MECHANISMS OF COMPLIANT SHELLS

Victor Charpentier

A DISSERTATION

PRESENTED TO THE FACULTY

OF PRINCETON UNIVERSITY

IN CANDIDACY FOR THE DEGREE

OF DOCTOR OF PHILOSOPHY

RECOMMENDED FOR ACCEPTANCE BY

THE DEPARTMENT OF

CIVIL AND ENVIRONMENTAL ENGINEERING

Adviser: Sigrid Adriaenssens

June 2019

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© Copyright by Victor Charpentier, 2019. All rights reserved.

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Abstract

Thin shells are three dimensional curved solids with a thickness that is small compared to

the two other dimensions. For structural engineers, traditional rigid thin shells are some of

the most efficient structural typologies. The curvature and continuity that characterizes

them are the source of their high stiffness under loading. They are also the basis for a new

typology of structures: compliant shells. Used for Their ability to deform elastically under

compressive buckling loads suggests that the shape of thin compliant shells can be tailored

to produce mechanism-like kinematics. In this dissertation, a case is made for thin

compliant shells as ideal candidates for tailored large deformation mechanisms.

Biology (and more particularly plants) is rich with compliant mechanisms. In this thesis a

comprehensive categorization of the structural mechanics in plant movements

demonstrates that shells are the most efficient mechanism to amplify actuation. The

increase in scale of the movements of those shells for engineering purposes is shown to be

limited by the influence of earth’s gravity. A non-dimensional analysis of mechanical

characteristic properties of thin shells leads to the identification of a size constraint for

gravity independence. A geometry-based method for the identification of compliant shell

mechanisms is then presented that relies on the computation of the low frequency

eigenmode. The methodology is applied to the design of a spherical motion mechanism

based on the geometry of a negative Gaussian curvature toroidal shell under two degree-

of-freedom compressive actuation, aimed for solar shading applications. Finally, a novel

methodology for the design of dynamic external shading systems, based on compliant shell

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mechanisms, is presented that substantiates the use of more comprehensive comfort-centric

performance-based for the reduction of energy demand in buildings.

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Acknowledgments

This dissertation would have never been possible without the support of many wonderful

and talented people. During my time at Princeton University they expanded the boundaries

of what I considered scientifically and humanly possible. Thanks to them I found an

unquenchable source of knowledge and intellectual challenges. I would like to

acknowledge them here.

To my thesis adviser, Sigrid Adriaenssens. Thank you for welcoming me in the Form

Finding Lab six years ago and providing such a fertile environment for growth. Thank you

for your trust in my ability to complete this diverse research work, for your constant support

in the good and less good times, and for guiding me on this scientific and human journey.

To Olivier Baverel, thank you for lighting the spark of this adventure at Ecole des Ponts

and for your long-range scientific vision over the last six years. Your ethics and positive

energy are inspiring.

To Forrest Meggers. I am grateful to have been an unofficial member of CHAOS, where

the power of teamwork dazzles. Thank you for your generosity, curiosity and brilliant

guidance.

To Sasha Eisenman. Thank you for having the boldness and curiosity to venture outside of

your field to guide me in the wonderful world of plant structures.

To Branko Glisic and Stefano Gabriele. I have been fortunate to learn from your extensive

knowledge. You shared it with me patiently and generously. Thank you.

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To Tim, Olek, Andrew, Ted, Lionel, Yousef, Hannah. Colleagues at the Form Finding Lab,

friends in life, you are the best of both. Thank you for making the daily work exciting and

challenging.

To my friends in France, in Princeton and over the world. I feel fortunate to know such

extraordinary people as you.

To Tracy. Thank you for your creativity and for challenging me to be more. Your influence

will last.

To Isabel, Kelsey, Jack and Link. I could climb any mountain with you. Thanks for being

exceptional.

To my family. Distances feel shorter and winters warmer with your unwavering and loving

support. Thank you for passing some of your strength and resilience to me.

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À Jade, ma sœur chérie

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Table of Contents

Abstract ............................................................................................................................. iii

Acknowledgments ............................................................................................................. v

Table of Contents ............................................................................................................. ix

Chapter 1: Introduction ................................................................................................... 1

1.1 Background and motivation .......................................................................................... 1

1.2 Research objectives ....................................................................................................... 7

1.3 Significance of research ................................................................................................ 8

1.4 Dissertation organization .............................................................................................. 8

Chapter 2: Kinematic amplification strategies in plants and engineering ................ 10

2.1 Introduction ................................................................................................................. 10

2.2 Cellular material and plant actuation .......................................................................... 14

2.2.1 Cell structure and resulting material properties .................................................... 14

2.2.2 Actuation by water transport ................................................................................. 16

2.2.3 Material hierarchy ................................................................................................. 18

2.3 Amplification of actuation in plant kinematic mechanisms ....................................... 19

2.3.1 Bilayer action through the shrinking and swelling of cells................................... 19

2.3.2 Omnidirectional movements from pressure differentials in pulvini ..................... 20

2.3.3 Amplification by geometry and material anisotropy in passive movements ........ 24

2.3.4 Anisotropic layering in hydration-related expansion ............................................ 27

2.3.5 Fast release of stored strain energy ....................................................................... 30

2.3.6 Amplification of external load by beam mechanisms and optimization of stiffness

........................................................................................................................................ 32

2.3.7 Coupling of geometry and shell mechanics .......................................................... 34

2.4 The parallels between engineered and plant actuators ................................................ 40

2.5 Conclusion .................................................................................................................. 44

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Chapter 3: Physical limits of compliance in the scaling of thin shells........................ 49

3.1 Introduction ................................................................................................................. 49

3.2 Non-dimensional numbers characterizing the mechanical behavior of shells ............ 52

3.2.1 Quantification of bending vs. stretching deformation .......................................... 52

3.2.2 Influence of gravity body forces on shell internal forces ..................................... 55

3.3 Results ......................................................................................................................... 56

3.3.1 Thin shells have similar 𝛾𝐹𝑣𝐾 values across scales ............................................. 56

3.3.2 Difference in gravitational driving force across scales ......................................... 61

3.4 Discussion ................................................................................................................... 63

3.5 Conclusion .................................................................................................................. 67

Chapter 4: From isometries to mechanisms ................................................................. 70

4.1 Introduction ................................................................................................................. 70

4.2 Morphogenesis of compliant shell mechanisms ......................................................... 72

4.2.1 Detection of isometric modes of transformation using eigenfrequencies ............. 72

4.2.2 Natural frequencies of six shell geometries .......................................................... 80

4.2.3 Discussion ............................................................................................................. 83

4.3 Kinematic study of a compliant shell mechanism ...................................................... 85

4.3.1 From surface to mechanism .................................................................................. 85

4.3.2 Kinematics of the compliant shell mechanism ..................................................... 98

4.4 Conclusion ................................................................................................................ 112

Chapter 5: Daylight driven optimization of dynamic shading for building energy

demand ........................................................................................................................... 114

5.1 Introduction ............................................................................................................... 114

5.2 Choice of shading systems and optimization method ............................................... 117

5.2.1 Facade orientation and room geometry ............................................................... 118

5.2.2 Choice of external shading system: awning, venetian shades and spherical solar

tracking ........................................................................................................................ 119

5.2.3 Selected energy and daylighting control variables ............................................. 122

5.2.4 Interpolation of simulated results for behaviour modelling ................................ 126

5.2.5 Optimization system for control of shades ......................................................... 129

5.2.6 Methodology for design and assessment of shading performance ..................... 130

5.3 Results ....................................................................................................................... 131

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5.3.1 Mitigation of energy demand for the three types of shades ................................ 131

5.3.2 Daylight conditions for the optimized positions ................................................. 136

5.4 Discussions ............................................................................................................... 143

5.4.1 Performance of the methodology ........................................................................ 143

5.4.2 Comparison of shading devices .......................................................................... 144

5.4.3 On the choice of constraint values ...................................................................... 146

5.5 Conclusion ................................................................................................................ 146

Chapter 6: Conclusions and future research.............................................................. 148

6.1 Introduction ............................................................................................................... 148

6.2 Solutions to research questions ................................................................................. 148

6.3 Recommendation for future research ........................................................................ 151

6.3.1 Automated search for compliant shell typologies ............................................... 151

6.3.2 Life cycle assessment of materials used for compliant shells ............................. 152

6.3.3 Daylight-driven optimization to improve the kinematics of shades ................... 154

Appendix A: Sample of cross-scale rigid and compliant thin shells instances ... 157

A1 Rigid engineered thin shells ................................................................................... 157

A2 Compliant engineered thin shells ........................................................................... 160

A3 Compliant plant thin shells .................................................................................... 162

A4 Compliant micro-scale thin shells .......................................................................... 162

A5 Rigid egg shells ...................................................................................................... 163

References ...................................................................................................................... 165

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Chapter 1:

Introduction

1.1 Background and motivation

The research goal of this thesis is to define the behavior and design of thin shell

mechanisms for engineering applications. The word shell will bring different physical

structures to the mind of each reader, but it generally refers to enclosures. Whether they

describe an egg, a dome or a pressure vessel, they are mostly thought of as structures

providing shelter or containment. This intuition reflects two of the defining properties of

shells, their continuity and curvature [1].

To the structural engineer, thin shells are one of the most efficient load-bearing structural

system known. Engineered shells are often rigid large-scale structures. They are designed

to be stiff and perform load bearing functions. Large rigid shells span long distances and

cover courtyards, protect stadiums or cool down powerplants [2]. One of the first novel

uses of shells that took advantage of curvature and continuity was the Firth of Forth bridge

built in Scotland by 1889. This bridge with a span of 520m is still today the second longest

cantilevered bridge in the world [3]. All the members in compression are tubular shells up

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to over 3m in diameter [3]. Those members are curved (tubular section) and are made

continuous by riveting sheets of steel together.

As the use of shells for structural engineering applications increased during the 20th

century, their thickness was reduced to save material cost. To the point that a new failure

mode described as catastrophic [1], came to be a prominent focus of shell research [4-6]:

compressive buckling, or buckling. Buckling is a failure mode in which a structure under

compression loses its main load bearing capacity due to an instability of the geometry. An

often sudden deviation of the geometry due to compressive stresses leads to a reduction of

stiffness of the member and potentially to large displacements. In shells, this phenomenon

is described as catastrophic since the geometry in large part provides the stiffness of the

structure. The geometry of shells used for structural load bearing is carefully established

through for example a form-finding process, to limit bending forces that could lead to

deviation of the geometry upon loading. This is a challenging task since the energy

necessary to deform a thin shell in-plane is much higher than the energy necessary to bend

it out of plane. Structural load bearing shells aim to be under a state of in-plane stresses

only (this is called membrane behavior). When buckling occurs in a shell, the energy stored

in-plane is converted to bending energy [7]. The potentially drastic change of the geometry

that ensues, can lead to an abrupt loss of stiffness and rapid failure of the load bearing

structure [8]. In cylindrical tubes under compression, this failure mode takes the form of a

periodic pattern of geometric deviation of the surface (Figure 1.1a). Compressive buckling

is therefore strongly associated with instabilities and catastrophic failure of shells.

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Recently a renewed interest in shell structures has changed this established order by

embracing buckling. From a functional point of view, the continuity of shell structures

provide opportunity for generating enclosures. The mechanical instabilities of thin shells

are now sought after and attempted to be predictively understood so that they can be

exploited for mechanisms [9]. The large deformation of an enclosure, or of a continuous

curved surface can lead to new engineering applications. Thin shells designed for

controlled large deformations undergo a morphing of their geometry. All the transitory

states of deformation of the structure can be maintained in equilibrium with the external

applied force (Figure 1.1b). Thin shells are compliant structures, they can be deformed

elastically and reversibly. They can perform similar motion to an origami fold for instance

[10], but in a distributed, continuous manner, without the presence of local weakness points

such as creases or folds (Figure 1.1b). The new paradigm for thin shells is that they can

now be used as compliant mechanisms [11]. They are mechanisms; a defined input (such

as force or displacement) yields a known output in a force or displacement.

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Figure 1.1 Different forms of buckling of shells: (a) Cylindrical shell buckled by internal

vacuum (credit: Michael Nemeth, NASA Langley Research Center) (b) large scale

reproduction of closing movement of Aldrovanda vesiculosa [10]

Finding shell geometries that qualify as mechanisms is not a trivial task. In a parallel with

the form-finding process for rigid shells, there is a search for geometries that exhibit

reliable, repeatable movements. In this case, thin shells that can deform with bending

stresses only are favored for the large displacements they generate at a low actuation cost.

To generate shell shapes undergoing significant, controlled displacements, one needs

differential geometry and bio-inspiration. Mathematics are rich of rules and examples that

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describe the deformation of pure surfaces [12]. For instance, a common transformation of

surfaces is the isometry between a catenoid and a helicoid. In this transformation, the length

of an arc on the surface is preserved. Engineers also turn to nature for inspiration. Shell

buckling appears regularly in plants and can provide inspiration for engineering

applications [13-16]. Understanding the role of compliance in structures is core to

understand the generation of movement in plants. Therefore codifying bio-inspiration has

played a key role in popularizing the domain of compliant systems to a growing community

of engineers and designers [17].

In the built environment, dynamic external solar shading of building facades has been at

the forefront of implementation of large compliant deforming shells [17]. Buildings

consume 36% of the global final energy consumption in western countries [18]. There is

significant interest in morphing structures for solar applications [19-21] since those

structures can be deformed and maintain their deformed position for any stage of the

transformation. They can be deformed during the day following the two components of the

sun movements (Figure 1.2) in the sky: azimuth (east to west) and elevation/zenith (up and

down). Dynamic external solar shading is defined as the mechanism to control the sun

radiations (light or thermal) on the façade or envelope of buildings by surfaces placed on

the outside of the building that move in accordance with the sun’s motion [22]. The

building envelope acts as a barrier between the outside and the inside environment.

Mitigating the amount of solar radiation inside a building is a challenge as old as the first

human habitats. As early as the first century B.C. Vitruvius the preeminent Roman architect

counseled of the location of the rooms as to optimize the amount the heat received in the

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winter and increase the occupants comfort [23]. In modern buildings and since the

invention of air conditioning, solar radiation has been mitigated by removing heat from the

warm interior space. Typically, daylighting control is achieved with internal shades.

Dynamic external shading has the potential to increase the well-being of occupants by

providing controlled amount of daylight during the day. In addition, by being placed on the

outside of the façade, it reduces the energy demand for cooling [24-27].

Figure 1.2 Geometry of solar motion

In the case of external shading, shell structures that are continuously deformed into precise

continuous surfaces oriented with respect to the sun’s position throughout the day are

expected to provide significant improvements to building systems.

However, despite the potential for thin shells to be used as mechanisms and the existence

of proven demand for continuous structures capable of morphing, limited research has been

conducted on the justification, systemization and generation of thin shell geometries that

could be used for their compliance under compressive buckling loads. Therefore, the

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presented research creates strategies for the use of compliant thin shells as mechanism and

exemplifies the methodology with the design of external dynamic shading.

1.2 Research objectives

The overall research goal of this dissertation is to infer the rules and methodologies for the

use of thin shells as compliant mechanisms relevant to a spectrum of engineering

applications such as dynamic façade design.

This overall goal is achieved by addressing the following specific research objectives:

i. Categorize, rationalize and rank compliant mechanisms found in plants to find the

best system for actuation amplification.

ii. Establish scaling laws for the use of thin shell mechanisms at engineered scales

iii. Develop a methodology to identify shell geometries suitable for compliance (large

reversible displacements)

iv. Generate a bio-inspired thin shell mechanism capable of spherical tracking for

building solar shading

v. Devise an occupant-centered methodology to quantify and optimize the

performance of dynamic solar shading in buildings

vi. Establish future research directions and applications

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1.3 Significance of research

While thin shells used for their flexibility have been occasionally designed and implemented

in engineering applications, to the best of the author’s knowledge, a multi-level argument based

on the rationalization of the structural advantages of thin shells as mechanisms has not been

made. In this work, the author explores shell morphologies for compliant mechanisms. Rules

for adapting thin shells into mechanisms are created from the observation of nature and

mathematics. Each part of this work contributes to bringing new evidence of the relevance of

shells for novel compliant engineered applications in a variety of domains such as soft robotics

or aerospace. The development of a comprehensive mechanical comparison of compliance in

plant structures yields the justification for the mechanical advantage of thin compliant shells

(chapter 2). Slender structures can be successfully scaled up or down but for the first time, a

length scale that limit the scalability of compliant shells under earth’s gravity is introduced

(chapter 3). Furthermore, the creation of guidelines for the morphogenesis of thin shell

mechanisms based on differential geometry principles opens the field of possibilities for

designers of compliant shells (chapter 4). Finally, the user-focused optimization of dynamic

external shading of buildings provides novel solutions for the design of a building envelope, a

complex and critical part of building systems (chapter 5).

1.4 Dissertation organization

Given the diversity of topics treated in the individual chapters, a thorough literature review of

each subject is provided at the beginning of each chapter. This dissertation is organized as

follows. In chapter 2 an exploration of biology’s mechanical structures presents the concepts

used in this thesis and provides a mechanical comparison of plant movements. A structural

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classification is provided to quantify the mechanical efficiency of each system. In chapter 3,

instances of thin shell structures over 8 orders of magnitude of characteristic dimension are

analyzed to assess the scalability thin shells for compliance purposes. Furthermore, using non-

dimensional numbers measuring the propensity to deform in bending and the size sensitivity

of compliant shells to earth’s gravity, a limit on the characteristic dimension of compliant

shells, is introduced. In chapter 4, the morphogenesis of compliant shells is elaborated and

formalized into a method for the design of compliant shells for a broad range of applications.

This method is illustrated with the design of a spherical tracking mechanism. In chapter 5, a

novel methodology for daylight-driven optimization of energy demand in buildings is

introduced. Based on the quantification of performance of three types of dynamic external

shading systems, recommendations for the design of early stage building envelops are

proposed. In chapter 6 the main conclusions of the dissertation are presented and areas of

further research for the design and implementation of thin shells as mechanisms are identified.

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Chapter 2:

Kinematic amplification strategies in plants and

engineering

Some sections of this chapter have been published as the following journal article, but edits

have been made for continuity within this dissertation.

Charpentier, V., Hannequart, P., Adriaenssens, S., Baverel, O., Viglino, E., & Eisenman,

S. (2017). Kinematic amplification strategies in plants and engineering. Smart Materials

and Structures, 26(6), 063002.

2.1 Introduction

This chapter presents an organized comparison of natural movements and draws parallels

to nature from recent engineering studies to categorize strategies that have successfully

replicated or synthesized plants. Plant mechanics have often been described on a case-by-

case basis. By grouping the kinematic amplification types in plants, the objective is to

provide an approach to explore new possibilities in compliant mechanisms. By placing

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those amplification mechanisms in their context, a second objective is to the engineer

understand the origin of the solutions found in biology and to create the basis for new

innovative combinations of actuations and amplifications. The engineered structures are

not necessarily directly inspired by plant mechanics, but similarities organically appeared

between the two.

In section 2.2, an introduction to plant material and actuation is provided. In section 2.3,

the origins of plant movements and five identified kinematic amplification mechanisms are

presented. In section 2.4, a parallel is drawn between plant actuation and material, and their

equivalent in the engineering field. The essence of the mechanical principles in plants and

in-built actuators presented in the document is synthesized in section 2.5.

Biology serves as inspiration for engineering kinematics and has been doing so for a long

time. Designers tend to search nature for solutions to specific engineering problems. This

direct approach has provided many successful engineering products but has constrained us,

engineers, to the role of observers. As exposed by Vincent [28], the solutions provided by

biology to problems challenge our rational, rigorous problem solving approaches. Taking

note of the variety of mechanisms involved in plant movements, there is a renewed interest

in understanding the mechanical behavior of plants for structural morphings. As described

in chapter 1, structural morphings modify the structures with continuous shape changes

only, no movements of discrete parts are involved [29]. As such they are a subcategory of

adaptive structures, a structure is called adaptive as soon as it presents an alteration of its

geometry and/or material properties [30]. Plants are morphing structures in the purest sense

of the term. The continuity of the material is key to the existence of a living entity and to

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the coherent movement of autonomous structures. The diversity of environments and

situations where plants successfully adapted provides a breadth of problem-solving

examples. There is however a need to break down the mechanical strategies developed in

plants to ensure that identified kinematic mechanisms are made available to the designer.

Engineered adaptive structures and actuators increasingly implement compliance in their

design. As such, the observation of plant-generated movements can lead to novel morphing

schemes in engineered structures. The origins of active plant movements and plant

mechanics have been presented in detail in [31-39]. In light of material advances,

downscaling of manufacturing and computer aided design, many new interdisciplinary

applications appear that make use of a nature-inspired palette of solutions for engineering

problems. In engineering the interest for morphing is growing in many fields such as

aerospace industry [29, 40], building engineering [19], micro-scale actuation [41], wind

turbines [42, 43], automotive industry [44] or medicine [45].

Kinematic amplification in plants and in engineering is reviewed in this chapter. The

kinematic amplification, also called distance advantage, is defined as the ratio of output

displacement to input displacements in a kinematic system. The amplification ratio

measures the efficiency of structures at transforming the input actuation into large

displacements. The input displacement considered in this chapter encompasses both the

rigid bar displacements of classical mechanisms and the displacement resulting from

material strain over the whole length of the active material. Assessing the kinematic

amplification in mechanisms is essential to understanding and emulating their strategies in

adaptive structures. Parallel to kinematic amplification but not treated here is force

amplification, also referred to as mechanical advantage. It is the ratio of the output force

13

over the input force of the mechanism. Force amplification is not considered in this review

chapter, but it is not completely unrelated to kinematic amplification. In the simpler case

of lever mechanisms or gears, the force amplification ratio (FAR) is the inverse of the

kinematic amplification ratio (KAR)1.

The hypothesis guiding the biology study is that even the most basic generations of plant

movement involve some degree of kinematic amplification. In hydraulic powered

movements (active or passive), the input displacement considered for the kinematic

amplification is the total expansion of the active layer of tissue (intermediate-scale

assemblage of cells to create a rigid building material). Such mechanism can be so effective

that the kinematic amplification ratio can reach ∼200 in Aldrovanda Vesiculosa [46] and

∼16 in pine cones [47, 48] (see Figure 2.1). Similarly high KAR values have been

measured in bimetal actuators (∼140 in [49]), which gives ground for a comparison of plant

and engineering mechanisms. In animals this value is often lower due to the lever

mechanisms in musculoskeletal systems. For instance, the KAR reaches values of ∼2 in

the mantis shrimp raptorial appendage [50] or ∼6 in the human biceps-elbow joint [51].

1 Considering a lever of short arm length a and long arm length b, the FAR given by the equation of

equilibrium at the fulcrum is a/b while the KAR given by the ratio of arc length of the end points of the lever

is b/a. Considering a set of two gears A (input) and B (output) of respectively radii rA & rB and angular speeds

ωA & ωB, the FAR given by the gear reduction is rB / rA. while the KAR given by the equation of equal speed

at the point of contact is rA / rB

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Figure 2.1 Calculation of KAR for scales of pine cone as presented in [48], 𝐷𝑓 is the final

displacement and 𝐷𝑖 is the input displacement

2.2 Cellular material and plant actuation

Plant cellular structures can be regarded as hierarchical, living material and as such they

differ from man-made materials. In order to understand kinematic amplification

mechanisms, an introduction to the characteristics of this material provides grounds for

comparison between natural and engineered materials and actuations.

2.2.1 Cell structure and resulting material properties

Unlike the cells of animals, plant cells are surrounded by a stiff cell wall comprised of a

composite material with organized cellulose fibers ingrained in a pectin matrix [52, 53].

Due in part to the stiffness of the walls, growth and movement of plants are generated by

the variation of a large internal hydrostatic pressure inside the cells called turgor pressure

15

[54]. The orientation of the cellulose fibers largely determines the overall shape and

behavior of the plant cells and tissues. The degree of alignment of cellulose micro-fibrils

is correlated to the degree of anisotropic expansion of the cell under an increase of turgor

pressure [53, 55]. In addition, the angle of the aligned fibers with the cell axis determines

the magnitude of the anisotropic deformation [56]. At the material level, this anisotropic

behavior is caused by the global alignment of the fibers in the cell walls (see Figure 2.2.2)

[57, 58]. The directionality of the material properties plays a role in both irreversible (e.g.

growth motions) and reversible (e.g. turgor induced) motions of plants.

