analysis, construction and design of origami inspired structures

7/29/2019 analysis, construction and design of origami inspired structures

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Analysis, Design and Construction of Origami

Inspired Structures.

Rahul Vaish

School of Computing, Science and Engineering

University of Salford

This dissertation is submitted in partial fulfillment of the

requirements for the Msc degree in structural engineering.


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DECLARATION

I, Rahul Vaish declare that this dissertation is my own work. Any section, part or

phrasing of more than twenty consecutive words that is copied from any other work or

publication has been clearly referenced at the point of use and also fully described in the

reference section of this dissertation.


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Acknowledgements

I am grateful to the faculty of structural engineering at the school of computing, science

and engineering at the University of Salford for their support and guidance during the

course of my dissertation work.

I am particularly thankful to my advisor, Mr Neil Currie for his time, ideas and valuable

discussions that were a constant source of motivation and encouragement during the

period of my work.


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Abstract

The study aims to look at solutions for spatial designs in the realm of plated shell

structures. A systematic review into the Japanese art of paper folding reveals suitable

patterns that readily adapt to spatial requirement of a dome shaped structure.

An approach to model the pattern geometry over various forms has been devised andthree standard spatial forms the Paraboloid, Catenoid and the Hemisphere are compared.

Experimental investigation is thereby conducted to quantify the aspects of interest such as

stress and efficiency through a parametric study of the most suitable form with a chosen

pattern namely the Miura Ori.

Further research aims to look closely at the effects of the parameters, or the number ofcorrugations on the weight and stresses that govern their forms via a Finite Element

Formulation.

Intuitive notions of rigidity are preserved although certain behaviours such as stress

distribution with the parameters are not so straightforward. Recommendations towardsdesign and analysis have been made based on the findings of the experiments.

Keywords: Origami, Plated Shell, MiuraOri.


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List of Figures

Figure no description Page no

2.1 Cantilever barrel vaults by Diesto2.2 The hyperbolic Paraboloid2.3 Concrete Shell and formwork by Candela employing the Hypar geom-

etry2.4 Tensile membrane structure2.5 Tensile membrane millennium dome.2.6 Various configurations for a lattice grid2.7 Folded art form with its crease pattern2.3.1 Mountain and valley fold2.3.2 Origami wave2.3.3 Origami earwig2.4.1 Uniaxial base2.4.2 Flat foldability2.4.3 Non flat foldable as it self-intersects2.5.1 Spherical representation and Gaussian curvature2.5.2 Polyhedral vertex and Gaussian curvature2.5.3 Four fold origami vertex2.5.4 Developable and non -developable four fold vertexes2.6.1 The miura ori pattern2.6.2 Freeform variations in the miura ori pattern2.6.3 Fold mechanism and the parameters2.6.4 The Yoshimura pattern2.6.5 Foldable tube

2.6.6 The hexagonal pattern2.6.7 Axial crushing of PVC tube2.6.8 The pattern2.6.9 Vault with hexagonal pattern2.6.10 Vault with diamond pattern2.7.1 Reverse fold2.7.2 Reverse fold applied to create a frame2.7.3 Origami frame for the chapel of St Loup.2.7.4 Changing the pattern2.7.5 Changing the corrugation amplitude2.7.6 S shaped curves2.7.7 Barrel vault with diamond pattern

2.7.8 Vaults with varying sectional profile and height2.7.9 Miura ori vault2.7.10 Radially folded yoshimura pattern and crease pattern2.7.11 Radial miura ori pattern2.7.12 Radial diamond pattern2.7.13 Twist foldable diamond pattern2.7.14 Radially folded hexagonal cone


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2.7.15 Groin vault approximation2.7.16 Combination of radial and barrel diamond pattern2.7.17 Double curvature shapes2.7.18 Radial diagonal pattern (orientation of quadrilaterals)2.7.19 Radial diagonal pattern(crease alignment)2.7.20 Diagonal pattern(crease orientation)2.7.21 Radial hexagonal(orientation of quads)2.7.22 Hexagonal (size orientations)2.7.23 Hexagonal(shape of quads)2.7.24 Radial hexagonal(orientation of crease)2.7.26 Origami frame2.7.27 Parametric variation A2.7.28 Parametric variation B2.7.29 Yoshimura and miura ori vault with parametric variation2.7.30 Concrete folded plate structure2.7.31 IBM pavilion, Renzo Piano2.7.32 Deployable structure based on the yoshimura pattern

2.7.33 The Yokohoma cruise terminal, Japan2.7.34 The chapel of St Loup, Buri2.7.35 Assembly hall, university of Illinoi2.7.36 Artistic rendition of origami inspired market hall2.8.1 The Resch pattern2.8.2 Four fold mechanism and a detailed view of the hinge2.8.3 Triangle based designs allow more flexibility2.8.4 Proposals for rigid origami2.8.5 Fabrication of hinge using vacuumatics2.8.6 Cross laminated timber panels2.9.1 Modular origami(rigid and kinematic)2.10.1 Curved crease domes

2.11.1 Forces on folded plates2.11.2 Deformation of vertex2.11.3 coupling2.11.4 Forces and displacement at plate edges2.11.5 Two different prototypes2.11.6 Failure of hinges2.12.1 Membrane forces in a hemispherical dome2.12.2 Boundary conditions2.12.3 Variable thickness for an arch and a shell2.12.5 Hanging weight model and force polygon for an arch2.12.6 Dome of the mausoleum of Farag Ibn Barqak,Cairo2.12.6.1 Construction methodology2.12.7 Dome failure2.12.8 Vault at kings college London2.12.9 Mideaval timber vault2.12.10 Ribbed masonry dome2.12.11 Geodesic hemisphere2.12.12 Geodesic dome with plated vertex2.3.1 Structural optimisation


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3.1 Transfer of forces in a shear connection and a miura ori dome3.2.1 Radially folded hybrid dome3.2.2 Radial crease pattern3.3.1 Experimental procedure3.4.1 trapezium3.4.2 Spatial position of the plates3.4.3 Arc length discretization of the parabola3.4.4 Plate angles3.4.5 Plate areas3.5.1 Plate element capable of bending and membrane action

3.5.2 Displacement of a sector3.5.3 Forces at base3.5.5 Stress field for rotation free(P,H,C)3.5.6 Stress field for fixed rotations(P,H,C)3.5.7 Displacement field3.6.1

Boundary condition3.6.2 Range of plate boundaries3.6.3 Variations in maximum stress and deflections(case 1)3.6.4 Variations in maximum stress and deflections(case 2)3.6.5 Variation in weight and thickness with meridian segments

Variation in stresses and displacement with meridian segments

Variation in stresses(Nv=5,10,12,15,20)

Displacement field

Variation in thickness with P

Variation in stresses and displacemant

Variation in thickness and weight

Stress distribution(Nh=12,16,20,24)

Displacement fieldTypical deflection profile and the use of tie

Uneven discretization of vertical segments

Hybrid stiffenend dome

Connection details


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List of tables:


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Contents

List of Figures ............................................................................................................................. 5

CHAPTER 1 .............................................................................................................................. 12

INTRODUCTION ..................................................................................................................... 12

1.1 Structure of the Dissertation .......................................................................................... 13

1.2 Aims and Objectives............................................................................................................ 14

CHAPTER 2 .............................................................................................................................. 15

LITERATURE REVIEW ......................................................................................................... 15

2.2 ORIGAMI: Historical Development ................................................................................... 21

2.3 Origami folding: Basic Considerations .............................................................................. 25

