................................................. 87

7.1 Joint design ........................................................................................................ 88

7.2 Thickness in rigid-foldable origami ................................................................ 95

7.3 Actuators ............................................................................................................ 98

7.4 Locking systems .............................................................................................. 103

7.5 Design criteria ................................................................................................. 108

Part III. Uneven Sarrus Chains

8. Uneven Sarrus Chains ....................................................................................... 111

8.1 Basic module ................................................................................................... 111

8.2 Joint-to-joint arrays ......................................................................................... 116

8.2.1 Polar module with joint offset .......................................................... 118

8.2.2 Mobility of joint-to-joint arrays ......................................................... 123

8.3 Overlap arrays ................................................................................................. 127

8.3.1 Polar module with ellipse method ................................................... 129

8.3.2 Overlap factor ..................................................................................... 131

8.4 Parametric tool for regular array design ................................................... 132

8.5 Secondary structural systems ..................................................................... 135

8.6 Case studies ................................................................................................... 140

8.6.1 Case study 1: Pedestrian bridge ....................................................... 140

8.6.2 Case study 2: Barrel vault .................................................................. 148

8.7 Conclusions .................................................................................................... 157

References ....................................................................................................................... 159

Appendix A. Formal Studies .......................................................................................... 170

Appendix B. Transformable Designs ............................................................................ 176

vii

List of Symbols

Greek symbols

Operating angle [rad]

Operating angle between xy-plane and original bars in Sarrus modules [rad]

Operating angle between xy-plane and original bars in Sarrus modules [rad]

Deformation angle in xy-plane [rad]

Polar angle [rad]

Polar angle for joint offset method if only 2 different bar lengths are used [rad]

Maximum polar angle [rad]

Maximum polar angle for joint offset method [rad]

Maximum polar angle for ellipse method [rad]

Kink angle in angulated scissor-like elements [rad]

Apex angle in Yoshimura based rigid-foldable origami [rad]

η Amount of plate elements in curved direction for Yoshimura based origami

Fold angle, operating angle for rigid-foldable origami [rad]

Twist angle at joint of overconstrained mechanisms (I) [rad]

Twist angle at joint of overconstrained mechanisms (II) [rad]

Bar proportion between and bars in uneven Sarrus modules

Form factor for snow load

ξ Spatial deformation angle for rigid-foldable origami [rad]

Maximum spatial deformation angle for rigid-foldable origami [rad]

Air density [kg/m²]

viii

Lower-case Latin symbols

Damping coefficient of hyperbolic paraboloid surfaces

Probability factor for wind load

Overlap factor for Sarrus modules in overlap arrays

Roughness factor for wind load

Vertical height of Sarrus module in fully deployed state [m]

Relative structural height of Sarrus module in fully deployed state

Joint offset used to introduce polar angle in uneven Sarrus chains [m]

Joint offset if only 2 different bar lengths are used [m]

Joint offset for maximum polar angle

[m]

Bar length, by default assumed greater than bar length [m]

Adapted bar length (from ) for joint offset method [m]

Adapted bar length (from ) for maximum polar angle

[m]

Adapted bar length (from ) for ellipse method [mm]


[m]

Terrain factor for wind load

Bar length, by default assumed lesser than bar length [m]

Projected bar length of [mm]

Adapted bar length (from ) for ellipse method [mm]


[m]

Length of translation vector in generalized trajectory of deployables (I) [m]

Length of translation vector in generalized trajectory of deployables (II) [m]

Distributed imposed load [KN/m²]

Peak velocity pressure for wind load [KN/m²]

ix

Snow load [KN/m²]

Basic wind speed [m/s]

Basic wind speed [m/s]

Total wind pressure [N/m²]

Internal wind pressure [N/m²]

External wind pressure [N/m²]

Wind load in global x direction [N/m²]

Wind load in global y direction [N/m²]

Reference height of structure for wind load [m]

Upper-case Latin symbols

Exposure factor for wind and snow load

External wind pressure coefficient

Internal wind pressure coefficient

Temperature coefficient for snow load

DaP Dihedral angle preserving (joint)

DoF Degrees of freedom of a mechanism

Total number of links in a mechanism

Total load from combinations [N]

Number of grounded links in a mechanism

Turbulence intensity for wind load

ISA Instantaneous Screw Axis

Number of joints of order i in a mechanism

x

Foldline in rigid-foldable origami

Mobility of a system, equals to degrees of freedom for rigid link mechanisms

Internal moment around local y axis [Nm]

Internal moment around local y axis [Nm]

Internal axial force [N]

Concentrated imposed load [N]

Reaction force in global x direction [N]

Reaction force in global x direction [N]

Reaction force in global z direction [N]

Number of mountain folds around vertex in rigid-foldable origami

Deformation in global x direction [m]

Number of valley folds around vertex in rigid-foldable origami

Self-weight [kg]

1. Introduction

1

1. Introduction

“Are you really sure that a floor can’t also be a ceiling?”

- Escher M. C.

The term itself, transformable structure, is an oxymoron: structure is what gives static shape to

systems, while transformable is a word more at home in the world of the shifting and the

unstable. It’s between these two worlds that transformable structures strike a balance, looking for

a trade-off between the mechanical and static qualities. In the transformation phase, controlled

movements must be potentiated, but once the mechanism is locked in place, the resulting

structure must be rigid and secure in its use. The terms structure and mechanism (or linkage) will

hence be used freely in the thesis, sometimes referring to the very same geometries, depending

on the state they are in.

The fact that this field of study is located at an intersection point of many other domains makes it

very diverse, and many points of view need to be reconciled in any transformable design. Not

only structural and kinematical aspects, but also three-dimensional geometric patterning,

transport and actuation play a part.

In short, thinking about transformable structures inherently includes the fourth dimension of

time,a factor often minimalized in the building industry, where static and unchanging

constructions rule. Although transformable structures have been used throughout history -

mostly on the fringes of architectural culture - their more recent popularization inscribes itself

into a wider paradigm shift, where a dynamic lifestyle and durability are two key notions.

Transformable structures can offer dynamic answers to modern problems, such as deployment

for creating temporary spaces, responsiveness to climatic influences, and change of use.

The design of transformable structures then is the design of change.

Fig 1.1 Metamorphosis I (Escher M. C. 1937)

1. Introduction

2

1.1 Categorization

Classification into different groups is useful to gain insight into shared underlying principles.

The categorization used in this thesis, shown in Fig 1.2, is based on the work of Hanaor A. and

Levy R. (2001) who discern two main axes that divide transformable structures.

The primary axis, kinematics, describes the important difference in how transformability is

achieved: mechanisms can be made up of rigid links that are connected at joints that offer

controlled local motion. Structures made from deformable links can change shape due to the

elastic properties of materials. Because of the large discrepancy between these two groups and

the advantage of better control in the first group, only rigid-link mechanisms are discussed in this

dissertation.

The secondary axis, morphology, is more arbitrary and describes the basic shapes that make up

the transformables, whether they are bar elements (lattices) or surface elements.

A third axis added here, the mobility, denotes the freedom with which a transformable structure

moves. A transformable with high mobility offers more options but is less easily controlled.

For rigid link mechanisms, the mobility translates directly to degrees of freedom.

Fig 1.2 Categorization of transformable structures

1. Introduction

3

1.2 Basic mechanical concepts

Since transformable structures are inherently mechanisms, some basic notions about their

mechanical systems and connections are necessary to understand them. Here, the concepts of

degrees of freedom and joint classes are shortly explained, and a more expended analysis of the

geometric-kinematic characteristics of the structures is given in chapter 6.

Perhaps the most important thing to keep in mind when dealing with the mechanical aspects of

transformable structures is that they behave as a closed system, and not just separate

mechanisms joined together. However, since many of the transformables are made up of basic

modules, analysis of each of the categories will for simplicity often start with these smallest

building blocks.

1.2.1 Degrees of freedom

Any possible motion a mechanical system can undergo is bound by its Degrees of Freedom (DoF),

or Mobility (M). “The degree of freedom of a mechanism is equal to the number of independent

parameters (measurements) that are needed to uniquely define its position in space at any

instant of time.” Or in other words: “[…] the number of inputs that need to be provided in order

to create a predictable output” (Norton R. L. 1991).

A host of mechanisms, including the ones found inside the domain of architecture, are of a single

degree of freedom (1DoF), since they are easily driven and need only a single control parameter

to function. As such, most mechanisms described in this dissertation are of 1DoF as well.

However, with the rise of data-driven adaptability, the application of mechanisms with a higher

DoF in architecture becomes more thinkable and even desirable. Particularly of interest are

mechanisms in which each DoF is directly linked to a design parameter and can easily be locked

without affecting the other DoF (and such, design criteria). An example would be the design for a

singular façade structure that could independently regulate shade, heating, and ventilation,

directly by changes made within each DoF.

1.2.2 Joint types

Joints can defined as he motion-permitting connection of two or more links. There are several

ways of categorizing joints, the main and most basic distinction being made in the literature

being the one between lower pairs and higher pairs. As Reuleaux F. (1963) defines it, lower pairs

are joints with surface contact (one element encompasses the other, such as in a spherical joint).

Higher pairs are joints with point or line contact. Norton R.L. (1991) notes that, due to

unavoidable practical imperfections, every joint is in fact made possible thanks to discrete

contact points.

Joints are further defined by the degrees of freedom they give between their connecting

elements. The most-used and important joints in transformable structures are listed on the next

page.

1. Introduction

4

R-Joint P-joint Half-joint

S-joint C-joint

Fig 1.3 Joint types

Revolute joint (R-): 1DoF. Often called Pin joint. Fixes the joined links on a mutual axis around

which they have one rotational freedom. Most structures discussed in this thesis use solely R-

joints.

Prismatic joint (P-): 1DoF. The joined links have one relative translational freedom. Telescopic

systems are made out of these joints.

Half-joint: 2DoF. Links have one translational- and one coplanar rotational freedom. They are

referred to as ‘half’ because they limit half the DoF as the typical R- and P-joints.

Spherical joint (S-): 3DoF. Links have three independent rotational freedoms, but all translations

are bound.

Cylindrical joint (C-): 2DoF. Links have one translational- and one perpendicular rotational

freedom.

Part I. Literature Review

2. Scissor-Like Elements

7


Scissor-Like Elements (SLEs), sometimes denominated as scissor units, pantographs or Nuremberg

mechanisms are the most widely used mechanism type in larger-scale structures, thanks to their

reliable synchronous movement, their compactness and their economic use of material. A basic

SLE is formed by bars that are interconnected along their length by one or more revolute joints –

the Intermediate hinges - allowing one free revolution in their (common) plane.

By linking SLEs together through articulated joints at their end nodes, planar and spatial grids can

be formed that all possess a single DoF, being able to deploy easily from compact bundles to

space-encompassing frameworks.

The intermediate hinges that allow this synchronous movement of the bars are at the same time

an encumbrance on the static-structural level. The continuity of the bars makes for a bending

moment at the location of the central hinge, exactly where the material is at its least because of

the need for a physical joint axis. Hence, the largest deformations will happen at the location of

these weak spots. The deformability of SLE structures and the fact that they need to statically

comply not only in deployed, but also in the intermediate states, make the design process

iterative and often long-winded. As Gantes C. J. (2004) put it:

“From a structural point of view, deployable [SLE] structures have to be designed for two

completely different loading conditions, under service loads in the deployed configuration, and

during deployment. The structural design process is very complicated and requires successive

iterations to achieve some balance between desired flexibility during deployment and desired

stiffness in the deployed configuration.”

In this chapter firstly a geometric categorization and the different variations of SLEs are given,

based mainly upon the theoretical work done by Escrig F., Langbecker T. and De Temmerman N.

and the designs and patents of Hoberman C. Afterwards, a brief history of the architectural

applications is given, showing the way SLE structures have been used as space-encompassing

systems.


8

2.1 Geometrical possibilities

Based on variations in the basic SLE – the shape of the bars and placement of the intermediate

hinges – three general subgroups can be identified: translational-, polar-, and angulated elements.

2.1.1 Translational units

Defining the unit lines (dotted lines in Fig 2.1) as the imaginary lines

that connect the articulated end nodes of the bars, the translational

SLEs are characterized by the fact that these lines always stay

parallel to one another. The curves connecting the R-joints are

straight, but there are still many ways of varying within this group.

The most basic scissor structure that is the repeated linkage of a

symmetrical translation element, forming a straight framework, is

called the lazy-tong and is shown in Fig 2.1.

Fig 2.1 Translational unit

Fig 2.2 Lazy-tong scissor mechanism


9

Translational flat variations

To form spatial frameworks of SLEs, a multitude of options is open, starting by defining the shape

of the formed array. Fig 2.3 shows the results of choosing a square grid array formed by

connecting the edge nodes to each other at straight angles. This results in an unstable structure

in the projected-plane, needing additional bracing for the square shapes to make it function

optimally in static state.

The hexagonal grid in Fig 2.4 is made up of equilateral triangles and offers in plane stability and

greater strength, at the cost of compactability.

Fig 2.3a Square grid array Fig 2.3b (Escrig F. 1991b)

Fig 2.4a Hexa-triangular grid array Fig 2.4b (Escrig F. 1991b)


10

More complex framework shapes are derived when the planes of each of the lazy-tongs no

longer cut each other perpendicularly, but obliquely. By doing so, the intermediate hinges of 2 or

more SLEs intersect, and fewer bars meet at the end nodes. In Fig 2.5 the example of a triangular

oblique grid is given. The intermediate hinge (marked in orange) has to be specifically designed

for the three bars crossing each other.

Fig 2.5a Oblique triangular grid array Fig 2.5b (Escrig F. 1991b)

As Escrig F. (1986) observed, the perpendicular type of grids can be formed as a collection of

prismatic elements (Fig 2.6), and the oblique grids from the collection of anti-prismatic elements

(the bars cross through the center of the circumscribing prism) (Fig 2.7). For a more complete set

of prismatic variations, the 1986 article of Escrig F. is highly recommended.

Fig 2.6 Prismatic modules make up Fig 2.7 Anti-prismatic modules

make up straight grid array make up oblique grid array


11

Translational curved variations

Even though the unit lines of translational elements by definition stay

parallel, a curved grid can be formed by varying the point on the bars

that the intermediate hinge is connect. Fig 2.9 shows such a

translational curved variation, in which the array consists of two

mirrored halves so that a central peak is formed. It should be noted

that the compacted bundle retains the original height of the

completely deployed mechanism, making it less than ideal for the use

in transportable structures.

Fig 2.8 Translational unit

(curved)

Fig 2.9 Curved translational scissor mechanism

Translational multi – layered variations

By raising the amount of intermediate hinges for each bar, multi-layered systems can be made

from SLEs. As the amount of fixed points and the structural height per SLE are increased, the

whole will be subject to a smaller maximal bending moment. That is, however, at the cost of

more material and lesser compactness, and a multi-layered structural system is seldom used for

deployable structures where a compact bundle is key. However, the rhombic shapes as in Fig

2.10 can form the basis of elaborate spatial structures, such as ruled surfaces, a category of SLE

structures of which the applications are discussed further on in this dissertation.


12

Fig 2.10 Multi-layered translational scissor mechanism

All of the variations mentioned above for translational elements can also be applied to the

undermentioned polar- and angulated units, touching off even more possibilities of

different shapes of regular frameworks.


13

2.1.2 Polar units

By moving the intermediate hinge of the SLE away from the center by

a certain eccentricity, the unit lines will move from being parallel to

having a polar angle between them, which changes from being 0 at

the completely (theoretical) folded state to being at its maximum at

the fully-deployed state. This maximum angle is proportional to

the eccentricity.

By linking these polar SLEs together, a nearly planar compact bundle

can deploy into a structure with constant curvature, as shown in

Fig 2.12.

Fig 2.11 Polar unit

Fig 2.12 Polar scissor mechanism

Polar free-form variations

More randomly curved shapes can be made by varying the size of the bars within each of the

SLEs. If the condition of complete foldability (and thus, maximum compactness) is to be achieved,

the sum of the partial-bar lengths of two adjoining elements must be equal. This is defined in an

equation first given by Escrig F. (1988) that is from here on referred to as the compactability

equation (Fig 2.13):

[2.1]


14

Fig 2.13 Random bar length scissor mechanism

Equation [2.1] was extended by geometrical description to three-dimensional SLE structures by

Langbecker T. (1991, 1999). The constraint can also be unfulfilled for one of the adjoining SLEs,

while the connecting one can completely collapse into a linear state, hence they will be partially

foldable (De Temmerman N.). Finally, the compactability constraint forms the basis of the

design of deployable structures, and applies much more widely than for SLEs.

Polar singly curved variations

By repeating the arches formed by polar SLEs in a linear fashion and connecting them by

translational SLEs, cylindrical structures can be made. Barrel vaults have been researched by

Escrig F. (1986, 1996) and geometrically and structurally investigated by Langbecker T. (2000). Fig

2.15 shows an alternative, braced cylindrical vault that was used as a calculation model of

Langbecker.

Fig 2.14 Barrel Vault (Escrig F. 1986) Fig 2.15 Barrel vault with cable

substructure and X-bracing

(Langbecker T. 2000)


15

Polar doubly - curved variations

By using polar units in multiple directions, doubly – curved spatial structures such as domes can

be made. Escrig F. (1988) demonstrated different ways to form dome-shapes from polar units.

The domes in Fig 2.16 and Fig 2.17 can be made using respectively square and triangular grid.

The advantage of these regular grids is the modularity of the polar units. Domes can be

generated from oblique grids as well, as is discussed in the section 2.2, architectural applications

of SLEs.

Fig 2.16 Dome from square modules (Escrig F. 1988)

Fig 2.17 Dome from triangular modules (Escrig F. 1988)

The domes made from these regular grids deviate from a pure spherical form. To approximate a

more constant curvature, the edges of the polygons from any geodesic dome can be replaced by

modular polar elements. The completely folded and completely deployed state don’t show any

geometric problems, but in certain intermediate phases there can be geometric incompatibilities,

which have to be resolved by artifices that locally open more DoF or by material deformation

(see bi-stable structures in chapter 6). An example is the geodesic dome in Fig 2.18.


16

Fig 2.18 Geodesic dome from triangular modules (Escrig F. 1988)

A last category mentioned by Escrig F. are the domes generated from a rhombic pattern; so-

called lamella domes, which don’t show any incompatibilities, and are easily designed (Fig 2.19).

Fig 2.19 Lamella dome (Escrig F. 1988)

Anticlastic surfaces such as hyperbolic paraboloids can also be made by flipping the side of the

eccentricity of the intermediate hinge, or keeping a central hinge and using side-by-side

compatible translational elements as demonstrated by Langbecker T. (2000).

Fig 2.20 Anticlastic surface from translational units (Langbecker T. 2000)


17

2.1.3 Angulated units

Fig 2.21 Angulated units Fig 2.22 Angulated unit variations

(Hoberman C. 1990a)

Angulated elements, popularized by Hoberman C. (1990a), possess a

central kink of angle ε which causes a constant angle γ between the

unit lines throughout the whole transformation process. As

Hoberman also demonstrated, it is actually the relative location of the

intermediate hinge that is of importance, and differently shaped

figures can be used (Fig 2.22). The angle δ between the unit line and

the adjoining semi-bar is hereby bound by the equation:

[2.2]

This opens the possibility to make ring structures as in Fig 2.24 that

open and close around a central point O. This radial deployment has

a lesser compactness than a linear one, since the minimum phase is

constrained by the continuity of the chain of elements.

Fig 2.23 Multi-angulated

unit

Fig 2.24 Radial scissor mechanism from angulated units


18

By introducing more kinks, a multi-angulated element can be formed. Equation [2.2] still holds

and, as such, a radial kinematic system can be made that is denser and consequently will be able

to carry more loads and span greater widths. An example of triply angulated elements is

given in Fig 2.25.

Hoberman C. (1990, 2001) changed the size of the adjoining elements – always complying the

compactability equation [2.1] – to make radial, non – circle shaped elements. Likewise, You Z. and

Pellegrino S. (1997) developed a general angulated unit for non-circular closed radial geometries.

Fig 2.25 Radial scissor mechanism from multi-angulated units

Doubly - curved spatial variations

Approximately spherical surfaces can be created by using these closed-ring structures as the

circumferences of a sphere, and making them intersect in a triangular grid by using a geodesic

subdivision into hexagons and pentagons (Fig 2.26). Platonic solids such as the Icosahedron in

Fig 2.27 can be made combining translational and angulated SLEs. These figures were also made

famous by Hoberman C. These mechanisms all have a mobility of 1DoF.

Fig 2.26 Expansion of geodesic dome from angulated units (Hoberman C. 1990a)


19

Fig 2.27 Expansion of icosahedron from angulated units (Hoberman C. 1990a)

Gómez Lizcano D. E. (2013) has shown how a pendentive dome can be generated by grouping

different angulated modules based on the intersections of a sphere and pyramidal shapes of

different width.

A myriad of different anticlastic shapes can be created using angulated SLEs by changing both

their total length and kink angle, as Hoberman has demonstrated. In the work of Roovers K. et al.

(2013 the conversion of any arbitrary continuous surface to a scissor mechanism is described

geometrically. His approach makes it possible to optimize any form for maximum compactness.

One should note, however, that freeform design almost always go at the loss of modularity of the

composing units.

Fig 2.28 Expansion of anticlastic geometry from angulated units (Roovers K. et al. 2013)


20

2.2 Architectural application

The usage of the SLE mechanisms in their different variations has led to some architectural

applications in which the scale of the projects together with the kinematic behavior are nothing

less than awe-inspiring. However, realizations are few because of the mentioned difficulties in

structural calculation of each phase of the deployment. The fact that furthermore there is no

single regulatory body for deployable structures makes them unknown and unused by the

mainstream of designers. The examples used in this section consist only of real-life structures or

models that were used to this goal: intents to materialize the discussed geometries.

The ‘father of scissor-like structures’, and the original person responsible for their proliferation

and the widespread research in the academic world, is undoubtedly the Spanish architect Emilio

Pérez Piñero. The reason his work has not been mentioned until this point in the dissertation is

that his methods were focused on generating real structures: he was not at all an academic

figure, but analyzed the geometrical and structural principles behind his work just in order to

obtain patents and to further their realization.

Fig 2.29 Emilio Pérez Piñero and his design for a deployable theatre, oblique grid from polar units

(adapted from http://perezpinero.org/)


21

In Fig 2.29, a domical design with an oblique triangular grid is shown. It is deployable through the

manipulation of 6 central joints (Escrig F. 1991a). It was in its time - 1961 - a novel structural

concept and consequently won the London-based competition for a traveling theatre. Because

of the press coverage and the consecutive travels of Piñero in which he promoted his designs,

SLE structures were made known to many of his contemporary architects and engineers.

