Design and Analysis of Deployable Bar Structures for Mobile Architectural Applications

Transcription

1 FACULTY OF ENGINEERING Department of Architectural Engineering Sciences Design and Analysis of Deployable Bar Structures for Mobile Architectural Applications Thesis submitted in fulfilment of the requirements for the award of the degree of Doctor in de Ingenieurswetenschappen (Doctor in Engineering) by Niels De Temmerman June 2007 Promotor: Prof. Marijke Mollaert

2 Members of the Jury: Prof. Dirk Lefeber (President) Vrije Universiteit Brussel Prof. Rik Pintelon (Vice-President) Vrije Universiteit Brussel Prof. Marijke Mollaert (Promotor) Vrije Universiteit Brussel Prof. Ine Wouters (Secretary) Vrije Universiteit Brussel Prof. Sigrid Adriaenssens Vrije Universiteit Brussel Prof. John Chilton Lincoln School of Architecture Prof. W.P. De Wilde Vrije Universiteit Brussel Dr. Frank Jensen Århus School of Architecture

3 Acknowledgements My interest in the exciting field of deployable structures came about through the process of writing my master s thesis on the subject, under the supervision of Prof. Marijke Mollaert. This has been the inspiration and drive to delve deeper into this rich and rewarding research topic of which this dissertation is the final result. I would like to express my gratitude to my supervisor Prof. Marijke Mollaert for sharing her vast research experience and for her invaluable scientific guidance. Also, I have great appreciation for her warm and kind personality and her continuous encouragement throughout the course of this research. I would like to thank everyone who has contributed in making the past four years into an exciting and enriching experience: My most heartfelt sympathy goes out to my colleagues of the Department of Architectural Engineering, whom I thank for providing a kind and stimulating environment, and for their friendship and support: Maryse Koll, Tom Van Mele, Thomas Van der Velde, Lars De Laet, Lisa Wastiels, Anne Paduart, Caroline Henrotay, Michael de Bouw, Prof. Ine Wouters, Prof. Hendrik Hendrickx, Prof. José Depuydt, dr. Jonas Lindekens. At the department of Mechanical Engineering, I would like to thank Prof. Dirk Lefeber and Prof. Patrick Kool for their help on gaining an insight in the mobility of mechanisms. Prof. Patrick De Wilde, Prof. Sigrid Adriaenssens and Wim Debacker from the department of Mechanics of Materials and Constructions, and Prof. Rik Pintelon from the Department of Fundamental Electricity and Instrumentation, I would like to thank for their scientific advice and suggestions. I gratefully acknowledge the financial support extended to me by IWT- Vlaanderen (Institute for the Promotion of Innovation through Science and Technology in Flanders).

4 dr. Frank Jensen and Prof. John Chilton I would like to thank for sharing their expertise on the subject and providing much valued comments and suggestions. Also, many thanks to Wouter Decorte, for the fruitful collaboration, his enthusiasm on the subject and for sharing his excellent model-making skills. My parents, Eric and Monique, my sister Ilka and her husband Tom, and also Solange, deserve special thanks for their love and friendship and for their unconditional support and encouragement. Above all, I wish to express my love and sincerest gratitude to Els, my partner and friend, for her continuous love and support. Without her I would never have come this far. Vilvoorde, June 2007 Niels De Temmerman

5 Abstract Deployable structures have the ability to transform themselves from a small, closed or stowed configuration to a much larger, open or deployed configuration. Mobile deployable structures have the great advantage of speed and ease of erection and dismantling compared to conventional building forms. Deployable structures can be classified according to their structural system. In doing so, four main groups can be distinguished: spatial bar structures consisting of hinged bars, foldable plate structures consisting of hinged plates, tensegrity structures and membrane structures. Because of their wide applicability in the field of mobile architecture, their high degree of deployability and a reliable deployment, two sub-categories are studied in greater detail: scissor structures and foldable plate structures. Scissor structures are lattice expandable structures consisting of bars, which are linked by hinges, allowing them to be folded into a compact bundle. Foldable plate structures consist of plate elements which are connected by line joints allowing one rotational degree of freedom. A wide variety of singly curved as well as doubly curved structures are possible. Although many impressive architectural applications for these mechanisms have been proposed, due to the mechanical complexity of their systems during the folding and deployment process few have been constructed at full-scale. The aim of the work presented in this dissertation is to develop novel concepts for deployable bar structures and propose variations of existing concepts which will lead to viable solutions for mobile architectural applications. It is the intention to aid in the design of deployable bar structures by first explaining the essential principles behind them and subsequently applying these in several cases studies. Starting with the choice of a suitable geometry based on architecturally relevant parameters, followed by an assessment of the kinematics of the system, to end with a structural feasibility study, the complete design process has been demonstrated, exposing the strengths and weaknesses of the chosen configuration. IV

6 Contents Acknowledgements Abstract List of Figures List of Tables List of Symbols 1. Introduction Deployable Structures Aims and scope of research Outline of thesis 5 2. Review of Literature Introduction Deployable structures based on pantographs Translational units Polar units Deployability constraint Structures based on translational and polar units Angulated units Closed loop structures based on angulated elements Foldable plate structures 29 V

7 3. Design of Scissor Structures Introduction Design of two-dimensional scissor linkages Method 1: Geometric construction Method 2: Geometric design Interactive geometry Three-dimensional structures Linear structures Plane grid structures Single curvature grid structures Double curvature grid structures Conclusion Design of Foldable Plate Structures Introduction Geometry of foldable plate structures Geometric design Regular structures Right-angled structures Circular structures Alternative configurations Conclusion Introduction to the Case Studies Introduction Geometry Structural analysis of the proposed concepts General approach Load cases Conclusion 132 VI

8 6. Case Study 1: A Deployable Barrel Vault with Translational Units on a Three-way Grid Introduction Description of the geometry Geometric design From mechanism to architectural envelope Deployment and kinematic analysis Structural analysis Open structure (single curvature) Closed structure (double curvature) Conclusion Case Study 2: A Deployable Barrel Vault with Polar and Translational Units on a Two-way Grid Introduction Description of the geometry Open structure Closed structure Deployment From mechanism to architectural envelope Deployment and kinematic analysis Structural analysis Open structure (single curvature) Closed structure (double curvature) Conclusion Case Study 3: A Deployable Bar Structure with Foldable Articulated Joints Introduction Description of the geometry From plate structure to foldable bar structure Deployment Alternative geometry Kinematic analysis 231 VII

9 8.4 Structural analysis Open structure (single curvature) Closed structure (double curvature) Conclusion Case Study 4: A Deployable Tower with Angulated Units Introduction A concept for a deployable tower Description of the geometry Geometric Design First approach: Design of the undeployed configuration Second approach: Design of the deployed configuration From mechanism to architectural structure Mobility analysis The erection process Alternative configuration Simplified concept: prismoid versus hyperboloid Structural analysis Conclusion Conclusions Novel concepts for deployable bar structures Case study 1: A Deployable Barrel Vault with Translational Units on a Three-way Grid Case study 2: A Deployable Barrel Vault with Polar and Translational Units on a Two-way Grid Case study 3: A Deployable Bar Structure with Foldable Articulated Joints Case study 4: A Deployable tower with Angulated Units Comparative evaluation of the proposed concepts Architectural evaluation Kinematic evaluation Structural evaluation Further work 303 VIII

10 References 305 List of Publications 313 IX

11 List of Figures Figure 1.1: Mobile deployable bar structure ( Grupo Estran)...2 Figure 1.2: Classification of structural systems for deployable structures by their morphological and kinematic characteristics [Hanaor, 2001]..3 Figure 2.1 : Translational units Figure 2.2: The simplest plane translational scissor linkage, called a lazy-tong Figure 2.3: A curved translational linkage in its deployed and undeployed position Figure 2.4: Polar unit...12 Figure 2.5: A polar linkage in its undeployed and deployed position Figure 2.6: The deployability constraint in terms of the semi-lengths a, b, c and d of two adjoining scissor units in three consecutive deployment stages Figure 2.7: Piñero demonstrates his prototype of a deployable shell [Robbin, 1996] Figure 2.8: Planar two-way grid with translational units and cylindrical barrel vault with polar units [Escrig, 1985] Figure 2.9: Top view and side elevation of a two-way spherical grid with identical polar units [Escrig, 1987] Figure 2.10: Top view and side elevation of a three-way spherical grid with polar units Figure 2.11: Top view and side elevation of a geodesic dome with polar units [Escrig, 1987] Figure 2.12: Top view and side elevation of a lamella dome with identical polar units Figure 2.13: Deployable cover for a swimming pool in Seville designed by Escrig & Sanchez ( Performance SL) Figure 2.14: Bi-stable structure before, during and after deployment Figure 2.15: Collapsible dome and a single unit, as proposed by Zeigler [1976] Figure 2.16: Bi-stable structures: elliptical arch and geodesic dome [Gantes, 2004]

12 Figure 2.17: Positive curvature structure with translational units in two deployment stages [Langbecker, 2001] Figure 2.18: Negative curvature structure with translational units in two deployment stages [Langbecker, 2001] Figure 2.19: Plane and spatial pantographic columns by Raskin [1998] Figure 2.20: Pantographic slabs by Raskin [1998] Figure 2.21: Deployable ring structure [You & Pellegrino, 1993] Figure 2.22: Angulated unit or hoberman s unit Figure 2.23: A radially deployable linkage consisting of angulated (or hoberman s) units in three stages of the deployment Figure 2.24: Multi-angulated element Figure 2.25: A radially deployable linkage consisting of multi-angulated elements in three stages of the deployment Figure 2.26: Multi-angulated structure with cover elements in an intermediate deployment position Figure 2.27: Model of a non-circular structure where all boundaries and plates are unique [Jensen, 2004] Figure 2.28: Computer model of an expandable blob structure [Jensen & Pellegrino, 2004] Figure 2.29: Reciprocal plate structure Figure 2.30: Swivel diaphragm in consecutive stages of deployment Figure 2.31: Reciprocal dome proposed by Piñero [Escrig, 1993] Figure 2.32: Iris dome by Hoberman [Kassabian et al, 1999] Figure 2.33: Retractable dome on Expo Hannover (courtesy of M. Mollaert) Mechanical curtain Winter Olympics Salt Lake City 2002 [Hoberman, 2007] Figure 2.34: Retractable roof made from spherical plates with fixed points of rotation Figure 2.35: Novel retractable dome with spherical plates with modified boundaries Figure 2.36: Basic layout of Foster s module [Foster, 1986] Figure 2.37: Combination of different modules [Foster, 1986] Figure 2.38: Simplest building with 90 apex angle [Foster, 1986] Figure 2.39: Building formed by two 90 -modules joined at their ends [Foster, 1986]

13 Figure 2.40: Building with apex angle of 120 [Foster, 1986] Figure 2.41: Building formed by two 120 -modules joined at their ends [Foster, 1986] Figure 2.42: Structure with 90 -, 60 - and 30 -elements [Foster, 1986] Figure 2.43: Temporary stage shell with 120 modules - Tension cables used to provide bulkhead [Foster, 1986] Figure 2.44: Double curvature variable shape (hyperbolic type), plane pattern [Tonon, 1993] Figure 2.45: Fold pattern with different individual plate angles, but a constant sum throughout the plate geometry, guaranteeing full foldability.35 Figure 2.46: Doubly curved folded shapes [Tonon, 1993] Figure 2.47: Linear and circular deployable double curvature folded shapes [Tonon, 1993] Figure 2.48: Folding aluminium sheet roof for covering the terrace of the pool area of the International Center of Education and Development in Caracas, Venezuela [Hernandez & Stephens, 2000] Figure 2.49: (a) Fold pattern; (b) Fold pattern with alternate rings to prevent relative rotation during deployment [Barker & Guest, 1998] Figure 3.1 : Translational and polar scissor unit Figure 3.2: An ellipse as the graphic representation of the deployability constraint for translational units, determining the locus of the intermediate hinge Figure 3.3: A circle as the graphic representation of the deployability constraint for polar units, determining the locus of the intermediate hinge Figure 3.4: Circular arc as base curve, determined by the design values H (rise) and S (span) Figure 3.5: The arc is divided in equal angular portions. Circles intersecting the arc determine the loci of the intermediate hinge points Figure 3.6: The arc is divided in unequal angular portions. Variable circles intersecting the arc determine the loci of the intermediate hinge points Figure 3.7: An inner and outer arc determine the constant unit thickness r 307

14 Figure 3.8: For the same span and rise, a pluricentred arc can offer increased headroom compared to a single-centred arc Figure 3.9: Example of a pluricentred base curve consisting of three arc segments with decreasing radius Figure 3.10: Each arc segment (with a different radius) is divided in equal angular portions. Identical circles ensure a constant bar length Figure 3.11: The original and double ellipse representing the deployability constraint intersection points M and M are the midpoints of the unit thickness t Figure 3.12: Double ellipses impose the deployability constraint on a translational linkage with constant unit thickness Figure 3.13: Two differently sized, but compatible ellipses representing the deployability constraint intersection points M and M are the midpoints of the unit thickness t 1 and t Figure 3.14: Ellipses of different scale determine the location of the intermediate hinges on the base curve to form a translational linkage with varying unit thickness Figure 3.15: The parameters used in the description of the geometry of the circular arc: rise (H r ) and span (S) Figure 3.16: Parameters needed for the geometric design of a polar linkage.. 60 Figure 3.17: Parameters for the geometric design of a translational linkage with four units (U=4), of which two are shown Figure 3.18: The relation between the original and the double ellipse in terms of semi-axes a and b and the unit thickness t Figure 3.19: Translational linkage with U=2 fitted on a parabolic base curve. 67 Figure 3.20: Screenshot of interactive geometry file in Cabri Geometry II [2007] software for designing arbitrarily curved translational linkages with constant unit thickness (base curve marked in black) Figure 3.21: Deployable landscape consisting of one arbitrarily curved translational linkage repeated in an orthogonal grid. Linkage designed using the interactive geometry tool (Aluminium, 4.5 m x 3 m, (photo: courtesy of Wouter Decorte) Figure 3.22: Possible shapes for three-dimensional stress-free deployable structures, which can be designed using the tools presented

15 Figure 3.23: Two-way grid with directions A and B Figure 3.24: Three-way grid with directions C, D and E Figure 3.25: Linear elements prismatic columns arches Figure 3.26: Parallel linear structures connected by non-deployable elements Figure 3.27: Plane translational units on a two-way grid Figure 3.28: Plane translational units on a three-way grid Figure 3.29: Plane translational units on a four-way grid Figure 3.30: Plane and curved translational units on a two-way grid Figure 3.31: Polar and translational units on a two-way grid Figure 3.32: Plane and curved translational units on a three-way grid Figure 3.33: Polar units on a three-way grid (variation 1) Figure 3.34: Polar and translational units on a three-way grid (variation 2) Figure 3.35: Translational units on a two-way grid (synclastic shape) Figure 3.36: Two variations for translational units on a two-way grid Figure 3.37: Translational units on a lamella grid Figure 3.38: Polar units on a lamella grid Figure 4.1: Typical foldable plate structure Figure 4.2: Fold patterns of type A and B for the smallest possible regular structure (p=5) Figure 4.3: Unfolded and fully folded configuration of patterns A and B (p=5) Figure 4.4: Elevation view of the compactly folded and fully deployed configuration for a regular structure with five plates and an apex angle of Figure 4.5: Right-angled fold pattern: altering one apex angle to 90 enables a compacter folded configuration (introduction of quadrangular plates near the sides) Figure 4.6: Elevation view of compactly folded and fully deployed configuration for a right-angled structure with five plates and an apex angle of Figure 4.7: Three stages of deployment for a basic regular foldable structure (p=7; β=120 ): completely unfolded, erected position and fully compacted for transport

16 Figure 4.8: Plate element, compactly folded configuration and fully deployed configuration (front elevation) for the first three compactly foldable structures (p=5, p=7, p=9) Figure 4.9: Side elevation of the fully deployed configuration of the first three compactly foldable structures (p=5, p=7, p=9) Figure 4.10: For a chosen number of panels p the apex angle β can be altered at will, affecting the width of the structure and the compactly folded state Figure 4.11: Parameters used to characterise a foldable structure: length L, span S, width W, apex angle β and the deployment angle θ Figure 4.12: A foldable plate and its parameters: length L, height H, H1, H2, apex angle β, the deployment angle θ and angles α, α 1, α Figure 4.13: Perspective view and side elevation of the vertical projection of a plate linkage for empirically determining the relationship between α 1 and p Figure 4.14: The relationship between the apex angle β and the deployment angle θ for regular structures with p=5, p=7 and p= Figure 4.15: Elevation view and perspective view of the deployment of a regular five-plate structure with β= Figure 4.16: The parameters associated with the polygonal contour of the flatly folded configurations with p=5, p=7 and p=9 and the expressions for the area in terms of the edge length L edge Figure 4.17: The relationship between the apex angle β and the deployment angle θ for right-angled structures with p=5, p=7 and p= Figure 4.18: Plate element, fold pattern, compactly folded configuration and fully deployed configuration (front elevation and side elevation) for three compactly foldable five-plate right-angled structures (drawn to scale) Figure 4.19: Only for p=5 can any regular and any right-angled structure be interconnected along a common edge, regardless of the value for β Figure 4.20: Top view and perspective view of circular foldable structure Figure 4.21: Fold pattern and a single sector of a circular structure with q=

17 Figure 4.22: Horizontal projection of a plate linkage for empirically determining the relationship between α 2 and q Figure 4.23: Connecting a regular module with two half-domes leads to an alternative fully closed configuration with high plate uniformity 110 Figure 4.24: Circular structure with q=6, q=8 and q=10 (top view) and its respective combination with a compatible regular structure (perspective view) Figure 4.25: Some examples of alternative configurations Figure 5.1: Some of the concepts for mobile structures presented in the following chapters Figure 5.2: Front elevation view of cases studies shows the mutual similarity of the geometry. Case study 1, 2 and 3 are based on the same shape (semicircle with radius of 3 m) Figure 5.3: Overall geometry for the case studies: single curvature shape (open) and double curvature shape (closed) Figure 5.4: Perspective view of the single and double curvature geometries.118 Figure 5.5: Perspective view and side elevation of case study Figure 5.6: Wind and snow action on the open and closed structure Figure 5.7: Schematic representation of considered wind loads on the closed and Figure 5.8: Schematic representation of snow loads on the closed and open structures Figure 5.9 : Method of accumulate damage [Eurocode 3, 2007] Figure 6.1: Deployable barrel vault with translational units on a triangular grid: scissor structure and tensile surface Figure 6.2: Plan view and perspective view of the same double curvature structure with translational units on a quadrangular grid Figure 6.3: Translational scissor module with only single units Figure 6.4: Translational scissor module with a double unit Figure 6.5: Plan view, perspective view and side elevation of a planar structure with a triangulated grid Figure 6.6: Plan view, perspective view and side elevation of a barrel vault with a triangulated grid

18 Figure 6.7: Perspective view and plan view of three different triangular modules Figure 6.8: OPEN structure: perspective view and plan view Figure 6.9: CLOSED structure: perspective view and plan view (double scissor marked in red) Figure 6.10: Front elevation, top view and perspective view of a portion of the barrel vault with four modules in the span: the projected versions (marked in red) of the scissor units U1 and U2 determine the real curvature Figure 6.11: Developed view of units U1, U2 and U3: graphic representation of the deployability condition by means of ellipses Figure 6.12: An ellipsoid representing the geometric deployability condition in three dimensions Figure 6.13: Vertical section view of the small and big ellipsoid, imposing the geometric deployability condition Figure 6.14: A scissor linkage fitted on a circular curve, with all relevant design parameters and the global coordinate system Figure 6.15: Developed view of the scissor linkage from Figure 6.14, showing a chain of double ellipses Figure 6.16: Perspective view of the scissor linkage from Figure Figure 6.17: Perspective view, front elevation and top view of the deployment process of the barrel vault with translational units OPEN structure Figure 6.18: Proof-of-concept model of (half of the) closed structure (aluminium, scale 1/10) Figure 6.19: Two double scissors in partially (left) and fully deployed (right) position Figure 6.20: Perspective view, front elevation and top view of the deployment process of the barrel vault with translational units CLOSED structure Figure 6.21: From scissor mechanism to the equivalent hinged plate linkage for mobility analysis of the open structure (idem for closed structure) - minimal constraints Figure 6.22: Fixing all lower nodes to the ground by pinned supports

19 Figure 6.23: An active cable (marked in red) runs through the mechanism, connecting upper and lower nodes along its path. After deployment it is locked to stiffen the structure Figure 6.24: Top view and perspective view of one scissor unit, its intermediate hinge and its end joints and their offset position relative to the theoretical plane Figure 6.25: Concept for an articulated joint, allowing the fins which accept the bars to rotate around a vertical axis, to cope with the angular distortion of the grid Figure 6.26: Partially and undeployed state: as the structure is compactly folded, the imaginary intersection point of the centrelines travels on the vertical centreline through the joint Figure 6.27: Perspective view and top view of OPEN structure with integrated tensile surface Figure 6.28: Perspective view and top view of CLOSED structure with integrated tensile surface Figure 6.29: Top view and perspective view of the skeletal scissor structure (left) and the boundary geometry for the compatible membrane (right) Figure 6.30: Views of the equilibrium form for the membrane Figure 6.31: Typical stresses in the membrane range from 4 to 5.5 kn/m Figure 6.32: FEM-model of six bars attached to a node Figure 6.33: An intermediate pivot hinge connects two scissor bars Figure 6.34: Local coordinate system of a bar element (left) and global coordinate system (right) Figure 6.35: Typical pattern of load vectors for transverse wind + pre-stress of the membrane Figure 6.36: Typical pattern of reaction forces under transverse wind Figure 6.37: Bending moments My under transverse wind Figure 6.38: Typical deformation under transverse wind: Figure 6.39: Perspective view of the resulting structure with rectangular sections of 120x60mm Figure 6.40: Reactions in the global coordinate system: the maximal reaction force occurs under ULS 2 (pre-stress + snow + transverse wind)

20 Figure 6.41: The critically loaded bar is located at the top. Summary of the stresses occurring in the critically loaded bar (positive stresses indicate pressure, negative values mean tension) Figure 6.42: Axial forces, transverse forces and bending moments in the local coordinate system of the bars Figure 6.43: Maximal nodal displacements in the global coordinate system.169 Figure 6.44: Continuous cable zigzagging through the structure, connecting upper and lower nodes and contributing to the structural performance Figure 6.45: Resulting structure after optimization, with cable elements Figure 6.46: Summary of the determining stresses and forces for the strength, stability and stiffness of case study 1: OPEN structure Figure 6.47: Perspective view of case study 1: CLOSED structure: with sections after structure design and total weight Figure 6.48: Summary of the determining parameters for the strength, stability and stiffness of case study 1 _ CLOSED structure Figure 6.49: Case study 1: Single curvature OPEN structure (barrel vault) Figure 6.50: Case study 1: Double curvature CLOSED structure Figure 7.1: Deployable barrel vault with polar units on a quadrangular grid: scissor structure and tensile surface Figure 7.2: Plan view and perspective view of a planar structure with a quadrangular grid Figure 7.3: Plan view and perspective view of a barrel vault with quadrangular grid Figure 7.4: Series of polar linkages with 3, 4, 5 or 6 units in the span, Figure 7.5: Geometric construction of the four-unit linkage Figure 7.6: OPEN structure: perspective view and top view Figure 7.7: Perspective view and developed view of units U1 (plane translational) and U2, U3 (polar): graphic representation of the deployability condition by means of ellipses Figure 7.8: Adding an end structure based on parallels and meridians to the main structure Figure 7.9: A lamella dome has a stress-free deployment

21 Figure 7.10: The main structure is provided with half of an adapted lamella dome Figure 7.11: CLOSED structure: perspective view and plan view Figure 7.12: Perspective view, front elevation and top view of the deployment process of the polar barrel vault OPEN structure Figure 7.13: Perspective view, front elevation and top view of the deployment process of the polar barrel vault CLOSED structure Figure 7.14: Proof-of-concept model (half of the structure) in three deployment stages Figure 7.15: Deployment sequence of a polar linkage Figure 7.16: Polar linkage in an intermediate deployment stage: 0 <ψ < Figure 7.17: Graph showing the relation between the deployment angle θ and the span S for the polar linkage with U= Figure 7.18: Deployment sequence of the polar linkage (U=4) Figure 7.19: Open barrel vault: scissor structure and equivalent hinged plate structure Figure 7.20: Geometry of a hinged plate module Figure 7.21: Fixing all inner lower nodes to the ground by pinned supports..195 Figure 7.22: Closed barrel vault: scissor structure and equivalent hinged plate structure Figure 7.23: Geometry of the occurring plate modules Figure 7.24: Closed barrel vault: scissor structure and equivalent hinged plate structure Figure 7.25: Fixing all inner lower nodes to the ground by pinned supports..197 Figure 7.26: Joint connecting four bars (no rotation of the fins of the joint around a vertical axis, as is the case for the translational barrel vault in Section 6.4.1, Figure 6.24) Figure 7.27: Top view and perspective view of one scissor unit and its intermediate and end joints Figure 7.28: Perspective view and top view of OPEN structure with integrated tensile surface Figure 7.29: Perspective view and top view of CLOSED structure with integrated tensile surface Figure 7.30: Result without any measures taken to improve structural performance

22 Figure 7.31: Improved result by inserting vertical cable ties Figure 7.32: Additional diagonal bars triangulate the grid Figure 7.33: Double diagonal cross bars offer no real advantage structurally Figure 7.34: Perspective view of case study 2 OPEN structure, with sections after structure design and weight/m Figure 7.35: Summary of the determining parameters for the strength, stability and stiffness for case study 2 OPEN structure Figure 7.36: Main structure and additional end structures with no additional measures to improve structural performance Figure 7.37: Perspective view of case study 2 CLOSED structure:, with resulting sections after structure design and total weight Figure 7.38: Summary of the results for the structural analysis of case study Figure 7.39: Case 2: OPEN structure Figure 7.40: Case 2: CLOSED structure Figure 8.1: Foldable bar structure based on the geometry of foldable plate structures Figure 8.2: Typical foldable plate structure Figure 8.3: Design parameters for a basic regular foldable plate structure Figure 8.4: For a chosen number of panels p the apex angle β can be altered at will, only affecting the width of the structure Figure 8.5: Graph showing the relation between the deployment angle θ and the apex angle β in the fully deployed configuration for p= Figure 8.6: The resulting regular geometry for the case study: two extreme deployment states and the fold pattern Figure 8.7: Top view and a perspective view of a circular plate geometry with six sectors arranged radially Figure 8.8: The resulting circular geometry for the case study: two extreme deployment states and the fold pattern Figure 8.9: A combination of a regular and a circular geometry Figure 8.10: Dimensions in plan view of the shapes Figure 8.11: A foldable plate structure (p=7) and its similar counterpart, a foldable bar structure

23 Figure 8.12: Pattern 1: double bars present Figure 8.13: Pattern 2: double bars removed Figure 8.14: Pattern 3: double bars and diagonal bars removed, without affecting the original kinematic behaviour Figure 8.15: Foldable 3 D.O.F.-joint derived directly from the fold pattern, therefore mimicking its kinematic behaviour Figure 8.16: Deployment sequence for the foldable joint: from the undeployed to the fully deployed position Figure 8.17: The (regular) open structure complete with bars and joints: Figure 8.18: Detailed view of bars and three variations of foldable joints occurring in the structure Figure 8.19: Deployment sequence for the open structure perspective view, front elevation and top view Figure 8.20: Proof-of-concept model of the regular structure (with scissors) in four stages of the deployment Figure 8.21: Deployment sequence for the dome structure perspective view, front elevation and top view Figure 8.22: Proof-of-concept model of the foldable dome (with additional scissor units) in six deployment stages Figure 8.23: Deployment sequence for the closed structure: 1 regular module + 2 semi-domes Figure 8.24: Six stages in the deployment of the closed structure (top view)227 Figure 8.25: Kinematic joint allowing all necessary rotations (3 D.O.F.) and the resulting bar structure Proof-of-concept model to verify the mobility Figure 8.26: Integration of the membrane beforehand by attaching it to the nodes Side elevation and perspective view of the undeployed and deployed position Figure 8.27: Right-angled geometry with its own set of joints Figure 8.28: Deployment sequence of a concept model of a right-angled structure with aluminium bars and resin connectors [De Temmerman, 2006a] Figure 8.29: Several regular and right-angled structures connected together after deployment Figure 8.30: The two loops and their common fold line

24 Figure 8.31: A foldable open structure with a compatible integrated scissor linkage one bar of each scissor unit doubles up as an edge the foldable bar structure Figure 8.32: Top view and perspective view of the finite element model of the foldable joint from Figure 8.15 (hinges are represented by dashed lines) Figure 8.33: Model with the middle bars in the rhombus-shaped modules still present Figure 8.34: Same model as in Figure 7.33, but with cross-bars Figure 8.35: Bars are grouped in pairs and joined by a fixed connection in their apex angle Figure 8.36: Adding struts again only increases the weight, while the section remains identical Figure 8.37: Summary of the determining parameters for the strength, stability and stiffness for case study 3 OPEN structure Figure 8.38: Resulting section and weight for the foldable dome Figure 8.39: Perspective view of case study 3 CLOSED structure with sections after structure design and total weight Figure 8.40: Summary of the determining parameters for the strength, stability and stiffness for case study 3 CLOSED structure Figure 8.41: Case 3 OPEN structure Figure 8.42: Case 3 Foldable DOME structure Figure 8.43: Case 3 CLOSED structure Figure 9.1: Design concept for a tensile surface structure with a deployable central tower Figure 9.2: Mobile structure with membrane surfaces arranged around a demountable central tower ( The Nomad Concept) Figure 9.3: The top of the tower is accessible to visitors, allowing them to enjoy the view Figure 9.4: Side elevation of the tower and canopy Figure 9.5: Top view of the structure showing the three tensile surfaces arranged radially around the central tower Figure 9.6: Dimensions of the tower and a single angulated bar

25 Figure 9.7: Comparison between a linkage with angulated SLE s and its polar equivalent Figure 9.8: Imposed condition on the length of the semi-bars a and b (a<b), in order to make the linkage foldable along the vertical axis Figure 9.9: Initial unfolding of the compacted linkage to its polygonal form253 Figure 9.10: Six stages in the deployment of a hexagonal tower: elevation and top view Figure 9.11: Design parameters of a two-module tower with angulated SLE s (three states) Figure 9.12: Perspective view: design parameters of a two-module tower with angulated SLE s Side elevation showing the non-coplanarity of the angulated elements (marked in red) Figure 9.13: Illustration of the influence of the apex angle β on the geometry of a linkage with angulated SLE s with two modules (n=2) in the undeployed (top) and fully deployed configuration (below) Figure 9.14: Illustration of the influence of the apex angle β on the geometry of a linkage with angulated SLE s with three modules (n=3) in the undeployed (top) and fully deployed configuration (below) Figure 9.15: A schematic representation of the relative rotations of the quadrilaterals around imaginary fold axes during deployment Figure 9.16: Kinematic joint connecting the angulated elements at their end nodes Figure 9.17: The kinematic joint and the axes of revolution for the seven rotational degrees of freedom Figure 9.18: The scissor linkage in its deployed state and its equivalent hinged plate structure for mobility analysis (left) Fixing the structure by pinned supports (right) Figure 9.19: Deployment of proof-of-concept model Figure 9.20: Deployment sequence (A, B and C) for the tower with the membrane elements attached Figure 9.21: Design for a deployable hexagonal tower with angulated elements Figure 9.22: Initial unfolding of the compacted linkage to its polygonal form

26 Figure 9.23: Three stages in the deployment of a hexagonal tower with 5 modules: elevation and top view Figure 9.24: Hyperboloid geometry (as proposed in previous sections) angulated elements do not remain coplanar during deployment..278 Figure 9.25: Prismoid geometry (simplified alternative to the previously described geometry) - angulated elements remain coplanar during deployment Figure 9.26: Non-symmetrical identical angulated elements result in a fully compactable configuration: hyperboloid solution Figure 9.27: Symmetrical identical angulated elements cannot be fully compacted Figure 9.28: Symmetrical and non-identical angulated elements result in a fully compactable configuration: prismoid solution Figure 9.29: Symmetrical and non-identical angulated elements result in a fully compactable configuration: prismoid solution Figure 9.30: Three consecutive stages in the deployment of a prismoid geometry Figure 9.31: Three consecutive stages of the corresponding planar closed-loop structure Figure 9.32: Perspective view of the deployment of a triangular tower Figure 9.33: Top view and side elevation of the prismoid tower Figure 9.34: Detailed view of the simplified hinge connecting four scissor bars Figure 9.35: Triangular and quadrangular prismoid solution and their respective equivalent hinged-plate structure, providing an insight in the kinematic behaviour Figure 9.36: Top view and perspective view of the structure with indication of the global coordinate system and the vector components of the wind action Figure 9.37: Side elevation of the equilibrium form of the membrane Figure 9.38: Top view of the equilibrium form of the membrane Figure 9.39: Horizontal cable ties to improve structural performance Figure 9.40: Perspective view, top view and side elevation of deployable mast

27 Figure 9.41: Summary of the determining parameters for the strength, stability and stiffness of case study Figure 9.42: Case 4 A temporary canopy and its deployable tower with angulated units Figure 10.1: Case study 1 (Chapter 6) Translational barrel vault Figure 10.2: Case study 2 (Chapter 7) Polar barrel vault Figure 10.3: Case study 3 (Chapter 8) Deployable bar structure with foldable joints Figure 10.4: Case study 4 (Chapter 9) Deployable mast

28 List of Tables Table 4.1: The first eight values for β in terms of p for compactly foldable regular structures Table 4.2: Minimum and maximum possible apex angles for regular structures with 5, 7 or 9 plates Table 4.3: The span S and rise R for a given number of plates p of regular foldable structures in terms of the plate length L Table 4.4: The area of the compact configuration for (p=5, β=90 ), (p=7, β=120 ) and (p=9, β=135 ) in terms of the plate length L Table 4. 5: The area of the sectional profile of the deployed configuration for (p=5, β=90 ), (p=7, β=120 ) and (p=9, β=135 ) in terms of the plate length L Table 4. 6: The expansion ratio λ for (p=5, β=90 ), (p=7, β=120 ) and (p=9, β=135 ) Table 4.7: Minimum and maximum possible apex angles for right-angled structures with 5, 7 or 9 plates, as can be read from the graph in Figure Table 4.8: Values for β and θ for a chosen q (circular structure), combined with a regular structure (p=5) Table 5.1: Values for the wind pressure w per zone Table 5.2: The seven load cases used for calculations in EASY Table 9.1: Characteristics of the hyperboloid geometry Table 9.2: Characteristics of the prismoid geometry Table 9.3: Load combinations for wind and snow