2.3.1 Crease, Crease Pattern, Folding Pattern, Mountains and Valleys ................................ 25

2.4 Origami Design: .................................................................................................................. 28

2.4.1 Flat Foldability: ............................................................................................................ 30

2.5 Origami and Curvature: ....................................................................................................... 32

2.6 Few Common Patterns ........................................................................................................ 35

2.6.1 The Miura- Ori pattern ................................................................................................. 35

2.6.2 The Yoshimura Pattern: ................................................................................................ 38

2.6.3 The Diamond Pattern: ................................................................................................... 402.7 Origami for architectural considerations: ............................................................................ 42

2.7.1 Radial origami: ............................................................................................................. 48

2.7.2 Parametric variations in origami: ................................................................................. 54

2.7.3 Some Notable Examples: ............................................................................................. 61

2.8 Rigid Origami: ..................................................................................................................... 66

2.8.1 Kinematics: ................................................................................................................... 66

2.9 Modular origami: ................................................................................................................. 71

2.10 Curved crease: ................................................................................................................... 72

2.11 Behavior under Load: ........................................................................................................ 73

2.12 Axisymmetric Domes: ....................................................................................................... 79

2.13 Optimization of Shell Structures: ...................................................................................... 90

2.14 Conclusion: ........................................................................................................................ 93


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CHAPTER 3 .............................................................................................................................. 94

DESIGN OF EXPERIMENT .................................................................................................... 94

3.1 Choice of pattern: ................................................................................................................ 94

3.2 The Miura - Ori radial pattern: Relationship between 2D and 3D form............................. 96

3.2.1 Range of parameters: .................................................................................................. 100

3.3 Design of Experiment: ......................................................................................................... 99

3.4 Geometric modeling of the Miura-Ori pattern .................................................................. 103

3.4.1 Results ........................................................................................................................ 109

3.5 Finite Element Modeling ................................................................................................... 111

3.5.1Restraints ..................................................................................................................... 112

3.5.2 Loading ....................................................................................................................... 112

3.5.3 Results ........................................................................................................................ 113

3.5.4 Discussion .................................................................................................................. 116

3.6 Boundary conditions:......................................................................................................... 117

CHAPTER 4 ............................................................................................................................ 121

RESULTS AND DISCUSSION.............................................................................................. 121

4.1 Experiment Stage 1: .......................................................................................................... 121

4.2 Experiment Stage 2: .......................................................................................................... 122

4.2.1 Discussion: ................................................................................................................. 124

4.3 Experiment stage 3: ........................................................................................................... 125

4.3.1 Discussion: ................................................................................................................. 125

4.4 Experiment stage 4: ....................................................................................................... 126

4.4.1 Discussion: ................................................................................................................. 128

CHAPTER 5 ............................................................................................................................ 130

CONCLUSION AND FUTURE WORK ................................................................................ 130

5.1Conclusion: ......................................................................................................................... 130

5.2 Future Work: ..................................................................................................................... 133

List of References: ................................................................................................................... 146


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CHAPTER 1

INTRODUCTION

Modern engineering is advancing at a rapid pace today. New paradigms emerge as wepush our limits across boundaries which are constantly being challenged as our

understanding of the world around us increases.

With the advent of computers, the modern engineer is empowered just like a biologist is

with a microscope. Today we see a rather interdisciplinary approach to problem solving

and new discoveries in materials and computation has seen a rapid modernization in the

field of structural engineering.

The modern engineer today has variety of tools at his disposal and structures are

becoming more slender, economic and cheap. We look towards a solution to explore the

use of plated structural forms to create efficient, lightweight, modern yet aesthetically

pleasing structures as it is an important responsibility as designers and engineers of the

modern fabric of society to explore interesting forms which will have a pleasant visual

impact as well as an efficient structural mechanism.

Naturally we look towards nature as it never fails to awe and inspire. Creating efficient

utility based structure which also appeal to us aesthetically can be a topping on the cake

and Origami can help us reassert this lacking aesthetic aspect in modern designs as the

structural benefits of folding are well known.

It is with this aim in mind that we proceed to undertake a study to reveal as to what

extent are we justified to use this ancient art as an inspiration to construct modern

structures.


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1.1Structure of the Dissertation

The study begins with a brief overview of the class of structures considered, spatial

structures, their coming of age, the problems associated and some possible alternatives

which are being successfully implemented today. Then some attention is paid to the

history and development of the art of paperfolding and how it can help us realize efficient

solutions for architecture and the problems associated with transferring the concept from

paper to a buildable rigid material with regard to the difficulties associated with their

analysis and design. Some successfully implemented designs have been mentioned and

discussed briefly.

Consideration then is given to axisymmetric domes and possible solutions in origami are

considered to arrive at a suitable pattern. Next it is of interest to take a closer look at the

structural properties after deciding upon a suitable finite element model which is obtained

after deciding upon the nature of constraints and boundary conditions to be used.

Experiment is designed to quantify the aspects of interest for structural engineering such

as stresses and deflections and define an efficient structure with regard to behavior and

weight. Origami designs, being repetitive and controlled by a few parameters are suitable

for a parametric investigation and hence the experiment compares three different forms,

namely the hemisphere, paraboloid, catenoid. The best form is forwarded to be studied

for its parameters in four stages.

The dissertation concludes with a discussion and recommendations based on the findings

of the experiment suggesting possible future work in the field.


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1.2 Aims and Objectives

Aims: To arrive at a suitable pattern for a dome like spatial structure and investigate its

structural properties via a suitable experiment.

Objectives:

Undertake a comprehensive literature review of various solutions for spatialstructures to understand their behavior.

Undertake a literature review into existing patterns of folded forms to rationalize asuitable design with consideration to the behavior of the pattern.

Undertake a numerical experiment to understand the nature of stresses to compareand shortlist the best form for the design via a comparative study.

Experimentally investigate into the parameters of the pattern to quantify thestructural aspects of interest to better understand the behavior of these structures.

Suggest recommendations as per findings.


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CHAPTER 2

LITERATURE REVIEW

2.1 Spatial Structures: An Overview

Spatial structures are designed to cover large spaces. Over the years they have evolved

considerably and most modern structures today are fabricated as single or double layered

lattice grids. Over the years the art of creating space structures has seen considerable

experimentation and advancement and today we see have a number of varieties such as

apace frames, cable and strut models, air inflated, tension membranes, cable net, geodesic

domes, plated shells, folded plates etc.

The earlier designs were made from reinforced concrete while today mainly tubular

members are preferred for the lattices for they do not contribute to planar problems.

Shells can fail catastrophically so a good judgment of the forces is required for their

design. Moreover it is mostly form that contributes to strength.

Earlier designers relied on basic intuition of shell behavior and many designed the form

such that the compression field tends to neutralize the tension field in the structure. Most

notable of such designs are works of Felix Candela, Diesto and contemporaries.

Candela exploited the geometry of the hyperbolic paraboloid to create very efficient

forms (figure no 2.2). A lot many designs have been based on using a section of standard

space geometries although the configurations are infinite. Figure 2.1 shows a cantilever

barrel vaults by Diesto which has been prestressed at the crown to avoid tension.

Figure shows the millennium dome which works under tension. The principle is not new

as the use of tents made by animal skin was as early as prehistoric times. The membrane

of these structures is in a state of tension (prestress) supported by masts which go in

compression due to the pull. These are efficient, lightweight, economical structures,

highly form dependent. It is because the geometric stiffness of the membrane arising

from change in geometry and membrane prestress is more significant than the extensional


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stiffness of the material.(R.Bradshaw) These are usually optimised for dynamic

amplification. As the tensile strength of fabrics is greater for uniaxial than in biaxial

loading, the failure is mainly due to tear propagation rather than tensile rupture.