Fig 2.30 Compacted bundle for transport (adapted from http://perezpinero.org/)

In Fig 2.30, the compact bundle for an oblique-grid dome structure can be seen. One of the most

important inventions that made the realization his structures possible was the design of the

central joints that connect the spatially intersecting SLEs, demonstrated in small scale metal

models in Fig 2.31 for both quadrangular and triangular grids. Pérez Piñero went on to design

both planar and curved scissor-like structures until his early death in 1972.

Fig 2.31 Joint details in deployable test models done by Pérez Piñero (Cruz J. P. S. 2013)

Having been inspired after seeing the work of Pérez Piñero, Spanish architect and engineer

Escrig F. carried on his legacy to the academic world. Together with Sánchez J. and Valcárcel J. he

not only geometrically and structurally analyzed SLE systems, but realized many new

pantographic typologies. In Fig 2.32, their design for the deployable cover of the San Pablo

swimming pool in Seville, based on a polar quadrangular grid, is shown. Notice the diagonal

bracing elements used to stabilize the grid.


22

Fig 2.32 Deployable swimming pool from rectangular modules (Escrig F. 2012)

As seen in Fig 2.33, Escrig and Sánchez also made use of a curved grid. This allows for large spans

to be covered with efficient material use, but goes at the cost of the compactness of the elements

and thus requires large-scale transportation.

Fig 2.33 Deployable swimming pool from multi-layered curved bars (Escrig F, 2012)

Aforementioned Hoberman C., a multidisciplinary designer, used his patented angulated units to

build several pantographic structures at sculptural-architectural scale. All of them are radially

deployable, either compacting to a central position as in the geodesic dome in Fig 2.34, or along

their circumference as in the Iris dome in Fig 2.35.


23

Fig 2.34 Expansion of triangulated geodesic dome from angulated units (Hoberman C. portfolio)

Fig 2.35 Edge-to-center deployment of Iris dome from angulated units (Hoberman C. portfolio)

As mentioned before, Gantes C. J. (1997) meticulously investigated stress-related effects of SLEs,

in particular to make self-locking structures (see bi-stable structures in chapter 6). As

mathematical analysis combined with finite-element modeling is of great importance before even

attempting to construct these structures full-scale, few of them have been built.

Raskin I. (1998) proposed and analyzed more simple systems such as deployable slabs and

columns (Fig 2.36) for regular use in construction. He describes in detail how it is possible to go

from the mechanical spatial grid to a stable structure by adding boundary conditions such as a

top layer of rigid plate elements.

Fig 2.36 Deployable column and slabs from translational units (Raskin I. 1998)


24

A system based on the angulated ring mechanism of Fig 2.25 was proposed by Kassabian P. E.

(1997) to form a retractable roof by adding plates to the bar structure. Jensen F. V. and Buhl T.

(2004) later improved upon this system by using only the plates as rigid elements, abolishing the

need for any secondary bar system. A stadium designed by Lake Associates used this very system

(Fig 2.37).

Fig 2.37 Retractable stadium roof based on angulated units (Lake Associates)

Another way of covering pantographic systems consists of using separate foldable plate elements

that join in completely deployed state. This method was first used by Pérez Piñero E. in the

design of glazing panels for the Dalí museum in 1970 and later improved upon by Valcárcel V. P.

(Escrig F. 2012). Fig 2.38 displays the ‘fish’ fold designed by the latter to cover a deployable dome.

Naturally, by introducing two-dimensional elements into the pre-existing grid of quasi

one-dimensional bars, the mechanism becomes less compact, as seen in the right-side image.

Fig 2.38 Use of additional plate elements on translational grids (Escrig F. 2012)


25

De Temmerman N. (2007) combined the know-how of pantographic systems and tensile

substructures to design quickly deployable structures in which the fabric works actively on a

structural level. Fig 2.39 and Fig 2.40 demonstrates some new tent typologies designed by De

Temmerman in this fashion.

Fig 2.39 Textile substructure in barrel vault Fig 2.40 Textile structure attached to

from variable polar units central mast from angulated units

(De Temmerman N. 2007)

Alegria Mira L. (2010) bridged the gap between translational-, polar- and angulated units by

designing a Universal Scissor Component (USC). By giving the possibility to vary the central

hinge, the eventual angles between the SLEs can be changed, and many singly- and doubly

curved shapes can be made. The price paid for this is the maximum compatibility, especially of

the translational and polar arrangements. Nevertheless, a higher structural strength of the units

is gained in the zones where the largest bending moment is introduced, making it a versatile and

effective design. In Fig 2.41, the basic USC is shown, together with an icosahedral variation.

Fig 2.41 Universal Scissor Component (USC) and icosahedral variation (Alegria Mira L. 2010)


26

A last group that can be shaped by the use of SLEs is that of the ruled surfaces. A one-sheet

hyperboloid type structure has been proposed by Escrig F. and Sánchez J. (2012) (Fig 2.42).

The compactability of these hyperboloids is too low to be considered a potent alternative to

single-layered pantographic structures, but their transformability can be useful e.g. for climatic

adaptation.

Fig 2.42 Deployable hyperboloid from multi-layered grid (Escrig F. 2012)

Hyperbolic paraboloids - as being investigated by Maden F. and Teuffel P. (2013) - can also be

formed by using multi-layered scissor elements. However, as the joints of the bars undergo a

relative translation, these mechanisms cannot be formed merely by revolute joints, and a

combination of P- and half-joints need to be specifically designed to allow for their mobility. For

this reason, their design is still troublesome. Fig 2.43 demonstrated the possible use of

6 independently mobile saddle-surfaces used for shading in a public space.

Fig 2.43 Adaptable hypar surfaces from multi-layered grid (Maden F. and Teuffel P. 2013)

3. Rigid-foldableOrigami

27

3. Rigid-Foldable Origami

Rigid-foldable origami elements, Foldable Plate Elements (FPE’s), or hinged plate elements form the

basis of kinetic surfaces that are traditionally continuous, but may just as well be discontinuously

connected. The rigid plate elements are connected by R-joints along their outer edges. The

amount and the relative in-plane angle of these edges will determine the boundary conditions of

the resulting kinetic surface. As such, by simply changing the relative inclination of the edges,

systems can sometimes gravely shift their kinematic behavior.

Because of the fact that the basic element is a surface – unlike with the quasi-linear SLE’s – and

because of the fact that these surface elements oftentimes need to be continuously connected,

the maximum compactness achieved in foldable plate structures is much lower, and so the value

of these structures is sometimes sought in the different qualities of their analogue phases: the

advantages of adaptive structures versus those of typically deployable ones.

On the flip side of the coin, they typically allow for an efficient distribution of forces because of

their continuous and corrugated surfaces. The continuity of the hinges is of importance here, and

special care has to be taken not to introduce local concentrated loads that unnecessarily change

the static behavior of the structure for the worse. Additionally, the waterproofing of these hinges

is an interesting theme to be addressed in the materialization process.

Almost all foldable plate structures are based upon the ancient Japanese art of origami.

Designing them often starts with a single sheet that is folded - without any cutting - into a three-

dimensional figure. When it is possible to generate an FP surface from this process, it is said to be

developable (Solomon et al. 2012). Sometimes developability and continuity of the surface are

unnecessary constraints when thinking of architectural applications. Nevertheless, the work done

on paper origami models when designing FP structures proves to be invaluably for fully

understanding the transformation processes.

In this chapter, firstly the general geometric possibilities and variations are described. This part is

supported by the works of major figures in the field of origami such as Miura K., Resch R., Tachi T.

and Hull T. Subsequently, the limited body of noteworthy architectural applications is given,

focusing only on the buildings or components that are truly transformable: static buildings

inspired by origami structures are left aside.


28

3.1 Patterns and tessellations

Both modularity and homogenous mechanical behavior are of value to FP surfaces. An efficient

way to design them is therefore to start with the basic repeated patterns made up of copied or

similar elements. These are named tiling patterns or tessellations. All of the discussed patterns are

based on foldable, developable surfaces. The following collection of origami tessellations is not

meant to be exhaustive, but to offer a view on the most interesting patterns for application in the

fields of engineering and architecture. Some have proven their use in large-scale application,

while other still wait to be used.

All of the patterns are made up of Mountain (M-) and Valley (V-) folds, folds around which the FPE’s

rotate clock- and counterclockwise, relative to the orientation of the surface. For each of the flat-

folded patterns in this chapter, as in Fig 3.1, mountain folds are shown in orange lines and valley

folds in blue dotted lines.

3.1.1 Miura-ori pattern

Fig 3.1 Miura-ori fold pattern

The name Miura-ori is a contraction of the Japanese ori – ‘fold’ – and the name of the inventor of

the pattern, Japanese astrophysicist Miura K. He devised the compactly foldable surface to use it

as a deployable solar sail for a space unit (Fig 3.3). Another big advantage of the mechanism is

the fact that it has only 1 DoF, making it easy to actuate. For these reasons, the Miura-ori pattern

is one of the most widely studied patterns in contemporary engineering.

Fig 3.2 Closed and semi-deployed Fig 3.3 Solar sail application of Miura-ori fold

Miura-ori mechanism (Miura K. and Natori M. 1985)


29

In Fig 3.2, the expanding mechanism is shown. It consists of interconnected quadrilateral plate

elements. The regular pattern is made up of identical parallelograms and unfolds into a

(corrugated) planar geometry. However, variations are possible that disturb the planar pattern

into a singly or doubly curved one. For any variation around a single internal vertex of the

pattern, Tachi T. (2009) quotes the relationship between the fold angles from and Hull T. C.

(2006) as:

and [3.1]

The minus value in the first equation is due to the fact that the fold line of is a mountain fold,

while the other ones are valley folds. The relationship between these two pairs of angles is then

given by:

( ) ( ) ( )

[3.2]

In which ξ is the angle between and , in any configuration except for the folded one given

by:

( ) [3.3]

which reaches its maximum when the maximum operating angle is reached. thus equals

only for flat-folded patterns. Using these equations parametric models of connected

(repeating) vertices can be set up.

Fig 3.3 Variation of Miura-ori fold around single internal vertex (adapted from Tachi T. 2009)

Tachi continues his 2009 paper by giving the necessary condition for rigid foldability of any

quadrilateral pattern in function of the lateral ( and ) and longitudinal fold angles (( and

), which in their turn can be defined in terms of the angles from equations [3.2] and [3.3].

Using this method, any rigid-foldable freeform Miura-ori pattern can be created. An example is

given in Fig 3.4.Tachi furthermore developed software, called Freeform Origami, for the easy

manipulation of (amongst others) Miura-ori patterns without affecting their foldability.


30

Fig 3.4 Expanding Miura-ori variation (Tachi T. 2009)

Another operation on Miura-ori patterns is the removal of facets that are unnecessary for

mobility, i.e. the single DoF of the pattern is left unaffected by this operation. Beatini V. and

Korkmaz K. (2013) give the boundary conditions for this operation. An example of their work is

seen in Fig 3.5.

Fig 3.5 Flat Miura-ori variation with removed facets

(Beatini V. and Korkmaz K. 2013)

A final note on the geometrical characteristics of the Miura-ori pattern is the existence of a bar-

shaped counterpart of the basic 4-facets tile in the form of a spherical linkage, as demonstrated

in Fig 3.6 (Abdul-Sater K. et al. 2014). Analogously, any origami pattern can be replaced by an

array of spherical linkages by replacing the foldlines by the rotational axes of compact R-joints.

This method is not only interesting for the analysis of origami mechanisms, but also gives the

possibility to develop more compact, simple, mechanisms based on plate elements.

Fig 3.6 Corresponding bar linkage of single Miura-ori vertex (Abdul-Sater K. et al. 2014)


31

3.1.2 Yoshimura pattern

Fig 3.7 Yoshimura fold pattern

The Yoshimura pattern is made up of triangular facets, and typically folds into a singly curved

corrugated surface, although also doubly-curved surfaces can be reached when the pattern is not

flat-foldable. It has multiple DoF, but can be stabilized fairly well and shows a high rigidity when

done so, which makes it suitable for engineering and architectural purposes.

Fig 3.8 Barrel vault from Yoshimura pattern (De Temmerman N. 2007)

The basic shape of interest that can be developed is the barrel vault, as shown in Fig 3.8

(De Temmerman N.). It results from a regular pattern as in Fig 3.7. For the pattern to be rigid-

foldable, the apex angle of each triangular facet has to lie in between ⁄ and . Furthermore, for

each amount of facets in the curved direction, there exists a certain apex angle for which the

mechanism can be fully folded into its most compact form, i.e. the form in which the semi-facets

touch each other, as in the second diagram in Fig 3.8. De Temmerman gives the clear formula:

( )

( ) [3.4]

In which: : apex angle

η: amount of plate elements in curved direction


32

E. g. for the pattern in Fig 3.8, where η = 7, an apex angle of ⁄ gives the compact form.

For a comprehensible design method based on the parameters of height, span, structural height

of barrel vaults based on semi-regular Yoshimura patterns, the doctoral thesis of De

Temmerman N. (2007) is highly recommended.

Irregular tessellations can also be generated easily from the Yoshimura pattern, their only

necessary condition for rigid-foldability being that the sum of 2 adjoining deformation angles

stays constant (Fig 3.9). Following this rule, interesting shapes such as the ones in Fig 3.10 can be

designed.

Fig 3.9 Yoshimura variations retain rigid-foldability when the sum of two adjacent

deformation angles is constant (Tonon O.L. 1993, as adapted by De Temmerman N. 2007)

Fig 3.10 Rigid-foldable Yoshimura variations (Tonon O. L. 1993)

By taking away the semi-facets at the top of the pattern and connecting the remaining full facets

with each other, polar, doubly curved geometries can be made, as in Fig 3.11. Different

combinations of regular singly- and doubly-curved patterns are shown by De Temmerman for a

maximum of 5 facets in the curved direction. It has to be noted that these geometries can only

exist in the erected state shown, i.e. they can’t be form a fully closed loop and at the same time

stay rigid-foldable.


33

Fig 3.11 Doubly-curved Yoshimura variation in compacted and fully deployed state

Fig 3.12 Combination of single- and doubly curved mechanism into static structures


Another possible geometry that applies the Yoshimura pattern is the fully closed cylinder that

folds upon itself, as investigated by Guest S. D. and Pellegrino S. (1994) (Fig 3.13). Later research

to use these cylinders structurally as inflatables has been done by Barker R. J. P. and Guest S. D.

(2000).These cylindrical mechanisms are also referred to as origami booms in the literature.


34

Fig 3.13 Inflatable booms based on Yoshimura pattern (Guest S. D. and Pellegrino S. 1994)

3.1.3 Waterbomb pattern

Fig 3.14 Waterbomb fold pattern

Another multi-DoF pattern is the waterbomb pattern, being made up triangular facets with

straight-angled apexes. The pattern typically introduces a double curvature in its surface, as seen

in its folded state in Fig 3.14. Its high mobility makes it less suitable for large-scale applications,

but may prove its use in smaller design applications.

Fig 3.15 Folded waterbomb pattern (Tachi T.)


35

3.1.4 Resch patterns

Fig 3.16 Resch fold pattern

The Resch patterns were developed by the geometrist and artist Ronald D. Resch during the

1960’s. Many variations on the basic pattern in Fig 3.15 have been proposed both by Resch and

others, but the all share some characteristics: they typically have 2DoF per module, one ‘twisting’

and one folding one. In the patterns two facet layers can be detected. Firstly, there is the front

layer of which the facets barely undergo any out-of-plane rotation, but which rotate around

normal axes through their centroids. Secondly, there is the back layer of which the facets

undergo a complete relative rotation of ⁄ between folded and unfolded state. This way, the

back facets are ‘tucked in’ between the front layers (Tachi T. 2013). The back layer is invariably

made up of triangular facets, while the facets of the front layer can take on different

complementing shapes.

The partially folded pattern is shown in Fig 3.16. In Fig 3.17 Resch and his arts- and architecture

students are shown next to a large-scale test of the same regular triangular Resch-pattern, still

from the documentary Paper and Stick Film.

Fig 3.17 Hexagonal Resch pattern Fig 3.18 Large-scale folded Resch pattern

(Resch R. D. and Armstrong E.)


36

Fig 3.17 Variations on Resch pattern (Piker D. 2009a)

Some regular variations of the Resch pattern are seen in Fig 2.17, respectively with a square,

triangular and hexagonal facets in the top layer. Because of the high mobility, the Resch patterns

are able to be made irregular more easily than - for example - the Miura-ori and the Yoshimura

patterns without affecting the foldability. Tachi T. (2013) makes use of this characteristic to

generate freeform rigid-foldable origami of nearly any shape, as exemplified in Fig 3.18.

Note that the compact foldability and the transformative motion are not of interest here, but the

final static shape is.

Fig 3.18 Random foldable shapes with Resch pattern

(Tachi T. http://www.tsg.ne.jp/TT/)


37

3.2 Flat-foldability

The compactability of many origami mechanisms will often be directly related to their flat-

foldability, their ability to reach the compact folded state in which the plate surfaces are all

parallel to each other. Take a single vertex with a surrounding crease pattern made up of

mountain folds m (orange) and valley folds v (blue). The creases are named , , …, with

. Let the angle between any two creases and be .

Three criteria for their flat-foldability will then be (Bern M. and Hayes B. 1996):

- The sum of alternate angles around the vertex equals π (Kawasaki’s theorem)

- | | (Maekawa’s theorem)

In which: U: number of mountain folds

V: number of valley folds

- if

, then and must have be opposite folds (U, V)

Fig 3.19 Flat-foldability around a single vertex

Kawasaki’s theorem alone is however sufficient to predict flat-foldability of a single vertex, as the

other theorems will follow directly out of this. Proof of this, and further explanation of the other

theorems is given in the 1996 article of Bern M. and Hayes B. and the 2002 article by Belcastro S.

and Hull T. (2012). Expanding to a global flat-foldability, i.e. of a multi-vertex plate structure,

Kawasaki’s theorem is a necessary but not sufficient condition. As of now, no algorithm is known

to solve the global problem, since it has been proven to be a problem of NP complexity.


38


Application of rigid-foldable origami mechanisms in architecture is limited, due the lack of know-

how and tradition in materialization. When the problems of hinges, plate thickness,

waterproofing and compactability are researched on a larger scale, many interesting designs will

become possible. In what follows, some examples from the limited body of work are given,

starting with the large scale theoretical projects and moving towards local design elements.

Some more theoretical origami structures have been proposed by Tachi T. Fig. 3.20 shows a

structure based on the waterbomb pattern. It consequently has multiple DoF, making it

deformable to the user’s needs or follies. The high mobility however also makes it structurally

highly unstable, and a strong connection to the ground plane would be needed here to stabilize

it. Another project by Tachi is the Miura-ori variation used to make a temporary and deployable

connection between two buildings in a museum complex (Fig 3.21).

Fig 3.20 Shape-shifting pavilion from Yoshimura pattern (Schenk M. 2012, project by Tachi T.)

Fig 3.21 Deployable passageway between buildings, from Miura-ori pattern

(Tachi T. http://tsg.ne.jp/TT/)

De Temmerman N. (2007) developed a deployable shelter based on the Yoshimura fold (Fig 3.22).

The plate elements here are substituted by bars lying on the perimeter of the facets, after which

half of the bars are removed in locations where they were doubled. The joints used here are

based directly on the vertices of origami folds. To give structural height to the resulting barrel

vault a fabric screen is added as a tensile layer.


39

Fig 3.22a Corresponding bar structure Fig 3.22b Joint detail

of Miura-ori pattern (De Temmerman N. 2007)

A large-scale project that was really materialized is the retractable roof system developed by the

Venezuelan architect Hernandez C. H. (2013) It was used first in the expo of 1992 in Sevilla and

later in projects such as a pool cover in Venezuela (Fig 3.23a). It is a regular Miura-ori pattern with

trapezoidal facets. Materialization was done in thin metal sheets, and the joints were designed

especially for stabilizing and waterproofing the cover (Fig 3.23b). Note that the roof is not

self-supporting, but carried by light-weight trusses.

Fig 3.23a Deployable roof from Miura-ori pattern Fig 3.23b Joints detail

(Hernandez C. H. 2013)

A mobile bamboo pavilion was proposed by architect Tang M. The first steps in the opening

process are shown in Fig 3.24. The basic pattern is fairly simple, but it needed to be triangulated

in order to be even mobile. The elegance of circular origami-fold mechanisms is that they can be

locked and made static by simply fixing the opposing ends to each other.


40

Fig 3.24 Radial shelter from variable pattern (Schenk M. 2012, project by Ming Tang)

Rigid-foldable origami has been applied successfully to kinematic facades. Most often the

materialization here is less cumbersome, since the active folding angles are not big and designs

taking into account the plate thickness are more easily achieved. A first and simple example

thereof is the shading device designed by Ernst Giselbrecht + Partner (Fig 3.25), where single-fold

and sliding mechanisms can each be actuated separately to ensure a pleasant inside climate. A

second example is the iconic façade designed by Aedas Architects, where 1DoF modules of 6

elements can closely regulate the solar gains (Fig 3.26).

Fig 3.25 Hinged facade (Ernst Giselbrecht + Partner) Fig 3.26 Triangulated foldable façade

(Aedas Architects)


41

On a smaller scale, the acoustic panels designed by RVTR make for a very interesting project: a

simple Resch pattern has been applied where the front layer of bamboo facets function as

reflectors, while the tucked in facets work as absorbers (Fig 3.27a). By increasing and decreasing

the operating angle, different acoustic atmospheres can be created. A central electronic panel

with sensors can adapt the different actuators in real-time. The actuators themselves are simple

P-joints between the facets, three per module (Fig 3.27b)

Fig 3.27a Acoustic panels from Resch pattern Fig 3.27b Actuation system (RVTR)

On a smaller scale yet, the kinematic characteristics of origami can be applied to create more

strong and rigid meta-materials, in particular sandwich panels. Schenk M. and Guest S. D.’s 2010

paper makes for a good introduction on the Miura-ori fold from the perspective of structural

engineering.

Engineers such as Miura K. (1972) introduced the sandwich panel (Fig 3.28) and modern

engineering firms such as Tessellated Group (Fig 3.29) are intending to commercialize their

origami sandwich products. The main advantage that these panels offer is their controlled

deformability which makes them suitable for impact resistance.

Fig 3.28 Miura-ori sandwich panel Fig 3.29 Corrugated sandwich panel

(Miura K. 1972) (Tessellated Group)


42

To conclude, rigid-foldable origami can also be found applied more trivially in design elements

such as the wood fabric created by Elisa Strozyk (Fig 3.30). The wooden facets are glued onto an

underlying textile. It is an interesting concept to apply on a larger scale, where discrete hinges

could be replaced by a continuous surface material connecting rigid facets.

Fig 3.30 Wooden facets on fabric (Elisa Strozyk)

What is interesting about smaller-scale projects is simply the fact that they are materialized, and

during the design process they have likely gone through some difficult iterations that bare the

problems involved in modeling rigid-foldable origami for real-world use. They can serve as

stepping-stones to popularize know-how about its employment. Together with the academic

studies of the origami thickness problem and the structural behavior in static state, they can

form a matrix for new and better designs.