29 List of Symbols Chapter 2 θ Deployment angle p.10 a, b Semi-bars p.11 γ Unit angle p.11 a, b, c, d Semi-lengths p.12 β Kink angle p.21 α Angle p.21 Chapter 3 θ Deployment angle p.40 γ Unit angle p.40 a, b, c, d Semi-lengths p.40 M Intermediate hinge P.41 H r Rise p.42 S Span p.42 t Unit thickness p.43 2δ Total unit angle p.43 O Centrepoint p.43 M Centrepoint p.43 C Intermediate point p.43 M Centrepoint p.43 t 1, t 2 Unit thickness p.45 O 1, O 2,,O 3 Centrepoint p.48 P, Q, S, T End nodes p.51 t Unit thickness p.54 K Intermediate hinge p.54 ϕ Quarter of total sector angle p.57 α n Angle p.57 P Base point of arc p.57 n P 0 Apex point of arc p.57 O Centrepoint p.57 R Internal radius p.57 in

30 U Number of units p.59 ω Sector angle p.59 R e External radius p.60 L Bar lenght p.60 E 0, E 1 Ellipse p.63 Chapter 4 P Basic plate element p.88 M Module p.88 p Number of plates p.88 β Apex angle p.89 L Plate length p.95 W Module width p.95 H r Rise p.95 S Span p.95 θ Deployment angle p.95 H Plate height p.96 H 1 Horizontal projection of plate height p.96 H 2 Vertical projection of plate height p.96 α, α 1, α 2 Angle p.96 L edge Edge length p.101 L Plate length p.102 λ Expansion ratio p.103 t p Thickness of a single plate element p.103 T p Total thickness of the compactly folded configuration p.104 q Number of sectors p.107 Chapter 5 t Unit thickness p.117 ρ Air density p.123 v Reference velocity p.123 ref c Altitude factor p.123 ALT c Direction factor p.123 DIR c Temporary factor p.123 TEM q Reference wind pressure p.123 ref w Total wind pressure p.123

31 w e Pressure on the external surfaces p.123 w i Pressure on the internal surfaces p.123 µ Opening ratio p.123 A L,W Total area of openings at the leeward and wind p.124 parallel sides A T Total area of openings at the windward, leeward and p.124 wind parallel sides C pi Internal pressure coefficient p.124 C pe External pressurecoefficient p.124 C pi,a Permeability p.124 s Characteristic snow load on the ground p.127 k C Temperature coefficient p.127 t C e Exposure coefficient p.127 µ i Form factor for the snow load p.127 G Permanent loads p.128 Q Mobile loads p.128 γ Safety factor p.128 D Damage p.130 n i Number of cycles p.130 N i Critical amount of load cycles p.130 σ i Fluctuating stresses p.130 σ c Resistance against fatique p.130 τ i Shear stresses p.132 Chapter 6 M1 Plane module p.139 M2 Slightly curved module p.139 M3 Highly curved module p.139 U1, U2, U3 Linkage p.142 t Unit thickness p.144 a, b Semi axes p.144 U Number of units in the span p.145 R Radius of the circular arc p.145 α 2 Angle p.145 A, A Circular arc p.145 a Distance between parallel arcs p.145 2

32 P 2, P 0, P 1, P 1, Intersection point p.145 P 2 E 0, E 1 Ellipsoid p.146 φ Angle p.146 n Node p.152 γ 2, γ 3 Angle p.155 f y Yield stress p.164 S max Maximum stress p.168 Chapter 7 U Units p.180 O Centrepoint p.180 P, Q, R Intersection points p.180 h Unit height p.180 U1, U2, U3 Linkage p.181 S Span p.188 H r Rise p.188 t Unit thickness p.188 a, b Semi-bar p.188 θ Deployment angle p.188 θ Deployment angle for which the maximum span is p.188 S max reached θ Deployment angle in the fully deployed p.188 design configuration S max Maximum span p.188 S design Span of the deployed configuration p.188 ψ Deployment ratio p.189 R in Internal radius p.189 R e External radius p.189 β Unit angle p.189 γ Sector angle p.189 S e External span p.192 Chapter 8 p Number of plates p.214 β Apex angle p.214

33 θ Deployment angle p.214 L Plate length p.214 S Span p.214 W Module width p.214 q The amount of sectors arranged radially p.216 m Number of modules p.231 R Degree of statical determinacy p.232 b Number of bars p.232 j Number of joints p.232 r Number of restraints p.232 N joints Number of continuous joints p.233 N links Total number of links p.233 N loops Number of loops p.233 Chapter 9 a, b Semi-bar length p.253 β Kink angle p.255 U Number of scissor units p.255 n Number of modules p.255 E Edge length p.255 φ Sector angle p.255 θ Deployment angle p.257 h Height of the undeployed position p.257 H Total height p.257 ψ Deployment ratio p.258 R Radius p.259 h Unit height p.261 L Base length p.261

34 Chapter 1 Introduction Chapter 1 Introduction 1.1 Deployable structures A large group of structures have the ability to transform themselves from a small, closed or stowed configuration to a much larger, open or deployed configuration. These are generally referred to as deployable structures though they might also be known as erectable, expandable, extendible, developable or unfurlable structures [Jensen, 2003]. Although the research subject of deployable structures is relatively young being pioneered in the 1960 s, the principle of transformable objects and spaces has been applied throughout history. Applications range from the Mongolian yurts, to the velum of the Roman Coliseum, from Da Vinci s umbrella to the folding chair. At present day, the main application areas are the aerospace industry, requiring highly compactable, lightweight payload and architecture, requiring either mobile, lightweight temporary shelters or fixed-location retractable roofs for sports arenas. Mobile shelter systems are a type of building construction for which there is a vast range and diversity of forms and structural solutions. They are designed to provide weather protected enclosure for a wide range of human activities. The main applications are exhibition and recreational structures, temporary buildings in remote construction sites, relocatable hangars and maintenance facilities and emergency shelters after natural disasters. Enclosure requirements are generally very simple, with the majority needing only a weather protecting membrane or skin supported by some form of erectable structure. In all applications, both the envelope and structure need to be capable of being easily moved in the course of normal use, which very often requires the building system to be assembled at high speed, on unprepared sites [Burford & Gengnagel, 2004]. An example of an easily erectable temporary exhibition structure is shown in Figure

35 Chapter 1 Introduction Figure 1.1: Mobile deployable bar structure ( Grupo Estran) Mobile deployable structures have the advantage of ease and speed of erection compared to traditional building forms. Because they are reusable and easily transportable, they are of great use for temporary applications. However, the aspect of deployability is associated with a higher mechanical complexity and design cost compared to conventional systems. This increased cost has to be balanced by the structure s potential to be suitable for the particular application. Deployable structures can be classified according to their structural system. In doing so, four main groups can be distinguished: Spatial bar structures consisting of hinged bars Foldable plate structures consisting of hinged plates Tensegrity structures Membrane structures It is noted that these deployable structural systems only constitute a portion of the possible applications in their respective field. The majority of spatial bar structures, plate structures, tensegrity structures and membrane structures is non-deployable and has a permanent location. What is referred to here are those specific applications which exhibit a certain ability to transform their shape, therefore adapting to changing circumstances and requirements. Hanaor [2001] has classified the aforementioned structural systems used in deployable structures by their morphological and kinematic characteristics (Figure 1.2). Because of their wide applicability in the field of mobile architecture, their high degree of deployability and a reliable deployment, two subcategories will be studied in greater detail (marked in red in Figure 1.2): 2

36 Chapter 1 Introduction Figure 1.2: Classification of structural systems for deployable structures by their morphological and kinematic characteristics [Hanaor, 2001] 3

37 Chapter 1 Introduction scissor structures and foldable plate structures. Scissor structures are expandable structures consisting of bars linked together by scissor hinges allowing them to be folded into a compact bundle. Although many impressive architectural applications for these mechanisms have been proposed, due to the mechanical complexity of their systems during the folding and deployment process, few have been constructed at full-scale [Asefi, 2006]. Foldable plate structures consist of rigid plate elements which are connected by continuous joints allowing one rotational degree of freedom. In their undeployed configuration they form a flat stack of plates, while a corrugated surface is formed in their fully deployed configuration. Singly curved as well as doubly curved surfaces are possible, characterised by a linear or radial deployment. 1.2 Aims and scope of research Although many different deployable systems have been proposed, few have successfully found their way into the field of temporary constructions. A cause for this limited use can be found in the complexity of the design process. This entails detailed design of the connections which ensure the expansion of the structure during the deployment process. Therefore, not only the final deployed configuration is to be designed, but an insight is required in the mobility of the mechanism, as a means to achieve that final erected state. Also, designing deployable structures requires a thorough understanding of the specific configurations which will give rise to a fully deployable geometry. The aim of the work presented in this dissertation is to develop novel concepts for deployable bar structures and propose variations of existing concepts which will lead to architecturally as well as structurally viable solutions for mobile applications. It is the intention to aid in the design of deployable bar structures by first explaining the essential principles behind them and subsequently applying these in several case studies. Starting with the choice of a suitable geometry, followed by an assessment of the kinematics of the system, to end with a structural feasibility study, the complete design process is dem- 4

38 Chapter 1 Introduction onstrated. By doing so, the strengths and weaknesses of the chosen structural system and geometric configuration, are exposed. Ultimately, the designer is provided with the means for deciding on how to cover a space with a rapidly erectable, mobile architectural space enclosure, based on the geometry of foldable plate structures or employing a scissor system. A review of previous research concerning scissor structures and foldable plate structures is given, offering an insight in the wide variety of possible shapes and configurations. An understanding of their geometry is crucial, because it greatly influences the deployment behaviour of the structure. The design principles behind these structures and several construction methods are explained and novel geometric design methods are proposed, based on architectural parameters such as the rise and span of the structure. These principles are then used in four case studies, which cover the key aspects of the design and are an application of novel proposed concepts for mobile deployable bar structures. 1.3 Outline of thesis In Chapter 2, previous work and a literature review of scissor structures and foldable plate structures is presented and the main researchers active in this field are discussed. The first part focuses on translational and polar scissor units employed in spatial structures and angulated elements applied in closed loop retractable structures. The second part is concerned with past developments within the field of foldable plate structures and their possible configurations. In Chapter 3 the basic principles needed for the design of deployable scissor structures are clarified. As a simple means of obtaining a deployable scissor linkage, several construction methods for translational and polar arches are explained. A geometric design method is proposed, for which the derived equations are based on the rise and span of the deployed configuration. This method allows the design of polar linkages of circular curvature and translational linkages of any curvature. It is shown how these can be used to obtain three-dimensional grid structures which are stress-free deployable. 5

39 Chapter 1 Introduction Chapter 4 is concerned with the design of foldable plate structures. Some basic single curvature or double curvature foldable configurations are identified which are compactly foldable for maximum transportability. The formulas needed for designing single curvature and double curvature configurations are derived. It is shown that a single plate element can be obtained from which domes and barrel vaults or combinations thereof can be composed. These design principles are applied in Chapter 8, in which a concept for a deployable bar structure is proposed based on a foldable plate geometry. Chapter 5 serves as an introduction to the case studies which will bring into practice the design methods discussed in Chapters 3 and 4. The geometry for the case studies is presented as well as the general approach for the structural analysis and design. Also, the considered load combinations are discussed. In Chapter 6 case study 1 is designed, which is a novel type of single curvature deployable structure composed of translational units on a three-way grid. A geometric design approach is proposed which is then brought into practice for designing a translational triangulated barrel vault with a circular base curve. Also, based on this barrel vault, a fully closed double curvature shape is proposed as an alternative configuration. An insight is provided in the kinematic behaviour during and after the deployment. The concept is structurally analysed according to the method specified in Chapter 5. In Chapter 7 a barrel vault with polar and translational units is designed. For the second case study a novel way of providing an open barrel vault with a compatible stress-free deployable end structure is proposed, making use of half of a slightly modified lamella dome. Analogous to case study 1, the kinematics of the system are discussed and a structural analysis is performed. In Chapter 8 an innovative concept for a mobile shelter system, based on the kinematics of foldable plate structures, is proposed. For case study 3 a basic foldable barrel vault, as well as a foldable dome are designed, based on the principles presented in Chapter 4. By combining these two basic shapes a closed doubly curved foldable geometry is obtained. The transition from plate 6

40 Chapter 1 Introduction structure to bar structures is discussed and a novel foldable articulated joint, serving as a connector for the bars, is proposed. The mobility of the mechanism is discussed and the concept is analysed structurally. Chapter 9 is concerned with the design of case study 4, which is a deployable tower with angulated scissor units. In the proposed concept the structure serves as a tower or truss-like mast for a temporary tensile surface structure and doubles up as an active element during the erection process. A comprehensive geometric design method is proposed and the influence of the design parameters on the geometry and the deployment process are discussed. Finally, the kinematic behaviour is explained and the structural feasibility is checked. Chapter 10 concludes the study by discussing the proposed concepts in a comparative evaluation. Also, a number of suggestions for further work are provided. 7

41 Chapter 1 Introduction 8

42 Chapter 2 Review of Literature Chapter 2 Review of Literature 2.1 Introduction In this chapter the main contributors to the field of deployable structures are discussed. A review is given of existing deployable scissor structures (or pantograph structures) and foldable plate structures for architectural applications. The first part is concerned with an explanation of the characteristics of translational and polar units, and the deployability condition they have to comply with when used in a scissor linkage, in order to guarantee deployability. Further, angulated elements, which are used to form closed loop structures, are discussed. These are characterised by a radial deployment, allowing the structure to retract towards its perimeter. The second part discusses the application of foldable plate structures, including single and double curvature configurations. An explanation is given of the possible plate linkages which generate compactly foldable configurations. Also, the condition which foldable plate configurations have to satisfy in order to be compactly foldable is mentioned. 2.2 Deployable structures based on pantographs Scissor units, otherwise called scissor-like elements (SLE s) or pantographic elements, consist of two straight bars connected through a revolute joint, called the intermediate hinge, allowing the bars to pivot about an axis perpendicular to their common plane (Figure 2.1). By interconnecting such SLE s at their end nodes using revolute joints, a two-dimensional transformable linkage is formed, as shown in Figure 2.2. Altering the location of the intermediate hinge or the shape of the bars gives rise to three distinct basic unit types: translational, polar and angulated units. 9

43 Chapter 2 Review of Literature Translational units The upper and lower end nodes of a scissor unit are connected by unit lines. For a translational unit, these unit lines are parallel and remain so during deployment. In Figure 2.1 a plane and a curved translational unit are shown, the plane unit being the simplest translational unit having identical bars. When these units are linked, a well-known transformable single-degree-of-freedom mechanism is formed, called a lazy-tong, shown in Figure 2.2. Unit line Intermediate hinge End node θ θ Plane unit Curved unit Figure 2.1 : Translational units The curved unit named such because it is commonly used for curved linkages has bars of different length. When the latter is linked by its end nodes, a curved linkage is formed, pictured in Figure 2.3. By varying the deployment angle θ a linkage is transformed from its most compact configuration (a compact bundle) to its fully deployed position, as shown in Figure 2.2 and Figure

44 Chapter 2 Review of Literature Figure 2.2: The simplest plane translational scissor linkage, called a lazy-tong Figure 2.3: A curved translational linkage in its deployed and undeployed position Polar units When in a plane translational unit the intermediate hinge is moved away from the centre of the bar, a polar unit is formed with unequal semi-bars a and b (Figure 2.4). It is this eccentricity of the intermediate hinge which generates curvature during deployment. The unit lines intersect at an angle γ. This angle varies strongly as the unit deploys and the intersection point moves closer to the unit as the curvature increases. In Figure 2.5 a polar linkage is shown in its undeployed and deployed configuration. 11

45 Chapter 2 Review of Literature a θ b γ Figure 2.4: Polar unit Figure 2.5: A polar linkage in its undeployed and deployed position Deployability constraint Crucial to the design of deployable scissor structures is the deployability constraint. This is a formula derived by Escrig [1985] which states that in order to be deployable, the sum of the semi-lengths a and b of a scissor unit has to equal the sum of the semi-lengths c and d of the adjoining unit. This translates theoretically into the ability of the bars to coincide in the compact state. Practically, this means that the scissor linkage is foldable into a compact bundle of bars. For the linkage in Figure 2.6, the deployability constraint is written as: a b = c + d + (2.1) 12

46 Chapter 2 Review of Literature c a d b Figure 2.6: The deployability constraint in terms of the semi-lengths a, b, c and d of two adjoining scissor units in three consecutive deployment stages It should be noted that scissor linkages which do not comply with Equation 2.1 can still be partially foldable: one unit might be fully compacted, while the adjoining unit might still be partially deployed. However, since this dissertation is concerned with the design of compactly foldable scissor structures, the deployability constraint is treated as a minimum requirement Structures based on translational and polar units In the early 1960 s, Spanish architect Emilio Perez Piñero [1961, 1962] pioneered the use of scissor mechanism to make deployable structures. He was among the first in modern times to employ the principle of the pantograph for use in deployable architectural structures, such as his moveable theatre (Figure 2.7). This particular model consisted of rigid bars and wire cables, which would become tensioned to provide the structure with the necessary stabilisation. The members remain unstressed in the compact, bundled configuration and the deployed state, except for their own dead weight. Furthermore, the structure is stress-free during the deployment, effectively behaving like a mechanism. Piñero was very productive in the field of deployable scissor structures, until all this was brought to an end by his tragic death in Another Spanish architect became one of the most prolific researchers on the subject. Felix Escrig [1984, 1985] presented the geometric condition for deployability (Section 2.2.3) and demonstrated how three-dimensional structures 13

47 Chapter 2 Review of Literature could be obtained by placing scissor units in multiple directions on a grid. Further, it was shown how curvature could be introduced in such a grid by varying the location of the intermediate hinge of the scissor units. Figure 2.7: Piñero demonstrates his prototype of a deployable shell [Robbin, 1996] Escrig has also investigated, in collaboration with J. Sanchez and J.P. Valcarcel, spherical two-way scissor structures based on the subdivision of the surface of a sphere. These two-way grids require measures, such as cross-bars or cables, to stabilise the structure in its deployed configuration, due to in-plane instability caused by non-triangulation. A myriad of geometric models has been proposed by Escrig [1985,1987] based on two-way and three-way grids with no curvature, single curvature or double curvature. An example of each category is given in Figures 2.8 to

48 Chapter 2 Review of Literature Figure 2.8: Planar two-way grid with translational units and cylindrical barrel vault with polar units [Escrig, 1985] Figure 2.9: Top view and side elevation of a two-way spherical grid with identical polar units [Escrig, 1987] Figure 2.10: Top view and side elevation of a three-way spherical grid with polar units [Escrig, 1987] 15

49 Chapter 2 Review of Literature Figure 2.11: Top view and side elevation of a geodesic dome with polar units [Escrig, 1987] Figure 2.12: Top view and side elevation of a lamella dome with identical polar units [Escrig, 1987] Besides constructing several models, Escrig has also designed a cover for a swimming pool in Seville. The design consists of two identical rhomboid grid structures with spherical curvature. The subdivision of the spherical surface is executed in such a way, that straight edges emerge, allowing several structures to be mutually connected along these edges (Figure 2.13). Figure 2.13: Deployable cover for a swimming pool in Seville designed by Escrig & Sanchez ( Performance SL) 16

50 Chapter 2 Review of Literature Some of the proposed geometric configurations for three-dimensional grid structures demonstrate a snap-through effect during the deployment. This means that they do not deploy as mechanisms and are no longer stress-free during expansion (apart from their own dead weight). This snap-through effect is caused by geometric incompatibilities between the member lengths associated with the way they are contained within the grid. Because they are in a stress-free state before and after deployment, but go through an intermediate stage with deployment induced stresses, they are called bi-stable deployable structures. Figure 2.14 illustrates the snap-through effect on a square module with diagonal units. The diagonal units (marked in red) are subject to elastic deformation in the intermediate deployment stage, while the undeployed and fully deployed configuration are stress-free. Figure 2.14: Bi-stable structure before, during and after deployment Figure 2.15: Collapsible dome and a single unit, as proposed by Zeigler [1976] 17

51 Chapter 2 Review of Literature Zeigler [1981, 1984] was the first to exploit this phenomenon as a self-locking effect, effectively making extra stabilisation after deployment (which is necessary for stress-free deployable structures) obsolete. He proposed, on these grounds, a partial triangulated spherical dome as shown in Figure Charis Gantes [1996, 2001] has thoroughly investigated bi-stable deployable structures and has developed a geometric design approach for flat grids, curved grids and structures with arbitrary geometry. Also, he has researched the structural response during deployment, which is characterized by geometric non-linearities. Simulation of the deployment process is, therefore, an important part of the analysis requiring sophisticated finite element modelling. The material behavior, however, must remain linearly elastic, so that no residual stresses reduce the load bearing capacity under service loads. Two of his proposals for bi-stable structures, an elliptical arch and a geodesic dome, are depicted in Figure Figure 2.16: Bi-stable structures: elliptical arch and geodesic dome [Gantes, 2004] A geometric and kinematic analysis of single curvature and double curvature structures has been performed by Travis Langbecker [1999, 2001]. He has used translational units to design several models of positive (Figure 2.17) and negative (Figure 2.18) curvature structures. By using compatible translational units and by keeping the structural thickness (unit thickness) constant throughout the whole structure, these configurations are always stress-free deployable. 18

52 Chapter 2 Review of Literature Figure 2.17: Positive curvature structure with translational units in two deployment stages [Langbecker, 2001] Figure 2.18: Negative curvature structure with translational units in two deployment stages [Langbecker, 2001] Pantographic deployable columns are linear deployable structures composed of translational or polar units and were researched by Raskin [1996, 1998]. His work focussed on pantographs behaving as mechanisms during deployment, which are to be stabilised in the deployed configuration by additional boundary conditions. First, plane linkages were investigated, which were subsequently used to form prismatic columns (Figure 2.19). Expanding his findings, deployable pantographic slabs that can be packaged in different arrangements were proposed. Figure 2.20 shows two variations of such a deployable slab, consisting either of prismatic modules or an arrangement of prismatic columns. 19

53 Chapter 2 Review of Literature Figure 2.19: Plane and spatial pantographic columns by Raskin [1998] Figure 2.20: Pantographic slabs by Raskin [1998] Under the guidance of Dr. Sergio Pellegrino, a research group called the Deployable Structures Laboratory, emerged at the Cambridge University in 1990 as a driving force in the field of deployable structure research. One of their proposals constituted a deployable pantographic ring structure developed as the edge beam of a deployable antenna. Together with Zhong You, the conditions for strain-free deployment of such a structure were derived [You & Pellegrino, 1993]. Structures of this type consist of translational linkages on the perimeter ring and inner ring, mutually connected by radially placed polar units. As an example, Figure 2.21 shows a structure based on a twelve-sided polygon. 20

54 Chapter 2 Review of Literature Figure 2.21: Deployable ring structure [You & Pellegrino, 1993] Angulated units Unlike common pantograph units with straight bars, angulated units consist of two rigidly connected semi-bars of length a that form a central kink of amplitude β. Because they were invented by Hoberman [1990] they are commonly denoted as hoberman s units. The major advantage is that, as opposed to polar units, angulated units subtend a constant angle γ during deployment (Figure 2.22). For this to occur, the bar geometry has to be such that α= γ/2. This implies that angulated elements can be used for radially deploying closed loop structures, capable of retracting to their own perimeter, which is impossible to accomplish with translational or polar units, which demonstrate a linear deployment. (Figure 2.23) shows a circular linkage with angulated elements in its undeployed and deployed configuration. a θ a α β γ α = γ/2 Figure 2.22: Angulated unit or hoberman s unit 21

55 Chapter 2 Review of Literature Figure 2.23: A radially deployable linkage consisting of angulated (or hoberman s) units in three stages of the deployment The structure shown in Figure 2.23 is formed by two layers of identical angulated elements, of which one layer is formed by elements in clockwise direction (marked in gray), while the other is arranged in counter-clockwise direction (marked in red). As the structure deploys, each layer undergoes a rotation, equal in magnitude but opposite to each other Closed loop structures based on angulated elements You & Pellegrino [1996, 1997] extended the previous concept to multiangulated elements, which are elements with more than one kink angle, as can be seen in Figure They found that two or more such retractable structures can be joined together through the scissor hinges at the element ends. Two angulated elements from layers that turn in the same direction of two such interconnected structures, were found to maintain a constant angle and could therefore be rigidly connected, thus forming a multi-angulated element. The deployment of such a structure, composed of two layers of twelve identical multi-angulated elements with three kinks, is depicted in Figure

56 Chapter 2 Review of Literature α α α α γ/2 γ/2 γ/2 α = γ/2 Figure 2.24: Multi-angulated element Figure 2.25: A radially deployable linkage consisting of multi-angulated elements in three stages of the deployment This concept was extended by You & Pellegrino [1996, 1997] to include generalised angulated elements (GAE) which allow non-circular structures to be generated. Depending on which type of GAE is used, such structures form patterns of either rhombuses of parallelograms. By providing this type of structure with cover elements, Kassabian et al. [1997, 1999] has shown it possible to employ them as a retractable roof (Figure 2.26). 23

57 Chapter 2 Review of Literature The cover elements provide, both in the open and closed position a gap-free, weatherproof surface. Figure 2.26: Multi-angulated structure with cover elements in an intermediate deployment position Jensen [2004] has found that, instead of covering a bar structure with plates, it is possible to remove the angulated elements and connect the plates directly by means of scissor hinges at exactly the same locations as in the original bar structure. Thus, the kinematic behaviour of the expandable structure remains unchanged. He has developed general methods and the conditions for connecting expandable structures of any plan shape (Figure 2.27), leading to the possibility of creating plane or stacked assemblies composed of individual expandable structures. This has led to the development of transformable freeform or blob -structures, as shown in Figure 2.28 [Jensen & Pellegrino, 2004]. Figure 2.27: Model of a non-circular structure where all boundaries and plates are unique [Jensen, 2004] 24

58 Chapter 2 Review of Literature Figure 2.28: Computer model of an expandable blob structure [Jensen & Pellegrino, 2004] Several other types of closed loop structures have been developed by Escrig et al. [1996], Chilton et al. [1998], Wohlhart [2000], You [2000] and Rodriguez & Chilton [2003]. Retractable reciprocal plate structures have been developed by Chilton et al. [1998], of which an example is shown in Figure The structure shown consists of six triangular rigid plate elements which each slide against each other as the structure is retracted, hence providing a continuous surface throughout the deployment process. Figure 2.29: Reciprocal plate structure Rodriguez & Chilton [2003] have proposed a novel retractable structure called the swivel diaphragm. It forms a ring of congruent parallelograms between angulated elements by using the fixed points of the structure together with straight bars. In Figure 2.30 a swivel diaphragm is shown in several stages of the deployment. As opposed to the multi-angulated elements proposed by Kassabian et al., the support points can always be directly connected to the angulated elements, which allows the angulated elements to swivel around the fixed points. The angulated elements can be replaced by rigid plate elements to form a continuous surface in both the open and closed position. Rod- 25

59 Chapter 2 Review of Literature riguez et al. [2004] has also developed methods for interconnecting several individual swivel diaphragms to form larger retractable assemblies. Figure 2.30: Swivel diaphragm in consecutive stages of deployment Several researchers have proposed dome shaped structures that can retract towards their perimeter. Piñero pioneered a diaphragm retractable dome with a number of wedge-shaped plates which are able to rotate about an axis normal to the sphere [Escrig, 1993]. When the plates are rotated, an aperture is created at the centre of the dome. During opening and closing, all plates show an overlap, except in the fully closed position where the plates form a gapfree spherical cap. To create a single-degree-of-freedom mechanism, pairs of adjacent plates are mutually connected through a revolute joint at the apex of one plate. This joint is then run along a certain path on the other plate, as illustrated in Figure Figure 2.31: Reciprocal dome proposed by Piñero [Escrig, 1993] Angulated elements connected by scissor hinges not only subtend a constant angle in a plane surface, but also on a conical surface. This was discovered by Hoberman [1991], who has proposed the Iris Dome shown in Figure The structures consists of five concentric rings of angulated elements, connected with axes of rotation tangential to the circular plan of the rings, and thus a 26

60 Chapter 2 Review of Literature retractable dome is formed. The dome uses rigid plates as cladding material, attached to the individual angulated elements. When the dome is closed, a continuous surface is formed, while in the open configuration the plates are stacked upon each other. Hoberman exhibited a model for a dome, which was continuously retracted by an actuator, at the Expo 2000 in Hannover. In 2002 he designed and built a semi-circular retractable mechanical curtain for ceremonial purposes at the Winter Olympics in Salt Lake City. Both structures are shown in Figure Figure 2.32: Iris dome by Hoberman [Kassabian et al, 1999] Figure 2.33: Retractable dome on Expo Hannover (courtesy of M. Mollaert) Mechanical curtain Winter Olympics Salt Lake City 2002 [Hoberman, 2007] Extending his work on two-dimensional retractable plate structures, Jensen [2004] has proposed an elegant solution for a retractable dome structure, using only plate elements, instead of a combination of multi-angulated elements and cover plates. One of his proposals is a novel type of retractable dome, described as a self-supporting reciprocal mechanism, similar to that proposed by Piñero [Escrig, 1993]. Unlike the concept developed by Piñero, the current structure does not have any overlaps, and hence friction between neighbour- 27

61 Chapter 2 Review of Literature ing plates is removed, which makes it better suited for large scale applications. Other advantages are the possibilities of modifying the plate boundaries and the location of the fixed points about which the plates rotate. Figure 2.34 shows a retractable dome with plates having fixed points of rotation. The plates provide a gap-free surface in the open and closed position. Figure 2.35 shows a retractable dome with modified boundaries. Figure 2.34: Retractable roof made from spherical plates with fixed points of rotation [Jensen, 2004] Figure 2.35: Novel retractable dome with spherical plates with modified boundaries [Jensen, 2004] 28

62 Chapter 2 Review of Literature 2.3 Foldable plate structures A family of foldable, portable structures, based on the Yoshimura buckle pattern for axially compressed cylindrical shells, has been presented by Foster and Krishnakumar [1986/87]. These structures have considerable shape flexibility (multiple degrees of freedom), but once erected they possess significant stiffness. Figure 2.36: Basic layout of Foster s module [Foster, 1986] Foldable plate structures consist of a series of triangular plates, connected at their edges by continuous joints, allowing each plate to rotate relative to its neighbouring plate. The plates can fold into a flat stack and unfold into a predetermined three-dimensional configuration, with a corrugated surface. Determining the actual shape of the deployed configuration, is an origami-like pattern formed by intersecting mountain folds and valley folds. The foldable plate linkage consists of a repetition of the basic plate element which is of triangular shape and must possess one angle of at least 90, in order to gener- 29

63 Chapter 2 Review of Literature ate a configuration which is compactly foldable into a flat stack of plates. All elements within sections (modules) of the structures must have the same shape and equal apex angles. A module, as Figure 2.36 shows, is essentially a strip of plates joined in such a way that opposite sides are parallel when fully deployed. The module pictured contains four plates: three full and two halves (which are counted as one full plate). Separate modules, which must each be capable of being compactly foldable, are joined together by continuous hinges along the parallel sides to form the complete structure. The apex angle of the triangular plate is the main parameter which determines the fold pattern, and therefore the geometry of the final deployed shape. As can be seen from Figure 2.36, some apex angles give rise to folded configurations, which are not fully compacted, i.e. there is a gap in the folded configuration. This can be avoided when the apex angle is a sub-multiple of 360. The simplest foldable shape is a four-plate linkage with an apex angle of 90. Now if all plates are isosceles triangles, the compact folded configuration has a square shape but when all plates are right-angled triangles, the compactly folded configuration is of rectangular shape. Figure 2.37: Combination of different modules [Foster, 1986] 30

64 Chapter 2 Review of Literature When two modules, one with isosceles triangles and one with right-angled triangles, are connected, their flatly packed shape is an overlapping square and rectangle, as shown in Figure This shows that not all elements in a structure have to be identical, but, on the other hand, the usefulness and the element uniformity of such a mixed structure is decreased. There is a maximum number of plates associated with each apex angle which will give a foldable configuration. Using a higher number of plates would make folding of the configuration physically impossible, due to plate overlap. For instance, a 90 apex angle corresponds with a maximum of four full plates, a 108 apex angle with five plates, a 120 apex angle with six plates, and so on. Figure 2.38: Simplest building with 90 apex angle [Foster, 1986] Consider the minimal foldable configuration: a four-plate linkage with an apex angle of 90, shown in Figure This linkage is the most compactly foldable 31

65 Chapter 2 Review of Literature configuration, as it folds into a compact square shape. Despite its advantageous folding properties, a major shortcoming of practical use is that clear headroom is quite low. A possible solution is to make a longitudinal field joint between two modules, as depicted in Figure Of course, the joined substructures will have to be disassembled in order to be foldable. Figure 2.39: Building formed by two 90 -modules joined at their ends [Foster, 1986] A plate linkage with appealing characteristics is the one with an apex angle of 120. From Figure 2.40 it can be seen that the width of the collapsed configuration is identical to the plate length. If, for example, the length is made 2.4 m, the headroom is 2.08 m and this would make it suitable for human habitation, as opposed to the 90 -elements, where the headroom would only be 1.7 m. As Figure 2.41 shows, by connecting two modules by means of a field joint, the resulting deployed structure would be employable as a temporary field service hangar for light aircraft and similar applications [Gantes, 2001]. Figure 2.40: Building with apex angle of 120 [Foster, 1986] 32

66 Chapter 2 Review of Literature Figure 2.41: Building formed by two 120 -modules joined at their ends [Foster, 1986] Employing isosceles triangular plates leads to skewed deployed configurations, which can equally be combined along their parallel, straight edges. Figure 2.42 shows a structure with 90 -, 60 - and 30 -elements. Figure 2.42: Structure with 90 -, 60 - and 30 -elements [Foster, 1986] The previously discussed structures are all characterised by a linear deployment. As the structure deploys, it undergoes a linear expansion in the longitudinal direction and a variation of the curvature in the transverse direction. The 120 -structure can also be circularly deployed, by holding the bottom elements together and only deploying the middle section. The example in Figure 2.43 could be used for a temporary stage shell. 33