Early form finding techniques were mainly empirical in nature with limitations when

nonlinear behavior is incorporated.

Today with the advent of computers we have sophisticated techniques which can be

accommodated to run various kinds of optimization to arrive at a suitable form.

Advancement in the field of computational geometry lets us analyse the geometrical

considerations like discretizations, segmenting etc. with ease.

However shells do not lend themselves to accurate analysis very easily .When the

thickness becomes small, the shell behaviour falls into one of two dramatically different

categories; namely, the membrane-dominated and bending-dominated cases. The shell

geometry and boundary conditions decide into which category the shell structure

falls, and a seemingly small change in these conditions can result into a change of

category and hence into a dramatically different shell behaviour.(Bathe 1997)

Numerous attempts have been made over the decades to model shell behavior and one of

the most reliable model today seems to be where the bending and membrane surfaces are

treated as two different surfaces.(CR 1983). See reference for a complete discussion.

However we also have the powerful numerical technique, FEM that most modern

designers use to analyse these structures.

There are problems of buckling associated with spatial structures. For positive Gaussian

curvature models, the surface is in compression and they tend to buckle while anticlastic

surfaces have problems with material failure. Modern lattice grids have problems such as

snap through buckling which is the major concern for these types.


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One major disadvantage of all dome shaped space structures can be poor acoustic and

light capabilities.

Origami domes will be an efficient solution as they can be designed to work under

membrane action and can have certain advantages such as better acoustic performance

and light modulation capabilities.

However as far as the structural properties are concerned we would be interested to know

the nature of forces arising in the plates.

Figure no 2.1: Cantilever barrel vaults by Diesto


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Figure no 2.2: The Hyperbolic Paraboloid

Figure no 2.3: Concrete Shell and formwork by Candela employing the Hypar geometry.


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Figure no 2.4: tensile membrane structure

Figure no 2.5: Tensile membrane millennium dome


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Figure 2.6: Various configurations for a lattice grid


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Origami requires one to make a succession of folds, creating a complex pattern of creases

that turns the piece of paper into a form made up of polygonal facet, usually in three

dimensional space.

Origami has begun to find relevance beyond merely an art form and is continuing to

expand in intricacy. In the last two decades, significant technical and artistic

developments have been made in the field, attributable to a growing interest and

development of a systematic aproach to its study.

Margherita Piazzolla Beloch gave the first set of axioms in 1936 to analyse the geometric

construction of origami. This being possibly the first contribution to origami

mathematics. Later contributions wereby Huzitas when he proposed hissix axioms.

Fundamental theorems on local crease patterns around a single flat-folded vertex were

established by Jun Maekawa, Toshikazu Kawasaki, and Jacques Justin. Over the years,

several studies have explored the use of origami folding to prove geometrical

constructions and theorems. Moreover it has been demonstrated that folding technique is

more efficient in explaining geometrical proofs than traditional methods and this has

given origami a successful space in teaching and education. (Boakes.N 2010)

Mathematical Origami or technical folding as it began to be referred to, was christened

Sekkei in Japanese. Thomas Hull extended this work into the area of flat -foldability.

Robert Lang one of the pioneers in origami design, developed an algorithm around 1993

and a software program thereafter for designing origami, which he called the Tree Maker,

because it is based on graph theory that uses trees. He goes on to describe how the theory

can convert the problem of finding efficient crease pattern into one of many nonlinear

constrained optimization problems which is well researched by the computer science

community.

Lang also describes techniques such as circle packing to design efficient art forms.

He has recently published Origami Design Secrets ((Lang.R (2003)) unfolding a

computational approach to origami design.


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Mathematical origami research generally revolves roughly around two foci -foldability

and design (Demaine .E.D 2007).The first focusorigami foldability, generally asks

which crease patterns can be folded into origami that uses exactly the given creases.

Figure no 2.7: A folded art form with its crease patterns.

The simplest forms can be creases emanating from a single vertex. In these cases, we can

completely characterize which folding sequence will lead to a successful flat folded state.

The problem arises with crease patterns which have many vertices. This is where origami

sekkei or technical folding begins to play a part.

The second focusorigami design is, generally, the problem of folding a given piece of

paper into an object with certain desired properties, for example, a particular shape, and

specifically in the case of this paper, into architectural forms which can be dealt with

using the tree maker or other algorithmic procedures.


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Mathematicians today are involved in the study of the various contributions that Origami

can make to a better understanding of phenomena in the physical world. For instance a

mechanism in nature, from which we can learn much, is how leaves of some plants are

folded or rolled when un-blossomed inside the bud and how they unfurl thereafter during

blossoming. (Kobayashi H 1998)

Cedar or Beech tree leaves have simple and regular corrugated folding patterns. These

patterns can recommend ideas for the design of deployable forms and structures such as

solar panels and light-weight antennae of satellites, or for the folding of membranes such

as tents, clothes or other coverings such as large scale parasol umbrellas, which need to

be tightly packed and reduced to a small size during transportation / pre-deployment and

then, to expand to their full size at the site.

These concepts have successfully adapted by NASA to unfold satellites, car

manufacturers for airbags, cardiologists for heart stents, researchers for optic systems,

and genome scientists for DNA exploration etc. It has also been demonstrated that

origami shaped box, has better absorption characteristics and a higher buckling load than

a conventional tubular or box structure. This may have potential application in design for

impact. (Ma J and You Z (2011))

Robert Lang has used the principles of folding to create a 5m wide aperture, foldable

Fresnel transmissive telescope lens which is a prototype for an eventual 100m wide

aperture lens for a telescope. The lens is designed to be spin stabilized.

The possibilities are only increasing with each passing year as it is a very efficient way to

manipulate spatial forms, the utility of which being limited only to human imagination.


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2.3 Origami folding: Basic Considerations

Certain basics of folding are presented here which are the fundamental building blocks of

the art form.

2.3.1 Crease, Crease Pattern, Folding Pattern, Mountains and Valleys

A crease is a line segment or a curve on a piece of paper. Creases may be folded either as

a mountain or valley fold forming a ridge or a trough. (figure3)

Figure no 2.3.1: Mountain and valley fold

A Crease Pattern (CP) is a means of conveying a folding instruction.(R). It may be seen

as a collection of lines drawn on a square of paper, meeting only at common endpoints,

which is usually a graph. A crease on a crease pattern may or may not be folded.

Moreover there may not be a unique step by step folding sequence to reach the desired

form.

Many modern origami designs, particularly if they were designed using tree theory,

circle packing, box pleating, or any of the other tools of modern design, are designed in

an all-or-nothing way. The creases all work together when they are fully folded, but it


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is often the case that there are no intermediate statesno subsets of the creasesthat

can be folded together, which would form the individual steps. For such a model, the only

way to assemble the model is to precrease all of the creases, and then gently bring them

altogether at once. That method of assembly, as it turns out, is most efficient and is the

approach used for folding a model from a Crease Pattern in most cases.

And thus, a crease pattern provides the folder insight into the thought processes of the

orgami composer in a manner that a step by step sequence cannot. The figure below

shows two kinds patterns and their final folded form. In figure 2.3.2, all the creases are

precreased and folded together in cohesion to obtain the desired form.

