PhD Dissertation Giulia Curletto

Ph.D. Program in Civil, Chemical and Environmental Engineering Curriculum in Structural and Geotechnical Engineering, Mechanics and Materials Department of Civil, Chemical and Environmental Engineering Polytechnic School, University of Genoa, Italy.

Rigid Deployable Systems: Concepts for Structures and Integrated Parametric Design Giulia Curletto

Origami

Adviser: Prof. Luigi Gambarotta – Department of Civil, Chemical and Environmental Engineering, University of Genoa

External Reviewers: Prof. Giuseppe Ferro – Department of Structural and Geotechnical Engineering, Politecnico di Torino Prof. Tomohiro Tachi – Department of General Systems Studies, University of Tokyo

Ph.D. program in Civil, Chemical and Environmental Engineering Curriculum in Structural and Geotechnical Engineering, Mechanics and Materials Cycle XXIX

RIGID DEPLOYABLE SYSTEMS: CONCEPTS FOR ORIGAMI STRUCTURES AND INTEGRATED PARAMETRIC DESIGN

BY GIULIA CURLETTO

Dissertation discussed in partial fulfillment of the requirements for the Degree of

DOCTOR OF PHILOSOPHY in Structural and Geotechnical Engineering, Mechanics and Materials, Doctoral Program in Civil, Chemical and Environmental Engineering, Department of Civil, Chemical and Environmental Engineering, University of Genoa, Italy

ABSTRACT

Deployable structures, a relatively young topic pioneered in the 1960s, are becoming increasingly popular in Engineering and Architectural applications for their structural, spatial and plastic qualities. They exhibit the core advantage of combining two significant mechanical and structural properties, the deployability and the increased stiffness and strength. They are suitable for the development of transformable spaces, whose shape evolves from a closed or stowed state to a much open or deployed state. In the service configuration, they are stiffened by a series of folds, creating a corrugation acting as load bearing element. In order to reach a suitable and efficient corrugation and deployability, this research investigates Origami tessellations, which consist on the repetition of the same module or pattern following a specific algorithm, where no cutting and gluing are involved. They can be transformed by a folding process, reaching a final configuration characterized by a great stiffness, due to the corrugation of the pattern. This property makes Origami tessellations of particular interest to be applied in the field of deployable structures.

Although Origami principles applied to deployable structures have a great potentiality, geometric and structural issues may result a complex task to integrate and successful applications are not so common.

Moreover, a remarkable difficulty may be observed in the application of the geometric concepts into structures, which include a series of kinematical, structural and mechanical aspects. For these reasons, the folding principles and the mathematical approaches, characterizing deployable structures, are here tackled, focusing the attention on the design and analysis of the structural behavior and proposing variations of existing concepts and novel solutions for the development of applications. The kinematic simulation, the mechanical and structural aspects are here investigated, exemplifying the concepts into three meaningful cases: Miura-Ori, Waterbomb base and Yoshimura Origami tessellations. The aim is to provide a parametric modeling process available for the design, mechanical and structural analysis of rigid Origami, with different corrugations and shapes. A quantitative assessment of the influence of the connection system adopted among adjacent plates on the structural behavior of deployable structures is provided, discussing the two limit cases of rigid connections and rotational hinges.

The application of the concepts on the design of a deployable mobile shelter allows integrating the geometric and the structural principles with technological issues (material for low thickness plates, connection system between adjacent plates, assembly and manufacturing, locking system etc., may interfere with the realization of an efficient deployable structure). The novel mobile system is designed to provide weather protection and to guarantee maximum flexibility in planning. Innovative technological solutions are adopted, testing the composite material HyliteÂŽ for its core advantage of guarantying a hinge function, with structural and aesthetic benefits. All elements are factory made and easily transported and installed on site, providing costs benefits and ensuring high quality standards and rapid delivery. A reduced scale physical model enables to demonstrate the feasibility of the project, testing the solutions proposed in terms of material, fabrication, folding process and locking system.