67 Chapter 2 Review of Literature Figure 2.43: Temporary stage shell with 120 modules - Tension cables used to provide bulkhead [Foster, 1986] Due to the fact that these structures are basically a mechanism, a number of constraints have to be considered to make them statically determinate. Figure 2.43 shows how tension cables can be integrated in the span of a 120 structure to provide bulkhead. Further expanding this concept, Tonon [1991, 1993] has studied the geometry of single and double curvature foldable plate structures, such as domes, conics, paraboloids and hyperboloids. To obtain these shapes, variations of the apex angle and/or the plate dimensions are imposed in subsequent modules. Although they are compactly foldable into a compact stack of plates, some fold patterns cannot be developed in a plane, as opposed to the configurations previously described by Foster. Figure 2.44: Double curvature variable shape (hyperbolic type), plane pattern [Tonon, 1993] 34

68 Chapter 2 Review of Literature As can be seen from Figure 2.44, which has a variable double curvature (hyperboloid shape), such a pattern cannot be developed in a plane. However, once the corresponding edges are connected by a field joint in a partially deployed state, the structure can be further folded until it reaches its fully compacted shape. Tonon has formulated the condition, which the plate geometry has to satisfy, in order to guarantee full foldability: the sum of the individual base angles of two neighbouring plate elements has to be constant throughout the plate geometry. This is illustrated by Figure 2.45, from which it can be seen that the size of the individual plate angles varies, although their sum remains constant throughout the pattern. It is noted that Tonon uses the base angle of the triangular plates to describe the fold pattern, as opposed to the apex angle used by Foster. Plates with variable base angles Constant sum Figure 2.45: Fold pattern with different individual plate angles, but a constant sum throughout the plate geometry, guaranteeing full foldability. A selection of doubly curved paper models is shown in Figure 2.46 and Figure Figure 2.46: Doubly curved folded shapes [Tonon, 1993] 35

69 Chapter 2 Review of Literature Figure 2.47: Linear and circular deployable double curvature folded shapes [Tonon, 1993] Using a plane foldable geometry, Hernandez & Stephens [2000] have proposed a folding aluminium sheet roof for covering the terrace of a pool area. The fold pattern consists of trapezoidal plate elements which give rise to a plane corrugated surface (Figure 2.48). Because no curvature is introduced by folding, the roof retracts on a supporting steel structure consisting of seven parallel trusses, to provide the necessary headroom. The retraction, during which the sheets are supported by wheels running on a rail, is realised by a motor driven system of wire cables. Special attention has been given to the joint design, providing a waterproof sealing between consecutive plates. Figure 2.48: Folding aluminium sheet roof for covering the terrace of the pool area of the International Center of Education and Development in Caracas, Venezuela [Hernandez & Stephens, 2000] Another interesting idea, although not of immediate use for an architectural application, is brought forward by Guest and Pellegrino [1994], treating the folding of triangulated cylinders, later expanded by Barker & Guest [1998]. Such triangulated cylinders consist of identical triangular panels, placed on a 36

70 Chapter 2 Review of Literature helical strip of the cylinder. All cylinders made of isosceles triangles fold down to prisms and are packaged as a compact stack of plates. They are strain-free in their compacted and fully deployed configuration. However, in intermediate folding positions, some deformation of the surface is required. The concept has been tested and applied to inflatable, thin walled metal cylinders. Figure 2.49: (a) Fold pattern; (b) Fold pattern with alternate rings to prevent relative rotation during deployment [Barker & Guest, 1998] 37

71 Chapter 2 Review of Literature 38

72 Chapter 3 Design of Scissor Structures Chapter 3 Design of Scissor Structures 3.1 Introduction This chapter is concerned with explaining the basic principles needed for the design of deployable scissor structures, composed of translational or polar scissor units, of which the characteristics have been discussed in Sections and Retractable closed loop structures consisting of angulated elements (Section 2.2.5) are not discussed in this chapter. As can be seen from Figure 2.14, such structures are not foldable into a compact bundle of bars in their undeployed position. In the context of highly compactable, easily transportable single curvature or double curvature grid structures, they offer therefore no added value over translational or polar units. However, a specific design approach will be proposed to employ angulated elements in a different setting: a linear deployable tower for case study 4, where they contribute substantially to the concept. It will be shown how translational and polar units can be connected to form the simplest of scissor mechanisms: a two-dimensional linkage, which can be designed by either of two methods: pure geometric construction or a novel geometric design method, using equations to obtain the complete geometry in its deployed configuration. Further, it is shown how these two-dimensional linkages can be combined to form stress-free deployable three-dimensional grid structures of single or double curvature, which will form the basis of the design of case studies 1 and 2. In Section (Figure 2.14) the difference between stress-free deployable and bi-stable structures has been explained. Because the aim of this dissertation is to provide the means for designing easily deployable scissor structures, the focus is on stress-free deployable structures only. Finally, the grid struc- 39

73 Chapter 3 Design of Scissor Structures tures known from literature (Section 2.2.4) which are guaranteed stress-free deployable, are identified and classified according to their grid type (two- or three-way) and their curvature (single or double). 3.2 Design of two-dimensional scissor linkages Two methods for obtaining a deployable scissor linkage will be presented. Whether it be pure geometric construction, or a geometric design method employing a series of equations, both approaches are based on the deployability constraint (Section 2.2.3). The two-dimensional and three-dimensional structures of which the design is explained in this chapter, are based on the translational or the polar unit (Figure 3.1). t θ t θ γ Figure 3.1 : Translational and polar scissor unit It should be noted that scissor linkages which do not comply with equation 3.1 can still be partially foldable: one unit might be fully compacted, while the adjoining unit might still be partially deployed. However, since this dissertation is concerned with the design of compactly foldable scissor structures, the deployability constraint is treated as a minimum requirement. The deployability constraint can be made more tangible by using its graphic representation. Consider the scissor unit with semi-lengths a and b in Figure 3.2. When a compatible unit is to be linked with the first unit, the location of 40

74 Chapter 3 Design of Scissor Structures the intermediate hinges has to be determined, in such a way that semi-lengths c and d satisfy the deployability constraint. The locus of all valid intermediate hinges M that comply with the deployability constraint is an ellipse, with the common end nodes of both units as its foci. This method of representation is valid for all general scissor linkages. c M a d b Figure 3.2: An ellipse as the graphic representation of the deployability constraint for translational units, determining the locus of the intermediate hinge When polar units are used, however, the locus of valid intermediate hinges can be represented by a circle, as shown in Figure 3.3. a b c d M Figure 3.3: A circle as the graphic representation of the deployability constraint for polar units, determining the locus of the intermediate hinge 41

75 Chapter 3 Design of Scissor Structures In the following section the deployability constraint and its geometric representation will be used to design two-dimensional deployable linkages composed of translational and polar units Method 1: Geometric construction Apart from a plane (or rectilinear) curve, the most common curve for a scissor linkage is a circular arc (Figure 3.4). From an architectural point of view, it makes sense to describe this base curve in terms of a given rise (Hr) and span (S). Once the geometry of the base curve is determined, it can be fitted with a series of compatible scissor units, each obeying the geometric constraint (or deployability constraint). The construction of polar and translational units will now be discussed in greater detail. H r S Figure 3.4: Circular arc as base curve, determined by the design values O H r (rise) and S (span) 42

76 Chapter 3 Design of Scissor Structures Polar linkages Method 1: the base curve contains the intersection points of the scissor bars (intermediate hinges). Linkages can be constructed with either a constant or a variable unit thickness t. The unit thickness is defined as the distance between the internal and external end nodes of the scissor units, as shown in Figure 3.1. Constant unit thickness (Figure 3.5) A polar linkage with a constant unit thickness has bars of identical length. All intermediate points C of the scissor units lie on the base curve. Construction: A: the base curve is divided in equal angular portions by the polar unit lines, which all intersect in centre O. Segment MC is tangent to the base curve, perpendicular to OC, which is the bisector of a segment. Point M is the centre of the circle which represents the geometric constraint for deployability. This circle intersects the base curve in the intermediate point C. B: now a scissor unit can be drawn, in such a way that MC is the bisector of unit angle 2δ. Now the unit thickness t is determined. C: the line through MC intersects the next unit line in point M. A circle can now be drawn with centre M and M C as radius. The intersection point of this circle with the base curve is the intermediate point of the next unit. D: proceeding this way leads to a complete linkage 43

77 Chapter 3 Design of Scissor Structures t M δ δ C M A B C D Figure 3.5: The arc is divided in equal angular portions. Circles intersecting the arc determine the loci of the intermediate hinge points 44

78 Chapter 3 Design of Scissor Structures Variable unit thickness (Figure 3.6) A polar linkage with a variable unit thickness has different units comprising of bars of variable length. All intermediate points C of the scissor units lie on the base curve. Construction: A: the base curve is divided in unequal angular portions by the polar unit lines, which all intersect in centre O. Point C is arbitrarily placed on the base curve between two consecutive unit polar lines. Segment MC is tangent to the base curve, perpendicular to OC. Point M is the centre of the circle which represents the geometric constraint for deployability. This circle intersects the base curve in the intermediate point C. B: now a scissor unit can be drawn, in such a way such that MC is the bisector of unit angle 2δ. Now the unit thickness t 1 is determined. C: the line through MC is intersected with the next unit line in point M. A (differently sized) circle can now be drawn with centre M and M C as radius. The intersection point of this circle with the base curve is the intermediate point of the next unit. The extended scissor bars of the first unit intersect with the unit line through M by which unit thickness t 2 is obtained. D: proceeding this way leads to a complete linkage 45

79 Chapter 3 Design of Scissor Structures t 1 M δ δ C M t2 A B C D Figure 3.6: The arc is divided in unequal angular portions. Variable circles intersecting the arc determine the loci of the intermediate hinge points 46

80 Chapter 3 Design of Scissor Structures Method 2: an inner and outer curve, with a constant distance t (unit thickness) between them, contain the inner and outer end nodes of the polar units. This is a simple method for constructing a linkage with constant unit thickness. All units have identical bars. t C A B C Figure 3.7: An inner and outer arc determine the constant unit thickness D 47

81 Chapter 3 Design of Scissor Structures Constant unit thickness (Figure 3.7) Construction: A: the inner curve is divided in equal angular portions by the polar unit lines, which all intersect in centre O. B: an outer arc is offset a distance t from the inner base curve, by which the unit thickness is determined and kept constant. C: the intersection points of the polar unit lines with the inner and outer curve determine the position of the inner and outer end nodes of the units. Constructing the units is simply a matter of connecting the appropriate nodes D: proceeding this way leads to a complete linkage A pluricentred polar linkage Using a pluricentred arc as base curve for the 2D-linkage (as opposed to the previously discussed single-centred arc) can increase the overall headroom, while maintaining the same span and rise. Figure 3.8a shows a single-centred arc with centre O 1. The segmented arc in Figure 3.8b has an identical span and rise, but consists of three arcs: one arc with centre O 2 and two arcs with centre O 3 and a decreased radius. Figure 3.8c shows the difference in headroom between the single-centred and the pluricentred arc. O 1 O 3 a b c Figure 3.8: For the same span and rise, a pluricentred arc can offer increased headroom compared to a single-centred arc O 2 48

82 Chapter 3 Design of Scissor Structures Figure 3.9: Example of a pluricentred base curve consisting of three arc segments with decreasing radius Constant unit thickness (Figure 3.10) In a polar pluricentred linkage with a constant unit thickness all bars are identical. The base curve is a concatenation of arc segments, each of which has its own centre. Because of the varying curvature of the base curve, there is a difference in unit geometry per segment, although all bars are identical. This means that the location of the intermediate hinge of the units differs per segment. The geometry is determined by the end nodes lying on the inner and outer curve. Construction: A: the first segment of the inner base curve with centre O 1 is divided in equal angular portions by the polar unit lines, which all intersect in centre O 1. A second arc segment with centre O 2 has a smaller radius 49

83 Chapter 3 Design of Scissor Structures B: an outer curve is drawn, with an offset distance t from the inner curve C: the intersection points of the polar unit lines with the inner and outer curve determine the position of the inner and outer end nodes of the units. One bar of the first unit is constructed by connecting the appropriate end nodes. A circle, which represents the geometric constraint is drawn with centre M and radius MM. D: now the same circle is drawn, but with centre M. The intersection point with the outer curve is the centre for the next circle. This is repeated until the end of the complete base curve is reached. The repetition of the circles ensures a constant bar length. E, F: analogous to phase D, the second chain of bars is now constructed, effectively completing the scissor linkage M t M R 1 R 2 O 2 A O 1 B 50

84 Chapter 3 Design of Scissor Structures C D E F Figure 3.10: Each arc segment (with a different radius) is divided in equal angular portions. Identical circles ensure a constant bar length Translational linkages As opposed to polar linkages, which are commonly based on a circular curve, translational linkage are easily based on any arbitrary curvature. Therefore, the general method for base curves with arbitrary curvature will be explained, by which all possible curves are covered. Linkages with variable or constant unit thickness are possible. Method: the base curve contains the midpoints M, M of the segments PQ and ST representing the unit thickness t, shown in Figure Points P, Q and S, T are also the foci of the ellipses representing the deployability constraint. 51

85 Chapter 3 Design of Scissor Structures Points M and M are found by intersecting a double ellipse (twice the size of the original ellipse) with the base curve. Once these intersection points are found, completing the scissor units is simply a matter of appropriately connecting the end nodes by segments PT and QS. Constant unit thickness (Figure 3.12) A translational linkage with a constant unit thickness has bars of variable length. All midpoints M lie on the base curve. Construction: A: a vertical segment of length t is placed twice on the base curve, a chosen distance removed from each other, in such a way that midpoints M and M are located on the base curve. This determines the geometry of the first translational unit. The small ellipse (deployability constraint) can be drawn and subsequently the double ellipse is derived. B: now the double ellipse and the unit thickness t are placed repeatedly on the base curve, each time using the intersection point from the previous ellipse. C, D: now the end nodes are appropriately connected to form the complete linkage. P t M M Q S M M Figure 3.11: The original and double ellipse representing the deployability constraint intersection points M and M are the midpoints of the unit thickness t T 52

86 Chapter 3 Design of Scissor Structures A B 53

87 Chapter 3 Design of Scissor Structures C D Figure 3.12: Double ellipses impose the deployability constraint on a translational linkage with constant unit thickness Variable unit thickness (Figure 3.13 and Figure 3.14) A translational linkage with a variable unit thickness {t 1, t 2,, t n } has bars of variable length. The intersection points K of the (small) ellipses with the base curve determine the location of the intermediate hinges. This approach is somewhat laborious because for each new unit a separate ellipse has to be drawn to determine the location of the next intermediate hinge. Because of the varying unit thickness, the method of the double ellipses cannot be used. Construction: A: a vertical segment of length t 1 is placed on the base curve. The location of the intermediate hinge K of the first unit is arbitrarily chosen on the base curve. P, Q and K are determined. Now the ellipse can be drawn. Lines through PK and QK are intersected by an arbitrary 54

88 Chapter 3 Design of Scissor Structures vertical unit line. These intersection points S and T determine the unit thickness t 2. Now the second ellipse can be drawn. B: the intersection point of the second ellipse and the base curve is determined and the previous steps are repeated C, D: now the end nodes are appropriately connected to form the complete linkage. P t 1 M Q K S M t 2 T Figure 3.13: Two differently sized, but compatible ellipses representing the deployability constraint intersection points M and M are the midpoints of the unit thickness t 1 and t 2 55

89 Chapter 3 Design of Scissor Structures A B C 56

90 Chapter 3 Design of Scissor Structures D Figure 3.14: Ellipses of different scale determine the location of the intermediate hinges on the base curve to form a translational linkage with varying unit thickness Method 2: Geometric design Parameterisation of the base curve based on the rise and the span This section is concerned with determining the geometry of polar and translational linkages based on architecturally relevant design parameters: the rise H r and the span S of the circular base curve, as shown in Figure The equations for a geometric design approach will be presented. Design values: Unknowns: H r, S R in,o, P 0, P n, α,ϕ n 57

91 Chapter 3 Design of Scissor Structures R in α n Figure 3.15: The parameters used in the description of the geometry of the circular arc: rise (H r ) and span (S) First ϕ, R, α are found through the following relationships: in n H r 1 2H r tan = ϕ = tan S 2 S S 2 1 S sin( 2ϕ ) Rin = R 2 sin 2ϕ ϕ (3.1) = (3.2) in π α = 2ϕ 2 ( ) n (3.3) The general polar equation of the circular arc is given by Eqn (3.4), with α α π : n α n x = R y = R in in cosα sinα (3.4) 58

92 Chapter 3 Design of Scissor Structures By joining Eqns (3.1), (3.2) and (3.4) we can now write the equation in terms of H and S: r x = y = S 1 2H sin 2 tan S S 1 2H sin 2 tan S r r cosα sinα (3.5) Point P 0(0, Rin ) is the midpoint of the curve and Pn ( Rin cosα n, Rin sinαn ) is an endpoint. Now the base curve has been derived from two design parameters S and H r, only two more parameters are to be given a value, in order to fully determine the complete linkage: the number of units U and the unit thickness t. This is shown for both a polar and a translational linkage in the following section. Geometric design of a polar linkage The base curve can now be fitted with a chosen number of polar units U. Therefore, the sector angle ω has to be divided into equal angular portions γ, as shown in Figure From Figure 3.15 and Figure 3.16, we know that ω = U γ (3.6) and ω = 4ϕ (3.7) Therefore, γ 4ϕ U = (3.8) 59

93 Chapter 3 Design of Scissor Structures R in obtained from Eqn (3.2), a With the inner radius (radius of the base curve) unit thickness t is chosen, which determines an outer curve with radius R R e : = R t (3.9) e in + Because the scissor units from Figure 3.16 are polar units, each bar length L is divided into two unequal semi-bars a and b. Now the geometry of the linkage can be totally derived by finding values for semi-bar lengths a, b and the deployment angle θ. From Figure 3.16 and from Sastre [1996] we can find the following relations: L a + b = (3.10) With γ known from Eqn (3.8) and R in, Re known from Eqns (3.2) and (3.9) we can use the cosine rule to obtain L : L = R + R 2 R R cosγ (3.11) in e in e n θ S t θ a m b ω γ R in R e γ Figure 3.16: Parameters needed for the geometric design of a polar linkage 60

94 Chapter 3 Design of Scissor Structures Through similarity of triangles is R R e m n in = (3.12) and m a = (3.13) n b Equating (3.12) and (3.13) results in Rin a a R R b e b R = e = in (3.14) Substituting Eqn (3.10) in Eqn (3.14) gives L Re b R + R and a = (3.15) in L R e in = (3.16) Rin + Re Also, m = a sin θ 2 (3.17) and m = R sin γ in 2 (3.18) Finally, by equating (3.17) and (3.18), an expression for the deployment angleθ is obtained: 1 R γ θ = 2sin in sin (3.19) a 2 The complete geometry has now been derived from design parameters H r, S, U and t. 61

95 Chapter 3 Design of Scissor Structures Geometric design of translational linkages This section is concerned with the geometric design of a translational linkage with constant unit thickness on a base curve. As opposed to the approach previously described for the polar linkage, a translational linkage is not characterised by the angular portions of the sector angle of the arc. This is because the unit lines for a translational unit are parallel and not radially arranged as is the case with polar units. Therefore, another approach is used for subdividing the base curve, which will involve solving a system of equations. First, a circular arc is used as base curve, but the approach is easily extended to other curves as well. Let H r and S determine a circular arc as shown in Figure Again, a semicircle or an arc segment can be derived, determined by the angle α. n The additional design parameters that will fully determine the geometry of the linkage are the number of units U, the unit thickness t. The number of desired units U will determine how many intersection points between the unit lines and the base curve will have to be calculated. To explain the method, a linkage with U=4 is used. Due to symmetry, only two units are shown. As shown in Figure 3.17 the location of P 0 and P 2 is known: P 0 is the midpoint and P 2 the endpoint of the curve. Essentially, the only solution needed is the position of point P 1 on the base curve. 62

96 Chapter 3 Design of Scissor Structures E 0 E 1 t P 0 P e P1 P in P 2 α2 α 1 Figure 3.17: Parameters for the geometric design of a translational linkage with four units (U=4), of which two are shown Analogous to the geometric construction method previously discussed, the double ellipse is used for imposing the geometric constraint and determining the intersection points P 0, P 1 and P 2 with the base curve (the original, small ellipses in Figure 3.17 are merely there to illustrate the geometric constraint of the units, but serve no further purpose in the calculation). Figure 3.18 shows the relation between the semi-axes of the double ellipse and the unit thickness. Their relation is given by t = b a (3.20) 63

97 Chapter 3 Design of Scissor Structures For the given design parameters (U and t) there is only one solution for which the array of double ellipses in Figure 3.17, fits exactly on the curve, i.e. the position of P 0, P 1 and P 2 is exactly as pictured. b t t a Figure 3.18: The relation between the original and the double ellipse in terms of semi-axes a and b and the unit thickness t The general polar equation for a circular arc is given by x = R cosα y = R sinα (3.21) The general parametric equation for an ellipse, with x = a cosθ y = bsinθ 0 θ 2π, is given by (3.22) For brevity, the radius of the circular arc is denoted by R, but as usual it can be expressed in terms of the rise and span using Eqn (3.2). The point P 0 (x 0, y 0 ) has coordinates (0, R) and P 2 is determined by α 2, which in turn can be expressed in terms of H r and S. 64

98 Chapter 3 Design of Scissor Structures The coordinates of P 1 are determined by the unknown angle α 1. P 1 (x 1, y 1 ) is located on the base curve and has coordinates: x 1 = R cosα 1 y 1 = Rsinα 1 (3.23) P 1 also lies on ellipse E 0, with centre P 0 (x 0, y 0 ): x + 1 = a cosθ 0 x0 1 = bsinθ 0 y0 y + (3.24) Equating (3.23) and (3.24) gives: R cos 1 = acosθ0 + x R sin 1 = bsinθ0 + y α (3.25) 0 α (3.26) 0 Analogously, we can write the coordinates for point P 2 (x 2, y 2 ), lying on the base curve: x 2 = Rcosα 2 (3.27) y 2 = Rsinα 2 and on ellipse E 1, with centre P 1 (x 1, y 1 ): x + 2 = a cosθ 1 x1 2 = bsinθ 1 y1 y + (3.28) Equating (3.27) and (3.28) gives: R cos 2 = acosθ1 + x R sin 2 = bsinθ1 + y1 α (3.29) 1 α (3.30) 65

99 Chapter 3 Design of Scissor Structures Eqns (3.25), (3.26), (3.29), (3.30) and (3.20) form a system of five equations in five unknowns. With R, α 2 and t as input parameters the system can be solved for α 1, a, b, θ 0 and θ 1 Rcosα1 = acosθ0 Rsinα1 = bsinθ0 + R Rcosα2 = acosθ1 + Rcosα1 Rsinα2 = bsinθ1 + Rsinα t = b a (3.31) Two different cases can be distinguished: α 2 = 0 : this evidently means that P 2 lies on the X-axis and coincides with the endpoint of the circular arc. Therefore, a semi-circle is used as the base curve. π 0 < α 2 < : in this case P 2 lies somewhere on the arc and therefore 2 only a user-defined portion of the circular arc is used for carrying the linkage. Now that a solution for α 1 has been found, the coordinates of P 1 is given by Eqn (3.23). The y-coordinate of the external end nodes P e of the scissor units are easily found by adding an amount t / 2 to the y-coordinates of P 0, P 1 and P 2. Analogously, by subtracting an amount t / 2 the internal nodes P in are found. Calculating higher numbers of units is simply a matter of adding the appropriate equations to the system. Per additional angle α (extra intersection points P on the base curve) to be calculated, two equations are added. Generally, for n units (U=n) and R, α n and t as design parameters, the system with 2n+1 equations for 2n+1 unknowns can be written as 66

100 Chapter 3 Design of Scissor Structures R cosα1 = a cosθ 0 Rsinα1 = bsinθ 0 + R... R cosα n = a cosθ n 1 + R cosα n 1 Rsinα n = bsinθ n 1 + Rsinα n t = b a which gives solutions for a, b,{ α 1..., α n 1 } and { θ 0..., θ n 1 }.,, (3.32) The approach is not limited to circular arcs only. Any equation for a curve can be used, such as the parametric equation for a parabola. Eqn (3.33) stands for an inverted (open towards negative y-values) parabola with parameter v. Figure 3.19 shows such a parabola, of which the focus lies in the origin, as base curve for a four-unit linkage, of which two are considered due to symmetry. x = 2 q v 2 y = q v (3.33) Y P 0 q o P 1 X P 2 Figure 3.19: Translational linkage with U=2 fitted on a parabolic base curve 67

101 Chapter 3 Design of Scissor Structures The base curve is determined by assigning a value to q. Analogous to the approach for circular arcs, values for U and t are chosen. The endpoint of the base curve P 2 is found by determining v 2. Now the coordinates of P 1 can be calculated from a system of five equations in five unknowns: 2q v1 = a cosθ 0 2 q v1 = bsinθ 0 + q 2q v2 = a cosθ1 + 2q v 2 q v2 = bsinθ1 q v t = b a 1 2 (3.34) which gives solutions for v 1, a, b, θ 0 and θ 1. The value obtained for v 1 will suffice for determining the position on the parabola of P 1, by using Eqn (3.33) Interactive geometry The geometric construction of scissor linkages can be automated by using a program called Cabri Geometry II [2007]. This allows the creation of interactive drawings responding in real-time to changes made by the user. For example, the geometry of a translational linkage with constant unit thickness has been drawn up using the method of the ellipses (Figure 3.20) After altering the geometry at will, all necessary values for the bar lengths and angles can be read. This tool has been used by Wouter Decorte [2007], a designer of kinetic art, for the formfinding of a deployable landscape. It consists of translational units with constant unit thickness on an arbitrary base curve. Figure 3.21 shows three consecutive deployment stages. 68

102 Chapter 3 Design of Scissor Structures Figure 3.20: Screenshot of interactive geometry file in Cabri Geometry II [2007] software for designing arbitrarily curved translational linkages with constant unit thickness (base curve marked in black) Figure 3.21: Deployable landscape consisting of one arbitrarily curved translational linkage repeated in an orthogonal grid. Linkage designed using the interactive geometry tool (Aluminium, 4.5 m x 3 m, (photo: courtesy of Wouter Decorte) 69

103 Chapter 3 Design of Scissor Structures 3.3 Three-dimensional structures In this section a description will be given of possible configurations for threedimensional scissor structures composed of translational and polar units. These can be designed by utilising the methods discussed in this chapter. There is a myriad of possible configurations for such structures, all obeying the deployability constraint, some of which have been reviewed in Section However, only part of those will be stress-free deployable, meaning that the scissor members will be in a stress-free state the stresses induced by selfweight left aside - before, during and after deployment, effectively behaving like a mechanism. Two-dimensional scissor linkages obeying the deployability constraint will always be stress-free deployable, but it is when they are placed into a three-dimensional grid that additional effects can come into play. Linear/Flat Single curvature Double curvature Figure 3.22: Possible shapes for three-dimensional stress-free deployable structures, which can be designed using the tools presented Although, theoretically, the configurations which will be discussed are linkages (or mechanisms) they will be referred to as structures, because ultimately they will be used as such in an architectural environment and become stabilised in order to be able to carry loads. They are classified according to their overall curvature (plane, single curvature or double curvature), the grid directions 70

104 Chapter 3 Design of Scissor Structures (two-, three-, or four-way grid) and the type(s) of units used (translational, polar or both). The characteristic shapes which can be designed using the methods previously described are shown in Figure In Figure 3.23 and Figure 3.24 a two-way and a three-way grid are shown with their respective directions, which will be used to contain the plane or curved translational and polar linkages. Figure 3.23: Two-way grid with directions A and B Figure 3.24: Three-way grid with directions C, D and E 71

105 Chapter 3 Design of Scissor Structures Linear structures The most elementary three-dimensional shape is the linear structure, which is often used as a prismatic column, masts or beam. It consists of translational or polar units, or a combination thereof. As these linear structure deploy, their length increases while their width decreases (Figure 3.25, left). Also, combinations of scissor units and non-deployable elements are possible, in which case the non-deployable elements give the structure a certain initial width, which is invariable throughout the deployment (Figure 3.25., right). Square and triangular beam Square and trapezoidal arch Triangular column Quadrangular column Figure 3.25: Linear elements prismatic columns arches (left): linear elements consisting solely of scissor units (right): Parallel scissor units connected by non-deployable elements 72

106 Chapter 3 Design of Scissor Structures This concept can be extended to the structures shown in Figure Here, the non-deployable elements are of such length, that a space enclosure is formed. By laterally connecting arches of increasing size by non-deployable elements, numerous variations become possible with single or double curvature (Figure 3.26, bottom right). Also, three-dimensional linear structures can be placed parallelly and become mutually connected (Figure 3.26, top left). Parallel triangular 3D-arches Parallel 2D- arches Polar and translational linkages Scaled 2D-arches Figure 3.26: Parallel linear structures connected by non-deployable elements 73

107 Chapter 3 Design of Scissor Structures Plane grid structures These are structures with zero curvature (zeroclastic) and consist of plane (rectilinear) translational linkages placed on either a two-, three- or four-way grid. Two-way grid Translational units This is the simplest of geometries. Plane translational units are placed in directions A and B of a two-way grid with square of rhombus-shaped cells (Figure 3.27). Plan view Perspective view Elevation view Figure 3.27: Plane translational units on a two-way grid 74

108 Chapter 3 Design of Scissor Structures Three-way grid Translational units By placing plane translational linkages in three directions C, D and E of a three-way grid, a plane grid is formed with equilateral grid cells (Figure 3.28). Plan view Perspective view Elevation view Figure 3.28: Plane translational units on a three-way grid 75

109 Chapter 3 Design of Scissor Structures Four-way grid Translational units By adding two more directions diagonally to the already present orthogonal directions A and B of the grid in Figure 3.29, a four-way grid structure is formed with isosceles grid cells. Per grid cell, the four units contained in the diagonal directions are basically identical to those in directions A and B, except that they are shortened, to fit inside the grid cells. It should be noted that for the grid to be stress-free deployable, all units have to be plane. Using curved units on the diagonals would render the geometry bi-stable. Plan view Perspective view Elevation view Figure 3.29: Plane translational units on a four-way grid 76

110 Chapter 3 Design of Scissor Structures Single curvature grid structures Cylindrical grids are monoclastic shapes, often referred to as barrel vaults. They are obtained by curving one direction of a two-way grid, or two, respectively three directions of a three-way grid. Two-way grid Translational units A combination of plane and curved translational units is used to form a translational barrel vault with an orthogonal grid. Grid direction A is kept plane, while curvature is introduced in direction B (Figure 3.30). Plan view Perspective view Elevation view Figure 3.30: Plane and curved translational units on a two-way grid 77

111 Chapter 3 Design of Scissor Structures Polar and translational units Rows of polar units are now contained in curved direction B, while direction A contains plane translational units. This structure is referred to as a polar barrel vault (Figure 3.31). Plan view Perspective view Elevation view Figure 3.31: Polar and translational units on a two-way grid 78

112 Chapter 3 Design of Scissor Structures Three-way grid Translational units Plane and curved translational units are placed in a three-way grid. Direction C contains plane translational units, while D and E contain curved units (Figure 3.32). The result is a translational barrel vault with isosceles grid cells. During deployment the angles inside the grid cells vary slightly, a phenomenon called angular distortion. This will be explained in greater detail in Section Plan view Perspective view Elevation view Figure 3.32: Plane and curved translational units on a three-way grid 79

113 Chapter 3 Design of Scissor Structures Polar and translational units (variation 1) Now direction C, which contains polar units, is curved and placed in the direction of the span (transverse direction). Directions D and E are also curved, but run diagonally over the span and contain translational units (Figure 3.33). The rows of translational units warp out of their common plane during deployment. In order for this structure to be deployable, the bars of each translational unit have to be able to move apart by sliding along their intermediate pivot hinge. Plan view Perspective view Elevation view Figure 3.33: Polar units on a three-way grid (variation 1) 80

114 Chapter 3 Design of Scissor Structures Polar and translational units (variation 2) Direction C contains translational units and is placed perpendicular to the span (longitudinal direction). Directions D and E are also curved, but run diagonally over the span and contain polar units (Figure 3.34). The rows of translational units in direction C warp out of their common plane during deployment. In order for this structure to be deployable, the bars of each translational unit have to be able to move apart by sliding along their intermediate pivot hinge. Plan view Perspective view Elevation view Figure 3.34: Polar and translational units on a three-way grid (variation 2) 81

115 Chapter 3 Design of Scissor Structures Double curvature grid structures Two-way grid Translational units Translational units are placed in a two-way grid. A linkage of any curvature can be repeatedly placed in the grid in one direction (A). When the same linkage is also placed in the perpendicular direction (B), a synclastic shape is obtained (Figure 3.35). When the linkage in direction B is inverted, an anticlastic shape is obtained (Figure 3.36a). As long as a constant unit thickness is used throughout the structure, any two arbitrarily curved linkages can be combined in a grid (Figure 3.36b). Using translational units of constant unit thickness in a two-way grid is a very powerful method of creating structures with positive or negative Gaussian curvature [Langbecker, 1999, 2001]. Plan view Perspective view Elevation view Figure 3.35: Translational units on a two-way grid (synclastic shape) 82

116 Chapter 3 Design of Scissor Structures Variations Anticlastic shape (a) Arbitrary curvature (b) Figure 3.36: Two variations for translational units on a two-way grid 83

117 Chapter 3 Design of Scissor Structures Lamella grid Spherical lamella grids have rhombus-shaped cells which are arranged circularly around a pole of the structure. They are deployed radially, either from the edge toward the centre or vice versa, in which case an opening exists in the middle. Translational units Curved translational units with constant unit thickness are placed in radial direction, to form a translational lamella dome (Figure 3.37). Plan view Perspective view Elevation view Figure 3.37: Translational units on a lamella grid 84

118 Chapter 3 Design of Scissor Structures Polar units Identical polar units are arranged radially to form a polar lamella dome (Figure 3.38). Plan view Perspective view Elevation view Figure 3.38: Polar units on a lamella grid 3.4 Conclusion In this chapter the basic principles behind the design of deployable scissor structures composed of translational or polar scissor units have been explained. It has been demonstrated how two-dimensional scissor linkages, with either translational or polar units, can be obtained by pure geometric construction, based on the geometric deployability constraint (Section 2.2.3). Several methods for obtaining polar linkages based on a circular base curve and translational linkages based on any curve have been discussed. 85