Figure 2.2 Microfibrillar arrangement influencing the growth of plant cells – with

parallel, horizontal microfibrillar orientation the cell grows longitudinally (left); with

randomized orientations the growth becomes isotropic (right)

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The mechanical properties of the tissue result from cell characteristics such as wall

thickness, cell geometry and turgor pressure magnitude [59, 60]. Plant materials exhibit

elasto-plastic behavior with a plastic strain limit around 2% [61, 62]. Reversible, turgor-

based motions involve the elastic regime of the cell walls while irreversible motions of

growth require elongation of the cells and therefore involve the plastic regime [63]. The

elastic modulus of extensible and reversible specialized cell walls varies between orders of

magnitude of 100 MPa [64] and 1 GPa [61, 65] while it is much higher in cells found in

woods, usually between 15 to 25 GPa for the hardest woods [66, 67]. Interestingly, the

turgor pressure has a direct influence on the stiffness of the material: in parenchyma cells

(cells with thin, non-specialized cell walls) a linear correlation has been reported between

the turgor pressure and the elastic modulus of tissues [39, 68]. However, this correlation

appears only when large strains occur [69]. Comparing a sample of carrot material with

internal cellular pressure preserved (fresh tissue) to a sample without internal pressure

(cooked for 1 min), Warner et al. [60] underline that relationship with qualitative

measurements of differing elastic moduli in large strains. Studies of fluid filled cellular

material highlight the need for nonlinear elastic modulus models to capture the behavior of

the plant material [56, 60, 69].

2.2.2 Actuation by water transport

Turgor pressure in cells drastically influences the mechanical properties of plant tissues.

Water potential gradients, largely resulting from changes in ion concentrations are the main

driver of many plant movements, with the variation of water uptake and loss by cells being

17

the basis for movements of plant organs [31]. Specialized motor organs like pulvini contain

motor cells with thin walls; when the thickness of the wall increases so does its flexural

stiffness. Cells presenting a wall thicker than 20% of their radius become insensitive to

variations of turgor pressure [70]. The difference of water chemical potential among cells

is the cause of water transport, and this transport occurs both at the cellular scale and at the

tissue scale [31]. Both scales of water transport can be explained with assumptions of flow

through semi-permeable membrane [71] and approximation of Darcy’s law linking water

flux and water-potential gradient through a porous material [72]. At the tissue scale, the

transport of water combined with the conservation of mass causes the water-receiving area

to increase in volume while the water-emitting area shrinks. This unbalance creates

bending to accommodate the strains (see Figure 2.3).

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Figure 2.3 Schematic variation of cell volume in the tissue due to water transport - Vtop

and Vbot represent the volume of the top and bottom cells respectively - L is the length

of the sample and ΔL is its variation.

2.2.3 Material hierarchy

Plant tissues are complex materials that implement an organization of multiple hierarchical

structural levels. The construction of these natural materials operate at scales ranging from

102m in tree trunks [73] to 10-7m in cell walls [36]. Between these two extreme values, it

19

is possible to distinguish between five to twelve structural levels [17]. In the case of the

structure of wood stems, this hierarchal material organization translates into seven levels:

tissue, cell, laminated cell walls, individual walls, cellulose fibers, microfibrils and

protofibrils [74]. This organization is found in many biological materials such as bones

[75], tendons [76] or hexactinellid (glass) sponges [77]. In essence, this organization allows

the construction of very large structures based on a limited number of building blocks. It

also creates the possibility of introducing combinatory variability at each level of the

hierarchy. The hierarchical organization is adapted for each situation through the

optimization of the arrangement of each structural level, and allow for material repair

mechanisms by relying on lower levels of material hierarchy [78, 79].

2.3 Amplification of actuation in plant kinematic mechanisms

Plant kinematic structures are actuated by hydraulics, passive or active, but also by external

forces, such as the visit of a pollinator. However, a number of those actuation sources

provide only small magnitudes of displacements. To answer the need of large

displacement, plants have evolved efficient kinematic amplification mechanisms to

increase the magnitude of the actuation. In this section, the amplification mechanisms in

plants are reviewed and analyzed based on mechanical principles.

2.3.1 Bilayer action through the shrinking and swelling of cells

The mechanical basis of this kinematic amplification strategy is the strain continuity

imposed to rigid bodies under differential tissue expansion (in active or passive hydraulics).

20

The compatibility condition in solids implies that strains are continuous [80]. Therefore the

difference of swelling/shrinking magnitude between layers creates large deformations of

the structure via bending, to accommodate strains within the material [33]. The input

displacement in this kinematic amplification is the total expansion of the expanding layer,

while the output displacement is the resulting bending. This section focuses on planar

bending created in pulvini.

2.3.2 Omnidirectional movements from pressure differentials in pulvini

Many heliotropic (solar tracking) plants such as Phaseolus vulgaris (regular string beans)

or Ranunculus adoneus implement reversible tracking movements [81-83]. Heliotropic

movements are dynamic, directional and involve short diurnal cycling as opposed to

general irreversible phototropic movements associated with growth [84]. The pulvinus is

an appealing structure for plant biomimetic endeavors. The pulvinus is a cylindrical motor

organ in its most general form. It has a length of a few millimeters and is usually located

at the base of the petiole (stalk attaching the leaf blade to the stem) [85]. In pulvini

presenting omni-directional movement, motor cells radially surround a rigid core (Figure

2.4A) made of thicker-walled, parenchyma cells [86, 87]. At the rest position, the motor

cells all have the same volume around the core. When the pulvinus is activated for a certain

orientation (β), the cells at an angle opposite to the desired deformation direction swell and

the pulvinus bends in-plane to perform the movement. The magnitude of the swelling

determines the bending angle (θ). With this hydraulic bending, plants can follow the

21

elevation and the azimuth of the sun throughout the day with a precision of up to 5-6°[85,

88].

Figure 2.4 (A) Cross section of omnidirectional pulvinus with active cells at azimuth

angle β – (B) Plane of deformation P – (C) In-plane deformation of the pulvinus and

elevation angle θ

In simpler mechanisms, the pulvini present more limited planar movements [89, 90]. In

this case, the structure of the pulvinus is similar to the omni-directional system but the

distribution of motor cells is limited to the plane of the movement. In this plane, the motor

cells are located on both sides of the rigid core. Such are the leaves of the Albizzia

julibrissin that consistently fold at night but do not implement multidirectional solar

tracking during the day [89]. This type of amplification carries strong similitudes with

bilayer thermostats and pneumatic actuation, which are presented in [91].

22

2.3.2.1 Versatility and adaptability of the bilayer effect in Mimosa pudica

In addition to diurnal folding due to variations of light intensity, Mimosa pudica exhibits

defensive folding motions following touch or vibration stimuli. These movement are

triggered by rapid changes of temperature, electric stimulation and wounding [92, 93]. This

non directional (thigmonastic) movement triggered by touch or vibration is relatively fast

compared to other plants [94]. However M. pudica does not implement elastic instabilities

to increase the speed of the folding movement (duration around 1s [95]). The movement is

solely generated by swelling and shrinking and amplified by a bilayer effect. Four

structural levels can be identified in the structure of M. pudica (see Figure 2.5) (1) the

principal structure is the stem, (2) petioles are attached to the stem, (3) pinnae (usually two

pairs, carrying the leaflets) are connected to each petiole, and finally (4) each pinna has 10-

20 pinnules (leaflets) attached directly by their base to the pinna [96, 97]. Structures (2),

(3) and (4) compose a leaf. Three levels of movements are observed in M. pudica. They

are generated by three pulvini, each is located at the junction of two previously mentioned

structural entities [92].

23

Figure 2.5 Touch stimulated movements during day and at night for the three pulvinus

levels of Mimosa pudica – P# denotes the following pulvini - P1 or primary: between the

stem and the petioles, P2 or secondary: between the petioles and the pinnae and P3 or

tertiary: between the pinna and the pinnules – The activated pulvini are written in darker

font.

The movements of M. pudica are not only triggered by touch but also follow the diurnal

cycles with the pulvini folding in absence of light [92]. Touch related movements involve

the primary and tertiary pulvini while the sleep movements involve all the pulvini [98].

The three types of pulvini allow two different typologies of planar deformation: upward or

downward. The primary and secondary pulvini deform downward while tertiary pulvini

deform upward [92, 93, 98]. This planarity has been confirmed for the tertiary pulvinus by

observation of the cellular layout. The unidirectional orientation of the motor cells in the

plane of the movement in both layers of the tertiary pulvinus confirms that only uni-

24

directional movement is possible [92]. In the case of the primary pulvinus however, recent

evidence suggests that its cellular structure is similar to an omnidirectional pulvinus with

a xylem (central fluid transport core) allowing bending in all directions [95]. In case of

stimulation by light (generating slower movement than touch), both horizontal and vertical

movements of the primary pulvinus are reported [95]. When only planar displacements of

the primary pulvinus are considered, the observation of the cellular displacement in the

upper and lower sections confirms that the upper cells are expanding and the lower cells

are contracting [95].

M. pudica is part of the Fabaceae (or Leguminosae), which contains a number of other

species that implement similar pulvinus-generated movements. Among them are Samanea

saman [99], Paraserianthes lophantha [100], Albizzia julibrissin [89], Phaseolus vulgaris

[90] and Codariocalyx motorius [101]. The pulvinus is a very polyvalent actuator allowing

either two or one rotational degrees-of-freedom.

2.3.3 Amplification by geometry and material anisotropy in passive movements

Flowering plants develop fruits to aid in seed dispersal. During their development those

structures are connected to the vascular system of the plant. However, after maturation and

separation from the vascular system, most seeds and many fruits become autonomous

structures. Similarly, pollen grains also act autonomously once released. The focus of this

section is on autonomous structure and their kinematic mechanisms created from

hydration. The external actuation imposes the actuation cadence of the autonomous

structure. Relying on a regular external actuation is therefore key to carrying out the

25

function of the structure. Interestingly, the highly-regular diurnal variation of air humidity

[102] is used as a reliable actuator for many seed cells rehydration. Hygroscopic plant

tissues shrink and swell largely as a function of variations in air humidity [33]. The majority

of reacting cells in seed tissues are dead dry cells. In general seeds contain between 5 and

15% of water [54] and therefore present great potential for rehydration.

2.3.3.1 Folding of shells

Shells are 3D continuous structures presenting one dimension considerably smaller

(thickness) than the other two (width and length). Shells can be flexible or stiff depending

on their boundary conditions. In the case of autonomous structures, pollen grains [103,

104] or seedpods [105, 106] provide valuable examples of flexibility.

Pollen grains are structures carrying genetic information for reproduction. The grains

present an outer surface divided in two surface types: one or several apertures (straight

zones on surface in Figure 2.6a) constituted of hydrophilic material and impermeable

surfaces for the rest of the grain [107]. After being released pollen grains desiccate. The

rate is variable but can reach 20% of the initial weight within an hour in most grass pollens

[108]. This water is typically lost through the apertures and the lost volume is compensated

by a deformation of the grain shell. Not all grains present apertures but for aperturate pollen

grains, the apertures fold inward during the drying process (c.f. Figure 2.6a) therefore

sealing the grain and halting the desiccation [108]. This process is referred to as

harmomegathy [109]. A large variety of closing typologies was presented in Halbritter and

Hesse [110]. Mechanically this thin shell deformation is achieved by bending action. When

allowed, bending is preferred to stretching in thin shell deformation. Energetically this

preference is explained by the stretching energy scaling as the cube of the bending energy

26

[111]. Couturier [103] presented an analytical model of pollen grain deformation with the

assumption that a minor amount of stretching is tolerated and introduces a new type of

mathematical surfaces useful for the analysis of thin shell morphings. This spherical

folding mechanism represents an efficient amplification of the actuation created by the loss

of water (Figure 2.6c).

At a larger scale, the seedpod of Vachellia caven exhibits an irreversible movement that

involves a similar desiccation process of a doubly curved envelope [105]. The banana

shaped pod presents a negative Gaussian curvature along the back of the pod. At the saddle

point, the longitudinal and meridional curvature therefore have opposite signs. During

desiccation the longitudinal curvature increases, and the meridional curvature decreases

(Figure 2.6d). This decrease triggers the opening of the pod and the release of the seeds.

Similarly, to the pollen grain, the mechanism has been modeled by isometric deformations

of shells. Those analytical models have the same kinematic properties as the plant seedpod

[105]. The analytical model could further inform the study of such pods (e.g., determining

the amplification ratio of the mechanism).

27

Figure 2.6 (a-b) Folding of pollen grains in hydrated (left) and dry (right) states – the

pollen grain of (a) Euphoria milii folds on itself along aperture during desiccation

(harmomegathy) while (b) Aristolochia gigantean shrinks by reflection of the surface

(one of many possible ones) [103] – (c) Two bending-only, mathematical deformations of

spheres correspond to pollen grain deformations [103] – (d) Variation of curvature in

seedpod of Vachellia caven upon drying, meridional curvature k1 decreases as

longitudinal curvature k2 increases. [105]

2.3.4 Anisotropic layering in hydration-related expansion

The second type of hydration deformation is controlled by differential anisotropic

expansion within tissues. In pulvini, local water gradients create a bulging of targeted

active cell layers. In contrast, dry cells of autonomous structures rehydrate uniformly when

the water content in the environment increases. Strong anisotropies are present in

autonomous structures creating in-plane bending, out-of-plane bending and torsion. A

simple example of such construction is a strip of bi-layer laminate with the first layer

composed of fibers in random orientations (isotropic) and the second layer with consistent

fiber orientation across the width of the strip (anisotropic) (Figure 2.7a). The result of the

28

swelling is an in-plane bending of the strip. If instead of random orientations the top layer

also receives an anisotropic, specific orientation, the strip of material can experience

double curvature (Figure 2.7b) or even twisting (Figure 2.7c). Many seedpods implement

such anisotropy to release the seeds [106, 112-114].

Figure 2.7 Laminate paper models of monomorph (a) and bimorph (b and c) hydraulic

actuators. reproduced from [33]

This hygroscopic mechanism is the basis for the movement of three common seeds. The

wheat seed (Triticum turgidum) has two awns on the seed dispersal unit, executing a

uniaxial bending/straightening movement correlated to the diurnal air humidity cycle

29

[115]. The awns are covered with upward oriented hair, which cause the seed to penetrate

the ground during the cycles (see Figure 2.8a). This uniaxial bending movement is also

found in pine cones (Pinus radiata [36, 47]). In pine cones, this mechanism produces a

remarkable KAR of about 160 [47].Two distinguishable material arrangements make up

the large cone scales. The tissue presents a uniaxial bending fiber distribution with well-

ordered straight fibers in the bottom layer and winding irregular fibers in the top layer

[116]. The result is a closed pine cone when humidity is high and an opening movement as

the air dries (see Figure 2.8b). The variation of angle is significant with approximately 50°

of range measured between the high and low extreme humidities [47]. A similar

mechanism is observed in the geranium Erodium cicutarium whose seeds also present a

self-burial movement due to the rotation of the awns. Due to a specific orientation of the

material fibers, the awns coil when air humidity is low, and straighten when hydrated [117,

118]. This cyclical movement generates the burial of the seed (Figure 2.8c).

Figure 2.8 (A) Wheat awns cyclical burial movement – reproduced with permission from

[115] (B) Drying gymnosperm cone– reproduced with permission from [48] and (C)

Coiling Erodium seeds – reproduced with permission from [31]

30

2.3.5 Fast release of stored strain energy

In species presented by Skotheim & Mahadevan [94] the time scale of this typology of

movements is found between 10-3 and 10-5s. For similar smallest dimensions of structures,

seed dispersal by fracture mechanism is faster than hydration-based and elastically instable

movements [94]. Most explosive plants rely on storage of elastic energy in bent, pre-

stressed or stretched members and its rapid release for dispersal [119]. The same principle

of energy storage was used in bent wood in archery or in certain rock-throwing catapults

before explosive-based weapons became the norm [76].

Several species in the genus Impatiens are known to present an interesting example of such

an instant elastic energy release mechanism [120, 121]. One species, Impatiens

glandulifera (Figure 2.9a) is an invasive species originally found in the Himalayas [122].

It is able to spread seeds up to two meters away from the mother plant via explosive

dehiscence [123]. An external touch trigger causes the movement. The loculicidal seedpod

is 1.5 to 3.5 cm long with longitudinal septa (seams) partitioning five valves. Each pod

contains 4 to 16 seeds [124]. The valves are maintained in a straightened position by the

connecting seam to the other valves. They are coiled in the relaxed state (see Figure 2.9b).

The elastic energy stored in the seedpod originates from the bending deformation

straightening the valves in the stressed position. This energy amounts to approximately

1mJ [120]. In comparison, the energy stored in a bow before shooting the arrow is on the

order of magnitude of 10 J [125]. The kinetic energy transferred to the seed represents

about 70% of the total stored elastic energy [120]. This high proportion of kinetic energy

transferred to the seeds makes the seed dispersal mechanism in I. glandulifera highly

efficient.

31

Figure 2.9 (a) Impatiens seedpods (credit: H.Zeil) (b) explosive release of seeds [120]

The energy necessary to propagate the fracture along the whole length of the seam between

the lobes is more than the total stored energy. However a finite element model of the system

reveals that only about 30% of seam length between the valves is needed to actually hold

the pod closed [120]. The system therefore gradually fractures along the seam before

reaching the decisive part of the seam. At this instant the pod is the most sensitive to touch

for explosion. The reason for this smaller fracture length is due to the shape of the valves

that are linearly tapered longitudinally and transversally. This singular geometry influences

how the valves are packed in the closed configuration. The elastic model of the valve as

beams of varying cross sections indicates that they should interpenetrate themselves in the

closed configuration [120]. Since this is physically not possible, the valves rest upon one

another in the closed position, which makes the system extremely sensitive to

modifications of any of its member. As soon as a valve deforms due to drying or to an

32

external trigger, the geometry of the four other valves is modified, the seams break and the

elastic energy is instantaneously released [120].

2.3.6 Amplification of external load by beam mechanisms and optimization of stiffness

In certain cases, plant organs connected to the plant vascular system produce movements

from the amplification of external forces created by pollinators or by repeated

environmental stimuli2. They are passive movements. In flowers, the pollinator’s weight

generates actuation during visits. In these cases, pollinators of flowers have predictable

landing sites on the flower. Strategies as evolved as ultraviolet guides [126], sexual

deception [127] or precise guiding through the flower [128] are implemented to insure the

spatial positioning of the pollinator. Such passive movements can be found in Calopogon

tuberosus [129], Salvia pratensis [130] and Strelitzia reginae [131] for instance. The

passive movement of Salvia pratensis is caused by a lever-type mechanism in which an

insect attempting to reach the nectar cavity pushes the shorter arm of the lever with its

head. In response, the longer arm is pushed on the pollinator’s back and spreads the pollen

on the insect [132].

2 Pollinators can also be the source of active movements in flowers when sensors signal a presence. This

phenomenon is discussed in Section 2.3.7.2

33

In Strelitzia reginae the movement of the flower is produced by a very different mechanism

involving torsional buckling [133]. The flower has three orange sepals and three blue

petals, with the latter having two fused together to form an arrow-shaped structure on which

pollinators such as weaver birds come to land. The nectaries are located in the short corolla

tube at the base of the three petals. The petals are attached to a long, fibrous rib [131] and

when this rib bends down under the weight of the pollinator, the petals accommodate the

deformation by rotating along the axis of the rib in a lateral torsional buckling mode, which

exposes pollen [134]. This mechanism can be triggered many times without fatigue and

has been used as basis for a bio-mimetic shading system [135].

Finally, passive amplification of external load can originate from a fine tuning of stiffness

properties. The carnivorous pitcher plant Nepethens gracilis uses the impact force of rain

droplets to propel ants from the underside of the lid covering the pitcher to the fluid-filled

pitcher itself [136]. The lid is short and stiff and when hit by a droplet, it pivots around the

hinged neck connecting it to the body of the pitcher. As a result of the form and structural

compliant hinge, the lid produces peak inertia forces 19 times greater than the

morphologically similar Nepenthes rafflesiana [137]. The underside of the lid, where ants

shelter in rain events, is coated with low friction wax crystals, finalizing the trapping

mechanism [137].

34

2.3.7 Coupling of geometry and shell mechanics

2.3.7.1 Bilayer action coupled with complex shell geometry: Aldrovanda vesiculosa

Aldrovanda vesiculosa is a small, rootless, aquatic carnivorous plant that floats freely in

shallow, nutrient poor waters from eastern Europe to Australia [138]. The plant exhibits

repetitive growth patterns with 6 to 8 leaves radially attached to the stem in a spoke-like

geometry [139]. Each leaf terminates in a trap measuring 3 to 6 mm long and consisting of

two lobes connected by a midrib. The traps elastically deform in a hinge-less mechanism.

The deformation is caused by the bending of the central part of the shell (midrib), with the

actuation produced by motor tissues located in both lobes along the midrib [140]. Iijima

and Sibaoka [140] reported that the motor cells have their longer axis oriented

perpendicular to the midrib, indicating that their radial expansion is the source of actuation.

Additionally, the motor cells in the motor zone are distributed in three layers: inner

epidermis, outer epidermis and a middle layer. Sibaoka [141] hypothesizes that during

firing the inner layer of the motor zone becomes flaccid while the outer layer remains

turgid. As a result of this variation in stiffness the midrib bends toward the inside of the

trap. This is a first amplification of movement implemented from bilayer effect.

Interestingly, the actuation of the trap is amplified a second time by the double-curved shell

geometry of the trap itself. The marginal zones located away from the midrib are non-motor

but their doubly curved geometry accommodates the bending of the midrib by creating a

folding motion of the lobe on itself. Numerical models of the geometry and actuation [46]

confirmed the hypothesis that the fast closing (100ms) of the trap is due purely to swelling

and shrinking and does not involve shell buckling. Poppinga and Joyeux [46] showed that

the amplification ratio of the trap closing displacement (output) over the reduction of

35

distance between the ends of the midrib (input) nears 200. This extremely efficient

amplification demonstrates both the high potential of combining several kinematic

amplification strategies and the potency of shells for kinematic amplification.

2.3.7.2 Progressive change of curvature: pollination of flowers

The flowers of certain plant species implement active movements to perform efficient

cross-fertilization. In the case of the orchid family (Orchidaceae), about one third of species

implement pollination by deception [132, 142], which consists of pollinator attraction

without providing a reward. The principle of this movement resembles the passive

displacement of the lever found in Salvia pratensis. However, by actively generating the

movement, the flower does not depend on the action of the pollinator beyond triggering

[143]. Examples of such movements can be found in species of the Australasian orchid

genus Pterostylis such as Pterostylis sanguinea [127] or in Pterostylis longifolia [144,

145]. In the fastest cases, the time-scale for those movements is ten milliseconds [146]. In

general, the resetting time is several orders of magnitudes longer than the trigger time, as

seen in Pterostylis [127]. In addition, a large number of repetitions tend to damage the

mechanism [127].

A relevant example of this strategy is seen in another genus, Stylidium, which carries the

common name triggerplant. The column that carries both stamens and style of the flower

is mobile. In a large rotational movement of the column triggered by touch, pollen is

deposited on the insect. This insect will in turn visit other flowers effecting in cross-

pollination. [146]. The active column consists of two distinct parts: the active and the

passive part (see Figure 2.10a). The active part, termed the bend, is easily identified due to

the curved rest state, different color and change of sectional curvature during the firing.

36

The passive part of the column remains straight during the movement and does not vary in

cross section during the movement [146]. Upon triggering, the active bend of the column

changes longitudinal curvature that creates a rotation of more than 180°, carrying the

anthers towards the pollinator (see Figure 2.10b). The entire movement happens in a time

span of 10 to 30 ms [146] which is among the fastest movements reported in [94]. The

mechanics of this movement are based on turgor pressure variations but have not yet been

clearly described.