The entire design process has been developed applying a new interactive parametric system, based on the definition of a series of steps and algorithms and the identification of the associated tools. Geometric, kinematic and structural aspects are included in a continuous analysis process that provides a strong control of the design, in which it is possible to generate rapid cycles of analyses and to variate the attributes to assess the sensitivity of the structural response, with the aim of finding the best morphology. This approach integrates of the visual scripting software Grasshopper (Rhinoceros plug-in) with the FEA package Ansys.

The first enables the drawing of the geometry, the rapid simulation of the kinematic motion and the control of the leading configuration parameters, while the second allows analyzing the structural behavior of the Origami deployable models. In order to connect and integrate these aspects, GeometryGym (Grasshopper plug-in) has been investigated as particularly interesting to define the structural features and to transfer geometric and kinematic attributes in a FE software.

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INDEX

1. INTRODUCTION ................................................................................................................... 7 1.1. MOTIVATIONS AND SCOPE ................................................................................... 7 1.2. OUTLINE .................................................................................................................... 11 REFERENCES .................................................................................................................. 11

2. ORIGAMI INSPIRED STRUCTURES ............................................................................. 13 2.1. DEPLOYABLE STRUCTURES ............................................................................... 13 2.1.1. Definition of deployable .................................................................................. 13 2.1.2. Classification ................................................................................................... 14 2.1.3. Advantages in deployable structures ............................................................... 18 2.2. ORIGAMI FOLDING PRINCIPLES ....................................................................... 25 2.2.1. Terminology .................................................................................................... 25 2.2.2. Curvature and developability........................................................................... 27 2.2.3. Flat-fodability .................................................................................................. 30 2.2.4. Rigid-fodability ............................................................................................... 35 2.3. ORIGAMI TESSELATIONS ENGINEERING APPLICATIONS ....................... 43 2.3.1. Space applications ........................................................................................... 44 2.3.2. Medical devices ............................................................................................... 45 2.3.3. Mobile footbridge ............................................................................................ 46 2.3.4. Sandwich panels .............................................................................................. 46 2.3.4. Temporary shelters .......................................................................................... 47 REFERENCES .................................................................................................................. 52 3. PARAMETRIC METHODOLOGY.................................................................................... 57 3.1. INTEGRATED ORIGAMI MODELING: GEOMETRIC AND MATHEMATICAL APPROACHES.................................... 57 3.1.1. Origami modelers and simulators .................................................................... 58 3.1.2. Parametric design and generative algorithms .................................................. 61 3.2. PARAMETRIC DESIGN USING GRASSHOPPER .............................................. 63 3.2.1. Computational tools for structural analysis .................................................... 63 3.3. IDENTIFICATION OF THE METHODOLOGY .................................................. 74 3.3.1. Application to the case of a longitudinal pattern ............................................. 75

INDEX

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REFERENCES .................................................................................................................. 80 4. ORIGAMI MODELING AND KINEMATICS.................................................................. 83 4.1. PLANAR AND SPATIAL ORIGAMI ...................................................................... 83 4.2. PARAMETERIZATION OF THREE CASE STUDIES ........................................ 83 4.2.1. Miura-Ori ........................................................................................................ 84 4.2.2. Waterbomb base .............................................................................................. 89 4.2.3. Yoshimura ....................................................................................................... 95 4.3. PARAMETRIC MODELING AND SIMULATION OF THE FOLDING MOTION................................................................................ 100 4.3.1. In-plane deployment Origami ....................................................................... 102 4.3.2. In-space deployment Origami ....................................................................... 106 APPENDIX 4.1. MULTI-DOFS ORIGAMI: A CASE STUDY .................................. 113 APPENDIX 4.2. ALGORITHMIC SEQUENCES ....................................................... 117 4.2.1. Quadrangular and triangular modules – geometry ........................................ 117 4.2.2. Replication of modules – compatibility condition ........................................ 118 4.2.3. Miura-Ori geometry ...................................................................................... 119 4.2.4. Waterbomb base geometry ............................................................................ 120 4.2.5. Yoshimura geometry ..................................................................................... 122 REFERENCES.................................................................................................................. 123 5. ANALYSIS OF ORIGAMI STRUCTURES..................................................................... 125 5.1. MECHANICS OF DEPLOYMENT ....................................................................... 125 5.1.1. Analytical solution ........................................................................................ 125 5.1.2. Numerical solution of three case studies ....................................................... 127 5.2. STRUCTURAL RESPONSE OF ORIGAMI DEPLOYABLE STRUCTURES 135 5.2.1. Three Origami models ................................................................................... 136 5.2.2. Finite element discretization.......................................................................... 138 5.2.3. Miura Origami structure ................................................................................ 140 5.2.4. Waterbomb base Origami structure ............................................................... 145 5.2.5. Yoshimura Origami structure ........................................................................ 150 5.2.6. Factors affecting the structural behaviour ..................................................... 154 REFERENCES ................................................................................................................ 158