119 Chapter 3 Design of Scissor Structures A geometric design method, for which the equations have been derived, has been proposed, based on the key design parameters for a space enclosure: the rise and span. It has been shown how based on these parameters, a base curve is determined, which is subsequently translated into a scissor geometry in its deployed configuration. Two-dimensional scissor linkages which comply with the deployability constraint are always stress-free deployable, since the bi-stable deployment is caused by geometric incompatibilities associated with a three-dimensional configuration. For three-dimensional grid structures, the deployability constraint is therefore a necessary, but not a sufficient condition for stress-free deployability. It has been found that, whether or not a certain configuration will be stress-free deployable, fully depends on the specific combination of the grid type (two-, or three-way), unit type (translational and/or polar) and curvature (plane, single or double). To assist the designer in the choice of a suitable configuration, the stress-free deployable configurations known from literature have been identified and discussed. It was found that designing single curvature structures (plane grids, barrel vaults) is quite straightforward when translational (plane or curved) and/or polar units are used on a two-way or three-way grid, provided that a constant unit thickness is imposed throughout the structure. But when it comes to using these units in doubly curved deployable grids (domes, saddle shapes, arbitrarily curved) some care must be taken in preserving the stress-free deployability. As opposed to polar units, for which the only valid stress-free deployable geometry is a lamella dome, translational units can be used for a myriad of arbitrarily curved double curvature grids, provided that a constant unit thickness is used throughout the structure and the sums of the semi-lengths of the scissor bars is constant. This has lead to the conclusion that translational units are a powerful means for designing stress-free deployable bar structures with positive or negative curvature. 86

120 Chapter 4 Design of Foldable Plate Structures Chapter 4 Design of Foldable Plate Structures 4.1 Introduction This chapter extends the work of Foster [1986/87] and Tonon [1991, 1993] who have proposed a variety of geometric shapes for foldable plate structures such as beams, barrel vaults and some doubly curved surfaces, as discussed in Section 2.3. However, not all configurations are foldable: some are merely demountable while others are constructed from smaller foldable sub-structures that have to be joined together afterwards [Hanaor, 2001]. Since the focus of this research is on rapidly erectable structures, only fully foldable configurations are discussed. An insight in the basic fold patterns and the impact they have on the geometry of the final deployed configuration is offered. The individual plate geometry as well as the complete fold pattern in both the flatly folded and the final deployed configuration are thoroughly described. An advance is made in the design process by developing a geometric design method for which the necessary equations are derived. These equations are used to geometrically describe regular structures, which are single curvature structures (barrel vaults) characterised by a linear deployment. Additional parameters, such as the expansion ratio, as a measure for the increase in size between the compacted, stowed configuration and the fully deployed configuration, are introduced. The geometric design method is then extended to right-angled structures a variation of the regular structures and circular structures. The latter are foldable domes characterised by a circular deployment. It is shown that it is possible, with appropriately chosen design values, to connect single and double curvature modules to form alternative foldable shapes, with maximum plate uniformity. 87

121 Chapter 4 Design of Foldable Plate Structures Figure 4.1: Typical foldable plate structure 4.2 Geometry of foldable plate structures Foldable plate structures consist of a series of triangular plates, connected at their edges by continuous joints, allowing each plate to rotate relative to its neighbouring plate. The plates can fold into a flat stack and unfold into a predetermined three-dimensional configuration, as the one shown in Figure 4.1. The origami-like fold pattern consists of intersecting mountain folds (continuous lines) and valley folds (dashed lines), as shown in Figure 4.2. A fold pattern consists of a repetition of the basic plate element marked P, which is of triangular shape and must possess one angle of at least 90. An array of several interconnected plates in the directrix direction is called a module, marked M. Now, the fold pattern can be expanded by repeating the module M any number of times in the generatrix direction. In this case both patterns consist of two modules (m=2), which leads to the smallest symmetrical and usable solution. Foldable plate structures can be characterised by the number of plates (p) they have in the span, which is found by adding up the number of whole and half elements in one module. The possible values for p are all odd integers equal to or greater than five. The minimal configuration (p=5) with two modules (m=2) is depicted in Figure 4.2. Even values for p would make the structure asymmetrical and are therefore discarded. Another important parameter is 88

122 Chapter 4 Design of Foldable Plate Structures the apex angle β at which the mountain fold lines intersect. The apex angle strongly influences the way curvature is introduced during deployment. So together, the three main design parameters p, m and β of the fold pattern, determine the overall morphology of the structure and its behaviour during deployment. β Figure 4.2: Fold patterns of type A and B for the smallest possible regular structure (p=5) The apex angle can range from 90 to 180 and can be identical or variable throughout the structure. When all β s are identical the structure is called regular, while changing one of the apex angles makes it irregular. Countless variations of shapes are possible but generally speaking the more exotic the fold pattern becomes, the less useful the resulting structure will be as a foldable space enclosure. Figure 4.5 and Figure 4.6 show that minor changes, such as altering the outer most apex angle of pattern A to 90 (the minimum), can give it an interesting quality: the fully folded configuration is much compacter that the one of pattern A and B, shown in Figure 4.2, although all patterns have p=5 and share the same dimensions. Another consequence is that the triangular plate located at the base becomes quadrangular which makes incorporating an entrance in the side panel easier than would be the case with triangular panels as in Figure 4.3 and Figure 4.4. Also, this so called rightangled structure has increased headroom near the sides when compared to its regular counterpart. 89

123 Chapter 4 Design of Foldable Plate Structures Figure 4.3: Unfolded and fully folded configuration of patterns A and B (p=5) Figure 4.4: Elevation view of the compactly folded and fully deployed configuration for a regular structure with five plates and an apex angle of

124 Chapter 4 Design of Foldable Plate Structures Figure 4.5: Right-angled fold pattern: altering one apex angle to 90 enables a compacter folded configuration (introduction of quadrangular plates near the sides) Figure 4.6: Elevation view of compactly folded and fully deployed configuration for a rightangled structure with five plates and an apex angle of 120 A specific class of foldable structures which are of interest are the regular structures with their apex angle being a multiple or sub-multiple of 360. In their most compact, fully folded configuration there is only one corresponding apex angle β for each specific number of plates p for which the edges of the end plates meet to form a closed circle. When designing a mobile structure, looking for the most compact folded configuration for a regular structure can only improve transportability. The example shown in Figure 4.7 is a regular structure with seven panels in the span (p=7), which will fold to its most compact form, only when the apex angle is

125 Chapter 4 Design of Foldable Plate Structures Figure 4.7: Three stages of deployment for a basic regular foldable structure (p=7; β=120 ): completely unfolded, erected position and fully compacted for transport A greater apex angle would cause the folded configuration to be less compact and show a gap, while a smaller apex angle would cause an overlap, therefore making the configuration unable to fold. Eqn (4.1) returns the apex angle β (degrees) for a given number of plates p and the first eight pairs of p and β are shown in Table 4.1. ( p 3) ( p 1) β = π (4.1) 92

126 Chapter 4 Design of Foldable Plate Structures p β [ ] , , Table 4.1: The first eight values for β in terms of p for compactly foldable regular structures The first three configurations from Table 4.1 (p=5, p=7, p=9) are pictured in Figure 4.8 and Figure 4.9. p=5; β=90 p=7; β=120 p=9; β=135 Figure 4.8: Plate element, compactly folded configuration and fully deployed configuration (front elevation) for the first three compactly foldable structures (p=5, p=7, p=9) 93

127 Chapter 4 Design of Foldable Plate Structures p=5; β=90 p=7; β=120 p=9; β=135 Figure 4.9: Side elevation of the fully deployed configuration of the first three compactly foldable structures (p=5, p=7, p=9) It is interesting to note what influence the apex angle has on the overall geometry. Figure 4.10 shows two structures with similar fold patterns and the same number of panels (p=5). Although they have a different apex angle, their projection (or silhouette) in the erected position is identical. This means that for a certain number of elements the width of the resulting structure can be dramatically increased, simply by increasing the apex angle. As a result, material can be used more economically, since the structure with an increased apex angle will reach more width with the same amount of connections. This explains an important characteristic concerning the design of such structures. From an architectural point of view, the span and the rise of a structure can be treated as key design parameters, as they determine the silhouette of the structure in vertical projection. This implies that for a certain chosen number of plates and a chosen rise and span, a myriad of possible configurations exist, all equal in projection, but with different actual plate dimensions. Since these plate dimensions, together with the apex angle β, determine whether or not the configuration will be compactly foldable, it is important to choose the appropriate plate geometry that will lead to the required deployed shape. 94

128 Chapter 4 Design of Foldable Plate Structures Figure 4.10: For a chosen number of panels p the apex angle β can be altered at will, affecting the width of the structure and the compactly folded state 4.3 Geometric design In order to design these structures, a parameterisation of a single module and the structure as a whole is needed. This provides us with a description of all relevant design parameters such as the apex angle (β), bar length (L), the span (S), rise (H r ), width (W) and the number of plates (p) in one module. In order to understand the way the various design parameters affect the deployment behaviour and the geometry of the various configurations, we need to establish their mutual relationships. A significant parameter is the deployment angle θ, measured between a triangular face and the vertical axis, as shown in Figure The deployment angle θ determines to what degree a structure is unfolded with values ranging from 0 (fully compacted configuration) to 90 (flat, completely unfolded position). In between these extremes there is a unique value for θ that corresponds with the fully erected position, i.e. a semicircular shape for the regular structure when viewed in elevation. For an irregular structure such as the one depicted in Figure 4.5 we are looking for the configuration whereby the side plates are standing perfectly vertical. 95

129 Chapter 4 Design of Foldable Plate Structures H r Figure 4.11: Parameters used to characterise a foldable structure: length L, span S, width W, apex angle β and the deployment angle θ β/2 H 1 H α 1 H 2 θ α L α 2 Figure 4.12: A foldable plate and its parameters: length L, height H, H1, H2, apex angle β, the deployment angle θ and angles α, α 1, α 2 The following relationships can be derived from Figure 4.12: 96

130 Chapter 4 Design of Foldable Plate Structures π β α = 2 (4.2) L H = tanα 2 (4.3) H 1 = H cosθ (4.4) L H 1 = tanα1 2 (4.5) H 2 = H sinθ (4.6) L H 2 = tanα2 2 (4.7) Substituting (4.3) in (4.4) and (4.6), and equating (4.4) to (4.5) and (4.6) to (4.7) gives: tanα 1 = tanα cosθ (4.8) tanα 2 = tanα sinθ (4.9) Equations (4.8) and (4.9) will be used to determine the relationship between the apex angle, the number of plates and the deployment angle for regular, right-angled and circular structures Regular structures The geometric relationship between the apex angle, the number of plates and the deployment angle can now be derived. There is a unique value for the deployment angle that corresponds with the fully erected position of the regular structure (although in reality the deployment angle can range from 0 to 90, when θ is mentioned hereafter, it is this unique value for the fully deployed configuration that is being referred to). As Figure 4.1, Figure 4.3 and Figure 4.4 show, in this deployed position the bottom most (half) panels touch the ground along their bottom edge and in elevation view the silhouette of the structure is a perfect semi-circle. It can be described as a barrel vault with cylindrical curvature. 97

131 Chapter 4 Design of Foldable Plate Structures Eqn (4.8) allows α 1 to be written in terms of α and θ: α tan 1 tanα cosθ 1 ( ) = (4.10) Figure 4.13 represents the projection in the vertical plane of a regular plate linkage with p plates. Empirically, it can be observed that α 1 is a sub-multiple of π/2. This results in the following equation: π ( p 1) α 1 = (4.11) 2 α 1 Figure 4.13: Perspective view and side elevation of the vertical projection of a plate linkage for empirically determining the relationship between α 1 and p Substituting (4.2) and (4.11) in (4.10) gives: π β ( 1) tan tan cosθ = π p (4.12) Finally, Eqn (4.13) gives the value of θ in terms of p and β for regular structures: ( ) = cos 1 π β θ tan tan (4.13) 2 p 1 2 The relationship between these parameters for geometries with p=5, p=7 and p=9 is plotted in the graph of Figure For example, looking at the regular structures of Figure 4.10 we can see that, while both configurations have p=5, 98

132 Chapter 4 Design of Foldable Plate Structures the one on the left has β=120 and the one on the right has β=90. From the graph we can read the appropriate deployment angle θ for both structures: and respectively. Looking at the graph from Figure 4.14 we can draw the following conclusions: The higher the number of plates, the blunter the apex angle can be. There is, for a given number of plates, a minimum and maximum value for the apex angle. These values are given in Table 4.2. Lower or higher values generate configurations that cannot be made into a foldable structure For a fixed number of plates, an increased apex angle means a decreased deployment angle For a fixed apex angle, increasing the number of plates also increases the deployment angle Relationship between the apex angle β and the deployment angle θ for REGULAR STRUCTURES Deployment angle θ [deg] p=5 p=7 p= Apex angle β [deg] Figure 4.14: The relationship between the apex angle β and the deployment angle θ for regular structures with p=5, p=7 and p=9 99

133 Chapter 4 Design of Foldable Plate Structures Number of plates p βmin[ ] βmax [ ] Table 4.2: Minimum and maximum possible apex angles for regular structures with 5, 7 or 9 plates Figure 4.15: Elevation view and perspective view of the deployment of a regular five-plate structure with β=120 Crucial to the design of space enclosures are the span and rise, which are influenced by the number of plates and their length. The relationship between the number of plates p, the plate length L, the span S (defined between the outer points) and the rise H r for regular structures is expressed by Eqns (4.14) and (4.15). Table 4.3 shows the span S and rise H r for a given number of plates p for regular foldable structures in terms of the plate length L. π L = S tan p 1 (4.14) 100

134 Chapter 4 Design of Foldable Plate Structures Because in projection the profile is a semi-circle, the rise is given by S H r = (4.15) 2 p S (x L) H r (x L) Table 4.3: The span S and rise R for a given number of plates p of regular foldable structures in terms of the plate length L. An expression can be found for the relationship between the width w of a single module and the plate length L, the apex angle β, the deployment angle θ for both regular and right-angled structures. From Figure 4.11 it can be seen that the width w of a single deployed module is equal to H 2. By substituting Eqn (4.9) in (4.7) we obtain the following expression for w: L w = tanα sinθ (4.16) 2 Substituting Eqn (4.2) in (4.16) gives an expression for w in terms of β and θ: π β w = L tan sinθ (4.17) 2 2 Or in other form: L β w = cot sinθ (4.18) 2 2 Eqn (4.19) gives the total width W in terms of the number of modules m. W = m w (4.19) An important characteristic in transportable structures is the compactness of the flatly folded plate linkage in its stowed condition. The space required will be the volume determined by the area of the polygonal footprint of the folded configuration and the total thickness of the stack of plates, determined by the individual plate thickness. For the fully foldable configurations from Figure 4.8, the area of the polygon is expressed in terms of the edge length L edge.. Figure 4.16 shows the relationship 101

135 Chapter 4 Design of Foldable Plate Structures between the apex angle β, the plate length L and the edge length L edge which allows the following relationship to be written: L L edge = β (4.20) 2sin 2 Now that the edge length L edge of the polygon is known, the area can be expressed. Figure 4.16 shows the expressions for the area of a polygon in terms of its edge length [Mathworld, 2007]. p=5; β=90 p=7; β=120 p=9; β=135 L edge β L 2 Ledge edge 2 2 L ( ) L 2 2 edge Figure 4.16: The parameters associated with the polygonal contour of the flatly folded configurations with p=5, p=7 and p=9 and the expressions for the area in terms of the edge length L edge Using Eqn (4.20) and the expressions from Figure 4.16, the area of the compact configuration is expressed in terms of L edge for the three basic fully foldable configurations (Table 4.4). Configuration p=5, β= L 2 Area compact p=7, β= L 2 p=9, β= L 2 Table 4.4: The area of the compact configuration for (p=5, β=90 ), (p=7, β=120 ) and (p=9, β=135 ) in terms of the plate length L 102

136 Chapter 4 Design of Foldable Plate Structures The area of the sectional profile of the fully folded configuration gives us information on the space the fully deployed configuration will occupy. The span S has been defined as the distance between the lower extremities of the circular profile (Figure 4.11). The area of the sectional profile equals the area of a semi-circle with radius Hr. Using the expressions for H r in terms of the plate length L from Table 4.3, we can now express the area of the sectional profile for the deployed configurations, for which the results are given in Table Configuration p=5, β= L 2 Area deployed p=7, β= L 2 p=9, β= L 2 Table 4. 5: The area of the sectional profile of the deployed configuration for (p=5, β=90 ), (p=7, β=120 ) and (p=9, β=135 ) in terms of the plate length L Now, the ration between the compacted shape and the fully deployed configuration can be expressed as the expansion ratio λ: Areacompact λ = (4.21) Area deployed The values for λ are given in Table It can be seen that the smallest configuration demonstrates the largest expansion. Configuration Expansion ratio λ p=5, β= p=7, β= p=9, β= Table 4. 6: The expansion ratio λ for (p=5, β=90 ), (p=7, β=120 ) and (p=9, β=135 ) The plate thickness has a great impact on the compactness of the stowed configuration. As can be observed from Figure 4.3, Figure 4.7 and Figure 4.15, the geometry of these structures is such, that each module (as defined in Figure 4.2) in its flatly folded configuration consists of two overlapping layers of plates. When t p represents the thickness of a single plate element, then the 103

137 Chapter 4 Design of Foldable Plate Structures total thickness T p given by: of the compactly folded configuration with m modules is Tp = 2 m t p (4.22) The expressions for the area and the thickness of the compactly folded configuration allow to determine the volume in the stowed position, simply by multiplying Area compact by the total plate thickness T p. Also, the volume occupied by the fully deployed configuration can be determined by multiplying the area of the sectional profile Area deployed by the total width W from Eqn (4.19) Right-angled structures Analogously, when p and β are known, the deployment angle for a rightangled structure such as the one in Figure 4.5 can be calculated by solving Eqn (4.23) for θ: ( 3) tan β 1 β cosθ cot + 2 tan cosθ tan = π p (4.23) The approach used for deriving Eqn (4.23) is identical to the empirical method used for regular structures and will therefore not be repeated. When looking at the graph from Figure 4.17, it is clear that altering the first and last apex angles to 90 generates a completely different behaviour as compared to regular structures: The higher the number of plates, the less sharp the apex angle can be. There is, for a given number of plates, a minimum and maximum value for the apex angle, which is given in Table 4.7. Lower values generate configurations that cannot be made into a foldable structure For a fixed number of plates, an increased apex angle means an increased deployment angle For a fixed apex angle, increasing the number of plates also increases the deployment angle 2 104

138 Chapter 4 Design of Foldable Plate Structures Relation between apex angle β and deployment angle θ for RIGHT- ANGLED STRUCTURES 90 Deployment angle θ [deg] p=9 75 p=7 70 p= Apex angle β [deg] Figure 4.17: The relationship between the apex angle β and the deployment angle θ for rightangled structures with p=5, p=7 and p=9 Number of plates p β min [ ] β max [ ] Table 4.7: Minimum and maximum possible apex angles for right-angled structures with 5, 7 or 9 plates, as can be read from the graph in Figure 9 Although the apex angle can range from βmin to 180, the highest values will generate quite useless structures since their structural thickness converges to zero. As opposed to the regular structures from Figure 4.8, there is no unique optimal value for the apex angle β for which a right-angled structure with p=5 will be compactly foldable without overlap or gap. All valid values for β (from the graph in Figure 4.17) will lead to compactly foldable five-plate configurations. This can be seen in Figure 4.18, which shows three configurations for p=5, each of which has a different apex angle. No matter which apex angle 105

139 Chapter 4 Design of Foldable Plate Structures is chosen (within bounds imposed by Eqn (4.23) and Figure 4.17), all resulting configurations are compactly foldable. p=5; β=90 ; θ=65.5 p=5; β=120 ; θ=68.1 p=5; β=135 ; θ=71.5 Figure 4.18: Plate element, fold pattern, compactly folded configuration and fully deployed configuration (front elevation and side elevation) for three compactly foldable five-plate rightangled structures (drawn to scale) To form an alternative configuration, a five-plate configuration of a rightangled structure can be connected to a regular structure, when a type B pattern is used (Figure 4.2). When five plates are used, such patterns will always have an identical vertical edge, for any value for β. By connecting along this common edge, linear arrays of regular and right-angled structures are possible, as shown in Figure

140 Chapter 4 Design of Foldable Plate Structures Identical vertical edge Figure 4.19: Only for p=5 can any regular and any right-angled structure be interconnected along a common edge, regardless of the value for β Circular structures Formulas for the geometric design in terms of the apex angle and the number of plates have been proposed. Not only single curvature structures with linear deployment are possible: dome-like structures with circular deployment can also be used as foldable space enclosures. As opposed to regular structures, their fold pattern cannot be developed in a plane. To flatten the shape, radial incisions have to be made. The foldable dome from Figure 4.20 and Figure 4.21 consists of a radial pattern of sectors joined together along their common edge. When this configuration is cut along one radius, it can be folded into a compact stack of plates by rotating its sectors around a vertical axis. These configurations are characterised by the number of sectors q that make up a full circle in plan view: in this case q=8. Empirically, it has been found that its value can be any even integer equal to or greater than six (due to apex angle restrictions for β min ). Values below six would give rise to non-foldable solutions. An interesting application is to connect half of a dome to an array of regular structures, making a fully closed space enclosure, as shown in Figure It would make the design process and fabrication easier when a single plate size could be used for both the half domes and the regular structure. Therefore, once the number of sectors q is chosen, the resulting apex angle β for the circular structure is also used for the regular structure, leading to a geometry with uniform plate elements. 107

141 Chapter 4 Design of Foldable Plate Structures Figure 4.20: Top view and perspective view of circular foldable structure Fold pattern developed in a plane Figure 4.21: Fold pattern and a single sector of a circular structure with q=8 Sector module Let q be the number of sectors in the circular structure, which can be freely chosen. Again, the geometric relationships derived from Figure 4.12 are used. α 2 Figure 4.22: Horizontal projection of a plate linkage for empirically determining the relationship between α 2 and q 108

142 Chapter 4 Design of Foldable Plate Structures Eqn (4.9) allows α 2 to be written in terms of α and θ: α 2 tan 1 ( tanα sinθ ) = (4.24) Figure 4.22 represents the horizontal projection of half a circular plate linkage. Empirically, it can be observed that α 2 is a sub-multiple of π. This results in the following equation: q α 2 = π (4.25) Substituting Eqn (4.2) and Eqn (4.24) in Eqn (4.25) gives the relationship between q, β and θ for circular structures: Or in other form π β tan 1 sinθ tan = π 2 q (4.26) = sin 1 π β θ tan tan q 2 (4.27) When a number of units q is chosen, solving Eqns (4.13) and (4.27) simultaneously will return a unique value for β and the corresponding deployment angle θ. Table 4.8 gives for a chosen q the appropriate β and θ. For example, the combined regular-circular structure (p=5, q=8) from Figure 4.23 has β=119.3 and θ=45 throughout the entire structure. 109

143 Chapter 4 Design of Foldable Plate Structures For p=5 q β [ ] θ [ ] , Table 4.8: Values for β and θ for a chosen q (circular structure), combined with a regular structure (p=5) Figure 4.23: Connecting a regular module with two half-domes leads to an alternative fully closed configuration with high plate uniformity Figure 4.24 shows the first three configurations (q=6, q=8 and q=10) for circular structures which are calculated with Eqns (4.13) and (4.27). They are combined with a compatible regular structure (p=5) with identical plate elements. Although Eqn (4.13), when used separately, allows the calculation of regular structures with any number of plates (p>5), the combined calculation for the fully closed configuration using Eqns (4.13) and (4.27) is only specifically applicable to regular structures with five plates, combined with a circular structure with any q equal to, or greater than six. Empirically, it has been found that regular structures with a higher number of plates (e.g. p=7) cannot be mutually connected with a seven-plate circular counterpart, if all plates remain identical. With uniform plates, no common edge for connecting the 110

144 Chapter 4 Design of Foldable Plate Structures structures was found. At this stage, it is unclear whether an irregular geometry with variable apex angles can be found which would solve this problem. q=6; β= ; θ=54.34 q=8; β= ; θ=44.9 q=10; β= ; θ=38.11 Figure 4.24: Circular structure with q=6, q=8 and q=10 (top view) and its respective combination with a compatible regular structure (perspective view) Alternative configurations By using the previously discussed regular, right-angled and circular plate structures as building blocks, certain alternative configurations can be composed. Figure 4.25 shows a few possible variations. 111

145 Chapter 4 Design of Foldable Plate Structures Figure 4.25: Some examples of alternative configurations 4.4 Conclusion In this chapter the findings of Foster [1986/87] and Tonon [1991, 1993], who have proposed a number of geometric shapes for foldable plate structures have been extended. Some foldable configurations are merely demountable while others are constructed from smaller foldable sub-structures that have to be joined together afterwards [Hanaor, 2001], thus compromising the transportability and the speed of erection. Therefore, a geometric design method has been developed which entails the design of fully foldable and rapidly erectable configurations with single or double curvature. A comprehensive study of the design parameters (deployment angle, apex angle, plate length, number of plates) has provided a thorough insight in the geometry of individual plates and the fold patterns they constitute, as well as 112

146 Chapter 4 Design of Foldable Plate Structures the impact on the compactly folded shape and the final deployed configuration. An advance has been made in the design process by developing a geometric design method for which the necessary equations have been derived. These equations are used to geometrically describe regular structures, which are single curvature structures (barrel vaults) characterised by a linear deployment. By introducing the deployment angle θ, it was made possible to derive equations - expressed in terms of the apex angle β and the number of plates p - which explain the relationship between the plate geometry and the fully deployed configuration. By analysing these formulas and the resulting graphs, an understanding has been provided on what minimum and maximum values for β and p give rise to compactly, fully foldable configurations. Additional characteristics, such as the expansion ratio a measure for the increase in size between the compact and the deployed state - have been studied and expressed in terms of the key parameters, i.e. the plate length and the apex angle. It has been shown that, by slightly altering the regular plate geometry, an interesting variation arises: a right-angled structure. This variation provides, in its deployed configuration, increased headroom near the sides of the structure and has quadrangular side panels, providing the option of a larger entrance space. It has also been found that, for a five-plate geometry, this type of structure is always compactly foldable. The design method has been extended to include right-angled structures, as well as circular structures. The latter are foldable domes characterised by a circular deployment. Further, it has been proven possible that, by solving the equations for domelike configurations and regular configurations simultaneously, a single plate element is found which can be used for both regular (single curvature) and circular (double curvature) shapes with five plates. Because of the plate uniformity, it was shown possible to combine these shapes into alternative double curvature shapes. 113

147 Chapter 4 Design of Foldable Plate Structures The current geometric design method is based on basic foldable shapes with a circular sectional profile and with high plate uniformity. The approach can be extended to include single and double curvature shapes with variable curvature. Although the method provided allows the design of regular foldable structures with any number of plates, the equations for circular structures, as well as those for the combined shapes, are currently valid for the smallest configuration (p=5). Further study is required to include configurations with a higher number of plates. 114

148 Chapter 5 Introduction to the Case Studies Chapter 5 Introduction to the Case Studies Figure 5.1: Some of the concepts for mobile structures presented in the following chapters 115

149 Chapter 5 Introduction to the Case Studies 5.1 Introduction This chapter serves as an introduction to the concepts proposed in the following chapters, in which a number of case studies will be discussed, bringing into practice the findings and methodologies presented in the previous chapters. All concepts presented are mobile structures for architectural applications that use either scissor-hinged bars or bars connected by foldable joints combined with a tensile surface for weather protection (Figure 5.1). An advance is made in the field of mobile structures by proposing novel concepts for deployable bar structures or by making use of existing concepts in an alternative, novel way. Four case studies, in which these concepts are put into practice, are chosen in such a way that a variety of structural systems is used: bars, cables and membranes in single and double curvature shapes, built from translational and polar scissor units on a triangular (case study 1) or quadrangular grid (case study 2), or based on the geometry of foldable plate structures (case study 3), or angulated (hoberman s) units connected in a linear way (case study 4). The geometry of the case studies is chosen in such a way that mutual comparison becomes possible. Per case study, a general description is given and the overall geometry, together with the dimensions, is discussed. As part of a feasibility study in which the proposed concepts are evaluated architecturally, kinematically and structurally, this chapter elaborates on the approach taken for the structural analysis of the case studies. It is explained how a simplified approach is used for the preliminary design and what consecutive steps are taken in the structural analysis. By using Eurocode 1 [2007], the wind and snow loads are calculated and it is shown how these will effect the different load zones of the structure. 116

150 Chapter 5 Introduction to the Case Studies 5.2 Geometry The first three case studies are concepts for small, easily erectable mobile shelters of semi-cylindrical shape (barrel vault). Deriving all three geometries from the same basic shape - a semicircle with a radius of 3 m - makes comparison in terms of architectural and structural qualities more convenient (Figure 5.2). The length of the barrel vaults is approximately 10 m, which does not exceed the maximum length/width ratio for barrel vaults of 2:1. In case several structures are connected in the longitudinal direction, and therefore exceeding the length/width ratio, diaphragm walls or some kind of stiffening cable arrangement should be introduced. Also, the structural thickness t is chosen to be identical for each case. These barrel vaults are open structures with no back nor front, as shown in Figure 5.3. Then, using the same structural elements from the single curvature structures, alternative configurations are proposed, to form double curvature shapes with a fully closed surface. The single and double curvature shapes will be referred to as open, respectively closed structures. t t t Case study 1 Case study 2 Case study 3 Figure 5.2: Front elevation view of cases studies shows the mutual similarity of the geometry. Case study 1, 2 and 3 are based on the same shape (semicircle with radius of 3 m) 117

151 Chapter 5 Introduction to the Case Studies Figure 5.3: Overall geometry for the case studies: single curvature shape (open) and double curvature shape (closed) The first case study consists of translational scissor units placed on a threeway grid (Figure 5.4). Scissor structures with translational units can be easily made into single or double curvature structures with a quadrangular grid. Two-way grids have quadrangular grid cells which make them susceptible to skewing (angular distortion of the grid cells). The proposed concept is a variation: a single curvature shape with translational units on a three-way grid (triangular grid cells). This makes triangulation of the grid cells, to counter the skewing effect, obsolete. Figure 5.4: Perspective view of the single and double curvature geometries for cases studies 1, 2 and 3 118

152 Chapter 5 Introduction to the Case Studies Polar scissor arches, combined with translational units on a two-way grid are used for the second case study (Figure 5.4). Using polar scissor arches is an effective way of introducing curvature in a quadrangular scissor grid, therefore, these polar barrel vault structures are relatively common. Identical polar units can also be used to make a deployable dome-like shape with rhombusshaped grid cells, otherwise called a lamella dome. As a variation, two halves of this dome are added to the open structure to turn it into a closed, double curvature shape. The third case study has its geometry and kinematic behaviour based on that of foldable plate structures. Instead of plates, bars are used as structural components to hold up a membrane surface (Figure 5.4). The reason for using a continuous membrane instead of separate plates is explained in Section 8.1. The bars are connected by foldable joints, which behaves like a miniature foldable plate mechanism. Optionally, scissor units can be added to influence the kinematic behaviour. The fourth and last case study is a deployable mast of approximately 8.5 m high which, when deployed, holds up three circularly arranged membrane canopies, each measuring 10 m x 5 m. It is made from scissor modules, stacked upon each other to form a vertical linear deployable truss-like structure. During deployment the mast - to which the membrane elements are attached - expands vertically. By doing so the canopy becomes gradually tensioned until maximal deployment is reached. The three scissor modules which make up the structure consist of angulated scissor elements. Angulated scissor units were not discussed in the context of the design of deployable scissor arches to form single curvature barrel vaults, for reasons specified in Section 3.1. However, in this particular case, as they are connected in a linear manner to expand vertically, they demonstrate a particular deployment behaviour which is used to the advantage of the proposed concept (Section 9.1). 119

153 Chapter 5 Introduction to the Case Studies Figure 5.5: Perspective view and side elevation of case study Structural analysis of the proposed concepts General approach Since all considered case studies are mobile architectural shelters, a canopy serving as a climatological protection is an integral part of the design. Because this tensile surface is subject to wind and snow loading and it is physically attached to the bar structure, its behaviour under load is to be determined first. Subsequently, the actions of the membrane are transferred to the bar structure which allows the members to be sized as part of a preliminary structural design. So first, the formfinding of the tensile surface is done with EASY-software [Technet, 2007] which uses the force-density method [Mollaert, Forster, 2004]. All membranes are attached to the nodes of the bar structure. These nodes become the boundary points for the membrane geometry that is used as input for the formfinding process. A moderate level of pre-tension is introduced in the membrane (1 kn/m in warp and weft direction). For mobile structures as the ones presented here this seems reasonable, as it provides a fair balance between the tension in the membrane and the resulting forces transferred to the bar structure. Also, avoiding the need for highly sophisticated material for introducing the pre-tension is in accordance with the low-tech nature of the structures. After the equilibrium form has been calculated, snow and wind 120

154 Chapter 5 Introduction to the Case Studies loads are applied, of which the latter are determined according to the Eurocode 1 [2007]. Seven load cases are used for the structural analysis of the tensile surface which returns the reaction forces of the membrane on the boundary points. These vectors are then transferred to a FE-model (Finite Element Model) of the bar structure which is used for a preliminary structural design performed in ROBOT [Robobat, 2007]. This simplified approach is chosen because these software packages are readily available and perform their respective tasks (tensile surface design and structural analysis and design) well and because this is a preliminary design to test the feasibility of the proposed concepts, this method seems well-suited for the purpose. The three loads (transverse wind, longitudinal wind and snow) cannot be separately applied in EASY and the subsequent load vectors (action of the membrane on the boundary points) combined into several load cases, which will be imposed on the bar structure in ROBOT. After all, this would mean that the load vectors, obtained from the non-linear calculations on the membrane, would become superimposed, which leads to false results. Instead, the loads are combined into several load cases (pre-stress combined with wind and/or snow) which are applied in EASY, and the non-linear response of the membrane is measured. Besides, combining the separate load vectors obtained from EASY into load cases in ROBOT would falsely take the pre-tension several times into account, because it would be an integral part of each calculated load case in EASY. Therefore each possible load case has to be manually determined and applied on the membrane model in EASY, after which the reaction forces are transferred to the FE-model of the bar structure. There, the load cases are combined with the self-weight of the bar structure. While the wind and snow load are live loads, the pre-tension in the membrane is treated as a permanent load, with all safety factors applied accordingly. It should be noted that this method is a simplification in the sense that it does not take into account the effect the bar structure has on the membrane, since its displacements under load would alter the boundary conditions of the membrane. If a more profound and accurate analysis would be required, an integrated, more detailed model of both the tensile surface and the skeletal structure - which takes into account the mutual response - would be a better 121