Figure no 2.3.2: origami wave (sourcehttp://www.langorigami.com/art/gallery/wave24)


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Figure no 2.3.3: origami earwig (source:

http://web.mit.edu/chosetec/www/origami/earwig/)

Figures 2.3.2 and 2.3.3 show paper models and their respective crease pattern. However

they are different in certain aspects. All creases in figure 4 have been folded while certain

lines in figure 5 are merely for understanding purposes.

A Folding Pattern (FP) is an identification of which creases should be folded as

mountains and which as valleys. Together, a Crease Pattern (CP) and a Folding Pattern

(FP) describe a MountainValley Assignment (MVA).


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2.4 Origami Design:

To design an origami model, one mainly needs a crease pattern and folding assignment

that will allow us to reach the final model. For various different types of models there

may be a particular approach that may be most suitable. To understand the process one

needs to be familiar with the fine interplay between the 2D representation of the fold

lines and the corresponding enclosed facet that is sent in the 3D representation upon

folding. Although difficult to a first reader, the subject is quite friendly and interesting.

The subject has received methodological treatment by pioneers in the field and one of the

most sophisticated technique may be the tree maker developed by Lang to fold insects

and figures.The main idea is to decompose the folding process into two steps: first fold a

base that roughly distributes the paper in the right places, and then shape the base

into the actual Origami model.

The input for Tree Maker is a two-dimensional stick figure that captures the essential

features of the target object. Theoretically, the stick figure is a weighted tree, the weights

being the lengths of the various edges. The computer works out a way to fold a square of

paper so that the result, which is called the base, sits exactly over the stick figure. In the

final (nonalgorithmic) step, standard origami techniques are used to flesh out the base,

making the model aesthetically satisfying and realistic.

Consider, for example, the design of a starfish where the task for Tree Maker, in effect,is

to pack five circles of equal radius inside a square.


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Figure no 2.4.1: Uniaxial base (source (Shenk.M 2008))

It does so by formulating the design challenge as a problem in nonlinear constrained

optimization, turning the weighted tree into a set of algebraic equations.

It finds a local maximum, Lang says. And with a little bit of human intelligence, you

can convince yourself that youve found the global optimum.(Lang.R (2003))

Two important conditions that need to be satisfied for a 2D surface to be able to go in a

3D folded state when referring to a CP or an FP are, that the conditions of Isometry and

Non-Crossing.(Demaine .E.D 2007)

Isometry here implies that the distances between two points, measured by the shortest

path on the surface of the paper, are preserved by the mapping, i.e., the mapping does not

shrink or stretch the paper. The Non-Crossing condition specifies that the paper does not

cross through itself when mapped by the folded state. Portions of paper are allowed to

come into geometric contact as multiple overlapping layers, but the layers must not

penetrate each other, i.e., the mapping must not tear or cut through the paper.

In folding an assignment one may have to twist one or more facet or tuck one facet

beneath other. This may not be possible if one is working with a rigid material like

cardboard.


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Also one may wonder, given any arbitrary crease pattern, is it possible to find a mountain

valley assignment that sends it to a flat folded state?

Origami foldability is governed by certain interesting rules and axioms which will be

presented below.

2.4.1 Flat Foldability:

One can speak of local and global flat foldability. For a structure to be deployable it

needs to be flat foldable and therefore this condition is of prime interest. Two important

theorems which provide a necessary but not sufficient condition for flat foldability are

(Hull.T 2010):

Theorem 1: Kawasakis theorem: A single-vertex crease pattern defined by angles 1 +

2 + n = 2 is flat foldable iff n is even and the sum of the odd angles (2i+1) is

equal to the sum of the even angles (2i), or equivalently either sum is equal to .

Figure no 2.4.2: Flat foldability


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Theorem 2.Maekawa-Justin theorem:In a flat-foldable single-vertex mountain-valley

pattern defined by angles 1 + 2 +. ..+ n = 2 , the number of mountains and the

number of valleys differ by 2. (MV= 2)

These two theorems although necessary are not sufficient a crease pattern can be found

that satisfies the above two theorems but is not flat foldable.

Figure no 2.4.3: non flat foldable as it self-intersects.


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2.5 Origami and Curvature:

For a curved surface, Gaussian curvature is useful to represent the curvature as an

intrinsic property of the surface. (Cohn-Vossen.S 1952))

The Gaussian curvature can be estimated by using the spherical representation as

illustrated in figure no 9. A closed loop is drawn on the curve around the point under

consideration and its spherical image is mapped on a unit sphere by mapping the normals

at selected points. It the sense of traversal of the points is same it is said to have a positive

Gaussian curvature, otherwise negative.

Figure no 2.5.1: spherical representation and Gaussian curvature.

Mathematically it can be expressed as the limit

0FGk LimF

or1 2

1 1kR R

Where1

R and 2R are the principle curvatures for the point considered.


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The idea can be extended to creased surfaces as well. Miura in his paper, demonstrates

the use of the principle to validate the existence of a set of creases with a given mountain

valley assignment. Since a folded surface is developed from a sheet, the Gaussian

curvature of the folded form should exhibit zero Gaussian curvature as the curvature of

the flat sheet is zero. For a single and two fold lines at a vertex, this is easily shown in

figureto be zero as the spherical representation is an arc enclosing nil area.

For three vertices, if all the normals have different orientations, then the enclosed area

cannot be zero and this case is not admissible for origami.

For four fold vertex, first consider a roof vertex with all mountain folds. The spherical

representation has a positive area and hence it cannot be developed. Furthermore

Calladine demonstrates that the solid angle subtended by the roof is merely the spherical

excess which is given by1

2n

i

i

, and the Gaussian curvature is given byA

, where

A is the area associated with the vertex. (CR 1983). This does not change by addition of

creases and the curvature is entirely contained in the vertex. The Gaussian curvature is

invariant under in extensional deformation.

Figure no 2.5.2: polyhedral vertex and Gaussian curvature


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Figure no 2.5.3: Four fold origami vertex

Miura demonstrates that the only developable configuration with four vertices is three

mountains and one valley. It is the simplest origami. Furthermore they cannot be

orthogonal otherwise the folding cannot be simultaneous. A fourfold inclined vertex is

also the generic case when a piece of paper is crushed. It possess only a single degree of

freedom and this remarkable property makes the miura ori pattern, which is a repetition

of identical four fold units(see later) simultaneously deployable in two directions.

The figure below shows two paper works make by the four fold vertex. The first uses the

foldable vertex and is developable while the second work cannot be developed.

Figure no 2.5.4: Developable and non-developable four fold vertex designs.


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2.6 Few Common Patterns

Literature review has suggested that few patterns are more useful than others. These

patterns arise from a systematic study of buckling behavior of plates and cylindrical

shells and are therefore flat foldable and can be readily adapted to suit architectural

needs. Three common patterns are described below namely the herringbone or Miura-

Ori, diamond and hexagonal pattern.

2.6.1 The Miura- Ori pattern

Take a piece of paper and crumple it. Unfold it and observe the pattern of the folds that

appear. Normally these will be a superimposition of various modes of buckling of the

sheet but the problem, devised as an experiment to study the buckling modes of an

infinite elastic plate uniformly compressed from all sides has been studied thoroughly and

various modes have been isolated and studied for their energy index. i:e the amount of

energy taken to deform in a particular mode. It was found that the miura ori pattern has

the lowest energy index and is the mode that requires the least energy for distortion

(Koryo.M 2009).

One of the most unique properties of this pattern is its deployable property. (Koryo.M

1989)

1) It can be deployed simultaneously in two orthogonal directions.

2) It possesses only a single degree of freedom.

3) The deployment and retraction follow the same path.