Figure 2.10 (a) Diagram of flower of Stylidium graminifolium in median longitudinal

section with scale of 5 mm – (b) Photography of the firing of the column of Stylidium

crassifolium - reproduced with permission from [146]

With a mechanism which allows for motion faster than dictated by the poroelastic timescale

described by [94], the Stylidium species implement an elastic instability. Overall, there are

37

three mechanical levels involved in the rotational movement of the bend [147]. Cell

expansion is the first level: turgor pressure increases in specialized section cells of the bend

that deform longitudinally. In the second structural level, the heterogeneity of the section

with cells that either expand or do not, create a bilayer effect. Finally in a third level, the

curvature of the bend’s section is reversed during firing.

Levels 1 and 2 comply with the poroelastic theory since they are based solely on fluid

transport in the material. The main process speeding up the motion of the column occurs

at level 3. This multilevel mechanism illustrates the coupling of several mechanical

principles in a natural material, the bi-layer effect (structural levels 1 and 2) present in

many pulvinus-based plant movements (as seen in [94]), typically Mimosa pudica [95] or

Codariocalyx motorius [148], and an efficient elastic mechanism (level 3). This latter

mechanism is not clearly understood. The shell structure model proposed in [149]

hypothesized a sudden and complete loss of stiffness of the bend of the column as source

of its rotation. However, this reasoning can be challenged. The turgor pressure in the active

cells of the bend increases during the deformation [147]. As reported in section 2.2, the

elastic modulus of the material increases with an increase of turgor pressure. A quasi-

spherical deformation occurs in the cells of the bend tissue in Stylidium species [147]. The

active cells of the bend appear to elongate radially as well as longitudinally [149] in the

posterior layer of the section’s core. The result is a change of transversal (or sectional)

curvature in the bend. This change of curvature of the section from concave to convex

during the swelling of the cells is due to the increase of radial cell volume on the concave

side of the bend (see Figure 2.11) [147]. The progressive change of this section’s curvature

along the bend coupled with the curved longitudinal geometry of the bend creates the fast

38

movement of the column. Interestingly, the layout of Stylidium motor cells with their long

axis perpendicular to the plane of the sectional curvature is similar to the midrib cellular

layout in Aldrovanda vesiculosa. In that plant, the cells’ long axis is oriented perpendicular

to the midrib. The radial expansion of the cells over the length of the midrib creates an in-

plane inversion of the midrib curvature. This mechanism is similar to the mechanism at

play in the posterior layer of Stylidium’s active bend section (Figure 2.10) this sees a

curvature inversion in the section’s core.

Figure 2.11 Layers in the structural core of transverse section of the bend of Stylidium

based on physiology described in [146] – The section’s concave curvature appears in the

posterior (P), central (C) and anterior (A) of the core.

2.3.7.3 Snap-through buckling in carnivorous plants

Pursuing the goal of caching prey to compensate poor nutritional habitats, Dionaea

muscipula has evolved adaptive digestive enclosures based on a shell-buckling

phenomenon. This movement is not unique in carnivorous plants. Utricularia inflata also

has enclosures triggered by prey [150-152]. The mechanics of D. muscipula have been

extensively documented in literature in [46, 153-159].

39

D. muscipula implements a type of elastic instability called snap-through buckling

characterized by a rapid change of geometry of a system between two equilibrium states

[4]. The bi-lobe traps are located at the end of the leaf blades, and are triggered by repeated

stimulation of three sensing hairs on the internal surface of each lobe [157]. They are set

off by repeated contact of a prey with a trigger hair on the internal surface of the lobe [156].

The number of contacts of the prey with the hair has been shown to have a direct impact

on the behavior of the trap. Two contacts at 30s apart are necessary to cause the closing of

the trap, five contacts in total are required to fully start the costly digestive process [159].

After the first two contacts, the trap closes. It will remain in a semi-closed state with the

cilia (long hair-like structure) on the edge of the trap forming an enclosure holding any

larger prey securely [157]. More contacts signal the presence of an nutrient-rich insect and

the plant eventually start the digestive process [159].

From a mechanical point of view, the mechanism driving the rapid closing of the trap has

been described in a number of studies [46, 154, 156]. The elastic phenomenon that

dominates at the larger structural level, is the buckling of a doubly curved shell [46, 154].

This buckling is caused by a bilayer effect [156]. Geometrically, a two-step sequence is

observed. When the traps are in the rest position, the mirroring lobes remain in a state of

positive average curvature. After being triggered, the lobes undergo a radical change of

geometry, shifting from a positive to a negative average curvature [154]. Each lobe presents

two main curvatures, which are described with respect to the midrib of the trap joining the

two lobes. The curvature transversal to the midrib, changes sign during closing of the trap

while the longitudinal curvature does not [154]. The transversal change of curvature drives

the closing movement of the trap [154]. This change of curvature is generated by a bilayer

40

effect in the two-layer structure of the lobe tissue, and has been described in a number the

studies [153, 156, 160]. The inside of the lobe (called the “upper layer”) remains under

high turgor-pressure at the rest state, creating a convex lobe geometry. The closing

movement is triggered by the gradual de-pressurization of the upper layer and the increase

of turgor-pressure in the exterior side (“bottom layer”) of the lobe [156, 160]. The bottom

layer experiences an increase of length in the transversal direction [154]. This elongation

is supported by observations of the microstructure of the bottom layer material, which is

comprised of cylindrical cells. These cells have their long axis oriented in the direction

transversal to the midrib and are reinforced with hoop-like, cellulose reinforcements [160].

From an energy perspective, the lobes of the plant produce a large release of elastic energy

during the closing [46]. This release coincides with the buckling phenomenon [46] and the

rapid change of curvature. As measured in [154], 60% of the total displacement happens

during this rapid phase (1/10 of the movement’s duration).

2.4 The parallels between engineered and plant actuators

In this section a parallel is drawn between the actuation of plant and engineered

mechanisms. The full extent of the comparison between kinematic amplification strategies

in plant and engineering can be found in the full version of this study [91]. Manmade,

automatically moving devices usually include a deployment system and an actuating

element, which converts an input energy into mechanical energy. Those systems are often

discrete and composed of rigid moving parts (e.g. pantograph) [134]. But in some cases,

the actuator can be embedded into the system: a subcategory of movements called

41

morphings presents the characteristics of continuous change of shape not by involving

discrete part movements but with deformation of the entire structure upon actuation [29].

Various transduction mechanisms based on physical phenomena, such as

electromagnetism, combustion, thermal expansion, static electricity, Joule effect, fluid

pressure, or phase changes, can be at the origin of the actuation [134]. A non-exhaustive

list of actuator types used in engineering or research application includes: electrical motors

(electromagnetic actuators), combustion motors, pneumatic actuators, hydraulic actuators,

electrostatic actuators, thermal actuators (thermal expansion and bimetal effect), electro-

thermo-mechanical actuators (thermal actuators heated by Joule effect), piezoelectric

actuators, shape memory alloys, dielectric elastomer actuators, magnetostrictive actuators,

chemical actuators and capillary force actuators [161].

Features of an actuator, depending on the physical phenomenon involved and on the

materials used, include stroke / rotational amplitude, torque / force, blocking force, speed

of response, operation speed, precision, power consumption, compactness, energy density,

silent operation, and lifetime span [134, 162]. This section focuses on actuator

displacement characteristics and in particular stroke or rotational amplitude. In order to

study the parallels between engineering device movements and plant movements, it is

crucial to select relevant actuation typologies. Combustion motors and electrical motors

rely on wheels and rotation axes: these elements are purely human engineered designs and

are not present under this specific form in the plant world. For this reason, traditional

motors will not be discussed here. The discussed actuators operate without complex

mechanisms and transmission systems including hinges, articulations and gears, which

again are not present in plants. Physical phenomena triggering the actuation are closely

42

linked to the properties of the materials constituting these actuators. Such phenomena

include electric conductivity, thermal expansion, fluid compressibility and thermo-

mechanical behavior. Some morphing systems already implement merged structure and

actuation. The seamless embedding of the actuation in the morphing structure is one of the

characteristics of plants movements.

Such engineered structures are very common at the micrometer scale, in micro-electro-

mechanical structures (MEMS) because it is difficult to manufacture hinges, articulations

and gears, and even impossible to manufacture electric motors at such small sizes [163].

One aspect shared by MEMS and actuators in nature is their frequent ability to both sense

and actuate, because the physical phenomena behind actuation often have a direct and a

converse effect: the Peltier effect and the Seebeck effect, the direct and converse

piezoelectric effect. The engineered device thus becomes simultaneously an actuator, a

structure and a sensor. Tremendous progress has been made in this area in the last decades,

along with the advances in materials science and often with bio-inspired principles [164].

However, microscale structures cannot always be reproduced at a mesoscopic or

macroscopic scale. In the case of a simple beam, the strength varies linearly with a linear

dimension while the volume has a cubic variation with this same dimension. As a

consequence, insects are often much stronger than bigger animals in terms of force per

body weight [163]. The comparison between plants and engineered smart structures also

relies on the similarity of the materials involved. Ashby’s material property charts [165]

particularly illustrate the proximity of the Elastic modulus versus Density values for natural

materials, and for polymers and composites, which are frequently used in morphing

structures (Figure 2.12).

43

Figure 2.12: Material property chart (Elastic modulus versus density), reproduced from

[165] with permission from Elsevier

Most of the small- and large-scale actuators rely on the same physical phenomena and have

a common characteristic; they generate small movements, and the output displacement –

or the output rotation – sometimes needs to be amplified. Various movement-amplification

strategies can thus be identified by reviewing different scales (from the micrometer to the

meter) and different material properties.

44

2.5 Conclusion

The complete study of plant movements involves a breadth of biophysical and

biochemistry sciences. In plant mechanics, kinematics eminate from the entanglement of

material and structure at the tissue scale. From the hierarchical assemblage of cells stems

an extremely evolved active, living material capable of sensing, actuating and carrying

loads.

The mechanism implemented in plants structures have been classified in five distinct

categories: differential expansion through controlled, localized swelling and shrinking of

active cells, differential expansion of a highly anisotropy layered body under re-hydration,

fast release of stored elastic energy, rigid-bar and beam-buckling mechanisms and finally

the coupling of geometry and shell mechanics. Plants implement purely mechanical

strategies to amplify the movements. First and foremost, differential expansion within a

layered-solid is the most common strategy for movements of plants.

In active movements, the organ called pulvinus performs differential expansion based on

turgor pressure variations. In its most advanced form, the pulvinus achieves spherical, two

degree-of-freedom, movements. In simpler variations, pulvinus movements are planar, one

degree-of-freedom. In passive movements of fruits and seeds, anisotropic expansion of dry

cells caused by external environment parameters such as diurnal cycle hygrometric

variations generate large plant displacements. Mechanical displacement-amplification

strategies such as lever action and torsional buckling enable flower passive movements.

Pollinators actuate the mechanism by their weight or attempt to collect nectar. The

actuation in passive movements therefore comes at no metabolic cost for the plant. In

45

functions requiring speed, such as predatory nutrition, pollination by deception or release

of seeds, elastic instabilities and geometric effects allow the fast release of energy powering

the movement. The compliance of the material allows the large variations of shape between

the different states of the mechanism.

The mechanics of carnivorous plants have been extensively studied and their movements

documented. However, many other plant movements are still unexplored. For example,

only a limited number of fast flower movements have been mechanically characterized.

The Australian genus Pterostylis, known for its hinged column/labellum, has an estimated

400 species alone [127] none of which have been studied for their mechanics. Countless

other genera with hinged columns/labella have been reported [166-171]. The mechanical

characterization of those movements certainly constitutes a large source of unexplored

information about plant movements.

Actuators for manmade mechanisms rely on the material properties and geometry of their

constituents for actuation. Motorless, hingeless and wheel-less actuation is based on

material deformations (expansion, incompressibility, etc.) resulting in small amplitude

displacements. These mechanisms therefore frequently resort to a movement amplification,

and parallels can be drawn with plant amplification strategies presented in this chapter. For

manufacturing reasons, these deformation-based actuation patterns are widespread at

micrometer scale, which is why various examples are drawn from the field of micro-electro

mechanical structures. One can distinguish four displacement amplification strategies: a

geometrical strategy, controlling the material properties like stiffness, anisotropy or

thermal expansion coefficients, the swelling and shrinkage, and the storage of energy.

46

These strategies can be found in plant movements. Plants and manmade mechanisms often

implement various displacement strategies at the same time, and each one can have its own

function for the output movement. The combination of amplification strategies can also

allow other mechanical features such as for example the stability of some zero-energy

positions. These couplings of strategies have been assembled into table 2.1. Some

kinematic amplification ratios (KAR) have been cited in this chapter, although a lack of

quantitative data has been remarked in the literature, especially in the biological field:

either the papers studied by the authors focused on other indicators describing the

performance of the deployable elements, or the kinematic data was not available at all.

Nevertheless, the efficiency of the bilayer amplification can be highlighted in relation to

other manmade mechanisms: it reaches a KAR of 140, an order of magnitude higher than

the other listed strategies. This provides a parallel between engineered moving devices and

moving plants: the bilayer effect caused by the differential expansion of cells or by the cell

anisotropy in hygroscopic tissues is also the most widespread amplification mechanism in

plants, probably because it is the most efficient.

47

Couplings of strategies Variation of material

properties Geometrical strategies Energy storage

Fluid incompressibility /

Swelling and Shrinkage

Variation of material properties

Aldrovanda vesiculosa

[46]: Turgor bilayer

action coupled with

double curved shell

bending

Explosive seed

dispersal in Impatiens

[120]: material

incompressibility and

prestressed beams

Fruits and seeds detached

from the vascular system:

increase of material volume

by hygroscopic material

directed by strong anisotropy

and geometry effects [33,

104]

Geometrical strategies

Flectofin [133]:

variation of the

composite properties

and amplification

through shape

changes.

Dionaea muscipula

[46]: Elastic

instabilities and snap-

through mechanisms

Fast movement of Stylidium

[146]:

Geometric effect of

progressive reversal of

transversal curvature in a

longitudinally curved beam

Energy storage Bistable steel tape

[172]

Snap-through

mechanism [173]:

amplification through

shape changes &

energy storage

Pulvinus [95]: actuator

implementing bilayer effect

for two degrees-of-freedom

Fluid incompressibility /

Swelling and Shrinkage

Hygroscopic

pavilion [174]:

bilayer mechanism

& hygroscopic

swelling

Array of inflatable cells

connected by valves

[175]

Fluidic origami [176]

Table 2.1 Plant and engineering mechanisms classified with the strategies identified in [91]

48

The plant’s microscale mechanical properties (mainly a stiff cell wall and the turgor

pressure) and the entanglement of appropriate structural organizations at each scale make

of plant tissues a genuine “smart material”. They account for a variety of advanced

structural features, some of them being usually avoided in engineering. Mechanical

couplings like torsional buckling, and elastic instabilities like the snap-through

phenomenon, testify to the great diversity of mechanisms that can be encountered in the

plant’s world. Their movements come with different speeds, kinematics, functions (e.g.

sun-tracking, growth, reproduction, predation), and can be passive or active with numerous

different triggers.

Most of the mechanical devices presented in [91] are not directly bio-inspired, but many

parallels have been drawn with plant movements. The systems are based on multi-

functionality and an optimized material hierarchy: they work with what may be called

“deformation-based” actuation patterns. These enable precise, reliable, repeatable and low-

energy movements. Even though there is still a lot to understand from the mechanics of

plant movements, inspiration from the identified plant strategies combined with new

manufacturing techniques, powerful modelling tools and advances in materials science can

lead to promising applications in industrial fields as different as microscale medicine and

construction.

49

Chapter 3:

Physical limits of compliance in the scaling of thin

shells

Some sections of this chapter are being prepared for publication at the time of writing, but

edits have been made for continuity within this dissertation.

Charpentier, V., Adriaenssens, S., Physical limits of compliance in the scaling of thin

shells. (2019). Under preparation.

3.1 Introduction

A compliant shell is a single continuous surface that can be deformed elastically to replace

several rigid parts for an equivalent motion, as described in chapters 1 and 2. In the physical

realm, thin shells whether rigid or compliant span over 10 orders of magnitude. They are

structures that make the most use of their material properties and their curved geometry.

Thin shells can be used to describe the shape transitions of viruses [177], they describe the

mechanical behavior of red blood cells [178] and are central mechanisms to some of the

fastest repeatable plant movements as seen in chapter 2. Thin shells have been identified

50

as one of the 5 mechanisms used by plants to amplify the water transport actuation (chapter

2). Hence, the way shells are implemented in biology can inform the design of mechanisms.

To be able to emulate and then master the large deformation of shell structures, there is a

need to assess whether the movements of compliant plant and micro-scale shells can be

replicated. Determining the limits of scalability of shell structures would allow designers

to expand the use of those structures, using the geometry of small-scale thin shells for large-

scale applications. The main results of this chapter are that thin shells have a similar

mechanical behavior across length scales. However, as the characteristic dimension of the

shells increases, the gravitational force tends to limit their use as compliant structures by

creating hard to overcome mechanical challenges. The limit of scalability below which

independence from gravity’s pull is guaranteed, is found to be around 0.1m.

We use two non-dimensional indicators of the mechanical properties of isotropic thin shell

structures to determine their scalability limits. The Föppl-von-Kármán number [179], 𝛾𝐹𝑣𝐾,

is an indicator of the type of deformations found in thin shells. This number is the ratio of

stretching elastic energy to bending elastic energy of a thin shell. A high value of 𝛾𝐹𝑣𝐾

indicates that the structure has a propensity to deform in bending rather than stretching.

The second number, introduced in this study, is the gravity impact (Gi) number. It is the

ratio of the elastogravity length scale [180] to the characteristic dimension of the shell. The

length scale determines the overhang scale at which bending deformations due to gravity

tend to appear in shells.

In this chapter, 5 types of thin shells are surveyed across 10 orders-of-magnitude of

characteristic dimensions. The 64 shells observed in the study and their dimensions are

51

recorded in Annex A. They were selected in literature on the criterion that they had been

the subject of a mechanical study. The five types of shell are:

• Engineered rigid shells [181]. 25 large scale reinforced concrete (high Young’s

modulus) thin shells used in buildings and architecture, their shape is fixed and can

carry external applied loads. Their characteristic dimension (R) is in the

[6 100𝑚; 8 101𝑚] range, while their thickness H is in the [5 10−2𝑚; 4 10−1𝑚]

range

• Engineered compliant shells [10, 133, 182-192]. 18 shells designed for use as

mechanisms, they are very flexible. Materials are varied but all have high Young’s

modulii. 𝑅 in [2 10−2𝑚; 8 100𝑚] and 𝐻 in [1.2 10−4𝑚; 9 10−4𝑚]

• Plant compliant shells [46, 146, 193]. 8 plant structures that can be described as

thin shells and exhibit fast and repeated motions. The material is a living tissue of

low Young’s modulus (~ 106 𝑁/𝑚2). 𝑅 in [1.5 10−4𝑚; 1 10−2𝑚] and

𝐻 in [3 10−5𝑚; 4 10−4𝑚]

• Avian egg rigid shells [194-196]. 8 rigid bird egg shells. The geometry is rigid and

the material is carbone silicate of various mechanical properties detailed in [196].

𝑅 in [3 10−2𝑚; 1.55 10−1𝑚] and 𝐻 in [2.2 10−4𝑚; 2.55 10−3𝑚]

• Micro-scale compliant shells [177, 179, 197-200]. 5 types of shells from red blood

cell to virus. They have been described mechanically as a shell and deform

significantly in operation. They are highly flexible. 𝑅 in [2 10−8𝑚; 5 10−4𝑚] and

𝐻 in [2 10−9𝑚; 1 10−6𝑚]

52

3.2 Non-dimensional numbers characterizing the mechanical behavior of shells

In this study the mechanical characteristics of rigid and compliant thin shells are compared.

A total of 64 thin shells are compared, of which 33 are rigid shells and 31 are compliant

shell of characteristic dimension’s order of magnitude smaller than 101 m. The mechanical

behavior of those shells can be described by the Föppl-von-Kármán (𝛾𝐹𝑣𝐾) and the gravity

impact (Gi) numbers. 𝛾𝐹𝑣𝐾 quantifies the types of deformation that will dominate the

behavior of a thin shell. The Gi number, introduced for the first time in this chapter,

characterizes the influence of the gravitational force on the shell. The thin shells are

considered isotropic elastic in this chapter. This assumption will be discussed in Section

3.4.

3.2.1 Quantification of bending vs. stretching deformation

Shells used as mechanisms rely on the property of very thin curved bodies to deform

without distortion of their surface metric, i.e. without stretching. This type of deformation

known as inextensional bending, or isometric bending [201-203] tends to minimize the

strain energy for thin shell deformations because it does not involve stretching the material.

The search for minimal elastic strain energy is the leading objective of compliant

mechanisms. Smooth deformations without stretching are geometrically possible if the

shell has free edges and only exceptionally if the surface is closed [202]. For thin shells,

the strain energy density 𝑊 includes stretching and bending (bending and torsion, equation

3.1). The stretching energy density 𝑊𝑠𝑡𝑟𝑒𝑡𝑐ℎ𝑖𝑛𝑔 is proportional to the thickness H, while the

bending energy density 𝑊𝑏𝑒𝑛𝑑𝑖𝑛𝑔 is proportional to the cube of the thickness H3. For equal

53

energy levels, bending deformations can be much larger than stretching deformation.

Therefore, bending allows the structure to deform with less effect on the overall strain

energy compared to stretching. Since isotropic thin shells are considered, the general form

for the surface strain energy density 𝑊 is given by the following equations [204]:

𝑊 = 𝑊𝑠𝑡𝑟𝑒𝑐ℎ𝑖𝑛𝑔 + 𝑊𝑏𝑒𝑛𝑑𝑖𝑛𝑔 (3.1)

With the stretching and bending energy scaling as

𝑊𝑠𝑡𝑟𝑒𝑡𝑐ℎ𝑖𝑛𝑔 ~ 𝑌𝐻

(1 − 𝜈2) 𝜖2

(3.2)

𝑊𝑏𝑒𝑛𝑑𝑖𝑛𝑔 ~ 𝑌𝐻3

12(1 − 𝜈2) 𝜅2

(3.3)

, where 𝜖 is the average in-plane strain (unitless), 𝜅 is the average variations of curvature,

Y the Young’s modulus, 𝜈 the Poisson’s ratio and H the thickness of the shell.

In order to measure and compare the propensity of bending-only-deformations in thin

shells, the dimensionless Föppl-von-Kármán number [179] is used. This number measures

the ratio of stretching to bending strain energy in a shell.

𝛾𝐹𝑣𝐾 =𝑌𝐻𝑅2

𝐷

(3.4)

, with 𝑅 the characteristic length of the thin shell (in general of same order of magnitude

as the principal curvature radii of the shell [205]) and 𝐷 the bending modulus (also named

flexural stiffness) of the shell,

𝐷 =𝑌𝐻3

12(1 − 𝜈2)

such that after simplifications the Föppl-van-Kármán number becomes proportional to

𝑅2/𝐻2:

54

𝛾𝐹𝑣𝐾 =𝑌𝐻𝑅2

𝐷

= 12(1 − 𝜈2)𝑅2

𝐻2

(3.5)

This number predicts the type of deformation a shell will experience. Very large values of

𝛾𝐹𝑣𝐾 indicate that the shell behaves similarly to a membrane. It will accommodate elastic

strain by wrinkling and if very thin, crumpling [206]. The shells with high values of 𝛾𝐹𝑣𝐾

will present large bending forces and low stretching forces. Lower values of 𝛾𝐹𝑣𝐾

correspond to thicker shells that have a very high bending stiffness. Such shells will have

both bending and stretching deformations and require large applied loading to be deformed.