4. Jitterbug-Like Linkages

43


The first Jitterbug-like linkages were nothing more than theoretical models professed firstly by

Buckminster Fuller. It was in fact a single transformative octahedral model that sparked Fuller’s

interest and that he dubbed the Jitterbug, which was the name of a popular ballroom dance of the

1940’s that the movement of the mechanism reminded him of. A later-made physical model of

this first Jitterbug throughout its transformation is shown in Fig 4.1.

The mechanism in itself is a closed spatial loop that has a single DoF. For Fuller, the discovery of a

mechanism that moved through different polyhedrons was paramount: In the mechanism “the

elementary geometric forms that have stood together since Plato’s time as a set of regular solids

are shown now to be a phase transition in a single process of metamorphosis.“

(Krausse J. and Lichtenstein C. 1999)

Fig 4.1 Transformation of Jitterbug mechanism (http://wvutoday.wvu.edu)

Fullers’ interest in the Jitterbug-like mechanisms was mainly intellectual, but he “wrote extensively

on […] how it could help understand the abstracted sciences of chemistry and physics by allowing

us to see movements that are normally occurring invisibly all around us.”

(Krausse J. and Lichtenstein C. 1999). This would later prove to be true at least for the field of

virology: certain viruses have been discovered using the expansive movement of Jitterbugs to

negotiate their environments (Shim J. et al. 2012). The discovery and further research of Jitterbug-

like mechanisms by Fuller has caused them often to be named Fulleroid-like linkages in the

modern literature by prominent researchers in this field such as Röschel O. (2012).

Because of the specific movements of the rigid elements relative to each other, the joints are a

study of careful design, as will be discussed later in this chapter. This difficulty in synthesizing the

joints led to the early physical models made by Fuller and his collaborators to be “mechanically

very unstable structures, requiring a supporting armature to keep them from collapsing.”

(Schwabe C. 2010) One of the original built models is displayed in Fig 4.2.


44

Fig 4.2 Model array of jitterbug-like mechanisms

stabilized by cables (http://popularmechanics.com)

4.1 Geometrical possibilities

The basic octahedral Jitterbug is taken here as an example for later generalization of the Jitterbug

movement. The transformation each triangular element of the octahedral mechanism undergoes

is a helical screw movement along an Instantaneous Screw Axis (ISA) that is normal to the plane of

the element and goes through the circumcenter of the triangle. Since for the octahedron there

exists an inscribed sphere that is tangent to the triangular facets at their circumcenters, the ISAs

of all of the facets pass through the center of the inscribed sphere.

Fig 4.3a Closed Jitterbug mechanism with Fig 4.3b Dilation of the base polyhedron

Instantaneous Screw Axes (ISAs) during transformation


45

The ISAs themselves are fixed in space, and hence the resulting movement of the whole

octahedron is a dilation. This becomes more obvious when the underlying octahedrons are seen

for each configuration of the Jitterbug : since all of the facets retain their original relative angles

and they dilate at the same rate, planes parallel with each element intersect along the edges of a

dilating octahedron, the base polyhedron. Fig 4.3a shows the ISAs of each facet of the octahedron,

and an inscribed circle tangent to some of the facets at the intersection with their ISAs.

Fig 4.3b shows the same geometry going through the Jitterbug transformation, in which the base

octahedron dilates around its center, and in each facet of this base polyhedron the real triangular

facets that define its edges undergo a rotation. The dilation ratio of the base polyhedron is

( ) .

Fig 4.4 Dilation of base polygons Fig 4.5 Maximum deployment

during transformation with base polyhedron

When the maximal configuration is reached, all the vertices of the triangular facets intersect the

edges of the base octahedron in the centers of these edges, as in the hexahedron in Fig 4.5. For

this case, the maximal value for is /3, and so the maximal deformation dilation

is ( ( )) . For the octahedral Jitterbug, the dilation is homogeneous, in other words it is

a homothetic transformation. There exist Jitterbug mechanisms for which the transformation is

not homogeneous, such as the cuboctahedron Jitterbug in Fig 4.6.

Fig 4.6 Dilation of cuboctahedral Jitterbug-like mechanism

(Kiper G. 2010, adapted from Röschel O.)


46

Kiper G. shows in his 2010 doctoral thesis that a mobile homothetic Jitterbug can be obtained

from any polyhedron of which the homothety centers of adjacent facets are in symmetrical

positions relative to their common edge, i.e. when the ISAs (which intersect the homothety

centers of their respective facets) of neighboring facets also intersect each other.

The movement between two adjoining facets along the Jitterbug transformation is made

physically possible thanks to the joints that link together those facets (Fig 4.7). For the purpose of

visibility they are exaggerated here. Dreher D. reportedly developed the first prototypes of said

joints when working as a student under Fuller (Schwabe C. 2010).

The joints allow each facet to rotate inside its plane, while maintaining the dihedral angle

between the facets. For this reason they are often referred to as Dihedral Angle Preserving (DaP)

joints. In the literature, they are also sometimes referred to as gussets, double rotary joints

(Wohlhart K. 1995), spherical double hinges (Röschel O. 2012) or because of their shape simply as

V-joints (Kiper G. 2010).

These joints obviously have two rotational DoF, and as such it is necessary to link the facets of the

Jitterbug together into a closed loop in order to get a 1DoF system.

Fig 4.7a Dihedral Angle Preserving (DaP) joint Fig 4.7b Joint detail (Verheyen H. 1989)

This means that the normals of two adjacent faces have a constant angle between them, even

though the plane which they define may undergo a rotation during the Jitterbug transformation.

(When their plane on the other hand undergoes a pure translation, this implies that the Jitterbug

transformation is a homothety).

The Jitterbug transformation in this way is applicable to different polyhedron groups, the most

obvious ones belonging to the Platonic- and Archimedean solids. As an example, in Fig 4.8 an

icosidodecahedral geometry is shown transforming to a rhombicosidodecahedral geometry.


47

Fig 4.8 Dilation of icosidodecahedral Jitterbug-like mechanism (Verheyen H. 1989)

4.1.1 Odd-valent vertices

It is important to note that the helicoidal movement of each facet of a Jitterbug is of a rotation

opposite to that of its adjacent facets. Because of this, it was once presumed impossible to

construct a mobile Jitterbug from a base polyhedron that has an uneven amount of facets

coming together at any of its vertices. This would imply that a single facet has to be able to

undergo both a clockwise- and a counterclockwise rotation. There are however some loopholes –

tricks that allow for a mobile Jitterbugs to be made out of base polyhedrons that have an odd

number of facets intersecting at their vertices. These will be discussed in the following section of

the chapter.

Double facets – dipolygonids

A first and most important technique in making odd-valent vertex Jitterbugs mobile is simply

doubling all of the facets and connecting them in an alternate manner. This doubles the valence

of each of the vertices, thus making any odd-even vertex polyhedron possibly mobile. The facet

doubles are attached to their original facets by means of a single R-joint whose axis is the ISA of

the original facet. Using this method thus implies making a type of spatial Scissor-like elements,

and any loop of four facets (two original ones and their doubles) form a single DoF mechanism. In

other words, there is no need for a closed group of elements to gain the single DoF, and more

stable Jitterbug-like mechanisms are the result.

The idea was first proposed by Clinton J., a former student of Fuller. It was picked up and

researched later in depth by Verheyen H. who named these mechanisms dipolygonids. Some

examples from Verheyen’s 1989 paper are shown in Fig 4.9a-c, showing respectively a cube-,

dodecahedron- and icosahedron dipolygonids, and Fig 4.9d showing the movement of the

icosahedron variation.


48

Fig 4.9a Hexahedral dipolygonid

Fig 4.9b Dodecahedral dipolygonid

Fig 4.9c Icosahedral dipolygonid Fig 4.9d Dilation of Icosahedral

dipolygonid (Verheyen H. 1989)


49

Multiple elements per facet

A more efficient way of making odd-valent vertex Jitterbugs mobile is to subdivide the facets of

the base polyhedron in an even amount of smaller subfacets, so that at each vertex there are an

even amount of them. This changes up the geometric transformation of the whole, and will

almost never result in a homothetic transformation. In Fig 4.10 examples of this method for the

tetrahedron and cube are given by Kiper G. (2010) (who adapted the figures to make them more

readable from the work of Wohlhart K. 2001).

Fig 4.10a Multi-facetted icosahedron Fig 4.10b Multi-facetted cube

(Kiper G. 2010)

In his thesis, Kiper G. also applies this technique to gain ring-like structures from dipyramidal

base polyhedrons. An example is given here for a subdivided geometry of an octagonal

dipyramid in Fig 4.11. These shapes have great interest because their facets behave very much

like SLEs, and hence have greatly varying dimensions in their open and closed states.

Fig 5.11 Multi-facetted dipyramid (Kiper G. 2010)


50

Offset elements

A third and last known technique for obtaining mobile Jitterbugs is using offset elements

between the vertices of connected polyhedron facets. This way, the total amount of elements

around each vertex gets doubled, and a clockwise-counterclockwise movement becomes

possible. An example is given by Kovács F. et al. The model they used in their research for the

behavior of micro-organisms is an expendable dodecahedron with offset elements.

Fig 4.12 Offset elements in dodecahedron (Kovács et al.)

Also in the work of Wohlhart (2001) these offset Jitterbug-like mechanisms are described into

detail. He uses the type mechanism to synthesize both regular and irregular dilating shapes.

Examples are the expanding icosahedron in Fig 4.13 and the cylinder in Fig 4.14.

Fig 4.13 Offset elements Fig 4.14 Offset elements in cylinder (Wohlhart, 2001)

in icosahedral geometry

(Wohlhart, 2001)


51

4.1.2 Planar variations

When the ISAs of adjacent facets are taken to intersect at ∞, they come to lie in the same plane.

The Jitterbug movement now is degenerated into a two-dimensional one. An example of

hexagonally connected triangles is seen in Fig 4.15.

Fig 4.15 Hexagonal dipolygonid pattern

Edges of these planar mechanisms can be connected to create larger polyhedral mechanisms.

In fact, this is exactly what is being done when a facet is subdivided to obtain even-valent

vertices.

When turning the edges, the movement of the facets on each side needs to be compatible with

its subdivided neighbour. Making each edge congruent is a simple way of achieving this, as in the

pyramidal dipolygonid by Verheyen H. (1989) in Fig 4.16.

Fig 4.16 Pyramidal dipoligonid with facetted faces (Verheyen H. 1989)


52


The relative scarceness of designer familiarity with the Jitterbugs and the difficulty in synthesizing

the DaP joints has led them to be underused at bigger-than-human scale in general, and the field

of architecture in particular. Notwithstanding, they show potential for larger-scale structures,

especially in the arts. Some of the few materializations of the Jitterbug mechanism are typically

gadgets and furniture, such as the coffee table fashioned by Verheyen H. in Fig 4.17.

Fig 4.17 Coffee table from cube Jitterbug-like mechanism (Verheyen H. 1989)

Further they have been used in the dancing arts by Tomoko Sato in her ‘Synergetics’ performance

(Schwabe C. and Ishiguro A. 2006), and other performers as seen in Fig 4.18 and Fig 4.19.

4.18 Free body expression with Jitterbug geometry (Schwabe C. and Ishiguro A. 2006)

4.19 Free body expression with Jitterbug geometry (Schwabe C. 2010)


53

A first attempt to apply the Jitterbug mechanism on large scale was the Heureka project (Fig 4.20)

that was the symbol of the Swiss national research exhibition in Zürich, 1991. It was a mobile

sculpture with the original octahedron Jitterbug geometry and 8m side lengths of the triangular

facets (Michaelis A. R. 1991).

It was through the initial lobbying and design done by Schwabe C. that this sculpture would be

realized. “When it opened, its height doubled and the volume five-folded. After three months in

operation it collapsed into the tetrahedral position, because the steel hinges and the connections

to the triangles made of composite polyester were not well enough engineered.”

(Schwabe C. 2010)

Fig 4.20 (Schwabe C. 2010)

A last example of the use of dipolygonid Jitterbugs in architectural applications is by

Gómez Lizcano D. E. (2013), who used a triangular grid applied to the polygons of an icosahedron

Fig 4.2. The polygons are connected to their sides by means of P-joints. The joint connections

between the facets here are more complex and require special maintenance. Although a

prototype of one facet with sliding sides has been made, the whole project remains unbuilt.


54

Fig 4.21a Triangular dipolygonid facet Fig 4.21b Icosahedral pavilion from

with sliding edges dipolygonid facets

Fig 4.21c Expandable dipolygonid pavilion in use (Gómez Lizcano D. E. 2013)

Many more possibilities remain undiscovered, not in the least façade application for the opening

and closing of planar variations, and public sculptures for the polyhedral Jitterbugs. Research in

the development of sturdy DaP (Dihedral Angle Preserving) hinges is a definite requirement

before designing these mechanisms on a large-scale can be made possible.

5.Overconstrained Linkages

55

5. Overconstrained Linkages

In this chapter, different 1 DoF spatial mechanisms with a minimum of elements are discussed.

The term overconstrained refers to the fact that they have more DoF than the analytically

determined constraints predict. This will be discussed in more detail in chapter 5, Kinematic

studies. Most of these mechanisms are well-studied in the fields of kinematics, but more recently

have found their way into the field of deployable structures. As of yet, ways of chaining of these

mechanisms together to form compactable wholes are unsatisfactory, but advances have been

made in the last decade.

Firstly three similar groups of mechanisms, namely the Bennett-, Myard-/Goldberg- and Bricard

linkages, are analyzed as single- and multi-loop systems. They are respectively 4R, 5R and 6R

mechanisms. The geometrical conditions for their mobility are given, clarifying their conceptual

connection to one another. “The three groups of linkages are strongly inter-related, so that it is

convenient, if not necessary, to treat them all at once” (Baker E. J. 1979).

Secondly parallel manipulators, a group of mechanisms popular in contemporary literature, are

discussed. The subgroups of Wren platforms and Sarrus linkages are analyzed further because of

their interest in the field of deployable architecture. Especially the second group will show to be

the basis for many known deployables that have been discussed in previous chapters.

5.1 Bennett linkages

Forming a closed loop of four elements, each connected to one other by a total of four R-joints,

there were formally only two known ways to construct a mechanism from the four parts. The first

one was by making the axes of the R-joints intersect at ∞, gaining a planar rhombus mechanism.

The second way was making each of the four R-axes intersect at a common center point, which

results in a spherical mechanism. In the early 1900’s, Bennett introduced a new single DoF

mechanism of four looped elements that doesn’t adhere to the last two classes of 4R

mechanisms: a new spatial mechanism was born. (Bennett G. T. 1903) A Bennett loop is shown in

Fig 5.1.

Fig 5.1 Deployment of Bennett linkage (Gan W. W. Pellegrino S. 2003)


56

For the 4-bar loop to be mobile, certain geometrical conditions between the revolute joints and

their axes must be satisfied. Here they are quoted from Bennett G. T. (1914) and Chen Y. (2003).

Fig 5.2 Bennett linkage (adapted from Chen Y. 2003)

- Two opposing elements have the same length:

- Two opposing joints have the same twist:

-The relationship between twists and lengths is fixed by:

[5.1]

- By these relationships, the operating angles become mutually dependent parameters,

as they are bound by three equations, and so a single DoF mechanism is the result.


57

[5.2]

[5.3]

( )

( )

[5.4a]

- In the special case of and , the resulting mechanism will be equilateral

and equation [5.4a] then becomes [5.4b]. Many of the studied Bennett linkages in the

field of deployable structures take on this form for the symmetry and simplicity of

reproducing the bar elements.

[5.4b]

-Furthermore, if and , all the elements are congruent and the motion

becomes discontinuous, since is no longer uniquely defined when .

- In the degenerate case of and , the mechanism becomes a 2D rhombus.

Chen Y. (2003) goes on to prove that a completely flat-folded state of Bennett linkages exists.

Furthermore, she makes both mathematical and physical models of chained Bennett linkages,

resulting in compactable Bennett chains of both planar (Fig 5.3) and cylindrical variations (Fig 5.4),

as shown by You Z. and Chen Y (2011).

Fig 5.3 Flat array of Bennett linkages (Chen Y. 2003)


58

Fig 5.4 Singly-curved array of Bennett linkages (You Z. and Chen Y. 2011)

Melin N. O. is involved in the same field of study, attempting to materialize a long-span structure

made of foldable Bennett units, as shown in Fig 5.5. These and other studies in the connecting of

Bennett units reveal a difficulty in compacting the mechanism when multi-layer structures are

desired in deployed state. More research on the connecting joints between units appears to be

necessary to make these deployables competitive with existing SLE structures. Their small

structural height further makes them less than ideal for large-scale applications.

g 5.5a Deployment of curved array of overlapping Bennett linkages

Fig 5.5b Diagram of deployed curved array (Melin N. O. 2004)


59

5.2 Goldberg and Myard linkages

The 5R Goldberg linkage is a single DoF five-bar mechanism that can be generated by joining two

Bennett linkages together through a mutual element, as in Fig 5.6 (Huang Z. et al. 2013).

If the first of the joined Bennett linkages has a pair of elements with length and twist , and

another pair of elements with length and twist , then the second Bennett linkage will have a

pair of links which share the link length and twist , and a last pair with length and twist .

Adding these linkages together will then give a single DoF mechanism with 6 bars, of which the

shared one is redundant. The relationships between the lengths and twists for this mechanism

then become:

[5.5]

Fig 5.6 Synthesis of 5R Goldberg mechanism from 2 Bennett modules

(adapted from Huang Z. et al. 2013)

What was formerly seen as a separately discovered mechanism by Myard in 1931, was actually a

plane-symmetric variation of the 5R Goldberg linkages. “Two Bennett linkages are mirror images

of each other, the mirror being coincident with the plane of symmetry of the resultant linkage”

(Baker E. J. 1979). So for this mechanism the twists and the remaining twist ⁄ .

A schematic Myard linkage is seen in Fig 5.7.

Fig 5.7 5R Myard linkage

(adapted from Chen Y. 2003)


60

These Myard linkages can be combined into circular arrays in which two bars and one revolute

joint of any linkage is shared with each of its neighbors. The left mechanism shown in Fig 5.8 is

such a combination of two linkages, the system on the right shows a circular array of six linkages.

Since the movements between all the composing linkages are congruent, the resultant system

also has 1DoF.

Fig 5.8 Array central connection of Myard linkages (adapted from Huang H. et al. 2012)

Another way of chaining together Myard linkages is by connecting them on the peripheral points

of the loops. This is done in Fig 5.9: three linkages are connected together to form a 1 DoF

mechanism that forms a planar triangle in its deployed state. Fig 5.9a shows this system for three

connected Myard linkages, while Fig 5.9b shows a combination of both the central and peripheral

connections to make a planar hexagon when deployed.

Fig 5.9a Array from peripheral connection Fig 5.9b Array from both central and peripheral

of Myard linkages connection of Myard linkages (Qi X. et al. 2013)

The multiple-loop systems here are new advances in space engineering (antennae are the mainly

proposed structures), but a good application of them may well lie in deployable architecture. If

singly and doubly curved surfaces are tessellated in a way that allows these mechanism patterns

to overlay them, efficient deployables might be distilled.


61

5.3 Bricard linkages

Between 1897 and 1927, Bricard R. established 6 new classes of 6R mechanisms that have 1DoF.

The classes are divided according to the orientation of the members throughout the motion, but

share basic geometric relationships. Here, a planar-symmetric variation (Chen Y. et al. 2005) is

discussed, since it shows the most applicability to deployable structures. Link lengths in this

variation are all equal, and twist angles of adjacent joints are supplementary:

and

The relationship between the operating angles is fixed:

and

Fig 5.10 Bricard linkage (Chen Y. et al. 2005)

When these hybrid Bricard linkages fold flat to triangular shape, they can be chained together

using two common elements with a central joint, as with SLEs. (Fig 5.11). This way, the 1DoF

property is maintained. However, since the connections made into a single-layer mechanism

need alternating orientations to function, it is not so trivial to fashion curved surfaces out of

them.

Fig 5.11 Flat array of Bricard linkages (Chen Y. 2003)


62

5.4 Parallel manipulators

In kinematics, “a parallel manipulator typically consists of a moving platform that is connected to a

fixed base by several limbs or legs in parallel.” (Li Y. and Xu Q. 2007) An advantage of parallel

manipulators is that geometric errors in one bar are normally compensated for by the others.

5.4.1 Modified Wren platforms

Kiper G. (2010) discusses the known Wren platform mechanism that uses spherical joints to

connect bars to the platforms (Fig 5.12). In the case where the legs are skew relative to each

other, the Dof of the system is 1. If the mechanism moves to a state where the legs are all

parallel, a 2DoF mechanism is the result.

Fig 5.12 Wren platform in its two mobilities (Kiper G. 2010)

To lift the possibility of the mechanism moving into a 2DoF state, Kiper G. (2010) proposed using

the DaP connections of Jitterbug mechanisms to connect the legs and platforms. These modified

Wren platforms indeed have 1DoF, but are difficult to chain together in deployable mechanisms,

since they have multiple DoF when using the platforms as common elements (Fig 5.13a and

Fig 5.13b).

Fig 5.13a Modified Wren Fig 5.13b Array of modified Wren

platform with DaP joints platforms (Kiper G. 2010)


63

5.4.2 Sarrus linkages

The first overconstrained linkage ever to be studied was the 6R linkage published by Sarrus P. T.

in 1853. It is a 1DoF parallel manipulator mechanism in which the legs consist of revolute pairs,

connected to the bases again by revolute joints. The result is that the connected platforms

undergo a straight line translation, which makes this mechanism interesting for a myriad of

applications.

For the mechanism to work, the minimum amount of leg pairs is 2. In the examples in Fig 5.14,

the legs are perpendicular, but the necessary condition is simply that the axes connecting them

to the base are not parallel.

Fig 5.14a Sarrus linkage Fig 5.14b Diagram of Sarrus linkage (Chen Y. 2003)

(http://es.wikipedia.org)

The Sarrus mechanism is so elegant in its simplicity that it has often unknowingly found its use in

many of the transformable structures known today. A first group of transformables that is

actually a chain of interconnected Sarrus linkages are the SLE deployables: often, they are

described as being interconnected two-dimensional mechanisms. But another point of view

shows that in three dimensions, their single DoF is due to the basic module of the Sarrus linkage

contained in them. Fig 5.16a shows a rectangular, planar SLE grid, made up of Sarrus units as in

Fig 5.15b.