6. DESIGN AND ANALYSIS OF A DEPLOYABLE SHELTER ...................................... 159 6.1. OBJECTIVES ........................................................................................................... 159

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6.1.1. Design criteria for deployable structures ....................................................... 159 6.2. COMPOSED ORIGAMI-INSPIRED MODEL ..................................................... 160 6.2.1. Geometric and kinematic parametric model .................................................. 160 6.2.2. Definition of the shelter size and mechanics of deployment ....................... 165 6.3. MANUFACTURING ASPECTS ............................................................................. 168 6.3.1. Material, thickness and connection system ................................................... 168 6.3.2. Fabrication, assembly and erection ............................................................... 172 6.4. STRUCTURAL ANALYSIS OF THE ORIGAMI SHELTER ............................ 175 6.4.1. General approach ........................................................................................... 175 6.4.2. Structural behavior of a module .................................................................... 176 6.4.3. Structural behavior of the shelter................................................................... 186 6.5. DEVELOPMENT OF A PHYSICAL MODEL ..................................................... 188 6.5.1. Design and components ................................................................................. 188 6.5.2. Fabrication and erection ................................................................................ 190 6.6. DISCUSSION ............................................................................................................ 194 APPENDIX 6.1. LOAD CASES ACTING ON THE SYSTEM................................... 195 APPENDIX 6.2. PARAMETRIC MODEL ALGORITHMIC SEQUENCE ............. 196 REFERENCES ................................................................................................................ 199 7. CONCLUSIONS AND FUTURE WORKS ...................................................................... 201

INDEX

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1. INTRODUCTION

Deployable structures, a relatively young topic being pioneered in the 1960s, are becoming increasingly popular in Engineering and Architectural applications for their structural, spatial and plastic qualities. They exhibit the core advantage of combining two important structural and mechanical properties: the increased stiffness and the deployability. Deployable structures in the service configuration are stiffened by a series of folds acting as load bearing elements. They are suitable for the development of transformable spaces, whose configuration evolves from a closed or stowed state to a much larger, open or deployed state (deeply discussed in 2.1.3).

In order to obtain a suitable and efficient corrugation and deployability, Origami has been analyzed as the best technic to be applied in the field of the deployable structures. Origami is the technic of paper folding that derives by the Japanese tradition and it enables to transform a sheet of paper with a series of folds, which nowadays is applied in many disciplines such as engineering, architecture, mathematics, fashion, computation, biology and medicine (see applications in 2.3). More appropriately, the category of Origami tessellations has been here analyzed, which consist on the repetition of the same module or pattern following a specific algorithm, where no cutting and gluing is involved. They can be transformed by a folding process, reaching a final configuration characterized by a great stiffness, due to the corrugation of the pattern (described in section 2.1.3). This property makes Origami tessellations of particular interest to be applied in the field of deployable structures. Various models of Origami tessellations have been created using regular patterns composed of periodic modules based on hexagonal, square and triangular geometries.