It follows that when we deploy a sheet from a flat folded state to become a planar surface,

the miura pattern will strain the material least. Combined with the 1DOF mechanism, this

pattern has been successfully used in space solar panels. Also given the fact that the

resulting planar quadrilaterals form a good criterion for rigid foldability (Naohiko and

Ken-ichi 2009), its variations can be explored to design rigid architectural origami. The


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flexibility of the pattern can be made clear from the following figure where a parametric

variation allows a flexible adaptation to an arbitrary surface. The figure below shows the

miura ori sheet and its crease pattern with variations of the angles (also referred to as

miuras). The flexibility of the pattern to adapt to various surfaces can be indicated by

figure no 21. The miura ori is the simplest and perhaps the most useful pattern.

Figure no 2.6.1: The miura ori pattern


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Figure no 2.6.2: Freeform variations in miura ori pattern (source (Sternberg.S)


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2.6.2 The Yoshimura Pattern:

This pattern was observed to appear on the surface of a cylinder while buckling under a

twisting moment. The transformation happens via bending only at the folds and not

stretching i: e the process is purely an origami. The pattern parameters, the fold angle and

the number of segments were sensitive to the length of the cylinder made to buckle and

the pattern provided post buckling strength. The pattern is also called as the Yoshimura

pattern as it was first discovered by Yoshimura.

Figure no 2.6.3: Fold mechanism and the parameters.


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Figure below shows the crease pattern and its application to create a folded tube

structure.

Figure no 2.6.4: The Yoshimura pattern

Figure no 2.6.5 : Foldable Tube


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2.6.3 The Diamond Pattern:

This pattern arises out of the buckling of a cylinder under axial compression as shown in

the figure below.

Figure no 2.6.6: The hexagonal pattern. Figure no 2.6.7: axial crushing of PVC tube

There are two possible representations of the pattern as shown below. The hexagonal and

the diamond case. They differ in the valency of the vertex. Under repetitive and

symmetrical conditions, both the patterns are flat foldable. These patterns are very

flexible and a large amount of variations is possible to produce varied shapes and designs

in space. The diamond case is very similar to the Yoshimura pattern but the difference

lies in the orientation of the folds.


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Figure no 2.6.8: The crease patterns

Figure no 2.6.9: Vault with hexagonal pattern

Figure no 2.6.10: Vault with diamond pattern


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2.7 Origami for architectural considerations:

For architectural applications we want the folded configuration to enclose a certain

desired space and have some structural rigidity to withstand loads.

It should be noted that all folds are derived from origami. Although folded plate models

for roof spans are well implemented, complex variations in architecture and structural

engineering have not been very popular, probably due to demanding geometries and

complex requirements of the design layout.

Before proceeding further it is desirable to introduce a basic kind of fold which has been

exploited by spatial designers. A figure 9 describes the reverse fold which can be made in

two ways and the spatial position is controlled via the angle shown.

Figure no 2.7.1: Reverse Fold


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Figure no 2.7.3: Origami frame for the chapel of S.t loup.

The cross sectional profile can be modelled over various curved profiles to create barrel

shaped vaulted geometries, example figure no 2.6.9.

In the patterns thus produced, each vertex is a four valent vertex i: e there are four creases

meeting at a given vertex. If the distance between the apex and the next base is collapsed

to zero we get a changed pattern with a six valent vertex as demonstrated in figure no

2.7.4.

Figure no 2.7.4: changing the pattern


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This also has certain implications on the depth of the plates. Figure no 2.7.5 shows the

side view of the pattern, from the gable direction, where one can define a term called the

amplitude of the corrugations denoted by A.

When this amplitude acquires the maximum available value, we change the pattern and

the valence of the vertex with an increase depth of the plates.

Figure no 2.7.5: Changing the Corrugation Amplitude.

The pattern changes from the hexagonal to the diamond pattern. Figures no 2.7.6 show

certain S shaped curves modelled with a hybrid pattern formed with the hexagonal and

diamond pattern. As the number of divisions increase, the curve is modelled more

closely. If we change the sectional profile to vary, we can map curvatures of double

curvature and hence a lot of flexibility can be achieved by manipulating the sectional and


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the corrugation profile. Figure 2.7.8 show two different, maximum amplitude patterns

modelled over varying section and varying height.

Figure no 2.7.6: S shaped curves.

Figure no 2.7.7: barrel vault with diamond pattern.


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Figure no 2.7.8: Profiles with varying section and heights with diamond pattern.

. Figure below shows a barrel vault made from the herringbone pattern.

Figure no 2.7.9: Miura Ori vault

The crease pattern can be interpolated from the space configurations by taking

appropriately scaled projections on the plane below. This gives an easy way to fabricate

the sizes of the panels required for the implementation of the design.


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2.7.1 Radial origami:

Although until now we have produced geometries of double curvatures, slightly different

configurations are achieved by using the radial form of the above stated patterns.

Axisymmetric domes of revolution can be obtained by folding the radial patterns. Figures

below show the radially folded diagonal pattern and miura ori pattern.

Figure no 2.7.10: Radially folded yoshimura pattern and crease pattern.(dashed lines

are valleys)


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Figure no 2.7.11: radial miura ori pattern and crease pattern.

By changing the inclination of the diagonals in figure 2.7.10, we produce geometries with

varying curvatures or the climb angles of the pattern. It suggests that by varying the

angles different geometrical configurations can be approximated.

We can only approximate the curves since these surfaces are not developable.

Figures no 2.7.12 and 2.7.13 show the radial diamond pattern and its variation. The first

dome can be folded along its circumference, like a curtain while the second dome can be


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folded by applying a twisting moment at the top. Notice the difference in the seam

arrangements of the variation.

Figure no 2.7.12: radial diamond pattern.

Figure 2.7.13: twist foldable diagonal pattern

In an architectural study (Mitra.A 2008-09), the author has tabulated a list of various

architectural geometries that can be mapped by various arrangements of crease patterns.


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Several interesting shapes have been made possible and the figures below are few

important illustrations taken from the study.

Figure 2.7.14: Radially folded hexagonal cone.

Figure no 2.7.15: Groin vault approximation with diamond pattern


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Various interesting shapes can be achieved by combining the radial pattern with others.

Figure no 2.7.18 shows a combination of radial and barrel diagonal pattern.

O.Tonon in his paper geometry of spatial folded form describes the formation of different

shapes using the combination of the above patterns. His approach is somewhat different

as he begins with the crease pattern in 2D state. Figure no 2.7.19 shows a picture taken

from his study.(O.Tonon 1993)

He remarks about the rigidity of the pattern with different sectional profiles. As the

profile closely models the curve, the rigidity begins to decrease and vice versa. There is

an intermediate form where the rigidity is proper.

Figure no 2.7.16: combination of radial and cylindrical pattern.


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Figure no 2.7.17: Double curvature shapes (source Tonon)


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Figure no 2.7.20: diagonal pattern (crease orientation)

At times it becomes quite puzzling to classify the pattern, especially while considering

parametric variations as Yoshimura or diagonal pattern as they are very similar.

Figure no 2.7.21: Radial hexagonal (orientation of quads)


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Figure no 2.7.22: Hexagonal (size, orientation)

In the above figure the quadrilaterals have to be twisted to accommodate the shape. They

will not be planar.

Figure no 2.7.23: Hexagonal (shape of quads)


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Figure no 2.7.24: Hexagonal (shape of quads)

Figure no 2.7.25: Radial Hexagonal (orientation of crease)

Various other shapes can be realised and a complete list can be found in the study. These

tubes have larger applications in robotic systems.