155 Chapter 5 Introduction to the Case Studies route. Or, in the current approach, going back and forth between programmes, each time using the obtained results from one calculation as new boundary conditions for the next, in an iterative way, would lead to more accuracy, but with a huge effort and little return for this study. In summary, these are the steps to be taken in the structural analysis: - Formfinding of the tensile surface: calculation of the equilibrium form of the membrane under pre-stress - Statical analysis of the tensile surface: determining the reaction forces of the membrane on the boundary points under different load cases - Structure design of the skeletal structure: the bar structure is dimensioned with the action of the membrane and the selfweight of the structure applied Figure 5.6: Wind and snow action on the open and closed structure Load cases Three different live loads are considered: longitudinal wind, transverse wind and a snow load (Figure 5.6). For this simplified approach, only two wind directions are used, for mutual comparison of the different configurations. In a more profound analysis, the 45 wind direction should also be included, since its effect, especially on the open structure, could prove significant. The closed 122

156 Chapter 5 Introduction to the Case Studies structure as well as the open structure are divided into three zones or load areas, allowing a differentiated wind load application: pressure or suction. Whenever assumptions are made or simplifications are done, the most unfavourable value is chosen. The reference wind pressure is expressed as [Eurocode 1]: ρ q ref v 2 2 = ref (5.1) with the air density ρ =1.25 kg/m3 and the reference velocity v ref determined from: v ref = cdir. ctem. calt. vref,0 (5.2) =26.2 m/s, the altitude factor c ALT For Belgium v ref, 0 =1, the direction factor c DIR corresponds with a one month exposure (November). This gives a value for q = kn/m 2. ref =1 and for the temporary factor c TEM a value of 0.8 is chosen which The total wind pressure acting on the surfaces is obtained from: w = (5.3) w e w i with we the pressure on the external surfaces and wi on the internal surfaces, given by: ( Ze ) C pe ( Zi ) C pi w. e = qref. Ce (5.4) i = qref. Ce (5.5) w. As terrain category, the most severe is chosen: category I, flat country. The external pressure coefficient is determined by the geometry of the structure, which means the surface is divided into several zones, each with their own C pe. The internal pressure depends on C pi and is calculated by taking the influence of openings in the surface into account, by means of a factor µ (opening ratio), which is given by: 123

157 Chapter 5 Introduction to the Case Studies A L, W µ = (5.6) A T Here, A L,W stands for the total area of openings at the leeward and wind parallel sides and A T stands for the total area of openings at the windward, leeward and wind parallel sides. The closed structure is assumed to have an opening (entrance) on both the front and back (windward and leeward side for the longitudinal wind) which will allow internal wind pressure to be generated. By calculating the µ s for both closed and open structures, values for C pi of -0.5 (suction under transverse wind) and 0.14 (pressure under longitudinal wind) are obtained. The permeability C pi,a of the membrane has not been taken into account. Figure 5.7 shows the different load zones in plan view for both the open and closed structures. The wind pressure and suction are shown schematically in a transverse and longitudinal section view. Table 5.1 gives a summary of all obtained values for the external and internal wind pressure and exposure coefficients and the resulting total pressure per load zone and wind direction. Although the schematic representations of the sectional profiles are shown as semi-circles, the loads are applied to the actual non-smooth shape of the membrane model in EASY. 124

158 Chapter 5 Introduction to the Case Studies Plan view Section view A B C CLOSED structure Transverse wind D E F OPEN structure Transverse wind I H G CLOSED structure Longitudinal wind L K J OPEN structure Longitudinal wind Figure 5.7: Schematic representation of considered wind loads on the closed and open structures 125

159 Chapter 5 Introduction to the Case Studies OPEN CLOSED OPEN CLOSED Transverse wind Zone C pe w e [kn/m 2 ] C pi w i [kn/m 2 ] w [kn/m 2 ] A B C D E F Zone C pe Longitudinal wind w e [kn/m 2 ] C pi w i [kn/m 2 ] G w [kn/m 2 ] H I J K L Table 5.1: Values for the wind pressure w per zone for the closed and open structures 126

160 Chapter 5 Introduction to the Case Studies Transverse section view Longitudinal section view CLOSED structure OPEN structure Figure 5.8: Schematic representation of snow loads on the closed and open structures The snow load is determined with the following formula [Eurocode 1]: s µ. C. C. s = (5.7) i e t k with s k the characteristic snow load on the ground in kn/m2, the temperature coefficient C t, the exposure coefficient Ceand the coefficient µ i (form factor for the snow load). All factors C t, C e and µ i are given the value 1 and for the characteristic snow load a value of 0.5 kn/m 2 is chosen. This leads to a snow load of 0.5 kn/m 2, which is a typical value. Because surfaces with an inclination of 60º and above are assumed to be without snow, the snow load is only applied to the zones at the top of the canopy, as shown in Figure 5.8. Based on the three mobile loads transverse wind, longitudinal wind and snow - seven load cases are compiled, according to the prescribed combination fac- 127

161 Chapter 5 Introduction to the Case Studies tors, with the most severe conditions chosen (weather conditions statistically occurring once every 50 years). It is assumed that the two wind loads, each having a distinct direction, cannot occur simultaneously. With G the permanent loads and Q the mobile loads and γ their respective safety factors, the total load can be written in its general form: S = Σ i γ G G k,i + γ Q Q k,1 + Σ j γ Q Ψ 0,1 Q k,j (5.8) For the ultimate limit state (ULS) γg=1.35 and γq=1.50. These values become equal to 1 when the service limit state (SLS) is considered. In ULS the strength (determination of sections) and stability (buckling analysis) of the structure are verified, while SLS is used for verification of the stiffness (displacements). Table 5.2 shows the resulting seven load cases for both ULS and SLS. For example, ULS 1 and SLS 1 are a combination of pre-stress, transverse wind and snow and are given by: ULS 1=1,35 pre-stress transverse wind snow SLS 1=1 pre-stress + 1 transverse wind snow Load case Permanent load Main solicitation Additional solicitation ULS/SLS 1 Pre-stress transverse wind snow ULS/SLS 2 Pre-stress snow transverse wind ULS/SLS 3 Pre-stress longitudinal wind snow ULS/SLS 4 Pre-stress snow longitudinal wind ULS/SLS 5 Pre-stress snow ULS/SLS 6 Pre-stress transverse wind ULS/SLS 7 Pre-stress longitudinal wind Table 5.2: The seven load cases used for calculations in EASY These load cases are used to determine the actions of the membrane on the boundary points in EASY. Therefore, the self-weight of the bar structure cannot be included at this stage. These load vectors will be transferred to the FE- 128

162 Chapter 5 Introduction to the Case Studies model in ROBOT, to analyse and design the bar structure under the considered load cases. With all load cases determined, a structural analysis of each case study will be performed, following the earlier described approach. At this stage the selfweight of the bar structure (with a safety factor of 1.35 applied) is included in the load cases. The structural analysis of the first case study the three-way grid barrel vault with translational scissor units will be discussed in detail. Of the remaining case studies, which are calculated analogously, a concise summary of the results will be given. Structural analysis in the following chapters will show that either buckling or strength is the dimensioning ULS criterion. Although (static) stiffness standards are not stringent for this kind of foldable structures, other SLS criteria, such as fatigue should be considered. Fatigue is defined as a progressive, localised structural damage that occurs when a material or a component is subjected to cyclic or fluctuating strains at nominal stresses. The values of stresses when fatigue occurs are less than the static yield strength of the material. This means that structural failure can happen at stresses below ultimate tensile stress (ULS). Although this phenomenon has not been included in the structural feasibility study, the designer should be aware of its possible impact on the structural performance. Since fatigue is caused by dynamic (and/or moving) loads, an obvious cause in the case of deployable structures could be the repeated folding and unfolding of the structure. However, as this will amount to a relative low number of cycles compared to the lifespan of the construction, another phenomenon is likely to influence the structural performance, namely fluctuating wind loads (on the erected configuration). Therefore the calculation methodology used in the European standards [Eurocode 3] will be explained. 129

163 Chapter 5 Introduction to the Case Studies In case of an applied load with variable amplitude, the accumulative damage D has to remain smaller than 1. D is expressed by the Palmgren-Miner equation [Eyrolles, 1999]: ni D = (5.9) N with n i the number of cycles of the extent of fluctuating stresses σ i during a reference period and N i the critical amount of load cycles during the same reference period and under the extent of fluctuating stresses γ M σ i which the structure or the material can withstand. Figure 5.9 shows the consecutive steps to determine all the necessary parameters [Eurocode 3, 2007]: i a) Based on existing knowledge of similar constructions the typical solicitation sequence can be determined. Herein a realistic fluctuation of wind pressure is difficult to define. A conservative way to solve this issue is to consider a constant amplitude of the design wind pressure. b) A stress history can be set up for a particular structural detail. A stress-time graph can also be determined; either by measuring on similar constructions or using dynamic calculations of the response of the construction. c) Stress histories may be evaluated by using one of the following known techniques to determine the extent of stresses and their number of cycles: i. Rain drop method ii. Reservoir method d) The range of stress variations are determined by drawing them, in combination with the number of cycles, in a descending way. e) Using the Wohler or S-N curves, the number of cycles N i can be read of. For typical structural details these curves already exist and can be plotted in families. Each construction detail family is defined by a reference value of the resistance against fatigue σ c at a constant amplitude and a number of cycles of cycles. f) The accumulative damage caused by the different stress variations can be calculated using the Palmgren-Miner equation 130

164 Chapter 5 Introduction to the Case Studies a b c d e f Figure 5.9 : Method of accumulate damage [Eurocode 3, 2007] 131

165 Chapter 5 Introduction to the Case Studies A similar way of working can be followed to determine the damage created by varying shear stresses τ. i Further experimental research has to be conducted to see if fatigue is a dimensioning criterion. The determination of the resistance against fatigue of the specific joints used in the presented cases requires experimental data, which can not be given at this stage of research. Furthermore, a more realistic stress history due to fluctuating wind loads requires wind tunnel testing. 5.4 Conclusion In this chapter the concepts have been introduced which will be presented in the subsequent chapters. These case studies use either scissor-hinged bars or bars connected by foldable joints combined with a tensile surface for weather protection. The proposed concepts contribute to the field of mobile deployable bar structures by being either novel, or by making use of existing ideas in a new way. To facilitate mutual comparison, an overall shape has been determined on which the single curvature geometry from case studies 1, 2 and 3 is based: a barrel vault shape with a radius of 3 m and an approximate length of 10 m. A fourth concept has been introduced, which uses an 8.5 m high linear deployable truss-like structure to support its architectural envelope. The general approach for the structural analysis and the steps it involves have been clarified in this chapter. It has been explained how the considered load cases [Eurocode 1, 2007], for both ULS and SLS, have been compiled and applied to the numerical model of the membrane. Subsequently, the obtained reaction forces on the boundary points are transferred to the corresponding nodes of the FE-model of the primary structure. This simplified approach seems suited for a preliminary design within the framework of a feasibility study. 132

166 Chapter 5 Introduction to the Case Studies It has been noted that this approach does not take into account the mutual response between the bar structure and the membrane. Therefore, it is suggested that, for a more profound analysis, an integrated model of both the primary structure and the tensile surface should be used. Also, the joints which are currently excluded from the structural design, will require detailed analysis. It is unclear at this stage whether strength will be the governing design criterion, or fatigue, caused by fluctuating wind loads. This fatigue calculation falls out of the scope of this dissertation, but requires further study. Therefore, the consecutive steps to determine the needed parameters for the calculation have been discussed [Eurocode 3, 2007], [Eyrolles, 1999]. 133

167 Chapter 5 Introduction to the Case Studies 134

168 Chapter 6 Case study 1 Chapter 6 Case study 1: A Deployable Barrel Vault with Translational Units on a Three-way Grid Figure 6.1: Deployable barrel vault with translational units on a triangular grid: scissor structure and tensile surface 135

169 Chapter 6 Case study Introduction The present chapter is concerned with the development of a new concept for a mobile deployable shelter. As described in Section 5.2, a semi-circular single curvature shape (barrel vault) is designed, consisting of plane and curved translational units on a three-way grid. Using translational scissor units is a very powerful method for developing positive and negative curvature surfaces [Langbecker, 1999]. This statement applies to two-way scissor grids (Sections and 3.3.4) but is no longer valid for three-way (or triangulated) scissor grids. Langbecker [1999] states that for a curved three-way scissor geometry to be stress-free foldable, double scissors units have to incorporated, as shown in Figure 6.4. In this chapter it is shown, however, that it is possible to obtain a single curvature triangulated grid without the use of double scissor units, which leads to the development of a novel scissor geometry. A novel geometric design method is developed based on equations nased on the rise and span of the base curve and the number of units in the span. Also, it is investigated how the semi-cylindrical shape can be equipped with suitable double curvature end structures to obtain a fully closed architectural space. A simple, yet elegant solution is proposed which does not alter the original deployment behaviour. The implications of the scissor geometry and the effect of the deployment on the joints is discussed. It is shown that the phenomenon of angular distortion, as a consequence of the deployment behaviour, can be dealt with by a custom joint design. By introducing the concept of an equivalent hinged-plate model, an interesting way of gaining insight in the kinematic behaviour of the deployed configurations is proposed. To test the structural feasibility of the concept, both the open barrel vault and the closed double curvature configuration are structur- 136

170 Chapter 6 Case study 1 ally analysed, using the method explained in Section and the results are discussed. 6.2 Description of the geometry Consider the saddle-shaped (anticlastic) structure depicted in Figure 6.2. As can be seen in both the plan and perspective view, it consists of identical translational scissor linkages placed in two directions of an orthogonal grid. Using translational units on an orthogonal grid, while maintaining a constant unit thickness throughout the structure, is a powerful method for obtaining a wide variety of singly or doubly curved structures (Sections and 3.3.4). Figure 6.2: Plan view and perspective view of the same double curvature structure with translational units on a quadrangular grid But this statement is no longer valid for three-way grids with single or double curvature. If a triangulated grid is to be designed, with single or double curvature, the introduction of double scissors is needed, in order to obtain a stressfree deployable configuration [Langbecker, 1999]. Figure 6.3 shows a scissor module composed of single scissor units, while Figure 6.4 shows a module with a double unit. The effect of the integration of a double unit is that the module is no longer triangular, but quadrangular. This non-triangulation of the grid can lead to in-plane instability, resulting in swaying or skewing of the structure. Although double scissor units are inevitable for triangulated double curvature structures, it will be shown that single curvature grids can be designed with only single units. 137

171 Chapter 6 Case study 1 Single unit Double unit Figure 6.3: Translational scissor module with only single units Figure 6.4: Translational scissor module with a double unit Consider Figure 6.5 which represents a planar translational scissor grid. The two-dimensional scissor linkages A, B and C contain plane units with constant unit height throughout the grid with equilateral triangular cells. B A C Figure 6.5: Plan view, perspective view and side elevation of a planar structure with a triangulated grid Now, by introducing curvature in direction B and C a cylindrical shape is obtained, as shown in Figure 6.6: - direction A (longitudinal direction) contains parallel rows of identical plane units - direction B and C run diagonally over the span and contain rows of nonidentical curved translational units 138

172 Chapter 6 Case study 1 B A C Figure 6.6: Plan view, perspective view and side elevation of a barrel vault with a triangulated grid Curving these scissor units has a particular consequence: for the scissor module to have its unit height unaltered while the units become curved, the equilateral grid cells have to turn into isosceles ones, i.e. their apex angle increases while their projected area decreases. Figure 6.7 shows a plane module M1 and its curved derivatives M2 and M3, which are both used to compose the barrel vault from Figure 6.6. γ 3 γ 1 γ 2 M1: plane M2: slightly curved M3: highly curved Figure 6.7: Perspective view and plan view of three different triangular modules 139

173 Chapter 6 Case study 1 In Figure 6.7 three different scissor modules are shown in their deployed position (not one single unit in consecutive deployment stages). So the only way to make a deployable barrel vault with translational scissor units on a triangular grid without the use of double scissor units - is to use isosceles scissor modules of constant unit height (Figure 6.7). But a double scissor can also prove itself useful. The barrel vault described earlier is an open structure, which means the membrane canopy forms only a roof, while the front and back side remain open (Figure 6.8). However, by using a novel way of providing the open barrel vault with double curvature end structures, the open structure can be fully closed. This is done by adding two modules which are identical to those used in the rest of the structure to both the front and back, as shown in Figure m 9.4 m Figure 6.8: OPEN structure: perspective view and plan view For the doubly curved end structures, the integration of double scissors cannot be avoided. In order for the structure to be able to deploy, each added module is provided with its own separate double scissor unit. The effect this has on the deployment is discussed in Section

174 Chapter 6 Case study 1 6 m 10.6 m Figure 6.9: CLOSED structure: perspective view and plan view (double scissor marked in red) Now that the composition of the grid has been discussed, a closer look is drawn to the structure itself: as shown in Figure 6.10 it has a circular curvature when viewed in front elevation and consists of four modules in the span. As can be seen in Figure 6.10, the curved scissor units which make up the modules are not coplanar, nor do they lie in the direction of the span. Therefore, the curvature of the structure - which is of importance to the design - is different from that of the scissor linkages B and C, because these find themselves at an angle with the span direction. The eventual curvature is determined, not by the actual scissor units, but by their projected counterparts, marked in red in Figure 6.10, with the projection plane lying in the direction of the span. 141

175 Chapter 6 Case study 1 U1 U2 U3 Figure 6.10: Front elevation, top view and perspective view of a portion of the barrel vault with four modules in the span: the projected versions (marked in red) of the scissor units U1 and U2 determine the real curvature All scissor units obey the geometric deployability condition (Section 3.3), which is graphically represented by ellipses on the linkage U1, U2 and U3, developed in a common plane, as shown in Figure Now there are two ways to design such a barrel vault, depending on how much control over the eventual curvature is desired. The first approach is based purely on geometric constructions and involves no numeric computation. It is simply a matter of constructing a two-dimensional linkage such as the one shown in Figure Subsequently, the scissor units are rotated in 3D until they form closed triangular scissor modules, as depicted in Figure 6.10, which are repeated to form a complete grid. As mentioned earlier, the definitive curvature can be evaluated by projecting the units on the plane lying in the direction of the span. 142

176 Chapter 6 Case study 1 U1 U2 U3 Top view of units U1, U2 and U3: rotated until coplanar U1 U2 U3 Figure 6.11: Developed view of units U1, U2 and U3: graphic representation of the deployability condition by means of ellipses It is evident that, when a designer wants total control over the final curvature, another, more precise approach is to be used. When, for example, a circular curve with a certain rise and span for the barrel vault is desired, numeric computation has to be used to find a geometry that will satisfy this requirement. The second approach will now be discussed in greater detail. 6.3 Geometric design Using the rise and span as initial design parameters, a circular arc is determined, which represents the actual curvature of the barrel vault. The goal of the numeric approach is to find, based on a number of design parameters, a series of scissor modules which will fit on the desired base curve. For this ex- 143

177 Chapter 6 Case study 1 ample a circular arc is chosen to facilitate comparison between the barrel vaults from case study 1, 2 and 3 - but any curve, in its parametric form, can be used. For this approach, the method of the ellipses for imposing the geometric deployability condition onto two-dimensional linkages (Chapter 3) is extended to three dimensions. By simply revolving the ellipses around their vertical axis, an ellipsoid is obtained with three semi-axes, of which two are identical, as shown in Figure Where the ellipse determines the locus of all valid positions for the intermediate hinges in two dimensions (Figure 6.13), the ellipsoid fulfils the same role in three dimensions. Instead of the small ellipsoid, marked in red in Figure 6.13, an ellipsoid of double size is used to determine the locus of the intersection points between the unit lines of a scissor unit and the curve. The small unit has unit thickness (t) which is also half the distance between the foci of the double ellipsoid. The relation between the design parameter t and the length of the semi axes a and b is expressed by Eqn (6.1): t = b a (6.1) K b K φ t θ t a Figure 6.12: An ellipsoid representing the geometric deployability condition in three dimensions Figure 6.13: Vertical section view of the small and big ellipsoid, imposing the geometric deployability condition 144

178 Chapter 6 Case study 1 When the following four design parameters are given a value, the geometry is fully determined: U: the number of units in the span R: the radius of the circular arc t: the unit thickness α 2 : the angle which determines the position of point P 2 and whether the curve π is a semi-circle (α 2 =0) or an arc segment ( 0 α2 ) - see Section Now consider Figure 6.14 which shows a circular arc A and a parallel circular a arc A with a distance of between them. The location of endpoint P 2 of the 2 arc is determined by α 2. P 0 P 0 P 1 α2 P 2 A R A Figure 6.14: A scissor linkage fitted on a circular curve, with all relevant design parameters and the global coordinate system A solution has to be found for P 1 (x 1, y 1, z 1 ). Its position on the circular arc is such that it determines a translational SLE (scissor-like element) which is 145

179 Chapter 6 Case study 1 compatible with the plane SLE, as depicted in the developed view of the linkage in Figure This is the intersection point of ellipsoid E 0 with the circle arc A, determined by the angle α 1 (which determines the position of P 1 on the circular arc). E 0 E 1 P 0 P 0 Plane unit P 1 Compatible curved units P 2 Plane unit Figure 6.15: Developed view of the scissor linkage from Figure 6.14, showing a chain of double ellipses The general form of the parametric equation for an ellipsoid with centre (x 0,y 0, z 0 ), semi-axes a and b and the axis of revolution parallel to the Z-axis (see Figure 6.12), with 0 θ 2π ; 0 φ π becomes: x = a cosθ sinφ + x y = a sinθ sinφ + y z = b cosφ + z (6.2) The general equation of a circular arc parallel to the ZX-plane (y=cte) with centre around the origin with 0 α π is x = R cosα z = R sinα (6.3) 146

180 Chapter 6 Case study 1 From Figure 6.14 the following relations between ellipsoids E 0, E 1 and points P 0, P 1, P 2 are derived: π The location of endpoint P2 of the arc A depends on α 2 with 0 α2 2 x2 = Rcosα2 z = Rsinα (6.4) 2 2 Figure 6.16: Perspective view of the scissor linkage from Figure 6.15 The coordinates for point P 1 on A are given by x = Rcosα 1 a y1 = 2 z = Rsinα (6.5) 147

181 Chapter 6 Case study 1 We can now write the parametric equation for ellipsoid E 0 with centre P 0 (0, 0, R): x = a cosθ sinφ y = asinθ sinφ 0 z = bcosφ + R (6.6) P 1 (x 1, y 1, z 1 ) is the intersection between E 0 and the circular arc A, which is expressed by joining Eqns (6.5) and (6.6) in: a cosθ sinφ = Rcosα 0 a asinθ0 sinφ0 = 2 bcosφ + R = Rsinα (6.7) (6.8) (6.9) P 1 is also the centre for ellipsoid E 1 (size identical to E 0 ) which intersects with circular arc A in point P 2. We can write the parametric equation for ellipsoid E 1 with centre P 1 : x = a cosθ sinφ + x y = asinθ sinφ + y z = bcosφ + z (6.10) Joining Eqns (6.4), (6.5) and (6.10) leads to the following relations: a cosθ sinφ + R cosα = R cosα a a sinθ1 sinφ1 + = 0 2 b cosφ + R sinα = R sinα (6.11) (6.12) (6.13) Equations (6.1), (6.7), (6.8), (6.9), (6.11), (6.12) and (6.13) lead to a system of seven equations in seven unknowns which is solved numerically. By assigning a, b, α, θ, θ, φ φ values to R, α 2 and t a solution is found for { } , 1 148

182 Chapter 6 Case study 1 a cosθ0 sinφ0 = R cosα1 a a sinθ0 sinφ0 = 2 b cosφ0 + R = R sinα1 a cosθ1 sinφ1 + R cosα1 = R cosα2 a a sinθ1 sinφ1 + = 0 2 b cosφ1 + R sinα1 = R sinα t = b a (6.14) The previous parameterisation determines a geometry of four units (U=4) in the span. More units can be calculated by adding the appropriate equations for the ellipsoids in an analogous manner, but as the number of units rises, the complexity of the system of equations increases. 6.4 From mechanism to architectural envelope Deployment and kinematic analysis A two-dimensional scissor linkage has a single rotational degree of freedom (D.O.F.). When such linkages are placed on a grid, this rotational D.O.F. is, depending on the grid geometry, either preserved or removed. All this depends on the type of scissor units used (translational, polar, angulated), the type of grid they form (two-way, three-way, four-way) and the curvature (plane, single or double). An insight in the mobility of the mechanism is needed to understand to what degree constraints have to be added after deployment to turn it into a load bearing structure. During deployment the rotational degree of freedom is used to expand the scissor mechanism. Figure 6.17 shows the different stages in the deployment. As is typical for translational scissor units, the mechanism expands gradually in two directions in the horizontal plane, while maintaining its overall height during deployment. The deployment itself can be performed by pulling the four 149

183 Chapter 6 Case study 1 lower corner nodes away from the centre. When on even terrain, wheels at the corners could aid in the deployment. Alternatively, small lifting equipment - which would usually be already present for loading and unloading during transport can be used to lift the mechanism at a central node. Figure 6.17: Perspective view, front elevation and top view of the deployment process of the barrel vault with translational units OPEN structure 150

184 Chapter 6 Case study 1 Figure 6.18: Proof-of-concept model of (half of the) closed structure (aluminium, scale 1/10) As mentioned before, the custom end structures which turn the open structure into a closed one, have double scissor units. There are two of these, one for each side, as shown in Figure In the partly deployed position there is a gap in the structure that will close during deployment. That is the reason for providing each module with its separate double scissor. In the fully deployed state these modules meet and the double scissors coincide. The deployment process of the closed structure is depicted in Figure A proof-of-concept model of the closed structure has been constructed, which has shown that the kinematic behaviour is as desired. Two double units, side by side Two double units theoretically coincide Figure 6.19: Two double scissors in partially (left) and fully deployed (right) position 151

185 Chapter 6 Case study 1 Figure 6.20: Perspective view, front elevation and top view of the deployment process of the barrel vault with translational units CLOSED structure A novel way of gaining an insight in the mobility of the mechanism and the restraints which are needed to turn it into a structure in the deployed configuration, is to represent the configuration by an equivalent hinged plate structure. This structure shares the same kinematic properties as the scissor geometry would have when its rotational degree of freedom (scissor action) is removed. Removal of this rotational D.O.F. is a minimum requirement for a scissor mechanism to act as a structure, otherwise it is allowed to collapse and return to its initial undeployed state. Since the triangulated geometry with translational units is a single-d.o.f.-mechanism, it is sufficient to block the movement of a single unit in order to remove the rotational D.O.F., e.g. by fixing the distance between two end nodes n 1 and n 2 (as shown in Figure 6.21) by means of a bar or cable element. Alternatively, two lower end nodes n 3 and n 4 152

186 Chapter 6 Case study 1 at opposite sides of the structure can be kept at the same distance in the same way (Figure 6.22). The reason the equivalent hinged plate model is a valid representation of the scissor grid with the rotational D.O.F. removed, is illustrated in Figure A scissor unit with its D.O.F. removed can be replaced by a rigid plate of which the vertices correspond with the end nodes of the scissor. The original parallel unit lines (imaginary lines connecting the upper and lower nodes of the scissor units) now become fold lines acting as continuous joints for the plates. Each module or grid cell consists of three plates, which is a rigid body. And because the complete configuration consists of such three-plate modules, the grid has no in-plane mobility (no skewing or swaying). This means we can form an idea of the extra restraints needed in the deployed configuration, apart from constraining the scissor action, to turn the mechanism into a structure. A minimum of seven translations need to be constrained (including constraining the scissor action), as Figure 6.21 shows. In practice however, it is suggested to fix all nodes touching the ground by pinned supports (Figure 6.22). Standard solutions for fixing mobile, lightweight structures the ground can be used. An overview of recoverable lightweight anchors is given by Llorens [2006]. n 2 n 1 Figure 6.21: From scissor mechanism to the equivalent hinged plate linkage for mobility analysis of the open structure (idem for closed structure) - minimal constraints 153

187 Chapter 6 Case study 1 n 4 n 3 Figure 6.22: Fixing all lower nodes to the ground by pinned supports Undeployed state: cable at full length Fully deployed state: cable shortened and locked Figure 6.23: An active cable (marked in red) runs through the mechanism, connecting upper and lower nodes along its path. After deployment it is locked to stiffen the structure Crucial to any deployable structure are the joints. Every unit consists of two bars connected by a revolute joint (intermediate hinge). At their ends, the bars are connected by another revolute joint. In reality, the members of the scissor units and the joints have discrete dimensions, unlike the theoretical geometric line models which have zero thickness. In theory, both bars of a scissor unit lie in a common plane. As opposed to the theoretical one-dimensional coplanar scissors, the physical bars are not in the same plane. A scissor unit has an imaginary centreline, which separates the two scissor bars lying on either side of that axis (Figure 6.24). The size of the joint is influenced by the number of bars it has to connect and by the dimensions of the bar. The wider the section of the scissor bars becomes, and the higher the number of bars that to be connected in the joint, 154

188 Chapter 6 Case study 1 the larger the radius of the joint becomes, in order to accommodate all elements without interference during deployment. Some joint solutions have been proposed by Escrig [1984]. The joints will have to allow every possible movement, imposed by the grid geometry, to ensure a stress-free deployment. It has been shown that, in the deployed position, the grid cells are isosceles triangles. But during deployment of the mechanism proposed here, the apex angles inside the scissor modules increase in size. Take the angles γ 2 and γ 3 from Figure 6.7: when the maximal deployment is reached, γ 2 =63º and γ 3 =95º. This angular distortion has to be allowed by the specially constructed joint, which adds some complexity to the design. A joint is proposed, of which the fins can freely rotate around the cylindrical hub. Figure 6.25 shows a top view of the designed joint connecting six bars. The bars are provided with wedge-shaped end pieces, which allow them to be as compactly arranged as possible, without interfering with one another during deployment. The centrelines of all units connected by a certain joint have a single intersection point G, lying on a vertical axis through the joint, as Figure 6.26 shows. In the fully deployed position, this intersection point of centrelines of the scissor units lies on the vertical axis through the joint. As the structure is compacted towards its undeployed state, point G moves further upward until all centrelines become parallel in the undeployed position. 155

189 Chapter 6 Case study 1 Figure 6.24: Top view and perspective view of one scissor unit, its intermediate hinge and its end joints and their offset position relative to the theoretical plane Figure 6.25: Concept for an articulated joint, allowing the fins which accept the bars to rotate around a vertical axis, to cope with the angular distortion of the grid 156

190 Chapter 6 Case study 1 Joint centreline Unit centreline G Figure 6.26: Partially and undeployed state: as the structure is compactly folded, the imaginary intersection point of the centrelines travels on the vertical centreline through the joint The bars are connected to a hinge point which is not at the intersection point of bar centrelines (different from the scissor unit centreline from Figure 6.24). This eccentricity has an effect on the structural performance of the structure in the sense that this geometric imperfection induces a 2 nd order effect, i.e. bending in the bars. The hollow, cylindrical hub around the fins of the joint rotate, can accept an active cable (Figure 6.23), guided by a pulley system, for stiffening the structure after deployment (or for raising the membrane and bringing it under pretension. In the deployed state, after the structure is appropriately fixed to its supports, the cable can also be used to influence the stiffness by introducing more or less tension. After deployment, the membrane is raised toward the inner end nodes of the bars, after which a basic level of pre-tension is introduced. In Figure 6.27 and Figure 6.28 the resulting mobile shelters (open and closed) are shown together with the area they cover. Alternatively, instead of attaching the membrane to the inner nodes of the structure, the tensile cover could function as an outer layer, attached to the external nodes, providing weather protection for the structure as well. 157

191 Chapter 6 Case study 1 Perspective view Top view Covered area 60 m 2 Figure 6.27: Perspective view and top view of OPEN structure with integrated tensile surface Perspective view Top view Covered area 58 m 2 Figure 6.28: Perspective view and top view of CLOSED structure with integrated tensile surface 158

192 Chapter 6 Case study Structural analysis Open structure (single curvature) Formfinding of the tensile surface Before any load can be applied to the membrane surface, an equilibrium form under pure pre-tension is to be found first. The boundary geometry used for generating the membrane in the formfinding process in EASY consists of a pattern of rhombi and triangles. A pre-tension of 1 kn/m in both directions is introduced in the net with a mesh length of 0.2 m. A force of 2kN is introduced in the boundary cables which have a stiffness of kn, while the membrane is given a basic stiffness of 400 kn/m in both directions. A typical PVC-coated polyester fabric is chosen for all case-studies. In Figure 6.29 a top view and perspective view of the skeletal structure and the matching boundary geometry for the membrane are shown. After formfinding, the equilibrium form for the tensile surface is found, as depicted in Figure Typical values for stresses in the membrane vary from 4 to 5.5 kn/m (Figure 6.31). Figure 6.29: Top view and perspective view of the skeletal scissor structure (left) and the boundary geometry for the compatible membrane (right) 159

193 Chapter 6 Case study 1 Figure 6.30: Views of the equilibrium form for the membrane as a result of the formfinding process Figure 6.31: Typical stresses in the membrane range from 4 to 5.5 kn/m 160

194 Chapter 6 Case study 1 Statical analysis under load combinations With all load combinations (Section 5.2.2) applied, the actions of the membrane on the outer boundary points are determined. In accordance with the hypothesis, the pre-tension is treated as a permanent load and is therefore multiplied with its appropriate safety factor. In the case of the ultimate limit state (ULS) the pre-tension is increased to 1.35 kn/m and the force in the boundary cable is raised to 2.7 kn. For the service limit state calculations (SLS) the membrane pre-tension and the force in the boundary cable are left at their initial values of 1kN/m and 2 kn respectively. Now the seven load combinations are applied and the resulting forces in the boundary points are calculated. Structural design of the scissor structure The resulting forces are applied to the corresponding nodes of the FE-model of the scissor structure in ROBOT [2007]. The lower nodes touching the ground are fixed with pinned supports. This removes the mobility from the mechanism and enables it to act as a structure and transfer loads. In accordance with the physical model, the joints which connect the scissor bars consist of separate elements that share one rotational degree of freedom around an axis through their common point and perpendicular to their common plane. Figure 6.32 shows six bars (black lines) attached at their ends (P) to six node elements (PQ) and the attributed rotational degrees of freedom: the bars of the scissor units are allowed to rotate around their local Y-axis (Figure 6.34) at their ends (P) as well as at their intermediate pivot hinge (RS). For modelling purposes, each physical bar is represented in the FE-model by two (half-length) separate lines, joined together with a fixed connection (R and S): 3 translations and 3 rotations constrained. 161