The ideal behavior for a thin shell mechanism is characterized by a low actuation applied

load, large bending deformations and preservation of the smoothness of the surface (i.e. no

crumpling). This ideal behavior can be found within a range of 𝛾𝐹𝑣𝐾 values that will be

described in this study. The upper value of the Föppl-von-Kármán number is found to be

𝛾𝐹𝑣𝐾 ≈ 1014 for a 200𝜇𝑚 square graphene sheet [207]. Graphene is a very flexible, one-

atom thick membrane with high in plane Young’s modulus (𝑌2𝐷𝑔𝑟𝑎𝑝ℎ𝑒𝑛𝑒= 500 𝐺𝑃𝑎 ).

Since there is nothing thinner than a single layer of atoms, graphene constitutes the limit

of physically feasible structures. In comparison and for reference, the Föppl-von-Kármán

number for a piece of paper is 𝛾𝐹𝑣𝐾 ≈ 106 [207]. The graphene sheet is prone to wrinkling

and crumpling (membrane behavior). A piece of paper can act as a very thin shell.

55

3.2.2 Influence of gravity body forces on shell internal forces

We hypothesize that the compliant behavior of shells can be scaled across several orders

of magnitude characteristic dimension and thickness. Gravitational pull needs to be

considered in the analysis of compliant thin shells as a limiting factor for movements. A

non-dimensional number, called the gravity impact (Gi) number is introduced for the first

time to quantify this impact on the shell. The Gi number is defined as the ratio of the

elastogravity length scale 𝑙𝑒𝑔 as defined in [180] to the characteristic dimension of the shell

𝑅.

The gravitational potential energy density (𝑊𝑔𝑟𝑎𝑣𝑖𝑡𝑦) scales as

𝑊𝑔𝑟𝑎𝑣𝑖𝑡𝑦 ~ 𝑔𝜌𝛿2 (3.6)

With g the acceleration of gravity, 𝜌 the volumetric mass density of the material and 𝛿 the

deformation due to gravity at a given point. From a dimensional point of view, the variation

of average curvature 𝜅 can be expressed as a function of 𝛿 as 𝜅 ~ 𝛿/𝑅2. The gravitational

pull will cause the shell to bend when the bending energy and the gravitational potential

energy are of the same order of magnitude, 𝑊𝑏𝑒𝑛𝑑𝑖𝑛𝑔 ≈ 𝑊𝑔𝑟𝑎𝑣𝑖𝑡𝑦. This situation occurs for

𝑅 ~ 𝑙𝑒𝑔. Equating equations (3.3) and (3.6) yield

𝑙𝑒𝑔 ~ (𝐷

𝑔𝜌)

1/4

Therefore, the nondimensional Gi number for a thin shell is

𝐺𝑖 =𝑙𝑒𝑔

𝑅= (

𝐷

𝑔𝜌𝑅4)

1/4

(3.7)

Therefore, if Gi is more than unity, the characteristic dimension of the thin shell will be

smaller than the length at which the shell deforms: the shell is unconstrained by gravity.

56

The nondimensional Gi number determines the propensity of a compliant shell to be

affected by the gravitational pull as a function of its scale. Values of Gi lower than unity

indicate that gravitational forces will exert a large influence on the shell behavior. In

opposition Gi values over one, signal the gravitational forces will not be the leading driving

force in the deformation. As discussed in Section 3.2.1, thin shells with low bending energy

density will be more subject to bending deformation. They are ideal candidates for large

deformations mechanisms granted that 1) their in-plane deformation be low enough for the

elastic material to sustain them and 2) the bending forces creating the motion of the

mechanism can overcome gravity forces. The gravitational pull increases as the

characteristic dimension of the shell increases. Compliant thin shells of large dimensions

are rare but there are many examples of such shells where the characteristic dimension is

in the order-of-magnitude of 100m or below. Therefore, the Gi number is used in this study

to detect and highlight the scaling limits of compliant thin shell mechanisms.

3.3 Results

3.3.1 Thin shells have similar 𝛾𝐹𝑣𝐾 values across scales

The 64 thin shells included in the study are plotted by thickness and characteristic

dimension in Figure 3.1. Thin shells are defined by the ratio of characteristic dimension R

to thickness H. In this study the ratio R/H for a thin shell is [208]

20 ≤ 𝑅

𝐻≤ 100 000

(3.8)

57

In comparison, thick shells have a larger ratio [208]

8 ≤ 𝑅

𝐻𝑡ℎ𝑖𝑐𝑘𝑆ℎ𝑒𝑙𝑙 ≤ 20

(3.9)

All rigid thin shells and compliant engineered thin shells in this study fall within the range

of size to thickness ratios defined in equation (3.8) and (3.9) (Figure 3.1). 3d solids are

structures that cannot be described as having two spatial dimensions much larger than the

third one, they are not shells and the equations cited above do not apply. Membranes are

extremely thin shells; therefore, they are too thin to have any bending stiffness and only

involve stretching forces. They cannot work in compression. They are out of the scope of

this study since they are not self-supporting. The majority (95%) of the 64 thin shells

recorded has values of 𝛾𝐹𝑣𝐾 between 103 and 108 (Figure 3.1 and Figure 3.2). Lower

values of 𝛾𝐹𝑣𝐾 correspond to instances of compliant plant shells. Those same instances fall

out of the range of R/H ratios defined in equation 3.8. 𝛾𝐹𝑣𝐾 describes the type of

deformation (stretching or bending) that control the deformed state of the shell. Being a

non-dimensional number, it applies to any shell, independent of the magnitude of its

characteristic dimension. As such it allows to compare the controlling deformation modes

for shells across different scales. Thin shells have similar values of 𝛾𝐹𝑣𝐾 despite having up

to 10 orders of magnitude between their characteristic dimensions.

For example, the Algeciras Market Hall reinforced concrete shell [181] has similar

mechanical characteristics than those of a red blood cell as described by 𝛾𝐹𝑣𝐾~105. For

both shells, this high value of 𝛾𝐹𝑣𝐾 indicates a high in-plane stiffness compared to the out-

of-plane bending stiffness. Therefore, the likely deformation mode for both structures will

be bending deformations.

58

The average value of 𝛾𝐹𝑣𝐾 is 3.95 × 106 for rigid engineered shells (indicated in purple

in figure 3.1), 1.33 × 107 for compliant engineered shells (red), 1.98 × 105 for rigid egg

shells (orange), 3.84 × 103 for plant compliant shells (yellow) and 2.54 × 104 for

compliant micro-scale shells (green). These values indicate a mechanical behavior

dominated by bending deformation for both rigid and compliant shells. However, despite

being higher than those values are lower than for the engineered shells by about 103.

Overall since thin shells have R/H ratios in the [20; 100 000] range, their 𝛾𝐹𝑣𝐾 values are

also contained by lower (𝛾𝐹𝑣𝐾 ~103) and upper bounds (𝛾𝐹𝑣𝐾 ~108). This observation

indicate that thin shells – whether they are engineered rigid or compliant, plant compliant,

micro scale compliant or egg rigid- exhibit similar mechanical behavior dominated by

bending deformations across scales.

59

Figure 3.1 Geometric properties and 𝛾𝐹𝑣𝐾 values for rigid and compliant thin shells, plant

compliant thin shells and compliant micro-scale shells. The scale for both axes is

logarithmic.

60

Figure 3.2 Föppl-von-Kármán number, 𝛾𝐹𝑣𝐾, in thin shells as a function of the

characteristic dimension. The scale for both axes is logarithmic.

61

3.3.2 Difference in gravitational driving force across scales

Thickness and characteristic dimension are related by the ratio expressed in Section 3.1.

Therefore, the characteristic dimension is taken as the reference indicator of geometry

going forward. The relationship between the Gi number and geometry is shown in Figure

3.3. It appears that rigid engineered thin shells have the largest values of Gi, while micro-

scale compliant shells have the lowest values.

As expected from equation (3.7), the gravitational pull is higher as the scale of the shell

increases (Figure 3.3). This is shown by thin shell of larger characteristic dimensions

having lower values of Gi such as façade shading shells [10, 183, 188]. Shells with their

characteristic dimension lower than 0.1m tend to have 𝐺𝑖 > 1. For these shells, large

deformation caused by gravity does not occur. The relationship 𝐺𝑖 > 1 only occurs for 1/3

of compliant engineered shells, which means that most engineered shells must deal with

the influence of gravity. All the rigid engineered shells have their elastogravity length scale

lower than their characteristic dimension. This indicates that the gravitational forces (self-

weight) dominate the elastic bending forces in the thin shells at that scale. The average

value of Gi is 0.109 for rigid engineered shells, 0.610 for compliant engineered shells,

2.465 for plant complaint shells, 0.822 for the egg shells and 7.739 for the micro-scale

complaint shells.

62

Figure 3.3 Gravitational force density impact in thin shells as a function of the

characteristic dimension. The scale for both axes is logarithmic. The horizontal dotted

line indicates values Gi=1 for which the gravitational force becomes predominant in the

equilibrium of the shell. The red dotted line at R=0.1m represent the approximate limit at

which thin shells start to be constrained by gravity.

63

There is not a clear divide created by Gi between compliant and rigid thin shells. Some

engineered complaint thin shells are used as mechanisms but have lower Gi than the rigid

thin shells. A high value of Gi can also originate from having a very thick shell. In this

case, the corresponding 𝛾𝐹𝑣𝐾 will be low.

The plant compliant shells have relatively high values of Gi. The structure does not sag

under the influence of gravity. The larger plants have Gi values comparable to rigid

engineered shells. For the living tissues, the ratio of volumetric mass density to Young’s

modulus is ~103 times lower than that for engineered shells, which could explain some of

the low values of Gi despite small characteristic dimensions.

3.4 Discussion

From our analysis, gravity has a large influence on rigid and compliant shells. Compliant

shells have unsupported overhangs that are large compared to their characteristic

dimension. For that reason and due to their low Gi value, large scale engineered shells do

not seem suited for compliant applications under gravity loads.

We have shown that thin shells have a propensity to deform in bending rather than in

stretching across scales. Most thin shells observed have values of the Föppl-von-Kármán

number between 103 and 108. This non-dimensional number is significant because it

allows to unify the behavior of shell across scales. This conclusion validates the bio-

inspiration approach that consists of observing the geometry of a plant shell (e.g. the ones

of the Aldrovanda Vesiculosa) or micro-scale shell (e.g. blood cell geometry) and scaling

it up to engineered scales. If the ratio of characteristic dimension R over thickness H is kept

64

high, the behavior of the inspired shell mechanism should be the same at the larger

engineered scale as the one at the observed biological scale. The five types of thin shells

(i.e. engineered rigid, engineered compliant, plant compliant, micro-scale compliant and

egg rigid) have instances with 𝛾𝐹𝑣𝐾 in the range 104 to 105. This exemplifies the trans-

scale nature of shell mechanisms. They can have a similar mechanical behavior dominated

by bending deformation across 10 orders of magnitude of their characteristic dimension

(Figure 3.4).

65

Figure 3.4. Classification of compliant and rigid thin shells. The nondimensional

gravitational force density Gi is plotted as a function of the Föppl-von-Kármán number

𝛾𝐹𝑣𝐾. The dotted line indicates values Gi=1 for which the gravitational force becomes

predominant in the equilibrium of the shell.

66

Very rigid shells can be found across all scales. Rigid thin shell are stiff systems in part

due to their curvature. Engineered rigid shells are very stiff structures. At a smaller scale

shells can also be very rigid, a good example is the avian egg. The ultimate axial

compression force of an ostrich egg (~150mm tall, ~65 mm radius and 2.5mm thick) can

be up to 5000N for instance [196]. As demonstrated by this example, a thin shell with a

low impact of gravity can still be very rigid. One must exert caution when scaling up a

shell. A closed shell, such as an egg, can rest on a plane without being damaged at a small

scale. However, when scaled up the self-weight of the structure increases. Under the same

support conditions the structure could be subject to localized deformation such as buckling,

as described in [203]. An appropriate set of boundary conditions must be chosen when

scaling up a shell (whether rigid or compliant).

In contrast to rigid shells, compliant shells used for their flexibility are not found across all

scales. As shown in Figure 3.1 and 3.2, the thin shells used for flexibility have in general

a maximum characteristic dimension of ~1m. That is the scale of the unsupported moving

section of a shell for the largest cases of shell compliance. This study excluded several

large scale compliant shells [209] for space applications since gravity is limited around

earth’s orbit. In general, compliant shells are very thin to maximize the bending

deformation and lower the stretching deformation. As a result, they are more prone to

sagging under gravity since the bending rigidity D is proportional to shell thickness

cubed, 𝐻3 . Consequently, tradeoffs need to be made between the amount of obtainable

bending deformation and sagging in the applications of compliant shells as the scale of the

shell increases. Specifically, when a shell is used as a mechanism, the main function of the

shell should not be impaired by gravity.

67

Finally, compliant shells found in plants are made from living tissues, a multileveled

arrangement of the basic building blocks: the cells. In this “material”, the transport of water

generated by electro-chemical reactions increases the water pressure in select part of the

tissue, thus generating actuation. Plant tissues are a self-actuating material. Plant

mechanisms were classified as thick shell or almost 3D solids. This classification is due to

their material. Their geometry shows that the elastogravity length scale is large compared

to their characteristic dimension (Figure 3.4). In the genus Stylidium, the characteristic

dimension of the mechanism is 4.3 times larger than the elastogravity length scale, which

indicates a movement independent of gravity by this plant shell. In general, plants can

move without having the deformed geometry influenced too much by gravity, i.e. the

orientation of their mobile parts does not obstruct or favor the shell mechanism.

3.5 Conclusion

Upscaling rigid and compliant shells means they cannot be abstracted from the pull of

gravity. Large-span rigid engineered shells use engineered materials with high Young’s

moduli and are designed to have a fixed shape that minimizes bending stresses and can

thus be made very thin. In contrast, compliant shells must be able to perform a repeated

mechanical deformation reliably at a low actuating cost. The scale at which shells’

compliant deformations start to be constrained by gravity is 𝑅 ~ 0.1 𝑚. Below that scale,

shells that experience compliance tend to have high Gi values (~1), which indicates the

gravity-independent operation. Compliant shells of a larger scales (𝑅 > 0.1 𝑚, Figure 3.3)

have elastogravity length scales smaller than their characteristic dimensions, i.e. Gi < 1.

68

They are prone to self-weight deformation under gravity load. Therefore, the maximum

size of a compliant shell depends on the application it is intended for. For instance, the

adaptive air inlet for aeronautic applications described in [187] must be able to function

under any orientation of the airplane. In contrast, a compliant shell mechanism for adaptive

shading of buildings does not have the same constraints and can therefore be up scaled to

a larger size [10, 133, 183]. There are strategies that can be used for compliant thin shells

to circumvent 𝐺𝑖 < 1 while having a high 𝛾𝐹𝑣𝐾. The strategies are the following:

• The mobile part of the shell should possess enough stiffness to be

cantilevered. In most compliant shell building shading systems [10, 133,

183] this stiffness is provided by curvature and built-up stresses.

• The mechanism should be oriented to limit the increase of cantilevered

length during the movement. For example, the façade of the Yoesu Expo

2012 Pavilion was designed so that the flexible shell elements do not create

large overhangs during the out-of-plane buckling deformation [188]. The

longest elements are 8m tall and still able to be elastically deformed

repeatedly.

• The bending deformation of thin shells can be predicted by studying the

possible isometric deformation of their geometry [202]. The deformation of

very curved surfaces could lead to mechanisms being able to withstand

gravity better due to their doubly-curved geometry [185].

• The final strategy to create large scale thin shell mechanisms is to operate

in outer space. The behavior of shells is similar across scales. Bending

deformation modes dominate stretching modes when shells are thin enough.

69

Being able to remove gravity forces could lead to large shells being used as

compliant mechanism.

70

Chapter 4:

From isometries to mechanisms

Some sections of this chapter are being prepared for publication at the time of writing, but

edits have been made for continuity within this dissertation.

Charpentier, V., Baverel, O., Adriaenssens, S., From isometries to mechanisms. (2019).

Under preparation.

4.1 Introduction

Compliant shells create movement by the elastic deformation of their continuum. They

have gained appreciation due to instances of common rigid-bar mechanisms adapted into

the compliant domain [210]. These instances include a micro-gripper with no mechanical

hinges [211], staplers made from one piece instead of a dozen [212] or even a new

methodology to turn any existing bar-hinges mechanism into a compliant one [213]. These

systems are all adaptations of rigid bar mechanisms. They localize their elastic deformation

at the spots of previous mechanical hinges, their compliance is lumped not distributed. As

71

noted in [214], “distributed compliance is essential for building flexible machines that have

to do heavy work”. Lumped compliance systems do not take advantage of the full range of

mechanical opportunities that compliance has to offer. The benefit of distributed

compliance is supported by our observations of plant compliant continua discussed in

chapter 2. Plant structures can alter their entire geometry in real time. This enables

biological machines to operate with efficiency. Whether they are biological or engineered,

thin shells have a tendency to accommodate strain by bending and by altering their

geometry. They are the perfect structural typology candidate to exhibit “distributed

compliance”.

Thin shells have already been used for their flexibility in some engineering applications,

as seen in chapter 3. But the geometries found in these examples often emerge from direct

bio-inspiration or from practical engineering experience. Currently no systematic way

exists to generate geometries of shell mechanisms that take advantage of the natural

propensity of shells to deform in bending and consequently lower the energy needed for

actuation.

The focus of this chapter is the deformation of surfaces called isometric transformations,

i.e. inextensional transformations, and the use of these transformations to identify new thin

shell typologies for compliant mechanisms. The approach presented consists of two steps:

(1) identify eigenfrequencies to single out isometric deformations of the shell and (2) apply

linear actuation to replicate the isometric deformations as buckling modes of the

mechanism. complete a parametric study of geometry on the principle radii of curvature

and length of an elastic compliant shell mechanism to evaluate the influence of geometry

72

on the output displacements. Finally, a kinematic relationship between actuation and

displacement is provided for this example.

4.2 Morphogenesis of compliant shell mechanisms

The focus of this section is on geometric rationality with the purpose to find thin shell

forms that will provide large elastic displacements at a low actuation cost. Due to their

small thickness, shells have similarities with deformation modes of mathematical surfaces.

4.2.1 Detection of isometric modes of transformation using eigenfrequencies

4.2.1.1 Isometric deformations of the middle surface of a shell

The study of thin shell deformations is a question of geometry, more so than for other

structures: a shell can be seen as a mathematical surface that has been given a small

thickness. This mathematical surface defines the middle surface of a shell. In chapter 3, the

behavior of shells was described as favoring bending deformations to stretching

deformations. As seen in chapter 3, the Föppl-von-Karman number of thin shells is in the

range of 103 to 107 which indicates that the behavior of shells is dominated by bending

deformation causing large curvature variations, rather than stretching, which causes in-

plane extensions of the middle surface. As early as 1849, Rev. Jellet mentions the study of

inextensible surfaces as paramount “to determine the possible displacement of a membrane

very slightly extensible and whose thickness is very small compared to its other

dimensions” [215]. But it was really Lord Rayleigh at the end of the 19th century, who used

inextensional deformation of shells to define his theory of sound and vibrations [216-218].

73

Differential geometry for infinitely thin, curved surfaces provides a valuable approach for

the study of compliant thin shells. Specifically, isometric (or length-preserving)

transformations of surfaces are useful to study thin shells as they do not involve any

stretching. A surface deformation is said to be isometric if the length of any arc on the

initial surface is the same as the length of its image on the deformed surface. In this

approach, only smooth transformations are considered, no kinks or wrinkles should appear

on the surface as a result of the transformation.

Let us consider a smooth surface 𝑆 with the set of curvilinear coordinates 𝑢, 𝑣. The metric

properties of the surface, such as length element area element is found using the first

fundamental form of differential geometry. In the curvilinear coordinate system, each point

𝒙 on the surface is defined by its coordinates 𝑢, 𝑣. The length 𝑠(𝑡) of an arc 𝐶 on 𝑆 is given

by Equation 4.1.

𝑠(𝑡) = ∫ √𝑑𝒙

𝑑𝑡.𝑑𝒙

𝑑𝑡 𝑑𝑡

𝑡

𝑡0

(4.1)

With the vectors 𝒙(𝑡) = (𝑎1𝑓1(𝑡), 𝑎2𝑓2(𝑡), 𝑎3𝑓3(𝑡)) defining the arc in 3D in the cartesian

notation, 𝒙′ = 𝑑𝒙/𝑑𝑡 the tangent vector to the arc at 𝑡 and 𝑡0 the initial point of the arc.

Equation 4.1 leads to writing the length element 𝑑𝑠 as a function of 𝑑𝒙 as shown in

Equation 4.2.

𝑑𝑠2 = 𝑑𝒙. 𝑑𝒙 (4.2)

In curvilinear coordinate the position vector is written 𝒙(𝑢, 𝑣) such that with the chain

derivation, the tangent vector 𝒙’ has the form given in equation 4.3.

74

𝒙′ =𝑑𝒙

𝑑𝑡=

𝜕𝒙

𝜕𝑢

𝑑𝑢

𝑑𝑡+

𝜕𝒙

𝜕𝑣

𝑑𝑣

𝑑𝑡= 𝒙𝑢

𝑑𝑢

𝑑𝑡 + 𝒙𝑣

𝑑𝑣

𝑑𝑡 (4.3)

Finally, by combining Equation 4.2 and 4.3,

𝑑𝑠2 = 𝑑𝒙 ∙ 𝑑𝒙 = (𝒙𝑢𝑑𝑢 + 𝒙𝑣𝑑𝑣) ∙ (𝒙𝑢𝑑𝑢 + 𝒙𝑣𝑑𝑣)

= 𝒙𝑢 ∙ 𝒙𝑢 𝑑𝑢2 + 2 𝒙𝑢 ∙ 𝒙𝑣 𝑑𝑢 𝑑𝑣 + 𝒙𝑣 ∙ 𝒙𝑣 𝑑𝑣2

Let’s set

𝐸 = 𝒙𝑢 ∙ 𝒙𝑢 𝐹 = 𝒙𝑢 ∙ 𝒙𝑣 𝐺 = 𝒙𝑣 ∙ 𝒙𝑣

Therefore,

𝑑𝑠2 = 𝐸 𝑑𝑢2 + 2 𝐹 𝑑𝑢 𝑑𝑣 + 𝐺 𝑑𝑣2 (4.4)

The quadratic form 𝑑𝑠2 is called the first fundamental form [219]. It defines the length of

a line element on a surface. In differential geometry, a continuous deformation of a surface

(or a portion of it) is called a bending if this deformation preserves the length of every arc

on the surface, i.e. during this transformation, the first fundamental form remains

unchanged [219]. A bending corresponds to a family of surfaces that can be continuously

deformed while remaining unstretched [220].

For example, such a family is the developable surface family. They are the only surfaces

that can be transformed isometrically into a plane. In that family, a rectangular surface can

be rolled into a cylinder without changing the length of an arc on its surface (Figure 4.1).

Similarly, the plane can also be rolled into a cone (Figure 4.1). This family provides a set

of examples to understand isometric transformations.

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Figure 4.1 Isometric transformations of developable surfaces: two planar surfaces are

rolled into a cone or a cylinder respectively.

The fact that in some transformation, the arc length on the deformed surface remains

constant, has significant implications for the study of surfaces. Most notably, it is the basis

for the Theorema egregium, proven by Gauss [221] in 1828, that states that the Gaussian

curvature (product of the principal curvatures, noted K) only depends on the coefficients

of the first fundamental form. A direct consequence is that surfaces resulting from

isometric transformations have the same Gaussian curvature at each point. Thin shells

cannot abstract themselves completely from in-plane strains since they have a thickness.

Even if the middle surface of the shell deforms according to an isometry, its top and bottom

surfaces will experience some in-plane strains.

The nature of the boundary of mathematical surfaces limits the existence of isometric

transformations of surfaces. Those properties of surfaces inform the design of boundary

conditions of all types of shells. A surface with an empty boundary (i.e. no free edge) is

known as closed. Closed convex surfaces are geometrically rigid [202] : surfaces of

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positive Gaussian curvature without free edges do not admit any isometric transformations.