Origami structures, as they suggest lend themselves to parametric analysis and design.

For any given architectural functionality requirement a number of forms can be derived

by choosing to vary the parameters that may govern the design. For example in a frame

design we can choose to vary the corrugation amplitude, corrugation profile or the cross

sectional profile to produce a variety of different forms.


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Figures below illustrate an example of a frame, showing the original sectional and

corrugation profile and few of the variations.

Figure no 2.7.26: origami frame

Figure no 2.7.27: parametric variation A


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Figure no 2.7.28: parametric variation B

Naturally we are faced with a problem of selecting the best possible configuration that

serves our purpose most efficiently. Little literature is available regarding the design and

analysis of these kinds of structures and a general framework is still lacking.

Efficiency is usually described as the strength to weight ratio of a structure. Origami

structures are an arrangement of plates that are stiff due to their orientations and folded

plates are considered a lightweight solution with a high span to depth ratio.

But the structure has various hinges at odd angles and as it may be quite intuitive that the

fabrication of hinges will play an important role in the design of these structures.

Change in parameters most often changes the plate configuration as well as the hinge

characteristics (the angle and their number). For example in the above example if we

choose to use the maximum amplitude of corrugations, we will deepen the plates,

increase the total length of creases and increase the valency of any joint. Additionally, if

we flatten the corrugation amplitude, we will flatten the plates and change the angle of

hinges. Which one of these will produce a more efficient design is not very easy to judge

by intuition alone.

However certain obvious parametric changes can be readily seen and implemented to

strengthen our design. Figure below shows a barrel vault with a sharp variation in


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corrugation profile as compared to the inner part. This produces plates with larger

inclination, hence stiffer. This helps minimise the deflection observed at the edges.

Also compare the total no of hinge lines and their load bearing characteristics. The miura

vault seems to be heavier as it has finer corrugations. But then the load carrying

mechanism is different and hence the load to be borne at hinges also differs.

Figure no 2.7.29: Yoshimura and Miura Ori vault with parametric variation.


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2.7.3 Some Notable Examples:

Figures below depict some successfully implemented folded plate designs. The list is not

exhaustive but several different types of ideas are captured.

Figure 2.7.32 shows a concrete folded plate building The construction on site of the

structure was complicated and needed time consuming guidance of the designing

engineer. The sheet thickness was just 7cm. Yet, this approach enabled a material saving

construction.

Figure no 2.7.30:Concrete folded plate structure in Neuss-Weckhofen, Polny/Schaller, 1969

In contrast, Figure 2.7.30 shows theIBM Pavilion which was assembled and

disassembled various times for a touring exhibition in twenty European cities. In this

example folding represents a practicable light-weight construction principle. (Trautz. M

2011)


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Figure no 2.7.31 shows yet another deployable structure based on the yoshimura pattern used for

a travelling fair. The crease lines have been replaced by linkages to produce a truss like

mechanism.

Figure no 2.7.31: IBM-Pavilion, Renzo Piano

Figure no 2.7.32: Deployable structure based on yoshimura pattern


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Figure no 2.7.33: The Yokohoma cruise terminal, Japan.

Figure above shows the modern Yokohoma cruise terminal, Japan which used a hybrid

structural system of steel trussed folded plate and concrete girders especially adequate in

coping with the lateral forces generated by seismic considerations.

Figure no 2.7.33 shows an origami structure made of cross laminated timber panels. The

joints are fabricated from nail using steel plates and several prototypes are studied in the

study (Buri H 2009). The roof plates are 60mm thick and span 9 meters (slenderness of

1/150) while the wall plates are 40mm thick. The enormous efficiency of folded plate

structures is evident by this very high slenderness. Figure 2.7.34 shows a concrete folded

roof which uses the principles of origami. Figure 2.7.36(a, b) shows a Miura Ori shell

made from timber plates which have been stiffened at the folds from inside.


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Figure no 2.7.34: the chapel of St Loup. (Buri)

Figure no 2.7.35: Assembly hall, university of Illinoi.


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Figure no 2.7.36(a,b): Miura Ori shell structure with stiffened plates, Germany.


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2.8 Rigid Origami:

Rigid origami is when the design is realized through rigid, thick panels and not

necessarily developed from a sheet. It is essentially a plates and hinge model for origami

where plates do not stretch or bend and there is synchronized motion between all the

plates. A classic example is the everyday shopping bag which can only be folded by

bending the facets and not otherwise. Rigid foldability deals with the question of

existence of a route from an unfolded state to a final folded state.(Naohiko and Ken-ichi

2009)

For architectural application the main problems encountered are to accommodate the

finite thickness of plates. It is especially nice if we develop a 1DOF model so that the

deployment can be semi-automatic.

2.8.1 Kinematics:

In order to apply kinetic rigid origami to various architectural and other engineering

purposes, one must consider the geometry of the plates in motion and providegeneralized methods that produce controlled variations of shapes that suit the given

design conditions.

Currently most deployed constructions are based on the use of textile materials as the

deployable element or completely rigid building elements, which can be removed

entirely. Deployable structures using folded plate constructions are rarely realized despite

the fact that it is possible to create wide variety of high performances structures with

enclosing and formative character. The articulated design of the folds allows the structure

to provide kinematical properties. Thus a structure can be designed which combines the

advantages of folded plate structures with the possibility of a reversible building element

through folding and unfolding [(Trautz.M 2009)]


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Several designs of rigid-origami structures have been proposed from around 1970s. For

example, the developable double corrugation surface (DDC) Miura-ori pattern. Resch and

Christiansen have proposed a kinetic plate mechanism, folded from a planar sheet that

forms a three-dimensional existing in two different configurations, one with and other

without curvature.(figure 2.8.1).(Resch.R (1971))

This is particularly useful if we are looking towards morphing structural forms to allow

modulation of light, acoustics etc. A number of such patterns exist and are studied under

the context of origami tessellations which can exist in two as well as three dimensional

states. An engineering approach based study can be found in (Shenk.M 2008).

Figure no 2.8.1: The Resch pattern

.

Rigid origami transforms in a synchronized motion based on multiple non-linear

constraints and the design of rigid origami is not a trivial problem given by an arbitrary

design approach without geometric considerations remarks Tachi who has developed the


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rigid origami simulator which can allows us to simulate the motion of an origami and

detect collisions. (Tachi.T (2009))

As for every deployable structure, the loads bearing capacity is reduced due to the

influence of the characteristics resulting from the associated structural design. The

principle load bearing characteristics of folded plate structures normally include two

aspects which cannot be generated.

One is the bending resistance between the plates in the area of the fold. Since the hinges

are articulated and we cannot allow any rigidity, this characteristic is disabled. The

second aspect results from allowing translational hinge deformations depending on the

chosen folding pattern. Most often, hinges will not allow adequate transfer of shear forces

in the folds or unbalanced live loads on the structure, which will reduce the load bearing

capacity.

The deformation of the folded plate structure under dead load results in a self-

deployment and therefore in further hinge translation. The figure below illustrates the

difficulties encountered at a typical hinge in a quadrilateral based design of rigid

origami.Rigid origami can be realized through triangle based or quadrilateral based

design approach where the former are more flexible.(Tachi.T 2010). Figure no 2.8.3

illustrates this fact via hypar designs which cannot exist in these states if they are not

triangulated.


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Figure no 2.8.2: Four fold mechanism and a detailed view of the hinge

Figure no 2.8.3: Triangle based design allow more flexibility

The problem of hinges has been dealt by Tachi and he proposes several articulation

techniques to allow for realizing thick origami with finite thickness. (Tachi.T 2011).