195 Chapter 6 Case study 1 P Q R S Figure 6.32: FEM-model of six bars attached to a node Figure 6.33: An intermediate pivot hinge connects two scissor bars The line elements that make up the nodes rotate around their local Z-axis. The global coordinate system and the local coordinate system of a bar are shown in Figure LOCAL Coordinate System GLOBAL Coordinate System Figure 6.34: Local coordinate system of a bar element (left) and global coordinate system (right) When, for example, load case 6, which is transverse wind is considered (a full description of the load cases is given in Section 5.3.2), the distribution of the load vectors and their relative size is shown graphically in Figure The resulting reaction forces in the supports are depicted in Figure The load vectors from Figure 6.35 represent the action of the membrane on the nodes of the structure, as a result of transverse wind load and priestess and the resultant load vectors are pointed inward. The reaction forces in the support points of the structure are shown in Figure As can be seen from the schematic, the action on the middle section induces a reaction force pointed 162

196 Chapter 6 Case study 1 upward and outward, while the reaction forces in the sections near the open ends are pointed inward and downward. Front elevation Side elevation Figure 6.35: Typical pattern of load vectors for transverse wind + pre-stress of the membrane Front elevation Side elevation Figure 6.36: Typical pattern of reaction forces under transverse wind Moments around the local Y-axes of the bars are shown in Figure 6.37, which is a typical diagram for a scissor arch. Here the maximal and minimal values for My are 2.21 knm and knm respectively. Figure 6.37: Bending moments My under transverse wind 163

197 Chapter 6 Case study 1 A typical deformation pattern (not to scale) under transverse wind load is shown in Figure Figure 6.38: Typical deformation under transverse wind: undeformed [grey] and deformed [red] configuration no scale Ultimate Limit State (ULS) With all load cases and combinations applied, a structural analysis is performed to examine the structure s strength (section design) and stability (buckling analysis), in accordance with Eurocode 3 [2007] in the deployed configuration. The extreme values for the reactions, forces and stresses will be discussed. When a load case is mentioned, it is assumed that the selfweight of the structure is included in the combination, with its appropriate safety factor applied, depending on whether ULS or SLS is considered. Aluminium is chosen as the material for the bars which are tubular and have a rectangular section. Because the joints will not be designed in this study, their influence on the structure design is made as small as possible. Therefore, their size is limited and they are awarded a very large section together with a high stiffness and yield stress (f y ). The structure design is performed by means of an iterative static analysis with minimum weight as the optimization criterion. When three possible tubular sections for the bars are suggested round, square and 164

198 Chapter 6 Case study 1 rectangular - the algorithm evaluates the latter as being the most structural efficient. Rectangular sections not only seem to be appropriate from a structural point of view, but they are preferred anyway over round or square sections because of the higher degree of compactness they provide in the undeployed position. A buckling analysis is performed with all bars having a buckling length coefficient of 1. After optimization, the proposed section is rectangular: TREC 120x60x3.2 mm. This leads to a total weight of 856 kg, which gives, for a total covered surface of 63 m 2, a weight-per-square-meter of 13.6 kg. Figure 6.39 shows the optimized structure with the resulting weight/m 2 and the material characteristics. Because the joints are not structurally analyzed and designed, their weight is not taken into account. Therefore, the structural performance is only a measure of the main members. Also, the weight the membrane is neglected and no analysis is performed in the partially deployed state. 165

199 Chapter 6 Case study 1 Section hxbxt [mm] Total weight [kg] Weight/m 2 [kg/m 2 ] TREC 120x60x3.2 mm Material E-modulus [GPa] f y [MPa] Aluminium Figure 6.39: Perspective view of the resulting structure with rectangular sections of 120x60mm In ROBOT positive stresses indicate pressure, negative values mean tension. The reaction forces in the global coordinate system are mentioned in Figure Induced by load combination ULS 2 (pre-stress + snow + transverse wind), the largest reaction force is 52 kn in the Z-direction. Marked in red is the support in which Fz max occurs. 166

200 Chapter 6 Case study 1 REACTIONS FX (kn) FY (kn) FZ (kn) MAX 13,36 44,50 52,03 Case ULS 5 ULS 1 ULS 2 MIN Case ULS 5 ULS 1 ULS 2 Figure 6.40: Reactions in the global coordinate system: the maximal reaction force occurs under ULS 2 (pre-stress + snow + transverse wind) STRESSES Section TREC mm Bar S max [MPa] S max(my) [MPa] S max(mz) [MPa] Fx/Ax [MPa] S min [MPa] S min(my) [MPa] S min(mz) [MPa] Figure 6.41: The critically loaded bar is located at the top. Summary of the stresses occurring in the critically loaded bar (positive stresses indicate pressure, negative values mean tension) 167

201 Chapter 6 Case study 1 The governing load combination for the design for strength is ULS 5 (pre-stress + snow). In the critically loaded bar a total stress level of MPa occurs, part of which is caused by My, inducing a bending stress of MPa (Figure 6.41). The bending stress around the local Z-axis is a fraction of S max (My). When the stability is checked, ULS 2 (pre-stress + snow + transverse wind) is the governing load combination, which induces an axial force Fx of kn (Figure 6.42). The bar, which is most susceptible to buckling, is located near the bottom of the structure. During buckling analysis in ROBOT, second order effects are taken into account. FORCES Bar FX (kn) FY (kn) FZ (kn) Load Comb. MX (knm) MY (knm) MZ (knm) ULS Figure 6.42: Axial forces, transverse forces and bending moments in the local coordinate system of the bars 168

202 Chapter 6 Case study 1 Service limit state (SLS) The stiffness of the structure is analysed under service limit state by checking the displacements of the nodes. The nodes which are subject to the largest displacements are marked in Figure 6.43: 2.0 cm in the X-direction (SLS 2), 1.0 cm in the Y-direction (SLS 5) and -1.4 cm along the Z-axis. Because this is a concept for a mobile structure, the same strict requirements as for permanent buildings do not apply. These values seem perfectly acceptable for a structure of this type and the displacements will not degrade the serviceability of the structure. The maximum deflection occurring in the structure is approx. 1/100. DISPLACEMENTS UX (cm) UY (cm) UZ (cm) Load Comb. SLS 2 SLS 5 SLS 5 Figure 6.43: Maximal nodal displacements in the global coordinate system The obtained section of mm causes the total weight to mount up to 856 kg which makes the structure bulky and heavy for transport. The section can be reduced by connecting the lower and upper nodes with a continuous cable running over pulleys. Figure 6.44 shows the cable and its path through the structure in both the undeployed and deployed state. After deployment, the cable is locked from which moment it can contribute to the overall structural performance. The effect is a decrease in section of the scissor 169

203 Chapter 6 Case study 1 bars and an increase of the structural stiffness. The steel cable with a section of 28 mm 2 has been assigned a calculation strength of 1500 MPa (safety factor of 1.5 applied). For practical reasons, the continuous cable is modelled as discrete cable fragments, each of which connects an upper and lower node. Although this is an approximation of the real situation, the results of the analysis indicate a noticeable positive effect on the structural behaviour (Figure 6.46). Figure 6.44: Continuous cable zigzagging through the structure, connecting upper and lower nodes and contributing to the structural performance Scissor bars Cable tie Tension cable Section hxbxt, d Total weight/section [kg] Weight/m 2 [kg/m 2 ] TREC 60x40x3.2mm 459 Cable 6 mm 6 Total weight: Figure 6.45: Resulting structure after optimization, with cable elements 170

204 Chapter 6 Case study 1 RESULTS for case study 1: OPEN structure (with cable elements) STRESS [MPa] S max S max S max S max Section [mm] (My) (Mz) Fx/Ax Load comb. TREC 60x40x ULS 5 Cable d= ULS 5 FORCE Section [mm] Fx [kn] My [knm] Mz [knm] Load comb. TREC 60x40x ULS 5 Cable d= ULS 5 REACTIONS [kn] FX Load comb. FY Load comb. FZ Load comb ULS ULS ULS 5 DISPLACEMENTS [cm] Ux Load comb. Uy Load comb. Uz Load comb. 3.3 SLS SLS SLS 6 Uz Uy Ux Figure 6.46: Summary of the determining stresses and forces for the strength, stability and stiffness of case study 1: OPEN structure 171

205 Chapter 6 Case study 1 When considering Figure 6.40 (no cable) and Figure 6.46 (with cable), a noticeable difference in values for Fy and Fz can be observed. This is due to the fact that only the highest peak value occurring in the structure, under a certain load case, is represented. The sums of all values of Fy (and equally for Fz) are identical for both structures, only the structure without cables demonstrates higher peak values. The effect of incorporating a cable is an alteration of the stiffness, resulting in a redistribution of forces and a decrease in peak values. 172

206 Chapter 6 Case study Closed structure (double curvature) Section hxbxt, d[mm] Total weight/section [kg] Weight/m 2 [kg/m 2 ] TREC 60x30x Cable 6 6 Total weight: Figure 6.47: Perspective view of case study 1: CLOSED structure: with sections after structure design and total weight The closed structure is provided with the same cable elements as in the open structure. The added modules lead to a slight decrease in section width (30 mm as opposed to 40 mm for the open structure). The weight/m 2 for both structures is approximately 7.5 kg/m

207 Chapter 6 Case study 1 RESULTS for Case study 1: CLOSED structure (with cable elements) STRESS [MPa] S max S max S max Load S max Section [mm] (My) (Mz) Fx/Ax comb. TREC 60x30x ULS 2 Cable d= ULS 2 FORCE Section [mm] Fx [kn] My [knm] Mz [knm] Load comb. TREC 60x30x ULS 2 Cable d= ULS 2 REACTIONS [kn] FX Load comb. FY Load comb. FZ Load comb ULS ULS ULS 2 DISPLACEMENTS [cm] Ux Load comb. Uy Load comb. Uz Load comb. 3.1 SLS SLS SLS 1 Ux Uy, Uz Figure 6.48: Summary of the determining parameters for the strength, stability and stiffness of case study 1 _ CLOSED structure 174

208 Chapter 6 Case study Conclusion It is known from literature that using translational scissor units on a two-way grid allows the design of stress-free deployable positive and negative curvature surfaces, but this does not apply to three-way (triangulated) scissor grids. For non-flat three-way grids to be stress-free deployable, the integration of double scissors is required [Langbecker, 1999]. Figure 6.49: Case study 1: Single curvature OPEN structure (barrel vault) In this chapter an advance has been made by developing a stress-free deployable scissor geometry of single curvature with translational units on a threeway grid. It has been shown that the curved triangulated grid can be solely composed of single scissor units, therefore making the integration of double scissor units obsolete. By avoiding double scissors, the number of connections is kept to a minimum. Also, the inherent triangulation of the grid provides inplane stability. As a consequence, the deployed configuration does not require additional cross-bracing of the grid cells to prevent the structure from skewing. A geometric design method has been developed and the necessary equations have been derived, based on architecturally relevant design parameters. Although this design method allows the design of barrel vaults with a semicircular section, this approach can easily be extended to other base curves as well. To test the feasibility of the concept, a barrel vault with four units in the span has been designed, of which two variations have been presented: an open, single curvature structure (barrel vault) (Figure 6.49) and a closed double curvature structure (Figure 6.50). In case of the double curvature structure, it 175

209 Chapter 6 Case study 1 has been found indeed inevitable, as opposed to the single curvature structure, to incorporate double scissors to guarantee stress-free deployability. Figure 6.50: Case study 1: Double curvature CLOSED structure An explanation has been given for the angular distortion, a phenomenon associated with the deployment of the proposed geometry. A design for a joint has been proposed, which takes this effect into account. By introducing the concept of an equivalent hinged-plate model, an interesting way of gaining insight in the kinematic behaviour of the mechanism has been offered. It has been shown that the proposed geometry is a singledegree-of-freedom mechanism. As a result of the structural analysis, it has been found that, due to high inplane bending stress in the scissor members, the strength is the governing design criterion. By adding a continuous cable between the upper and lower nodes, a significant increase in structural performance and a reduction in weight has been achieved. The open and closed structure, both covering approx. 65 m 2, achieve a weight ratio of approximately 7.5 kg/m 2. It is noted that the joints are not included in the structural design and therefore no statement is done concerning their weight. 176

210 Chapter 7 Case study 2 Chapter 7 Case study 2: A Deployable Barrel Vault with Polar and Translational Units on a Two-way Grid Figure 7.1: Deployable barrel vault with polar units on a quadrangular grid: scissor structure and tensile surface 177

211 Chapter 7 Case study Introduction In this chapter the design of the second case study, a deployable barrel vault with polar and translational scissor units on an orthogonal two-way grid, is described. Using polar units is an effective way of introducing single curvature in an orthogonal grid. A number of examples have been presented in literature, such as single curvature shelters and scissor arches (Section 2.2.4). This case study is an illustration of how a basic stress-free deployable single curvature structure can be obtained by putting the construction methods and the geometric design methods, as described in Section 3.2, into practice. A structure with four units in the span is proposed, similar in shape and dimensions to case study 1. What sets this concept apart is the innovative way the open structure is fully closed by adding doubly curved end structures, which are compatibly deployable with the original structure. Previous solutions, such as the elegant proposal by Escrig [2006], also provide a doubly curved closed surface, but are, however, not stress-free deployable. Therefore, it is shown how an existing geometry, the lamella dome (Section 3.3.4) is adapted to fit the required purpose. Based on the geometric design method proposed in Section 3.3.2, the equations, which allow the study of the geometry in several stages of the deployment, are derived. These are then used to predict the maximum span the configuration will reach during deployment. Similar to case study 1, the mobility of the system is discussed by means of an equivalent hinged-plate model providing an insight in the stability of the deployed configuration. As quadrangular grids are prone to skewing due to a lack of in-plane stability, measures are taken such as the introduction of crosscables or cross-bars in several configurations, which are structurally analysed and discussed. 178

212 Chapter 7 Case study Description of the geometry Open structure Consider Figure 7.2 which represents a plan view and perspective view of a planar translational scissor grid. The two-dimensional scissor linkages A and B contain plane units with constant unit height throughout the grid with square cells. B A B A Figure 7.2: Plan view and perspective view of a planar structure with a quadrangular grid B A B A Figure 7.3: Plan view and perspective view of a barrel vault with quadrangular grid As mentioned in chapter 3, a polar unit is simply obtained by moving the intermediate hinge of a plane translational unit away from the middle of the bar. This eccentricity of the revolute joint creates curvature when the unit becomes deployed. Now, by introducing curvature in direction B, a cylindrical shape is obtained as shown in Figure 7.3: - direction A (longitudinal direction) contains parallel rows of identical plane units 179

213 Chapter 7 Case study 2 - direction B runs in the direction of the span and contains rows of identical polar units Figure 7.4 shows a series of polar linkages with 3, 4, 5 and 6 units in the span. From Figure 7.4, the linkage with four units in the span (U=4) is chosen, for reasons explained in Section 5.1: all case-studies (with single curvature shape) are based on the same circle arc with a radius of 3 m and share the same structural thickness of 1.25 m. This facilitates mutual comparison. 3 units 4 units 5 units 6 units Figure 7.4: Series of polar linkages with 3, 4, 5 or 6 units in the span, based on the same circular arc The geometric construction is based on a chosen circular arc, marked in red in Figure 7.5, which is divided into four segments (U=4) by radial lines through centre O. The intersection points on the arc, such as point Q, determine the location of the inner end nodes of the scissor units. Now a line is drawn through Q, tangent to the circle arc. P and R are the intersection points which mark the outer end nodes of the units, by which the unit height (h) is immediately determined. By symmetry, the complete geometry can now be constructed. 180

214 Chapter 7 Case study 2 P Q O h R Figure 7.5: Geometric construction of the four-unit linkage Subsequently, a curved four-by-four grid is composed by connecting parallel placed polar arches (direction B) by plane translational linkages (direction A). All bars, both translational and polar, are of identical length, the only difference between the translational and polar units being the location of the intermediate hinge, relative to the middle of the bars. The resulting configuration is shown in Figure m 6 m 10.9 m Figure 7.6: OPEN structure: perspective view and top view Evidently, all scissor units obey the geometric deployability constraint (Section 3.2), which is graphically represented by ellipses on the linkage U1, U2 and U3, developed in a common plane, as shown in Figure

215 Chapter 7 Case study 2 U1 U2 U3 U1 U2 U3 Figure 7.7: Perspective view and developed view of units U1 (plane translational) and U2, U3 (polar): graphic representation of the deployability condition by means of ellipses The translational unit U1 belongs row A in Figure 7.3, while the polar units U2 and U3 belong to the perpendicular direction B. When this linkage is developed in a common plane, the ellipses which impose the deployability constraint can be drawn, as shown in Figure 7.7. It should be noted that, although polar units are used here, an ellipse is required to impose the deployability 182

216 Chapter 7 Case study 2 constraint. This is because the polar units from the transverse direction are to be linked with the translational units from the longitudinal grid direction. Using circles is only valid for polar units, while the ellipse is valid for both unit types. Therefore, mixing polar and translational units requires the general method, hence, an ellipse is to be used Closed structure As an alternative to the open structure, a closed double curvature structure is proposed. For this, half of the open structure (called main structure ) is kept and provided with compatibly deployable end structures. Escrig [2006] has proposed an elegant solution for a deployable dome which consists of polar units placed on a grid of parallels and meridians. Figure 7.8 shows a single slice of such a dome added to part of the open structure, in three consecutive deployment stages. However elegant this solution may be, there is a slight snap-through phenomenon during deployment and, as can be seen in Figure 7.8, severe angular distortion in the end nodes, to allow the deformation of the grid cells. This proposal is discarded because a stress-free deployable solution is to be found. Consider the polar structure from Figure 7.9 (Section 3.3.4) called a lamella dome. It consists of identical polar scissor units placed on a spherical grid with six rhombus-shaped grid cells arranged radially, as can be seen in the top view. There is no snap-through phenomenon during deployment, so it is fully compatible with the stress-free deployment of the main structure. A minor adaptation to the lower scissor units is needed to ensure a completely closed surface. The dashed lines in Figure 7.10 represent the unaltered lower units of the lamella dome, of which the end nodes would float above ground level, if all units were to be identical. This is due to the fact that the lower lamellaunits meet the ground plane at an angle, as opposed to the lower units of the main structure, which are normal to the ground plane. For the simple reason of the longer distance these have to bridge to actually touch the ground, this requires them to be elongated. The width of the compact bundle of bars is not 183

217 Chapter 7 Case study 2 influenced by the longer bars, however, they protrude at the top and at the bottom, therefore increasing the height of the undeployed shape. Full dome (meridians and parallels) Adding a single row to the main structure During deployment there is a snap-through phenomenon and angular distortion of the grid cells Figure 7.8: Adding an end structure based on parallels and meridians to the main structure Figure 7.9: A lamella dome has a stress-free deployment 184

218 Chapter 7 Case study 2 Top view of full lamella In order to form a gapless enclosure, the lower units dome with six modules (red) of the lamella modules have to be elongated until arranged radially they touch the ground Figure 7.10: The main structure is provided with half of an adapted lamella dome The resulting closed structure is shown in Figure m 15.2 m 11.5 m Figure 7.11: CLOSED structure: perspective view and plan view 185

219 Chapter 7 Case study Deployment The deployment process of the open and closed structure is depicted in Figure 7.12 and Figure Figure 7.12: Perspective view, front elevation and top view of the deployment process of the polar barrel vault OPEN structure 186

220 Chapter 7 Case study 2 Figure 7.13: Perspective view, front elevation and top view of the deployment process of the polar barrel vault CLOSED structure Figure 7.14: Proof-of-concept model (half of the structure) in three deployment stages 187

221 Chapter 7 Case study From mechanism to architectural envelope Deployment and kinematic analysis The polar linkage with four units in the span (U =4) can easily be constructed as explained in Section Alternatively, the parametric design approach, described in Section can be used. Either way, the geometry can be fully derived from four design parameters: the span S and rise H r of the base curve, the number of units U and the unit thickness t. Ultimately, values for the semi-bar lengths a and b and the deployment angle θ design are obtained which fully determine the geometry of the linkage in its deployed position. Furthermore, these three parameters suffice to study the deployment behaviour of the polar linkage, as will be shown later. An expression for the span S as a function of the deployment angleθ, including the design constants a, b and U has been derived and is used to determine the value the maximum span is reached during the deployment. θ S max for which Now polar linkages have a peculiar property: during deployment, as the deployment angle θ increases from 0 (compact configuration) to θ design (deployment angle in the fully deployed configuration), the span S will not merely increase. Asθ increases, a maximum span (S max ) is reached, only to decrease again slightly until θ design is reached (S design ). Figure 7.15 shows a typical deployment pattern for a polar linkage. 188

222 Chapter 7 Case study (S max =maximum span) 5 (S design ) Figure 7.15: Deployment sequence of a polar linkage It is useful to know, for practical reasons, what the precise behaviour of the linkage will be during deployment, in terms of variation of the span, and at what point this maximum is reached. In order to locate this point in the deployment, an expression has to be found for the span, as a function of the constants a, b and U and the variable θ. With 0 θ θdesign the deployment ratio ψ can be defined as: θ ψ = with 0 1 θ design ψ (7.1) where ψ = 0 stands for the undeployed configuration and ψ = 1 for the fully deployed configuration. The constants U, a and b are known, while θ, t, unknown and vary throughout the deployment. R in, R e, β, γ and S are We can derive a few relations from Figure Using the cosine rule we can write the following expression: t = a + b 2abcos( π θ ) (7.2) 189

223 Chapter 7 Case study 2 which gives for t : 2 2 t a + b + 2a bcosθ = (7.3) S e b θ a m t η S φ t β γ 2φ R in R e Figure 7.16: Polar linkage in an intermediate deployment stage: 0 <ψ < 1 From Figure 7.16 the following can be readily written: R R t = (7.4) e in + R R i e a b a R b R = e = in (7.5) Thus, by substituting Eqn (7.4) in Eqn (7.5) an expression for R in a t b a Rin is obtained: = (7.6) 190

224 Chapter 7 Case study 2 Further, we know that γ = U β (7.7) and from Figure 7.16 we can see that γ = 4ϕ (7.8) From Eqns (7.7) and (7.8) we can find ϕ ϕ = U β (7.9) 4 Also, m = a sin θ 2 (7.10) and m = R sin β in 2 (7.11) from which an expression for β is found: β = 2sin sin 2 1 a θ R in (7.12) From Section (Eqn (3.3)) we know that S = 2 sin(2ϕ ) (7.13) R in Finally, substituting (7.3), (7.6), (7.9) and (7.12) in (7.13) results in an expression for the span S as a function of U, a, b and θ 1 a θ S = 2 R in sin U sin sin (7.14) Rin 2 a 2 2 with R in = a + b + 2 a b cosθ (7.15) b a Alternatively, an expression can be found for the external span S e as defined in Figure It can be observed that S e S + 2t cosη = (7.16) 191

225 Chapter 7 Case study 2 and π η 2ϕ 2 = (7.17) Substituting (7.17) in (7.16) gives π = S + 2t cos 2ϕ 2 Or in other form: S e S + 2t sin2ϕ S e (7.18) = (7.19) Now, the value for the deployment angle θ(s max ) at which the structure will have reached it maximum span S max, can be found. Using f(x) for Eqn (7.14), then θ(s max ) is found by ( θ ) = 0 f (7.20) When applied to the polar linkage from which the barrel vault is built, the key stages in the deployment sequence are obtained, as shown in Figure 7.18, marked A, B and C. The relation between the deployment angle and the span is represented by the graph in Figure In the undeployed configuration (A) both θ and S are evidently equal to 0. Between stages (A) and (B) the correlation betweenθ and S is virtually linear. The maximum span is reached in stage (B). Then the linkage further deploys until stage (C) is reached, for which it is designed, where a slight decrease in span occurs. Theoretically, the deployment can be continued until the extremities of the linkage meet to complete a circle and the span is reduced to 0, but as an architectural space enclosure this configuration is useless. Alternatively, the external span S e can be used (Eqn 7.19) to describe the maximum span the structure will reach at its extremities during the erection process. This could prove especially useful on sites with limited dimensions, to check whether S e remains within the limits allowed. 192

226 Chapter 7 Case study 2 Relation between the deployment angle (θ) and the span (S) of the polar linkage with U=4 S max S design Span (S) θ(s max ) Deployment angle (θ) θ design A B C Figure 7.17: Graph showing the relation between the deployment angle θ and the span S for the polar linkage with U=4 Stage A represents the undeployed position: both θ and S are equal to 0. The maximum span of approximately 7 m is reached for θ= Stage C is the final deployed position for which the linkage is designed and has θ=135 for a span of 6 m, which are evidently the chosen design values. A: θ=0, S=0 B: θ=111.6, S=701.7 cm C: θ=135, S=600 cm Figure 7.18: Deployment sequence of the polar linkage (U=4) Similar to the approach used for the translational barrel vault, a hinged plate model is made, based on the grid geometry. This eliminates the rotational degree of freedom of the scissors and allows to study the kinematic properties of 193

227 Chapter 7 Case study 2 the grid, in order to determine to what extent constraints have to be added to turn the mechanism into a structure. Figure 7.19 shows the equivalent hinged plate structure, consisting of plates that are mutually connected by a line joint, allowing one rotational degree of freedom. This model represents the mobility of the grid. Figure 7.19: Open barrel vault: scissor structure and equivalent hinged plate structure The complete structure is built from one module, pictured in Figure Because the fold lines that represent the line joints do not share a single intersection point, this module has no mobility. By deduction, the complete structure has no mobility. Therefore, there is no need for additional constraints, other than the one needed to eliminate the rotational degree of freedom of the original scissor mechanism. However, similar to the translational barrel vault (Section 6.4.1, Figure 6.21), all inner nodes touching the ground are fixed by pinned supports, as shown in Figure Figure 7.20: Geometry of a hinged plate module 194

228 Chapter 7 Case study 2 Figure 7.21: Fixing all inner lower nodes to the ground by pinned supports Figure 7.22: Closed barrel vault: scissor structure and equivalent hinged plate structure Figure 7.23: Geometry of the occurring plate modules 195

229 Chapter 7 Case study 2 2 deformed (red) and 1 undeformed 3 modules undeformed (black) modules Figure 7.24: Closed barrel vault: scissor structure and equivalent hinged plate structure Figure 7.22 and Figure 7.23 show the individual modules from which equivalent hinged plate structures can be built for the closed configuration. Because the middle part of the structure (half of the previously discussed open structure) has no mobility, the degrees of freedom of the complete structure are those of the added end structures alone. In Figure 7.24 the undeformed and deformed state of an end structure is shown. When one module is kept immobile, the two other can be simultaneously deformed. This leads to the conclusion that there are two degrees of freedom per end structure, which gives a total of four for the complete geometry. Again, in the deployed position all inner nodes touching the ground are pinned to the ground, effectively removing all mobility, as shown in Figure The resulting open and closed structure, with the tensile surface attached, are shown in Figure 7.28 and Figure 7.29, with a covered area of 66 m 2 and 60 m 2 respectively. The joint solution is similar to that of the translational barrel vault, as described in Section However, this is a simpler version because no rotation of the plate elements (or fins ) of the joint has to be allowed to cope with angular distortion of the grid. 196

230 Chapter 7 Case study 2 Figure 7.25: Fixing all inner lower nodes to the ground by pinned supports Figure 7.26: Joint connecting four bars (no rotation of the fins of the joint around a vertical axis, as is the case for the translational barrel vault in Section 6.4.1, Figure 6.24) 197

231 Chapter 7 Case study 2 Figure 7.27: Top view and perspective view of one scissor unit and its intermediate and end joints Perspective view Top view Covered area 66 m 2 Figure 7.28: Perspective view and top view of OPEN structure with integrated tensile surface 198

232 Chapter 7 Case study 2 Perspective view Top view Covered area 60 m 2 Figure 7.29: Perspective view and top view of CLOSED structure with integrated tensile surface 7.4 Structural analysis Open structure (single curvature) Analogous to case 1, the structural analysis will now be discussed. Without taking any measures to improve structural performance, the calculated section for the polar barrel vault is mm, which, for a covered surface of 66 m 2, translates into a weight-per-square-metre of 15.8 kg/m 2 (Figure 7.30). With steel cable elements (d=6 mm, design strength=1500 MPa) inserted between upper and lower nodes (cfr. Section 6.5.1, Figure 6.45), the section of the scissor bars is decreased to mm, leading to an improved 12.7 kg/m 2, as shown in Figure

233 Chapter 7 Case study 2 Element type Scissor bar T-cable Cross bar Cross cable Section [mm] Weight [kg] Total weight 1046 Weight/m Figure 7.30: Result without any measures taken to improve structural performance Element type Section [mm] Weight [kg] Scissor bar T-cable d=6 7 Cross bar Cross cable Total weight 835 Weight/m Figure 7.31: Improved result by inserting vertical cable ties Triangulation of the quadrangular grid can enhance the structural stability. Therefore, diagonal struts are added inside each quadrangular scissor module, as shown in Figure These crossbars can be added after deployment by fixing them to the nodes, which requires some extra manual labour. Alternatively, these crossbars could be equipped with an intermediate hinge which enables them to be integrated into the structure beforehand and allows them 200

234 Chapter 7 Case study 2 to deploy compatibly with the scissor modules. However, these hinged bars need to be locked at their intermediate hinge in order to fulfil their structural role. Also, they are the longest bars in the structure which makes them susceptible to buckling. The outcome of the analysis of this configuration is a section of mm for the scissor bars, while the crossbars have a tubular round section of mm, both made from aluminium (Young s modulus: 75 GPa, design strength: 180 MPa). The section of the scissor bars decreases to mm and although 100 kg is added through the crossbars, the weight-covered area ratio drops to 8.9 kg/m2 because of the reduced section of the scissor bars. Element type Section [mm] Weight [kg] Scissor bar T-cable d=6 7 Cross bar Cross cable Total weight 589 Weight/m Figure 7.32: Additional diagonal bars triangulate the grid By adding another series of diagonal crossbars, as shown in Figure 7.33, their section is slightly decreased to mm and the section of the scissor bars is further decreased to mm, leading to a weight ratio of 8.5 kg/m 2. This is too little a difference to be called an improvement, especially since a lot more connections have to be made. 201

235 Chapter 7 Case study 2 Element type Section [mm] Weight [kg] Scissor bar T-cable d=6 7 Cross bar Cross cable Total weight 560 Weight/m Figure 7.33: Double diagonal cross bars offer no real advantage structurally Due to the added complexity of the locking mechanisms in case of the foldable crossbars or the additional on-site assembly in case of non-folding bars, another option is chosen. Steel cables can instead be used to triangulate the scissor grid to become tensioned by deployment of the structure. As a result, the section of the scissor bars, and therefore the weight increases in comparison to the versions with cross-bars, but nonetheless this option is chosen because of its lower complexity. The resulting structure weighs 9.9 kg/ m 2 with a section for the scissor bars of mm. The resulting geometry and the weight is given in Figure It is noted that the membrane, if warp and weft directions are placed diagonally over the grid, can simulate the effect of cross-bracing with cables. In Figure 7.35 a summary is given of the resulting stresses, forces and displacements of the analysis of the open structure. The governing load case is ULS 6 (pre-stress + transverse wind). The nodal displacements are small and are comparable to those for the translational barrel vault from Chapter 6. The bar deflections (not shown) do not exceed 1/300 of the bar length. 202

236 Chapter 7 Case study 2 Diagonal cable mm Cable d=6 mm Section hxbxt, dxt [mm] Total weight/section Weight/m 2 [kg/m 2 ] [kg] TREC 100x50x Cable d=6 7 Diagonal cable d=6 30 Total weight: Figure 7.34: Perspective view of case study 2 OPEN structure, with sections after structure design and weight/m 2 203

237 Chapter 7 Case study 2 RESULTS for Case study 2 OPEN structure (with cable elements) STRESS [MPa] S max S max S max Load S max Section [mm] (My) (Mz) Fx/Ax comb. TREC 100x50x ULS 6 Cable d= ULS 6 Cross Cable d= ULS 2 FORCE Section [mm] Fx [kn] My [knm] Mz [knm] Load comb. TREC 100x50x ULS 6 Cable d= ULS 6 Cross Cable d= ULS 6 REACTIONS [kn] FX Load comb. FY Load comb. FZ Load comb ULS ULS ULS 5 DISPLACEMENTS [cm] Ux Load comb. Uy Load comb. Uz Load comb SLS SLS SLS 6 U y U z U x Figure 7.35: Summary of the determining parameters for the strength, stability and stiffness for case study 2 OPEN structure 204

238 Chapter 7 Case study Closed structure (double curvature) After structural analysis and design of the closed structure, without any additional measures taken to improve structural performance, a section of mm for the bars is obtained. This leads to a total weight of 1060 kg, which gives a massive 17.6 kg/m 2 for a covered area of 60 m 2 (Figure 7.36). When now diagonal steel cables (d=6 mm, design strength=1500 MPa) are added to cross-brace the quadrangular scissor modules, the structural performance is enhanced. The section is reduced to mm which gives, for a total weight of 798 kg and a covered area of 60 m 2, a ratio of 13.3 kg/m 2. Element type Structure bar Cable d=6 Section [mm] Weight [kg] Total weight 1060 Weight/m Figure 7.36: Main structure and additional end structures with no additional measures to improve structural performance Form Figure 7.38 it can be seen that ULS 3 (pre-stress + longitudinal wind + snow) is the governing load case. Again, displacements and deflections (not shown) are within limits and are comparable to those of previous calculations. 205

239 Chapter 7 Case study mm Diagonal cable Cable d=6 mm Section hxbxt, dxt [mm] Total weight/section Weight/m 2 [kg/m 2 ] [kg] TREC 120x60x Cable d=6 16 Diagonal cable d=6 27 Total weight: Figure 7.37: Perspective view of case study 2 CLOSED structure:, with resulting sections after structure design and total weight 206