Fig 5.15a Rectangular SLE grid Fig 5.15b Applied Sarrus linkage

http://es.wikipedia.org)/


64

The double-faced Jitterbug-like mechanisms (dipolygonids) are another mobile group build out

of Sarrus linkages. This can be seen when the 2DoF DaP joints are seen as kinked physical

elements which are connected by 2 revolute joints. So, another definition of the platforms in

Sarrus linkages is ‘dihedral angle preserving links’. The connected facets in Fig 5.16a are replaced

by bar structures that are contained within them in Fig 5.16b, in order to show the identity of the

two mechanism groups.

Fig 5.16a Dipolygonid connection with DaP joints Fig 5.16b Applied Sarrus linkage

Another place in transformable structures where the Sarrus linkages have found their use is in a

certain type of cupola structures described by Wohlhart K. In his 2007 paper he describes the

need to have a 1DoF building block in order to make 1DoF cupola mechanisms that aren’t closed

loops.

One of his solutions is doubling the links in order to have each of the facets move like Sarrus

linkages (Fig 5.17). This method implies physical interference between the links and hence low

compactability.

Fig 5.17 Application of Sarrus linkages in cupula mechanisms (Wohlhart K. 2007)


65

In his 1981 doctoral thesis, Calatrava Valls S. studied some overconstrained mechanisms,

amongst which some were based on the Sarrus linkage. Fig 5.18 shows multiple Sarrus units

joined together.

Fig 5.18 Hexagonal array of Sarrus linkages (Calatrava Valls S. 1981)

A last example of Sarrus chains is by Huang H. et al. (2012). In their paper, one of the proposed

mechanisms is a group of perpendicularly connected Sarrus mechanisms. Notice in Fig 5.18 that

the axes connecting the smaller (black) leg pairs are not normal to the surface of their connected

larger (red) bars, since this would give a planar 3DoF mechanism. This slight twist in joints makes

the model a 1DoF Sarrus chain.

Fig 5.18 Alternative rectangular array of Sarrus linkage (Huang H. et al. 2012)

The prevalence of the Sarrus linkages in many of the deployable structures is the proof of its

practical usefulness. Studying them when faced with existing transformables can help better

understand their behavior.

New transformable structures can and will very likely be distilled from the

overconstrained mechanisms mentioned here. The question of how to successfully link

them together into different shapes that allow their movement is one the frontiers of the

design of deployable structures.

Part II. Design Tools

6. Kinematic Studies

67


In this chapter fundamental concepts and methods for describing the geometric-kinematic

characteristics of transformable structures are explained, using concise examples and offering

references for further study. It is the understanding of these very basic methods that make it

feasible to design and later analyze the kinematic behavior of more complex mechanisms.

The concepts are generally applicable to any structural system that falls under the category of

rigid links, such as SLEs, rigid-foldable origami, Jitterbug-like mechanisms, etc.

Understanding these underlying concepts, the arbitrariness of the categorization and the

subdivision between these groups also becomes clearer, and crossovers between the different

transformable structure groups become thinkable.

6.1 Determining degrees of freedom

A simple formula, known as Grüblers equation (Grübler, 1917), can be used to define the DoF of a

planar mechanism. The formula stems from a few basic observations: Firstly, that the amount of

independent displacements of any planar link is threefold: one rotation and two mutually

perpendicular translations. Any free planar link thus has a DoF of 3. Restricting these three

displacements makes for a grounded link, reducing its DoF to 0.

Connecting two independent links by means of a joint takes away the independence of their

respective movements by a degree that depends upon the type of joint. The most used of all

joints, a planar R-joint, binds the two translational freedoms of the connecting links, thus

reducing the DoF of the whole by 2.

When a joint connects more than two links, it is said to be of a higher joint order or valence

(Norton R. L. 1991), and accordingly reduces the mobility of the connected links. The order is

equal to the amount of connected links minus one. For example, an R-joint connecting three links

is of order 2, and reduces the mobility of the system by 4. An R-joint connecting four links is of

order 3 and reduces the mobility by 6. Pouring these observations into a formula gives:

[6.1]

In which: M: mobility of the mechanical system

E: number of links, including the grounded link

G: number of grounded links

number of joints of order i

i: order of corresponding joint


68

As in the analysis of mechanisms there will always be a chosen frame in which the movement

occurs, normally one single link is held to be grounded, thus simplifying the equation to:

( ) [6.2]

Since joints of 2DoF, such as half joints which possess a translational and rotational freedom, also

exist in planar mechanisms, Kutzbach modified Grübler’s equation to account for these:

( ) [6.3]

In which: : joints with 1DoF and order i

: joints with 2DoF and order i

For the structure in Fig 6.1, equation [6. 1] gives a mobility of 0, the structure thus is static and

there is no movement possible.

Fig 6.1 Static structure

(2 x 1st

order, 3 x 2nd

order, 1 x 3rd

order)

( ) ( )


69

Fig 6.3a 1DoF system with 4 dependent rotations Fig 6.3b Dependent deformations

J = 6 (3 x 1st

order, 2 x 2nd

order, 1 x 3th

order)

( ) ( )

In Fig 6.3 one of the bars is removed, resulting in equation [6. 2] giving a single DoF: there are 4

joints in which a rotational movement is liberated. However, the rotation around one of the joints

completely determines the rotation around the remaining three, as fixed by .

The figure of the rhombus thus makes the movement of the structure in Fig 6.3b possible, and

since the left square is static due to the triangulating diagonal, the mechanism can be reduced to

its pure form. This rhombus system is one of the simplest (1DoF) mechanisms in existence. It is

also commonly referred to as a planar 4-bar mechanism and is used extensively, for example

forming the basis of the 1DoF scissor-like structures.

Fig 6.3c Planar 4-bar mechanism


70

To turn the system in Fig 6.3 into a structure again, it is necessary to add one more bar to make

its mobility M equal 0 (a static system), or a negative integer (a hyperstatic system). However,

adding an extra link in just any place might give the desired result to Grüblers equation; it will not

necessarily make a static structure out of the mechanism. The added bar in Fig 6.4 was not

inserted in the mobile part of the mechanism (the rhombus). This is a trivial exception to

Grüblers equation, since it can be taken as a rule that the DoF need to be spread evenly over the

analyzed mechanism and no superfluous links should be counted.

Fig 6.4 Superfluous links in a mechanism

However, inherent exceptions to Grüblers equations validity do exist, and the mechanical

systems that don’t follow the rules set by the equation are called kinematic paradoxes. One of the

simplest examples of these paradoxes is the triangle bar system with 3 (sliding) P-joints, as

shown in Fig 6.5.

Fig 6.5a Kinematic paradox (3P loop) Fig 6.5b Movement of 3P loop

E = 3

J = 3 (3 x 1st

order)

( )


71

Grüblers equation here gives what would be a static system, when it is clearly a mechanism. It is

important to be aware of the limitations that are still found in this (and any non-geometric

method) used to calculate DoF, since it cannot account for dimensional exceptions.

“Objectively speaking, there is a sharp contradiction between the theoretical formula [Grübler]

and practical applications. There is an urgent need to resolve this contradiction in order to

ensure the continuous invention and application promotion of new functional mechanisms.”

(Zhao J. et al. 2014).

Different methods have been proposed for calculating the actual DoF of systems, among which

the most successful are those based on screw theory. Hitherto no consensus has been reached

for adopting any of these methods and equations. This stresses the need for designers to

always test their ideas on both real prototypes and virtual models to get feedback on the

kinematics.

The method used to get to Grüblers equation can be applied to determine the DoF of spatial

structures, adapting the formula to take into account that each joint has a total of 6 independent

possible movements: one translation along each of the 3 independent axes, and one rotation

around each of these axes. Spatially, any 1DoF joint will bind 5 of the formerly independent

displacements of the connected links. The equations [6.2] and [6.3] become:

[6.4]

( ) [6.5]

In which: : number of joints with xDoF and order i

It’s useful to understand that the positive terms in equations [6. 1] to [6. 5] are – more general

than joints – the amount of independent movements that any part of a structure can undergo,

while the negative terms are – more generally than joints – the restricting equations, or

constraints, to these movements.

As an example of the spatial formula, analyzing an open 3-bar mechanism with 2 R-joints with

equation [6. 4] gives 2DoF, each being of course the independent rotations of the bars round the

grounded bar (Fig 6.6).


72

Fig 6.6 Spatial mechanism

L = 3

J = 2 (2 x 1st

order)

( )

There are also paradoxes to the equation for three-dimensional DoF. Fig 6.7 shows a 1DoF

Bennett mechanism (see chapter 5). Applying equation [6.4] here gives a DoF of -2. Again, the

general equation can’t account for dimensional exceptions such as this one.

As seen in chapter 5, these kinds of systems are referred to as overconstrained mechanisms, as

alternative name for kinematic paradoxes, since the Grüblers equation predicts more constrains

than are actually working on them.

Many times, overconstrained mechanisms can be perceived as made up from sub-mechanisms

with 1DoF each, joined together in a compatible way as to still produce a 1DoF system together.

The Bennett mechanism in this case can be dissected into two 2-bar mechanisms with 1DoF each

(Fig 6.). Points (A,B) and (A’,B’) have the same trajectories: trajectory curves c and c’ are identical

but displaced. When joining A with A’ and B with B’, the axes going through them can be chosen

as to always make c and c’ coincide, resulting in a 1DoF mechanism.

These complimentary sub-mechanisms that share the same trajectory curves were dubbed

cognate links by Hartenberg R. S. and Denavit J. (1959).


73

Fig 6.7a sub-mechanisms forming Bennett linkage Fig 6.7b Bennett

linkage

E = 4

J = 4 (4 x 1st

order)

( )

Overconstrained mechanisms give a kinematic advantage over regular mechanisms in the

sense that their mobility is only given within specific geometric bounds. This means that

unexpected degrees of freedom due to slight material deformations are less common in

them, making them more reliable particularly for 1DoF mechanisms. This is also the reason

why the most successful deployable structures are based on overconstrained mechanisms, giving

them more controllability on the kinematic level.

There is a flipside to this however, since for certain groups of overconstrained linkages, such as

the Myard 5-bar mechanism, small production imperfections may cause the geometric conditions

for mobility to not be met. This would cause the mechanisms to become practically static or

unable to be assembled (Huang H. et al.). For this reason, additional joint clearance can be added

to the mechanisms, for example giving additional rotational freedoms perpendicular to existing

R-joints.


74

6.2 Trajectories and envelopes

The continuous sequence of movements of a structure – its trajectory or displacement path – can

be displayed in different ways. A common way of doing so is by plotting out the complete range

of motion of its members. A two- or three- dimensional surface, called the envelope, is marked by

the linear trajectory described by the vertices. This envelope can then be used to predict the total

space the structure takes in throughout its transformation process. Choosing the origin of the

coordinate center is of importance for the eventual perception and vector equations of the

trajectory (Fig 6.8).

By choosing the origin on the vertex O results in the envelope being a regular ellipse, while the

trajectory for the origin on the vertex O’ will give a lopsided ellipse. In Fig 6.9, the spatial

trajectories of a Sarrus-based mechanism are shown for the origin on the side vertices of each of

the 4 modules.

Fig 6.8 Trajectories for: origin in O origin in O’

Fig 6.9 Trajectory for Sarrus-based mechanism (Calatrava Valls S. 1981)


75

6.3 Generalized trajectory of 4-bar deployable structures

Many of the known deployable structures based on overconstrained mechanisms (Bennett-,

Myard-, Sarrus- and thus all SLEs) have comparable trajectories: when looking at each of the

moving bar elements in these deployables, it can be observed that during the transformation the

bars coincide with the changing rulings of a hyperbolic paraboloid. Furthermore, for 4-bar

deployables, the bars will together form an inscribed hyperbolic paraboloid. For 1DoF

deployables, the inscribed hypar surface will vary its shape by one variable parameter. Fig 6.10

shows this surface for a rectangular SLE module.

Fig 6.10 Inscribed hypar surface of deployables

The perpendicular sets of rulings of this hypar surface are given by fixing either x or y in the

formula:

[6.6]

Depending on the damping coefficient c, a hypar surface will change shape between its two

degenerate cases of a plane ( ) and a line ( ), all rulings are parallel. This indeed

makes the hyperbolic paraboloid appear like the perfect base shape for deployables,

which require high compactability. This shape can then be combined in different ways to form

different modular arrays.


76

As stated, the trajectories of each of the bars are also hypar surfaces. For an operating angle

between each bar and the xy-plane, the bars move along the different rulings of a set. Fig 6.11a

shows the trajectory of one bar relative to the center, the bar itself is drawn perpendicular to the

x-axis. Fig 6.11b shows the combination of the 4 bars in planar state, i.e. for The equations

of these rulings as the bars rotate and move towards or away from the center of a rectangular

hypar surface in point { 0 ; 0 ; 0 } are given as in [6.7] as parametric vectors.

Fig 6.11a Trajectory of single bar Fig 6.11b Trajectory of all 4 bars in deployable

in 4-bar deployable

:

[

]

[

]

[6.7]

:

[

]

[

]

In which: : operating angle ⁄

: (joint to joint) bar length

As the opening angle moves through the interval [0 , ⁄ ], the hypar surface formed by all bars

shifts from open (planar) to closed position. The intersection points of the rulings (A, B, C, D) are

the joint nodes of the deployable and are given by:


77

A: :

[

]

C: :

[

]

[6.8]

B: :

[

]

D: :

[

]

As a check-up, these coordinates show that points A and B are indeed the symmetric images of

points C and D respectively.

Note that these equations don’t account for physical joint size, since the bars would have to meet

in a single point. Simple joint offsets can be inserted in the equations. Since the end points of the

bars no longer intersect after the offset, all points A, B, C, D are split into two, each point of the

pairs being the end point of one of the connected bars (see Fig 6.12). For joints as in Fig 6.12

these offsets give:

:

[

]

:

[

]

[6.9]

:

[

]

:

[

]


78

:

[

]

:

[

]

:

[

]

:

[

]

In which: p: joint offset from center in tangent direction

q: joint offset from center in perpendicular direction

Fig 6.12 shows diagrammatic top views of the 4-bar deployable, Fig 6.12a for a joint offset p in the

tangent direction of the bars, Fig 6.12b for a perpendicular offset.

Fig 6.12a Tangent joint offsets Fig 6.12b Perpendicular joint offsets

In physical joints p and q can’t both be zero. Mostly, there will be opted for giving the joint a

material thickness in either of the two directions: tangential while keeping the perpendicular

offset 0, or vice versa. However, an offset in both directions is also possible with this method.

More information on joint offsets is given in the ‘cantilever and straddle joint mounts’ section

under the joint design section in chapter 7.


79

Fig 6.13 Projected joint nodes Fig 6.14 In-plane deformation of deployable 4-bar

For non-rectangular deployables, the projections of the original intersection points A, B, C, D on

the xy-plane - A’, B’, C’, D’ respectively - will move along the and bisectors as the

in-plane deformation angle changes. The translated points along these bisectors, , , ,

, can then be given in function of , in relationship to the projections of the original

intersection points in the rectangular configuration, for which ⁄ .

For the projected points A’ and C’ the magnitude of the translation vectors is , for point B’ and D’

the length of the translation vectors is :

(

√

) [6.10]

(

√

)

In which: : projected bar length in xy-plane, , ⁄

Taking the parameter as variable, the maximum compactability of a deployable will be reached

when its corresponding hypar surface has its maximum area surface in planar state. This is

achieved when the cross product | | , in other words when or ⁄ . In a

strict sense, the most efficient 4-bar closed loop deployables that can be made are rectangular.

That is of course, neglecting the structural problems brought on by the lack of in-plane stability of

rectangular grids.


80

Since the translations due to a changed deformation angle only affect the original intersection

points A,B,C,D in the xy-plane, they can be applied easily to them as they were for their xy-

projections. The new intersection points, , , and , are then calculated. The difference in

signs before the addends is to compensate for the direction of the translational vectors m and n

in reference to the x- and y-axes.

[ √ ⁄

√ ⁄ ]

[

]

[ √ ⁄

√ ⁄

]

[ (

√

)

( √

)

]

[6.11]

and likewise:

[ √ ⁄

√ ⁄

]

[ (

√

)

( √

)

]

[ √ ⁄

√ ⁄

]

[ (

√

)

( √

)

]

[ √ ⁄

√ ⁄

]

[ (

√

)

( √

)

]


81

Here, the new point coordinates are written in function of and . The joint offset parameters

are used and the trigonometric formulae are substituted by applying the half-angle formulae.

Expanding these vector representations gives:

[

( √ )

( √ )

]

⇒ [6.12]

[

√

√

]

and likewise:

[

√

√

]

Fig 6.15a Joint A

[

√

√

]

[

√

√

]

Fig 6.15b Joint B


82

[

√

√

]

[

√

√

]

Fig 6.15c Joint C

The new bars between and , between and , between and

and between and will each have changed lengths from the original bar length | |

by an amount of , owing to the joint offset in their tangent direction. The center points of the

joints are found when both p and q assume a value of 0. [6.13] gives the final vector equations of

the bars.

Using expressions [6.12] and [6.13], the trajectories of closed-loop deployables can be described

for variables (deformation angle) and and (joint offsets). Notice that the deformation angle

should stay fixed to give a singular trajectory, in other words a 1DoF mechanism. The kinematic

design of some deployables may come down to controlling this parameter for all of its closed

loops.

[

√

√

]

[

√

√

]

Fig 6.15d Joint D


83

=

[

√ *

(√ √ )+

√ *

(√ √ )+

]

[6.13]

=

[

√ *

(√ √ )+

√ *

(√ √ )+

]

=

[

√ *

(√ √ )+

√ *

(√ √ )+

]

=

[

√ *

(√ √ ) +

√ *

(√ √ )+

]

In which: : operating angle ⁄

: deformation angle

: distance between adjacent joint center

actual bar length = -2p

: joint offset from center in tangent direction

: joint offset from center in perpendicular direction


84

6.4 Auxetic Geometries

An interesting geometric way of looking at transformable structures, is by defining them as a

collection of auxetic structures, a term that is normally reserved for material analysis on the

atomic scale. The term ‘auxetic’ signifies a negative Poisson ratio. In other words: when an auxetic

material is enlarged in one direction, it increases its size in all perpendicular directions. Likewise,

when compressed, it decreases in all directions (Álvarez Elipe M. D.; Anaya Díaz J. 2012, 2013).

The atomic and mathematical models used for auxetic structures coincide with many of the

hitherto developed 1DoF architectural structures. By being aware of the overlap between the

distinct disciplines on micro scale (material science and nanotechnology) and macro scale

(architecture), independently discovered models could be exchanged, leading to new kinematic

structures that are sized to human dimensions.

Fig 6.16a Auxetic square re-entrant pattern Fig 6.16b Rigid-foldable waterbomb pattern

(Tachi T. http://www.tsg.ne.jp/TT)

Fig 6.17a Auxetic triangular arrowhead pattern Fig 6.17b Spatial grid of pyramidal module

(http://www.strutpatent.com/)


85

Both Fig 6.16a and Fig 6.17a have a framework of bars that rely on the same basic principle: the

shifted repetition of a geometric figure is made foldable by inserting R-joints at the centers of the

common bars. In the waterbomb pattern in Fig 6.18b the bars are replaced by the fold lines of

the origami pattern, making the surfaces shift outside of their plane to allow for the rotation. In

6.19b the triangular pattern is extrapolated in 3D, devising a foldable spatial truss with load-

bearing qualities.

Fig 6.18a Auxetic rotational square pattern Fig 6.18b Square Resch pattern

(Tachi T. http://www.tsg.ne.jp/TT)

Fig 6.19a Auxetic rotational triangular pattern Fig 6.19b Dipolygon pyramid

(Verheyen H. 1989)

The main merit of the auxetic models is that they simplify the sometimes complex

kinematic structures into their most basic form. Oftentimes these auxetic patterns can be

found when the transformables’ geometry is projected unto one of its symmetry planes.

7. Materilization Challenges

87

7. Materialization Challenges

With the geometry of the transformable structure chosen, the materialization is the part of the

design process where many problems may arise. Conceptual lines are now translated into bars,

surfaces into panels and joints go from being simple meeting points of axes to complicated

design exercises. As such, the main reason why transformables aren’t more widely applied in

architecture is the lack of know-how about translating geometric models into real structures:

different parameters are at play than for designing static buildings and few people in the

conservative construction sector appear to possess the experience to work with them.

Furthermore, there is no industrial standardization as of yet in fields such as scissor structures

and rigid-foldable origami, which have nonetheless been studied academically.

If transformable structures are to be made more accessible to the mainstream,

standardized and even catalogued joints designed for each category will be important.

Concerning the choice of materials, transformable and deployable structures often rely on

lightness, both for ease of transportation and to be able to manipulate the mechanisms with as

little energy as necessary. Materials that can boast a relatively high structural resistance versus a

low volumetric weight are especially suited. In this category one will find diverse materials such

as aluminum, wood and its derivatives, cardboard, composite sandwich panels, textiles, etc.

depending on the proposes design.

For larger scale structures which have a longer life-span and less transformation cycles, the

lightness of course becomes increasingly less important, and sometimes even steel elements are

used in these cases.

Other important parameters to consider are the ease of reproduction and the modularity of the

elements: can they be constructed readily from industrially available parts, or even more easily

from (recycled) by-products?

In this chapter the design parameters and difficulties in materialization are addressed, in

particular the design of joints, actuators and blocking mechanisms. A small part is dedicated to

the different tactics available for solving the problem of thickness in rigid-foldable origami.


88

7.1 Joint design

As mentioned, the joint is where the design of a transformable structure can be rated to its

quality. To start with, different joint types might be able to give comparable mechanical results,

implying a first choice here. Afterwards, the many parameters of mechanical resistance,

maintenance, production, ease of montage, waterproofing, etc. come into play. This part of the

chapter offers some information about each of these.

Joint types

When one can choose between different types of joints, there are some practical considerations

concerning friction and maintenance. The prevalent and simple revolute pin joints are in a clear

advantage: they are easy to design and require little to no maintenance. They owe this last

characteristic to the fact that they can keep a lubricant film trapped in the cavity between pin and

whole by capillary working. This separation of the parts is called hydrodynamic lubrication (Fig 7.1).

If necessary a seal can easily be provided around them to keep the lubricant surplus in and dirt

parts out. Radial holes can allow replacement lubricant to be placed without disassembly

(Norton R. L. 1991).

Fig 7.1 Capillary influence on lubricant

film in R-joints (Norton R. L. 1991)

When choosing revolute joints in a design that is projected to have a long life span or

considerably forces acting on the joints, the use of bearings can be provident. Different kinds of

bearings offer different advantages. Ball bearings (Fig 7.2) are actually higher pair joints,

connecting the pin only through point contact. They are easily lubricated, require little

maintenance, but cannot withstand much load and are not impact-resistant.

Cylindrical (needle) bearings (Fig 7.3) are generally more expensive and require more

maintenance, but they can carry both radial and moderate axial loads while allowing for an

almost frictionless gyration, making them ideally suited for highly stressed joints. When

circumstances are not so demanding, the revolute joints can be fabricated with simply drilled

holes and bolts, preferable free of screw-thread at the contact surface.