Figure 2.7.42 shows few proposals to accommodate finite thickness rigid origami where

the hinges are put on the valley side and the edges are trimmed to accommodate folding

motion.


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Figure 2.8.4: Proposals for Rigid Origami

One of the latest studies involves exploring the use of vacuumatics to allow for the

complex and demanding stiffness characteristics of the hinges to allow for deployment

and strength. (Tomohiro Tachi 2011).This way a number of flexible configurations may

be possible by varying the boundary geometry and support conditions. The basic idea is

that of a double membrane structure where the compression forces on the valley side can

be manipulated via a vacuum. This varies the stiffness of the structure. This compressive

force is balanced by the tension force on the hinges due to external load.

Figure 2.8.5 illustrates such a hinge. The valley side of the hinge is filled with inflatable

material. When vacuumed, moment is generated this aids folding and can be controlled to

stiffen the hinge.

Figure no 2.8.5: fabrication of hinge using vacuumatics.

However an interesting development is the fabrication of origami inspired design of

folded plate geometries with using cross laminated timber panels. However only static

properties of these forms have been used where the main concern will be to design

efficient connection methodologies based on numerical studies.(Haasis. M 2008)

The problem is again not trivial as it requires calculating the offset based on the angle and

thickness into the cutting machine.


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The process is cumbersome if the angles and thickness vary.

Figure no 2.8.6: cross laminated timber panels

2.9 Modular origami:

Modular origami is about building shapes using a basic origami unit as a starting point.

These manufactured folded modules are built into spatial forms. The form variety reaches

over simple platonic bases such as cubes and tetrahedron up to complex polyhedrons.

The advantage is that the total form is always based on basic geometrical form and is

mathematically recordable. The number of different modules and detail connections are

limited here and are therefore organized for planners and engineers.

Rigid as well kinetic structures are possible in modular origami and several domes have

been realized.


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Figure no 2.9.1: modular origami (rigid and kinematic)

2.10 Curved crease:

Curve creases on paper also show promising possibilities in the domain of designs of

structures from paper models. Curved folds are characterized by larger DOF than regular,

straight line origami. Folding flat sheets of paper at a small scale along curved creases

results in shapes with remarkable strength and stiffness properties yet their deployment

trajectory is not understood. Curved folding is an advantageous product of folding and

bending: the surface consisting of developable surface patches and a relatively small

number of separate curved creases can have a single degree of freedom only.

Figure no 2.10.1: curved creased domes


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Let the total vertical deflection be with components 1

and 2

. Since the plate is very

rigid for in plane actions, 2

is small and correspondingly 1

is also small.

Consider the pure bending action where a reaction force is obtained at the hinge aligned

with W1 but in the opposite direction. This reaction has a component R2 which is resisted

by in plane action so the coupling is evident.

Figure no 2.11.3: Coupling

The hinges encounter shearing forces and transmit bending moments from one plate to

another. Ideal analysis will incorporate a rotational stiffness and a translational stiffness

at the hinge.

If the joints are rigid and deflections are small, a linear elastic solution under the

assumption of superimposition to hold can be used to arrive at an exact solution for

simply supported plate system using an elasticity formulation.

The method assumes four displacements and four forces at each edge of the plate which

are analyzed individually. The figure below illustrates the displacements (threetranslations and a rotation) and the corresponding forces for the system shown in the

local element coordinates.


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Figure no 2.11.4: Forces and displacement at plate edges

The displacements for the longitudinal direction(x) can be represented by a half range

Fourier series as shown below. These confirm to the boundary conditions and the analysis

essentially becomes one dimensional.

0

( )( , ) ( ) cosm

m

m xu x y u y

L

0

( )( , ) ( ) sinm

m

m xv x y v y

L

0

( )( , ) ( ) sinm

m

m xw x y w y

L

These are related by the equilibrium equations for plate bending and plane stress as:


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1)4 4 4

4 2 4 42 0

w w y

x x y y

2) 0xyx

x y

3) 0y xy

y x

Where

2

( )1y

E v u

y x

2

( )1x

E u v

x y

( )2(1 )xy

E u v

y x

E= Youngs modulus

= poisons ratio.

The analysis assumes that the forces are applied at the joints also expressed as a Fourier

series. The equations are solved by equating the in plane translations normal andtangential to plate edges, the normal translations of the edges and the rotations for

symmetric and anti-symmetric cases in the global coordinate system.

The author demonstrates that the solution thus produced matches well with a finite

element solution obtained by using elements capable of in plane and bending capabilities.

The idea can be extended to folds in both edges as well, although no attempt as such was

found available for the elasticity method.(Hassan.A 1971)


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Figure no 2.11.5 illustrates two different types of prototypes, the first folded from metal

sheet where the connections have a finite rigidity along their length. This is because the

behaviour of material at a fold is like a cylinder of very small radius in compression. It is

therefore difficult to crumple a sheet of paper beyond a certain limit due to the buckling

force required to break a fold increases with diminishing length of fold.

In the second type, timber plates are joined via nails and longitudinal and transverse shear

forces are resisted by the nails at the connection. Their behaviour and failure mechanism

are also different. The overall behaviour of a folded plate structure and the choice of a

failure mechanism depend on the distribution of stiffness of the folds and the plates and

the type of connections, if any.

Figure no 2.11.5: Two different prototypes(source Buri)

The results for the metal prototypes are not available but the timber prototype was studied

via the finite element method.

A test model of a single hinge was loaded to find the stiffness against rotation which was

fed into the FEM procedure. It was found that the prototype was much more flexible than

the computer model.


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Moreover three different softwares used showed different results for hinges with

rotational freedom but two of them aligned for rigid hinges suggesting that analysis for

rigid hinges is more reliable.

Figure below shows the prototype rupture. The folds have opened up indicating that the

joints needed strengthening.

Figure no 2.11.6: failure of hinges


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2.12 Axisymmetric Domes:

Axisymmetric domes are a class of spatial structures which are generated by revolving acurve about an axis and they are also called shells of revolution.

Dome structures have a long history and the earliest occurrences can be found in parts of

Greece, Egypt, China and India. These structures have served as roof spaces for places of

worship and were an inherent part of medieval architecture. Early designs utilized cut out

stone, sun dried mud, for their construction.

In modern days these shapes are usually utilized for storage, funerary, and utilitarian

purposes such as defense, storage, kilns etc. these are robust shapes but modern

architectural forms have seen several variation such as lattice formed, geodesic shapes

etc.

Elementary load carrying mechanism may be understood by visualizing the shape as a

number of arches in different planes. There is a horizontal thrust at the base which is

taken care of by providing a ring which works in tension. The structure then only

transmits vertical reaction forces. (Figure2.12.2 b). Figure 2.12.1 below shows typical

forces under equilibrium analysis in a spherical dome under symmetric loading. These

are membrane forces, assumed to act in the middle plane of the shell. The meridian forces

are compressive all along and increase as we move down and the hoop forces are

compressive at top and tensile at the bottom. For a hemisphere the angle of change is 51.5

degrees. The forces on a differential element and expressions for the meridian and hoop

forces are shown in the figure

Full shell analysis is rather complex and the reader is referred to (CR 1983)but under

large curvature the surface of the shell exhibits rigidity towards in plane stretching and

usually these forces are dominant if the thickness to radius ratio(h/R) is <


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closed surface. In addition some localized moments do occur near the edges(c) which are

taken care of by providing additional thickness (figure2.12.4). However, the response of a

shell to localized forces is by bending which can cause significant bending stresses.