In structural engineering terms, a shell derived from such a surface will be extremely stiff

(e.g. thin-walled pressure vessel). From a mathematical standpoint, those surfaces cannot

be deformed without being stretched [220]. They are not good candidates for use in

compliant shells. As an example, the ostrich egg, discussed in chapter 3, can withstand a

force of 5000N along its long axis [196]. The closed convex mathematical middle surface

of the egg does not admit any smooth isometric transformation. The consequence for the

physical shell is that any applied distributed external load will generate in-plane stresses,

rather than bending stresses. Since stretching creates high levels of strain energy (see

chapter 3), the external forces needed to cause in-plane deformations will be high. This

translates to a high breaking point for the ostrich egg. In addition, fixed boundary

conditions can provide similar geometric rigidity to a mathematical surface that is not

initially geometrically rigid. If well chosen, the boundary condition of such a surface will

make it geometrically inflexible, i.e. rigid. An unsupported half-spherical surface admits

smooth isometric deformations, but a half-spherical surface fastened along its edge does

not. The only isometric deformations in the context of differential geometry that exist for

a half-sphere will cause kinks in the surface. It then becomes a localized problem.

Examples of non-smooth isometries of surfaces are kinks on a ping pong ball [220], or the

mirroring of part of a half-sphere when pressed against a rigid plane [203].

Finding smooth isometric transformations of middle surface of a thin shell is the objective

of this study. In order to find shell that can be used as mechanisms, surfaces with preferred

deformations must be found that fulfil the kinematic constraint of the mechanism. For

instance, if a lever mechanism needs to be replaced by a compliant shell, a geometry that

77

presents a similar preferred mode of deformation must be found. Mathematically, it is

possible to find the isometric deformations of generic surface for cases called infinitesimal

bendings. Such bendings are normal perturbations of the surface that keep the surface

metric unchanged at the first order. For this type of transformations, the solutions to a

partial differential equation on the normal perturbation yields the possible mathematical

bendings of the surface [220, 222].

An alternative method to identify isometric modes of deformation of surfaces is to evaluate

the lowest natural frequencies of a shell geometry.

We mentioned that inextensional deformations of surfaces were at the root of Lord

Rayleigh’s theory of vibration. In 1888 he writes, “I have applied the theory of bending to

explain the deformation and vibration of thin elastic shells [...]. The validity of this

application depends entirely upon the principle that when the shell is thin enough and is

vibrating in one of the graver possible modes, the middle surface behaves as if it were

inextensible.”[217]. The lower modes of vibrations of shells are showing the inextensible

deformation modes of thin shells. Therefore, an eigenfrequency analysis can be used to

identify the natural frequencies of the structure and by proxy the isometric deformations of

shells.

4.2.1.2 Six common shell geometries

In order to exemplify the search for isometric modes of deformation for thin shells, the

eigenfrequency analysis of six common shell geometries is performed: a rectangular strip

(technically referred to as a plate), a cylindrical strip, a half-sphere, a quarter sphere, a

positive Gaussian curvature section of a quarter of a torus and a negative Gaussian

curvature of a quarter of a torus (Figure 4.2). These six relatively simple shells have been

78

chosen to illustrate that their preferred deformation modes (i.e. low eigenmodes) are

already used in nature and in engineering as the basis for compliant shell mechanisms.

We use ABAQUS [223] finite element commercial software to run an eigenmode analysis

of those thin shells. The analysis is a linear perturbation frequency analysis ran with the

default Lanczos eigensolver. The shells are modeled with standard S4R shell elements and

5 integration points across their section. The material used is PETG (Young’s modulus

5GPa, Poisson’s ratio 0.4 and volumetric mass density 1270 kg/m3). The choice of this

material comes from its use as base material for building prototypes of thin shells using

vacuum forming. The same material as for prototyping is used in order to inform the design

of the shading mechanism presented in chapter 5. The shells presented in Figure 4.2 are

1mm thick and their smallest and largest dimensions are in the 0.1m order of magnitude.

The rectangular and cylindrical strips are both 0.1m wide. They have the same length of

0.31m. This translates to a 0.1m radius for the cylindrical strip (Figure 4.2b). Two spherical

geometries are included, they are both sections of a sphere cut by horizontal plane. One is

a sphere cut in half (Figure 4.2c) and the other is a quarter of a sphere (Figure 4.2d). The

circular free edge of the quarter sphere has the same diameter as the free edge of the half-

sphere. The last two shells are section surfaces from a torus of large radius 0.30m and small

radius 0.07m. The Gaussian curvatures of these surfaces are positive, null and negative

(Figure 4.3), showing that this methodology can be used all types of shell geometries. As

no boundary condition were imposed, the deformed shapes for the natural frequencies of

the shells are uninfluenced by restriction of movement. This approach is essential to search

for favorable deformations that could be adapted into compliant shell mechanisms. In

4.2.1.1, the boundary conditions are shown to have a large influence on the deformability

79

of shell structures. They can drastically reduce the range of bending deformation of a shell

with free edges, to the point that they would be considered equivalent to mathematically

rigid surfaces.

Since only the lower modes of vibration are interesting for the identification of inextensible

deformations, only 10 frequencies are calculated in the analysis. Six of those frequencies

correspond to rigid body motions (all 0 Hz) since the shells are unrestrained in the 6

degrees-of-freedom of 3D space.

Figure 4.2 Six shell geometries used for eigenfrequency analysis.

80

Figure 4.3 Gaussian curvature (noted K) shell geometries used for eigenfrequency

analysis. (a) rectangular strip (b) cylindrical strip (c) half-sphere (d) quarter-sphere (e)

exterior (positive Gaussian curvature) of a quarter of a torus and (f) of a quarter of a torus

4.2.2 Natural frequencies of six shell geometries

Using the analyses described in section 4.2.1.2, eigenmodes studies for the 6 discussed

shell geometries are carried. As the first 6 modes refer to the rigid body motions, only

modes 7 to 10 are displayed for the 6 surfaces in Figure 4.4. Some of those modes display

known patterns of deformations for each of the shell geometries structures.

Most notably, the lowest mode for the rectangular strip (21.6Hz) bends the strip into a

cylindrical shape and the lowest mode for the cylindrical strip (17.3Hz) generates an

unrolling of the strip. The transformation of both shells reproduces the isometric

transformations shown in Figure 4.1, the middle surface of both of those shells is

developable. For each of the rectangular and cylindrical shells, mode 9 (62.0Hz) is a higher

mode of than mode 7. Much like the harmonic modes of vibrations of a rope under tension,

81

mode 9 represents a declination of the deformation occurring in mode 7 but with a shorter

wavelength. Mode 8 (40.6Hz) and 10 (86.4Hz) of the rectangular strip are twisting the

shell. Mode 8 (33.9Hz) of the cylinder is similarly a twisting of the cylinder. A similar

deformation of a prestressed rectangular cylindrical strip in [224] leads to a neutrally stable

shell.

The eigenmodes of the two spherical shells are identical. For both shells, mode 7 and 8 are

identical (Figure 4.4): the spherical shell is folding on itself. Two points of the free edge

are moving towards the center of the sphere while two points are moving outwards. Each

pair of points is located on opposite ends of a diameter of the circular free edge. This motion

is identical to that of deployable shell space reflectors [225], similar to the scaled-up

Aldrovanda vesiculosa mechanism [10] and is identified explicitly as bending modes in

[220]. Modes 9 and 10 of the spherical surfaces have very high frequencies (>100Hz) and

the displacements become more local. For these modes, three points of the free edge are

moving toward the center of the sphere while three other points are moving outward.

The toroidal shell of positive Gaussian curvature has lower modes of deformation than the

negative Gaussian curvature one. Since they are both extracted from the same torus, the

part of the torus with positive Gaussian curvature has a larger surface area. This larger

surface area contributes to making the frequencies lower since for the same material

thickness the shell is larger, hence less stiff. The eigenmodes are similar for both toroidal

shells. Mode 7 consists of a rotation of the crowns of the torus (lines at the top and bottom

of the torus with K=0) by opposite angles. As a result, the longitudinal ends of each shell

rotate (see Figure 4.5 for detail). In mode 9, for both shells one of the small radius of the

torus is reduced while the other one is increased. This motion is asymmetric. Mode 10 is

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above 100Hz for both toroidal shells. This mode involves more localized displacements

than the previous modes for both shells.

Figure 4.4 First 4 non rigid-body eigenmodes of 6 common shell geometries (see Figure

4.2 for details of the surfaces). The color scale (indicative) denotes the magnitude of

displacements (red=high, blue=low).

83

4.2.3 Discussion

The eigenmodes found in section 4.2.2. present the preferred modes of deformation for six

shells of similar largest dimension: a rectangular strip, a cylindrical strip, a half-sphere, a

quarter-sphere and two surfaces extracted from a torus, a positive Gaussian curvature

section and a negative Gaussian curvature section. Four modes were produced for each

structure. From the results of the natural frequency analysis, the higher modes involve more

localized deformations (such as the six moving points on the free edge of the spherical

shells). In addition, some of the lower frequency modes appear to have already been

applied in the engineering of existing compliant shells. In particular, the lower modes of

the spherical shells already appear in [10, 220, 225]. Such a spherical shell however, only

present one mode of deformation. This single mode limits the use of spherical surfaces to

applications demanding opening and closing motion of spheres. The geometric effects of

the rectangular and cylindrical shells are mostly known since those shells are developable

surfaces. In their lowest modes, the rectangular shell tends to roll up into a cylinder and the

cylindrical shell tends to flatten into a plane. Finally, both toroidal shells have diverse mode

shapes. Modes 7, 8 and 9 appear unrelated and involve movements of the entire shell. Mode

10 has a high frequency for both shells.

In order to gain some physical insights into each of the three modes of the toroidal surfaces,

the positive and negative Gaussian curvature surfaces were physically prototyped using the

vacuum forming technique discussed Appendix B (Figure 4.5). In Figure 4.5, each of the

7, 8 and 9 mode is visualized experimentally. This low thickness of the PETG shells (1mm)

facilitates the large deformations of the prototypes. The modes are most accurately

replicated with the negative curvature toroidal shell prototype. These modes appear as

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preferred modes of deformation of the shell and form the basis for the design of a compliant

shell mechanism discussed in section 4.3.

Figure 4.5 Experimental visualization of the eigenmodes for the toroidal K>0 (left) and

K<0 (right) shells.

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4.3 Kinematic study of a compliant shell mechanism

In this section the isometric deformations of the toroidal surface are translated into a shell

mechanism. In order to study the influence of the geometry on the kinematics of the

mechanism, a geometric design space is defined to be explored with finite element

mechanical analysis and external applied actuation forces. The aim of this section is to

show that the deformation of shells can be tailored to convert linear actuation into spherical

two degree-of-freedom motion of the shell.

4.3.1 From surface to mechanism

4.3.1.1 Choice of toroidal shell geometries for two degree-of-freedom mechanism

Knowing what the end use of the mechanism will be is useful to properly translate the

deformation of the chosen surface into a mechanism. In the case of this work, the

application is a solar tracking device for shading purposes. To perform solar tracking, a

mechanism must be able to achieve spherical motions. It must be able to follow the

elevation (up/down movement) and the azimuth (east to west movement) of the sun

throughout the day. The preferred deformation modes of the six shells indicate that, the

toroidal shells and the strip shells have low modes well adapted to this design challenge

(see Figure 4.4). Their two lowest non-rigid modes are very different, they are candidates

for translation into a two degree-of-freedom mechanism. One mode straightens / or rolls

up the shell (mode 7 for the cylindrical and rectangular strips and mode 8 for the toroidal

shells). This movement corresponds to the up and down movements of the sun during the

day. The next mode twists the shell. In strips (mode 8) and in toroidal shells (mode 7) the

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eigenmode produces a rotation of one of the free edges of the shell (see the toroidal

movement in Figure 4.6). This movement corresponds to the azimuth movements of the

sun.

Choosing one geometry over the other four follows four criteria. Together they aim to

assess the difficulty of adaptability of a pure surface to a compliant shell mechanism. Those

criteria are:

1. Complexity of the boundary condition for the number of modes required

2. Degree of geometry modification necessary to perform application

3. Criterion specific to the actuation: ability to actuate the mechanism with linear

actuators

4. Criteria specific to the application: stiffness under self-weight

The difficulty to fulfill each criterion is weighted as 1=low, 2=intermediate and 3=high.

The structure with the lowest total score is selected for use as a mechanism. The evaluation

of these criteria is based on the appreciation of eigenmodes by the designer at this stage of

the study.

Criteria 3 and 4 are specific to the application and actuation method. The actuation method

chosen (criterion 3) is linear actuation. Solar shading involves covering large areas of

glazing. Therefore, independently of which solution is chosen (a single large module per

window or a multitude of small modules per window), self-weight under gravity will

impact the shell (criterion 4). In chapter 3, 0.1m is determined to be the size limit for

independence from gravity. The result of this multi-criteria assessment is reported in Table

4.1.

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The rectangular and cylindrical strips can be fixed along one of their short edges because

those remain straight for mode 7 and 8. For both toroidal shells fixing the short edge would

interfere with the isometric deformations since the edge deforms in both mode 7 and 8.

This implies more difficult boundary conditions. The adaptation needed for each geometry

to perform as a shading device is related to their projected surface area onto a fictitious

window plane. The rectangular strip has a high projected area therefore doesn’t need

adaptation. The three other shells would each need to be modified to perform shading (e.g.

extrusion of an edge or other method of addition of surface area). Due to their double

curvature, the toroidal shells are the simplest to actuate. The linear actuators would have

lever arms to create moments in the shell right from the start of the actuation. Due to their

zero Gaussian curvature, the actuators would have to potentially overcome buckling load

to induce some small curvature before generating large moments (and movements). The

buckling load of a cylindrical shell was reported to be 3 to 5 times higher than any

experimental data suggest [5]. The reason for this discrepancy in results is that an

experiment on a physical shell always involves geometric imperfections, i.e. localized or

global non-zero curvature. The theoretical shell on the other hand is geometrically perfect.

This discrepancy illustrates the large influence of curvature on the generation of large

displacements. Due to its negative Gaussian curvature, the interior section of the torus

(Figure 4.2f) is the least subjected to the impact of gravity. The positive Gaussian curvature

section of the torus is credited with intermediate resistance to gravity. The rectangular and

cylindrical shells are not adapted for large overhang (K=0).

88

Criterion

complexity

of

boundary

conditions

difficulty of

adaptability

for

application

difficulty of

actuation

impact of

gravity Sum

rectangular 1 1 3 3 8

cylindrical 1 2 2 3 8

toroidal K>0 2 2 1 2 7

toroidal K<0 2 2 1 1 6

Table 4.1 Multi-criteria selection of shell geometry for spherical tracking

This multi-criteria analysis leads to select the negative Gaussian curvature toroidal shell

for the spherical tracking mechanism. Specifically, the aim is to harness the preferred

displacements highlighted by the eigenmodes of the negative Gaussian curvature toroidal

shell (Figure 4.6) into a functional two-degree-of-freedom mechanism.

Figure 4.6 (a) Negative Gaussian curvature surface of a quarter torus (b) Mode 7 (35.8

Hz) and (c) Mode 8 (86.7 Hz)

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4.3.1.2 Geometric definition of the mechanism

A torus is defined as a circular tube, by two radii (Figure 4.7). The major radius (𝑅ℎ) is the

distance between the axis of symmetry of the torus (called center of the torus) and the

closest point of the surface, i.e. the radius of the hole. The minor radius (𝑅𝑤) defines the

radius of the tube. The mechanism, identified for a solar tracking device, is a section of the

K<0 region of the surface. This surface possesses a mirror symmetry with the plane

containing the center of the torus and perpendicular to the axis of symmetry of the torus

(Figure 4.7). The size of the mechanism is limited by two parameters: the total width (𝑤)

and the length (𝑙𝑒) of the circular outer edge (Figure 4.7). For the purpose of the solar

tracking device, the width is kept constant at 𝑤 = 10𝑐𝑚, while the length 𝑙𝑒 is a variable

of the geometry. This mechanism’s geometry is cut from this larger torus. A short, curved

stiffening beam-like surface is added along each outer edge of this geometry to prevent

local buckling of the edges due to actuation (Figure 4.8).

Figure 4.7 Mechanism geometry (green) and relationship to torus surface (grey). The

parametrization of the torus surface appears in the perspective (a), cross-section (b) and

right (c) views. The dotted line is the axis of symmetry of the torus.

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Figure 4.8 Geometry and parametrization of the mechanism. The yellow surface is the

pure section of the torus. The stiffening beams are shown in dark grey.

4.3.1.3 Terminology for 3D rotations

The terminology of airplane principal axes is used in the remainder of chapter 4 to describe

the angular motion of the mechanism. The 3 rotations of the 3D space are yaw, pitch and

roll (Figure 4.9). Yaw is the angle that orients the nose of the airplane left or right. Pitch is

the angle that makes the nose of the airplane go up or down. Lastly, roll is the rotation

angle of the plane around its longitudinal axis.

Figure 4.9 Naming conventions for the 3D rotation of an object used in an aircraft.

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The angular motion of the deformed shell is labeled with these angles in Figure 4.10. The

components of the normal vector to the deformed control plane (Figure 4.8) are used to

find the yaw, pitch and roll angles. The angles are measured between the components of

the normal vectors of the deformed and undeformed planes (as seen in Figure 4.10). The

displacements are measured as angles between the normal of two control planes. Each of

those planes contains both ends of the free edge and its middle point. The angles are

measured between the components of the normal vectors of the deformed and undeformed

planes.

Figure 4.10 Use of the naming conventions to describe the deformation of the compliant

mechanism.

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4.3.1.4 Actuation of the mechanism and load cases

The mechanism, presented in 4.3.1.1, is actuated by two linear actuators connected to the

shell in a cross pattern (Figure 4.11). Actuator 1 (pink) and 2 (green) can be activated

simultaneously (symmetric load case) or individually (asymmetric load case). Actuator 1

and 2 are identical in initial length and material. The mechanism is fixed at the support

point, located at the origin (vertex #650). The mechanism translates the displacements of

those actuators into rotational motion via shell buckling.

The assumption is made that the actuators are shape memory alloy wires. This type of wires

has been available since the 1950’s and have shown potential for actuating mechanisms

[226]. In wire form, the material contracts by 6 to 10% when heated. After cooling down,

the wire needs to be stretched to regain its initial length. The mechanical characterization

of the material is beyond the scope of this thesis. Extensive modeling of the shape memory

alloy wire material model has been done in [227].

Figure 4.11 Detail of the actuation and support condition of the mechanism.

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The two actuators are controlled independently, and each actuator can be contracted

individually. On each outer edge, the distance between the nodes connecting the actuators

to the shell is 3cm. This distance was selected after a sensitivity study. The further apart

the cables are, the greater the output displacement. However, there is the physical issue

that when the cables’ connecting points are too far apart along the edge, they penetrate the

shell during deformation. The physical cables cannot intersect with the shell. The distance

of 3cm is chosen to keep a wide design space for the parametric study. The combination of

the possible contracted states of the actuators leads to three load cases. In state one the two

actuators are contracted the same amount at the same time, the deformation of the shell

mechanism is symmetrical (Figure 4.12a). The pitch angle increases but yaw and roll are

null. In the second state actuator 1 is activated (shortened) (Figure 4.12b). The loading is

asymmetric. The pitch and yaw angles increase. The roll angle decreases (Figure 4.10).

Finally, in the third state actuator 2 only is activated. The pitch and roll angles increase.

The yaw angle decreases. Since the shell has a vertical plane of symmetry and the actuators

are placed in a cross pattern, load cases 2 and 3 will result in the mirrored deformations. In

both load cases the displacements and associated strains and changes of curvature are

mirrored, they differ by having the sign of the yaw and roll angles be opposite. Therefore,

there are two load cases considered in this mechanism: symmetric (state 1) and asymmetric

(state 2 or 3) (Figure 4.12).

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Figure 4.12 Deformed shape of the mechanism under symmetric (a) and asymmetric (b)

load cases.

4.3.1.5 Finite element model of the shell and the actuators

The mechanical analysis of the shell is performed in the commercial finite element software

ABAQUS [223]. The shell is modeled with quadrangular shell elements (S4R) and 5

integration points in the section. The model is non-linear to account for the large

displacements of the shell. Each shell has 1500 elements and 1586 nodes.

For the shell’s material, the model implements a generic PETG-like material with a

Young’s modulus of 5Gpa and a Poisson’s ratio of 0.4.

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The actuators are T3D2 tension-only elements composed of a fictitious material with a

Young’s modulus of 210 GPa and a Poisson’s ratio of 0.3. The section surface area of the

cylindrical actuator is 0.0314 m2 with a radius of 0.1m. The actuator elements are extremely

stiff compared to the shell so that no actuator’s strain interferes with the shells’ movements.

In addition, the most important material property for the actuation element is the thermal

expansion coefficient. In order to generate a linear force that adapts to the large

displacements and changes of orientation of the shell surface during the movement, the

actuator contracts by a prescribed percentage of its length. This contraction is caused by an

increase in temperature. Therefore, the coefficient of thermal expansion is negative and

fictitiously tuned so that with a 100°C increase of temperature and an initial length of

0.104m, a 5% shortening of the actuator is obtained. For each geometry, a sequence of

actuation is defined to cover the three load cases presented in Table 4.2. The deformations

produced by load cases 2 and 3 are expected to be mirrored due the two identical actuators

being contracted independently. However, in the finite element model the sequence of

actuating couples covers the whole range of actuation to verify that in this complex non-

linear mechanical analysis the sequence of loading can be performed. The loading sequence

must be compatible with residual stresses and geometric nonlinearities to produce a viable

mechanism. The couple (𝛼1, 𝛼2) describe thepercentage of contraction with respect to the

original length in actuator 1 and 2 at each calculation step. The model does not restart from

a zero-stress state between each load case (Table 4.3).

Load case Actuation 𝜶𝟏 (%) Actuation 𝜶𝟐 (%)

1 −5.00 −5.00

2 0.00 −5.00

3 −5.00 0.00

Table 4.2 Load cases translated to thermal strain in the actuators

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Calculation step Actuation 𝜶𝟏 (%) Actuation 𝜶𝟐(%)

1 -5.00 -5.00

2 -3.75 -5.00

3 -2.50 -5.00

4 -1.25 -5.00

5 0.00 -5.00

6 -5.00 -3.75

7 -3.75 -3.75

8 -2.50 -3.75

9 -1.25 -3.75

10 0.00 -3.75

11 -5.00 -2.50

12 -3.75 -2.50

13 -2.50 -2.50

14 -1.25 -2.50

15 0.00 -2.50

16 -5.00 -1.25

17 -3.75 -1.25

18 -2.50 -1.25

19 -1.25 -1.25

20 0.00 -1.25

21 -5.00 0.00

22 -3.75 0.00

23 -2.50 0.00

24 -1.25 0.00

25 0.00 0.00

Table 4.3 Actuation steps in the FE model

4.3.1.6 Measure of curvature

The principal curvatures at different stages of the deformation are evaluated using the mesh

shown in Figure 4.11 and described in Section 4.3.1.5. For each shell, the mesh has 1500

4 node elements (quad elements) and 1586 vertices. The principal curvatures 𝜅1, 𝜅2 and

the associated principal vectors are calculated. These principal vectors and principal

curvatures are calculated using the a per-vertex normal algorithm [228] implemented in

Dodo [229], a plug-in for Rhino 6/Grasshopper [230, 231].

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The Gaussian curvature 𝐾 = 𝜅1𝜅2 is calculated to describe the type of transformation of

the mechanism as described in Section 4.2. The integral of Gaussian curvature is used as a

metric of surface stretching between the initial and deformed shapes and is defined as

𝐼𝜅 = ∫ 𝐾 = ∫ 𝜅1𝜅2 𝑑𝑆𝑆

(4.5)

In order to calculate 𝐼𝜅 over the mesh, the integral is approximated by a sum over the 1586

vertices of the mesh (Equation 4.6).

𝐼𝜅 = ∑ 𝜅1𝑖𝜅2𝑖

𝛽𝑖

1586

𝑖=1

(4.6)

With 𝜅1𝑖, 𝜅2𝑖

the principal curvatures at vertex 𝑖 and 𝛽𝑖 the area of influence of the vertex

𝑖. The stretching of the surface between the initial and deformed meshes is measured by

the ratio of deformed to undeformed integrals of Gaussian curvature 𝐼𝜅1/𝐼𝜅0

. In an

isometric deformation, the Gaussian curvature at each point is constant (section 4.2.1.1). If

such a property is verified, the ratio 𝐼𝜅1/𝐼𝜅0

should be equal to 1.