89

Fig 7.2 Ball bearings Fig 7.3 Cylindrical (needle) bearings

(http://www.nskeurope.com)

The sliding or prismatic joints have some notable disadvantages: they generally give much more

friction, are less forgiving in their imperfections and need more maintenance: the lubrication is

not geometrically fixed in place and needs to be resupplied by manual regreasing or running the

joint in an oil bath (Norton R. L. 1991). The rails they use to slide on need to be rigid and clean,

but open rails tend to accumulate dirt particles, building up the friction up to points that it can

grind the mechanism. The same applies to half-joints, where the sliding and rotational

movements are combined. Sliding ball bearings exist for both prismatic and cylindrical joints, but

require a near perfect surface to run on (Fig 7.4 and Fig 7.5). Simple wheel-rail systems are often

more pragmatic and able to take on higher loads, with more friction (Fig 7.6).

Fig 7.4 Sliding P-joints on rails Fig.5 Cylindrical joint ball bearings

(http://sdpsi.com) (http:/.nskeurope.com)

Fig 7.6 Variations of sliding P-joints (De Marco Werner C. 2013)


90

Cantilever and straddle mounted joints

Fig 7.7 Cantilever and straddle

mounted joints (Norton R. L. 1991)

Joint elements can be supported one- or two-sidedly, denominated cantilever and straddle

mounted respectively (Fig 7.7). The straddle mounting can avoid (excessive) bending moment

being taken up by the link by keeping the forces on the same plane (Norton L. R. 1991). The

double section of a straddle mount moreover means that both sides can take up shear force.

Since this makes the cantilever joint inherently weaker, it seems obvious to discard it. It can boast

its own advantages however: a more easy materialization and, especially for high valence joints,

higher compactness.

In deployable structures, the net sum of space saved by using cantilever joints can be

considerable for categories such as the SLEs where there are many R-joints. The axonometric

drawings and top plans of cantilever (left) and straddle mounted (right) joints are shown in

Fig 7.8. The joints are hatched grey in the top view. In both cases the joint offsets are the same

distance, but the cantilever offers a much higher compactness (it is not yet in its most compact

state in the top view). When using cantilevered joints like this, the torsion in the joints themselves

needs to be taken into account.

Fig 7.8 Comparison of compactness between cantilever and straddle mounted joints


91

Ease of production

Naturally, a reliable production process and the use of existing industrial parts is a main

parameter to consider when designing the joints. For the examples in Fig 7.8, the cross-shaped

joints on the right would have to be especially developed for a multi-step production process,

while the square joints on the left could be made easily by slicing a metal profile of standardized

cross-section.

It is always interesting to check if any of the commercially available joints from sectors outside of

construction might offer a valid alternative. E.g. Friction hinges or constant torque hinges are

industrialized revolute joints that apply a fixed amount of torque between two adjoining

elements, being able to (slightly) fix the parts in any position and thus making them especially

suited for transformables that have a wide range of desirable in-between positions.

When choosing aluminum as a material to work with, a complex two-dimensional drawing can

easily be extruded. This could serve to design versatile joints that fit in multiple connection

positions, joints that allow for the attachment of secondary systems such as cables or textiles,

allow for locally adding more links that form part of an actuator or blocking mechanism, or give

waterproofing to the connection.

Because of local stress concentrations and their size determining the overall compactness of the

system, it’s often decided to fabricate the joints in a stronger material, e.g. steel joints used in

aluminum systems or wooden block joints in cardboard structures.

Ease of montage and replacement

A quick and easy joint design will improve the efficiency in the montage of the whole structure.

Any standardized joint should be either symmetric or should denote differences in angles very

clearly. If (shoulder) bolts are used as revolute pin connections, they should be readily insertable

without overlapping (a common problem in compact revolute joints of higher valence).

Elements should be easily removable to change their orientation in the case of montage errors.

Depending on the projected life-span of the structure, the ease of replacement needs to be taken

into account. The material in the joints will typically suffer the most from extended normal

(non failure-related) use. This holds true especially in larger-scale projects where there is a

multitude of the same recurring joints, projects which are expected to last through many

transformation cycles. In systems like these, the joints where the local friction and other residual

stresses are higher will fail first and need replacement for the system to live out its time.


92

Waterproofing and sealants

Waterproofing is oftentimes a bothersome issue in transformable structures, since there needs

to be a continuous surface circumscribing an interior that might become entangled during the

deployment process. Special care should go to the connection to any textile with the joints.

An example of good detailing is the swimming pool designed by Escrig F. and Sánchez J. (Fig 2.34).

Here, the textile is connected to the interior of an SLE structure, protecting the metal bars and

joints from the erosive chlorine gasses.

On another note, hydro-induced deformations may occur in the elements and joints themselves

if they aren’t waterproofed. The summation of these deformations may cause the mechanism to

change its geometry to a point where it becomes unmovable. This goes especially for wood-

based elements.

The case of sealants is also a complicated one, since they need enough flexibility to not

compromise the movement during deployment, but enough rigidity as to ensure sealing and

even help the locking of parts (De Marco Werner C. 2013). Preformed joint sealants in the form of

strips, layers or prisms are commonly available in rubber, plastics and foams and the specific

characteristics such as mechanical resistance should be considered when choosing them.

Also, thin-layered coverings that can deform elastically are available commercially. Unlike sealant

strips, they are continuously connected to both of the linked pieces, making them highly

compactable, but sealed during all stages of transformation (Fig 7.9).

Fig 7.9 Different sealant techniques

(De Marco Werner C. 2013)

In some cases, sealants can be used efficiently as material joints. An example could be the use of

flexible neoprene or rubber in the fold lines of rigid-origami structures, ensuring mobility and

weatherproofing. Such a strategy could potentially solve the vertex difficulties that are typical in

materializing rigid-foldable origami.


93

Joint examples

Some example joints that try to incorporate different parameters are discussed below. While the

first one is generally applicable for structures of the same category, the others are developed as

highly individual for their projects and therefore are quite complex.

The light-weight and waterproofing of the aluminum R-joints in Fig 7.10, typically used for doors

in industrial spaces, make them very suitable and a tested solution in foldable architecture. The

added top profile gives the joint a higher mechanical strength and could be used for water-

proofing origami projects. The global compactness in some larger fold patterns might be

compromised.

Fig 7.10 Rigid-foldable origami joint

(http://www.abhmfg.com)

An attempt to link multiple Bennett linkages was undertaken by Piker D. (2009b) , who designed

joints in which the connecting joints each have additional interdependent rotational DoF to allow

for the compatible movement of paired Bennett modules. The author acknowledged that

unwanted DoF still exist in the joint.

Fig 7.11a Tent design from array of Bennett linkages Fig 7.11b Joint detail (Piker D. 2009b)


94

Fig 7.12 shows an S-joint with the possibility of reaching a very high DOF is currently being

developed by Yokosuka Y. and Matsuzawa T. (2013) with the particular aim of using it in

transformable architecture. The connected elements are being held together by a system that

exerts a multidirectional push towards a sphere that sits inside the center of the joint – a type of

ball-bearing. All of the elements have 3 completely free, independent rotations. The generic

name of the prototyped connection is the ‘Multilink Spherical Joint’, and allows for larger variable

structures, since there is a lesser accumulation of joint tensions due to physical inaccuracies.

Fig 7.12 Multi-DoF joint (Yokosuka Y. and Matsuzawa T. 2013)

A more conceptual joint is given by De Temmerman N. (2007) (Fig 7.13) as a possible solution to

the geometric incompatibilities in doubly-curved SLE grids. The fins that accept the bars are

joined around a central cylinder, so that they can rotate small amounts around a vertical axis.

Note that the primary function of the hollow cylinder is to conduct a cable that forms part of a

complementary structure system that can co-determine the rigidity by the introduced tension.

This joint tries to tackle different issues all in one design, but it would not be easy to materialize it

without compromising its stress resistance.

Fig 7.13 Joint with additional rotational freedom (De Temmerman N. 2007)


95

7.2 Thickness in rigid-foldable origami

Before making the leap to rigid-foldable origami applications on a larger scale, a necessary

stepping stone is modeling the plate elements with real thickness. This especially applies to

designs where the compactability is an issue and the maximum fold angle is to be reached. There

exist several strategies for dealing with thick elements, each of which – depending on the specific

design – has its advantages and disadvantages.

Axis shift

Hoberman C. (1988) proposed a method that alternates the axes of rotation on a facet and

introduces a removal of half of the material at the places where the different facet would

intersect (Fig 2.20). A complete folding motion of can be reached this was. The plate elements

here have 2 levels of thickness, but compared to the other proposed methods, it is one of the

easiest to geometries to materialize.

Another important downside is that the application of this technique is limited to regular fold

patterns, i.e. it only works for symmetrical and flat-foldable vertices (Tachi T. 2011).

Fig 7.14 Axis shift in thick origami panels (Tachi T. 2011)

Offset facets

Hoberman C. (1990b) introduced a second strategy for coping with thickness: the introduction of

increasingly bigger offset facets in triangular and trapezoidal shape (Fig 7.15 and Fig 7.16). It is in

particular applicable to high-frequency Miura-ori and the Yoshimura fold patterns, where the

addition of overlapping facets forms a real problem to the foldability.

It is a method very suitable for materials in which foldlines are easily introduced and as such no

separate hinges need to be added. All offset facets in a single row are by definition different,

which forms a problem for easy industrial production.


96

Fig 7.15 Axis shift in thick origami panels (Hoberman C. 1990b)

Fig 7.16 Expansion of model based on axis shift method for thick origami panels (Hoberman C. 1990b)

Slidable hinges

A third possible method is giving each of the revolute joints and extra translational degree of

freedom, letting them slide accros each other to reach a quasi-foldable solution (Fig 7.17). It was

proposed by Trautz M. and Künstler A. (2009) in order to materialize structures based on the

Miura-ori fold. This method is not generally applicable because of the global behavior, since the

summation of the translations can cause major geometric imperfections. It greatly complicates

the joint design.

Fig 7.17 Slidable hinges for thick origami

panels (Trautz M. Künstler A. 2009)


97

Tapered Panels

Tachi T. (2011) presented a fourth method: starting with the ideal (zero thickness) origami model,

the facets are thickened by the same amount on both sides. Then, folding the model until quasi-

folded state, the volume of each of the panels is trimmed by the volume of its direct neighboring

panels (Fig 7.18). Truncated pyramid shaped panels are the result. Depending on the thickness, a

better approximation of the completely folded state of the ideal facet model can be achieved.

Since this method is not based on any particular geometry, it is both locally and globally

applicable. The downside however, is that the tapering of the panels is a three-dimensional

process: an easy geometric design is traded for a more difficult panel production process.

Fig 7.18 Tapered edges for thick

origami panels (Tachi T. 2011)

Beveled Panels

A comparable method as the tapered panels has been proposed by Buffart H. C. and Traut M

(2013). Here the volumetric coincident of adjoining plates is trimmed away in closes state, giving

even more complex panels (Fig 7.9). The structural performance of the fine vertices is dubious.

Fig 7.19 Beveled edges for thick origami panels (Buffart H. C. and Traut M. 2013)


98

7.3 Actuators

For a transformable system, any change of shape requires a change of energy inside its system.

The ways energy is added to the system will depend on the scale and weight of the structure, as

well as the required precision of the intermediate stages. The possible actuators for

transformable systems are commented on below.

Gravity and manpower

The easiest solution to the actuation problem is using the energy sources that are readily

available. For 1DoF mechanisms up to a certain scale, gravity may be a reliable source for at least

the deployment phase. For singly or doubly curved bar mechanisms such as those based on SLEs,

the uppermost point of the geometry should be lifted in its desired position, leaving the

connected elements to unfold under their own self-weight. This process was actually used in the

first experimental works of Emilio Perez Piñero, who proposed using a telescopic truss on the

back of a truck to lift the transported SLE structure into the air, opening under its own weight.

Blocking systems would then be inserted, after which the structure becomes self-supporting and

the supporting truss and truck can be taken away (Fig 7.20).

Fig 7.20 Gravitational actuation

At a smaller scale, gravitational forces might not even be needed to unfold a structure. Many

small deployables, such as the commercially available SLE tents used by market vendors and for

small events, are easy to set up alone or with a few people. (Fig 7.21).

Fig 7.21 Manual actuation (http://rent-a-tent-web.com)


99

Using actuators in cases as these would be a waste of material and time. In the cases where

manpower can just barely serve to actuate the transformable, mechanical aids such as pulleys,

gear trains and leverages can of course be used to transfer the muscle energy more efficiently.

Considering leverages, the system needs a substructure to ground itself in order for any

momentum to be applied. The biggest leverage should be determined in completely closed

position, where the system is hardest to actuate since there is no momentum.

Cable and pulley systems

Either actuated by human muscle or on a motorized coil, cable systems are among the most

efficient for actuating deployable systems. In almost any transformable geometry there exist

points that come to lie closer to each other when the system changes shape. Fig 7.22 shows this

for a polar SLE structure. The cables between opposing joints are tensioned until the fully open

form is reached.

Fig 7.22 active cable system actuation

Fig 7.17 shows a transformable project done by Laboratoria de Arquitectura, in which a rigid

container is hinged around a floor plate. A counterweight is added and the manual coil is used

together with a pulley in order to slowly lift and lower in between its closed and ( ⁄ ) open state.

Fig 7.23 Cable coil actuation system (http://www.laboratoriodearquitectura.com, photo Pedro Kok)

http://www.laboratoriodearquitectura.com/


100

Cable systems for mechanism control can be given a secondary structural role, forming a

complementary tension network in a deployed structure. Intermediate tensors can be changed in

compact state as to give more or less rigidity to the structure in open state. The lower cables in

Fig 7.22 would serve exactly this purpose. Special joints could be designed which integrate a

guiding track for cables (as the hollow cylinder in Fig 7.13), and at the same time being actuated

locally so that by defining their position on the cable they open or close the deployable.

The dome-shaped SLE structures by Emilio Perez Piñero were not only supported by a structural

cable network on the interior convex side, but also on the outer side, as to offer resistance in the

case of wind load reversing the bending moment of the whole.

Some of the larger art pieces done by Hoberman Associates have been operated by the use of

cable systems wrapped around a motorized axis, giving reliable results for pieces in near

constant motion.

An example can be seen in Fig 7.24 for a hypar surface made up of angulated SLEs.

Fig 7.24 Model suspended from cables for actuation (Hoberman C. portfolio)

Motorization

For heavier structures, motors may be needed to actuate the systems. The rotational energy

gained from the motors can be directly used on an R-joint of a deployable structure, turning one

of the connected joints while keeping another fixed. Another option is to create a type of rocker-

crank subsystem or a cable system that is being operated by the motor. A motor is especially

useful in case of constantly adapting structures, as opposed to deployables which only have 2

states of interest.


101

Motor types should be decided on based on their torque-speed curves, determining their speed

sensitivity to load. Most of the time, slight variations on the movement speed in these type of

mechanisms will not be important, but for certain applications, closed-loop servomotors might

be a necessity for smooth movement. For more in-depth information and analysis of motors, the

referenced work of Norton R. L. (1991) is advised.

A notion to consider for motorized, mobile transformables is the availability of energy: will there

always be an energy output at the intended locations? If not, would a generator be a valid energy

provider, or do more efficient ways actuating the system exist? In most cases, motorization is

only a valid design choice if the transformable structure is of considerable scale or fixed in

location as a movable building part.

Hydraulics and pneumatics

Likewise useful for transformable structures that are fixed in location are hydraulic and

pneumatic systems to actuate them. Hydraulic systems need a supporting infrastructure of

pumps, tubes and pressurized containers. They are however efficient in both energy and space

use, since a small hydraulic cylinder can exert a relatively large force, and can serve at the same

time as a blocking mechanism.

Examples of the use of motorized hydraulic cylinders can be found in the transformable building

parts designed by Calatrava Valls S. The synchronous working of all of the cylinders is necessary

for a good functioning, so that a feedback loop has to be inserted in some way.

Fig 7.25a detail of hydraulic jack Fig 7.25b Hydraulic jacks in Hemisférico in operation

(http://calatrava.com)


102

Another example of the use of hydraulic cylinders is on the roof of the Merck Serono building in

Geneva, designed by Murphy/ Jahn Architects et al. for an international competition. The roof

located above an atrium can open up to 5 meters to allow for natural ventilation to occur.

Fig 7.26a Detail of hydraulic jacks Fig 7.26b Opened roof of Meck Serono building

(http://jahn-us.com)

Pneumatic jacks can be used for smaller scale projects and are more energy-efficient. The initial

energy needed can be inserted manually and they can take up to medium loads, which make

them more suitable for mobile and low-budget projects.

Heat deformation

Used in smaller scale projects, heat deformations in the materials themselves might cause

spectacular effects when the small differences are scaled up by summation or repetition. These

are interesting nearly exclusively for light-weight projects, such as the Smartwindow project by

Doris Sung, in which thermal bimetals are used to curl open or closed along a thermal curve, so

that shading is only applied at moments when solar gains are highest (Fig 7.27). The bimetal

surfaces are made up of continuously connected metal layers with different expansion

coefficients, making them bend out of plane when heated.

Fig 7.27 Smartwindow project actuated by heat deformation (http:// dosu-arch.com)


103

7.4 Locking systems

After the transformable structure has reached its desired phase, the geometry has to be locked

in place. There are different methods to reach this goal, all of which spring from the basic

concept of introducing a constraining relationship between two mobile links. They are listed here

in order of commonness.

Constraining elements

The most obvious method of materializing the constraining relationships is by turning them

straight into material components, forming fixed triangles with the existing geometry that reduce

the local and global DoF, and in doing so stabilize the structure. For example, the changeable

distance between two mobile bar link could be fixed by introducing a third bar link that has its

end points on the two original links.

Retaking the bar geometry of Fig 7.22, this constraining element would be the central orange bar,

as in Fig 7.28. This central bar prohibits the scissor system to close and to open further. However,

if the mechanism here formed part of a larger polar network, the central bar could be replaced

by a cable with the same effects.

Whenever a deployable structure’s movement is delimited to a certain phase by its own

geometry, using tensile geometries can offer the advantage of simplicity and a locking

mechanism less prone to fail suddenly under unexpected loading due to buckling of constraining

bar elements. Furthermore, tensile constraining elements can be included in a secondary

structural system.

Fig 7.28 Constraining bar element

There are other ways of dealing with unexpected loading (e.g. wind gusts can have a strong effect

on typically light-weight deployables). A tested method for preventing plastic deformation and

sudden failure is using spring-parts as constraining elements. An example from the industry of

small deployables is seen in Fig 7.29, where gable springs are used to stabilize a square-grid SLE

tent structure in a non-maximum position. When loading gets too high, the springs yield and

reduce pressure on the connected bars, avoiding any damage to the structure while also

guaranteeing enough rigidity of the system.


104

Fig 7.29a Constraining gable springs Fig 7.29b Joint detail

(http://pro-tent.ch)

Another option when using constraining elements is using the same hydraulic or pneumatic

cylinders used for actuation, locking them in place to turn the mechanisms into structures,

making them very efficient parts. An example of this can be seen in one of Calatrava’s mobile

designs, an entrance gate to the Valencian metro where a hinged hydraulic cylinder is fixed to

keep them open or closed (Fig 7.30).

Fig 7.30 Hydraulic jack for opening (http://structurae.net/)

A final important aspect to constraining elements is their frequency or spread: how many and at

what exact locations are they inserted into the mechanism. In 1DoF systems, the constraining

elements will have to resist the forces in all of the parts that would normally cause the system to

transform. Therefore high tensions will tend to form in and around the constraining elements,

prompting the designer to distribute them evenly over the structure. This way, the accumulation

of small local deformations will also be held in check. In designs for space-encompassing

structures, an array of constraining elements can be installed along the connection to the

ground, at the same anchoring the whole.


105

Toggle position

The term toggle is used in mechanics to describe a stationary configuration of a mechanism, in

which two elements connected by a revolute joint are collinear. For typical 4-bar linkages, the

toggle position will mean that a triangular shape is formed, with one of its side composed of the

two elements (Fig 7.31).

Since the connected elements are collinear, no coplanar force can mobilize the system, making it

pseudo-static. However, force eccentricities can make the system buckle, and therefore the

toggle position should not be used as the only locking system in transformable structures that

are intended for bearing load. It can serve as a place-holder for keeping the structure stable while

other constraining elements are being locked, or for designs where loads are small and coplanar

to the toggled elements.

Norton R.L. (1991) writes:

“In other circumstance the toggle is very useful. It can provide a self-locking feature when a

linkage is moved slightly beyond the toggle position and against a fixed stop. It must be manually

pulled “over center,” out of toggle before the linkage will move. You have encountered many

examples of this application, as in card table or ironing board leg linkages and also pickup truck

or station wagon tailgate linkages.”

So a way of using toggle to its maximum advantage is by introducing constraining elements that

keep the mechanism in its toggle position, as a safe-guard for when eccentricities cause the

mechanism to buckle. This way, lesser stress will be taken up by the constraining elements.

Fig 7.31 Mechanism in toggle position

(Norton R. L. 1991)


106

Snap-through effect and bi-stable structures

A third method for stabilizing transformables is using an inherent geometric incompatibility, which

happens when a mechanism is impeded in its foreseen motions because they are at odds with

the trajectory of a partnering mechanism. Most incompatibilities are found in complex systems,

in which the trajectories of multiple mechanisms intertwine spatially.

The transformable will undergo additional stresses when it is in phases of geometric

incompatibility. If these stresses occur only at intermediate phases of the deployment or

transformation, it is said that there is a snap-through effect: an additional amount of force is

needed to deform the system and cause it to snap from one phase to the next. This

phenomenon can even be of an added advantage, as Zeigler H. (1976) first used these stresses to

attain a self-locking mechanism: the structure is fixed easily into its deployed state.

Fig 7.32 Deformation of elements in bi-stable structure (De Temmerman N. 2007)

If the elements are in a stress-free state before and after deployment, but go through an

intermediate stage with deployment-induced stresses, they are called bi-stable structures. The SLE

structure in Fig 7.32 is such a bi-stable structure: the marked-red bars are most stressed in an

intermediate phase, locking the structure into a semi-stable state when fully deployed.

To successfully design and calculate the structures for a snap-through analytical and computer

models are needed to make sure the elements deform only elastically, not reaching yield point

and destroying the structure in the process.


107

Joining kinematically incompatible mechanisms

A last method of locking systems is joining generally incompatible mechanisms together in one

exact phase in which they are compatible with one another. Doing so makes them into a stable

structure in which the stresses due to constrains are not local, but spread globally. An advantage

over bi-stable structures is that with this method no deformations should occur during the

transformation process and hence its simples to calculate and materialize.