240 Chapter 7 Case study 2 RESULTS for Case study 2 _ CLOSED structure (with cable elements) STRESS [MPa] S max S max S max Load S max Section [mm] (My) (Mz) Fx/Ax comb. TREC 120x60x ULS 3 Cable d= ULS 4 FORCE [kn] Section [mm] Fx My Mz Load comb. TREC 120x60x ULS 3 Cable d= ULS 4 REACTIONS [kn] FX Load comb. FY Load comb. FZ Load comb ULS ULS ULS 4 DISPLACEMENTS [cm] Ux Load comb. Uy Load comb. Uz Load comb. 1 SLS SLS SLS 3 U x U z U y Figure 7.38: Summary of the results for the structural analysis of case study 2 CLOSED structure 207

241 Chapter 7 Case study Conclusion This chapter was concerned with the design of a deployable barrel vault with polar and translational scissor units on an orthogonal two-way grid. First, an open barrel vault with four units in the span has been designed (Figure 7.39), similar in shape and dimensions to case study 1, as described in Section 5.2. For this case study, the construction methods and the geometric design methods, as described in Section 3.2, have been put into practice, to obtain a basic, single curvature stress-free foldable solution. Figure 7.39: Case 2: OPEN structure Despite the simplicity of the design, providing the barrel vault with doubly curved end structures to form a fully closed envelope, while these added substructures are required to be stress-free foldable, has proven troublesome. A clever solution proposed by Escrig [2006] consisting of polar scissor units, placed in parallels and meridians, is characterised by an unwanted bi-stable deployment. An innovative solution was found in the lamella dome (Section 3.3.4). It has been shown that, with a slight modification to the bottom most scissors, half of such a dome can be connected to a polar barrel vault and demonstrate a compatible, stress-free deployment (Figure 7.40). Based on the geometric design method proposed in Section 3.3.2, the equations, which allow the study of the evolution of the span throughout the deployment, have been derived. These were then used to predict the maximum span the configuration reaches during deployment. This parameter has pro- 208

242 Chapter 7 Case study 2 vided information on the minimum space required for deployment and can prove useful during erection on a site with limited dimensions. Figure 7.40: Case 2: CLOSED structure The mobility of the system has been discussed by means of an equivalent hinged-plate model, a method proposed in Section It was found that the open barrel vault is a one-degree-of-freedom mechanism, while each added end structure has two D.O.F., leading to a total of five D.O.F. for the fully closed, doubly curved structure. It was suggested to fix all lower nodes to the ground by pinned supports, effectively eliminating all mobility from the structure. To prevent the quadrangular grids from skewing due to a lack of in-plane stability, several configurations were provided with either cross-cables or crossbars. From the structural analysis it was found that the configuration with double cross-bars provided the lightest solution (8.5 kg/m 2 ). However, on practical grounds, the slightly heavier solution with the cross-cables was opted (9.9 kg/m 2 ). Although this structure has larger sections for the scissor members, the connection of four extra cables to a joint, as opposed to four extra bars, is preferred because of simpler connection and a more compact undeployed configuration. The structural analysis of the fully closed configuration has led to a structure with a weight ratio of 13.3 kg/m 2. The increase in section is attributed to the longer bars of the doubly curved end structures and the resulting higher buckling sensitivity. A uniform section has been chosen for all bars in the structure and, hence, an increase in weight was observed. 209

243 Chapter 7 Case study 2 210

244 Chapter 8 Case study 3 Chapter 8 Case study 3: A Deployable Bar Structure with Foldable Articulated Joints Figure 8.1: Foldable bar structure based on the geometry of foldable plate structures 211

245 Chapter 8 Case study Introduction In this chapter a novel concept for a mobile shelter system is developed, based on the geometry and kinematic behaviour of foldable plate structures, of which the architectural application has been described in Section 2.3. Evaluated by Hanaor [2001], plates are found to require an substantial thickness because they are subject to compression and bending. The resulting overall weight may be higher than a structure surfaced with a membrane. Storage efficiency is low due to the plates dimensions and thickness and waterproofing of joints between individual plates poses a problem. Therefore, as an architectural envelope for the third case study, a textile membrane is proposed, as it provides a flexible, lightweight, translucent and continuous shelter surface. The membrane is held up and tensioned within the foldable bar structure, acting as the primary load bearing structure. As this case study is based on a foldable plate structure, the geometric design method proposed in Section 4.3 is used to design a single curvature shape (barrel vault, as shown in Figure 8.1), similar to the shape of case study 1 and 2. Also, a double curvature configuration (foldable dome) consisting of the same plate elements is designed. Further, it is shown how these basic shapes can be connected to form new, alternative configurations [De Temmerman, 2006b]. Next, it is explained how the transition is made from plate geometry to bar structure. An innovative foldable articulated joint, serving as a connector for the bars, is proposed [De Temmerman, 2006a]. It is demonstrated that the barjoint system preserves the kinematics of the plate structure it is derived from, even when certain bars are discarded from the structure. An insight in the mobility is offered [Foster1986/87], since these mechanisms typically possess multiple degrees of freedom [Kool, 2006], which are to be removed after deployment to stabilise the structure. The structural feasibility of the concept is assessed in a preliminary structural analysis, where several 212

246 Chapter 8 Case study 3 configurations are tested and mutually compared. Key aspects regarding the architectural use are discussed. Figure 8.2: Typical foldable plate structure 8.2 Description of the geometry As described in Chapter 4 foldable plate structures consist of triangular plates, hinged together along their edges by continuous joints which provides them with one rotational degree of freedom. These hinges allow the structure to fold according to a pattern of mountain and valley folds. By unfolding the initially flattened plate linkage, a three-dimensionally expanding corrugated surface arises, as shown in Figure 8.2. In Chapter 4 an overview is given of some typical configurations and the equations needed for parametric design are presented. For clarity, the key design parameters will be concisely explained. Although in the end a foldable bar structure is to be designed, the underlying geometry is no different from that of the plate structure it is derived from. Therefore the geometry will be described in terms of plates and all previously determined design parameters and equations remain valid. In Figure 8.3 the parameters are shown which completely determine the geometry of a foldable plate structure. This particular shape is called regular. The key parameters which determine the overall geometry are: the number of 213

247 Chapter 8 Case study 3 plates (p) in the span, the apex angle of the plates (β), and the deployment angle (θ) for the fully deployed configuration. What is called the fully deployed configuration corresponds with the position for which the edges of the plates nearest to the ground are perfectly horizontal, i.e. they touch the ground (Figure 8.2). In elevation view the silhouette of the structure is then a perfect semi-circle. The impact additional parameters such as the plate length (L), the span (S) and the module width (W) have, is limited to a variation of the dimensions of the structure. Fully deployed position Fold pattern Figure 8.3: Design parameters for a basic regular foldable plate structure. 214

248 Chapter 8 Case study 3 As stated in the introduction to the case studies (Chapter 5) a circular arc with a radius of 3m is used as base curve. This two-dimensional profile can be seen as a projection in the vertical plane of a three-dimensional plate geometry with five plates in the span (p=5). Taking a projected profile as a starting point, implicates that there is an endless collection of plate geometries which will fit this requirement. This is illustrated in Figure 8.4: the two pictured fully deployed configurations are both valid solutions for p=5, but have a different apex angle β and a different width W. Figure 8.4: For a chosen number of panels p the apex angle β can be altered at will, only affecting the width of the structure However, not all values for the apex angle give rise to a valid, foldable configuration which in addition can be folded into a compact configuration. It has been shown that the minimal apex angle for a five-plate linkage which is still compactly foldable is β min =90 while the maximum value is β max =135. This can also be seen in the graph from which, for a certain chosen β, the appropriate deployment angle θ can be derived for linkages with p=5 (the graph is drawn up using Eqn (4.13) from Section 4.3.1). 215

249 Chapter 8 Case study 3 Relation between the apex angle (β) and the deployment angle (θ) in the fully deployed configuration for p=5 70 Deployment angle (θ) [ ] Apex angle (β) [ ] Figure 8.5: Graph showing the relation between the deployment angle θ and the apex angle β in the fully deployed configuration for p=5 Because the 90 solution achieves - for a certain number of plates and connections - the greatest expansion (and therefore the greatest width), this seems a logical choice, but there are other factors that play a part in choosing a suitable geometry. The equations for designing circular structures have been proposed as well. It has also been shown that a common module geometry can be found which can be used in both regular and circular configurations, leading to a higher uniformity of plate elements. The amount of sectors arranged radially in the circular structure (q) can be freely chosen. Figure 8.7 shows such a sector of which there are six in this particular example, which will be the amount used for this case study. With chosen values p=5 and q=6, solving Eqn (4.13) and Eqn (4.27) simultaneously results in an apex angle β=109.2 and an accompanying deployment angle θ=54.3 (marked in red in the graph from Figure 8.5). This means that the original value for β has risen from 90 to 109.2, which translates in a reduced width in the deployed configuration. But the benefit is 216

250 Chapter 8 Case study 3 that a plate geometry is obtained which can be used to form both regular and circular configurations with the same plate elements and that these can be mutually connected to form new alternative configurations, as shown in Figure 8.9. Fully deployed Compactly folded Fold pattern Figure 8.6: The resulting regular geometry for the case study: two extreme deployment states and the fold pattern Figure 8.7: Top view and a perspective view of a circular plate geometry with six sectors arranged radially 217

251 Chapter 8 Case study 3 c a b Fully deployed Compactly folded c c a b Fold pattern Figure 8.8: The resulting circular geometry for the case study: two extreme deployment states and the fold pattern Figure 8.9: A combination of a regular and a circular geometry 218

252 Chapter 8 Case study 3 In Figure 8.10 the dimensions are given for three obtained geometries, which all share the same interior height at their highest point: 4.25 m. 8.5 m 8.5 m 8.5 m 10.4 m 10.8 m Open structure Closed structure Dome Figure 8.10: Dimensions in plan view of the shapes 8.3 From plate structure to foldable bar structure Now that the geometry has been fully determined, the transition from foldable plates to foldable bars can be made. The goal is to devise a foldable bar structure with a kinematic behaviour identical to that of its similar counterpart, the foldable plate structure. So instead of plate elements as cladding components, a continuous membrane supported by a skeletal structure will be used to form the architectural envelope. Figure 8.11: A foldable plate structure (p=7) and its similar counterpart, a foldable bar structure A first way of obtaining the bar structure is to cut away the middle sections of the plates until only thin borders remain, which represent the bars. Where 219

253 Chapter 8 Case study 3 there were originally two neighbouring plates meeting edge to edge, there are now two parallel bars, one on each side of the fold line. These double bars, however, lead to an inefficient use of material (Figure 8.12). Figure 8.12: Pattern 1: double bars present Therefore, as many superfluous bars as possible are removed, without affecting the original kinematics of the system. Of each pair of double bars, one is consequently discarded, as shown in Figure Figure 8.13: Pattern 2: double bars removed Additionally, provided that the bars are appropriately connected, some of the middle bars (valley between two triangles) can be left out, without altering the kinematic behaviour, provided that the two remaining V-shaped legs are joined by a fixed connection at the apex. The reason for discarding these bars, is to be able to incorporate a membrane, hung inside from the nodes. If the bars would remain in place, the membrane would be unable to reach its anticlastic shape inside the V-shaped folded rhombuses. The resulting bar pattern 220

254 Chapter 8 Case study 3 is shown in Figure 8.14, which is the type that will be used for the open structure. Figure 8.14: Pattern 3: double bars and diagonal bars removed, without affecting the original kinematic behaviour By connecting the bars by means of a custom foldable joint, shown in Figure 8.15 the bar-node system demonstrates the same kinematic behaviour as the plate structure it is originally derived from. It consists of six parts connected by hinges in such a way that the folding and unfolding is unhindered by the thickness of the bars (Figure 8.16). In this way a lightweight articulated bar structure is obtained which behaves like a foldable mechanism. Figure 8.15: Foldable 3 D.O.F.-joint derived directly from the fold pattern, therefore mimicking its kinematic behaviour 221

255 Chapter 8 Case study 3 Figure 8.16: Deployment sequence for the foldable joint: from the undeployed to the fully deployed position By appropriately connecting bars and foldable joints according to the fold pattern previously discussed, a foldable barrel vault is formed, as shown in Figure All joints are identical, except for those at ground level, which are slightly modified versions. The dome-shaped structure from Figure 8.8 and the combined geometry shown in Figure 8.9 can both be built in a similar manner, composed from the same elements. Figure 8.17: The (regular) open structure complete with bars and joints: perspective and top view side and front elevation 222

256 Chapter 8 Case study Figure 8.18: Detailed view of bars and three variations of foldable joints occurring in the structure Deployment The deployment process of the open structure is shown in Figure During deployment almost all the expansion occurs in the longitudinal direction (perpendicular to the span). In the transverse direction (parallel to the span) the variation of the geometry is not as significant. In the undeployed position, the outermost nodes, marked in Figure 8.19, touch the ground. These remain in contact with the ground throughout the deployment and can therefore be used to attach wheels to facilitate the deployment on even terrain. 223

257 Chapter 8 Case study 3 Figure 8.19: Deployment sequence for the open structure perspective view, front elevation and top view Figure 8.20: Proof-of-concept model of the regular structure (with scissors) in four stages of the deployment The foldable dome is deployed in a circular manner, by rotating all sector around a vertical axis through the highest point. The deployment process of the circular structure is shown in Figure

258 Chapter 8 Case study 3 Figure 8.21: Deployment sequence for the dome structure perspective view, front elevation and top view Figure 8.22: Proof-of-concept model of the foldable dome (with additional scissor units) in six deployment stages 225

259 Chapter 8 Case study 3 The deployment of the combined structure is similar to that of its regular and circular sub-structure. First the regular structure is deployed while the circular modules are kept compacted, as shown in Figure 8.23 and Figure 8.24 (stage A, B and C). Then, the circular modules are deployed until the structure becomes a fully closed envelope (stage D, E and F). D E F A B C Figure 8.23: Deployment sequence for the closed structure: 1 regular module + 2 semi-domes 226

260 Chapter 8 Case study 3 Top view A B C D E Figure 8.24: Six stages in the deployment of the closed structure (top view) An alternative way of translating the fold pattern into a bar structure has been looked at. As opposed to the previous method, a bar is placed exactly on every fold line of the pattern. The bars are connected by custom kinematic joints which allow all necessary rotations during deployment. Some bars are grouped into triangles to make connections simpler. The resulting geometry is shown in Figure A model has been constructed to evaluate the mobility of the mechanism which behaves in the desired way. However, fully folding or unfolding proves difficult because obstruction occurs in the joints, and therefore this solution cannot be folded as flat as the configuration with the foldable joint. Consequently, this proposition will not be further investigated. F 227

261 Chapter 8 Case study 3 Figure 8.25: Kinematic joint allowing all necessary rotations (3 D.O.F.) and the resulting bar structure Proof-of-concept model to verify the mobility Ultimately the foldable bar structure has to provide shelter by means of a tensile surface. Two approaches are proposed, depending on whether the membrane is integrated in advance or not. Because the deployment of the structure is characterised as folding, a membrane can well be integrated beforehand by attaching it to the nodes. Then, it is unfolded along with the structure and brought under tension as the structure reaches its fully deployed position. In Figure 8.26 the integrated membrane is shown in both the undeployed and deployed position. Another option is to deploy the bar structure first, after which the membrane can be pulled up to the nodes by cables, until it becomes sufficiently tensioned. 228

262 Chapter 8 Case study 3 Figure 8.26: Integration of the membrane beforehand by attaching it to the nodes Side elevation and perspective view of the undeployed and deployed position Alternative geometry As an alternative configuration, a right-angled structure as described in Chapter 4 has been designed using the same approach. Such a geometry, shown in Figure 8.27, can be very compactly folded, provided that the design parameters are appropriately chosen. Also, the vertical sides are quadrangular, which could make it easier to provide access to the structure. As a downside, the higher variation of the apex angles, when compared to the regular structure, translates in three different foldable joints, with another two variations at ground level. But the main concern is the relatively low structural thickness, due to the gentler corrugation of the surface. This is because a right-angled structure has a larger deployment angle θ and it is this increased deployment range that gives the roof its higher slenderness in the deployed configuration, making it susceptible to sagging or even snapping through. A model (Figure 229

263 Chapter 8 Case study ) with aluminium bars and resin connectors (not to scale) has been built to test the deployment behaviour, which proves to be as required. A B C A B C Figure 8.27: Right-angled geometry with its own set of joints Figure 8.28: Deployment sequence of a concept model of a right-angled structure with aluminium bars and resin connectors [De Temmerman, 2006a] Right-angled and regular structures with a type B pattern (Section 4.2) and with p=5 have identical edges in the vertical plane. This implies that they can be linked together along that edge to form a chain of structures. However, the 230

264 Chapter 8 Case study 3 difference in shape between the regular and the right-angled structure in the undeployed configuration prevents a simultaneous deployment. Therefore, they are to be coupled after deployment (Figure 8.29). Figure 8.29: Several regular and right-angled structures connected together after deployment Kinematic analysis The devised articulated bar system and the plate system it is derived from, are mechanisms with multiple degrees of freedom. The mobility of foldable plate linkages can be determined from the layout of the plates. In chapter 4, the difference between a type A and type B pattern has been explained. For all patterns similar to pattern A the formula for the total number of degrees of freedom is derived by Foster [1986/87]: p + 2 m + 2 (8.1) where p stands for the number of plates in one module and m for the number of modules in the pattern. Analogous to Eqn (8.1), a similar formula can now be derived for patterns of type B: p + 2 m + 4 (8.2) 231

265 Chapter 8 Case study 3 From these two expressions it is immediately apparent that patterns of type B posess, for an identical number of plates and modules, a higher mobility. Because we want to limit the mobility of the mechanism, in order to gain control over the deployment, the patterns used in this case study are therefore chosen to be of type A. Applying Eqn (8.1) to the open structure (Figure 8.19) with p=5 and m=6, the mechanism has 19 D.O.F. Now it is necessary to determine what constraints have to be added in order to turn the linkage into a structure. Pinning eight lower nodes of the linkage to the ground adds 24 constraints (8 3 translations removed). This ensures stability for the structure which is now 5 times overconstrained. Now, for comparison, the number of constraints for the articulated bar linkage from Figure 8.25 is determined. Eqn (8.3) for pin-jointed truss systems gives the degree of statical determinacy (R) in terms of the number of bars (b), the number of joints (j) and The number of restraints (r) on the structure [Calladine, 1978]: R = b ( 3 j r) (8.3) When the pattern for the open structure is built up using the structural system shown in Figure 8.25, the number of bars b=53 and the number of joints j=24. With again 8 pinned supports to constrain the structure (8 3 constraints), the statical determinacy R=-5. This means the structure is 5 times overconstrained and therefore stable. These identical results demonstrate the similarity between a foldable plate system and the articulated bar structure in terms of kinematic behaviour. It can be asked to what extent the predicted degree of mobility represents the actual kinematics of the system. Often, formulas such as Kützbach-Grübler [Hiller, 1991] for predicting the mobility of a system based on counting the number of joints and links, fail due to singularities occurring in mechanisms with high symmetry. Therefore Kool [2006] has devised a method for deter- 232

266 Chapter 8 Case study 3 mining the mobility of foldable plate structures which uses a joint matrix for each kinematic loop in the structure. The number of kinematic loops can be easily determined with Eqn (8.4), attributed to Euler. N loops = N joint s N links + 1 (8.4) Here, N joints is the number of continuous joints between plate elements, while N links is the total number of plates in the configuration. In short, once the number of loops has been established, the mobility can be predicted with a structure matrix. The columns of the joint matrix are the line vectors representing the hinges between the plates. The joint matrices can be assembled into a larger structure matrix. The folding patterns can be calculated by using a svdcomposition of this matrix and the zero singular values which are indicative for the rank-deficiency of the matrix correspond to the number of possible patterns. For a detailed explanation of the method please refer to [Kool, 2006]. Applying this approach to the foldable plate configuration from Figure 8.7, the number of loops is found using Eqn (8.4): N loops = =2. These two loops are shown in Figure Figure 8.30: The two loops and their common fold line By applying the mentioned approach, five degrees of freedom are found for the foldable plate configuration from Figure 8.30: two in each loop and one angle around the common fold line. Practically, this means that, when the an- 233

267 Chapter 8 Case study 3 gle around the fold line is determined (deployment angle θ), another angle in each loop can be chosen to place the bottom edges on the ground. Finally, a second angle in each loop can be chosen to force the side edges in a vertical plane in order to be able to mutually connect several modules. Possibly, this high degree of mobility of the mechanism can cause undesired movement. Therefore, incorporating a scissor linkage a one D.O.F-mechanism could be a solution. Figure 8.30 and Figure 8.31 show the deployment of a dome and a regular structure, both with an integrated compatible scissor linkage. A foldable dome with a compatible integrated scissor linkage only the upper nodes of the scissors are connected to the dome Figure 8.31: A foldable open structure with a compatible integrated scissor linkage one bar of each scissor unit doubles up as an edge the foldable bar structure 234

268 Chapter 8 Case study Structural analysis Open structure (single curvature) The bars and foldable joints are modelled in ROBOT as shown in Figure The foldable articulated joints are entered as six separate entities, each consisting of fixed bar elements. The bars, in turn, are connected to these joint elements by fixed connections. By releasing the rotations, represented by the axes in Figure 8.32, the joints have the same kinematic behaviour as the actual joint model, shown in Figure Bars Joint element Figure 8.32: Top view and perspective view of the finite element model of the foldable joint from Figure 8.15 (hinges are represented by dashed lines) The load combinations, specified in Chapter 5, are applied to the model as nodal forces. These load vectors are distributed over the six elements of the nodes. The joints elements are excluded from the design process. Therefore, their size is limited and they are awarded a very large section together with a high stiffness and design strength. Again, some variations of the configurations have been structurally analysed. First, a configuration where the middle bars of each rhombus-shaped module are still present, has been calculated. All lower nodes are fixed to the ground by pinned supports. The resulting section and the weight ratio is shown in Figure As material for the bars, aluminium is chosen (Young s modulus: 75 GPa, design strength: 180 MPa). This results in a tubular round section of mm, with a weight ratio of 5 kg/m

269 Chapter 8 Case study 3 Element type Section [mm] Weight [kg] Structure bar Strut Total weight 375 Weight/m 2 5 Figure 8.33: Model with the middle bars in the rhombus-shaped modules still present When cross-bars are added to the top layer of the system, the section remains unaltered. Instead, the added bars only increase the weight which has mounted to 403 kg, leading to a weight ratio of 5.4 kg/m 2 (Figure 8.34). Element type Section [mm] Weight [kg] Structure bar Strut Total weight 403 Weight/m Figure 8.34: Same model as in Figure 7.33, but with cross-bars The configuration from Figure 8.35 is the one which is presented in the previous sections. It has no middle bars in the rhombus-shaped modules and the bars are joined, in pairs, at their apex angle by a fixed connection. This is also shown in Figure 8.14, Figure 8.15 and Figure The result is again a round tubular section of mm, but with fewer bars, which gives a total weight of 296 kg, resulting in a weight ratio of 3.9 kg/m 2, which is the lightest solution of all proposals. 236

270 Chapter 8 Case study 3 Element type Section [mm] Weight [kg] Structure bar Strut Total weight 296 Weight/m Figure 8.35: Bars are grouped in pairs and joined by a fixed connection in their apex angle When struts are added, the section isn t decreased. Adding struts only adds to the weight, which mounts up to 335 kg, resulting in a weight ratio of 4.5 kg/m 2 (Figure 8.36). Element type Section [mm] Weight [kg] Structure bar Strut Total weight 335 Weight/m Figure 8.36: Adding struts again only increases the weight, while the section remains identical Figure 8.37 shows a summary of the results of the structural analysis of the proposed configuration. For the strength, the governing load case is ULS 5 (pre-stress + snow). The stresses are quite low and buckling sensitivity is the determining phenomenon here. Also, the snow action (in SLS 5) is the governing load for the displacements, which stay within acceptable bounds. Deflections (not shown here) do not exceed 1/

271 Chapter 8 Case study 3 RESULTS for Case study 3 _ OPEN structure STRESS [MPa] S max S max S max S max Load Section [mm] (My) (Mz) Fx/Ax comb. TRON 88x ULS 5 FORCE Section [mm] Fx [kn] My [knm] Mz [knm] Load comb. TRON 88x ULS 5 REACTIONS [kn] FX Load comb. FY Load comb. FZ Load comb ULS ULS ULS 5 DISPLACEMENTS [cm] Ux Load comb. Uy Load comb. Uz Load comb. 0.6 SLS SLS SLS 5 Uy Ux Uz Figure 8.37: Summary of the determining parameters for the strength, stability and stiffness for case study 3 OPEN structure 238

272 Chapter 8 Case study Closed structure (double curvature) Element type Section [mm] Weight [kg] Structure bar Total weight 208 Weight/m Description: Foldable dome: bar structure without middle bars and no struts Figure 8.38: Resulting section and weight for the foldable dome First, the proposed dome structure is analysed (Figure 8.38). The same results as for the open structure are obtained: tubular round aluminium sections measuring mm. The weight is 208 kg leading to a weight ratio of 4.5 kg/m

273 Chapter 8 Case study 3 Section hxbxt, d[mm] Total weight/section [kg] Weight/m 2 [kg/m 2 ] TRON 88x Total weight: Figure 8.39: Perspective view of case study 3 CLOSED structure with sections after structure design and total weight Next, the closed structure is analysed (Figure 8.39). Again, the same sections are obtained and a comparable result in terms of weight ratio is obtained: 5 kg/m 2. Figure 8.40 gives an overview of the strength, stability and displacements. ULS 5 (pres-stress + snow) is the governing load case for the strength and stability. Displacements are still acceptable without harming the serviceability of the structure, but have risen in comparison with the open structure. Also, deflections (not shown) have risen (compared to the open structure) to 1/

274 Chapter 8 Case study 3 RESULTS for Case study 3 CLOSED structure STRESS [MPa] S max S max S max S max Load Section [mm] (My) (Mz) Fx/Ax comb. TRON 88x ULS 5 FORCE [kn] Section [mm] Fx My Mz Load comb. TRON 88x ULS 5 REACTIONS [kn] FX Load comb. FY Load comb. FZ Load comb ULS ULS ULS 5 DISPLACEMENTS [cm] Ux Load comb. Uy Load comb. Uz Load comb. 0.6 SLS SLS SLS 5 Uz Ux Uy Figure 8.40: Summary of the determining parameters for the strength, stability and stiffness for case study 3 CLOSED structure 241

275 Chapter 8 Case study 3 Transverse forces and bending moments are high in the elements which constitute the joints (bodies of the foldable articulated joint) and are low in the bars. This structural behaviour is comparable with that of a truss with fixed nodes (no releases for bars in the nodes). The fact that the transverse forces and the shear stresses are high in the nodes requires further, detailed analysis in which the joints are structurally designed. The governing phenomenon is buckling of the bars. The relatively great length of the bars especially the inverted V-shapes at the front and back of the open structure makes them susceptible to buckling. This explains the relatively low stresses in the bars, which leads to the conclusion that the stability is determining for the design, and not the strength. 8.5 Conclusion In this chapter a novel concept for a rapidly erectable shelter system, based on the geometry and kinematic behaviour of foldable plate structures, has been proposed. First a single curvature foldable plate geometry (barrel vault Figure 8.41) has been designed using the geometric design method proposed in Section A five-plate geometry was chosen, since its sectional profile is similar to that of case study 1 and 2, as explained in Section 5.2. Also, a double curvature shape (dome Figure 8.42) has been designed using the same approach. Figure 8.41: Case 3 OPEN structure 242

276 Chapter 8 Case study 3 Due to the plate uniformity, achieved by using the equations derived in Section 4.3, it has been shown possible for both these shapes to be combined into a closed doubly curved foldable geometry (Figure 8.43). It has been shown that the transition from plate structure to bar structures can be made, while preserving the kinematic behaviour of the original plate system. To simplify the structure, it was found that some excess bars could be discarded from the configuration, while still maintaining the same deployment behaviour. Figure 8.42: Case 3 Foldable DOME structure For connecting the bars, an innovative foldable articulated joint has been developed. Through a proof-of-concept model, it has been proven that the barjoint system preserves the kinematic properties of the plate structure it is derived from. By counting the number of loops, it was found that a five-plate module has five D.O.F. s. To ensure a controlled deployment, it has been proposed to strategically incorporate scissor mechanisms, turning the configuration in a single-d.o.f. mechanism. 243

277 Chapter 8 Case study 3 Figure 8.43: Case 3 CLOSED structure From the structural analysis it was found that transverse forces and bending moments were high in the joints and low in the bars. Buckling of the bars has been recognised as the governing phenomenon. Therefore, other configurations with a higher number of units (for the same span), leading to shorter bars, should be investigated. Currently, all three configurations (joints excluded) have achieved a weight ratio of approximately 5 kg/m 2. While the proposed concept seems promising and the architectural and kinematic feasibility have been demonstrated, more profound and detailed structural analysis is needed on an integrated model consisting of the bar structure, the joints and the tensile surface. 244

278 Chapter 9 Case study 4 Chapter 9 Case study 4: A Deployable Tower with Angulated Units Figure 9.1: Design concept for a tensile surface structure with a deployable central tower 245

279 Chapter 9 Case study Introduction This chapter is concerned with the design of the fourth and final case study, which is an innovative concept for a deployable hyperboloid tower with angulated scissor elements. Its purpose is two-fold: serving as a mast for a tensile surface structure while acting as an active element during the erection process. Angulated scissor elements have been extensively investigated and this has yielded a wide range of concepts and applications in the field of deployable scissor structures [Hoberman, 1991], [You & Pellegrino, 1996, 1997], [Rodriguez & Chilton, 2003], [Jensen, 2004]. Although primarily intended for radially deployable closed loop structures, it is shown in this chapter that angulated elements can also prove valuable for use in a linear three-dimensional scissor geometry. It is explained how angulated elements offer, for the proposed application, an advantage over polar units in terms of deployment behaviour and a reduction of the number of connections. A comprehensive geometric design method is proposed for which the equations, expressed in terms of relevant design parameters, are derived, enabling the design of any hyperboloid shape. Two different approaches are introduced, offering the designer a choice between designing the undeployed or the deployed configuration. As the deployment is an integral part of the design, an insight in the relationship between the geometry of the structure and its subsequent kinematic behaviour is offered. The mobility of the system is assessed through use of the equivalent hinged-plate model, as introduced for case study 1 (Section 6.4.1) and case study 2 (Section 7.3.1). An innovative design for a joint is proposed, allowing all necessary rotations between subsequent elements. Finally, the tower is structurally analysed under wind and snow action and conclusions are drawn on the structural feasibility of the proposed design. 246

280 Chapter 9 Case study A concept for a deployable tower This case study is quite different from the previous three. Although a deployable bar structure and a tensile surface are once again involved, the way it provides an architectural enclosure is what sets it apart. For the barrel vaults of case studies 1, 2 and 3 the bar structure and the membrane surface follow the same curvature (monoclastic in the case of the open structure and synclastic for the closed structures). In this case on the other hand, the scissor structure is a central vertical linear element, used to hold up the anticlastic membrane canopies at one of their high points. The question was raised whether it would be possible to design such a deployable tower for a temporary tensile structure and to use it as an active element during the erection process. In addition, the pantographic tower allows visitors to access several platforms to enjoy the views, under or above the different membrane elements (Figure 9.1). The proposed concept is based on a design by The Nomad Concept [2007], a company active in the field of tensile surface structure design. Figure 9.2 shows the original (undeployable) tower (or mast), consisting of several modules which are assembled and dismantled on-site by stacking them vertically, for which a lifting device is needed. After assembly the membrane would have to be attached to the top, after which the pre-tension in the membrane can be introduced. 247

281 Chapter 9 Case study 4 Figure 9.2: Mobile structure with membrane surfaces arranged around a demountable central tower ( The Nomad Concept) By making the tower deployable, all connections could be made on ground level, while the mechanism is in its undeployed, compact state, therefore eliminating the need for additional lifting equipment. After connections between the membrane elements and the tower have been made, the mechanism is deployed until the required height is reached and the membrane elements become tensioned. The tower could be deployed to such an extent that a sufficient amount of pre-tension is introduced in the membrane, ensuring the ability to withstand external loads. Since the tower is basically a mechanism additional bracing is needed after full deployment to turn it into a load-bearing structure. 248

282 Chapter 9 Case study 4 Figure 9.3: The top of the tower is accessible to visitors, allowing them to enjoy the view ( The Nomad Concept) 9.2 Description of the geometry The deployable tower is horizontally divided in several modules, which are closed-loop configurations of identical hoberman s units or otherwise called angulated SLE s. Figure 9.4 shows an example of a tower with triangular modules, of which three are stacked vertically. Because every bar in the structure is identical, the sums of the semi-lengths are evidently constant, therefore, the geometric deployability constraint is automatically fulfilled. The dimensions of the structure are shown in Figure 9.4, Figure 9.5 and Figure 9.6. The tensile surfaces are identical and measure 10 m along their longest diagonal. The top of the second module, at which the membrane elements are attached, is located at 5.2 metres above ground level. The other high point of the membranes is held 4 m above ground by additional masts. 249

283 Chapter 9 Case study 4 4 m 5.2 m Figure 9.4: Side elevation of the tower and canopy 10 m Figure 9.5: Top view of the structure showing the three tensile surfaces arranged radially around the central tower 250

284 Chapter 9 Case study 4 Figure 9.6: Dimensions of the tower and a single angulated bar It could be argued that a tower with a broad base and a narrow top can equally be built with polar units with decreasing size as they are located nearer to the top. In Figure 9.7 two linkages one with angulated elements, another with polar units - are shown, with identical height and width, but with varying number of units U and different bar lengths. Using the angulated elements offers an advantage: while the linkage with angulated elements is built from only 3 SLE s with 11 hinges and nodes, the equivalent polar mechanism needs 8 units with 26 connections to reach a similar deployed geometry. The effect that the angulated elements have on the modules is that, during deployment, the top of a module becomes narrower than its base. The radius of the top of a certain module becomes equal to the radius of the base of the next, higher located module. This means that the narrowing effect is enhanced and passed on through the mechanism, from module to module, from bottom to top. 251

285 Chapter 9 Case study 4 Linkage with angulated SLE s: Equivalent linkage with polar SLE s: Number of SLE s: n=3 Number of hinges and end nodes: 11 Number of SLE s: n=8 Number of hinges and end nodes: 26 Figure 9.7: Comparison between a linkage with angulated SLE s and its polar equivalent The dimensions of the individual bars of the scissor units are such, that the horizontal projection of b is equal to a, as shown in Figure 9.8. The imaginary vertical axes connecting the end nodes of the bars can act as fold lines, used to further flatten the linkage. Therefore, the modules are cut open along one 252