In the 1960s and 70s Resch (1970) experimented interesting rotations and translations that a fold can assume and he started diagramming and regularizing paper creases. His approach was highly dependent on regular geometries but, with the help of computational modeling, currently it is possible to manipulate patterns in such a way that they resemble freeform surfaces, as experimented by Tachi (2009, 2010).

1.1. MOTIVATIONS AND SCOPE Although the application of Origami technic in the field of deployable structures has a great potentiality, successful applications are not so common. A remarkable difficult may be observed in the application of the geometric concepts into structures, which include a series of kinematical, structural and technological issues. For this reason, the methods, the mathematical principles and the advantages characterizing deployable structures inspired by Origami are here investigated, proposing variations of existing concepts and solutions for the development of applications in deployable structures.

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I.

The primary problem affecting the design of Origami deployable structures is the insufficiency of tools for the integration of the modeling, the analysis and the realization of complex models. This research stems from the belief that a design method that rapidly generates and modifies the shape, easily simulates the folding motion and efficiently analyzes the structural behavior of a deployable structure can substantially improve the productivity in this field of research. The identification of a method to analyze efficiently deployable structures is basic to a correct interpretation of its behavior, especially since Origami have a very complex geometry that requires detailed studies. For this reason, the Part I of the research is dedicated to the development of a new efficient and interactive system, denominated integrated parametric methodology, for realizing cycles of modeling, simulation and structural analyses.

The new parametric methodology is focused on the definition of a series of steps and algorithms and the identification of the associated tools, which enable the development of a single analysis process, from the geometry to the simulation of the folding motion, until to the structural behavior, in the same workflow. The approach is based on the integration of the visual scripting software Grasshopper (Rhinoceros plug-in) with the FEA package Ansys. The first has been selected for the drawing of the geometry, for the rapid simulation of the kinematic motion and for the possibility of controlling the leading configuration parameters, while the second enables to analyze the structural behavior of the Origami deployable models. In order to connect and to integrate these phases of design, GeometryGym (Grasshopper plug-in) has been investigated as particularly interesting to define the structural features and to transfer geometric and kinematic attributes in a FE software.

The aim of this phase of research is to provide a complete mastery and a strong control of the project, in which it is possible to rapidly generate, modify and analyze structural models, which is an essential strength for realizing such complex deployable systems. It has the core advantage of providing a design that continuously evolves and is simultaneous updated at the variation of its geometric, kinematic or structural data. Moreover, the system provides information about the functionality of the model, warning when non-functional designs are considered. These properties allow quickly experimenting the structural behavior of articulated structures, simply by defining an efficient algorithm and by altering the variables governing the model (topic discussed in chapter 3 and 4).

II.

A second relevant problem affecting Origami deployable structures regards the difficulty in the definition of an efficient process that enables to interpret the kinematics, the mechanics and the structural response of these complex models. In order to acquire a deep knowledge of folding principles it seems necessary to define a geometric parameterization to generate Origami tessellations, characterized by various corrugations and to simulate their kinematics when different internal mechanisms occur.

The realization of a deployable structure requires a further step of investigation of great importance, which focuses on the mechanics of deployment and the structural response of the model in the service (or operating) configuration. The deployment of Origami structures implies the study of the mechanism that enables the transition from a flat configuration to the operating state, where a locking system restraints the structure kinematically determined. In this contest, the assessment of forces, which have to be applied to the external nodes along the deployment, as well as the associated power

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expended have to be evaluated. Once the structure is locked in the operating configuration, a critical issue that arise regards the definition of the structural model that has to be analyzed and the connection system adopted among adjacent plates has an influence on the structural response. A possibility is to consider the operating configuration as a folded state, with relative rotations between adjacent faces locked and rigid connections imposed. However, in general the concept of deployable implies that relative rotations are allowed between adjacent panels. The difference between assuming rigid connections or rotational hinges is important and it requires further investigations.

The kinematic simulation and the mechanical and structural aspects are here investigated, focusing the attention on the definition of the principles characterizing rigid Origami patterns. The geometric parameterization, the folding motion and the algorithmic sequences of various corrugations are discussed and exemplified into three meaningful cases: Miura-Ori, Waterbomb base and Yoshimura.