4.3.1.7 Parametric space of the geometry

Three parameters were selected to assess the influence of geometry on the magnitude of

the shell deformation. These three parameters defined the geometric design space of the

analysis and are the transversal radius of 𝑅𝑤, the longitudinal radius of curvature 𝑅ℎ and

the length of the outer edge 𝐿𝑒 of the shell. Since there are three variables in the parametric

study, the design space is 3-dimensional.

In this study, the transversal radius of curvature 𝑅𝑤 varies from 0.1 to 0.2m. The

longitudinal radius of curvature 𝑅ℎ varies from 0.03 to 0.08m. The last parameter, the

length of the outer edge of the shell 𝐿𝑒 varies from 0.05m to 0.06m. In total, the 3d design

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space is discretized in 24 geometries presented in Figure 4.12. The range of variation of

the three parameters 𝑅𝑤, 𝑅ℎ and 𝐿𝑒 is discretized in 24 geometries based on the formalism

of multidimensional grids [232, 233]. The black line on the surfaces show 𝑅ℎ, the white

line shows 𝑅𝑤.

Figure 4.12 Parametric design space of the mechanism.

4.3.2 Kinematics of the compliant shell mechanism

In Section 4.3.1, the analysis framework for the mechanism has been set up. In this section

the results of the FE simulation are presented. These results leads to the introduction of the

kinematic relationships between actuation and displacements as a function of the geometry

of the mechanism.

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4.3.2.1 Some variation of the Gaussian curvature

For the symmetric and symmetric load cases (1,2 and 3), the analysis of the 24 shells

indicate some amount of stretching of the middle surface for all geometries analyzed. The

average value of 𝐼𝜅1/𝐼𝜅0

is 1.0565 for the symmetric load case and 1.0408 for the

asymmetric load case. The average value of the ratio is within 6% margin of the target

value of 1. The hypothesis of constant value of the integral of gaussian curvature seems

verified. However, the relationship between those integrals presents some variability

(Figure 4.13). The standard deviation of the ratio is relatively high compared to the ratio’s

target value of 1. It is equal to 0.2827 for the symmetric load case and to 0.3190 for the

asymmetric load case. The coefficient of determination 𝑅2 is calculated at 0.8997 and

0.8733 for the symmetric and asymmetric load cases. The coefficient of determination

explains the goodness of the fit of a model. It indicates how well the linear relationship

𝑦 = 𝑥 fits the real data points. A perfect fit yields a value of 1 but values about 0.87 are

indicators that the relationship is a strong predictor of the real behavior of the system. The

quality of the hypothesis that the integral of curvature remains unchanged after the shell is

deformed is mostly verified. There are however geometries with high values (>1.50) of the

ratio of integrals (Figure 4.18) in the case of asymmetric load.

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Figure 4.13 Variation of the value of integral of Gaussian curvature over the deformed

(∫ 𝐾1) and undeformed (∫ 𝐾0) shell geometries for the symmetric and asymmetric loads.

The case of no variation of the integral of Gaussian curvature before and after actuation is

located on the identity (dotted line)

4.3.2.2 Influence of the geometry on the displacement of the shells

The analysis of the 24 shell geometries indicates that geometry does have a significant

influence on the magnitude of the movements generated by the mechanism. The two target

motions of the mechanism are pitch and roll. The yaw motion is not required to perform

solar tracking, yet it is a by-product of the geometry.

The pitch angle varies the most under symmetric load. The maximal values of the pitch

angle that the shell mechanism can generate depends on the geometry of the shell. For the

symmetric load case, the maximum values of the pitch are in the range [ 10.6°, 71.4°]

(Figures 4.14a, 4.15a and 4,16) over the whole design space. For the asymmetric load cases

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(2 and 3), the range of maximum values of pitch is [9.2°, 51.0°] (Figures 4.14b and 4.15b)

over the whole design space.

The roll angle is null for the symmetric load case. The roll angle varies only for the

asymmetric load cases (2 and 3). The maximal values of the roll angle are in the range

[3.8°, 6.6°] (Figures 4.14b, 4.15b and 4.16). The range of variation of the roll angle is very

narrow, which implies that the geometry has little influence on this degree-of-freedom.

Finally, the yaw angle is null for the symmetric load case as well. The yaw angle only

varies for the asymmetric load case. The maximal values of the yaw angle are in the range

[1.1°, 9.9°] (Figures 4.14b and 4.15b). Yaw and roll angles seem to be interconnected. In

the asymmetric load case, the increase of roll angle is complementary with the increase of

yaw angle (Figures 4.14b and 4.15b). Specifically, the shells that create the most roll (low

𝑅ℎ and low 𝑅𝑤) also create the most yaw.

The maximum pitch angle is reported at 71.4° for the symmetric load. In comparison the

maximum roll angle found in the design space is 6.6° (Figure 4.14). An order of magnitude

separates the extreme values of the ranges of motion of these two degrees-of-freedom.

Therefore, the mechanism in its present form is more suited for symmetric loads (load case

1) than asymmetric ones (load case 2 and 3). The geometry has more influence over the

variations of pitch angle than roll and yaw angles. In addition, the yaw and roll angles are

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Figure 4.14 Interpolated relationships between geometry and output displacements for

symmetric and asymmetric load cases for the 𝑙𝑒 = 0.05𝑚 instances of the mechanism

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Figure 4.15 Interpolated relationships between geometry and output displacements for

symmetric and asymmetric load cases for the 𝑙𝑒 = 0.06𝑚 instances of the mechanism

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Figure 4.16 Deformed shells (blue) and undeformed shells (black) show the maximum

pitch angle variation for the symmetric load (load case 1). The shells are viewed from the

side.

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Figure 4.17 Deformed shells (blue) and undeformed shells (black) show the maximum

roll angle variation for the asymmetric load (load case 3). The shells are viewed from the

front.

4.3.2.3 Variation of strain energy

The shells are deformed under the action of the actuation. The strain energy resulting from

this deformation and stored in the shell gives an indication of the magnitude of the

deformation in the shell (Figure 4.18). Bending and stretching contribute to the strain

energy. The hypothesis of no stretching was asserted to be valid, high values of strain

energy indicate large bending deformations of the shells. Unsurprisingly, those high values

of the shell strain energy correspond to large angular displacements (mostly pitch angle as

seen in Figure 4.14 and 4.15). Therefore, the values of the geometric parameters (being

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high 𝑅𝑤, low 𝑅ℎ and high 𝐿𝑒) that lead to high angle changes also lead to high strain energy

in the system. The maximum strain energy is reached for shell 15 (𝑅𝑤 = 0.2𝑚, 𝑅ℎ =

0.03𝑚 and 𝐿𝑒 = 0.06𝑚).

Figure 4.18 Strain energy as a function of the geometric parameters 𝑅𝑤 , 𝑅ℎ and 𝑙𝑒 for

the symmetric load case (a) and the asymmetric load case (b)

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4.3.2.4 Discussion

The results of this analysis lead to the conclusion that the three principal angles of rotation

yaw, pitch and roll cannot be activated independently. Only the pitch angle can be activated

independently in the case of both actuators being shortened simultaneously (load case 1).

In the case of an asymmetric load, increasing the roll angle is goal for the application of

solar tracking device. However, in addition to increasing the roll angle, that load case also

activates the yaw and pitch angles. Several factors can be considered to understand the

interdependency between the roll and yaw (such as actuation methodology). It appears the

7th eigenmode of the negative Gaussian curvature torus section introduced in Section 4.2

has a slight amount of yaw, as seen on Figure 4.19. The variation of this angle appears in

the preferred deformation mode of the toroidal surface. The asymmetric load case assumes

that the 7th eigenmode can be replicated to allow the shell mechanism to perform solar

azimuth tracking. This eigenmode contains some amount of yaw (the precise measure is

not available on the eigenshape) in addition to the torsion deformation of the toroidal shell.

Therefore, it is consistent with our design assumption that the shell mechanism would

produce an interdependent relationship between yaw and roll when the asymmetric load is

applied.

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Figure 4.19 Top view of the 7th mode (35.8 Hz) of the negative Gaussian curvature

section (Section 4.2). The eigenmode has a small amount of yaw (angle between the

vectors)

The shells with the highest strain energy also have the ratio of deformed (∫ 𝐾1) to

undeformed (∫ 𝐾0) integrals of Gaussian curvature that diverges the most from the average

value (Figure 4.20). Shell 15 has been cited in 4.3.2.3 as having the highest strain energy,

it is also the geometry that has the highest value of 𝐼𝜅1/𝐼𝜅0

(Figure 4.20). This observation

indicates that large values of strain energy include both bending and stretching components

since a large variation of the integral of Gaussian curvature signifies that the transformation

might include some stretching of the surface, i.e. the transformation is not isometric.

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Figure 4.20 Strain energy as a function of the ratio of deformed to initial Gaussian

curvatures for the symmetric load case and the asymmetric load case. 𝜇 is the average

value of the ratio and 𝜎2 is its variance

The shells that produce the largest pitch variations are geometries 3 and 15. Those shells

also have large variations of the integral of Gaussian curvature between the initial and

deformed states. These large variations are not an issue for the shell mechanism if the

transformation remains elastic. The benefit of using the eigenmodes as described in Section

4.2 and in this section is to facilitate the design of mechanisms that implement the preferred

deformation modes of the shell geometries. Such modes have a lower stiffness than other

modes of deformation and thus require less actuating force. The actuators in this study are

designed to be very stiff (𝑟 = 0.1𝑚). The force they apply is enough to actuate the system.

The actuation forces can be calculated using the conservation of energy that states the

external work (i.e. the work of the actuators 𝑊𝑒) and the internal work (the strain energy

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𝑈𝑒) must be equal (Equation 4.7). The hypothesis is made that no energy is lost in the form

of heat.

𝑊𝑒 = 𝑈𝑒 (4.7)

The work of the actuator is the given by

𝑊𝑒 = ∑1

2𝐹𝑖Δ𝑖

𝑖∈𝑎𝑐𝑡𝑢𝑎𝑡𝑜𝑟𝑠

With, 𝐹𝑖 the force in the actuator and Δ𝑖 the linear displacement of the actuator. In the

symmetric case, the two actuators are simultaneously activated. They contract by 5% upon

activation. Therefore, the force and linear displacement of both actuators will be equal

respectively. In the asymmetrical case, only one of the actuators is activated. Therefore,

the external work for each load case is given in Equation 4.8.

𝑊𝑒𝑠𝑦𝑚= 𝐹Δ 𝑊𝑒𝑎𝑠𝑦𝑚

=1

2𝐹Δ (4.8)

The initial length of the actuators is given by the geometry of the undeformed shells. Using

Equations 4.7 and 4.8, the actuating forces are given as follows

𝐹𝑠𝑦𝑚 =𝑈𝑒𝑠𝑦𝑚

Δ 𝐹𝑎𝑠𝑦𝑚 = 2

𝑈𝑒𝑎𝑠𝑦𝑚

Δ

(4.9)

The actuation forces are the force per active cable at the given actuation state. The

symmetric load case has two active cables. The value of the forces is reported in Figure

4.21. The forces are higher in the asymmetric load case. In addition, since the cables all

have similar initial length and all contract by 5%, the variations of the forces are similar to

those of the strain energy as a function to the geometric parameters of the analysis. The

difference is that the forces in the actuators for the symmetric load case are lower than in

the asymmetric load case.

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Figure 4.21 Actuation forces per active actuator as a function of the geometry for

symmetric (a) and asymmetric (b) load cases

Finally, the geometric parameters that have the most influence to create large movements

depend on the degree-of-freedom considered. The pitch angle is favored by high values of

𝑅ℎ and 𝐿𝑒, and with small values of 𝑅𝑤 (Figure 4.18). In contrast, the roll angle is higher

with small values of all three parameters 𝑅ℎ, 𝑅𝑤 and 𝐿𝑒 (Figure 4.19).

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4.4 Conclusion

In this chapter the preferred modes of deformation of a geometric surface to produce a

compliant shell mechanism were identified. The use of eigenfrequencies as the

corroboration for inextensional modes of deformations of random surfaces has been

justified from a mathematical standpoint. In addition, the Gaussian curvature is identified

as an invariant of a surface under such transformation.

We performed the eigenfrequency analysis of six common shell surfaces: a rectangular

strip, a cylindrical strip, a half-sphere, a quarter-sphere, a positive Gaussian curvature

toroidal surface and a negative Gaussian curvature toroidal surface. From this analysis, the

negative Gaussian curvature toroidal surface was selected as candidate to serve as a two

degree-of-freedom mechanism capable of independently activating pitch and roll angles to

perform spherical tracking. The shading compliant shell based on the toroidal mechanism

is presented in chapter 5. A parametric study involving 24 iterations of the shell geometry

was performed to compare the influence of the three geometric variables on the

displacements. It appeared that for this mechanism, one of the degrees-of-freedom offers

range of motion than the other. With a symmetrical actuating load, the pitch angle is

increased up to about 71.4° for the best geometry. The roll angle on the other hand “only”

increases up to 6.7° for the best geometry.

This chapter establishes the process of eigenmode-inspiration for compliant shell

mechanisms. The methodology proposed for the search of suitable surfaces is based

repetitive inputs and future work could focus on automating the search by optimization

methods such as heuristic search optimization that has proven to be efficient for this type

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of applications [234]. As shown in chapters 3 and 4 of this thesis, compliant shells at

different scales have the same bending dominated behavior when they are thin enough.

Basing the developments of shell compliant mechanisms on bio-inspiration only will lead

to a diversity bottle-neck as instances of flexible shells in nature are relatively rare.

However, the preferred modes of deformation of shell surfaces can be tied to an eigenmode

of a natural frequency. Making the connection between geometry and eigenmodes leads to

a new approach to generate compliant shell typologies from the vibrational physical

properties of structures, a step beyond the narrow imitation of nature’s structures.

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Chapter 5:

Daylight driven optimization of dynamic shading

for building energy demand

This chapter is largely based on following journal publication, which was under review at

the time of writing the thesis manuscript:

Charpentier, V., Meggers, F., Baverel, O., Adriaenssens, S. (2019). Daylight-driven

optimization of building energy demand for dynamic shading. Building Research and

Information , Under Review

5.1 Introduction

In this chapter, a novel methodology is proposed based on interpolation of simulations and

optimization of environmental performance to substantiate the use of more advanced

shading systems. Linear interpolation of simulation results bridges is implemented to create

multivariate function included in an analytical optimization system. Dynamic shading

systems can reach higher standards of performance than static, both for the control of solar

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gains and for the visual comfort of users. To this end, single and double degrees-of-freedom

(d.o.f.) shading systems are analyzed. Solar tracking is defined as the capacity of the shades

to be oriented to follow solar azimuth and elevation. For the purpose of studying the

benefits of spherical solar tracking as shading systems, a simplified model of a shading

device is used based on previous work from chapters 2 and 4.

The range of motion of dynamic shades provides solutions that balance the need for

daylight in buildings with the demand for lower energy consumption. Such systems make

better use of the environmental resources than fixed systems. In contrast to fixed shading,

dynamic shades can often be retracted if they are not needed. A study [27], showed that

external dynamic shading can reduce the cooling energy by 50% annually when a control

algorithm measures the illuminance level to activate the shades. The overall energy

reduction was 12%: heating and lighting consumption were negatively impacted by the

algorithm (respectively by +32% and +38%). Indeed, the operation the shade tends to limit

the level of natural daylight penetrating the space hence increase the use of artificial

lighting. Whether they are external or internal, the operation of shading systems is subject

to user comfort. Thermal comfort and visual discomfort in the form of glare and

illuminance levels, strongly impact the way shades are operated. The demand of comfort

by occupants tends to reduce the overall energetic performance of the shades. Reduction

of available daylight and free winter heating need to be compensated [27]. From the user’s

point of view, the operation of shades is dominated by improving visual comfort. In a

survey of user interaction with manual blinds, reduction of direct light and glare on

electronic display is the goal of 80% of the users [235]. In addition, views and connection

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to the outside represent major concerns for occupants [236]. Visual variables alone can

explain the operating actions on shade in buildings with efficient use of daylight [236].

Therefore as noted in [25], a reduction of solar gain and user visual comfort turn out to be

competing performance criteria in certain cases.

While adaptability increases the environmental performance of the building’s façade, not

all dynamic systems perform equally in all orientations and climates. Differentiating

between existing systems is challenging. In addition, new shading systems might perform

better but they must be informed by performance-based indices. The added flexibility of

operation gained from dynamic shading needs to answer specific demands of both the

building energy and the user comfort, to be deemed appropriate. Control strategies

contribute to the increase of the façade’s efficiency by insuring the optimal positioning of

the dynamic shades throughout the operating period. As a result, both a performance

evaluation and design approach are needed to measure the performance of dynamic shades.

Control strategies of internal blinds focus on the optimization of daylight [237-239]. Those

for external blinds often include both thermal and daylighting parameters [25, 27, 240-

242].

The tracking shades introduced in this chapter consist of an active and a passive part. The

active part generates the 2 d.o.f. motion by twisting and bending of the material, orienting

the passive part for maximal shade efficiency. The performance of the shades is measured

as the capacity to satisfy both an objective of minimization of energy demand and precise

daylighting constraints. The performance of the three shading systems are compared. As

detailed in the article, the methodology is applied for annual sun hours in Princeton, NJ,

USA. The results presented in this study are specific to the case chosen. They are

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representative of a design case one might encounter. The methodology, however, can be

applied to many cases. It can be used in early design phases as well as for in-depth

simulations. The methodology will find use in early design phases of façade systems

because it allows to compare several design cases for a specific environmental context. It

can also be used to refine the design of a specific shading system by quantifying effects of

parametric variations on the system’s performance.

5.2 Choice of shading systems and optimization method

An optimization analysis was performed to investigate choice strategies in early stage

design of dynamic shading systems. Three types of shading systems are evaluated and

compared to a baseline, non-shaded scenario for three facade orientations. Previous studies

[243] have demonstrated the usefulness of design choices with thermal and lighting

objectives. A study of model-based control of shading [244] applied daylighting objectives

to adequately position the roller shades. The present study focuses on mediating the energy

(heating, cooling and lighting) demand under daylighting constraints for three types of

shading systems. The room is a typical (4.5 m x 5 m x 3.2 m high) perimeter office space

in Mercer County NJ (USA, latitude 40.3573° N) and the window-to-wall-ratio is fixed by

design at 65%. Results express in this study the adequacy of a shading system through

optimal control of its position during the whole year. Weather data relative to Mercer

County comes from the typical meteorological year (TMY3) dataset [245]. In the dataset

4306 hours of sunlight are recorded for this position. Three factors were selected to enter

the optimization: (i) the heating and cooling energy demand, (ii) the lighting energy

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demand and (iii) the average and maximum illuminance on an 80-cm high work plane. The

theoretical nature of this study implies that assumptions have been made on material

properties, room dimensions and building orientations of the cases.

5.2.1 Facade orientation and room geometry

East, south-east and south orientations are considered. By symmetry, the western

orientation is expected to behave similarly to the eastern orientation. One of the main

differences being that thermal mass carryover tends make to make afternoons worse on

west than east where the night was cool before exposure.

The simulated perimeter office is 5 m deep, 4.5 m wide and 3.2 m high (Figure 5.1) with a

65% window-to-wall ratio. The window is 2.2 m high (ℎ𝑤) and 4 m wide (𝑤𝑤), it seats at

0.5 m from the ground (Figure 5.1). The shading system covers the entire window. The

work plane is defined by a plane 0.80 m above the interior ground and offset by 0.50 m

from the window (Figure 5.1). The choice of dimensions for the work plane will have an

influence on the outcome of the study. The work plane chosen starts at 0.5m from the

window which will lower the average illuminance factor in the analysis. In our case, this

work plane was designed in agreement with precedents in the literature [246]. An outside

ground plane (30x24m) is added to the model (Figure 5.1) to add ground reflections in the

daylighting study.

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Figure 5.1 (a) Perimeter office space – window (ww) in blue, work plane (wp) in

yellow - exterior / interior ground (grd), walls (wll) and ceiling hinted – (b) daylighting

grid

5.2.2 Choice of external shading system: awning, venetian shades and spherical solar

tracking

5.2.2.1 Spherical tracking dynamic shades and simplified model

Spherical movements derive from spherical coordinates in which the position of a point in

space is described not by (𝑥, 𝑦, 𝑧) but by (𝑟, 𝜙, 𝜃),with 𝑟 the radial distance from the origin,

𝜙 the azimuthal angle and 𝜃 the zenith angle. The shading system proposed is capable of

spherical tracking motion. It follows both the elevation and the azimuth angle of the sun so

that the shade surface can always be oriented perpendicularly to the sun vector, if needed.

Such spherical system has been proposed for photovoltaic collection on facades [247]. The

geometry and range of motion of the spherical tracking shades are modeled after the

ongoing work of the authors to design a spherical tracker (see Figure 5.2). The top section

of the shell presented in Figure 5.2a is the negative Gaussian curvature toroidal surface

presented in chapter 4. The long passive surface connected to it is created but extruding the

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edge of the mechanism downward. Two d.o.f. increase the range of motion of the spherical

tracker but add mechanical complexity. The spherical solar-tracking shade is two degree-

of-freedom by default and is inspired by previous research on plant solar tracking

movements seen in chapters 2 and 4. The simplified model of the shade is constructed in

two parts (Figure 5.2). The smaller active part generates the movement and the larger

passive part produces the shade. The passive part is a flat surface (Figure 5.2). In a case of

perfect solar tracking, the passive surface would always remain perpendicular to the sun

vector.

Figure 5.2 Spherical tracking dynamic shades prototype (a, b and c) and simplified model

(d, e and f) – The two systems in the initial configuration (a and d), symmetric load case

(b and e) and asymmetric load case (c and f)

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5.2.2.2 Simplified models of the three shading systems

Three categories of external shades are implemented: typical awnings, venetian blinds and

spherical solar tracking shades presented in 5.2.2.1. The unshaded window is evaluated in

the analysis for a baseline comparison. The spherical tracking shades are controlled 2

actuators. Therefore, this system has two degrees-of-freedom. Awnings and external

venetian blinds controlled by a single actuator. They are single-degree-of-freedom systems

(see Figure 5.3). They have been modelled to resemble commercially available systems

[29-32]. Since they are all dynamic, the three shading systems can be described as tracking

the sun movements to some extent. However, the typical awning and the venetian shades

do not track azimuthal movements of the sun. This reduces the dimensionality of the

tracking and hypothetically, limits its performance.

The typical awning shades is controlled by the rotation of the roller carrying the fabric. The

range of motion of the roller allows the shade to cover the full height of the window ℎ𝑤

(d.o.f. 1). The shades angle with the façade with a circular arc motion as the roller is rotated

(see Figure 5.3). The 8 slats of the venetian blinds individually angle (d.o.f. 1)

between 5° and 70° from the vertical window plane. Each slat is 27.5 cm deep such that

they cover the whole window when closed. Finally, the spherical solar tracking shade

angles with the façade to follow sun elevation in the range [0°, 70°] (d.o.f. 2) and rotate

longitudinally to track the sun azimuth (d.o.f. 1) with the range [−45°, 45°]. Each element

of the tracking system is 20 cm wide and 27.5 cm long. Only the awning system allows the

window to be fully open and fully closed. The venetian shades and the spherical tracker

system always remain in front of the glazing.

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Figure 5.3 Three types of shades and their associated d.o.f. (a) Awning with roller

extension (d.o.f. 1), (b) Venetian shades with slat angle (d.o.f. 1), and (c) Solar tracker with

azimuth angle (d.o.f. 1) and elevation angle (d.o.f. 2) – Straight arrows denote linear

actuators, while circular arrows denote rotational actuators

5.2.3 Selected energy and daylighting control variables

Three factors were selected to act as controls for the optimization: (i) the heating and

cooling energy demand, (ii) the lighting energy demand and (iii) the average and maximum

illuminance on an 80-cm high work plane. The energy and daylighting simulations are

performed with DIVA [248] in the program Rhino3D/Grasshopper[230]. The thermal state

of the test room is simulated with EnergyPlus [249]. The daylighting and electric lighting

analysis are performed by Radiance [250].