286 Chapter 9 Case study 4 fold line, after which the whole can be flatly folded for easier transport. Such a fold sequence is shown in Figure 9.9. This way of further compacting is presented as an option and could be ignored, provided that the dimensions in the undeployed state are kept reasonable. a b Figure 9.8: Imposed condition on the length of the semi-bars a and b (a<b), in order to make the linkage foldable along the vertical axis Step 1 (Compacted for transport) Step 2 Step 3 Step 4 (ready to deploy) Figure 9.9: Initial unfolding of the compacted linkage to its polygonal form 253

287 Chapter 9 Case study 4 The deployment sequence of the tower is presented in Figure 9.10, showing a top view and a side elevation for each stage. The maximum deployment is reached when the upper end nodes of the top module meet in one point. Step 5 (Undeployed) Step 6 Step 7 Step 8 Step 9 Step 10 (Fully deployed) Figure 9.10: Six stages in the deployment of a hexagonal tower: elevation and top view 254

288 Chapter 9 Case study Geometric design Two different design approaches are presented here, depending on what the determining design criteria are: The first method from unit to linkage allows to design the mechanism in its compact, undeployed configuration, after which it is deployed into the desired position. This offers control over the geometry of the scissor elements themselves, rather than the overall final deployed result. This is a straightforward, linear design approach that allows each subsequent parameter to be derived from the previous one. The second approach from linkage to unit allows the overall geometry of the deployed configuration to be determined from a set of design values. Finding solutions for the remaining parameters (geometry of the SLE s) relies on numerical calculations. In this section the two approaches and all relevant design parameters are discussed. All equations are valid for any n-sided polygon (n>3) as basic shape for the tower. For simplicity, the parameterisation is applied to a two-module mechanism, but the approach is easily extended to more modules First approach: Design of the undeployed configuration (Unit linkage) The tower is designed by means of parameters which determine the geometry of the linkage in the undeployed state (Figure 9.11b) and remain constant throughout the deployment: semi-bar lengths a and b, the kink angle β of the angulated SLE s, the number of scissor units U in one module and the total number of modules n stacked vertically. The number of units U with the minimum obviously being three in the closed loop determines the sector angle φ (Figure 9.11e). The edge length E (a parameter specific to the undeployed state) of the base polygon determines the length of the semi-bars a and b, with a being half the length of E and b s horizontal projection is equal to a (Figure 9.11d). (As mentioned earlier, this extra requirement makes it possible to further compact the whole linkage in its folded configuration). 255

289 Chapter 9 Case study 4 L a β L b (d) Elevation view of angulated element θ 2 θ 1 φ E (a) Intermediate state (0 ψ 1); (0 θ1 θ2 θmax) (e) Top view showing the edge length E and the sector angle φ L b L a L a L b θ max β θ 1 γ (b) Undeployed state (ψ=0); (θ1=θ2=0) (c) Fully deployed state (ψ=1); (0 θ1 θ2=θmax) Figure 9.11: Design parameters of a two-module tower with angulated SLE s (three states) 256

290 Chapter 9 Case study 4 The design parameters a, b, β, U and n remain constant throughout the deployment which is achieved by letting the deployment angle θ vary over a certain interval. As the module deploys, its width decreases while its height increases.. E, β, U and n completely determine the geometry of the linkage in the undeployed configuration. The sector angle φ is a submultiple of 2π and depends on the number of units U, corresponding with the number of sides of the base polygon: ϕ = 2π (9.1) U The shortest semi-bar a measures half of the length of the edge of the base polygon E: E a = (9.2) 2 The length of semi-bar b is given by: a a b = = (9.3) cos ( π β ) cos β In the undeployed position the height of one module equals: h undeployed = a tan β (9.4) while the total height of all modules is: H undeployed = n h undeployed (9.5) Now that all necessary design parameters are given a value, we can unfold the linkage by increasing the deployment angle θ. Figure 9.11a shows two modules (n=2) in an intermediate deployment position in which the bottom and top module both have their separate deployment angle θ 1 and θ 2 respectively. 257

291 Chapter 9 Case study 4 We can define to what degree a linkage is deployed by introducing the deployment ratio ψ, with 0 ψ 1: θ ψ = (9.6) θ max The uppermost (n th ) module will have reached its maximal deployment before any other module. Therefore its deployment angle θ n shall be used for determining the deployment of the whole structure. When θ max is reached, the uppermost end nodes meet in one point, rendering further deployment physically impossible. We can express θ max in terms of the kink angle β, via definition of an extra angle γ or it can be found directly from the drawing in Figure 9.11: π γ 2 γ π β = θmax (9.7) = (9.8) which gives for θ max : π θmax = β (9.9) 2 There are three distinct deployment stages: undeployed (ψ=0), intermediate (0 ψ 1) or fully deployed stage (ψ=1). These will now be discussed in greater detail. 258

292 Chapter 9 Case study 4 L 2 M 2 R 2 p 2 h 2 p 2 p 1 H 2 p 0 M 1 R 1 L 1 h 1 p 1 M 0 H 1 R 0 L 0 p 0 Intermediate state (0 ψ 1); (0 θ1 θ2 θmax) Figure 9.12: Perspective view: design parameters of a two-module tower with angulated SLE s Side elevation showing the non-coplanarity of the angulated elements (marked in red) The general, intermediate configuration (Figure 9.11a) is used to draw up the geometric relations between the design parameters. The undeployed and fully deployed states can be seen as special cases of the intermediate one. From Figure 9.12 a number of geometric relations can be drawn: ( sinθ cosθ tan β ) 1 = a h (9.10) 259

293 Chapter 9 Case study 4 ( sinθ cosθ tan β ) h (9.11) 2 = a L 0 = 2a cosθ1 (9.12) L 1 = 2a cosθ2 (9.13) = a cosθ sinθ tan β ( 1 ) ( cosθ sinθ tan β ) L (9.14) = 2a R 1 sinϕ L (9.15) H H L = (9.16) L 2 = 2 R 2 sinϕ (9.17) ϕ M 0 = R0 cos 2 (9.18) ϕ M 1 = R1 cos 2 (9.19) ϕ M 2 = R2 cos 2 (9.20) H = H 1 + H 2 (9.21) ( M ) h1 0 M1 = (9.22) ( M ) h2 1 M 2 = (9.23) a) The intermediate position (0 ψ 1) (0 θ 1 θ 2 θ max ) The linkage is designed from top to bottom, by assigning a value to θ 2 with 0 θ 2 θ max. Together with the previously determined design parameters n, U, β, a, b and φ, all dimensions can be derived by subsequently solving Eqn (9.24) to (9.31). The parameters used in this section are pictured in Figure For the top module: ( 2sinθ2 cosθ2 tan β ) ( cosθ sinθ tan β ) h (9.24) 2 = a 2 = 2 a L (9.25) L2 R 2 = (9.26) 2sinϕ 260

294 Chapter 9 Case study 4 H ϕ M 2 = R2 cos 2 (9.27) L 1 = 2 a cosθ2 (9.28) L1 R 1 = 2sinϕ (9.29) ϕ M 1 = R1 cos 2 (9.30) ( M ) h2 1 M 2 = (9.31) For the bottom module: θ 1 can be calculated from this equation: = a cosθ sinθ tan β ( ) L (9.32) Alternatively, θ 1 can be expressed in terms of L 1, a and β: cos β a L1 cos β a L1 4 a sec β sin β 1 = 1 cos 2 2 a θ (9.33) The unit height h 1, base length L 0, radius R 0 and projected height H 1 are given by: = a sinθ cosθ tan β ( ) h (9.34) H L 0 = 2 a cosθ1 (9.35) L0 R 0 = 2sinϕ (9.36) ϕ M 0 = R0 cos 2 (9.37) ( M ) h1 0 M1 = (9.38) When the total linkage consists of more modules (n 2), these can be calculated by repeating Eqns (9.32) to (9.38) for each extra module. 261

295 Chapter 9 Case study 4 b) Fully deployed position (ψ=1) (0 θ 1 θ 2 =θ max ) With θ 2 = θ max Eqns (9.24) to (9.31) become: For the top module: ( sinθ cosθ tan β ) h (9.39) 2 = a Knowing that Eqn (9.39) becomes π θ2 = θmax = β (9.40) 2 ( 2 cos β sin β tan β ) h (9.41) 2 = a + Alternatively, a simpler form can be found: cos 2 β = a cos β h (9.42) Because the upper end nodes meet in one point: = R = M 0 L (9.43) = Also, L 1 = 2 a cosθ2 (9.44) L1 (9.45) R 1 = 2sinϕ ϕ M 1 = R1 cos (9.46) 2 With M 2 =0 Eqn (9.31) becomes H h2 M1 = (9.47) 262

296 Chapter 9 Case study 4 For the bottom module: The calculation is identical to that for 0 ψ 1, idem as Eqns (9.32) to (9.38): θ 1 can be calculated from this equation: ( cosθ sinθ tan β ) L (9.48) 1 = 2 a Alternatively, θ 1 can be expressed in terms of L 1, a and β: cos β a L1 cos β a L1 4 a sec β sin β 1 = 1 cos 2 2 a θ (9.49) The unit height, base length, radius and projected height are given by: ( sinθ cosθ tan β ) h (9.50) 1 = a H L 0 = 2 a cosθ1 (9.51) L0 R 0 = 2sinϕ (9.52) ϕ M 0 = R0 cos 2 (9.53) ( M ) h1 0 M1 = (9.54) When the total linkage consists of more modules (n>2), these can be calculated by repeating the last set of Eqns (9.48) to (9.54) the appropriate number of times. c) Undeployed position (ψ=0) Evidently, in the undeployed position both modules are equal. Deployment angles θ 1 =θ 2 =0 in which case L 0 =L 1 =L 2 =E (9.55) R 0 =R 1 =R 2 (9.56) h 1 =h 2 = h undeployed a tan β = (9.57) 263

297 Chapter 9 Case study 4 H total = n h undeployed (9.58) This position is after all the one from which the mechanism was designed Second approach: Design in the deployed configuration (Linkage unit) By following the reverse approach, it is possible to impose overall dimensions on the linkage, after which the remaining parameters such as the bar length and the kink angle are derived. The governing equations for a simple onemodule linkage are derived first, after which the geometry for two modules will be simultaneously derived. For determining the global geometry n, U, R 0, R 1, H 1 are chosen and can be awarded any value (Figure 9.12 ). Depending on the value of R 1, this can be an intermediate position (R 1 >0) or an fully deployed position (R 1 =0), in which case the top nodes coincide. The sector angle φ is found from ϕ 2π U = (9.59) while L 0 and M 0 are given by L 0 = 2 R 0 sinϕ (9.60) and L 1 and M 1 are given by ϕ M 0 = R0 cos (9.61) 2 L 1 = 2 R 1 sinϕ (9.62) ϕ M 1 = R1 cos 2 (9.63) 264

298 Chapter 9 Case study 4 which gives for h 1 : h ( M ) H1 + 0 M1 = (9.64) A system of four equations (9.10), (9.12), (9.14) and (9.3) in four unknowns is used to calculate the remaining parameters. ( 2sinθ cosθ tan β ) h1 = a 1 L 0 = 2 a cosθ1 L1 = 2 a 1 a b = cos β ( cosθ + sinθ tan β ) 1 1 (9.65) Solving this system numerically gives two sets of solutions { a, b, β, θ 1 } and { a, b, β, θ 1 } which means that two different geometries meet the imposed design requirements. It is up to the designer to make a choice on which one to use. It is likely that the most extreme configurations, although strictly a correct solution, will not be of great use as a deployable structure with a large deployment range. We can use a similar approach for simultaneously calculating two modules. Again we impose dimensions on the top and bottom extremities of the linkages. Choose n, U, R 0, R 2. The first parameters are calculated from: ϕ = 2π U (9.66) L 0 = 2 R 0 sinϕ (9.67) ϕ M 0 = R0 cos 2 (9.68) L 2 = 2 R 2 sinϕ (9.69) ϕ M 2 = R2 cos 2 (9.70) 265

299 Chapter 9 Case study 4 Now, the remainder of the parameters can be found from a system of twelve equations (9.71) in twelve unknowns. Solving this system numerically gives two sets of solutions { a, b, β, θ 1, θ 2, h 1, h 2, H 1, H 2, R 1, L 1, M 1 } and { a, b, β, θ 1, θ 2, h 1, h 2, H 1, H 2, R 1, L 1, M 1 }: a b = cos β h1 = a h 2 = a 2 L0 = 2 a cosθ1 L1 = 2 a cosθ2 L 1 = 2 a L2 = 2 a 2 L1 = 2R1 sinϕ ϕ M1 = R1 cos 2 2 H1 = h1 2 H 2 = h2 H = H1 + H 2 ( 2sinθ1 cosθ1 tan β ) ( 2sinθ cosθ tan β ) ( cosθ1 + sinθ1 tan β ) ( cosθ + sinθ tan β ) ( M M ) 0 ( M M ) (9.71) Again, a choice can be made between the two obtained geometries. Compared to the calculation for one module, the complexity has risen. For each additional module to be calculated (simultaneously) an extra set of six equations is needed. a) Influence of parameters Minimal changes in the design values can have a significant effect on the overall geometry. The two parameters with the strongest impact on the geometry are the kink angle and the number of modules in the linkage. Figure 9.13 shows the undeployed and fully deployed position for six different con- 266

300 Chapter 9 Case study 4 figurations with various combinations of two or three modules with values for β of 135, 150 and 165. All configurations have the same edge length. n=2 β=135 n=2 β=150 n=2 β=165 Figure 9.13: Illustration of the influence of the apex angle β on the geometry of a linkage with angulated SLE s with two modules (n=2) in the undeployed (top) and fully deployed configuration (below) 267

301 Chapter 9 Case study 4 n=3 β=135 n=3 β=150 n=3 β=165 Figure 9.14: Illustration of the influence of the apex angle β on the geometry of a linkage with angulated SLE s with three modules (n=3) in the undeployed (top) and fully deployed configuration (below) 268

302 Chapter 9 Case study 4 We can draw the following conclusions from Figure 9.13 and Figure 9.14: n=cte, β cte: a fixed number of modules, but different kink angles. As β increases, the overall height of the deployed configuration also increases, while the radius decreases. The biggest impact however is noticeable in the undeployed configuration. By increasing β from 135 to 165, the height in the stacked position is reduced to a third. So blunter kink angles lead to linkages which are more compact easier transportable - in their undeployed state. n cte, β=cte: Since all configurations are maximally deployed, increasing the number of modules actually means adding more modules at the bottom of the linkage, each of which will be less deployed than the previous one. As a consequence although it is obvious that adding more modules leads to an increased height in both the undeployed and deployed position the contribution of each subsequently added module becomes less. The top module in the linkage is the determining factor for the deployment range. Units with sharp kink angles tend to quickly reach their maximal deployment, therefore halting the deployment of the remaining modules. So if a substantial expansion in height is desired, it would be a better option to choose a blunt kink angle in combination with a higher number of modules: the blunt kink angle makes the undeployed configuration more compact and increases the deployment interval (0 to θ max ). A choice will have to be made concerning the optimal number of modules that will suit the design, taking all relevant parameters into consideration. 269

303 Chapter 9 Case study From mechanism to architectural structure Mobility analysis Figure 9.15 shows a schematic representation of an undeployed and an intermediate deployment position of the same linkage. As the deployment progresses, the angulated SLE s of each module tilt inward at the top. The dotted lines are imaginary fold lines around which mobility has to be allowed in order to complete the deployment. Through connection of the end nodes, each scissor unit can be represented by a trapezoid, of which the contour changes constantly during deployment. Between quadrilaterals ABDC and CDFE and between CDFE and EFHG there is a relative rotation which causes them not to remain coplanar. The joints connecting the end nodes of the units will have to take into account all aspects of this mobility. In Figure 9.16 a proposal for such a joint is pictured, showing the seven rotational degrees of freedom needed for the deployment, as well as for the linkage to be compactly folded (Figure 9.17). G H E F D C B Figure 9.15: A schematic representation of the relative rotations of the quadrilaterals around imaginary fold axes during deployment A 270

304 Chapter 9 Case study 4 Figure 9.16: Kinematic joint connecting the angulated elements at their end nodes Figure 9.17: The kinematic joint and the axes of revolution for the seven rotational degrees of freedom In order for the mechanism to be usable as a structure, the mobility will have to be constrained. Analogous to the previous case studies, an equivalent hinged plate model is presented. Figure 9.18 represents the linkage with the rotational degree of freedom of the scissor linkage removed. After removal of this D.O.F. the remaining mobility determines to what extent constraints have to be added. Due to triangulation of the modules, there is no additional mobility which means it is basically a single D.O.F.-mechanism. Therefore, it is sufficient to constrain the movement of the rotational degree of freedom of the 271

305 Chapter 9 Case study 4 scissor units. As usual, fixing two appropriately chosen nodes is enough to remove the rotational D.O.F from the scissor linkage. But for using the tower as a load bearing structure, all three lower nodes have to be fixed to the ground by pinned supports. Figure 9.18: The scissor linkage in its deployed state and its equivalent hinged plate structure for mobility analysis (left) Fixing the structure by pinned supports (right) The erection process Figure 9.19: Deployment of proof-of-concept model A proof-of-concept model has been constructed to verify the deployment behaviour (Figure 9.19). A short description is given of how the erection process could be executed, as shown in Figure 9.20: 272

306 Chapter 9 Case study 4 A: the tower is in its undeployed form. The membrane elements are attached to the nodes of the mechanism and fixed by their low points to the ground B: As the tower gradually deploys, the membranes are raised. When sufficient height is achieved, the additional masts are inserted and gradually put in their right location. Then, the cables fixing the secondary masts to the ground are brought under tension. C: Finally, the tower is slightly deployed further to add pre-tension in the membrane. Then, the tower is fixed to the ground by pinned supports and additional horizontal ties (cables or struts) can be inserted at the appropriate level. 273

307 Chapter 9 Case study 4 Figure 9.20: Deployment sequence (A, B and C) for the tower with the membrane elements attached After deployment horizontal ties are added to enhance structural stiffness (see structure analysis). Several solutions are possible: cable ties could be used, 274

308 Chapter 9 Case study 4 which are already present before deployment and are shortened as the structure deploys and becomes narrower. Struts could be added afterwards to brace the structure. An active cable can run over appropriately chosen nodes along a path and can be shortened to aid in the deployment. This needs to be further investigated Alternative configuration As mentioned earlier, deployable towers with a base polygon of more than three sides are possible. Shown here is an alternative configuration with hexagonal modules (Figure 9.21). The upper two modules are identical, but mirrored versions of the two modules located just below. Figure 9.22 shows how the six-unit modules can be compacted for transport and Figure 9.23 shows three stages in the deployment. Figure 9.21: Design for a deployable hexagonal tower with angulated elements 275

309 Chapter 9 Case study 4 Step 3 Step 4 Step 5 (ready to deploy) Figure 9.22: Initial unfolding of the compacted linkage to its polygonal form Step 6 (Undeployed) Step 7 Step 8 Figure 9.23: Three stages in the deployment of a hexagonal tower with 5 modules: elevation and top view 276

310 Chapter 9 Case study Simplified concept: prismoid versus hyperboloid The previously described geometry is an exact translation of the static hyperboloid geometry of the tower, as originally proposed by The Nomad Concept [2007], into a deployable version. In its undeployed state the scissor linkage has a prismatic shape and all angulated elements per vertical row (or lateral face of the prism) are coplanar. During deployment the shape gradually changes into a hyperboloid, which means that the angulated elements per vertical row are no longer coplanar, i.e. they experience relative rotation, as can be seen in the triangular example of Figure As a consequence, the articulated hinges (Figure 9.17) will have to allow an extra rotational D.O.F around the horizontal axes between modules to cope with this movement which adds to the complexity of the joint design. The described deployment behaviour is caused by the particular geometry of the angulated elements which - as explained in Figure 9.8 and Figure is such that in its undeployed state the linkage can be further compacted by folding along the vertical fold axes. This means that the angulated elements consist of two differently sized semi-bars a and b, turning the angulated elements non-symmetrical. The overall geometry of this solution shall be referred to as hyperboloid. 277

311 Chapter 9 Case study 4 Non-symmetrical angulated elements Axis of rotation between modules Module edges Undeployed Partially deployed Figure 9.24: Hyperboloid geometry (as proposed in previous sections) angulated elements do not remain coplanar during deployment Now, an alternative concept is proposed, which is similar in setup to the hyperboloid version, but has simplified joints for interconnecting the modules. If the angulated elements within a vertical row can be kept coplanar, then the hinges between modules would not have to allow an extra rotational D.O.F. around the horizontal axes between modules, effectively decreasing the mechanical complexity. Also, the end nodes of the angulated elements remain collinear, as shown in the triangular example of Figure The effect on the overall shape is that it resembles a prism before, during and after deployment. More precisely, such a shape is known in geometry as a prismoid [Mathworld, 2007]. 278

312 Chapter 9 Case study 4 Symmetrical angulated elements End nodes are collinear Undeployed Partially deployed Figure 9.25: Prismoid geometry (simplified alternative to the previously described geometry) - angulated elements remain coplanar during deployment The particular geometry of Figure 9.25 can only be achieved if symmetrical angulated elements are used. Figure 9.26 and Figure 9.27 show the difference between non-symmetrical and symmetrical elements. As described earlier, non-symmetrical elements have differently sized semi-bars a and b and are used for the hyperboloid solution. Non-symmetrical and identical angulated elements a b Figure 9.26: Non-symmetrical identical angulated elements result in a fully compactable configuration: hyperboloid solution The symmetrical elements used for the prismoid solution consist of two identical semi-bars a, as Figure 9.27 shows. 279

313 Chapter 9 Case study 4 Symmetrical and identical angulated elements a a Figure 9.27: Symmetrical identical angulated elements cannot be fully compacted In both cases the angulated elements are identical throughout the entire structure. An angulated scissor linkage built from non-symmetrical identical elements is fully compacted in its undeployed position, as Figure 9.26 shows. But a linkage consisting of symmetrical identical elements (Figure 9.27) is not fully compactable: when the bottom most SLE is fully compacted, the above SLE is still partially deployed. To overcome this issue, an extra condition is imposed on the scissor geometry, as Figure 9.28 shows. The angulated elements become smaller near the top and their geometry is such that their lower end nodes and intermediate hinge are collinear in the undeployed condition. This configuration ensures the highest degree of compactness. 280

314 Chapter 9 Case study 4 Symmetrical and non-identical angulated elements a 1 a 1 a 0 a 0 Lower end nodes and intermediate hinge are collinear Figure 9.28: Symmetrical and non-identical angulated elements result in a fully compactable configuration: prismoid solution The relationship between the lengths of semi-bars of consecutive angulated elements is depicted in Figure The length of the semi-bar a 1 can be expressed in terms of a 0 as follows: ( π β ) a = cos (9.72) 1 a 0 With the kink angle β and the length of the semi-bar of the bottom most angulated element a 0 chosen as design parameters, the length semi-bar of the n th element can be written as: a n ( ( π β )) n = a cos 0 (9.73) 281

315 Chapter 9 Case study 4 a 1 a 0 β β a 1 a 0 a 1 Figure 9.29: Symmetrical and non-identical angulated elements result in a fully compactable configuration: prismoid solution To demonstrate what happens during deployment, three consecutive stages - undeployed, partially deployed and fully deployed - are shown in Figure During the course of the deployment, a constant angle (marked by the red lines), is subtended by each vertical linkage. This characteristic is precisely what makes the design of radially deployable closed loop structures possible. To illustrate this, Figure 9.31 shows the corresponding closed loop structure which uses the same vertical linkage, but arranged radially in a common plane. Undeployed Intermediate Fully deployed Figure 9.30: Three consecutive stages in the deployment of a prismoid geometry 282

316 Chapter 9 Case study 4 Corresponding closed loop structures Figure 9.31: Three consecutive stages of the corresponding planar closed-loop structure Figure 9.32 shows a three-dimensional model of a triangular prismoid tower in three deployment stages, while Figure 9.33 shows a top view and a side elevation. Figure 9.32: Perspective view of the deployment of a triangular tower 283

317 Chapter 9 Case study 4 Figure 9.33: Top view and side elevation of the prismoid tower The simplified solution for the articulated hinge - which connects four bars at once - is shown in Figure Figure 9.34: Detailed view of the simplified hinge connecting four scissor bars Note that the rotational D.O.F. around the vertical axis - marked with the dashed line in Figure is not necessary during the deployment of the tower. Its purpose is to allow the structure to be further compacted in the undeployed position, by means of folding, analogous to the method described in Figure 9.9 and Figure

318 Chapter 9 Case study 4 In Figure 9.18, an equivalent hinged-plate structure for the hyperboloid geometry was introduced, which has shown that the only D.O.F. in the system is the rotational D.O.F. of the scissors. When the same method is applied to the prismoid solution, it can be seen that this holds no longer true. Figure 9.35 shows a triangular and a quadrangular geometry and their equivalent hingedplate structure. The triangular shape cannot be folded along the vertical axes between neighbouring plates, while the quadrangular solution can be flatly folded. It can be concluded that the prismoid solution is apart from the triangular geometry a multiple D.O.F.-mechanism. To turn the mechanism into a structure, and therefore removing all D.O.F. s, all lower nodes are fixed to the ground by pinned supports. Figure 9.35: Triangular and quadrangular prismoid solution and their respective equivalent hinged-plate structure, providing an insight in the kinematic behaviour Table 9.1 and Table 9.2 contain a comparison between the hyperboloid and the prismoid solution in terms of the geometry of the angulated elements, the D.O.F. during deployment and the mechanical complexity of the hinges between modules. 285

319 Chapter 9 Case study 4 Hyperboloid geometry - Identical, non-symmetrical angulated elements - Deployed shape is hyperboloid: angulated elements per vertical row do not remain coplanar - End nodes of angulated elements are not collinear during deployment: 1 D.O.F.-mechanism - Articulated hinges between modules require extra D.O.F.: increased mechanical complexity Table 9.1: Characteristics of the hyperboloid geometry Prismoid geometry - Non-identical, symmetrical angulated elements - Deployed shape is prismoid: angulated elements per vertical row are coplanar - End nodes of angulated elements remain collinear during deployment: multiple D.O.F. (except for triangular geometry) - Articulated hinges between modules do not need an extra D.O.F.: decreased mechanical complexity Table 9.2: Characteristics of the prismoid geometry 286

320 Chapter 9 Case study Structural analysis For this case study, a nominal wind load of 0.6 kn/m 2 is combined with a snow load of 0.5 kn/m 2, applied on the whole surface, so no load zones are defined. For this analysis, only a transverse direction is considered for the wind load. The alternative wind direction at 60 was not included in the analysis. The resulting load vectors are calculated automatically by EASY, the snow load is treated analogously. The considered load combinations are given in Table 9.3. Load combination Permanent Main solicitation Secondary ULS/SLS 1 pre-stress + snow ULS/SLS 2 pre-stress + snow + wind ULS/SLS 3 pre-stress + wind ULS/SLS 4 pre-stress + wind + snow Table 9.3: Load combinations for wind and snow Figure 9.36: Top view and perspective view of the structure with indication of the global coordinate system and the vector components of the wind action 287

321 Chapter 9 Case study 4 The approach discussed in Chapter 5 is used for determining the actions of the membrane on the structure under wind and snow load. Shown in Figure 9.36 are the load vectors in the global coordinate system. Figure 9.37 and Figure 9.38 show the equilibrium form of the membrane calculated in EASY-software. Figure 9.37: Side elevation of the equilibrium form of the membrane Figure 9.38: Top view of the equilibrium form of the membrane First, a finite element model of the tower with additional cable ties (steel cable d=8 mm, design strength = 1500 MPa) is analysed under the specified load 288

322 Chapter 9 Case study 4 combinations. The resulting section is mm, with a weight of 187 kg (Figure 9.39). For a height of the tower of 8.43 m, the weight ratio (the weight of the membrane itself is neglected as in the previous case studies) is 23 kg/m. Element type Section [mm] Weight [kg] Structure bar Tension cable d=8 7 Total weight 194 Weight/m 23 Figure 9.39: Horizontal cable ties to improve structural performance When the horizontal cable ties are replaced by aluminium bars, the structural performance increases. Figure 9.40 shows the results for which ULS 4 (prestress + wind + snow) is the governing load combination (Figure 9.41). The section is decreased to 90x50x3.2 mm and the additional struts are assigned a tubular round section of 42x2.6 mm. The total weight drops to 160 kg which results in a weight ratio of 2.2kg/m 2. Although ULS 4 is the determining load combination for the strength as well as the stability, the greatest displacements are a result of SLS 2 (pre-stress + snow + wind). The displacements are 289

323 Chapter 9 Case study 4 of an acceptable magnitude, especially when the nature of the structure is considered. The deflections (not shown) do not exceed 1/200. Section hxbxt, d[mm] Total Weight/m [kg/m] weight/section [kg] TREC 90x50x TRON 42x Total weight: Figure 9.40: Perspective view, top view and side elevation of deployable mast 290

324 Chapter 9 Case study 4 RESULTS for Case study 4 STRESS [MPa] S max S max S max Load S max Section [mm] (My) (Mz) Fx/Ax comb. TREC 90x50 x ULS 4 TRON 42x ULS 4 FORCE Section [mm] Fx [kn] My [knm] Mz [knm] Load comb. TREC 90x50 x ULS 4 TRON 42x ULS 4 REACTIONS [kn] FX Load comb. FY Load comb. FZ Load comb. 27 ULS ULS ULS 4 DISPLACEMENTS [cm] Ux Load comb. Uy Load comb. Uz Load comb. 0.3 SLS SLS SLS 4 Uy Uz Ux Figure 9.41: Summary of the determining parameters for the strength, stability and stiffness of case study 4 291

325 Chapter 9 Case study Conclusion In this chapter, a novel idea has been put forward for a deployable hyperboloid tower, used for the deployment of a membrane canopy, without the need for additional lifting equipment. The two-fold purpose of the tower, namely holding up the membrane elements in the deployed position and serving as an active element during the erection process, has been demonstrated. It has been found that the proposed linear structure offers an advantage over existing solutions: using angulated elements instead of polar units for the same deployed geometry, has lead to a significant reduction of the number of scissor members and connections. Figure 9.42: Case 4 A temporary canopy and its deployable tower with angulated units A geometric design method has been proposed for which the equations were derived and expressed in terms of relevant design parameters, such as the shape of the base polygon, the bar length or the kink angle of the angulated element. Although the equations were derived for a two-module geometry, the design method is equally applicable to configurations with a higher number of modules. However, it was found that the number of equations to be solved simultaneously was equally raised, therefore significantly increasing the complexity of the calculation. Also, the method has been devised such, that by 292

326 Chapter 9 Case study 4 simply changing the kink angle to 180, a polygonal tower with translational units is obtained. It has been found that the deployment behaviour is heavily influenced by the kink angle of the angulated element: the blunter the kink angle, the larger the expansion range is before the fully deployed position is reached. The influence of these parameters on the geometry and the deployment process has been discussed. It has been shown that the hyperboloid shape of the tower causes the angulated units, per vertical row, not to remain co-planar during deployment. A novel articulated joint which allows this relative rotation has been proposed and found to work well. Further, an alternative shape, called a prismoid geometry, has been proposed. This has proven to be a simpler solution compared to the hyperboloid geometry in terms of kinematic behaviour, therefore allowing the use of greatly simplified joints. Through the use of an equivalent hinged-plate model, the mobility of both the hyperboloid and prismoid geometry has been assessed. It was found that the hyperboloid configuration is always, regardless of the polygonal shape, a one-degree-of-freedom mechanism. The prismoid solution, on the other hand, is always, apart from the triangular geometry, a multiple-d.o.f.- mechanism. Finally, the structural feasibility has been checked under wind and snow action. The resulting structure has a weight/height ratio of 19 kg/m, excluding the weight of the joints, which have not been structurally designed. This fourth case study has made innovative use of angulated elements in an original application. Although the concept has been proven to work, more detailed analysis, including structural design of the joints, is needed. 293

327 Chapter 9 Case study 4 294

328 Chapter 10 - Conclusions Chapter 10 Conclusions The aim of the work presented in this dissertation was to develop novel concepts for deployable bar structures and propose variations of existing concepts, leading to architecturally and structurally viable solutions for mobile applications. Equally, it was the objective to provide the designer with the means for deciding on how to cover a space with a rapidly erectable, mobile architectural space enclosure, based on the geometry of foldable plate structures or employing a scissor system. By presenting a review of previous research on scissor structures and foldable plate structures, an insight is given in the wide variety of possible shapes and configurations. The design principles behind scissor structures have been explained and it has been shown that using geometric constructions is a straightforward way of designing deployable scissor linkages of any curvature. The design of scissor structures, as well as foldable plate structures, has been advanced by proposing novel geometric design methods based on parameters which are relevant to the designer, such as the rise and span of the structure. These principles were then used in four case studies, which cover the key aspects of the design, including a kinematic and preliminary structural analysis of novel concepts for deployable bar structures. 295

329 Chapter 10 - Conclusions 10.1 Novel concepts for deployable bar structures Case study 1: A Deployable Barrel Vault with Translational Units on a Three-way Grid (a) OPEN structure (b) CLOSED structure Figure 10.1: Case study 1 (Chapter 6) Translational barrel vault With the first case study, an advance has been made by developing a stressfree deployable scissor structure of single curvature with translational units on a three-way grid. It has been shown that the curved triangulated grid can be uniquely composed of single scissor units, therefore avoiding the integration of double scissor units. By doing so, the number of connections could be kept to a minimum and the grid demonstrated an inherent triangulation, therefore providing in-plane stability. It has been shown that, as a consequence, the deployed configuration does not require additional cross-bracing of the grid cells to prevent the structure from swaying. 296

330 Chapter 10 - Conclusions The geometric design method developed for this case study is not limited to circular base curves only, but it has been devised such, that it can be extended to structures of any curvature. It has been proven possible to provide the open barrel vault with suitable end structures, providing a fully closed envelope and greatly enhancing the architectural applicability of the concept Case study 2: A Deployable Barrel Vault with Polar and Translational Units on a Two-way Grid (a) OPEN structure (b) CLOSED structure Figure 10.2: Case study 2 (Chapter 7) Polar barrel vault For the second case study, a deployable barrel vault with polar and translational scissor units on an orthogonal two-way grid, has been proposed. This case study is an illustration of how a basic stress-free deployable single curvature structure can be obtained by putting the construction methods and the 297