The process has been developed by using a geometric approach for the drawing and the simulation of the folding motion of Miura-Ori, while an analytical approach is applied for the definition of Waterbomb base and Yoshimura Origami. The mechanics of deployment is investigated in terms of forces and power that has to be expended during the deployment. Several technological solutions have been proposed in literature to deploy the structure from the flat configuration to the operating state. Here, it has been assumed to deploy a structure by assuming a set of horizontal forces impressed at the base supports. To get a simple evaluation of the forces involved in the deployment the virtual work theorem has been applied under the hypothesis of omitting both elastic and internal dissipative mechanisms in the linear hinges connecting adjacent surfaces, such as frictional or viscous effects. First, the analytical solution has been given, then the three cases are discussed and compared. Finally, the structural behavior of low thickness structures in the operating configuration has been analyzed, focusing the attention on the factors that most influence the structural response, as the corrugation of the pattern and the connection system adopted.

The aim of this phase of research is to extract the geometric, mechanical and structural principles characterizing Origami deployable structures. Through the exemplification of the concepts with the three case studies, one aim is to propose a parametric modeling process available for the rigid Origami modeling, having different corrugations and shape (quadrangular and triangular periodic modules) and different replication conditions (translations and rotations). The second aim is to provide information on the mechanics, comparing the results obtained from the analysis of structures having a in-plane or in-space deployments. The third aims focuses of the evaluation of the factors that most influence the structural response of a deployable structure and the identification of the most interesting patterns to be adopted into applications. A quantitative assessment of the influence of the connection on the structural behavior of deployable structures will be provided, discussing the two limit cases in which the displacement and stress fields must be evaluated: rigid connections, with relative rotations locked, and rotational hinges, with relative rotations between adjacent panels allowed.

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III.

Although the wide scientific research activity on Origami deployable structures, the application of foldable principles on real structures is currently under-explored (Schenk, 2013). This circumstance depends on difficulties in transforming geometric models to real structures, where the geometric, mechanical and structural aspects, previously discussed, have to be integrated with technological issues. In this context, the selection of the material for low thickness plates, the connection system between adjacent plates through hinges, the assembly and manufacturing, the deployment mechanisms and the locking system may interfere with the realization of an efficient deployable structure. A lightweight material is required for ease of both transportation and manipulation. Various are the solutions (concrete, wood, cardboard, textiles, etc.), which have a considerable impact on the model in terms of structural behavior and aesthetic value. Low thickness of the panels is crucial because when the structure is approaching the closure, contacts between the adjacent faces take place.

The hinges have to be integrated with the whole system by guaranteeing the simultaneous folding motion of adjacent plates and they have a strong impact on the final perception of the structure.

In order to investigate these aspects, the development of an application for a deployable mobile shelter is proposed, in which the geometric and the structural principles are integrated with technological issues. It has been estimated the whole structure’s life cycle, aiming to meet criteria related to the design (architectural flexibility and component uniformity), the storage and the transport (compactness of the stowed state) the site operation (low complexity and high speed in the assembly) and the erection process. A new shape, inspired by various Origami pattern, has been modelled to provide maximum flexibility in planning and to facilitates variations in site configuration. An innovative composite material is tested: HyliteŽ has the core advantage to provide a hinge function without the application of additional components for cylindrical hinges realization, with structural and aesthetic benefits. All elements are factory made and easily transported and installed on site, providing costs benefits and ensuring high quality standards and rapid delivery.

The scope of this phase is to propose a new project testing and showing its feasibility. The novel mobile system is designed to provide weather protection for a wide range of outer activities.

Exhibition and recreational space, temporary refuge in remote construction sites, emergency shelter or relocatable temporary building are the most interesting applications for the project.

The last part of the dissertation focuses on the development of a reduced scale physical model, which enables to exemplify the concepts elaborated in the design of the mobile temporary shelter.