Figure no 2.12.1: membrane forces in a hemispherical dome

Figure no 2.12.2: boundary conditions


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1 cos

aqN

1

( cos )1 cos

N aq

Figure no 2.12.3: element equilibrium

Figure no 2.12.4: variable thickness for an arch and a shell

Historical methods of analysis are not available but the ancient builders had a command

over the forces and geometry. Studies available regarding the investigation of

construction techniques of the domes of Cairo suggest the ancient designs are indeed

quite elegant for example the figure 2.12.6 depicts the dome of the Mausoleum of Farag

Ibn Barquq which has stood for over hundreds of years. The dome is less than 15 inches

thick and spans 47 feet. (h/R=0.02) and sits on a 25 feet cylindrical wall. The dome has

no reinforcement despite the fact that modern analysis predicts it to be unstable without

tensile reinforcement.(.W.Lau)


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Earliest reported techniques have been mainly empirical in nature such as graphic statics,

utilizing the force polygon to find an equilibrium solution.

The method works on finding a thrust line in accordance with the geometry and the

weight distribution of the dome via a force polygon construction. The line of thrust is a

hypothetical path over which the internal forces transport the external loads to the

supports. A geometrical construction is demonstrated in figure 2.12.5 where an inverted

cable with distributed weights assumes a form which will stand in compression as

demonstrated by Poleni in 1748. He concluded that if this line lies within the effective

thickness of the structure, it will be safe.

This lower bound approach is quite safe as in neglects the hoop forces which have a

stabilizing effect on the dome.

Figure no 2.12.5: hanging weight model and force polygon for an arch.


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Figure no 2.12.6:Dome of the Mausoleum of Farag Ibn Barquq, Cairo.

This basic idea has been refined by researchers like Eddy, Heynmenn Wolfe to

incorporate hoop forces obtain upper and lower bound theorems for analysis of masonry

domes.

The methods of construction are illustrated in the figure below, from a study devoted to

the understanding of these domes. Masonry needs to be supported over a formwork

before it can acquire strength. Modern bricks are made from clay and dried at a furnace

before they acquire strength. Masonry can fail in shear by slipping and in tension by

rupture. However it is extremely efficient for compression. Ancient masons had a good

knowledge of forces for they aligned the joints in a staggered fashion to avoid lining the

joints with the flow of force. Modern masonry is usually reinforced by steel by using

especially manufactured bricks which provide grooves and holes in them.


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Figure no 2.12.6.1: construction methodology(source(Cipriani.B 2005))

Ancient and medieval architecture has withstood the test of time. Figure 2.12.8 depicts a

vault at kings college London where the thickness to span ratio is less than that of an

eggshell speaking of a very daring design.

Figure 2.12.7 shows a typical collapse mechanism for a dome where first cracks occur

along meridians, separating them into lunes (pie shaped arches). Then cracks along the

hoop develop forming a mechanism. The top part bends inwards while the bottom part

rotates outwards. This mechanism suggests that reinforcement elements along the

meridians will be important for a strong design.


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Figure no 2.12.7: Dome failure

Later designs saw domes to be reinforced along the meridian for example there are

several ribbed domes where the stiffest element is along the meridian. Figure 2.12.9

depicts a medieval timber vault. Figure 2.12.10 depicts a ribbed masonry dome. A large

number of designs have been successfully implemented by meridian reinforced elements.

Figure no 2.12.8: Vaults at the Kings College London


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Modern versions of domes are articulated lightweight lattice structures which have

elements along meridian and hoop directions and are analysed through computer

programs.

Figure no 2.12.9: medieval timber vault.


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Figure no 2.12.10: Ribbed masonry dome

In 1951 fuller introduced the geodesic dome which was a paradigm shift in dome

construction. The dome is formed by triangulating the facets of an icosahedron and

mapping them to the surface of a sphere. The points thus lie on the geodesics and this

proved to be a very efficient load carrying mechanism. The triangulation can be done in

two ways as shown and are referred to as class 1 or class 2 subdivisions.

Fuller claimed that geodesic domes built upon principles embodying force distributions,

similar to those of atoms, molecules, and crystals, would form the lightest, most efficient

forms of construction.


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Figure no 2.12.11: Geodesic hemisphere

While radial domes exhibit greater stiffness for uniform loads, geodesic domes exhibit

larger stiffness for non-uniform loads. The forces in a geodesic network are a

combination of tension and compression, tension forces being global and continuous,

while compression forces are local and discontinuous.(Kubik 2009)


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Figure below shows a geodesic surface formed with plated vertex and stiffened by

tubular members.

Figure no 2.12.12: Geodesic dome with plated vertex


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2.13 Optimization of Shell Structures:

Shell structures are known to be extremely parameter sensitive. even small changes of

the initial design, e.g., to the shape of the shell, may drastically change the internal

stress state or an initial imperfection may significantly affect the buckling load.(Ramm

1993). For a concrete shell the ideal case is a state of pure membrane state of

compression. In many situations where the ideal form is not obvious, form finding

methods are employed to arrive at a form to suit a required optimal condition.

A typical problem of structural optimization is characterized by an objective function

f(x) and constraints g(x) and h(x) which are non-linear functions of the

optimization variable x.

It can be stated as:

Minimize: f(x)

Subject to: h(x) = 0; g(x) = 0

Because of their general formulation, methods of structural optimization can tackle

problems with many load conditions, arbitrary design objectives and loads such as

changing boundary conditions and forces.

For example strain energy nay be chosen as the minimization function for shapes that act

in membrane state of stress i:e compression and tension but no bending.

Where the total strain energy given by:

1( ) . .

2 vF x dV

Where G(x) may represent a reliability constraint such as stress or displacement limit.

If a leveled state of stress is desired, the function may be chosen as

2( ) ( ) .av

F x dV


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If we want to maximize stiffness for a given mass, ( )h x may be used to represent a

constant mass otherwise unrealistic heavy solutions may be obtained. This is usually the

case when external loads are dominant.

The minimization function generally can be anything from a cost function, weight or

natural frequency or any parameter of interest.

Depending on the objective (strain energy, weight, etc.), the constraints

(equality, non-equality) and their combinations, the optimization problems can vary

from totally unconstrained (stress leveling) to semi constrained (strain energy

minimization with fixed mass, displacement limit) to highly constrained problems

like weight minimization, which tend to reduce mass until the limit of material

resistance is reached.

Sophisticated computer methods are employed as they involve extensive search

algorithms, structural analysis FEM, and design modeling (CADG) to work together.

Figures below illustrate an example taken from the study. The initial configuration is a

parabolic shape shell( uniform snow load of2

5 /KN m and hinged supports) and the

various forms it achieves under optimality of different conditions. 2s

and 1s

are the

heights at the middle and edge cross sections respectively.

Various conditions that were simulated are:

b) Strain Energy Optimization with fixed thickness and no stress constraint.

c)

Stress leveling (

2100 /

aKN m

)

d) Weight minimization with constraint on maximum von- mises stress(400 KN/m2)


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Only linear material properties and geometry formulations were considered but Nonlinear

relations can also be implied and the subject is under much research.

Figure no 2.13.1: Structural Optimization.


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2.14 Conclusion:

Origami inspired structures have been few despite the fact that a number of spatial forms

are possible using the patterns overviewed in the review. The major shortcoming may be

the demanding geometry and fabrication of elements and the problems associated with a

realistic analysis. Except for the pleated corrugation, which has been used f

analysis, construction and design of origami inspired structures

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