Examples of this method can be found in the rigid-origami Yoshimura pattern based structures,

in which a straight and curved mechanisms are joined together in erect state. This is the only

state in which the edges of both mechanisms are coincident, hence connecting them in this way

effectively makes for a stable structure (Fig 7.33). The method is however not exclusively useful

for rigid origami structures, and can find its application in most kind of deployable systems.

Fig 7.33 Joining of kinematically incompatible Yoshimura based origami mechanisms



108

7.5 Design criteria

With the knowledge of the kinematic characteristics and the materialization difficulties applied to

each of the transformable structure categories, a designer can make a well-founded evaluation of

any kinematic design.

Choosing one certain category and detailing method over another is sometimes very straight-

forward, while at other times a lot of iteration is needed. An example: for building a space-

encompassing structure, the design choice of using rigid-foldable plate elements over bar

elements can be justified by their properties of continuity, offering a weatherproof surface and a

self-supporting structure all at once. Then, it must be considered if these advantages weigh up

against the higher compactability of bar systems. Finally, ease of maintenance and durability of

the joints are important themes in repeated deployment. These and many more parameters

come into play when designing and choosing a suitable transformable structure.

To be able to methodologically evaluate any transformable structure design over another,

Hanaor A. and Levy R. (2001) proposed a system with 9 criteria to evaluate the effectiveness and

efficiency of any design. The criteria are geared towards self-supporting deployable structures

(low weight, high compactability and transportability), but they touch on some important aspects

for all kinematic structures. Fig 7.35 shows the criteria in a chart.

Fig 7.34 Evaluation chart for deployable structures (Hanaor A. and Levy R. 2001)

Part III. Uneven Sarrus Chains

8. Uneven Sarrus Chains

111


In chapter 5 it has been shown that the Sarrus mechanism forms the basis of many of the

existing deployable structures, and new ways of linking Sarrus mechanisms together have been

described both by Wohlhart K. (2007) and Calatrava Valls S. (1981) - although the latter wrongly

analyzed their kinematic properties.

In this chapter firstly a geometric and kinematic description is given of the basic Sarrus module.

Secondly, different ways of chaining these basic modules into arrays are presented, their

geometric possibilities (both flat and polar) shown and kinematic characteristics discussed. The

geometric analysis of the arrays and their composing modules is then used to create a

parametric design tool. Thirdly, two case studies using different array types are analyzed to their

structural properties.

8.1 Basic module

Fig 8.1 Deployment of uneven Sarrus

module

The degree of freedom of a minimal Sarrus mechanism, in Fig 8.2 formed by bars , , ,

and physical joint elements in and , is given by [6.4]:

( )

Which shows that indeed the Sarrus module is an overconstrained mechanism since it actually

moves with one degree of freedom. This holds true for any deformation angle .


112

Fig 8.2 Uneven Sarrus module

As the title implies, the basic Sarrus mechanism examined here is asymmetric. The bars

connected to the top joint are shorter than the bars connected at the bottom joint. The

proportion between the two bar lengths is from here on defined as:

[8.1]

The mechanism will open unevenly, since the uneven rhombus moves between a closed collinear

position and an open position where the shorter bars are coplanar, while the longer bars are

oriented diagonally. The nonlinear relationship between the operating angles and (Fig 8.3) is

given by:

=

( ⁄ )

[8.2]

The graphed version of this equation in Fig 8.3 shows the increasing divergence between the two

angles as approaches . On a kinematic level this means that small imperfections may cause

relatively large deviations of in the phase close to complete deployment, where the influence

of is small. Care should be taken in the design process to avoid potential movements, for

example by locking the shorter bars in their deployed position.


113

Fig 8.3 Relation between opening angles

In order to be able to fold closed completely, and thus reach maximum compactness, the sum of

lengths of a shorter and longer bar of a pair needs to be constant throughout a module (and any

connected modules). As shown in [2.1], this is called the compactability equation:

[8.3]

When completely deployed, the coplanarity of the bars makes for a toggle state in the

mechanism. As was discussed in chapter 7, the toggle mechanism can serve as a locking system

aid, as the bars are in a stationary configuration where no in-planar force can cause movement in

the mechanism.

From this toggle position, an inverted configuration (Fig 8.4) can be reached where .

Note that the concave quadrilaterals formed here are equal ( and ). The inverted

configuration can be desirable from the point of view that it offers greater compactness. Joints

will however need to accommodate for bar thickness, resulting in differently designed joints in

and .

0

0,314

0,628

0,942

1,256

1,57

0 0,314 0,628 0,942 1,256 1,57

𝛼k

𝛼𝑙

λ = 0.9

λ = 1

rad

𝜶𝒍

𝜶𝒌 =

𝒄𝒐𝒔 𝟏( 𝒍 )

𝒄𝒐𝒔 𝟏 ( 𝒍 )

rad rad rad rad rad rad rad


114

Fig 8.4 Inverted configuration

The lack of control to which configuration the bars will move after toggle position, in combination

with the greater deviation of to close to this deployed position, call for adapted design

solutions for the joint. Fig 8.5 shows an example of this where the joint is extended (here

shortened slightly for visibility) to form a central connection bar to lock the joint into place in

the unfolded position.

Fig 8.5 Extended bottom joint

The uneven Sarrus module forms the building block for different arrays that can eventually serve

to span distances and carry loads. Fig 8.6 shows three different ways of chaining the Sarrus

module together, for each of them the method(s) for introducing a polar angle is shown.


115

Fig 8.6 Array types and their methods for introducing polar angles

The first array method is the widely used scissor connection, in which the bars are continuous.

The second array – joint-to-joint – has no continuous bars, but introduces an intermediate

module. In the third array the bars of the modules overlap to form intermediate rhombi. In what

follows, the joint-to-joint array and overlap array are shown more extensively, the methods for

introducing a polar angle in each of them are given, and their peculiarities are discussed.

Scissor array

Joint-to-joint array

Overlap array

Polar angles are introduced by

changing the bar proportion .

The max structural height is

thus bound to the polar angle.

Polar angles are introduced by

keeping top bar lengths equal

and using a joint offset with

new bar lengths .

Polar angles are introduced

by using a joint offset and

or

using new bar lengths and

derived by ellipse method.


116

8.2 Joint-to-joint arrays

As the name implies, joint-to-joint arrays are made by chaining together modules at their joint

nodes so that a repetitive pattern of non-continuous bars is formed. They are one of the

deployable structures that Calatrava Valls S. (1981) mentions in his dissertation. The physical

model in Fig 8.7 and the render in Fig 8.8 both show a flat rectangular array. Note that there are

two joint types: those that are on the inside of the Sarrus modules, and those that are on its

edges, which receive the double amount of bars.

Fig 8.7 Deployment of flat rectangular joint-to-joint array

Fig 8.8 Flat rectangular joint-to-joint array


117

Fig 8.9 Deployment of flat hexagonal joint-to-joint array

Fig 8.10 Flat hexa-triangular joint-to-joint array

By changing the angles at which the bars meet each other in the joint nodes, different patterns

such as the hexagonal-triangular one in Fig 8.9 and Fig 8.10 can be formed. The method of joint-

to-joint arrays has an important limitation in that only patterns of polygons with an even amount

of sides can be formed. Triangulation in the xy-plane of the structure will thus need to be added

by other means.


118

8.2.1 Polar module with joint offset

Fig 8.11 Polar module with joint offset Fig 8.12 Introduction of polar angle by joint offset

By partitioning the physical joint in into one part at its original location and a second part

offset a vertical distance j upwards, two rhombi with different proportions are formed:

and . The resulting module is still a 1DoF overconstrained mechanism. The

compactability equation [8.3] is now expanded to:

[8.4a]

In which: j: joint offset

is constant for both rhombi types

While the rhombus reaches its fully opened state, the rhombus is partly opened.

I.e. while the bars of the first rhombus are parallel, the bars of the second rhombus are at an

angle , creating an asymmetrical polar Sarrus module suitable to array along singly curved

surfaces. To find a certain j and corresponding to any chosen polar angle , [8.4a] is expressed

in function of :

( ) ( ) [8.4b]

In a right-handed coordinate system with origin in , the coordinates of A, , and become:

[

] [

√ ] [

] [8.5]


119

Substituting [8.5] in [8.4b] and expanding within the square roots:

√ ( ⁄ ) √( )

[8.6]

For which has one root:

( )

( ( ⁄ )) [8.7]

In which: h = √ , the vertical distance between and when deployed

Finally, j and can be expressed for :

( ⁄ )

( ⁄ ) [8.8]

( ⁄ )

( ⁄ ) [8.9]

Fig 8.13 Maximum polar angle for joint offset method

The maximum polar angle that can be achieved for any chosen bar lengths and is

reached when the new bars can unfold to become collinear.

The values for , and are then:

(√ ( )

) [8.10]

[8.11]

[8.12]


120

Mostly, this maximum polar angle has little practical application, since it would mean a relatively

large joint offset, which could easily potentiate a bending moment in the joint that is too big to

compensate for.

Fig 8.14 Joint offset giving equal bar lengths

For reasons of simplicity and easier production, there can be opted for making the length of the

new bars , so that only 2 different bar lengths are used. The corresponding and are

fixed to:

(

) [8.13]

[8.14]

When placed in a joint-to-joint array, the polar module cannot go from toggle position to its

inverted configuration, since the toggle position is not reached for both rhombi and

(except for when is used). This characteristic can be exploited to effectively lock the

mechanism in its open position by tensioning a cable between the central joints and .

By using this method for generating polar modules, the minimal structural height h can be

determined from the values chosen for bar lengths and . Independently, the curvature can be

decided upon: using the polar angle , the joint offset j is calculated relative to the chosen bar

lengths. If the joint offset is perceived as too large, iterations can be made.

Fig 8.15 Polar modules with extended bottom joint


121

Taking a repetitive array of a rectangular polar module, cylindrical geometries can be made of

which the curvature can be determined independently of the structural height. An example of a

semi-arch consisting of modules with constant polar angle ⁄ can be seen in Fig 8.16. These

arrays of polar modules are completely flat-foldable since the compactability equation [8.4a] is

fulfilled. This also means that the changing polar angle converges to zero as the mechanism

closes.

Fig 8.16 Semi-arch joint-to-joint array from polar modules with joint offset

A polar module can also be introduced locally in an array of compatible flat and/or polar

modules. An example of this is given in Fig 8.17, where a double-sloped structure is created by

connecting two flat arrays by a linear chain of polar modules with polar angle = ⁄ .

Fig 8.17 Locally introduced polar modules connecting flat rectangular joint-to-joint arrays


122

8.2.2 Mobility of joint-to-joint arrays

The overconstrained joint-to-joint arrays typically have 1DoF, but some physical models show an

additional, dependent, DoF that can occur due to material imperfections. Firstly, the inherent

1DoF movement is analyzed, here described for a minimal array of two flat Sarrus modules as in

Fig 8.18.

Fig 8.18 Minimal joint-to-joint array of two flat 4 bar modules

In the regular DoF, the bars’ trajectory lies on the generalized hypar surface as described in

chapter 6. Accordingly, the location of joint nodes can be given for a variable operating

angle between each of the bars and the xy-plane, as the mechanism opens and closes:

A:

[

]

C:

[

]

[8.15]

:

[

]

:

[

]

In which: ⁄


123

A secondary hypar trajectory is described by the bars, sharing the coordinates of the A and C

nodes. In the same coordinate system, the locations of joint nodes and can then be

derived. The x- and y- values will remain the same, while the z-value compensates takes into

account the uneven bar proportion .

:

[

]

:

[

]

Using [8.2] allows for writing these solely in function of :

:

[

√ ]

:

[

√ ]

[8.16]

It can be checked that indeed for a completely deployed state where thus ,

the z-value is √ ; while for the completely folded state where ⁄ ,

the z-value becomes (

).

Additional mobility and solutions

Physical models show an additional mobility that is notably more lenient close to the folded state,

and lessens greatly to disappear as the mechanism approaches the deployed state. This shows

that this dependent mobility is not inherent to the geometry but rather originates from small

deformations that allow the two modules ( and ) to rotate slightly around the

diagonal axis passing through nodes A and C. Meanwhile, the same joints in nodes A and C rotate

around the perpendicular diagonals, turning outside of their original plane.

Fig 8.19 shows the same physical model as in Fig 8.7; here the additional mobility is

demonstrated. A small joint clearance allows for a small angular distortion near the opening

phase. Fig 8.20 shows an equivalent fold-line model with the normal mobility of the Sarrus chain

to the right and the additional mobility shown (exaggerated) in the bottom. The additional

mobility will completely disappear in models with absence of joint clearance.


124

Fig 8.19 Additional mobility of joint-to-joint array in physical model

Fig 8.20 Additional mobility for minimal joint-to-joint array


125

Since the additional mobility appears only during the deployment phase, the effect on the

deployed structure can be made minimal if it is fixed in multiple locations, especially when

extended joint-bars and secondary structural systems are used. However, in some cases the

deployment process will need to be strictly controlled, and measures should be taken to make

sure the mechanism doesn’t deviate too much from its 1DoF trajectory. Some example solutions

are given below.

A first solution method is to locally introduce rhombic elements which have a shared bar

connecting them, i.e. local pantographs. Fig 8.21a shows some possibilities for how to achieve

this in the xz-plane of two chained Sarrus modules. Only two connected rhombi need to be

formed; Fig 8.21b and Fig 8.21c show some examples of implementing this method with fewer

bars. As can be seen in Fig 8.21c, the bars can be scaled - making for a good location to install an

actuation mechanism.

Fig 8.21a Possible locations for introduction of local pantographs to inhibit additional mobility

Fig 8.21b Introduction of local pantograph at top bar layer

Fig 8.21c Introduction of local pantograph at bottom bar layer


126

Since the additional mobility makes the opposing joints rotate out of their plane, another method

is simply linking any 2 opposing joints by a 3R-mechanism that does not lie in a plane

perpendicular to the xy-plane. Fig 8.22 shows this method with the 3R-mechanism lying in the xy-

plane itself.

Fig 8.22 Introduction of transversal bars to inhibit additional mobility

In deployed state, the diagonal bars can also serve for the triangulation of a rectangular array.

The disadvantage of this method is that in compacted state, the bars that form part of the 3R-

mechanism won’t be collinear with the original bars, lowering the compactness of the whole.

A last method for removing the additional mobility in joint-to-joint arrays is to locally introduce

overlapping bar elements, also applying these added bars for the xy-triangulation of the

structure. This is demonstrated further in Fig 8.26


127

8.3 Overlap arrays

The overlap arrays are made by letting the bar pairs of the Sarrus modules cross with

corresponding bar pairs of other modules. This way, intermediate pantographic rhombi are

formed that fold-closed to a line and deploy to a triangle (due to the uneven bar proportion ).

The overconstrained overlap arrays have 1DoF and don’t display the additional mobility noted in

the joint-to-joint arrays.

Fig 8.23 shows a simple overlap array of 4 flat Sarrus modules that is in the midst of its

deployment. In Fig 8.24 a completely deployed array of the same modules can be seen.

Fig 8.23 Semi-deployed rectangular overlap array

Fig 8.24 Flat rectangular overlap array


128

As the hexa-triangular array in Fig 8.25 exemplifies, array patterns of odd joint count can be used,

making for an easier xy-triangulation and giving more options to both planar and curved

geometries.

Fig 8.25 Flat hexa-triangular overlap array

The joint-to-joint array and overlap array can be combined with locally overlapping bar pairs that

both remove the additional mobility and give xy-triangulation to a typically even-numbered joint-

to-joint array. The original bar lengths of the modules in the joint-to-joint array should be used as

to comply with the compactability equation [8.3]. After reaching the toggle position, this method

will also prevent the mechanism from going into its inverse configuration (Fig 8.4). Fig 8.26 shows

how two identical Sarrus modules with 3 equal pairs of legs each are fit together to get an

overlap connection in the diagonal sense.

Fig 8.26 Minimal joint-to-joint array with diagonal overlap bars


129

8.3.1 Polar module with ellipse method

Fig 8.27 Points for which the compactability Fig 8.28 Rotation of new top bar length

equation holds, lie on an ellipse around top joint node by polar angle

In the overlap system, a polar angle can also be introduced by using the joint offset method, as

was done for the joint-to-joint system and parametrized by equations [8.4] to [8.14].

However, a simpler method for the overlap array specifically consists of changing a pair of bar

lengths, while keeping their sum constant. For the compactability equation [8.4] to be fulfilled,

the new bar lengths and need to comply with:

[8.17]

Since both the original bars and , as the new bars and share the same end points and

, this relationship can be visualized by plotting an ellipse with its center point on ⁄ and its

two foci on said joint nodes and . Fig 8.27 shows some of the different bar pairs that comply

with [8.17], including the ones giving an inverted configuration. The length of the semi-major axis

of the ellipse then equals ( ) ⁄ and the lengths of the semi-minor axis √ ( ) ⁄ .

Making the origin of the coordinate system coincide with the center of the ellipse then gives its

general equation:

( ) ⁄ +

( ) ⁄ = [8.18]

To find the values of and corresponding to any chosen polar angle , a point at horizontal

distance from is introduced. Its coordinates as in Fig 8.28 are ( ⁄ ).


130

The rotation of this point around with a certain angle will then place it on the ellipse.

Substituting the x- and y- values of this rotated point into the ellipse equation [8.18] gives:

( ( ))

( ) ⁄ +

( ( ) ⁄ )

( ) ⁄ = [8.19]

for which:

solving for gives:

= ( )

(

( ) ( ))

[8.20]

and from [8.17] and [8.20]:

= ( )

(

( ) ( )) [8.21]

Fig 8.29 Maximum polar angle for ellipse method

The maximum polar angle

that can be achieved for any chosen bar values and is

reached when the new bars can unfold to become collinear. The values for , and

are:

( ) [8.22]

[8.23]

Overlapping rectangular modules which have a polar angle in of their directions, cylindrical

geometries can be generated. A physical test model as part of a singly curved overlap array like

this is shown in Fig 8.30. Doubly curved configurations which fold flat such as the lamella dome in

Fig 8.31 can also be made. Their geometric description is outside the scope of this dissertation.


131

Fig 8.31 Singly-curved overlap array

Fig 8.31 Lamella dome overlap array

8.3.2 Overlap factor

The modules analyzed above are then chained together by overlapping them by a certain

amount, the overlap factor , and connecting the bars and bars with each other at the center

of the overlap. The overlap factor is taken at zero when there is no overlap and at 1 if the two

modules completely overlap. In a larger flat array, the maximum overlap factor for practical

materialization is 0.5, since larger factors would make for a double overlap between 3 adjacent

modules. In a rectangular grid array, there are two independent overlap factors that can be

chosen.

Fig 8.32 Overlap factor


132

8.4 Parametric tool for regular array design

Using the equations established in this chapter for making flat and polar arrays, a software tool

was developed for parametrically designing scalable arrays of both joint-to-joint and overlap

type. The tool was implemented in the Grasshopper plug-in for Rhinoceros 3D, and allows for

parametrizing regular arrays in function of the parameters shown in the screen capture

in Fig 8.33.

Fig 8.33 Parameters Fig 8.34 Array-specific parameters and output

A simple Boolean toggle further allows the choice between said two array types, which each of

them having their output parameters numerically plotted as well, allowing comparison between

the maximum polar angle that can be used, bar lengths for a certain polar angle, and the

bounding dimensions of the arrays (Fig 8.34).

Since the paramatrized arrays are regular, the polar arrays result to cylindrical geometries, being

suited for generating the geometries of barrel vaults, such as the one analyzed in later

mentioned case study 2. The headroom output parameter in Fig 8.34 thus refers to the

maximum interior height of the barrel vaults.

Variations of regular arrays can be made by generating different but compatible regular arrays

and joining them together, for example a repeated polar array and flat arrays of the same basic

module. This technique was used for joining two flat arrays to a central polar array in later

mentioned case study 1.


133

A simplified step diagram of the tool is shown in Fig 8.35. In reality the main parameters refer to

almost all steps, and the geometries generated at intermediate steps also create output material.

Fig 8.35 Step diagram of parametric tool

The first step after entering the parameters is generating the Sarrus modules, which happens

separately for each of the array types, since the methods for generating polar modules are

different between them. Mostly, intersecting circles are used here in combination with the

equations for polar modules ([8.4] to [8.14] and [8.17] to [8.23]). This ensures that bar lengths

stay constant during deployment.

The geometry of the basic modules is halved and copied around to form the smallest chain that

is later copied into larger arrays. This intermediate step is mostly use to make certain that the

borders of the arrays stop on continuous lines of bars, not having the edge and corner bar pairs

sticking out. The overlap factors, parametrized separately for the flat and polar direction in the

array, is inserted in this step as well.

The smallest chains are then used to generate flat arrays, which in turn are copied into polar

arrays. Attention has been given here to avoiding doubled geometries of the copied elements.

Finally, the resulting arrays are translated and rotated to stand upon the xy –plane. Numerical

output from the mostly geometrically generated model is then given in data panels.

To return this numerical data for both of the array types in order to compare the two, the input

of the Boolean toggle that chooses between them is actually inserted at the end of the process,

hiding one of both of the arrays in the graphical representation.


134

Changing the operating angle , the parametrized mechanism opens and closes in real time.

The other parameters can be changed independently; e.g. for the joint-to-joint array in Fig 8.36

the changing bar proportion is shown, altering the structural height and the expandability. Flat

arrays can be made by choosing a zero polar angle, e.g. for the overlap array in Fig 8.37.

Fig 8.36 Parametric tool applied to create singly-curved joint-to-joint array

Fig 8.37 Parametric tool applied to create flat overlap array


135

8.5 Secondary structural systems

When a joint-to-joint or overlap array is locked in its fully deployed state, it can be said that the

bars form a compressive layer, while the bars function as diagonals. In most cases an additional

(tensile) layer may then be necessary to give structural height to the array. Firstly the relationship

between the structural height and the bar proportions is described and, secondly some

secondary structural systems are discussed here.

The relative structural height is defined as the vertical distance between the and nodes in

deployed phase, divided by the bar length:

=

√

= √ [8.24]

Plotting to naturally gives the first quadrant of a circle (Fig 8.38) The graph can simply be

read as the trade-off between the relative structural height and the compactability, since 𝒍 𝒌⁄

is directly proportional to how far the Sarrus module opens in the perpendicular direction.

Fig 8.38 Trade-off between relative structural height and compactability

This trivial relationship makes it easy for designers to decide the proportion , which mostly won’t

go lower than the √ ⁄ mark, since lower values give increasingly smaller returns to .

To visualize this point, the optimization of the sum of relative structural height and the bar

proportion is shown in Fig 8.39. -values between 0.6 and 0.8 give the best trade-off.