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5.2.3.1 Thermal energy

The goal of the thermal model is to produce energy demands for heating and cooling of the

space. The simulation engine is EnergyPlus. It is interfaced by DIVA/Archsim in

Grasshopper3D/Rhino3D. The model takes the climate data from the epw file of Mercer

County, NJ as weather input. In this model, the heating and cooling are running every day

of the week during the hours of occupancy of the space (Figure 5.4). The heating setpoint

is set at 20°C while the cooling setpoint is set at 26°C. The heat balance is performed with

4 timesteps per hour and implement the conduction transfer function method. The

calculation of solar radiation is performed with the detailed timestep integration method.

In this study, the thermal model in EnergyPlus is not set to integrate electric lighting and

daylighting simulations. Both of those are run independently in Radiance. Hourly internal

gains from people are modelled based on the office occupancy schedule in Figure 5.4 and

vary depending on the hour of the day (Figure 5.4).

Figure 5.4 Occupancy schedule for modelling the presence of people in the test

room. This schedule is only valid for weekdays. No occupants are present during the

weekend.

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The construction material of the space is detailed as follows: the ground (interior and

exterior) is an adiabatic 200 mm thick concrete slab and the walls and roof are a layering

of 120 mm thick insulation and 200 mm thick structural concrete. The total thermal

resistance for the walls and roof is 3.63 K·m²/W. The window is a clear double paned

window with a high thermal resistance of 13.3 K·m²/W. For the interior convection the

TARP algorithm is used. For the exterior convection the DOE-2 model is used. Finally, an

outside air infiltration of 0.2 air changes per hour (ACH) is implemented.

EnergyPlus uses the Sutherland Hodgman polygon clipping algorithm [251] to determine

projection of the shading modules on the window. This algorithm does not support concave

polygon shadows [252]. Since the spherical tracker is constituted of two parts, it can

potentially once projected be a concave polygon. The impact of this algorithmic limitation

is not studied here but should be investigated further.

5.2.3.2 Daylighting model

The illuminance on the work plane is used as metric for assessment of daylight quality. It

is calculated using Radiance with 25 sensors. They divide the work plane (20.25 m2) in 0.9

m x 0.86 m squares. The glazing is taken as a standard clear double pane with 80%

transmittance. The shades present a 4% transmittance as implemented in [253]. The inside

and outside ground have 20% reflectivity. The walls have a 50% reflectivity, while the

ceiling has an 80% reflectivity. These parameters are selected to be generic and should be

adapted to case specific studies. The window has a solar heat gain coefficient (SHGC) of

0.764 and a visible transmittance of 0.812. The calculations were performed for Mercer

County, USA (latitude 40.3573° N) for the entire year. The calculation in Radiance has

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been shown to overpredict the illuminance computed [254], so the results should ideally

be verified against experimental data for validation.

Two criteria are derived from the raw illuminance on the work plane to quantify the

daylight quality: the average illuminance (Eaverage) and the maximum illuminance (Emax).

An average illuminance of 500 lx is recommended for the work plane for paper work /

computer work [254-256]. This value can also be found between 300 and 500 lx in other

references [257, 258]. In the optimization the target illuminance for average illuminance is

set at 500 lx. Maximum illuminance over 2000 lx are likely to cause user visual or thermal

discomfort [259, 260]. Some studies on the matter of useful daylight illuminance (UDI)

correlate the occurrence of glare to values of illuminance over 3000 lux [261]. In this study,

however, the maximum value of illuminance of 2000 lux is considered as the upper limit

of daylight comfort.

5.2.3.3 Electric lighting energy

As described in previously, an objective of 500lx average illuminance constrains the

system. The average illuminance provided by daylight is not uniform over the work plane

and will sometimes not be sufficient to provide comfortable ambient light conditions.

Therefore, electric lighting is modelled to predict the demand of electric energy required

to complement the optimized daylight provided by each shading system. A typical lighting

system with 10.8 W/m2 of power is assumed for the simulation. The daylighting grid

divides the work plane in 0.9 m x 0.9 m grid with four sensors placed in two rows (see

Figure 5.1b). The control system considered is a dimming with occupancy on/off control.

The setpoint of the lighting system is 500 lx. The control algorithm dims the lights

proportionally to the difference between the actual average of the sensors and the objective.

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When the daylight is 500lx over sensor, the dimmer is set to its lowest value. The lighting

at a given moment is determined to provide enough light to reach the setpoint of the system

uniformly over the work plane. Given the geometry of the room and the presence of a large

window on one wall only, natural daylight is not uniform over the work plane.

The algorithm for electric lighting is open loop, it does not feed back into the daylighting

assessment. The electric lighting energy provided, therefore, informs on the demand at a

given moment but does not reflect what the final lighting demand would be. This final

lighting demand would be found by iterating over the daylight and electric lighting

contributions. The occupancy schedule is set to same hours as the thermal schedule (Figure

5.4), to the exception that it considers the daylight-saving time.

5.2.4 Interpolation of simulated results for behaviour modelling

Awnings, venetian shades and solar tracker are designed to continuously mitigate

irradiation and improve daylight comfort. In a two-step process, the range of motion of the

three shading systems is sampled and then interpolated. This method provides a

manageable way to predict their effect on the overall energy demand. The sampling is done

by discretizing the actuation in 9 steps for each d.o.f.. A sensitivity analysis on the sampling

of the actuation range was performed to determine the number of steps to be simulated to

reach a good approximation of the dynamic shades’ continuous behaviour. Performing the

optimization in a continuous domain of actuation allows designers to later tune the interval

between discrete steps, if necessary.

At each solar position and each orientation, there are 81 actuation steps for the spherical

tracker and 9 for the awning and the venetian shades (Figure 5.5). Once the simulation

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results are available, they are extrapolated to a continuous domain. This method allows the

optimization algorithm to search the approximate continuous domain with analytical

functions. Consequently, it predicts and compares the performance of the shading systems

in an analytical framework.

The interpolation is performed in MATLAB using a linear interpolant object

(griddedInterpolant function) for both awning and venetian single-degree-of-freedom

shades, and spherical tracking two d.o.f. shades. The interpolant is an interpolating function

that can be evaluated at query points. It is easily integrated in analytical optimization

systems. During simulation, the data is sampled at constant interval of the actuator’s range

of motion. Thus, the data is formatted to create uniform grids for interpolation. The

interpolation is chosen to be linear. It provides a 𝒞0 continuity to the function. This

continuity is satisfactory for this optimization. The linear interpolation function is less

smooth than a spline interpolation. It was selected because it has the benefit of keeping the

response to a local change local. For instance, if the simulation produces an abnormal result

a point of the actuation grid, the linear interpolation would not propagate the error to

neighboring points. The values of illuminance can peak sharply and locally to high orders

of magnitude (e.g. from 102 lx to 104 lx), it is therefore beneficial to constrain the response

locally, hence linear interpolation is preferred.

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Figure 5.5 Linear interpolation on heating, cooling and lighting energy and on average and maximum illuminance for 1D actuated

awning shades (a) and venetian shades (b) and for 2D actuated spherical tracker shades (c) data for July 6 at 12h00 – East orientation

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5.2.5 Optimization system for control of shades

The optimization system is set to minimize energy demand (heating, cooling and lighting)

under constraints on the daylighting quality of the space. In section 5.2.4, the five parameters

of the optimization are described with analytical relationships based on the simulated results.

These functions are used in an Augmented Lagrangian Genetic Algorithm (ALGA) [262] in

Matlab (ga function) to solve the constrained optimization system (𝒮). The formulation of

the optimization system (𝒮) in Equation 5.1 refers to a single solar position 𝑠 and the shading

system 𝛼.

(𝒮) {

𝒎𝒊𝒏𝒙∈𝐼

(𝑬𝒉𝒆𝒂𝒕.𝒔𝜶(𝒙) + 𝑬𝒄𝒐𝒐𝒍.𝒔

𝜶(𝒙) + 𝑬𝒍𝒊𝒈𝒉𝒕.𝒔

𝜶(𝒙))

𝑠. 𝑡. {𝐴𝑠

𝛼(𝑥) = 𝑙1

𝑀𝑠𝛼(𝑥) < 𝑙2

+

(5.1)

, with 𝛼 ∈ [awning, venetian, tracker] the shading system, 𝑠 the sun vector, 𝐼 the actuation

interval, 𝑬𝒉𝒆𝒂𝒕.𝒔𝜶 the interpolated heating energy for sun vector 𝑠 and shade 𝛼, 𝑬𝒄𝒐𝒐𝒍.𝒔

𝜶 the

interpolated cooling energy for sun vector 𝑠 and shade 𝛼, 𝑬𝒍𝒊𝒈𝒉𝒕.𝒔

𝜶 the interpolated lighting

energy for sun vector 𝑠 and shade 𝛼, 𝐴𝑠𝛼 the average illuminance interpolated function, 𝑀𝑠

𝛼

the maximum illuminance interpolated function, 𝑙1 is the target value for the average work

plane illuminance (500 lx) and 𝑙2+ the upper limit for the maximum work plane illuminance

(2000 lx).

The system is solved for both single- and two-degree-of-freedom shades the same way. The

difference between the two cases is the actuation interval 𝐼. In the 1D case, the interval 𝐼 is a

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segment 𝐼 = [0,1] while in the 2D case, 𝐼 is a plane 𝐼 = [0,1] × [0,1]. The constraints

tolerance is set to 25lx to speed up convergence of the optimization system.

The optimization methodology is adapted to both hot and cold periods since the objective

function is the sum of all the energy demand in the system at a point t in time. Similarly, the

constraints of comfort are the same for users throughout the year. The optimization system

must be solved for each shading system, for each orientation and for each sun hour

considered. In total, the optimization system was solved 38754 (= 3 × 3 × 4306) times.

5.2.6 Methodology for design and assessment of shading performance

The elements discussed in section 2 are combined into a five-step methodology for the

assessment of performance of shading devices. The success of the process is conditioned on

the prior definition of the objectives (e.g. minimization of energy demand) and constraints

(e.g. visual comfort, thermal comfort, glare) of the analysis. The methodology is sequenced

as follows.

(1) Design of the shading system / Selection of existing shading system

(2) Evaluate the metrics selected for the analysis at each actuation step, each sun hour

(3) Model the behavior of the shade with interpolation

(4) Optimize shade position under constraints

(5) Validation of results. Possible to loop back to step 1 for re-design.

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5.3 Results

The results of the simulations are presented below in two sections for the energy demand

(Section 5.5.3.1) and the daylight condition of the work-plane (Section 5.5.3.2). The results

are produced by the optimization energy demand with constraints of daylight comfort

presented in Section 5.2.5.

5.3.1 Mitigation of energy demand for the three types of shades

The overall combined impact of the optimization methodology on heating, cooling and

lighting annual energy demand is reported in Figure 5.6. For the three orientations observed,

the annual energy demand for the baseline case of no shading is significantly decreased by

the three shading systems (Table 5.1). The optimization results show a decrease of annual

cooling demand on the east of 26% for the awning, 47% for the venetian and 37% for the

spherical tracking shades. This decrease is more pronounced on the south east and south with,

on the south east - 35% for the awning, - 57% for the venetian and - 47% for the spherical

tracking shades, and on the south - 37% for the awning, - 57% for the venetian and - 52% for

the spherical tracking shades. The efficiency of the shading system on the east is about 10

percentage points lower than for the other orientations (Table 5.1).

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Figure 5.6 Annual energy demand in MJ (1 MJ = 0.28 kWh) for east, south east and

south orientation and for the baseline case, awning, venetian and spherical tracker shades.

Heating, cooling and lighting energies are to make the total energy demand.

Total annual energy demand variation

Orientation Awning Venetian Tracker

East -26% -47% -37%

South East -35% -57% -47%

South -37% -57% -52%

Table 5.1 Variations of total annual energy demand of the three shading systems to

the baseline case. The results are produced by the optimization process.

This significant decrease of annual energy demand is mostly due to a large reduction of the

cooling energy demand for all the cases (Table 5.2). The lighting (Table 5.3) and heating

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energies both increase, but they represent a small fraction of the total energy demand,

therefore the total energy demand is still decreasing overall.

Annual cooling energy demand variation

Orientation Awning Venetian Tracker

East -38% -68% -56%

South East -46% -76% -62%

South -50% -76% -67%

Table 5.2 Variations of annual cooling energy demand of the three shading systems

to the baseline case.

Annual lighting energy demand variation

Orientation Awning Venetian Tracker

East +57% +81% +84%

South East +62% +81% +73%

South +74% +94% +87%

Table 5.3 Variations of annual lighting energy demand of the three shading systems

to the baseline case.

The lighting energy needed for the awning shades increases as the façade is rotated from east

(+57%) to south (+74%). Simultaneously, the cooling energy demand for the awning system

decreases when moving from east (-38%) to south (-50%). For the two other shading system,

no correlation between lighting and cooling energy demands seem to appear.

The test building is located in Princeton, NJ with a 40° latitude and simulated with high

thermal resistance materials. The low values of the heating energy demand indicate the

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perimeter space is cooling dominated. The annual heating energy demand represents 3% of

the total annual energy in average for the four window configurations, 8% in average for the

venetian shades.

The total energy demand further observed in an hourly distribution (Figure 5.7). The hourly

energy demands higher than 1kWh are highlighted for contrast between cases. For the

awning, the high energy demand situations appear in indirect light (see Figure 5.9 for

direct/indirect light visualization). A similar pattern appears for the spherical tracking shade

to some degree but not for the venetian shading system that highly reduces the total hourly

energy demand throughout the year.

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Figure 5.7 Hourly total energy demand for east, south east and south orientation and for the baseline case, awning, venetian and

spherical tracker shades. Hourly demands over 1kWh are highlighted in red.

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5.3.2 Daylight conditions for the optimized positions

5.3.2.1 Maximum illuminance

All three shading systems satisfy the constraint of maximum illuminance set in the

optimization system. As represented on Figure 5.8, no maximum illuminance over 2000lx is

reported for the 9 shading cases simulated. The constraint of the optimization is satisfied for

all 4306 sun hours of each of the 9 cases.

The baseline case reveals patterns of hourly available daylight for each orientation (Figure

5.8). High illuminance values (>2000lx) occur dominantly in the morning on the east and

appear to be centered around the noon sun hour on the south. The south east orientation is an

intermediate case with both high values of illuminance in the morning and later in the day.

For those three orientations, the annual mean value of maximum illuminance is well beyond

the 2000lx comfort threshold (Table 5.4), which signals that if unshaded this room would be

subject to frequent and intense discomfort. In addition, the standard deviation of maximum

illuminance is close to the mean value itself, an additional indication (if one was needed) of

the extreme variability of environmental daylight.

It is therefore significant that the variability of maximum illuminance for the three shading

systems is so greatly reduced. The standard deviation of the maximum illuminance is an order

of magnitude lower for shaded cases than for the baseline unshaded case (Table 5.4). Over

the three orientations, the overall mean maximum illuminance is ~1100lx for the awning,

~520lx for the venetian and ~780lx for the spherical tracker shading systems. For venetian

and spherical tracker, the mean maximum illuminance is close to the 500lx target for the

work-plane average illuminance.

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Figure 5.8 Hourly maximum daylight illuminance resulting from the optimization for east, south east and south orientation and

for the baseline case, awning, venetian and spherical tracker shades. Average illuminances over 2000 lx are highlighted in red.

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Maximum illuminance

Orientation

Baseline Awning Venetian Tracker

mean

(lx)

std dev

(lx)

mean

(lx)

std dev

(lx)

mean

(lx)

std dev

(lx)

mean

(lx)

std dev

(lx)

East 2273 1827 1095 375 418 327 676 443

South East 4290 5569 1176 577 572 469 805 521

South 3492 3680 1135 389 581 455 862 458

Table 5.4 Statistical distribution of maximal illuminance received on the work plane

for the east, south east and south orientations and for the unshaded and shaded cases.

5.3.2.2 Average illuminance

High average values (Figure 5.9) coincide with the high maximum illuminance values

(Figure 5.8) described in Section 5.5.3.2.1. Similarly, as for the maximum illuminance, the

amount of daylight penetrating the space is excessive without shading. Most baseline average

work-plane illuminance values are superior to 600lx for the east (58%), for the south east

(64%) and for the south (65%) (Figure 5.10). The intervals below 600lx each represent about

3% of the number of baseline values for each orientation of the façade. As shown in Figure

8, those values occur on the fringe of the high daylight periods.

The mean value of average work-plane is 868lx on the east, 1103lx on the south east and

1006lx on the south. In addition, the variability of the unshaded average work-plane

illuminance is high. The standard deviation is of the same order of magnitude as the mean

value (Table 5.5).

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The 500lx average illuminance constraint can only be maintained in shaded cases if the

unshaded baseline case provides at least 500 lx of average illuminance. None of the hours

that initially provide less than 500lx of average illuminance (deep blue in Figure 5.8) see an

increase of illuminance once shaded.

The three shading systems meet this constraint with various levels of success. Over the three

orientations, the 500lx target is met on average 3% of sun hours for the baseline, 68% for the

awning, 16% for the venetian and 35% for the spherical tracking shades (Table 5.6). The

efficiency of each shading system is relatively similar for each orientation. The success of

the optimization for the annual hourly cases, established as repeatedly reaching the expected

500lx constraint value, translates as a uniformly colored hourly map (Figure 5.9) and a mean

average work-plane illuminance close to 500lx (Table 5.5). The awning shades perform the

best with almost entirely uniform maps (Figure 5.9) and mean values of the average

illuminance above 400lx for the three orientations. This good performance is confirmed in

Figure 5.10 with the awning shades only failing when the daylight does not provide 500lx in

the baseline case. In contrast, the spherical tracking and venetian shading systems do not

perform as well. The annual hourly map is less uniform than for the awning shades and the

mean values of the illuminance are contained in [220𝑙𝑥, 330𝑙𝑥 ] interval. This is confirmed

in Figure 5.10 by the small percentage of the successful cases for the venetian shades and to

a lesser extend for the spherical tracker.

The three shading systems have 0.1% of occurrences in the intervals above 525lx. The

optimization methodology implemented successfully limits the amount of daylight

transmitted by the shades to the space.

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Figure 5.9 Hourly average daylight illuminance resulting from the optimization for east, south east and south orientation and for

the baseline case, awning, venetian and spherical tracker shades. Average illuminances over 1000 lx are highlighted in red

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Average illuminance

Orientation Baseline Awning Venetian Tracker

mean (lx) std dev (lx) mean (lx) std dev (lx) mean (lx) std dev (lx) mean (lx) std dev (lx)

East 868 642 427 135 227 155 283 164

South East 1103 879 418 143 251 169 314 173

South 1006 666 432 135 266 167 336 174

Table 5.5 Statistical distribution of average illuminance received on the work plane for the east, south east and south orientations

and for the unshaded and shaded cases.

500lx constraint success rate (% of sun hours)

Orientation Baseline Awning Venetian Tracker

East 4% 70% 12% 25%

South East 3% 61% 18% 36%

South 3% 72% 17% 43%

Average 3% 68% 16% 35%

Table 5.6 Success rate of the optimization for the 500lx average illuminance constraint. The baseline case is not part of the

optimization and is given as an indicator.

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Figure 5.10 Annual distribution of average illuminance occurrences for the east, south

east and south orientations.

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5.4 Discussions

In all presented cases, the governing assumption is that user comfort should be the

controlling parameter to a dynamic shading system. Mathematically, this translates to

setting the energy demand parameters as the objective and the visual comfort as constraints.

This methodology is adaptable since the criteria selected for the energy or the visual

comfort can be modified or replaced. The inputs of the methodology are the numerical

values at each actuation step which makes the methodology approachable and flexible. The

results presented in this study are specific to the case chosen. They are representative of a

design case one might encounter. The methodology, however, can be applied to many

cases. It can be used in early design phases as well as for in-depth simulations. The

methodology will find use in early design phases of façade systems because it allows to

compare several design cases for a specific environmental context. It can also be used to

refine the design of a specific shading system by quantifying effects of parametric

variations on the system’s performance.

5.4.1 Performance of the methodology

We created a methodology to evaluate and compare the effect of dynamic shading on

energy gains and user comfort. This novel methodology is presented here in a theoretical

setting. Case studies and implementation for control are necessary to move forward.

The genetic algorithm solver used in this study provides a successful avenue to find global

maximum for results of environmental simulation. Using the constrained genetic algorithm

allows to find optimal positions of the shadings systems for 9 independent cases and annual

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span of analysis. The interpolated functions generated from the simulation results present

discontinuities, for instance maximum illuminance values can have a sharp step of several

orders of magnitude. In this case a robust global minimum method, such as genetic

algorithm, must be used. In our case, the maximum number of generations for the genetic

algorithm solver is set to 15. The algorithm converges in 3 to 4 generations for most solar

positions.

The presented methodology adds new possibilities of analysis from the results of the energy

and daylighting simulations. It increases and encourages designers to rely on a

mathematical modelling approach of environmental data to advance the design of dynamic

shading. Single or multidimensional interpolation widens the design field by adding a high-

level of abstraction to the design process. The presented method is applied to a two-degree-

of-freedom system and two single-degree-of-freedom shading systems and both are

successfully integrated in the framework. The approach can be treated as part as a recursive

process to design the most efficient shading system possible or to inform the operation of

a facade system for maximum user enjoyment.

5.4.2 Comparison of shading devices

The performance in this study are representative of the three shading systems as presented.

One of the main takeaways of the study is that providing improved daylight access can be

a barrier to shading energy performance. There is a necessary compromise between

blocking solar radiation from entering the space and providing daylight for occupants. The

awning shades exemplify that by providing very good daylight but being less good than

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the other shading systems on reduction of energy demand. All three shading systems limit

the maximum illuminance to the 2000lx set in the optimization system. But only the awning

shading system delivers the desired average work-plane illuminance conditions

consistently. For 68% of the annual sun hours (16% for venetian shades and 35% for

spherical tracker) the average illuminance equals the 50lx constraint. This good daylight

performance is balanced by a lower reduction of annual total energy demand than the two

other shading systems (Section 5.3.1).

The lack of daylight in the venetian and spherical tracking shading cases is due to the design

of the shading systems themselves. The methodology allows for a parametric iterative

approach to take place to design shading systems. The initial tested design might perform

adequately but present constructability challenges. Long overhanging awnings are more

sensitive to façade wind load than an external venetian shading system with 20cm wide

slats. In addition, if the durability of the shading system is a concern, using textile material

in the awning might not be the best choice. Those criteria would help refine the choice of

parameters for design of a shading system.

For the awning shades, the east and south orientations present the best overall results. The

south east orientation more difficult to tackle for this type of shade. This reinforces the

difficulty to treat south east orientations due to the solar vectors’ high incident angles with

windows. In that orientation, both results of the daylight and the energy demand are less

good for the awning.

146

5.4.3 On the choice of constraint values

The choice of comfort criteria to be used as input in the optimization system has an

influence on the outcome of the analysis. A choice of a lower average work-plane

illuminance constraint may have produced different results, for instance. The venetian

shadings and the spherical tracking shades produce average illuminances consistently in

the 220lx-330lx interval in our study. Setting the constraint within that range could have

shown one these two shading systems as the best overall system. However, this would have

certainly obstructed the fact that those are less versatile than the awning system in their

tested configurations. Similarly, the spherical tracking shades are intuitively better for

unobstructed views from the inside to the outside of the space, due to their ability to twist

out of plane. Selecting the unobstructed view as a comfort criterion could have favored the

spherical tracking shades. Setting adequate constraints for an optimization will shape the

type of shading that will perform best. Picking the right constraints is therefore essential.

Since in this methodology the constraints are based on human comfort, occupants are put

at the center of the study, in a position that promises to increase the wellbeing and

enjoyment in the building.

5.5 Conclusion

This study shows that the performance of shading systems can be precisely modelled and

tuned to reduce heating, cooling and lighting energy demands and increase user comfort.