Moreover, it aims to gain the three following objectives: to demonstrate the feasibility of the project, focusing the attention on the fabrication, folding process and locking system, to simulate the folding motion and to verify the ultimate configuration of deployment, and to examine the accurateness of the drawing developed using the parametric methodology.

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1.2. OUTLINE The dissertation proceeds as follows.

Chapter 2 focuses on the literature review that initially introduces the topic of deployable structures and then proceed to describe the principles and the properties associated with rigid Origami tessellations.

A proper definition of the term deployable is provided and a classification of deployable structures according the morphologic and kinematic aspects is proposed. The primary folding principles of Origami are discussed with a mathematical approach, focusing the attention on the concepts of curvature and developability, rigid-foldability and flat-foldability, for which the conditions and the evaluation methods are described. Finally, a selection of Engineering and Architectural applications of Origami deployable systems is proposed, distinguishing space and medical devices, sandwich panels and temporary shelters, and pointing out the structural and technological solutions adopted.

Chapter 3 makes up the Part I of the dissertation, presenting a new integrated parametric methodology for the design of Origami deployable models. It enables the development of cycles of rapid and efficient analyses, in which geometric, kinematics and structural variables are integrated in the same workflow.

After a brief overview on previous work in Origami simulators, a discussion of the most relevant concepts have been developed focusing the attention on the operating principles. The parametric modeling software Grasshopper is presented as particularly interesting modeling tool to be integrated with structural analysis instruments for FEA. The instruments and principles behind the new methodology are discussed, developing some examples for each steps: geometry, kinematics simulation and structural response.

Chapter 3 and 4 make up the Part II of the dissertation, the kinematic simulation and the structural analysis of Origami deployable structures.

Chapter 4 concerns the parameterization of the geometry and the simulation of the folding motion of Origami models, focusing the attention on three case studies: Miura-Ori, Waterbomb base, and Yoshimura.

Geometric and kinematic variables are discussed, proposing both a geometric and an analytical definition of the patterns. The kinematics of patterns having a single degree of freedom has been tackled, differing the principles characterizing in-plane and in-space deployment. In this phase, the new methodology developed in Chapter 3 has been applied for the drawing of quadrangular and triangular modular Origami and for the simulation of the folding motion, when translations and relative rotations are involved.

Appendix 4.1 is dedicated to the simulation of the folding motion of multi-DoFs Origami, discussing a specific case study of a 3DoFs, while appendix 4.2 illustrates the algorithms defined on Grasshopper for the drawing of the geometry and the kinematic simulation.

Chapter 5 discusses mechanical and structural issues, providing a new contribution on the influence that the connection system between adjacent panels has on the structural response of Origami deployable structures.

The mechanics of deployment is investigated, evaluating the assessment of forces that have to be applied to the free nodes along the deployment process from the flat configuration to the operating one, as well as the power expended, in the hypothesis of rotational hinges. Then, the problem of the influence of the

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connection system adopted between adjacent panels is investigating, providing analyses with rotational hinges and comparing the differences and similarities with rigid connections. The analysis is targeted to Origami structures in the operating configuration, assuming elastic plane plates with no imperfections and ignoring the second order geometric effects on the equilibrium and the buckling collapse mechanisms.

Chapter 6 makes up the Part III of this dissertation, describing the novel design and construction of a mobile temporary shelter, in which the new methodology of chapter 3 and the principles carried out in the chapters 4 and 5 are exemplified.

The transition from an Origami-inspired geometry to a deployable rigid structure is described, in which new solutions in the choice of the shape, material and connection system are proposed. The geometry, in which the cover and the supporting elements are integrated in a continuous surface, provides high flexibility in planning and great stiffness. A particular composite material and system connection between panels has been proposed in order to guarantee a lightweight structure, where hinges are integrated in the thickness of the panels. The structural response of the structure is tested under different load combinations. Finally, a physical model is developed to demonstrate the feasibility of the technological solution adopted and to test the deployment process.

Chapter 7 concludes the dissertation, discussing the original results and pointing out suggestions for which each of the concepts discussed could be advanced further in future works.