0

0,2

0,4

0,6

0,8

1

0 0,2 0,4 0,6 0,8 1

𝐡𝐫

Trade-off: Structural height (𝐡𝐫) vs Compactability ( )


136

Fig 8.39 Trade-off with compactability as primary goal

A first method for adding a secondary structural system to gain structural height is doubling the

bars and connecting this new layer to the bottom joints. Fig 8.40a shows a semi-deployed joint-

to-joint array where the top layer of bars is to be added to the lower joint nodes. Fig 8.40b

shows the same array where the lower layer of bars has been joined, here colored orange for

visibility.

Fig 8.40a Adding of lower bar layer Fig 8.40b Joint-to-joint array with lower

bar layer in semi-deployed state

A way of looking at the new mechanism created by adding the bottom bar layer is as two arrays

of Sarrus modules interlocking upside-down and sharing their bars (diagonals). In Fig 8.41 the

same array as in Fig 8.40 can be seen in its fully deployed state.

1

1,1

1,2

1,3

1,4

1,5

0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1

𝐡𝐫

Optimization of (𝐡𝐫 ) with as main parameter


137

Fig 8.41 Joint-to-joint array with lower bar layer in fully deployed state

Fig 8.42 shows how extended joints (here in blue) can be added to form vertical connecting struts

when the mechanism closes, effectively forming a square-on-square truss grid in this case. They

can also be introduced locally as constraining elements to lock the structure in deployed state.

Fig 8.42 Joint-to-joint array with lower bar layer and extended

bottom joints to form square-on-square grid truss

A downside of this secondary bar system is of course a less compact bundle when the

mechanism is closed. A design decision can then be to implement this secondary bar layer with

the inverted configuration (such as in Fig 8.4) to save space. When adding a secondary bar layer

to a polar array, the curvature will naturally make the bars in the inner curve smaller, but the

same strategy for adding them can be applied.

The advantage of the secondary bar layer method is that changing moments are accounted for,

since both the top and bottom layer can function in compression. When potentially changing

moments aren’t a concern, the introduction of a tensile layer is the lighter and more compact

solution. The tensile layer can exist of either cables or textile, or a combination of both.


138

Cables can be added either parallel to the upper layer of bars, or diagonally as to give structural

stability to the xy-plane. This second method is shown in Fig 8.43 for a joint-to-joint array and in

Fig 8.44 for an overlap array, where X-braces are formed.

Fig 8.43 Joint-to-joint array with triangulating cable substructure

Fig 8.44 Overlap array with X-brace cable substructure


139

In Fig 8.45 a polar joint-to-joint array model can be seen where a fabric is introduced to take up

tensile forces. This method naturally offers triangulation to a quadrilateral grid, which is an issue

for joint-to-joint arrays.

Fig 8.45 Singly-curves joint-to-joint array with textile substructure

A last secondary structure system mentioned here are added plate elements. They can be placed

in between and over the compression bar layer to add rigidity and lessen deformations to the

rest of the structure, serving also as constraining elements. Since they have to be placed on the

structure after the transformation, they add time to the deployment process.

Although added plate elements give stability in the xy-plane and increase the overall rigidity, no

structural height is reached with them, and a cable or bar layer can be combined with them to

this purpose. In Fig 8.46 a flat joint-to-joint grid is given a tensile layer and plate elements.

Fig 8.46 Joint-to-joint array with added plate elements


140

8.6 Case studies

In this section, some designs are used to show some of the real-world applications and

possibilities of the analyzed Sarrus arrays. A short geometric description of the designs is given

and they are structurally analyzed both to global and local stability and section strength, using

SCIA Engineer 14 software. Eurocode standards are used wherever applicable, and the unity

checks are done both for ULS (Ultimate Limit State) and SLS (Serviceability Limit State).

8.6.1 Case study 1: Pedestrian bridge

The first case study is the design for an easily deployable bride to span a small river of 10 meters

wide. It should be possible to quickly recover and compact the bridge during any floods, such as

in the monsoon season. A joint-to-joint array materialized in aluminum is chosen as the primary

structure, stainless steel cables give it structural height, and plywood plates form the walkway.

Fig 8.47 Deployable pedestrian bridge from joint-to-joint array (all dimensions in meters)


141

In order for local boats to pass underneath and to introduce less of a bending moment, the

bridge should be arched. However, the maximum inclination at any point of the bridge is chosen

to be ⁄ , so that it is accessible when materialized either with flat non-skid plates or stepped

plates. These conflicting interests are solved by linking together two inclined flat arrays and a

central polar array into one mechanism (Fig 8.48).

Fig 8.48 Joining of two flat arrays to central polar array

As can be seen in Fig 8.49, a central bar is fixed to the lower joint, while the upper joint slides

over it. This bar does not affect the mechanism, but serves both for receiving the handrail and for

triangulation in its lower part. A welded ledge on this central bar should stop the top joint just

before toggle position so that the mechanism folds closed more easily and does not enter the

inverted configuration.

Fig 8.49 Central bar with P-joint serves for triangulation and receiving the handrail


142

Fig 8.50 Deployment stages of pedestrian bridge


143

The deployment stages are shown in Fig 8.50. The compact package is delivered by either crane

or boat. The deployment can happen using the self-weight of the aluminum bars, and is

controlled by active cables running along the top joints (marked orange in the central drawing).

Later, these light cables can be used to compact the mechanism again. After opening, the last two

rows of joints on either side are pinned to premade foundations, making the structure static.

Lastly, the plywood plates are added from the sides inwards. The plates further rigidize the

structure once it’s in use, helping to remove any residual mobility in the system. This works well

with the joint-to-joint arrays, where an additional mobility was found to appear with larger joint

clearance. In this sense, the combination of joint-to-joint grids and plate elements can also be

suitable for deployable stage designs.

The two different kinds of modules used in the design are shown in Tab 8.1. The relative

structural height does not take into account the small offset that is necessary for attaching the

cables to the bottom joints. The values of the joint offset and the new bar length of the polar

module are calculated using [8.8] and [8.9].

Flat module

= 900 mm

= 1150 mm

= 0.78261

= 720 mm

Polar module

=

= 127.5 mm

= 1023 mm

Tab 8.1

Due to the eccentricity that is inherent in the use of a joint offset, a bending moment will be

introduced in the bottom joints of the polar modules. Care is needed when designing these

joints, so that they can offer a great enough resisting moment. In this case, a hollow joint section

is needed to connect the central bars anyway, and its relatively large dimensions are adequate

for resisting said bending moment.

Tab 8.2 shows the material properties used for each of the element types, along with their largest

and smallest dimensions occurring in the design. Short elements and a relatively high bar

proportion , together with the use of a medium-high strength aluminum alloy, allow the use of

small sections.


144

Material Elastic modulus

[Gpa]

Density

[kg/m³]

Dimensions

[mm]

Bar

elements

Aluminum

EN AW 6082 T5

70 2,720 50x50x3 x 1150

x 800

60x60x4 x 200

x 100

Cables

Stainless steel wire

EN 10270-3

190 7,850 ⌀8 x 1700

x 1250 Plate

elements

Plywood EN 636-3 7 550 24 x 1800x900

Tab 8.2

Tab 8.3 shows the different loads used in the calculation with SCIA Engineer software. The

primary structure – excluding the plate elements – has a total weight of 312.6 kg, which makes it

easy for transportation, and quite lightweight at 13.8 kg/m². The stainless steel cables make up

18.4 kg of this total weight, and they could be replaced with ropes if further reduction of weight

were required.

Type Load

Self-weight W Permanent Primary structure: 314.6 kg total

Plywood plates: 20 kg/piece

Distributed load 𝒌 Variable 4kN/m²

Concentrated load 𝒌 Variable

Local

4kN

Wind load in x

direction

(transverse)

Variable 1.02kN/m²

Wind load in y

direction

(longitudinal)

Variable 0.501kN/m²

Tab 8.3

The variable load of 4kN/m² is an overestimate, since the bridge would be used only for

pedestrians and light vehicles such as bikes and scooters. Snow load is omitted since the bridge

here is proposed in equatorial climates where snow tends not to occur.

Wind loads are calculated referring to EN 1991 1-4, particularly the section about bridge design.

The most important values and coefficients used for wind load calculation are shown in Tab 8.4.


145

Basic wind speed 25 m/s Roughness factor 0.758

Mean wind speed 19 m/s Turbulence intensity 0.25

Structure reference height 2.7m Exposure factor 1.59

Air density 1.25 kg/m³ Force coefficient 1.625

Terrain factor (II) 0.19 Force coefficient 0.9

Tab 8.4

The model done in SCIA Engineer describes the physical joints as short (120 and 200mm) hollow

section 60x60x5 profiles, receiving 4 or 8 bars orthogonally with one rotational freedom. The

joints receive the bars head-on, i.e. not on the sides in cantilever, in order to minimize axis

eccentricities. This goes at the cost of compactness, as shown in Fig 7.8.

Fig 8.51 Hinged supports

Hinged supports are modeled under both the bottom and side joints of the last row of flat

modules, as they would be fixed to a base plate after deployment (shown in Fig 8.50).

Four complete ULS load combinations exist, since the longitudinal and transverse wind loads are

mutually exclusive. EN 1990 gives the coefficients for the governing load combinations, which

simplify to:

ULS1: [8.25]

ULS2:

ULS3:

ULS4:

in which: : self-weight

: distributed load

: concentrated load

: transverse wind load

: longitudinal wind load


146

max

[kN]

max

[kN]

max

[kN]

21.52

ULS1

7.38

ULS1

27.58

ULS1

Tab 8.5

𝒍 bars 𝒌 bars central bars cables

max unity

check

0.192

ULS1

0.674

ULS1

0.278

ULS1

0.727

ULS2

max

[kN]

-11.9

ULS2

-22.39

ULS1

-7.62

ULS2

6.57

ULS2

max

[kNm]

0.0

ULS1

0.32

ULS1

0.45

ULS2

-

max

[kNm]

0.0

ULS1

0.0

ULS1

0.50

ULS2

-

Tab 8.6

The deciding bar section has a maximum unity check of 0.674, which means that the structure

could be further optimized. However, iterating over the next-in-range sections available

industrially gives a failed unity check (>1), hence it’s decided to use the 50x50x3 mm profile

section. The deformations are checked for the SLS, the governing load combinations and their

coefficients as given by EN 1990 become:

SLS1: [8.26]

SLS1:

SLS3:

SLS4:

max

[mm]

max

[mm]

max

[mm]

3.0

SLS1

3.3

SLS1

-8.9

SLS1

Tab 8.7


147

The deformations being lower than ⁄ , the structure is more than rigid enough for its

intended use. The rigidity is owed partly to the use of the plate elements and the network of

cables, them being necessary additions to the basic joint-to-joint array.

The largest local deformations are to be found at about ¼ of the length and half the width of the

structure, near the top of the deck, as shown in Fig 8.52. This is where the flat Sarrus modules are

not triangulated since there are no central bars used here to connect the top and bottom joints.

Fig 8.52 Largest deformations occur at unsupported top joints (shown for ULS for clarity)

Lesser rigidity around the top joint (connecting the bars) is inherent in joint-to-joint arrays,

where the angle of top bars becomes more independent near fully deployed state (Fig 8.3). A

possible solution would be to introduce an extended joint (as in Fig 8.4) that adds triangulation

here as well.

The design of this small bridge is meant as a first study of the possibility of using the joint-to-joint

array consisting of both flat and polar modules. The rigidization by the plates that serve as a

walkway and the central bars in the module that give triangulation are important design

elements. Similar bridges of larger scale (and most likely, built in steel) could be based on the

same design strategy.


148

8.6.2 Case study 2: Barrel vault

The second case study analyzed here is a barrel vault made from a polar overlap array. A fabric is

used as a secondary tensile system on the inside of the polar array, but is simplified in the

SCIA Engineer model by applying cables in the polar direction and X-braced cables in the edge

bays.

Fig 8.53 Deployable barrel vault from overlap array (all dimensions in meters)


149

The analysis done here is a direct reference to the work done by Alegria Mira L. (2010), who

structurally analyzed and optimized the Universal Scissor Unit she designed in the application of

a barrel vault. Since a comparative structural study is interesting, Alegria’s methodology was

largely followed for this case study. The main geometric characteristics were reproduced: the

amount of modules in the array in both polar and flat directions is the same. Due to the repeated

overlaps, the resulting vault is smaller. The folded bundle of the overlap array is relatively

compact because only one-dimensional bar elements are used. As this design focuses more on

compactness and transportability, instead of using steel (S235), the lighter aluminum (EN AW

6082 T5) is chosen for the bar elements.

Fig 8.54 Cantilever joints allowing high compactness

The use of square-sectioned bars was decided upon early, since these offer the most compact

configuration. In combination with a straddle-mounted joint, they form a very space-efficient

system of which the top view can be seen in Fig 8.54.

For this case, said method gives a compact bundle of 4 .9m³ that expands to 142 times its volume

during deployment. The joints are modeled in SCIA Engineer as placeholders with the same offset

as in reality (Fig 8.55) and they are not further analyzed in this case study. The circles in Fig 8.55

are SCIA Engineer’s notation of rotational degrees of freedom, where the bars meet the joint.

Fig 8.55 Modeled joint thickness


150

Fig 8.56 Deployment stages of barrel vault


151

From the parametric Grasshopper file the deployment of this barrel vault was analyzed, as shown

in Fig 8.56 for an opening angle of ⁄ up to complete deployment.

Active cables as in Fig 7.22 would be used to pull the top and bottom joint of each of the modules

together to prevent force concentrations. No active cables were included in the calculational

model: their effects are beneficial to the rigidity of the locked structure, but were chosen to be

ignored here.

The structure is primarily locked by placing constraining bar elements at the edges of the barrel

vault. These edge rows are also where the X-brace cables are located in the model, and hence

where SCIA engineer predicts the largest risk of member buckling. Thus the additional

constraining elements prevent overdimensioning these edge bars. After the constraining

elements are in place, the structure is locked further by pinned or weighted supports along all of

its bottom joints, which ensures a more even force distribution.

Only one polar module (Tab 8.8) was used in the design, simplifying the production process.

Calculating the polar bar lengths and was done in function of the desired polar angle ,

using equations [8.20] and [8.21] for overlap arrays. The overlap factor is taken constant for

both the flat and polar directions of the vault.

Polar module = 828 mm

= 1150 mm

= 0.720

= 798 mm

=

= 862 mm

= 1116 mm

= 0.3

Tab 8.8

Tab 8.9 shows the material properties used for each of the element types, along with their largest

and smallest dimensions occurring in the barrel vault. As noted, the membrane is simplified in

the SCIA Engineer model to a set of steel cables in the polar direction. The elastic properties of

the cables are used, but the weight of the fabric is used in calculating the total self-weight.


152

Material Elastic modulus

[Gpa]

Density Dimensions

[mm]

Bar

elements

Aluminum

EN AW 6082 T5

70 2,720

kg/m³

60x60x4 x 1150

x 828

Fabric

PVC coated Polyester 1.2 0.8

kg/m²

1050 - 1159

(between supports)

Cables

Stainless steel wire

EN 10270-3

190 7,850

kg/m³ ⌀8 x 1050

x 1159 Tab 8.9

In Tab 8.10 the different loads used in the calculational model are shown. The total structure,

including the joints and the membrane, has a weight of 3256.4 kg, making it quite lightweight at

23.5 kg/m². Combined with the compactness – the total package fits with the 2 x 2 x 1.3 m

container – the deployable barrel vault is suited for transportation and quick deployment.

Type Load

Self-weight W Permanent

Primary structure: 3100.5 kg total

Membrane: 155.9 kg total

Wind load in x

direction

(transverse)

Variable max 0.536 kN/m²

Wind load in y

direction

(longitudinal)

Variable max 0.305 kN/m²

Snow load s Variable 1 kN/m²

Tab 8.10

Wind loads are here again calculated referring to EN 1991 1-4. The coefficients used can be seen

in Tab 8.11. For the calculation of the peak velocity pressure, the seasonal factor is taken

into account to lower the statistically determined percentile rank, seeing that one use-cycle of the

structure would likely be less than three months.

Basic wind speed 25 m/s Roughness factor 0.783

Mean wind speed 19 m/s Turbulence intensity 0.243

Structure reference height 3.08 m Exposure factor 1.524

Air density 1.25 kg/m³ Probability factor 0.85

Terrain factor (II) 0.19 Peak velocity pressure 0.412 kN/m²

Tab 8.11


153

EN 1991 1-4 prescribes the different zones that need to be defined for obtaining the internal and

external pressure coefficients in the structure. The zonification of the cylindrical structures

analyzed by Alegria Mira L. (2010) and De Temmerman N. (2007) serve as references here. Fig

8.57 shows the zonification and Tab 8.12 – Tab 8.13 show the resulting wind loads in each of the

zones.

Fig 8.57 Zonification and wind load diagrams (adapted from Alegria Mira L. 2010)

External

pressure

coefficient

External

pressure

[kN/m²]

Internal

pressure

coefficient

Internal

pressure

[kN/m²]

Total pressure

w [kN/m²]

A 0.8 0.330 -0.5 -0.206 0.536

B -1.2 -0.494 -0.5 -0.206 -0.288

C -0.4 -0.165 -0.5 -0.206 0.0041

Tab 8.12

External

pressure

coefficient

External

pressure

[kN/m²]

Internal

pressure

coefficient

Internal

pressure

[kN/m²]

Total pressure

w [kN/m²]

D -0.6 -0.247 0.14 0.058 -0.305

E -0.3 -0.127 0.14 0.058 -0.185

F -0.2 -0.0824 0.14 0.058 -0.140

Tab 8.13


154

The snow load is shortly calculated from EN 1991 1-3, taking zero load for the zones of the barrel

vault that have an inclination greater than ⁄ . The coefficients applied here are shown in Tab

8.13.

Form factor 2

Exposure coefficient 1

Temperature coefficient 1

Characteristic snow load on ground 0.5 kN/m²

Tab 8.13

All loads except for the self-weight are introduced at the lower joints (through SCIA Engineer’s

load panels that redirect the forces to its vertices). Realistically, the supported membrane would

indeed be suspended from the lower joints of the modules.

Four complete ULS load combinations exist, since the longitudinal and transverse wind loads are

mutually exclusive. EN 1990 gives the coefficients for the governing load combinations, which

simplify to:

ULS1: [8.27]

ULS2:

ULS3:

ULS4: 𝟏 𝟏 𝒔

in which: : self-weight

: transverse wind load

: longitudinal wind load

: snow load

max

[kN]

max

[kN]

max

[kN]

16.62

ULS4

0.81

ULS3

4.41

ULS4

Tab 8.14

𝒍 bars (polar) 𝒌 bars (polar) 𝒍 bars (flat) 𝒌 bars (flat) Cables

max unity

check

0.967

ULS4

0.754

ULS4

0.1

ULS4

0.065

ULS4

0.92

ULS4

max

[kN]

-16.64

ULS4

-6.74

ULS4

0.1

ULS4

0.1

ULS4

12.18

ULS4

max

[kNm]

0.06

ULS4

0.04

ULS4

0.02

ULS4

0.02

ULS4

-

max

[kNm]

0.97

ULS4

0.73

ULS4

0.04

ULS4

0.03

ULS4

-

Tab 8.15

Results of SCIA Engineer’s calculational model are shown in Tab 8.15. The unity check for said

loads is barely complied with for the lowest bars at the edge bays, due to the accumulated forces


155

almost exceeding their buckling resistance. However, the unity checks for the other bars on the

lowest lines also result close to 1, meaning the structure is nearly optimized.

The deformations are checked for the SLS, the governing load combinations and their coefficients

as given by EN 1990 become:

SLS1: [8.28]

SLS2:

SLS3:

SLS4: 𝒔

max

[mm]

max

[mm]

max

[mm]

6.3

SLS4

0.6

SLS4

-16.8

SLS4

Tab 8.16

The vertical deformation occurring at the center of the barrel vault is relatively small, at less than

1/700 of the projected span. To further investigate how precise the SCIA model can describe the

deformations (and further, rigidity) of the overlap array, smaller physical sample models should

be analyzed under load and their deformations compared to the ones in the calculational tool.

Effects of higher and lower joint clearance could then be checked and the eccentricity of the bars

to each other and to the (straddle mounted) joints researched.

The basic rigidity the overlap gives to the structure ensures that a changing moment, such as

with upwards wind load during a storm, the structure can still offer a resisting moment without

deforming excessively. To check this in the SCIA model, the transverse wind – giving the biggest

deformation – is applied as a sole load on the model, ignoring the stabilizing self-weight and

snow load. The results of the joint displacements are shown in Tab 8.17, and are well in

acceptable range.

max

[mm]

max

[mm]

max

[mm]

6.4

transverse

wind load

0.4

transverse

wind load

+6.3

transverse

wind load

Tab 8.17 Fig 8.58 Deformation by transverse wind


156

Fig 8.59a Large maximum bending moment Fig 8.59b Spread of bending moment over

at central hinge of scissor arrays shorter elements in overlap arrays

One of the structural shortcomings of regular scissor arrays is that the central connection

introduces a bending moment in the connected bars at the location where the section is at its

weakest (shown diagrammatically in Fig 8.59a). With the overlap array method, the bars are

doubled in the central zone where more structural height and material is needed, lowering the

maximum moment (Fig 8.59b). This way, the overlap method can ensure increased rigidity and

strength both during and after deployment. The bars in an overlap array are shorter, making

them more resistant to buckling.

Furthermore, in an overlap array, the relative structural height in completely deployed state is

independent of the polar angle , allowing this important factor for structural behavior to be

determined freely; while in a scissor array the structural height and polar angle in deployed state

are dependent, sometimes prompting the array to not fully deploy in order to gain

structural height.

The downside of using an overlap array over a regular scissor array is a lowered compactness,

directly proportional to the overlap factor . Good detailing can guarantee that the top ( ) bars

and the bottom ( ) bars lie in the same plane, preventing the compact bundle of becoming more

bulky.

Fig 8.60 Hinge detail example for compact overlap arrays


157

8.7 Conclusions

In this chapter the variations of the Sarrus mechanism and ways of chaining them together were

researched. By primarily focusing on the basic mechanism as a module, important characteristics

were derived. These include the relationship between the bar proportion, the different opening

angles and the structural height.

Three different array methods were discerned: the well-known scissor array, the joint-to-joint

array and the overlap array. The latter two were investigated on further in this work.

The joint-to-joint type has been used in the work of Wohlhart K. and Calatrava Valls S. (1981), but

a more thorough analysis of the geometric possibilities and structural use of the arrays had been

lacking. The overlap array, in turn, is novel and shows promising structural advantages.

A novel way of introducing polar angles in a Sarrus module has been discovered. It uses a vertical

joint offset and can be applied both to the joint-to-joint array and overlap arrays. A second

method that uses changed bar lengths and the ellipse locus for introducing a polar angle in

overlap arrays was analyzed. Both methods started from the assumption that the compactability

constraint needs to be complied with, i.e. stating that the joined elements should be able to fold

flat into a linear state as to offer compact deployables.