This analysis opens a path to fulfil the currently underdeveloped potential for daylighting

improvement and substantiates the benefit of more advanced shading systems. The

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methodology is based on the interpolation of environmental simulation results for the

number of actuation states of the shades and the minimization of objective functions under

constraints. Three types of dynamic external shades have been analyzed and compared with

the aim of reducing the solar gains on a building office glazing, while maintaining a precise

work plane illuminance level. This study confirms the result of previous studies that

dynamic shading decreases the cooling loads dramatically but tends to increase the heating

and lighting loads. The critical path for the successful application of this analysis is

choosing the correct metrics of performance and comfort adapted to the environmental

context of the building. The 2 axis spherical tracking shades have the potential to increase

view to the outside but as tested in this study, providing improved daylight access can be

a barrier to shading energy performance but the design of better shading systems can

contribute to decrease the tension between both aspects. Typically, this analysis would take

place after the building is selected, after the user’s comfort demands have been formulated

and while the shading types most suited for the orientation, latitude, room geometry and

material properties are still under consideration.

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Chapter 6:

Conclusions and future research

6.1 Introduction

In this chapter the main conclusions of this dissertation are summarized, and it is shown how

the research objectives presented in chapter 1 have been accomplished. In addition,

recommendations for further research for the use of thin shells as mechanisms are discussed in

this chapter.

6.2 Solutions to research questions

In chapter 2, the mechanisms that amplify the actuation in plants structures have been

classified in five distinct categories: differential expansion through controlled, localized

swelling and shrinking of active cells, differential expansion of a highly anisotropy layered

body under re-hydration, fast release of stored elastic energy, rigid-bar and beam-buckling

mechanisms and finally the coupling of geometry and shell mechanics. Shell mechanics

with instances of amplification of actuation (KAR) up to ~200 for Aldrovanda vesiculosa

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have proven to be the most efficient actuation amplification strategy. This exploration of

plant movements has shown that plants implement purely mechanical strategies to amplify

the actuation. The plant’s microscale mechanical properties (mainly a stiff cell wall and

the turgor pressure) and the entanglement of appropriate structural organizations at each

scale make plant tissues a genuine “smart material”. However, it is the structural mechanics

that allows for a variety of advanced structural features, some of them being usually

avoided in engineering. Mechanical couplings like torsional buckling, and elastic

instabilities like the snap-through phenomenon, testify to the great diversity of mechanisms

that can be encountered in the plant’s world. Their movements come with different speeds,

kinematics, functions (e.g. sun-tracking, growth, reproduction, predation), and can be

passive or active with numerous different triggers. Yet plant movements are often at least

several of the following: precise, reliable, repeatable and low-energy.

In chapter 3, a limit was established for upscaling compliant shells by defining a scale at

which they cannot abstract from the pull of gravity anymore. Based on the Föppl-van-

Kàrman number calculated for over 60 instances of fixed and compliant shells, thin shells

across scales are confirmed to have the invariable tendency to deform in bending over

stretching. Compliant shells, specifically, must be able to perform a repeated mechanical

deformation reliably and at a low actuating cost. The newly introduced Gravity Impact (Gi)

number led to the determination that the scale at which shells’ compliant deformations start

to be constrained by gravity is R ~ 0.1 m. Below that scale, compliant shells tend to have

high Gi values (≥ 1), which translates the gravity-independent operation. Compliant shells

of a larger scales (R>0.1 m) have elastogravity length scales smaller than their

150

characteristic dimensions, i.e. Gi < 1. They are prone to self-weight deformation under

gravity load.

In chapter 4, a methodology to identify suitable compliant thin shell geometries was

devised. The preferred modes of deformation of thin shells were identified by searching

for their low modes of resonance. The use of eigenmodes and natural frequencies to

identify inextensional modes of deformations of surfaces has been justified from a

mathematical standpoint. The methodology proposed increases the diversity of shell

typologies available to engineers by expanding the relatively rare instances of flexible

shells found in nature. Making the connection between geometry and eigenmode

inaugurates a novel approach to generate compliant shell typologies from the vibrational

physical properties of structures, a step further than the ultimately narrow imitation of

nature’s structures. Additionally, in chapter 4, using this methodology a compliant shell

mechanism based on a section of negative gaussian curvature was introduced to perform

spherical tracking. The kinematic relationship between actuation and angle displacement

was derived using finite element non-linear analysis. Furthermore, it was shown that the

geometry of the mechanism (radii of curvature and length) had a significant impact on the

performance of the mechanism, confirming that geometry was a controlling design

parameter for compliant shells.

Finally, in chapter 5, a novel optimization methodology for the control of dynamic shades

was introduced. This chapter shows that the performance of dynamic shading systems can

be modelled and tuned to reduce heating, cooling and lighting energy demands throughout

151

the year and increase the comfort of the building’s occupants. This analysis opens a path

to fulfil the currently underdeveloped potential for daylighting improvement and

substantiates the benefit of more advanced shading systems. Based on the interpolation of

environmental simulation results for the number of actuation states of the shades, the

energy demand is minimized analytically under constraints of visual comfort. Three types

of dynamic external shades were analyzed and compared (spherical tracker, typical awning

and venetian shades). This study confirms the result of previous studies that dynamic

shading decreases the cooling loads dramatically but tends to increase the heating and

lighting energy demands.

6.3 Recommendation for future research

There are many opportunities to extend the work presented in this thesis and make further

contributions to the design of thin compliant shell structures as mechanisms. This final

section describes areas of research recommended by the author to explore.

6.3.1 Automated search for compliant shell typologies

While the six common geometries explored in chapter 4 led to the selection of a geometry

that could perform a motion with two degrees-of-freedom, the automatization of the search

for inextentional deformations of surfaces should be explored. The toroidal surface of

negative Gaussian curvature was selected along with 3 other surfaces because their 2 lowest

modes of deformation showed potential for spherical tracking. Although this surface was

selected, it does not mean that it is the best for the task of spherical tracking. In a sense, the

152

choice of the surfaces was good enough at first evaluation so that the shading mechanism

was designed based on that initial finding.

The formalization of the search for surfaces would alleviate the risk of selecting a shell

geometry that is shows satisfactory kinematic behavior but not optimal. The best surface

that can be used for a shell structure satisfying the kinematics of the mechanism shall be

found by an optimization process. The formulation of this optimization problem leads to

dividing the task in two main areas: quantify the appropriateness of a structure and generate

a diversity of open shell geometries. The metric for an eigenmode’s adequacy to assume a

specific degree-of-freedom should be defined. Such a metric would most likely be based

on curvature changes of the surface as it was shown in chapter 4 that they are indicators of

the state of the deformed shell. Once a metric is defined, it can be applied to an infinite

number of geometries. Generating large numbers of open shell geometries is a real

challenge. The choice of the surfaces to be tested, will impact the outcome of the analysis.

This problem is central in many controversies in data science today, such as the biases in

training data of decision-making algorithms employed in the judicial system or child care

services [263]. If artificial intelligence or heuristic methods are to be used for this

prediction task, the training data should be carefully prepared.

6.3.2 Life cycle assessment of materials used for compliant shells

As described in chapter 3, many of the compliant shells surveyed are built using novel

materials such as carbon fiber reinforced plastics, glass fiber reinforced plastics or other

materials such as copolymers (PET, PETG) or polycarbonate. The reason for choosing to

153

use fiber reinforced plastics (FRP) are several. They provide some of the highest Young’s

modulus to density ratios in engineered material (chapter 2). The low weight is an

advantage in compliant shell structures if size is a constraint, as shown in chapter 3. Since

FRPs are engineered materials they allow the designer to define their anisotropic

mechanical properties. A principal direction can have a higher stiffness than the other, or

a direction on the surface can be favored for deformation. Finally, FRPs can be molded to

precise shapes during the manufacturing process. The typical manufacturing of these

materials is a step-by-step process that generally starts by layering the fibrous plies and

then curing the material under a vacuum in an oven. The process allows a great freedom of

forms to be created. FRPs present two main downsides. As of now they remain practically

impossible to recycle. Separating the epoxy matrix and the fibers require intense energy

input and advanced techniques [264, 265]. In addition, the manufacture process is long and

requires highly skilled labor.

The comparison of compliant shell instances in chapter 3 has shown that the dynamic

behavior of shells can be obtained from a wide range of materials. Many instances of the

compliant shells surveyed in this chapter employ PETG or polycarbonate. Those materials

can be vacuum formed [266]. A method that allows a precise shell shape to be created from

a mold. The material properties of those two materials are not as impressive as those of

FRPs. However, as the analysis showed in chapter 3, compliance in shells can still occur

with less strong material. Plant material with low Young’s modulus can display shell

compliance. PETG is fully recyclable for reuse. Polycarbonate is more difficult to recycle

[267]. They are both industrial staples, however.

154

Therefore, an extension of the work of this thesis should consider the life cycle analysis of

the materials used in the design of the compliant shells. As plastic pollution becomes a

global issue for our planet [268] it seems vital that future applications of structures with

such large potential as compliant shells should be built in a manner that takes into account

the whole life of their materials.

6.3.3 Daylight-driven optimization to improve the kinematics of shades

The methodology introduced in chapter 5 was used to compare the optimal operation of

dynamic shading under the objective to lower the energy demand of building while

maintaining satisfactory daylight levels. While this methodology allowed us to compare

the performance of shading systems, it can also be used for perfecting the design of a

dynamic shading system by measuring the usefulness of each degree-of-freedom. To the

best of the author’s knowledge, there is currently no performance-based design

methodology available to design multi-degree-of-freedom novel dynamic shading

typologies.

An additional optimization ran with the venetian shades operated by two independent

degrees-of-freedom (addition of a vertical motion of the shades to the angle with the

facade) produced daylighting results on par with the tested awning shades. The addition of

the additional degree-of-freedom improved the performance of the shading system. Ideally

a shading system would be as simple to operate as possible, i.e. one degree-of-freedom

only. The novel methodology proposed in chapter 5 could be used to find whether other

high-performance shading typologies actuated by a one-degree-of-freedom actuation exist.

155

They should produce similar results as the awning shades minus the issues associated with

this type of shades (detailed in chapter 5). Further work should also focus on multi-degree-

of-freedom shading systems to evaluate which one should be maintained and which one

could be discarded. It would be almost impossible in practice to operate a three-degree-of-

freedom system, but the methodology could help inform the designer of shades which of

the degree-of-freedom to eliminate. This approach could lead to new shading and actuation

typologies to be discovered. In the spirit of “form follows function” the field of dynamic

shading needs more performance-based design methods to generate high value novel

shading typologies and keep reducing the energy demand in buildings.

156

157

Appendix A:

Sample of cross-scale rigid and compliant thin

shells instances

In chapter 3, the mechanical behavior of a large variety of thin shell structures is evaluated

using the nondimensional numbers presented in Section 3.2. This appendix references the

thin shells included in the study. Large scale thin shell structures found in literature are

mostly rigid. In contrast, compliant thin shell structures present a characteristic dimension

in the order-of-magnitude 100m or below.

A1 Rigid engineered thin shells

Most structures considered in this section were constructed with concrete or reinforced

concrete. This material gives a good order of magnitude for the material properties used in

large scale engineered thin shells. The material properties needed for the calculation of the

𝛾𝐹𝑣𝐾 and the Gi numbers are Young’s modulus, volumetric mass density and Poisson’s

Ratio. Those values are taken from the Eurocode 2 [269] and presented in Table A1. The

values reflect commonly used values of concrete in design. They are indicative of order-

of-magnitude for the parameters considered.

158

Material Property Value

Volumetric mass density (𝑘𝑔. 𝑚−3) 2500

Modulus of Elasticity (𝐺𝑃𝑎) 35

Poisson’s ratio 0.20

Table A1. Material properties used in the calculation of 𝜸𝑭𝒗𝑲 and Gi numbers for the

engineered thin shells.

The structures selected for this study have been built throughout the 20th and 21st centuries.

They have been extensively described in [181]. Their dimensions are reported in Table A2.

159

Id Name/Location Designer Ref. Span (m) Thickness (m)

Min. Max. Avg. Min. Max. Avg.

1 Aichtal Balz, Isler [181] 42.0 42.0 42.0 0.090 0.120 0.105

2 Algeciras Sanchew Arcas, Torroja [181] 47.5 47.5 47.5 0.089 0.457 0.273

3 Bacardi Candela [181] 36.8 36.8 36.8 0.040 0.040 0.040

4 Bundesgartenschau SBP [181] 10.0 26.0 18.0 0.012 0.015 0.014

5 Lomas De Cuernavaca Candela [181] 18.0 31.0 24.5 0.040 0.040 0.040

6 Milagrosa Candela [181] 11.0 21.0 16.0 0.040 0.040 0.040

7 San Jose Obrero Candela [181] 30.0 30.0 30.0 0.040 0.040 0.040

8 Cosmic Rays Candela [181] 12.0 12.0 12.0 0.015 0.050 0.033

9 Deitingen Isler [181] 31.6 31.6 31.6 0.090 0.090 0.090

10 Florelite Isler [181] 41.0 41.0 41.0 0.080 0.080 0.080

11 GiessHauss Henschel [181] 16.0 16.0 16.0 0.175 0.320 0.248

12 Gringrin Sasaki [181] 70.0 70.0 70.0 0.400 0.400 0.400

13 Heimberg Isler [181] 48.5 48.5 48.5 0.090 0.100 0.095

14 Hippo SBP [181] 29.0 29.0 29.0 0.040 0.060 0.050

15 Hyperthreads Zaha Hadid [181] 6.0 6.0 6.0 0.080 0.080 0.080

16 Jeronimo De Castillo, de Boitaca [181] 10.0 10.0 10.0 0.070 0.100 0.085

17 Kakamigara Ito, Sasaki [181] 20.0 20.0 20.0 0.200 0.200 0.200

18 Kitagata Isozaki, Sasaki [181] 25.0 25.0 25.0 0.150 0.150 0.150

19 Kresge Saarinen, B&H, A&W [181] 48.8 48.8 48.8 0.075 0.455 0.265

20 Los Manantiales Candela [181] 42.5 42.5 42.5 0.040 0.040 0.040

21 Mapungubwe Rich, Ochsendorf, Ramage [181] 5.0 14.0 9.5 0.300 0.300 0.300

22 Rolex SANAA, Sasaki [181] 80.0 80.0 80.0 0.040 0.080 0.060

23 Rio Warehouse Candela [181] 15.3 15.3 15.3 0.040 0.040 0.040

24 Sicli Hiberer, Isler [181] 58.0 58.0 58.0 0.100 0.100 0.100

25 Teshima Nishizawa, Sasaki [181] 43.0 60.0 51.5 0.250 0.250 0.250

Table A2. Dimensions of large scale engineered thin shells included in the study. The structures are described in [181]

160

A2 Compliant engineered thin shells

More than any other type of shell structures in this study, engineered compliant thin shells

have very different material properties from one to the other. Table A3 and A4 present

those material properties and the characteristic dimensions

id Description Ref. Material Poisson's

ratio

Young's

modulus

(N/m-2)

Volumetric mass

density (kg/m-3)

1 Aldrovanda Half Sphere [10] CFRP 0.3 7.60E+10 1800

2 Snap Curved Helicoid [182] Polycaprolactone 0.4 3.53E+08 1145

3 Snap Curved Cylinder [182] PET 0.4 5.00E+09 1380

4 Flectofin [133] GFRP 0.4 2.50E+10 1800

5 Flectofold [183] GFRP 0.4 1.15E+10 1100

6 Gravity Compliant Shell [185] PETG 0.4 2.35E+09 1300

7 Multistable-Corrugated Shells [186] copper–beryllium 0.3 1.31E+11 8950

8 Multistable Inlet [187] CFRP 0.3 7.60E+10 1800

9 Yoesu One Ocean [188] GFRP 0.4 2.50E+10 1800

10 Scoliosis Brace Helix [189] CFRP 0.3 7.60E+10 1800

11 Scoliosis Brace Cantilever [189] Polycarbonate 0.4 2.90E+09 1270

12 Tape Spring [191] Steel 0.3 2.10E+11 7800

13 Stiffness Study Shell 1 [192] Acrylic 0.4 3.20E+09 1180

14 Stiffness Study Shell 2 [192] PETG 0.4 2.06E+09 1270

15 Antenna Tape Spring [190] CFRP 0.3 3.56E+10 1440

16 Collapsible Booms [184] CFRP 0.3 7.60E+10 1800

17 Deformable Mirrors [270] CFRP 0.3 7.60E+10 1800

Table A3. Material properties of the compliant engineered thin shells included in the

study

161

id Description Ref.

Span (m) Thickness (m)

Min. Max. Avg. Min. Max. Avg.

1 Aldrovanda Half Sphere [10] 0.800 1.000 0.900 5.00E-04 8.00E-04 6.50E-04

2 Snap Curved Helicoid [182] 0.025 0.035 0.030 1.00E-03 1.00E-03 1.00E-03

3 Snap Curved Cylinder [182] 0.025 0.035 0.030 1.20E-04 1.20E-04 1.20E-04

4 Flectofin [133] 0.250 0.250 0.250 2.00E-03 2.00E-03 2.00E-03

5 Flectofold [183] 1.100 1.100 1.100 1.25E-03 1.25E-03 1.25E-03

6 Gravity Compliant Shell [185] 0.050 0.100 0.075 9.00E-04 9.00E-04 9.00E-04

7 Multistable-Corrugated Shells [186] 0.100 0.250 0.175 1.25E-04 1.25E-04 1.25E-04

8 Multistable Inlet [187] 0.040 0.100 0.070 2.50E-04 2.50E-04 2.50E-04

9 Yoesu One Ocean [188] 1.300 8.000 4.650 9.00E-03 9.00E-03 9.00E-03

10 Scoliosis Brace Helix [189] 0.050 0.050 0.050 3.50E-03 3.50E-03 3.50E-03

11 Scoliosis Brace Cantilever [189] 0.070 0.100 0.085 3.00E-03 3.00E-03 3.00E-03

12 Tape Spring [191] 0.021 0.050 0.036 2.00E-04 2.00E-04 2.00E-04

13 Stiffness Study Shell 1 [192] 0.100 0.150 0.125 2.00E-03 2.00E-03 2.00E-03

14 Stiffness Study Shell 2 [192] 0.015 0.075 0.045 5.00E-04 5.00E-04 5.00E-04

15 Antenna Tape Spring [190] 0.050 0.050 0.050 2.25E-04 3.00E-04 2.63E-04

16 Collapsible Booms [184] 0.011 0.036 0.023 2.00E-04 2.00E-04 2.00E-04

17 Deformable Mirrors [270] 1.000 1.000 1.000 2.00E-04 3.00E-04 2.50E-04

Table A4. Dimensions of the compliant engineered thin shells included in the study

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A3 Compliant plant thin shells

The material properties of plant mechanisms are presented in table A5. Their characteristic

dimensions are presented in table A6. In plants, parenchyma cells are alive and constitute

the bulk of tissue in the thin shell mechanisms [39].

Material Property Value

Volumetric mass density (𝑘𝑔. 𝑚−3) 1300

Modulus of Elasticity (𝑀𝑃𝑎) 5

Poisson’s ratio 0.50

Table A5. Material properties used in the calculation of 𝜸𝑭𝒗𝑲 and Gi numbers for the

compliant plant thin shells [46]

id Name Ref Span (m) Thickness (m)

Min. Max. Avg. Min. Max. Avg.

1 Stylidium crossocephalum [146] 1.00E-03 1.00E-03 1.00E-03 5.00E-04 5.00E-04 5.00E-04

2 Stylidium graminifolium [146] 1.00E-03 1.00E-03 1.00E-03 5.00E-04 5.00E-04 5.00E-04

3 Stylidium piliferum [146] 1.00E-03 1.00E-03 1.00E-03 5.00E-04 5.00E-04 5.00E-04

4 Aldrovanda vesiculosa [46] 2.60E-03 2.60E-03 2.60E-03 4.00E-05 7.00E-05 5.50E-05

5 Dionea muscipula [46] 1.00E-02 1.00E-02 1.00E-02 4.00E-04 4.00E-04 4.00E-04

6 Utricularia Sp. [193] 1.00E-04 2.00E-04 1.50E-04 2.00E-05 4.00E-05 3.00E-05

7 Utricularia vulgaris [193] 1.00E-04 3.00E-04 2.00E-04 2.00E-05 4.00E-05 3.00E-05

8 Utricularia australis [193] 3.30E-04 7.20E-04 5.25E-04 2.00E-05 4.00E-05 3.00E-05

Table A6. Dimensions of the compliant plant thin shells included in the study

A4 Compliant micro-scale thin shells

The material properties of micro-scale compliant shells are presented in table A7. Their

characteristic dimensions are presented in table A8.

163

id Description Ref. Poisson's

ratio

Young's

modulus

(N/m-2)

Volumetric

mass density

(kg/m-3)

1 Red Blood Cell [198] 0.5 3.10E+06 1000

2 Artificial Capsules [179,

197]

0.5 1.00E+09 1000

3 Virus [177,

179]

0.5 3.10E+06 1000

4 Vesicle 1 [199] 0.5 1.00E+09 1000

5 Vesicle 2 [199] 0.5 1.00E+09 1000

Table A7. Material properties of the compliant micro-scale thin shells included in the

study

id Description Ref.

Span (m) Thickness (m)

Min. Max. Avg. Min. Max. Avg.

1 Red Blood Cell [198] 4.00E-06 1.00E-05 7.00E-06 9.00E-08 9.00E-08 9.00E-08

2 Artificial Capsules [179,

197]

1.00E-06 1.00E-03 5.01E-04 1.00E-06 1.00E-06 1.00E-06

3 Viruses [177,

179]

1.50E-08 3.00E-08 2.25E-08 2.00E-09 2.00E-09 2.00E-09

4 Vesicle 1 [199] 2.40E-05 3.00E-05 2.70E-05 5.00E-07 5.00E-07 5.00E-07

5 Vesicle 2 [199] 3.20E-05 4.00E-05 3.60E-05 5.00E-07 5.00E-07 5.00E-07

Table A8. Dimensions of the compliant micro-scale thin shells included in the study

A5 Rigid egg shells

The material properties of micro-scale compliant shells are presented in table A9. Their

characteristic dimensions are presented in table A10.

id Description Ref Poisson's ratio Young's modulus

(N/m-2)

Volumetric mass

density (kg/m-3)

1 hen's egg [194,

195]

0.3 7.24E+10 2710

2 quail egg [196] 0.3 1.05E+10 2710

3 chicken pullet egg [196] 0.3 1.48E+10 2710

4 chicken white egg [196] 0.3 2.75E+10 2710

5 chicken organic egg [196] 0.3 1.80E+10 2710

6 chicken jumbo egg [196] 0.3 2.46E+10 2710

7 goose egg [196] 0.3 1.04E+10 2710

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8 ostrich egg [196] 0.3 6.60E+09 2710

Table A9. Material properties of the avian egg thin shells

id Description Ref Span (m) Thickness (m)

Min Mix Average Min Mix Average

1 hen's egg [194,

195]

4.54E-02 5.50E-02 5.02E-02 3.50E-04 5.00E-04 4.25E-04

2 quail egg [196] 3.00E-02 3.00E-02 3.00E-02 2.20E-04 2.20E-04 2.20E-04

3 chicken pullet egg [196] 5.45E-02 5.45E-02 5.45E-02 4.40E-04 4.40E-04 4.40E-04

4 chicken white egg [196] 6.04E-02 6.04E-02 6.04E-02 3.50E-04 3.50E-04 3.50E-04

5 chicken organic egg [196] 6.04E-02 6.04E-02 6.04E-02 4.10E-04 4.10E-04 4.10E-04

6 chicken jumbo egg [196] 6.31E-02 6.31E-02 6.31E-02 4.00E-04 4.00E-04 4.00E-04

7 goose egg [196] 8.74E-02 8.74E-02 8.74E-02 6.70E-04 6.70E-04 6.70E-04

8 ostrich egg [196] 1.55E-01 1.55E-01 1.55E-01 2.55E-03 2.55E-03 2.55E-03

Table A10. Dimensions of the avian egg thin shells

165

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