The numerical relationships between the elements lengths and polar angle of a Sarrus module

have been derived and, using the resulting equations, a parametric tool was developed in

Grasshopper software. This tool can be used to generate both flat and polar arrays.

An additional dependent mobility was noticed in physical models of joint-to-joint arrays and first

attempts at describing this mobility have been made. Furthermore, various solutions for

removing the dependent mobility in order to have a more controlled deployment have been

offered.

Information was given about the trade-off between the structural and kinematic aspects, and

secondary structural systems intended for giving structural height to the arrays were addressed.

The introduced double-bar system could be used to form deployable square-on-square

truss grids.


158

Further work

Some geometric possibilities of the joint-to-joint and overlap array were already given, but many

more remain undiscussed. In particular the doubly-curved variations of the overlap array should

be researched. Predictably, many of the doubly-curved geometries of the simple scissor arrays

will be translatable directly to the overlap array. Variable overlap factors might also allow for new

compactable geometries to be used.

A more precise description of the additional mobility in joint-to-joint arrays is needed. Physical

testing should reveal the influence of the joint clearance on this mobility. Test models should be

made to test the effectiveness of the proposed methods for inhibiting it.

Physical test models should also be made for both array types to check whether the structural

responses coincide with those that calculational software describes, specifically with regards to

the overall rigidity.

A comparative study between overlap arrays and regular scissor arrays would be interesting,

using compactness and structural strength and rigidity as main parameters, and plotting the

exchange between them.

Further study of the materialization of the joints is needed to make the real-world use of the

structures possible. Tests should be done on the influence of the joints eccentricities, including

the vertical joint offset, on the structural response.

References

159

References

Abdul-Sater K., Irlinger F., Lueth T. C. (2014): Four-Position Synthesis

of Origami Evolved, Spherically Constrained Planar RR Chains

Mechanisms and Machine Science, Vol. 26

63-71

Alegria Mira L. (2010): Design and Analysis of a Universal Scissor Component

for Mobile architectural Applications

Master’s thesis, Free University of Brussels, Faculty of Engineering

Álvarez Elipe J. C., Díaz Lantada A. (2012): Comparative studiesof auxetic

geometries by means of computer-aided design and engineering

Smart Materials and Structures 21.10: 105004

Álvarez Elipe M. D., Anaya Díaz J. (2013): Development of transformable structures and geometries

New Proposals for Transformable Architecture, Engineering and Design in the Honor of E. P. P.

Escuela Técnica Superior de Arquitectura, Sevilla

269 – 274

Baker J. E. (1979): The Bennett, Goldberg and Myard linkages—in perspective

Mechanism and Machine Theory, Vol. 14

239-253

Barker R. J. P., Guest S. D. (2000): Inflatable triangulated cylinders

IUTAM-IASS Symposium on Deployable Structures: Theory and Applications, Cambridge, UK

17-26

Beatini V., Korkmaz K. (2013): Shapes of Miura mesh mechanism with mobility one

International Journal of Space Structures, Vol. 28, No. 2

101

Belcastro, S.-M., Hull, T. C. (2002): A mathematical model for non-flat origami

Origami^{3}

39-51

Bennett G. T. (1903): A new mechanism

Engineering, Vol. 76

777-778

Bennett G. T. (1914): The skew isogram mechanism

Proceeding of London Mathematics Society 2, Vol. 13

151-173

Bern M., Hayes B. (1996): The complexity of flat origami

Annual ACM-SIAM Symposium on Discrete Algorithms, Atlanta, Georgia

175-183

http://link.springer.com/book/10.1007/978-94-015-9514-8

References

160

Bricard R. (1927): Leçons de cinématique

Tome II Cinématique Appliquée, Gauthier-Villars, Paris

7-12

Buffart H. C., Trautz M. (2013): Construction approach for deployable

folded plate structures without transversal joint displacements

New Proposals for Transformable Architecture, Engineering and Design in the Honor of E. P. P.,


331-336

Buhl T., Jensen F., Pellegrino S. (2004): Shape optimization of cover plate for retractable roof

structures

Computers & Structures, Vol. 82, No. 15-16

1227 - 1236

Calatrava Valls S. (1981): Zur Faltbarkeit von Fachwerken

Doctoral thesis, Eidgenoessischen Technischen Hochschule Zürich

Chen Y. (2003): Design of Structural Mechanisms

Doctoral thesis, St Hugh’s college, University of Oxford

Chen Y., You Z., Tarnai T. (2005): Threefold-symmetric Bricard linkages for deployable structures

International Journal of Solids and Structures, Vol. 42, No. 8

2287-2301

Cruz P. J. S. (2013): Structures and Architecture: New concepts, applications and challenges

CRC Press, Florida, US

De Marco Werner C. (2013): Transformable and transportable architecture

Master’s thesis, Escuela Técnica Superior de Arquitectura de Barcelona,

Universidad Politécnica de Cataluña

De Temmerman N. (2007): Design and analysis of deployable bar

structures for mobile architectural applications

Doctoral thesis, Free University of Brussels, Faculty of Engineering

EN 1990: Basis of structural design, European Standards

Eurocode

EN 1991: Actions on structures, European Standards

Eurocode

EN 1993: Design of steel structures, European Standards

Eurocode

EN 1999: Design of aluminium structures, European Standards

Eurocode

References

161

Escrig Palláres F. (1985): Expandable space structures

Space Structures Journal, Vol. 1, No. 2

79 – 91

Escrig Palláres F. (1986): Introducción a la geometría de las

estructuras espaciales desplegables de barras

Boletín Académico Escuela Técnica Superior de Arquitectura, Coruña, Vol. 3

48 -57

Escrig Palláres F. (1991a) Las estructuras de Emilio Peréz Piñero

Arquitectura Transformable, Escuela Técnica Superior de Arquitectura, Sevilla

9-32

Escrig Palláres F. (1991b) Geometría de las estructuras desplegables de aspas

Arquitectura Transformable, Escuela Técnica Superior de Arquitectura, Sevilla

93 – 124

Escrig Palláres F., Valcárcel V. P. (1988): Expandable curved space bar structures

IETCC, Escuela Técnica Superior de Arquitectura, Sevilla

Escrig Palláres F., Sánchez J. Valcárcel V. P. (1996): Two way deployable spherical grids

International Journal of Space Structures, Vol. 11, No. 1

257-274

Escrig Palláres F., Valcarcel V. P. (2012): Modular, ligero, transformable:

un paseo por la arquitectura ligera móvil


Fuller R. B., Applewhite E.J. (1975): Synergetics, Explorations in the geometry of thinking

Macmillan Publishing Co. Inc.

Gan W. W., Pellegrino S. (2003): Closed loop deployable structures

Structural Dynamics, and Materials Conference, Norfolk, Virginia

American Institute of Aeronautics and Astronautics

Gantes C.J. (1997): An improved analytical model for the prediction

of the nonlinear behavior of flat and curved deployable space frames

Journal of Constructional Steel Research, Vol. 44

129 - 158

Gantes C. J., Konitopoulou E. (2004) Geometric design of arbitrarily curved

bi-stable deployable arches with discrete joint size

International Journal of Solids and Structures, Vol. 41, No.20

5517-5540

References

162

Gómez Lizcano D. E. (2013): An expandable pendentive dome and

polyhedral mechanisms using angulated scissors with sliding elements



53-58

Grübler M. F. (1917): Getriebelehre

Springer Verlag, Berlin

Guest S. D., Pellegrino S. (1994): The Folding of Triangulated Cylinders, Part I: Geometric

Considerations

Journal of Applied Mechanics, Vol. 61, No. 4

773-777

Hanaor A., Levy, R. (2001): Evaluations of deployable structures for space enclosures

International Journal of Space Structures, Vol.16, No.4

211 - 229

Hartenberg R. S., Denavit J. (1959): Cognate links

Machine Design, 16/04/1959

Hernandez C. H. (2013): Experiences in deployable structures



41-46

Hoberman C. (1988): Four sided central zone and pair of triangular flap hinges

US patent 4 780 344 A

Hoberman C. (1990a): Reversibly expandable doubly-curved truss structure

US Patent, number 4 942 700

Hoberman C. (1990b): Reversibly expandable structures

US patent 4 981 732

Hoberman C. (2001): Connections to make foldable structures

EU Patent, number 1 219 754 A1

Hoberman C. : portfolio

http://www.hoberman.com/portfolio/

Accessed on 11/12/2013

Huang H., Deng Q., Li B. (2012): Mobile assemblies of large deployable structures

Journal of Space Structures, Vol. 5, No. 1

1-14

http://hoberman.com/portfolio/

References

163

Huang Z., Li Q., Ding H. (2013): Theory of parallel mechanisms

Springer, Mechanisms and Machine Science series, Vol. 6

Huffman D. (1976): Curvature and creases: a primer on paper

IEEE Transactions on Computers, Vol. 25, No. 10

1010–1019

Hull, T. C. (2006): Project Origami

A. K. Peters Press

Jensen F.V. (2004): Concepts for retractable roof structures

Doctoral thesis, University of Cambridge, Department of Engineering

Kassabian P.E. (1997): Investigation into a type of deployable roof structure

Final year project, University of Cambridge, Department of Engineering

Kiper G. (2011): Design methods for planar and spatial deployable structures

Doctoral thesis, Middle East Technical University

Krausse J. , Lichtenstein C. (1999): Your Private Sky R.Buckminster Fuller

Lars Müller Publishers

Li Y., Xu Q. (2007): Kinematic analysis of a 3-PRS parallel manipulator

Robotics and Computer-Integrated Manufacturing, Vol. 23(4)

395-408

Langbecker T. (1991): Kinematic analysis of deployable scissor structures

International Journal of Space Structures, Vol. 14

1–16

Langbecker T. (2000): Kinematic and non-linear analysis of foldable barrel vaults

Engineering Structures, Vol. 23

158–171

Maden F., Teuffel P. (2013): Development of transformable anticlastic

structures for temporary architecture



251-256

Melin N. O. (2004): Application of Bennett mechanisms to long-span shelters

Doctoral thesis, Magdalen College, University of Oxford

References

164

Michaelis A. R. (1991): ‘Heureka’ – the Swiss national research exhibition 1991, Zürich

Interdisciplinary Science Reviews, Vol. 16, No. 4

374-375

Miura K. (1972): Zeta-core sandwich – its concept and realization

ISAS Report, 37(6), University of Tokyo, No. 480

137-164

Miura K. (1980): Method of packaging and deployment of large membranes in space

31st IAF Congress, No. IAF-80-A31

Miura K., Natori M. (1980): 2-D array experiment on board a space flyer unit

Space Solar Power Review, Vol. 5

345-356

Myard E. F. (1931): Contribution à la géométrie des systèmes articulés

Societé mathématique de France, Vol. 59

183 -210

Norton R. L. (1991): Design of Machinery

McGraw Hill, University of Worcester

Piker D. (2009a): Origami Electromagnetism

https://www.spacesymmetrystructure.wordpress.com/


Piker D. (2009b): Networked Bennett linkages in Grasshopper

https://www.spacesymmetrystructure.wordpress.com/


Qi X., Deng Z., Li B., Liu R., Gua H. (2013): Design and optimization of large deployable mechanism

constructed by Myard linkages

CAES Space Journal, Vol. 5, No. 3-4

147-155

Raskin, I. (1998): Stiffness and stability of deployable pantographic columns

Doctoral thesis, University of Waterloo , Civil Engineering Department

Resch R. D. (1968): Self-supporting structural unit having a series of repetitious geometrical modules

US Patent 3407558 A

References

165

Resch R. D., Armstrong E. (1971): Paper and Stick Film

Reuleaux F. (1963): The Kinematics of Machinery

Dover Publications, New York

Translation by A. B. W. Kenedy

Roovers K., Alegria Mira L., De Temmerman N. (2013): From surface to scissor structure



275–280

Röschel O. (2012): A Fulleroid - like mechanism based on the cube

Journal for Geometry and Graphics, Vol. 16, No. 1

19-27

Sarrus P. T. (1853): Note sur la transformation des mouvements rectilignes

alternatifs, en mouvements circulaires et réciproquement

Académie des Sciences, 36

1036-1038

Schenk M. (2012): Origami in engineering and architecture

Presentation, University of Cambridge, Advanced Structures Group

Schenk M., Guest S. D. (2010): Origami folding: a structural engineering approach

Origami 5: Fifth International Meeting of Origami Science, Mathematics, and Education

291-303

Schwabe C., Ishiguro A. (2006): Geometric Art

Kousakusha publishers, Tokyo

Schwabe C. (2010): Eureka and Serendipity: The Rudolf von Laban

Icosahedron and Buckminster Fuller’s Jitterbug

Kurashiki University of Science and the Arts, Okayama

Shim J., Perdigou C., Chen E. R., Bertoldi K., Reis P. M. (2012) : Buckling-induced

encapsulation of structured elastic shells under pressure

Proceedings of the National Academic of Science of the United States of America, 109

5978-5983

Solomon J., Vouga E., Wardetzky M., Grinspun E. (2012): Flexible developable surfaces

Eurographics Symposium on Geometry Processing 2012, Vol. 31, No. 5

1567-1576

Tachi T. : Freeform Origami

http://www.tsg.ne.jp/TT/software/


References

166

Tachi T. (2009): Generalization of rigid foldable quadrilateral mesh origami

Evolution and Trends in Design, Analysis and Construction of Shell and Spatial Structures,

Universidad Politécnica de Valencia

173-179

Tachi T. (2011): Rigid-Foldable Thick Origami

Fifth international Meeting of Origami Science, Mathematics, and Education

253-263

Tachi T. (2013): Freeform Origami Tessellations by Generalizing Resch’s Patterns

ASME International Design Engineering Technical Conferences and Computers and Information in

Engineering Conference

Tonon O. L. (1993): Geometry of Spatial Folded Forms

4th

International Conference on Space Structures, University of Surrey, Guildford, UK

2042 – 2052

Trautz M., Künstler A. (2009): Deployable folded plate structures -

folding patterns based on 4-fold-mechanism

Evolution and Trends in Design, Analysis and Construction of Shell and Spatial Structures,

Universidad Politécnica de Valencia

2306 - 2317

Verheyen, H. F. (1989): The complete set of Jitterbug transformers and the analysis of their motion

The International Journal of Computers and Mathematics With Applications, Vol. 17

203-250

Wohlhart K. (1995): New overconstrained spheroidal linkages

9th world congress on the theory of the machines and mechanisms, Milan

149-155

Wohlhart, K. (2001): Regular polyhedral linkages

2nd workshop on computational kinematics, Seoul

239-248

Wohlhart K. (2007): Cupola Linkages

12th

World Congress in Mechanisms and Machine Science, Besançon, France, Vol. 1

319-324

Yokosuka Y. and Matsuzawa T. (2013): Shape-finding for variable

geometry structures formed by a Multilink Spherical Joint

New Proposals for Transformable Architecture, Engineering and Design in the Honor of E. P. P.,


86 -88

References

167

You Z., Pellegrino S. (1997): Foldable bar structures

International Journal of Solids and Structures, Vol. 34, No. 15

1825-1847

You Z., Chen Y. (2011): Motion structures : deployable structural assemblies of mechanisms

Taylor and Francis, Hoboken, NY

Zhao J., Feng Z., Chu F., Ma N. (2014): Advanced theory of

constraint and motion analysis for robot mechanisms

Elsevier Science & Technology Books, 2013

Zeigler H. (1976): Collapsible self-supporting structures

US Patent No. 3 968 808

References

168

Websites

https://www.1stdibs.com


http://www.aedas.com


http://www.abhmfg.com


http://www.calatrava.com


http://www.civ.eng.cam.ac.uk/dsl/roof/planar


http://www.dosu-arch.com


http://www.elisastrozyk.de


http://www.perezpinero.org/


http://www.giselbrecht.at


http://www.jahn-us.com


http://www.laboratoriodearquitectura.com

Accessed on 15 /11/2014

http://www.mcescher.com/


http://www.nskeurope.com


http://www.popularmechanics.com


http://www.pro-tent.ch


References

169

http://www.rent-a-tent-web.com


http://www.rvtr.com/


http://www.sdpsi.com


http://www.structurae.net


http://www.strutpatent.com/


http://www.tsg.ne.jp/TT/


http://es.wikipedia.org/wiki/Mecanismo_de_Sarrus


http://www.wvutoday.wvu.edu/


Software programs used for modeling

Dassault Systèmes (2014): Solidworks 2014

McNeel R. (2010): Grasshopper Generative Modeling with Rhino

Robert McNeel and associates, Seattle, WA, USA

McNeel R. (2014): Rhinoceros 3D, version 5.0

Robert McNeel and associates, Seattle, WA, USA

Nemetschek (2014): SCIA Engineer 14

http://www.nemetschek-scia.com/

Appendix A. Formal Studies

Appendix I. Formal Studies

171


172


173


174


175

Appendix B. Transformable Designs

Appendix II. Transformable Designs

177

Acoustical shell for street musicians

Categorization Rigid-foldable origami, Yoshimura pattern

A deployable, lightweight shell for performers in the public

space. Made out of a curved Yoshimura pattern, the identical

triangles not only give rigidity to the whole, but in the unfolded

state also act as acustical diffusers for the mid- to-high tones,

creating a more balenced sonic field.

Materialization challenges

The primary difficulty is, as in almost all of compactable

origami designs, how the detailing allows the folding of the

facets, which should take in account the cumulative thickness.

A second main design challenge is how to lock the structure

into static shape. Another aspect here is the reflective

properties on the interior of the facets, which will determine

the usability of the piece.


178 Design and detailing of this work where done in collaboration with Michiel Van Der Elst and Charlotte De Vreese

Façade plug-in

Categorization Rigid-foldable origami

This foldable facade plug-in is the result of a research on what

advantages a dynamic metal sheet-facade can bring. The used

parameters are all linked to interior comfort.

In fully closed state the plug-in offers of solar shading without

affecting the interior view. Indirect reflection ensures an

evenly spread out light inside. At the same time the enclosed

air in between the interior and exterior glass panes makes for

a greenhouse effect, which lowers the wind chill.

In opened state the plug-in allows for ventilation and offers

shading, easily adapting the angle to the height of the sun.

Boundary conditions

The structure moves with a single degree of freedom and thus

can be made functional by one actuator, possibly connected to

temperature and humidity sensors.


Detailing is done with thin sandwich aluminum panels. The

folds are made possible by fitting an elastic material as

neoprene between the ends of the panels.


179

Mobile padel court

Categorization SLEs

A mobile court designed for the increasingly popular racket

game padel. The main mechanism used is a simple SLE grid

that has a fixed maximum opening angle to lift the court some

50 cm of the ground. This increases the visibility of the players

and, more importantly, allows for the court to be placed on

uneven terrain, using the adaptable footings.

The side columns upon which the characteristic panels are

hung are unfolded by a 4R planar lever system. Like this, the

workers can easily hang the upper panels and hoist them up,

without lifting overhead or needing heavy machinery.

Boundary conditions

The end bars of this conventional SLE system slide down the

side columns when opening, until the end of the rail where the

desired maximum aperture is reached.

To fix the mechanism into place, stage panels are clicked into

the joints of the SLE. This way, the panels are supported each

meter to prevent sag.


All of the structural bars would be made from aluminum.

The difficulty here is to optimize the weight-stability

relationship, to make it easily transportable and at the same

time limit any deformations caused by the dynamic load of the

athletes.


180

Scissor gates

Categorization SLEs

A design for the acces gates to a deconstructionalist museum

building, the client wanted these 5m gates to be

dematerialized into 2 segments. In the final design, the lower

part of the gate is prolonged to close off a recessed part in the

building.

Boundary conditions

The gate is made up out of 2 parts supported by a central axis,

which is in turn supported by a (half-joint) wheel in a rail. To

ensure that the mechanism moves only in the desired way

(1DOF), the 2 parts are each fixed to the wall at their

extremities by a vertical (half-joint) wheel and a horizontal (R)

wheel.


181

Solar panel array, primary axis rotation

Categorization 6R Planar Linkages

Design for a solar panel array in which there is a 1DOF

rotation around the primary axis.

The main design concept is to make the panels rotate and at

the same time undergo a relative translation, which ensures a

greater distance between them as the sun changes angle. This

way, casting shadows on one another is avoided. For the same

reason of avoiding shadows, any structural pieces of the

mechanism are placed below the panels. Furthermore, this

movement has to be abled in both directions, to allow the

rotation to respond to all of the suns cycle. The whole could be

driven by a single actuator, powered directly by the energy

collected by the panels.

Boundary conditions

The system is made up of a row of 6R linkages that each

actuate a column of panels. The outer edges of the panels

themselves form part of the system as rigid links. The bottom

edges of the panels are centrally supported by half-joints that

slide along a rail.


The multitude of joints makes the system susceptible to more

friction. A simpler way of supporting the system can be

sought, in which the supporting half-joints is replaced by a

simpler solution.


182

Solar panel array, secondary axis rotation

Categorization Planar Jitterbug-like mechanism, SLEs

A 1DOF set-up of solar panels which can rotate around their

secondary axes. Any shadows on the solar cells themselves

are avoided by the translation taking place when the

translating the panels further apart with a greater aperture,

and by placing all the mechanism links underneath the

surfaces.

Boundary conditions

The mechanism is made up of triangular links of which the

panel border forms one edge. R-joints connects bars to the

center of the diagonal edge of the triangular link. In a scissor-

like array these elements are finally connected to a rail by

half-joints at their lower verteces.

Looking at the triangular link as half a rectangle and the

connecting bar as the diagonal of another rectangle, the

system can be described as a type of planar Jitterbug

mechanism.


The geometry is pretty straightforward and easy to

materialize. Rows of panels could be connected to each other,

making the rail in which the bars slide only necessary at both

ends


183

Foldable bar unit

Categorization Rigid-foldable origami

An easily compactable and deployable bar unit for temporary

events. The main geometry is the repetition of a Miura-ori fold,

while the edges are a variation that unfold perpendicularly.

The zigzag floor plan and the edges give a minimum of

transversal stability to the piece.

Boundary conditions

The single degree of freedom makes for easy deployment. The

particular geometry of the fold pattern doesn’t allow the

plates to fold any further than rad. In its fully deployed

position constraining elements are added underneath the top

plates in order to lock the whole into place and add strength

locally.


To account for the thickness of the plates, offset elements

have been introduced at the location of the hinges. Controlling

these offset distances, the top plates can be made to fold over

the vertical plates. This way, both a compact bundle and a

deployed state with fine edges can be secured.


184

Deployable truss network

Categorization Re-entrant auxetic pattern

Array based on the square re-entrant pattern forming

deployable trusses. Basic units of the compacted bars are

linked together on site in order to make transportation easier.

Diagonal bars with an R-joint are added for structural

performance.

Boundary conditions

While computer models assumed the system to behave with

one degree of freedom, physical test models have shown an

additional mobility because of joint clearance. Local

pantographs need to be introduced in order to control this

dependent mobility and add rigidity.

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